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Justified SMT 1: The Minikanren inside Z3
Z3 actually has a logic programming language inside it if you know how to look. This makes it one the easiest to pull off the shelf because Z3 has so much work put into it and excellent bindings. It
also is perhaps one of the most declarative logic programming languages available with very cool strong theory support.
Here I talked about how to use Z3 to make a minikanren, keeping the search in the python metalayer. https://www.philipzucker.com/minikanren-z3py/ This is still a useful and interesting idea. I
mention that the metalevel conj and disj can be replaced by z3’s And and Or but at the cost of a quantifier. This is still true.
I find myself revisiting these old ideas with hopefully more sophisticated perspective.
There is an old topic that I’ve encountered most strongly in the answer set programming community of what exactly is the logical semantics of prolog? Prolog is quasi operational.
If we take this prolog program that searches for pathes in a graph.
path(X,Y) :- edge(X,Y).
path(X,Z) :- edge(X,Y), path(Y,Z).
We might think the declarative semantics of this program are
from z3 import *
BV = BitVecSort(2)
edge = Function('edge', BV,BV, BoolSort())
path = Function("path", BV,BV, BoolSort())
x,y,z = Consts('x y z', BV)
base = ForAll([x,y], Implies(edge(x,y), path(x,y)))
trans = ForAll([x,y,z], Implies(And(edge(x,y), path(y,z)), path(x,z)))
∀x, y : edge(x, y) ⇒ path(x, y)
∀x, y, z : edge(x, y) ∧ path(y, z) ⇒ path(x, z)
In a loose intuitive sense this is true and is what the prolog syntax is alluding to.
In a stricter sense that z3 implements something akin to multi sorted first order logic, this is false.
The thing Z3 (or any smt solver) tries to do is return models that satisfy the constraints. It does not really have an operational semantics.
What the axioms actually say is that path is transitive with respect to edge. This is not the same as saying path is the transitive closure of edge. Transitive closure is inexpressible in a certain
generic sense inside first order logic https://math.stackexchange.com/questions/1286141/transitive-closure-and-first-order-logic . As with many no-go theorems, I’m not sure there isn’t perhaps a way
around achieving the spirit of the objective that avoids the preconditions of the theorem.
The transitive closure is the least transitive relation. Z3 is still free to overapproximate path. A simple useful test case is to consider whether path=True still works even when it’s not
solve(And(base,trans, edge(0,1), edge(1,2)))
[path = [else -> True],
edge = [else ->
And(Not(And(Var(0) == 1,
Not(Var(1) == 2),
Not(Var(1) == 1))),
Not(And(Var(0) == 2,
Not(Var(0) == 1),
Var(1) == 1)))]]
One attempt to patch this up is to note that really we want path to be true not only if one of the preconditions of a rule holds, but if and only if. This is the idea behind clark completion https://
www.inf.ed.ac.uk/teaching/courses/lp/2012/slides/lpTheory8.pdf .
You do the clark completion by gathering up every rule of a given head. You can turn it into a rule with a body that is a giant or of ands. To make all the heads unifiable, make them unique variables
and add the approriate equality constraints into each of the individual bodies.
Minikanren is intrinsically written in clark completion form by the nature of it’s abuse of the function call mechanisms of it’s host language. Every rule that produces it’s head is gathered up in
the body of that relations definition.
# a sketch. I don't have a working minikanren on my pc right now or loaded up in my head.
def path(x,z):
yield from disj(
fresh(lambda y: conj(edge(x,y), path(y,z)))
Ok well then how about
clark = ForAll([x,z], path(x,z) == Or(
Exists([y], And(edge(x,y), path(y,z)))
s = Solver()
s.add(ForAll([x,y], edge(x,y) == Or(
And(x == 0, y == 1),
And(x == 1, y == 2)))
m = s.model()
print("edge", {(x,y) for x in range(4) for y in range(4) if m.eval(edge(x,y)) })
print("path", {(x,y) for x in range(4) for y in range(4) if m.eval(path(x,y)) })
edge {(0, 1), (1, 2)}
path {(0, 1), (0, 2), (1, 2)}
This is still not correct. The Clark completion is not sufficient basically because it still allows circular reasoning in the form of loops. Consider the below reformulation of the same basic idea,
except that I write the transitive part of path differently. path can sort of dignify itself.
I’m not exactly sure the conditions under which clark completion alone is sufficient, but they seem subtle and can’t possibly always work. The edge-path form of transitivity I think is correct
because of a stratification and grounding of path with respect ot edge.
clark = ForAll([x,z], path(x,z) == Or(
Exists([y], And(path(x,y), path(y,z)))
s = Solver()
s.add(ForAll([x,y], edge(x,y) == Or(
And(x == 0, y == 1),
And(x == 1, y == 2)))
m = s.model()
print("edge", {(x,y) for x in range(4) for y in range(4) if m.eval(edge(x,y)) })
print("path", {(x,y) for x in range(4) for y in range(4) if m.eval(path(x,y)) })
edge {(0, 1), (1, 2)}
path {(0, 1), (1, 2), (0, 0), (1, 1), (0, 2), (1, 0)}
But there is a fix.
Earlier I said z3 is merely multi-sorted first order logic. This is a good first pass understanding, but it isn’t true. Ok, it does directly have support for transitive closure as a special relation
https://microsoft.github.io/z3guide/docs/theories/Special%20Relations/ , but actually even the uncontroversial addition of algebraic data types like Option/List/Nat has some kind of least fixed point
character that let’s you constrain the relations.
It’s actually quite fascinating. What you do is add an extra parameter to your relation that contains a proof tree ADT.
1. Add an extra proof parameter to the definition of the relation itself
2. Make a datatype with a constructor for each case in your minkanren program
3. (optional?) Put any existentials into the proof sturcture
This is the same thing as adding a tracing parameter to a datalog (provenance https://souffle-lang.github.io/provenance ), prolog, or minikanren program. The tracing parameter can record the call
tree that succeeds without using any extralogical funkiness. This is an instance of a general principle that the trace of any system you think is proving something is a proof object.
I stuck to bitvectors because then I knew the quantifiers wouldn’t go off the rails. But if you use the define-fun-rec facilities of z3, you don’t know generic quantifiers and you can get z3 to
return proof trees and results even in infinitary cases. define-fun-rec implements a different mechanism than general quantifiers, something like iterative deepening. A recursive function definition
is logically equivalent to using a quantified equality, but it is implemented differently. If you’re seeking unsat, maybe either works, but if you’re seeking models that contain that analog of
minikanren answers, define fun rec seems superior.
pathpf = Datatype("pathpf")
pathpf.declare("trans", ("y", BV), ("p1", pathpf), ("p2", pathpf))
pathpf = pathpf.create()
p = Const("p", pathpf)
path = RecFunction("path", BV, BV, pathpf, BoolSort())
RecAddDefinition(path, [x,z,p], Or(
And(pathpf.is_base(p), edge(x,z)),
And(pathpf.is_trans(p), path(x, pathpf.y(p), pathpf.p1(p)), path(pathpf.y(p), z, pathpf.p2(p))))
edge = RecFunction("edge", BV, BV, BoolSort())
RecAddDefinition(edge, (x,y),Or(
And(x == 0, y == 1),
And(x == 1, y == 2),
And(x == 2, y == 3)))
s = Solver()
#s.add(ForAll([x,y], edge(x,y) == Or(
# And(x == 0, y == 1),
# And(x == 1, y == 2)))
#pathpf.trans(1, pathpf.base, pathpf.base)))
m = s.model()
#print("edge", {(x,y) for x in range(4) for y in range(4) if m.eval(edge(x,y)) })
#print("path", {(x,y) for x in range(4) for y in range(4) if m.eval(path(x,y,p)) })
trans(1, base, base)
Bits and Bobbles
The performance of finding these models is a bit unstable.
Asnwer set programming has been described as justified smt https://www.weaselhat.com/post-1067.html
I’ve used this trick to embed static datalog like analyses into constraint solver. An over approximation of liveness is ok, so just clark completion is acceptable. Usually other objectives will tend
to push the liveness down to what is strictly needed anyhow. https://www.philipzucker.com/compile_constraints/
Z3’s transtivie special relation or using its optimiation functionality to get the “least” path are other options.
I think I can use this to encode inductive relations for knuckeldragger. More on this next time. Indcution principles, recursors.
path = Function("path", BV, BV, pathpf) is also interesting and maybe good? Uniqueness of proofs
It doesn’t have a notion of negation as failure. It’ll just hang.
I jibbered about this on twitter more than I realized. Well, it’s good to have it actually written up in some form
A trickier question that bore these ideas is how to do justified equality in z3. i wanted to mimic egglog / egg since the z3 model is kind of the egraph. Equality is very slippery. It’ll ruin your
justifications out from under your feet.
Mark Nelson comments that ASP calls the sufficiency conditions for ASP to be ok as “tight conditions” https://x.com/mm_jj_nn/status/1811131182228082804 First order logic inductive defintiions FO(ID)
may be related to the ideas above http://cs.engr.uky.edu/ai/papers.dir/VL65Final.pdf https://ojs.aaai.org/aimagazine/index.php/aimagazine/article/view/2679 First Order Logic with Inductive
Definitions for Model-Based Problem Solving
https://lawrencecpaulson.github.io/papers/Aczel-Inductive-Defs.pdf An Introduction to Inductive Definitions * PETER ACZEL
example of proof parameter tracking using DCGs https://x.com/SandMouth/status/1558473206239006720
%sequent( Hyp, Conc, Var )
:- use_module(library(clpfd)).
%:- table prove/2.
:- use_module(library(solution_sequences)).
%:- op(600, xfy, i- ).
prove(S, ax(S, id)) :- S = (A > A).
prove(S, ax(S, fst)) :- S = (A /\ _B > A).
prove( A /\ B > B, ax(A /\ B > B, snd)).
prove( S, ax(S, inj1 )) :- S = (A > A \/ _B).
prove( S, ax(S, inj2 )) :- S = (B > _A \/ B).
prove( false > _ , initial ).
prove( _ > true , terminal ).
prove( A > B /\ C, bin(A > B /\ C, pair, P1, P2)) :- prove(A > B, P1), prove(A > C, P2).
prove( A \/ B > C , bin(A \/ B > C, case, P1, P2)) :- prove( A > B, P1), prove( A > C, P2).
prove( A > C, bin(A > C, comp, P1, P2)) :- prove(A > B, P1), prove(B > C, P2).
height(ax(_,_), 1).
height(un(_,_,PX), N) :- N #> 1, N #= NX+1, height(PX,NX).
height(bin(_,_,PX,PY), N) :- N #> 1, N #= max(NX , NY) + 1, height(PX,NX), height(PY,NY).
% maybe explicilty taking proof steps off of a list. using length.
% use dcg for proof recording?
prove(A > A) --> [id].
prove(A /\ _ > A) --> [fst].
prove(_ /\ B > B) --> [snd].
prove(A > A \/ _) --> [inj1].
prove(B > _ \/ B) --> [inj2].
prove(false > _) --> [initial].
prove( _ > true) --> [terminal].
prove(A > B /\ C) --> [pair], prove(A > B), prove(A > C).
prove(A \/ B > C) --> [case], prove(A > C), prove(B > C).
prove(A > C) --> [comp], prove(A > B), prove(B > C).
:- initialization(main).
%main :- format("hello world", []).
%main :- between(1, 10, N), height(Pf, N), writeln(Pf), prove( w /\ x /\ y /\ z > w, Pf ), print(Pf), halt.
main :- length(Pf, _), phrase(prove(w /\ x /\ y /\ z > w \/ y),Pf), print(Pf), halt.
main :- halt.
Yes, the dcg form is very pretty. It’s a writer monad of sorts. It is recording a minimal proof certificate that is reconstructable to the full thing pretty easily.
G --> [tag], D1, D2 Should be read as
D1 D2
------------------ tag
prove(Sig , A > B) --> { insert(A,Sig,Sig1) }, [weaken(A)], prove(Sig1, A > B).
prove(A > forall(X,B)) --> , prove(weak(X, A) > B).
prove(A > forall(X,B)) --> {insert(X,Sig,Sig2) }, prove(Sig1, A > B).
prove(Sig, forall(X,A) > ) --> , prove(weak(X, A) > B)
Maybe start with implication prove((A > (B > C) ) –> [curry], prove( A /\ B > C). prove((A /\ (A > B) > B ) –> [eval]. | {"url":"https://www.philipzucker.com/minikanren_inside_z3/","timestamp":"2024-11-10T20:50:08Z","content_type":"text/html","content_length":"54560","record_id":"<urn:uuid:cbc86f2b-917a-4a9e-9900-d070e1490b6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00689.warc.gz"} |
(Log-Normal) heterogeneity priors for binary outcomes
TurnerEtAlPrior {bayesmeta} R Documentation
(Log-Normal) heterogeneity priors for binary outcomes as proposed by Turner et al. (2015).
Use the prior specifications proposed in the paper by Turner et al., based on an analysis of studies using binary endpoints that were published in the Cochrane Database of Systematic Reviews.
TurnerEtAlPrior(outcome=c(NA, "all-cause mortality", "obstetric outcomes",
"cause-specific mortality / major morbidity event / composite (mortality or morbidity)",
"resource use / hospital stay / process", "surgical / device related success / failure",
"withdrawals / drop-outs", "internal / structure-related outcomes",
"general physical health indicators", "adverse events",
"infection / onset of new disease",
"signs / symptoms reflecting continuation / end of condition", "pain",
"quality of life / functioning (dichotomized)", "mental health indicators",
"biological markers (dichotomized)", "subjective outcomes (various)"),
comparator1=c("pharmacological", "non-pharmacological", "placebo / control"),
comparator2=c("pharmacological", "non-pharmacological", "placebo / control"))
outcome The type of outcome investigated (see below for a list of possible values).
comparator1 One comparator's type.
comparator2 The other comparator's type.
Turner et al. conducted an analysis of studies listed in the Cochrane Database of Systematic Reviews that were investigating binary endpoints. As a result, they proposed empirically motivated
log-normal prior distributions for the (squared!) heterogeneity parameter \tau^2, depending on the particular type of outcome investigated and the type of comparison in question. The log-normal
parameters (\mu and \sigma) here are internally stored in a 3-dimensional array (named TurnerEtAlParameters) and are most conveniently accessed using the TurnerEtAlPrior() function.
The outcome argument specifies the type of outcome investigated. It may take one of the following values (partial matching is supported):
• NA
• "all-cause mortality"
• "obstetric outcomes"
• "cause-specific mortality / major morbidity event / composite (mortality or morbidity)"
• "resource use / hospital stay / process"
• "surgical / device related success / failure"
• "withdrawals / drop-outs"
• "internal / structure-related outcomes"
• "general physical health indicators"
• "adverse events"
• "infection / onset of new disease"
• "signs / symptoms reflecting continuation / end of condition"
• "pain"
• "quality of life / functioning (dichotomized)"
• "mental health indicators"
• "biological markers (dichotomized)"
• "subjective outcomes (various)"
Specifying “outcome=NA” (the default) yields the marginal setting, without considering meta-analysis characteristics as covariates.
The comparator1 and comparator2 arguments together specify the type of comparison in question. These may take one of the following values (partial matching is supported):
• "pharmacological"
• "non-pharmacological"
• "placebo / control"
Any combination is allowed for the comparator1 and comparator2 arguments, as long as not both arguments are set to "placebo / control".
Note that the log-normal prior parameters refer to the (squared) heterogeneity parameter \tau^2. When you want to use the prior specifications for \tau, the square root, as the parameter (as is
necessary when using the bayesmeta() function), you need to correct for the square root transformation. Taking the square root is equivalent to dividing by two on the log-scale, so the square root's
distribution will still be log-normal, but with halved mean and standard deviation. The relevant transformations are already taken care of when using the resulting $dprior(), $pprior() and $qprior()
functions; see also the example below.
a list with elements
parameters the log-normal parameters (\mu and \sigma, corresponding to the squared heterogeneity parameter \tau^2 as well as \tau).
outcome.type the corresponding type of outcome.
comparison.type the corresponding type of comparison.
dprior a function(tau) returning the prior density of \tau.
pprior a function(tau) returning the prior cumulative distribution function (CDF) of \tau.
qprior a function(p) returning the prior quantile function (inverse CDF) of \tau.
Christian Roever christian.roever@med.uni-goettingen.de
R.M. Turner, D. Jackson, Y. Wei, S.G. Thompson, J.P.T. Higgins. Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis. Statistics
in Medicine, 34(6):984-998, 2015. doi:10.1002/sim.6381.
C. Roever, R. Bender, S. Dias, C.H. Schmid, H. Schmidli, S. Sturtz, S. Weber, T. Friede. On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects
meta-analysis. Research Synthesis Methods, 12(4):448-474, 2021. doi:10.1002/jrsm.1475.
See Also
dlnorm, RhodesEtAlPrior.
# load example data:
# determine corresponding prior parameters:
TP <- TurnerEtAlPrior("surgical", "pharma", "placebo / control")
# a prior 95 percent interval for tau:
## Not run:
# compute effect sizes (log odds ratios) from count data
# (using "metafor" package's "escalc()" function):
crins.es <- escalc(measure="OR",
ai=exp.AR.events, n1i=exp.total,
ci=cont.AR.events, n2i=cont.total,
slab=publication, data=CrinsEtAl2014)
# perform meta analysis:
crins.ma01 <- bayesmeta(crins.es, tau.prior=TP$dprior)
# for comparison perform analysis using weakly informative Cauchy prior:
crins.ma02 <- bayesmeta(crins.es, tau.prior=function(t){dhalfcauchy(t,scale=1)})
# show results:
# compare estimates; heterogeneity (tau):
rbind("Turner prior"=crins.ma01$summary[,"tau"], "Cauchy prior"=crins.ma02$summary[,"tau"])
# effect (mu):
rbind("Turner prior"=crins.ma01$summary[,"mu"], "Cauchy prior"=crins.ma02$summary[,"mu"])
# illustrate heterogeneity priors and posteriors:
plot(crins.ma01, which=4, prior=TRUE, taulim=c(0,2),
main="informative log-normal prior")
plot(crins.ma02, which=4, prior=TRUE, taulim=c(0,2),
main="weakly informative half-Cauchy prior")
plot(crins.ma01, which=3, mulim=c(-3,0),
main="informative log-normal prior")
abline(v=0, lty=3)
plot(crins.ma02, which=3, mulim=c(-3,0),
main="weakly informative half-Cauchy prior")
abline(v=0, lty=3)
# compare prior and posterior 95 percent upper limits for tau:
## End(Not run)
version 3.4 | {"url":"https://search.r-project.org/CRAN/refmans/bayesmeta/html/TurnerEtAlPrior.html","timestamp":"2024-11-05T15:35:16Z","content_type":"text/html","content_length":"10555","record_id":"<urn:uuid:4b42a81a-2445-4969-b051-d1d38350dac6>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00680.warc.gz"} |
Why is 0.7%0.05 giving 0.05 when it should give 0?
Multiply both numbers by 100, or whatever power of 10 is needed to remove the decimal places.
Solution by @RamJoT
I have some code for a sliding GUI and need to check if the number entered into a text box is a multiple of 0.05 from 0 to 0.9.
I check for this by using modulus: if num % 0.05 == 0 then print(num) end
num being the input in the text box.
When I enter 0.3, 0.6 and 0.7, I am given an output of 0.05.
I also typed in a calculator 0.7%0.05, which still gave me 0.05.
I tried many other numbers such as 0.15, 0.5 etc. which still gave 0.05.
I’d love some help
1 Like
a%b is the same as:
a - math.floor(a/b)*b.
0.7 - math.floor(0.7/0.05)*0.05
0.7/0.05 = 14, and floor rounds down but 0.7/0.05 results in an integer anyways.
0.7 - 14*0.05 = 0
Yes, the result should be 0…
1 Like
Perhaps a work around might be to multiply both numbers by 100, to remove the decimal places, and then try the modulus.
1 Like
I’ll try that. Thanks for replying
It worked! Thanks again
You should post your solution so that future people who may have the same problem know how to fix it, and mark @RamJoT’s post as the solution
1 Like
It isn’t actually giving 0.05. This is a floating point imprecision problem.
local n = 0.7 % 0.05
--> 0.04999999999999993339
2 Likes | {"url":"https://devforum.roblox.com/t/why-is-07005-giving-005-when-it-should-give-0/411185","timestamp":"2024-11-09T13:11:11Z","content_type":"text/html","content_length":"35691","record_id":"<urn:uuid:6d99e47c-6b74-4fe9-8344-6c79eba4f78f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00217.warc.gz"} |
Potential issue?
hw4fme1.mod (2.1 KB)
hi, I have some issue dealing with TANK model.
I checked the mod file several times and struggling to find the solution,
it would be highly appreciated to let me know what is the problem with this file.
thanks in advance!
1 Like
Your Taylor rule must be
Try computing the steady state analytically. | {"url":"https://forum.dynare.org/t/potential-issue/26579","timestamp":"2024-11-12T17:23:03Z","content_type":"text/html","content_length":"13694","record_id":"<urn:uuid:92ecde4a-c3d5-42bf-b1b2-bb7fafd82234>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00242.warc.gz"} |
Math, Grade 7, Proportional Relationships, Identifying Proportional Relationships
Driving to the Park
Work Time
Driving to the Park
This table shows that Mr. Lee drove 72.5 miles in 1.5 hours on his way to the amusement park.
• Fill in the missing values in the table, assuming that Mr. Lee drove at a constant speed.
• Does the table represent a proportional relationship? Explain why or why not.
INTERACTIVE: Driving to the Park
What distance did Mr. Lee travel in 1 hour? How can you find out? | {"url":"https://oercommons.org/courseware/lesson/2956/student/?section=4","timestamp":"2024-11-08T07:14:13Z","content_type":"text/html","content_length":"36737","record_id":"<urn:uuid:f5798df2-f7ce-4e69-a639-4c2885d4ce9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00843.warc.gz"} |
The size of a ridge beam for a 24', 22', 20', 18', 16', 12' and 10-foot span - Civil Sir
The size of a ridge beam for a 24′, 22′, 20′, 18′, 16′, 12′ and 10-foot span
The size of a ridge beam needed for a 24′, 22′, 20′, 18′, 16′, 12′ and 10-foot span depends on several factors, including the type and load of the roof, snow load, dead load, live load, local
building codes & regulation, roof pitch, design and construction requirement.
A ridge beam is a horizontal structural member that runs along the peak of a roof, providing support for the roof rafters and helping to distribute the weight of the roof to posts evenly. To
determining the appropriate ridge beam size you should consider various factors.
A ridge beam in construction provides support and stability to the roof structure by connecting the tops of opposing roof slopes. It helps distribute the weight of the roof and prevents sagging or
spreading of the walls.
Here’s is the general guideline for ridge beam size and their span for Douglas Fir lumber:-
1. For a 24 foot span:- The size of a ridge beam for a 24-foot span need to be 3-1/2″×16″, made of Glulam or LVL or engineered wood or 4-2×16 dimensional lumber
2. For a 20 to 22 foot span:- The size of a ridge beam for a 20 to 22-foot span need to be 3-1/2″×14″, made of Glulam or LVL or engineered wood, or 4-2×14 dimensional lumber
3. For a 18 foot span:- The size of a ridge beam for a 18-foot span need to be 4-2×12, made of dimensional lumber or wood
4. For a 14 to 16 foot span:- The size of a ridge beam for a 14 to 16-foot span need to be 3-2×12, made of dimensional lumber or wood
5. For a 12 foot span:- The size of a ridge beam for a 12-foot span need to be 2-2×12, made of dimensional lumber or wood
6. For a 10 foot span:- The size of a ridge beam for a 10-foot span need to be 2-2×10, made of dimensional lumber or wood
7. For a 28 foot span:- The size of a ridge beam for a 28-foot span need to be 4-2×18 dimensional lumber or wood or 5-1/4″×14″ Glulam or LVL
8. For a 30 foot span:- The size of a ridge beam for a 30-foot span need to be 4-2×20 dimensional lumber or wood or 5-1/4″×16″ Glulam or LVL.
What size ridge beam do i need to span 30′, 28′, 24′, 22′, 20′, 18′, 16′, 12′ and 10-foot
Determining the size of a ridge beam for a span depends on various factors, including the type of wood, local building codes, and the load it needs to support.
What size ridge beam do i need to span 30′, 28′, 24′, 22′, 20′, 18′, 16′, 12′ and 10-foot
In general, the size of ridge beam needed to span a 30-foot should be 4-2″×20″ lumber, while the size of ridge beam needed to span a 24-foot should be 4-2″×16″ lumber. Likewise, the size of ridge
beam needed to span a 18-foot should be 4-2″×12″ lumber, while the size of ridge beam needed to span a 16-foot should be 3-2″×12″ lumber. In addition, the size of ridge beam needed to span a 12-foot
should be 2-2″×12″ lumber, while the size of ridge beam needed to span a 10-foot should be 2-2″×10″ lumber.
What size ridge beam do i need for a 24-foot span
The size of the ridge beam needed to span a 24 foot should be 16 inches deep and 8 inches wide. So, you would need something like 4-2″×16″ size of wood ridge beam or 3-1/2″×16″ size of engineered
lumber (LVL, Glulam) for a 24 foot span used for residential application.
What size ridge beam do i need for a 20-foot span
The size of the ridge beam needed to span a 20 foot should be 14 inches deep and 8 inches wide. So, you would need something like 4-2″×14″ size of wood ridge beam or 3-1/2″×11-7/8″ size of engineered
lumber (LVL, Glulam) for a 20 foot span used for residential application.
What size ridge beam do i need for a 28-foot span
The size of the ridge beam needed to span a 28 foot should be 18 inches deep and 8 inches wide. So, you would need something like 4-2″×18″ size of wood ridge beam or 5-1/4″×14″ size of engineered
lumber (LVL, Glulam) for a 28 foot span used for residential application.
What size ridge beam do i need for a 30-foot span
The size of the ridge beam needed to span a 30 foot should be 20 inches deep and 8 inches wide. So, you would need something like 4-2″×20″ size of wood ridge beam or 5-1/4″×16″ size of engineered
lumber (LVL, Glulam) for a 30 foot span used for residential application.
What size ridge beam do i need for a 16-foot span
The size of the ridge beam needed to span a 16 foot should be 12 inches deep and 6 inches wide. So, you would need something like 3-2″×12″ size of wood ridge beam or 3-1/2″× 9-1/2″ size of engineered
lumber (LVL, Glulam) for a 16 foot span used for residential application.
What size ridge beam do i need for a 18-foot span
The size of the ridge beam needed to span a 18 foot should be 12 inches deep and 8 inches wide. So, you would need something like 4-2″×12″ size of wood ridge beam or 3-1/2″× 11-7/8″ size of
engineered lumber (LVL, Glulam) for a 18 foot span used for residential application.
The size of a ridge beam for a 24-foot span need to be 3-1/2″×16″ engineered wood, a 3-1/2″×14″ for a 20-foot span, a 4-2×12 for a 18-foot span, a 3-2×12 for a 16-foot span, a 2-2×12 for a 12-foot
span, a 2-2×10 for a 10-foot span, and a 5-1/4×16 size ridge beam for a 30-foot span. | {"url":"https://civilsir.com/the-size-of-a-ridge-beam-for-a-24-22-20-18-16-12-and-10-foot-span/","timestamp":"2024-11-08T15:45:04Z","content_type":"text/html","content_length":"94095","record_id":"<urn:uuid:dbd32447-5957-4649-a944-fad0f8bef3aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00474.warc.gz"} |
Who's That Mathematician? Paul R. Halmos Collection - Page 25
For more information about Paul R. Halmos (1916-2006) and about the Paul R. Halmos Photograph Collection, please see the introduction to this article on page 1. A new page featuring six photographs
will be posted at the start of each week during 2012.
Halmos photographed logician William Howard in 1978 at a colloquium at the University of Illinois at Chicago Circle, where Howard was a professor. Howard earned his Ph.D. in 1956 from the University
of Chicago with the dissertation “k-Fold Recursion and Well-Ordering,” written under Saunders Mac Lane. He recalls that he first met Halmos, who was a professor at Chicago at the time, in the spring
of 1949 at a conference at the University of Illinois at Urbana-Champaign, where he was a graduate student. That summer, Halmos helped him transfer to the University of Chicago, but first he had to
pass a test:
He asked me: "What kind of mathematics do you like? A page full of formulas, or one symbol sitting in the middle of the page?" My answer: "The latter." Of course, I knew that that was the answer
he would like, but actually, that was my preference.
Howard was a researcher at Bell Labs from 1956 to 1959 and, from 1959 to 1965, was on the faculty at Pennsylvania State University, where he worked with the logician Haskell Curry, who retired from
Penn State in 1966. In 1965 Howard joined the faculty of the University of Illinois at Chicago Circle (University of Illinois at Chicago as of 1982), where he is now Professor Emeritus and lists his
interests as proof theory, foundations of mathematics, and history of mathematics. (Sources: Mathematics Genealogy Project, MacTutor Archive: Curry, UIC Mathematics)
Probabilist Gilbert Hunt (1916-2008) was photographed by Halmos in 1972. After giving up a promising tennis career and performing wartime service as a weather forecaster (1941-46), Hunt earned his
Ph.D. in 1948 from Princeton University with the dissertation “On Stationary Stochastic Processes,” written under Salomon Bochner. From 1946 to 1949, he also worked as an assistant to John von
Neumann at the Institute for Advanced Study in Princeton. He was a faculty member at Cornell University from 1949 to 1959 and 1962 to 1965 and at Princeton from 1959 to 1962 and from 1965 onward. He
is especially well known for his contributions to the theory of Markov processes. (Sources: MacTutor Archive, Mathematics Genealogy Project)
Halmos photographed functional analyst Robert C. James (d. 2004) at the AMS-MAA Joint Summer Mathematics Meetings in Amherst, Massachusetts, on August 26, 1964. James earned his Ph.D. in 1946 from
the California Institute of Technology (Caltech) with the dissertation “Orthogonality in Normed Linear Spaces.” He was the only mathematics professor among the seven founding faculty members at
Harvey Mudd College in Claremont, California, which opened its doors in 1957 to 48 mathematics, science, and engineering students, and he was a professor there at the time this photo was taken.
Harvey Mudd College annually awards the Robert James Prize to two outstanding first year mathematics students. According to his Ph.D. student Steve Bellenot, who also was an undergraduate at Harvey
Mudd College from 1966 to 1970, James left HMC in about 1968, spent a year at SUNY Albany, and then returned to Claremont to serve as a founding professor in the mathematics program at the Claremont
Graduate School (now the Claremont Graduate University), where he spent the rest of his career and where Bellenot earned his Ph.D. in 1974.
In his primary research field, Banach spaces, James was especially well known for "James' Theorem" and for his counterexamples. Bellenot, who is now a mathematics professor at Florida State
University, remembers that James named the solar-powered retirement retreat he built in the Sierra foothills of California "Trees and Bushes" after its surroundings and after a famous counterexample
called the James tree space and a collection of essays on James trees spaces titled The James Forest (LMS, 1997). James may have been even better known for the Mathematics Dictionary he co-authored
with his father, Glenn James. Founded by the elder James in about 1940, Bob James published its 5th and final edition in 1992. (Sources: Mathematics Genealogy Project, Harvey Mudd College History,
Harvey Mudd College Math Awards, MathSciNet)
Number theorists Ralph D. James, Leonard Tornheim, and Donald J. Lewis (left to right) were photographed by Halmos in about 1955.
Born in England and educated in Vancouver, British Columbia, Canada, Ralph James (1909-1979) earned his Ph.D. in 1932 from the University of Chicago with the dissertation “Analytical Investigations
in Waring’s Theorem,” written under Leonard Eugene Dickson. After studying with E. T. Bell for a year at Caltech and G. H. Hardy for a year at Cambridge, James was on the mathematics faculty at the
University of California, Berkeley, for five years and the University of Saskatchewan for another five years before becoming professor of mathematics at the University of British Columbia in 1943.
From 1961 to 1963, he served as president of the Canadian Mathematical Society. He retired from UBC in 1974 after coordinating the International Congress of Mathematicians held in Vancouver that
year. (Sources: MacTutor Archive, CMS Presidents)
Leonard Tornheim earned his Ph.D. in 1938 from the University of Chicago with the dissertation “Integral Sets of Quaternion Algebras over a Function Field,” written under advisor A. Adrian Albert. He
published in a wide variety of fields, including number theory, classical algebra, harmonic analysis, numerical analysis, and statistics, from 1941 to 1970. His publications include the book Vector
Spaces and Matrices, co-authored with Robert Thrall and published in 1957. He was a faculty member at the University of Michigan from 1946 to 1955 (as was Robert Thrall from 1937 to 1969). (Sources:
Mathematics Genealogy Project, MathSciNet, UM Faculty History Project: Tornheim, UM Faculty History Project: Thrall)
Donald J. (D.J.) Lewis (1926-2015) earned his Ph.D. in 1950 from the University of Michigan under advisor Richard Brauer. After holding positions at Ohio State University (1950-52), the Institute for
Advanced Study (1952-53), and the University of Notre Dame (1953-61), Lewis returned to the University of Michigan, where he has spent the rest of his career and advised at least 25 Ph.D. students.
He received the American Mathematical Society’s Award for Distinguished Public Service in 1995 and then continued to serve the mathematical community as director of the National Science Foundation’s
Division of Mathematical Sciences from 1995 to 1999. (Sources: “1995 Award for Distinguished Public Service to Mathematics,” AMS Notices 42:4 (April 1995); Mathematics Genealogy Project; UM Faculty
History Project)
Halmos photographed F. Burton Jones, Madeleine Jones, and Mary Ellen Rudin at the International Congress of Mathematicians in Vancouver, British Columbia, Canada, on August 22, 1974. (See page 7 of
this collection for a photograph of Enrico Bombieri and David Mumford, who were awarded Fields Medals at the 1974 ICM in Vancouver.)
F. Burton Jones (1910-1999) earned his Ph.D. in 1935 from the University of Texas at Austin with a dissertation in general topology written under advisor R. L. Moore. He was on the mathematics
faculty at UT Austin from 1935 to 1950 (except for 1942-44, when he did war work), at the University of North Carolina at Chapel Hill from 1950 to 1962, and at the University of California,
Riverside, from 1962 onward. Jones wrote important papers on point set topology and homogeneous continua, and was known as both an excellent researcher and an excellent teacher. (Source: MacTutor
Archive; see also the biography of Jones available from the California Digital Library or from UC Riverside as a pdf file.)
Madeleine Maire Jones (1918-2006) married F. Burton Jones in 1936. A mathematics and education major at the University of Texas at Austin, she assisted with the publication, Creative Teaching: The
Heritage of R. L. Moore (1972, 1993, 1999), now available online. In 1987-88, she and her husband endowed the F. Burton Jones Chair in Topology at the University of California, Riverside. (Sources:
ancestry.com, Creative Teaching: The Heritage of R. L. Moore at Topology Atlas)
Mary Ellen Estill Rudin (1924-2013) earned her Ph.D. in 1949 from the University of Texas at Austin with a dissertation in general topology written under R. L. Moore. She also was influenced by F.
Burton Jones. From 1949 to 1953, she was on the mathematics faculty at Duke University, and from 1953, when she married complex analyst Walter Rudin, to 1971, she held “temporary part time” positions
at the University of Rochester and the University of Wisconsin at Madison while her husband held full-time positions at these institutions. In 1971, she became Professor of Mathematics at Wisconsin,
in 1981 the first Grace Chisholm Young Professor there, and in 1991 Emeritus Professor. Throughout her career, Rudin published important papers in set-theoretic topology and was especially well known
for her skill in constructing counterexamples. She was selected to give the Association for Women in Mathematics Noether Lecture at the Joint Mathematics Meetings in Louisville, Kentucky, in 1984,
speaking on set-theoretic aspects of “Paracompactness.” (Sources: MacTutor Archive, Mathematics Genealogy Project, UW Mathematics, AWM Noether Lecturers)
Vaughan Jones, left, discoverer of the Jones polynomial in knot theory, is pictured together with the discoverers of the HOMFLY polynomial, a generalization of the Jones polynomial, on February 10,
1985. The HOMFLY discoverers are, left to right, Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter Freyd, W. B. Raymond Lickorish, and David Yetter. According to Millett, the photo was taken on the
University of California, Berkeley, campus during a pair of Mathematical Sciences Research Institute (MSRI) conferences / workshops on index theory and low-dimensional topology, and this was "the
only time that all of us were together." (Source: Wolfram MathWorld)
Born and educated in New Zealand through the M.Sc. degree, Vaughan Jones earned his Ph.D. in 1979 from the University of Geneva, Switzerland. After one year at the University of California, Los
Angeles, and four years at the University of Pennsylvania in Philadelphia, he moved to the University of California, Berkeley, in 1985. He received the Fields Medal in 1990 at the International
Congress of Mathematicians in Kyoto, Japan, for finding a new polynomial invariant for knots and links, a discovery arising from his proof of the Index Theorem for von Neumann Algebras. In 2011,
Jones became Distinguished Professor of Mathematics at Vanderbilt University in Nashville, Tennessee. (Sources: MacTutor Archive, Vanderbilt University Department of Mathematics)
Jim Hoste earned his Ph.D. in 1982 from the University of Utah in Salt Lake City with the dissertation "Sewn-up r-Link Exteriors." He is now professor of mathematics at Pitzer College in Claremont,
California. (Sources: Mathematics Genealogy Project, Pitzer College Mathematics)
Adrian Ocneanu earned his Ph.D. in 1983 from the University of Warwick, England, with a dissertation on von Neumann algebras written under Ciprian Foias (pictured on page 14 of this collection).
Ocneanu is now professor of mathematics at Pennsylvania State University in State College. (Sources: Mathematics Genealogy Project, Penn State Mathematics)
Ken Millett earned his Ph.D. in 1967 from the University of Wisconsin, Madison, with a dissertation on Euclidean bundle pairs. In 1998, he received the AMS Award for Distinguished Public Service. He
is professor of mathematics at the University of California, Santa Barbara. (Sources: Mathematics Genealogy Project, UC Santa Barbara Mathematics)
Peter Freyd earned his Ph.D. in 1960 from Princeton University with a dissertation on functor theory. He is now professor emeritus of mathematics and director of the logic and computation group at
the University of Pennsylvania in Philadelphia. (Sources: Mathematics Genealogy Project, University of Pennsylvania Mathematics)
W. B. Raymond Lickorish earned his Ph.D. in 1964 from the University of Cambridge, England, where he is now emeritus professor of geometric topology. He is the author of An Introduction to Knot
Theory in the Springer Graduate Texts in Mathematics series. (Sources: Mathematics Genealogy Project, amazon.com)
David Yetter earned his Ph.D. in 1984 from the University of Pennsylvania with the dissertation "Aspects of Synthetic Differential Geometry," written under advisor Peter Freyd. Vaughan Jones was also
on the U Penn faculty from 1981 to 1985. Yetter is now professor of mathematics at Kansas State University in Manhattan, Kansas. (Sources: Mathematics Genealogy Project, KSU Mathematics)
For an introduction to this article and to the Paul R. Halmos Photograph Collection, please see page 1. Watch for a new page featuring six new photographs each week during 2012.
Regarding sources for this page: Information for which a source is not given either appeared on the reverse side of the photograph or was obtained from various sources during 2011-12 by archivist
Carol Mead of the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas, Austin. | {"url":"https://old.maa.org/press/periodicals/convergence/whos-that-mathematician-paul-r-halmos-collection-page-25","timestamp":"2024-11-11T10:19:04Z","content_type":"application/xhtml+xml","content_length":"135307","record_id":"<urn:uuid:bac3fed0-5693-4901-bd33-c2d8c6829b5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00086.warc.gz"} |
Number and Operations (NCTM)
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Develop meaning for integers and represent and compare quantities with them.
Understand meanings of operations and how they relate to one another.
Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and
Compute fluently and make reasonable estimates.
Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.
Grade 7 Curriculum Focal Points (NCTM)
Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations
Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of
arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding,
subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these
equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of
equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation. | {"url":"https://newpathworksheets.com/math/grade-7/using-integers?dictionary=number+line&did=136","timestamp":"2024-11-13T04:54:39Z","content_type":"text/html","content_length":"44660","record_id":"<urn:uuid:6de17b59-1523-4b77-b51d-72190218bc79>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00001.warc.gz"} |
Moment capacity (section) (Columns: AS 4100)
The (section) moment capacity check is performed according to AS 4100 clause 5.1 for the moment about the x-x axis (M[x]) and about the y-y axis (M[y]), at the point under consideration.
For (member) moment capacity refer to the section Lateral torsional buckling resistance (Member moment capacity).
Note that for all section types, the effective section modulus about the major axis (Z[ex]) will be based on the minimum slenderness ratio considering both flange and web. Internally Tekla Structural
Designer will calculate the following:
• flange slenderness ratio, z[f] = (λ[ey] - λ[ef]) / ( λ[eyf] - λ[epf])
• web slenderness ratio, z[w] = (λ[eyw] - λ[ew]) / ( λ[eyw] - λ[epw])
For sections which have flexure major class either Compact or Non-compact, the effective section modulus about the major axis (Z[ex]) will then be calculated by:
• Z[ex] = Z[x] + [MIN(z[f], z[w], 1.0) * (Z[c] - Z[x])] where Z[c] = MIN(S[x], 1.5 * Z[x])
Note that for Channel sections under minor axis bending:
• if there is single curvature with the flange tips in compression then Z[ey] will be based on Z[eyR]
• if there is single curvature with the web in compression then Z[ey] will be based on Z[eyL]
• if there is double curvature then Z[ey] will be based on the minimum of Z[eyR] and Z[eyL]
Eccentricity Moments
Eccentricity moment will be added algebraically to the coincident real moment (at top or bottom of column stack) only if the resulting 'combined' moment has a larger absolute magnitude than the
absolute real moment alone.
The resulting 'combined' design moment (major and/or minor) will be that used in moment capacity, combined bending & shear, LTB, and combined actions checks. | {"url":"https://support.tekla.com/doc/tekla-structural-designer/2024/ref_momentcapacitysectioncolumnsas4100","timestamp":"2024-11-11T15:07:58Z","content_type":"text/html","content_length":"54637","record_id":"<urn:uuid:a9262cd6-db0e-4305-8246-a8758c72215a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00795.warc.gz"} |
--- title: "An Introduction to spmodel" author: "Michael Dumelle, Matt Higham, and Jay M. Ver Hoef" bibliography: '`r system.file("references.bib", package="spmodel")`' output: html_document: theme:
flatly number_sections: true highlighted: default toc: yes toc_float: collapsed: no smooth_scroll: no toc_depth: 3 vignette: > %\VignetteIndexEntry{An Introduction to spmodel} %\VignetteEngine
{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, include = FALSE} # # jss style # knitr::opts_chunk$set(prompt=TRUE, echo = TRUE, highlight = FALSE, continue = " + ", comment = "") # options
(replace.assign=TRUE, width=90, prompt="R> ") # rmd style knitr::opts_chunk$set(collapse = FALSE, comment = "#>", warning = FALSE, message = FALSE) # load packages library(ggplot2) library(spmodel)
``` # Introduction The `spmodel` package is used to fit and summarize spatial models and make predictions at unobserved locations (Kriging). This vignette provides an overview of basic features in
`spmodel`. We load `spmodel` by running ```{r, eval = FALSE} library(spmodel) ``` If you use `spmodel` in a formal publication or report, please cite it. Citing `spmodel` lets us devote more
resources to it in the future. We view the `spmodel` citation by running ```{r} citation(package = "spmodel") ``` There are three more `spmodel` vignettes available on our website at [https://
usepa.github.io/spmodel/](https://usepa.github.io/spmodel/): 1. A Detailed Guide to `spmodel` 2. Spatial Generalized Linear Models in `spmodel 3. Technical Details Additionally, there are two
workbooks that have accompanied recent `spmodel` workshops: 1. 2024 Society for Freshwater Science Conference: "Spatial Analysis and Statistical Modeling with R and `spmodel` available at [https://
usepa.github.io/spworkshop.sfs24/](https://usepa.github.io/spworkshop.sfs24/) 2. 2023 Spatial Statistics Conference: `spmodel` workshop available at [https://usepa.github.io/spmodel.spatialstat2023/]
(https://usepa.github.io/spmodel.spatialstat2023/) # The Data Many of the data sets we use in this vignette are `sf` objects. `sf` objects are data frames (or tibbles) with a special structure that
stores spatial information. They are built using the `sf` [@pebesma2018sf] package, which is installed alongside `spmodel`. We will use six data sets throughout this vignette: * `moss`: An `sf`
object with heavy metal concentrations in Alaska. * `sulfate`: An `sf` object with sulfate measurements in the conterminous United States. * `sulfate_preds`: An `sf` object with locations at which to
predict sulfate measurements in the conterminous United States. * `caribou`: A `tibble` (a special `data.frame`) for a caribou foraging experiment in Alaska. * `moose`: An `sf` object with moose
measurements in Alaska. * `moose_preds`: An `sf` object with locations at which to predict moose measurements in Alaska. We will create visualizations using ggplot2 [@wickham2016ggplot2], which we
load by running ```{r, eval = FALSE} library(ggplot2) ``` ggplot2 is only installed alongside `spmodel` when `dependencies = TRUE` in `install.packages()`, so check that it is installed before
reproducing any visualizations in this vignette. # Spatial Linear Models {#sec:splm} Spatial linear models for a quantitative response vector $\mathbf{y}$ have spatially dependent random errors and
are often parameterized as $$ \mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\tau} + \boldsymbol{\epsilon}, $$ where $\mathbf{X}$ is a matrix of explanatory variables (usually including a
column of 1's for an intercept), $\boldsymbol{\beta}$ is a vector of fixed effects that describe the average impact of $\mathbf{X}$ on $\mathbf{y}$, $\boldsymbol{\tau}$ is a vector of spatially
dependent (correlated) random errors, and $\boldsymbol{\epsilon}$ is a vector of spatially independent (uncorrelated) random errors. The spatial dependence of $\boldsymbol{\tau}$ is explicitly
specified using a spatial covariance function that incorporates the variance of $\boldsymbol{\tau}$, often called the partial sill, and a range parameter that controls the behavior of the spatial
covariance. The variance of $\boldsymbol{\epsilon}$ is often called the nugget. Spatial linear models are fit in `spmodel` for point-referenced and areal data. Data are point-referenced when the
elements in $\mathbf{y}$ are observed at point-locations indexed by x-coordinates and y-coordinates on a spatially continuous surface with an infinite number of locations. The `splm()` function is
used to fit spatial linear models for point-referenced data (these are often called geostatistical models). Data are areal when the elements in $\mathbf{y}$ are observed as part of a finite network
of polygons whose connections are indexed by a neighborhood structure. For example, the polygons may represent counties in a state who are neighbors if they share at least one boundary. The `spautor
()` function is used to fit spatial linear models for areal data (these are often called spatial autoregressive models). This vignette focuses on spatial linear models for point-referenced data,
though `spmodel`'s other vignettes discuss spatial linear models for areal data. The `splm()` function has similar syntax and output as the commonly used `lm()` function used to fit non-spatial
linear models. `splm()` generally requires at least three arguments: * `formula`: a formula that describes the relationship between the response variable and explanatory variables. * `formula` uses
the same syntax as the `formula` argument in `lm()` * `data`: a `data.frame` or `sf` object that contains the response variable, explanatory variables, and spatial information. * `spcov_type`: the
spatial covariance type (`"exponential"`, `"spherical"`, `"matern"`, etc). If `data` is an `sf` object, the coordinate information is taken from the object's geometry. If `data` is a `data.frame` (or
`tibble`), then `xcoord` and `ycoord` are required arguments to `splm()` that specify the columns in `data` representing the x-coordinates and y-coordinates, respectively. `spmodel` uses the spatial
coordinates "as-is," meaning that `spmodel` does not perform any projections. To project your data or change the coordinate reference system, use `sf::st_transform()`. If an `sf` object with polygon
geometries is given to `splm()`, the centroids of each polygon are used to fit the spatial linear model. Next we show the basic features and syntax of `splm()` using the Alaskan `moss` data. We study
the impact of log distance to the road (`log_dist2road`) on log zinc concentration (`log_Zn`). We view the first few rows of the `moss` data by running ```{r} moss ``` We can visualize the
distribution of log zinc concentration (`log_Zn`) by running ```{r} ggplot(moss, aes(color = log_Zn)) + geom_sf() + scale_color_viridis_c() ``` Log zinc concentration appears highest in the middle of
the spatial domain, which has a road running through it. We fit a spatial linear model regressing log zinc concentration on log distance to the road using an exponential spatial covariance function
by running ```{r} spmod <- splm(log_Zn ~ log_dist2road, data = moss, spcov_type = "exponential") ``` The estimation method is specified via the `estmethod` argument, which has a default value of
`"reml"` for restricted maximum likelihood. Other estimation methods are `"ml"` for maximum likelihood, `"sv-wls"` for semivariogram weighted least squares, and `"sv-cl"` for semivariogram composite
likelihood. Printing `spmod` shows the function call, the estimated fixed effect coefficients, and the estimated spatial covariance parameters. `de` is the estimated variance of $\boldsymbol{\tau}$
(the spatially dependent random error), `ie` is the estimated variance of $\boldsymbol{\epsilon}$ (the spatially independent random error), and `range` is the range parameter. ```{r} print(spmod) ```
Next we show how to obtain more detailed summary information from the fitted model. ## Model Summaries We summarize the fitted model by running ```{r} summary(spmod) ``` Similar to summaries of `lm()
` objects, summaries of `splm()` objects include the original function call, residuals, and a coefficients table of fixed effects. Log zinc concentration appears to significantly decrease with log
distance from the road, as evidenced by the small p-value associated with the asymptotic z-test. A pseudo r-squared is also returned, which quantifies the proportion of variability explained by the
fixed effects. In the remainder of this subsection, we describe the broom [@robinson2021broom] functions `tidy()`, `glance()` and `augment()`. `tidy()` tidies coefficient output in a convenient
`tibble`, `glance()` glances at model-fit statistics, and `augment()` augments the data with fitted model diagnostics. We tidy the fixed effects by running ```{r} tidy(spmod) ``` We glance at the
model-fit statistics by running ```{r} glance(spmod) ``` The columns of this `tibble` represent: * `n`: The sample size * `p`: The number of fixed effects (linearly independent columns in $\mathbf{X}
$) * `npar`: The number of estimated covariance parameters * `value`: The value of the minimized objective function used when fitting the model * `AIC`: The Akaike Information Criterion (AIC) * `AICc
`: The AIC with a small sample size correction * `BIC`: The Bayesian Information Criterion (BIC) * `logLik`: The log-likelihood * `deviance`: The deviance * `pseudo.r.squared`: The pseudo r-squared
The `glances()` function can be used to glance at multiple models at once. Suppose we wanted to compare the current model, which uses an exponential spatial covariance, to a new model without spatial
covariance (equivalent to a model fit using `lm()`). We do this using `glances()` by running ```{r} lmod <- splm(log_Zn ~ log_dist2road, data = moss, spcov_type = "none") glances(spmod, lmod) ``` The
much lower AIC and AICc for the spatial linear model indicates it is a much better fit to the data. Outside of `glance()` and `glances()`, the functions `AIC()`, `AICc()`, `BIC()` `logLik()`,
`deviance()`, and `pseudoR2()` are available to compute the relevant statistics. We augment the data with diagnostics by running ```{r} augment(spmod) ``` The columns of this tibble represent: *
`log_Zn`: The log zinc concentration. * `log_dist2road`: The log distance to the road. * `.fitted`: The fitted values (the estimated mean given the explanatory variable values). * `.resid`: The
residuals (the response minus the fitted values). * `.hat`: The leverage (hat) values. * `.cooksd`: The Cook's distance * `.std.residuals`: Standardized residuals * `geometry`: The spatial
information in the `sf` object. By default, `augment()` only returns the variables in the data used by the model. All variables from the original data are returned by setting `drop = FALSE`. Many of
these model diagnostics can be visualized by running `plot(spmod)`. We can learn more about `plot()` in `spmodel` by running `help("plot.spmodel", "spmodel")`. ## Prediction (Kriging) Commonly a goal
of a data analysis is to make predictions at unobserved locations. In spatial contexts, prediction is often called Kriging. Next we use the `sulfate` data to build a spatial linear model of sulfate
measurements in the conterminous United States with the goal of making sulfate predictions (Kriging) for the unobserved locations in `sulfate_preds`. We visualize the distribution of `sulfate` by
running ```{r} ggplot(sulfate, aes(color = sulfate)) + geom_sf(size = 2) + scale_color_viridis_c(limits = c(0, 45)) ``` Sulfate appears spatially dependent, as measurements are highest in the
Northeast and lowest in the Midwest and West. We fit a spatial linear model regressing sulfate on an intercept using a spherical spatial covariance function by running ```{r} sulfmod <- splm(sulfate
~ 1, data = sulfate, spcov_type = "spherical") ``` We make predictions at the locations in `sulfate_preds` and store them as a new variable called `preds` in the `sulfate_preds` data set by running
```{r} sulfate_preds$preds <- predict(sulfmod, newdata = sulfate_preds) ``` We visualize these predictions by running ```{r} ggplot(sulfate_preds, aes(color = preds)) + geom_sf(size = 2) +
scale_color_viridis_c(limits = c(0, 45)) ``` These predictions have similar sulfate patterns as in the observed data (predicted values are highest in the Northeast and lowest in the Midwest and
West). Next we remove the model predictions from `sulfate_preds` before showing how `augment()` can be used to obtain the same predictions: ```{r} sulfate_preds$preds <- NULL ``` While `augment()`
was previously used to augment the original data with model diagnostics, it can also be used to augment the `newdata` data with predictions: ```{r} augment(sulfmod, newdata = sulfate_preds) ``` Here
`.fitted` represents the predictions. Confidence intervals for the mean response or prediction intervals for the predicted response can be obtained by specifying the `interval` argument in `predict()
` and `augment()`: ```{r} augment(sulfmod, newdata = sulfate_preds, interval = "prediction") ``` By default, `predict()` and `augment()` compute 95% intervals, though this can be changed using the
`level` argument. While the fitted model in this example only used an intercept, the same code is used for prediction with fitted models having explanatory variables. If explanatory variables were
used to fit the model, the same explanatory variables must be included in `newdata` with the same names they have in `data`. If `data` is a `data.frame`, coordinates must be included in `newdata`
with the same names as they have in `data`. If `data` is an `sf` object, coordinates must be included in `newdata` with the same geometry name as they have in `data`. When using projected
coordinates, the projection for `newdata` should be the same as the projection for `data`. ## An Additional Example We now use the `caribou` data from a foraging experiment conducted in Alaska to
show an application of `splm()` to data stored in a `tibble` (`data.frame`) instead of an `sf` object. In `caribou`, the x-coordinates are stored in the `x` column and the y-coordinates are stored in
the `y` column. We view the first few rows of `caribou` by running ```{r} caribou ``` We fit a spatial linear model regressing nitrogen percentage (`z`) on water presence (`water`) and tarp cover
(`tarp`) by running ```{r} cariboumod <- splm(z ~ water + tarp, data = caribou, spcov_type = "exponential", xcoord = x, ycoord = y) ``` An analysis of variance can be conducted to assess the overall
impact of the `tarp` variable, which has three levels (clear, shade, and none), and the `water` variable, which has two levels (water and no water). We perform an analysis of variance by running
```{r} anova(cariboumod) ``` There seems to be significant evidence that at least one tarp cover impacts nitrogen. Note that, like in `summary()`, these p-values are associated with an asymptotic
hypothesis test (here, an asymptotic Chi-squared test). # Spatial Generalized Linear Models When building spatial linear models, the response vector $\mathbf{y}$ is typically assumed Gaussian (given
$\mathbf{X}$). Relaxing this assumption on the distribution of $\mathbf{y}$ yields a rich class of spatial generalized linear models that can describe binary data, proportion data, count data, and
skewed data. Spatial generalized linear models are parameterized as $$ g(\boldsymbol{\mu}) = \boldsymbol{\eta} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\tau} + \boldsymbol{\epsilon}, $$ where $g
(\cdot)$ is called a link function, $\boldsymbol{\mu}$ is the mean of $\mathbf{y}$, and the remaining terms $\mathbf{X}$, $\boldsymbol{\beta}$, $\boldsymbol{\tau}$, $\boldsymbol{\epsilon}$ represent
the same quantities as for the spatial linear models. The link function, $g(\cdot)$, "links" a function of $\boldsymbol{\mu}$ to the linear term $\boldsymbol{\eta}$, denoted here as $\mathbf{X} \
boldsymbol{\beta} + \boldsymbol{\tau} + \boldsymbol{\epsilon}$, which is familiar from spatial linear models. Note that the linking of $\boldsymbol{\mu}$ to $\boldsymbol{\eta}$ applies element-wise
to each vector. Each link function $g(\cdot)$ has a corresponding inverse link function, $g^{-1}(\cdot)$. The inverse link function "links" a function of $\boldsymbol{\eta}$ to $\boldsymbol{\mu}$.
Notice that for spatial generalized linear models, we are not modeling $\mathbf{y}$ directly as we do for spatial linear models, but rather we are modeling a function of the mean of $\mathbf{y}$.
Also notice that $\boldsymbol{\eta}$ is unconstrained but $\boldsymbol{\mu}$ is usually constrained in some way (e.g., positive). Next we discuss the specific distributions and link functions used in
`spmodel`. `spmodel` allows fitting of spatial generalized linear models when $\mathbf{y}$ is a binomial (or Bernoulli), beta, Poisson, negative binomial, gamma, or inverse Gaussian random vector.
For binomial and beta $\mathbf{y}$, the logit link function is defined as $g(\boldsymbol{\mu}) = \ln(\frac{\boldsymbol{\mu}}{1 - \boldsymbol{\mu}}) = \boldsymbol{\eta}$, and the inverse logit link
function is defined as $g^{-1}(\boldsymbol{\eta}) = \frac{\exp(\boldsymbol{\eta})}{1 + \exp(\boldsymbol{\eta})} = \boldsymbol{\mu}$. For Poisson, negative binomial, gamma, and inverse Gaussian $\
mathbf{y}$, the log link function is defined as $g(\boldsymbol{\mu}) = \ln(\boldsymbol{\mu}) = \boldsymbol{\eta}$, and the inverse log link function is defined as $g^{-1}(\boldsymbol{\eta}) = \exp(\
boldsymbol{\eta}) = \boldsymbol{\mu}$. As with spatial linear models, spatial generalized linear models are fit in `spmodel` for point-referenced and areal data. The `spglm()` function is used to fit
spatial generalized linear models for point-referenced data, and the `spgautor()` function is used to fit spatial generalized linear models for areal data. Though this vignette focuses on
point-referenced data, `spmodel`'s other vignettes discuss spatial generalized linear models for areal data. The `spglm()` function is quite similar to the `splm()` function, though one additional
argument is required: * `family`: the generalized linear model family (i.e., the distribution of $\mathbf{y}$). `family` can be `binomial`, `beta`, `poisson`, `nbinomial`, `Gamma`, or
`inverse.gaussian`. * `family` uses similar syntax as the `family` argument in `glm()`. * One difference between `family` in `spglm()` compared to `family` in `glm()` is that the link function is
fixed in `spglm()`. Next we show the basic features and syntax of `spglm()` using the `moose` data. We study the impact of elevation (`elev`) on the presence of moose (`presence`) observed at a site
location in Alaska. `presence` equals one if at least one moose was observed at the site and zero otherwise. We view the first few rows of the `moose` data by running ```{r} moose ``` We can
visualize the distribution of moose presence by running ```{r} ggplot(moose, aes(color = presence)) + scale_color_viridis_d(option = "H") + geom_sf(size = 2) ``` One example of a generalized linear
model is a binomial (e.g., logistic) regression model. Binomial regression models are often used to model presence data such as this. To quantify the relationship between moose presence and
elevation, we fit a spatial binomial regression model (a specific spatial generalized linear model) by running ```{r} binmod <- spglm(presence ~ elev, family = "binomial", data = moose, spcov_type =
"exponential") ``` The estimation method is specified via the `estmethod` argument, which has a default value of `"reml"` for restricted maximum likelihood. The other estimation method is `"ml"` for
maximum likelihood. Printing `binmod` shows the function call, the estimated fixed effect coefficients (on the link scale), the estimated spatial covariance parameters, and a dispersion parameter.
The dispersion parameter is estimated for some spatial generalized linear models and changes the mean-variance relationship of $\mathbf{y}$. For binomial regression models, the dispersion parameter
is not estimated and is always fixed at one. ```{r} print(binmod) ``` ## Model Summaries We summarize the fitted model by running ```{r} summary(binmod) ``` Similar to summaries of `glm()` objects,
summaries of `spglm()` objects include the original function call, summary statistics of the deviance residuals, and a coefficients table of fixed effects. The logit of moose presence probability
does not appear to be related to elevation, as evidenced by the large p-value associated with the asymptotic z-test. A pseudo r-squared is also returned, which quantifies the proportion of
variability explained by the fixed effects. The spatial covariance parameters and dispersion parameter are also returned. The `tidy()`, `glance()`, and `augment()` functions behave similarly for
`spglm()` objects as they do for `splm()` objects. We tidy the fixed effects (on the link scale) by running ```{r} tidy(binmod) ``` We glance at the model-fit statistics by running ```{r} glance
(binmod) ``` We glance at the spatial binomial regression model and a non-spatial binomial regression model by running ```{r} glmod <- spglm(presence ~ elev, family = "binomial", data = moose,
spcov_type = "none") glances(binmod, glmod) ``` The lower AIC and AICc for the spatial binomial regression model indicates it is a much better fit to the data. We augment the data with diagnostics by
running ```{r} augment(binmod) ``` ## Prediction (Kriging) For spatial generalized linear models, we are predicting the mean of the process generating the observation rather than the observation
itself. We make predictions of moose presence probability at the locations in `moose_preds` by running ```{r} moose_preds$preds <- predict(binmod, newdata = moose_preds, type = "response") ``` The
type argument specifies whether predictions are returned on the link or response (inverse link) scale. We visualize these predictions by running ```{r} ggplot(moose_preds, aes(color = preds)) +
geom_sf(size = 2) + scale_color_viridis_c(limits = c(0, 1), option = "H") ``` These predictions have similar spatial patterns as moose presence the observed data. Next we remove the model predictions
from `moose_preds` and show how `augment()` can be used to obtain the same predictions alongside prediction intervals (on the response scale): ```{r} moose_preds$preds <- NULL augment(binmod, newdata
= moose_preds, type.predict = "response", interval = "prediction") ``` # Function Glossary Here we list some commonly used `spmodel` functions. * `AIC()`: Compute the AIC. * `AICc()`: Compute the
AICc. * `anova()`: Perform an analysis of variance. * `augment()`: Augment data with diagnostics or new data with predictions. * `AUROC()`: Compute the area under the receiver operating
characteristic curve for binary spatial generalized linear models. * `BIC()`: Compute the BIC. * `coef()`: Return coefficients. * `confint()`: Compute confidence intervals. * `cooks.distance()`:
Compute Cook's distance. * `covmatrix()`: Return covariance matrices. * `deviance()`: Compute the deviance. * `esv()`: Compute an empirical semivariogram. * `fitted()`: Compute fitted values. *
`glance()`: Glance at a fitted model. * `glances()`: Glance at multiple fitted models. * `hatvalues()`: Compute leverage (hat) values. * `logLik()`: Compute the log-likelihood. * `loocv()`: Perform
leave-one-out cross validation. * `model.matrix()`: Return the model matrix ($\mathbf{X}$). * `plot()`: Create fitted model plots. * `predict()`: Compute predictions and prediction intervals. *
`pseudoR2()`: Compute the pseudo r-squared. * `residuals()`: Compute residuals. * `spautor()`: Fit a spatial linear model for areal data (i.e., spatial autoregressive model). * `spautorRF()`: Fit a
random forest spatial residual model for areal data. * `spgautor()`: Fit a spatial generalized linear model for areal data (i.e., spatial generalized autoregressive model). * `splm()`: Fit a spatial
linear model for point-referenced data (i.e., geostatistical model). * `splmRF()`: Fit a random forest spatial residual model for point-referenced data. * `spglm()`: Fit a spatial generalized linear
model for point-referenced data (i.e., generalized geostatistical model). * `sprbeta()`: Simulate spatially correlated beta random variables. * `sprbinom()`: Simulate spatially correlated binomial
(Bernoulli) random variables. * `sprgamma()`: Simulate spatially correlated gamma random variables. * `sprinvgauss()`: Simulate spatially correlated inverse Gaussian random variables. * `sprnbinom()
`: Simulate spatially correlated negative binomial random variables. * `sprnorm()`: Simulate spatially correlated normal (Gaussian) random variables. * `sprpois()`: Simulate spatially correlated
Poisson random variables. * `summary()`: Summarize fitted models. * `tidy()`: Tidy fitted models. * `varcomp()`: Compare variance components. * `vcov()`: Compute variance-covariance matrices of
estimated parameters. For a full list of `spmodel` functions alongside their documentation, see the documentation manual. Documentation for methods of generic functions that are defined outside of
`spmodel` can be found by running `help("generic.spmodel", "spmodel")` (e.g., `help("summary.spmodel", "spmodel")`, `help("predict.spmodel", "spmodel")`, etc.). Note that `?generic.spmodel` is
shorthand for `help("generic.spmodel", "spmodel")`. # Support for Additional **R** Packages `spmodel` provides support for: 1. The `tidy()`, `glance()` and `augment()` functions from the `broom`
**R** package [@robinson2021broom]. 2. The `emmeans` **R** package [@lenth2024emmeans] applied to `splm()`, `spautor()`, `spglm()`, and `spgautor()` model objects from `spmodel`. # References | {"url":"https://www.stats.bris.ac.uk/R/web/packages/spmodel/vignettes/introduction.Rmd","timestamp":"2024-11-14T07:19:26Z","content_type":"text/plain","content_length":"26653","record_id":"<urn:uuid:33de4f0a-c8d0-4e79-8b6e-7901e55a138d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00021.warc.gz"} |
Prior Election Czars' decision to count votes according to their own method, rather than using the Single Transferable Voting method recommended by the Senate Elections Handbook, disenfranchised many
student voters, an investigation by The Inquirer found.
The Student Body Elections Handbook recommends that all elections at Reed be counted by the Single Transferable Voting (STV) method, but gives Election Czars the power to change the counting method.
When asked if they used the Single transferable Voting method, former Election Czar Ares Carnathan, who oversaw the Fall 2022 and Spring 2023 student body elections, said, "I have no motivation nor
inclination to find the answers to the questions you ask. Please do not contact me again." Former Election Czar Aidan Mokalla, who oversaw the Fall 2023 election, did not respond to a request for
However, the data suggests that neither Election Czar used STV. The official results for all three elections show the total number of votes increasing after each round of counting, ultimately
exceeding Reed's student body size. This is impossible under STV. To understand why, let's consider a simplified example.
To get started, cast your vote by ranking the blue, red, and yellow candidates.
Your vote is awarded to your first choice, the candidate.
But this election won't be very interesting with only one voter. Let's add some more.
For the sake of simplicity, there will be only 21 voters in this model election, including you. The three candidates are running for two open seats.
Counting Round: 1
Each voter's ballot is awarded to their first choice candidate.
The minimum number of votes that a candidate needs to win is set based on the total number of votes and the number of seats up for grabs, using the Droop Formula. In this case, 21 votes have been
cast to fill 2 open seats, so candidates will need at least 8 votes to win.
The candidate has already met the quota, and is elected to the first open seat. With 10 votes, the candidate has two extra votes that they do not need to win. These 'surplus votes' are now
transferred to their supporters' second choices.
Counting Round: 2
The candidate's surplus votes are transferred to their supporters' second choices. Some versions of STV transfer surplus votes at a fraction of their original value to add weight to voters' rankings.
Others transfer whole votes unchanged. For simplicity, we will transfer whole votes.
Unfortunately, even after receiving surplus votes, neither the candidate nor the candidate has enough votes to win. Elimination begins.
Counting Round: 3
The candidate, your first choice, has the fewest votes, and is eliminated. Your vote is transferred to your second choice, the candidate. Other voters who picked as their first choice also have their
votes transferred to their second choice.
Now that the candidate's votes have been transferred, both the and candidates have enough votes to win, and are elected.
Through this process of transfer and elmination, STV promises to discourage extreme positions, and reward middle-of-the-road candidates who can win support from across the political spectrum.
Yet since votes are only transferred, never added, it is impossible for this counting method to have produced the results of recent Reed elections. Let's take a look at the method that the data
suggests was actually used, starting from the same ballots as in the first example.
Counting Round: 1
One whole vote is awarded to your first choice, the candidate.
Counting Round: 1
Likewise, one vote is awarded to each voter's first choice.
Counting Round: 2
One additional vote is awarded to each voter's second choice.
Counting Round: 3
And then to each voter's third choice.
The and candidates have the most votes, and are elected.
Despite starting from the same ballots, we have reached a different result by using this counting method. How is this possible?
The method used at Reed in recent elections — one similar to Bucklin Voting, according to the fall results — seems to save time on election night by simply taking the sum of each candidate's first,
second, and third choice votes.
This means that any candidate you rank, regardless of order, receives one vote.
For example, consider where your votes for the , , and candidates ended up.
Because you ranked all three candidates, your ballot was counted as one equal vote for each of them. Essentially, your ballot canceled itself out, and had no impact on the outcome of the election. We
can safely eliminate it without changing the results.
Likewise, we can safely eliminate the votes of all other voters who ranked each of the candidates, regardless of the ordering of their preferences. In both our simplified election and, the data
suggests, the last three Reed elections, this means eliminating a majority of the electorate.
The result remains unchanged, and we're left only with voters who ranked some, but not all, of the possible candidates.
Remember them? In the previous example, their ballots looked like this.
In the Fall 2022, Spring 2023, and Fall 2023 Reed elections, these voters, and these voters alone, likely controlled the outcome.
If you ranked all of the possible candidates in any of those election cycles, your vote was likely not counted in any meaningful way.
The opposite of STV, this counting method encourages extreme candidates whose supporters vote only for them, and withhold their lower ranking votes from all other possible candidates. Under this
system, encouraging such voting strategies is the only way to win an election.
Another side effect of this counting method? The additional votes we originally observed in Reed's election results.
Note that in our model, the total number of votes cast increased after each round of counting, ultimately exceeding the number of voters, exactly as it did in the official results for the last three
Spring 2024 Election Czars Eleanor Davis-Diver and Maya Hanser-Young have promised to use STV counting in the current election.
This is a developing story. The Inquirer will report election results as they are released. | {"url":"https://reedinquirer.org/ranked-choice.html","timestamp":"2024-11-04T06:07:23Z","content_type":"text/html","content_length":"22616","record_id":"<urn:uuid:f440b5bf-6de5-4d30-9440-eda9e5e8fb74>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00770.warc.gz"} |
Why Good Time Estimators Are Better at Math
Since most of us were never called on in class to answer a tough time-estimation question, or quizzed on the lengths of tones in milliseconds, we don't have a good grasp of our skill in this area.
It's kind of exciting. You could be a prodigy and not know it! But a cold dose of reality comes from new research saying skill in time estimation is tied to mathematical intelligence. If you're not
amazing at math, your temporal abilities probably aren't A-plus either.
Writing in PLos ONE, a group of Italian researchers describe a
done on 202 adults. The subjects listened to a series of tones through headphones, and estimated the length of each tone in milliseconds. ("We first made sure that participants knew that one
millisecond is a thousandth of a second.") Tones ranged from 100 to 3000 milliseconds long. That's a tenth of a second to three seconds, for those of you who are non-amazing at math.
Everyone became more accurate as tones got longer. Unsurprisingly, it's easier to guess that a sound lasts for one second or three seconds than 100 or 200 milliseconds.
Subjects were also tested on their arithmetic skills, general intelligence, and working memory. All these tests came from a standard set of IQ questions. Arithmetic problems ranged from very simple
("What's 5 apples plus 4 apples?") to more difficult ("If 8 machines can finish a job in 6 days, how many machines are needed to finish it in half a day?"). To gauge non-mathematical intelligence,
researchers gave subjects a verbal comprehension test (for example, "How are an orange and a banana similar?"). A challenge to remember strings of digits and recite them forward or backward tested
subjects' "working memory," which is the ability to hold things in the mind and process them.
People's accuracy at guessing the length of tones was closely tied to their mathematical IQ. Less accurate estimators had lower math scores, and better estimators were better at math. But this
connection didn't extend to general intelligence, or at least not to verbal intelligence: there was no relationship between subjects' estimation skills and their performance on the verbal
comprehension test.
The researchers also found no relationship between time estimation and working memory. This is a little unexpected, since judging how long something took seems like a task for the short-term memory.
And a previous study of time estimation did find a connection to working memory. But in that study, subjects did arithmetic problems
estimating times. The authors argue that making subjects do two things at once was a test of their working memory to begin with; subjects who excelled at estimating times while doing math problems
would necessarily have a good working memory. In the new study, tasks were taken one at a time, and skill at time estimation seemed to be separate from working memory.
Subjects were also asked to rate their own mathematical ability on a scale of 0 to 10. These ratings followed the same pattern as math IQ scores: people who considered themselves as better at math
were also better at estimating tone lengths. (Interestingly, out of the 202 Italian subjects, not one person rated himself or herself a 10. Does this indicate a general trepidation toward math? Some
sort of cultural modesty? Surely in the U.S. someone would have claimed to be the best.)
Your sense of time, then, seems to be tied not to your intelligence or memory, but to your sense of numbers. The authors believe the connection lies in lines--the timeline and the number line.
Previous research has shown that people use a mental number line to do math, sensing smaller numbers to the left and larger numbers to the right. People estimate lengths of time using another
left-to-right mental path: small intervals are on the left, and larger intervals are on the right. (How would these experiments play out in a culture that reads right-to-left, or vertically?)
If it's all about lines, then mathematical and temporal skills may come down to a person's ability to judge increments, to arrange items in a path. Working with your mental timeline or number line,
that is, may really be a spatial skill. And the best time estimators in the classroom might be the line leaders.
Kramer, P., Bressan, P., & Grassi, M. (2011). Time Estimation Predicts Mathematical Intelligence PLoS ONE, 6 (12) DOI: 10.1371/journal.pone.0028621 Image: James Laing/Flickr
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ptimal control methods for multibody systems
Efficient optimal control methods for multibody systems
Current problems require engineers to predict how specific parameters or control inputs will affect the behavior of the complex multibody system. The adjoint method is one of the more interesting
methods for systematically calculating the gradient.
Efficient indirect optimal control methods for multibody systems (Excellence Initiative, Research University in the field of Artificial Intelligence and Robotics, 2020-2022)
Current problems require engineers to predict how specific parameters or control inputs will affect the behavior of the complex multibody system. In the optimal control or design of MBS, an implicit
dependency exists between state and design variables, further adding complexity to the problem. This branch of computer-aided engineering is tightly combined with sensitivity analysis, i.e.,
efficient computation of the derivatives. To this end, various families of methods have been derived, each having its own set of benefits and drawbacks. The adjoint method is one of the more
interesting methods for systematically calculating the gradient. The concept behind this approach lies in invoking necessary conditions for the minimum of the optimized functional. Once obtained via
variational calculus, these conditions constitute the system adjoint to the dynamic equations of motion – an underlying model of the MBS. Solving the adjoint system yields so-called adjoint
variables, allowing for efficient gradient computation. The adjoint method can be applied to such problems as optimal control, optimal design, or parameter identification.
Furthermore, the indirect optimal control methodology provides offline (potentially online) trajectory generation tools that can be utilized as a feedforward signal in the control loop. A recent
research project involved a practical implementation of the adjoint method in a feedback-feedforward control architecture. A mathematical model has been derived, composed of an electromechanical
device with a five-bar multibody system and two DC motors with a gear transmission. Based on the derived outcome, the input control signal and corresponding trajectory predicted by the model were
synthesized. Subsequently, these signals are introduced as reference values to the hardware and compared with classical control algorithms.
The block diagram below presents a model-based control architecture. Symbol r denotes a reference signal that nominally must be enforced. Accordingly, this signal plays the role of an input to the
adjoint-based optimization procedure founded on the mathematical model of the MBS. The optimization algorithm yields a control signal u[ff] (theoretically) capable of carrying out the required
maneuver. The response generated by the model for u[ff] is depicted as y[d], becoming the actual reference trajectory for the system. Due to the discrepancy between the model and the plant, and the
presence of disturbances d and measurement noise n, it is required to introduce a feedback loop that generates minor corrections u[fb] during the execution on the hardware. | {"url":"https://ztmir.meil.pw.edu.pl/web/eng/Badania/Projekty-badawcze/Optimal-control-methods-for-multibody-systems","timestamp":"2024-11-08T16:00:10Z","content_type":"application/xhtml+xml","content_length":"33561","record_id":"<urn:uuid:0f60c7b8-9bce-4d7c-a752-3b52d2111716>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00479.warc.gz"} |
Briefly explain linear and logistic regression
linear regression:
In statistics, linear regression is a linear approach to modelling the relationship between a dependent variable(y) and one or more independent variables(X). In linear regression, the relationships
are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Linear Regression is one of the most popular algorithms in Machine Learning. That’s due to its
relative simplicity and well known properties.
logistic regression:
It’s a classification algorithm, that is used where the response variable is categorical . The idea of Logistic Regression is to find a relationship between features and probability of particular
outcome .
E.g. When we have to predict if a student passes or fails in an exam when the number of hours spent studying is given as a feature, the response variable has two values, pass and fail.
Linear Regression is used to predict continuous variables.
Logistic Regression is used to predict categorical variables (mostly binary)
Linear Regression outputs the value of the variable as its prediction
Logistic Regression outputs the PROBABILITY of occurrance of an event as its prediction
Linear Regression’s accuracy and goodness of fit can be measured by loss, R squared, Adjusted R squared etc.
Logistic Regression: Measuring the accuracy of categorical distributions can become tricky due to imbalance. So, we have to use a bunch of metrics to measure the model’s fit. Some of them are -
Accuracy, Precision, Recall, F1 score (harmonic mean of precision and recall), ROC curve (for determining probability threshold for classification), Confusion Matrix, Concordance, Gini and the list
goes on…
There are many other comparisons that can be drawn in addition to this. I have tried to keep it “practitioner friendly”. | {"url":"https://discuss.boardinfinity.com/t/briefly-explain-linear-and-logistic-regression/381","timestamp":"2024-11-06T18:25:50Z","content_type":"text/html","content_length":"20345","record_id":"<urn:uuid:74372f8c-df21-42c7-9c50-dd2824e34561>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00676.warc.gz"} |
quantum computer is
Photon quantum computer is impractical forever.
Quantum computer
Photon quantum computer (= still Not a computer ) is unrealistic, impossible due to massive photon loss.
Photons are too easily lost to become components of quantum computers.
(Fig.1) ↓ A photon or very weak light is too easily lost. which makes it impossible to scale up photon quantum computer.
A photon qubit is expressed as a photon (= weak light ) traveling in one path (= 0 ) or the other path (= 1 ).
Photon quantum computer tries to use a fictional photon particle or weak light as a quantum bit or qubit.
Each photon or light's polarization is often used as a qubit state (= vertically-polarized light is used as a qubit's 0 state, horizontally-polarized light is used as a qubit's 1 state ).
In a photon quantum computer (= still Not a computer ), when each photon or weak light splits at a beam splitter (= BS ) into the upper path (= upper waveguide ) or the lower path, these paths or
waveguides are used as a photon qubit's two states (= each qubit can take 0 or 1, a photon in the upper path means a qubit's 0 state, a photon in the lower path means a qubit's 1 state, this
p.1-Fig.1, this-Figure 1, this 2nd-paragraph ).
When a photon (= just weak divisible classical light wave ) magically splits into both upper and lower paths at a beam splitter, it is treated as quantum superposition where each photon can be both
in the upper and lower paths simultaneously using fictional parallel worlds ( this p.35-Fig.3.5 ).
A photon can split into two paths 0 and 1 at a beam-splitter using (fictional) quantum superposition or parallel universes !?
↑ Of course, this quantum superposition or parallel worlds are just fiction.
A photon is just a divisible weak light that can realistically split at a beam splitter.
This 9th, 11th paragraphs say
"Then, photons are moving inside a waveguide in one direction (could be compared to plumbing pipes where you send water in). Each waveguide is called a mode, the unit of light in each mode is used to
represent a qubit, two modes and one photon can encode 1 qubit. Imagine the modes as two parallel lines where the photon can be either on the upper line or the bottom line. At this point we can agree
that having a photon on the upper waveguide corresponds to qubit in state |0> and the lower waveguide corresponds to qubit in state |1>, and this is the general idea that defines our qubits. We call
it dual-rail encoding"
"For instance, assume we have |0> how do we end up in a quantum superposition ? ..
Once the single photon passes into a beam splitter it will move randomly in the upper or lower mode with a 50–50 chance. Furthermore, we might need different probabilities for example 30–70% or
40–60%, in that case we must have more sophisticated beam splitters using phase shifters (also called unbalanced beam splitters)."
Photon quantum computer is impossible because of easy photon loss.
The problem is a fragile photon or weak light so easily gets lost that quantum computers using such unstable photons as qubits are impractical forever ( this 4th-paragraph, this p.5-right-4.4, this
p.1-left-2nd-paragraph ).
This-middle Photonic networks say
"Unlike with other qubit technologies, two-qubit gates in photonic networks are probabilistic (= random, causing errors ), not deterministic."
"..However, fidelity at scale is a significant hurdle for photonic networks. The largest source of error for photonic networks is photon loss during computations."
This Challenges of Photonic Quantum Processors say
"Losses: Photons are easily lost, limiting the performance of photonic quantum processors.
Control: It is difficult to control photons' behaviour, making it challenging to perform complex quantum operations.
Error correction: Quantum computers are susceptible to errors, which means that error correction techniques need to be developed to ensure the accuracy of calculations."
This 2nd-paragraph says "One of the main challenges in photonic quantum computing is the loss of photons"
This-p.1-right-2nd-paragraph says
"The remaining criteria are harder to satisfy because photons don’t easily interact, making deterministic two-qubit gates a challenge (= still useless, because changing a qubit based on another qubit
state = two-qubit gate operation needed for computation is impossible ). Among the additional technical considerations is photon loss,.."
".. And although photons are always flying, computing and networking tasks may need them to be delayed or stored (= usually very long bulky light cables to confine the always flying photons or light
are needed ), so an extra device—an optical quantum memory (= still impractical ) —may sometimes be needed."
↑ The impractical photon quantum computer with easily-lost photons ( this p.1-right-1st-paragraph ) motivated physicists to focus only on detecting meaningless random photons (= just divisible weak
classical lights ) in fake quantum advantage.
Quantum computers using fragile photons suffer too high error rates.
Even a single logic gate by such a fragile photon has Not been built, a photon quantum computer is far more impossible.
A real computer needs logic gates which change the state of a bit (or qubit ) 0 ↔ 1 by external stimulus.
But a photon quantum computer still cannot realize even one two-qubit logic gate (= two-qubit gate operation means "change one qubit state 0 ↔ 1 based on another qubit state" ).
This p.3-A universal set of quantum gates say
"arbitrary single-qubit operations can be expressed as combinations of beamsplitters and phase-shifters—an optical interferometer (= for changing photon qubit state or path )...
The implementation of two-qubit gates is a challenge for photons. The fact that photons do not interact also means that it is difficult for the operation on one photon depend on the state of the
Each photon operation causes 76% error rate, which cannot be used for quantum computers.
Error rate of a photon quantum computer's logic gate is still high = more than 76%, which is useless.
This recent paper on (useless) photon's two-qubit logic gate (= one photon's state influences the other photon's state ) ↓
p.1-Abstract says "The experimentally achieved efficiency in an optical controlled NOT (CNOT) gate reached approximately 11% in 2003 and has seen no increase since (= photon's two-qubit gate
operation error rate is impractically-high = 89% = 100% - 11% efficiency )..
We demonstrate a CNOT gate between two optical photons with an average efficiency of 41.7% (= still error rate is impractically high = about 60% )"
↑ Efficiency is the success rate or the probability that a desirable photon was detected by a photodetector ( this p.1-3rd-paragraph ).
This success rate (= efficiency) was still very low = only 41.7%, which is useless.
↑ So the error rate (= more than 60% error rate ) of a photon two-qubit gate operation is impractical and far worse than error rates (= 1%) of even other (impractical) superconducting or ion qubits
Photon quantum computer's error rate (= more than 76% ) is far worse, higher and more impractical than other superconducting or ion qubits' error rate (= 1% ).
↑ The same paper ↓
p.2-Fig.1 shows photon's quantum computer's two-qubit logic gate consists of (classical) polarizing beam splitters and mirrors where two lights destructively or constructively interfering determine
the final qubit state (= which path photon is detected in the last ).
p.4-left-C. says "The efficiency of the gate is the probability that no photon is lost inside the gate, if one control and one target photon impinge on the gate"
p.9-left-1st-paragraph says "Multiplying this by the 41.7% average efficiency of the gate, one obtains a 24% probability of detecting a two-photon coincidence per incoming photon pair (= only 24%
photons could be detected, which means the remaining 76% photons were lost, or its error rate is extremely high = 76% )."
p.9-left-2nd-paragraph says "obtain a naive estimate for the two-photon coincidence rate of 10 s−1 (= only 10 pairs of photons or 10 qubits were detected per second, which is too slow and useless as
a computer due to massive photon loss )"
Photon quantum error correction is just hype and impossible
Even the probability of successfully preparing states needed for photon error detection is only 1%. ← Quantum error correction is impossible.
This introduction and lower-Challenge: Generation of error-correctable state — GKP state say
"However, a significant challenge lies in generating specialized states like the Gottesman-Kitaev-Preskill (GKP) state, crucial for error correction and achieving fault tolerance"
"indicating the successful projection of the last mode into a GKP state. Nevertheless, this approach is probabilistic, with a success rate as low as 1% or even lower (= photon error correction needs
to create GKP light superposition state, which success rate is only less than 1%, which is useless )"
Probability of generating the photon's GKP state for error correction is impractically low.
The recent research tried to generate this fault-tolerant photon's GKP state, but still No photon's error correction has been realized even in this latest research.
↑ This research paper ↓
p.2-right-2nd-paragraph says "Our generation method is based on the two-mode interference between cat states (= just multiple mixed weak classical light wave states )"
p.5-left-1st-paragraph says ". Our current setup has the success rate of about 10 Hz which is not sufficient for actual computation (= only 10 photon GKP-states per second were generated, which is
too few, too slow, useless, and this research did Not conduct error correction )"
This p.5-2nd-paragraph says
"The generation rate of each cat state in the GKP state generation is about 200 kHz. The duty cycle for the sample & hold is 40% and taken into account of the allowed time window for simultaneous
detection of ±0.6 ns, the simultaneous detection of the two cat state is expected to be (2 × 10^5 = 200 kHz )^2 × (1.2 × 10^−9 = detection efficiency ) × 0.4 = 19 Hz, which is on the same order with
actual simultaneous detection rate of 10 Hz (= only 10 photon target states per second were generated, which was too slow, too few to use as a practical computer )"
↑ Two photon simultaneous detection rate was extremely low = only 1.2 × 10^−9 due to massive photon loss.
As a result, the significant photon loss hampers the realization of a photon quantum computer (forever).
Photon quantum computer is just a joke due to extremely low photon detection efficiency.
To scale up photon quantum computer, physicists need to simultaneously detect as many photons (= each photon is used as a qubit taking 0 or 1 states ) as possible.
If they cannot simultaneously detect multiple photons or multiple qubits, it means photons or qubits relevant to calculations are lost (= so calculation using the easily-lost photons is impossible ),
or irrelevant background photons are mixed.
So increasing success rate of simultaneous detection of multiple photons or qubits is indispensable for scaling up the photon quantum computer.
But the present simultaneous photon detection rate is too bad and too low to scale up a photon quantum computer (= even success rate of detecting only small numbers of 4 ~ 8 photons or qubits is
unrealistically low and too bad ).
Success rate of detecting only 4 photons or 4 qubits simultaneously is extremely low = only 0.000000001, which is useless.
The recent research paper ↓
p.4-right-Experimental design says "pumped by a femtosecond ultraviolet laser (390 nm, 76 MHz = 76 × 10^6 photons or weak light pulses were generated )"
p.5-left-last paragraph says "the final fourfold coincidence rate (= rate of detecting only 4 photons simultaneously ) is about 0.03 Hz (= 1 Hz or Hertz means detecting one photon per second )."
↑ It means only 0.03 photons (= 0.03 simultaneous four-photon detection ) per second were detected (= only 0.03 × 4 = 0.12 photon bits or information per second can be utilized ), which is too few
and too slow to use as a computer's memory (qu)bits.
Success simultaneous four-photon detection rate was 0.03/76000000 = 0.000000001 = 99.9999999 % error rate, which is completely impractical.
Success rate of detecting only 6 photons or 6 qubits simultaneously is extremely low = 0.0000000001, which is impractical.
This another recent research paper ↓
p.2-left-3rd-paragraph says "pulse laser with a repetition rate of ∼76 MHz, the QD emits ∼50 MHz polarized resonance fluorescence single photons at the end of the single-mode fiber (= about 50 MHz or
50 × 10^6 photons or light pulses were generated from the light or photon source )"
p.5-Fig.4a shows the six-photon coincidence counts (= the total number of detecting 6 photons simultaneously ) were only less than 300 per 23 hours, which counts were too few and too slow to be
↑ Only 300 × 6 = 1800 bits (= each bit can take only 0 or 1 ) or 1800 photons could be used per 23 hours, which pace is too slow to use as practical computer's memory bits due to massive photon loss.
↑ the 6-photon simultaneous detection rate was only 300/(50 × 10^6 × 3600 × 23 ) = 0.0000000001, which is extremely low, and useless.
Photon quantum computer, which can Not detect even 10 photons or 10 qubits, is impractical forever.
This another recent paper ↓
p.1-right-last-paragraph says "That is, by combining a quantum dot (QD)–based photon source, from which we measure single-photon count rates at 17.1 MHz (= 17.1 × 10^6 photons were estimated to be
generated from light source per second )"
p.5-Fig.4c shows the coincidence 8 photon count rate drastically decreased to only less than 0.01 Hz (= less than one detection per 100 seconds ) from the original single photon rate of 17.1 MHz (=
17100000 Hz ), which means success probability of detecting 8 prepared photons simultaneously is only 0.01/17100000 = 0.000000001.
↑ Detecting only 8 photons or 8 qubits is too slow (= just 0.01 detections per second ), hence, building just 9 ~ 10 photon qubits is impossible, which is far from a practical quantum computer that
will require millions of qubits.
As a result, the probability of detecting multiple photons simultaneously is extremely low and disastrous, which massive photon loss makes it impossible to scale up photon's quantum computer forever.
2024/6/20 updated. Feel free to link to this site. | {"url":"http://www7b.biglobe.ne.jp/~kcy05t/photochip.html","timestamp":"2024-11-11T04:25:21Z","content_type":"text/html","content_length":"23403","record_id":"<urn:uuid:dadfd9f5-8341-4fc5-a740-a99c966d0363>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00473.warc.gz"} |
Analysing Secret Sharing Schemes for Color Images
Volume 03, Issue 01 (January 2014)
Analysing Secret Sharing Schemes for Color Images
DOI : 10.17577/IJERTV3IS10263
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Priyanka Chaudhari, Anuja Pardeshi, Priyanka More, Sayli Thanekar, 2014, Analysing Secret Sharing Schemes for Color Images, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume
03, Issue 01 (January 2014),
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• Authors : Priyanka Chaudhari, Anuja Pardeshi, Priyanka More, Sayli Thanekar
• Paper ID : IJERTV3IS10263
• Volume & Issue : Volume 03, Issue 01 (January 2014)
• Published (First Online): 17-01-2014
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Analysing Secret Sharing Schemes for Color Images
Priyanka Chaudhari1, Anuja Pardeshi2, Priyanka More3 and Sayli Thanekar4
1234 Department of Computer Engineering, Pimpri Chinchwad College of Engineering, Pune
University, India.
Images are of great importance in communication field for conveying messages. Using images we can convey these messages very easily to the audience and there is no need to read the text, hence
security of these images is of big concern. In recent years, many techniques were proposed to provide security to images and image secret sharing is one of the effective approaches for the same. This
paper analyses different image secret sharing schemes like Shamirs secret sharing scheme, Thien Lins secret sharing scheme and Lie Bais secret sharing scheme. Performances of these schemes are
analysed based on parameters like ideal, perfect, threshold based, accuracy, share size, image type etc. The comparative study shows that Lie Bais method of matrix projection is more effective and
reliable secret sharing method. This scheme also satisfies the security and accuracy conditions required by any image secret sharing scheme.
1. Introduction
Secret Sharing Schemes [1] (SSS) refers to a method for distributing a secret amongst a group of participants and each participant have allocated share of a secret. To reconstruct the original
secret sufficient number of
In the worldwide computer network environment, secure transmission of data is needed on a wide range. In many commercial, medical and military applications the effective and secure protections of
sensitive information which is mostly in the form of images are important. Image secret sharing is a better approach for these kinds of applications.
IMAGE SECRET SHARING: [2]
Image secret sharing operates directly on the bit planes of the digital input. The input image is decomposed into bit-levels which can be viewed as binary images. Using the {k,n} threshold
concept, the image secret sharing procedure encrypts individual bit-planes into the binary shares which are used to compose the share images with the representation identical to that of the input
image. Depending on the number of the bits used to represent the secret (input) image, the shares can contain binary, grey-scale or color random information. Thus, the degree of protection
afforded by image secret sharing methods increases with the number of bits used to represent the secret image.
shares needs to be combined together and individual shares are of no use on their own.
There are certain situations where the set of people
Secret grey scale
Share 1
Share 2
Decrypted image
needs to perform particular actions for executing it. Let us consider the example, suppose for accessing the confidential data or document minimum four authorize users needs to perform certain
actions and reveal the data. The scheme in which group of people come together and perform certain actions which regenerates the authorized or secret information is commonly
known as Secret Sharing Schemes.
The decryption operations are performed on decomposed bit-planes of the share images. Using the contrast properties of the conventional {k, n}-schemes, the decryption procedure uses shares' bits
to recover the original bit representation and compose the secret image. The decrypted output is readily available in a digital format, and is identical to the input image. Because of the
symmetry constraint imposed on the encryption and decryption process, image secret
sharing solutions hold the perfect reconstruction property. This feature in conjunction with the overall simplicity of the approach make this approach attractive for real-time secret sharing
based encryption/decryption of natural images.
Output: n shares are created in the form of an integer for the n participants to keep.
Step1: Choose a random prime number p larger than d0. Step2: select k-1 integer values d1, d2 dk-1 range of 0 through p-1.
Step3: Select n distinct real values x1, x2 xn
Step 4: Use the following (k-1) degree polynomial to compute n function values f(xj), For j=1, 2 n
2 j
F (xj) = d0 + d1xj + d x 2 ++ d (k-1) xj (k-1) (mod p)
Secret color image
Share 1
Share 2
Decrypted image
Step5: Deliver the secret shares as pairs of values (xi, f (xi)), 1<=i<=n and 0 < x1 < x2 ..< xn<p-1.
Color image secret sharing scheme supports the RGB color model. Red, Green, Blue are the primary color components of the RGB color space. All the other color can be obtained by using additive
color mixing of different RGB color components. The intensity of the primary color can be defined as the grey level in the grey-scale palette. A primary color will have an intensity range between
0 and 1, with 0 representing black and 1 representing the maximum possible intensity of that color. The RGB color palette is created from the grey-scale palette, which represents the intensity
palette 1for red, green, and blue. In real color system R,G,B are each represented by 8 bits, and therefore each single color based on R,G,B can represent 0-255 variations of scale.
2. Literature Survey
1. Shamirs Secret Sharing Scheme [3]
Shamir secret sharing scheme is explained in [4].Shamir developed the idea of a (k, n) threshold- based secret sharing technique (k <= n). The technique is to construct a polynomial function
of order (k – 1) as,
f(x) = d0 + d1x + d2x2 ++ d(k-1)x(k-1)(mod p)
Where the value d0 is the secret and p is a prime number.
Algorithm 1: (k, n)-threshold secret sharing
Input: Take secret d0 in the form of an integer, n is number of participants and threshold is k n.
The polynomial function f (xi) is destroyed after each server Pi possesses a pair of values (xi, f(xi)) so that no single server knows what the secret value d0 is. The following describes the
equation for solving the process of secret recovery.
Algorithm 2: Secret recovery of shares
Input: Select k shares from the n participants and the prime number p with both k and p
Output: Secret d0 is hidden in the shares and coefficients di used in (1) where i=1, 2, 3 d- 1.
Step1: Use the k shares (x1, f(x1)), (x2, f(x2)) (xk
,f(xk)) to set up
2 j
F (xj) = d0 + d1xj + d x 2 ++ d (k-1) xj (k-1) (mod p)
Step2: Lagrange interpolation formula [6] is commonly used to solve the secret value d0.Solve the k equations by Lagranges interpolation to obtain k as follows.
2. Thien and Lins Image Secret Sharing Scheme [4]
Thien and Lin proposed a (k, n) threshold-based image SSS by cleverly using Shamirs SSS to generate image shares. The essential idea is to use a polynomial
function of order (k – 1) to construct n image shares from an l x l pixels secret image (denoted as I) as,
Sx(i,j) = I (ik + 1,j) + I(ik+ 2, j) x.. + I(ik+ k, j) xk-1 (mod p)
where 0 I ( l/k) and 1 j l
This method reduces the size of image shares to become 1/k of the size of the secret image. Any k image shares are able to reconstruct every pixel value in the secret image.
An example of (2, 4) image secret share construction process is illustrated in Figure 1 where k = 2 and n = 4. According to the technique, a first order polynomial function can be created as
Sx (i,j) = (110 + 112x) (mod 251)
Where 110 and 112 are the first two pixel values in the Lena image. For our four participants, we can randomy
equal or close values. It is evident that the first two pixel values (110 and 112) are very close to each other. That creates the possibility that one image secret share may be used to
recover the secret image by assuming the neighbouring pixels have the same values in the first order polynomial function.
3. XOR secret sharing scheme [5]
The (n,n) threshold scheme which can be constructed based on XOR operation have no pixel expansion and the time complexity for constructing shared image is O(k1,n),where k1 is size of shared
image and this time complexity is excluding time needed for generating n distinct random matrices. This scheme also provides perfect secrecy. XOR color secret sharing scheme supports the RGB
color model.
Assume that 0, 1c is the set of all color appearing in an original image. Where 2 is the maximum color value of a color images.
pick four x values, and substitute them into the polynomial function by setting p value to be 251 which is the largest prime number less than 255 which is
A= [aij] j=1 n)
Where aij
{0 c-1}, (i=1, 2.m and
maximum grey image value.
Fig. Secret sharing process for Lena image
Four shares are computed as (1, 222), (2, 83), (3, 195) and (4, 56). They become the first pixel in four image shares. The second pixel is computed in the same manner by constructing another
first order polynomial function using next two pixels in the Lena image. This process continues until all pixels are encoded. Four image shares are the bottom right images shown in Figure 1,
and the size of each image share is half (1/2) size of the original image. None of the image shares appear to reveal information about the secret image. However, the pixel values in a natural
image are not random because the neighbouring pixels often have
Consider matrix A and matrix B and perform XOR and AND operation of matrices by using the following formula,
aijA, B, Aij
C=A B = [aij bbij] (i =1, 2 m; j=1.n) D=A & B= [aij&bij] (i =1, 2 m; j=1.n)
To express the model conveniently some assumptions were made which are as follows,
Assumption 1: The pixel matrix of secret image A is equal to secret image A.
Assumption 2: The matrix of secret image is n, Ai1Ain are used to denote n distinct matrices of A1An for convenience (n2).
If n2, then there must be n distinct matrices A1An satisfying the following conditions:
It means the XOR of any n-1 matrices cannot be used to obtain any information of matrix A.
It indicates that only the XOR of n matrices can be used to recover information from matrix A.
4. Lie Bais Matrix Projection scheme [6]
In this scheme the secret image will get divided into n image shares such that: i) any k image shares (k <=n) can be used to reconstruct the secret image in lossless manner and ii) any (k-1)
or fewer image shares cannot get sufficient information to reveal the secret image. Here, we briefly describe the procedure in two phases:
Construction of Secret Shares from secret matrix S
1. Construct a random matrix A of size m x k of rank k where m>2(k-1)-1
2. Choose n vectors of size (k x 1) where any k vectors are linear independent
3. Calculate shares vi=(A x xi) (mod p) for 1in
4. Compute projection matrix
5. $ =(A(ATA)-1AT)(mod p)
6. Solve remainder matrix R=(S- $)(mod p)
7. Destroy matrix A, xi, S, $
8. Distribute n shares vi to n participants and make matrix R publicly known
Secret Reconstruction
1. Collect k shares from any k participants, say the shares are v1, v2, ,vk and construct a matrix B=[v1 v2 vk]
2. Calculate the projection matrix
$ = (B(BTB)-1BT)(mod p)
3. Compute the secret S=($ + R) (mod p)
3. Comparative Analysis
In above section, we have studied different image secret sharing schemes. Comparative analysis of these
schemes is done based on certain parameters like ideal, perfect, accuracy, image share size etc.
Schemes XOR
Shamirs scheme Thien and Lins scheme Lie Bai scheme
Parameters Scheme
Ideal Yes Yes Yes Yes
Perfect Yes No Yes Yes
Yes Yes No Yes
Threshold Type (k,n) (k,n) (2,2) (k,n)
Share size Same 1/k Same 1/m
Secret Sharing Single Single Single Multiple
Secret sharing Matrix
Polynomial Polynomial polynomial
Based on Projection
Proactive No No No Yes
Accuracy More Less Less More
Image Type NA Grey Grey Color
Above table shows that Lie Bais Matrix projection method is more efficient and secure on the basis of parameters like accuracy, proactiveness and share size. Also this scheme is an applicable for
sharing multiple secrets.
Also the various extended capabilities [9] [10] are required in secret sharing schemes as per the need of an application.
4. Conclusion
In this paper several secret sharing schemes like Shamirs secret sharing scheme, Thien Lins secret sharing scheme, XOR secret sharing scheme, Lie Bais secret sharing scheme are discussed. Table 1
gives the Comparison of these schemes based on different
parameters like ideal, perfect, threshold based scheme, image type, image size, accuracy etc. This analysis shows that Lie Bais method of matrix projection is better secret sharing method for
5. References
[1]D. R. Stinson, Cryptography: Theory and Practice, CRC Press, Boca Raton 1995.Menezes, A., P. Van Oorschot, and
S. Vanstone, Handbook of Applied Cryptography,CRC Press, 1996, pp. 524-528.
1. http://www.colorimageprocessing.com/research
2. Shamir, A.,How to Share a Secret, Communications of the ACM, vol.22, no.11, 1979.
3. Thien and Lin,Secret image sharing, Computers &Graphics, vol. 26, no.5, pp. 765-770, 2002
4. Wang Dao-Shun, ZangLei, MaNingSecret color images sharing schemes based on XOR operation, 2013
5. Lie Bai A Reliable (k, n) Image Secret Sharing Scheme, 9th International Conference on Information Fusion, Sponsored by the International Society of Information Fusion (ISIF), Aerospace &
Electronic Systems Society (AES),
6. Kai Wang, XukaiZou and Yan Sui,A Multiple Secret Sharing Scheme based Matrix Projection, 33rd Annual IEEE
7. E. D. Karnin, J.W. Greene, and M. E. Hellman, On secret sharing systems, vol. IT-29,no. 1, pp. 35-41, Jan. 1983.
8. Sonali Patil and Prashant Deshmukh, An Explication of Multifarious Secret Sharing Schemes, International Journal of Computer Applications 46(19):5-10, May 2012.
9. Sonali Patil and Prashant Deshmukh, Analysing Relation in Application Semantics and Extended Capabilities for Secret Sharing Schemes, International Journal of Computer Science and Issues Volume
9, Issue 3, No. 1, 2012, 1694-0814
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Making Maths Exciting - Maths Scholars Celebratory Event
Making Maths Exciting - Games, Goats and Gold
Maths Scholars Celebratory Event - 22 September 2018
It was the first event in our scholarship year; the buzz in the air was palpable as the new scholars were set the task of introducing themselves and discussing how much money they would be willing to
accept from the banker when left with two boxes of 1p and £250,000 – and in a group full of mathematicians, the figure that came about was obviously unrealistically large and based upon averages. We
ploughed on through to our next game – the chase, could we finally have a mathematical probability of beating the chaser. This was a fascinating game than involved looking more at event trees and how
mapping the number of events was more intuitive than mapping the probability of an event; a useful tool for pupils who struggle with fractions and prefer integers. We looked at how to extend the game
for those pupils who need the challenge by changing probabilities and the number of rungs ahead of the chaser you start.
Monty Hall, everyone knew the problem, everyone knew the solution, and we all knew that we were going home with the greatest prize of all, a (goat) car. The age-old problem, shown beautifully through
a simple simulation, became more than just a logical but theoretical understanding of probability, it became a virtual, visual representation of justifying why the probabilities work, enough to
convince the greatest doubters of all, eleven-year-old pupils. I will add to this blog when I get around to teaching probabilities in the next few weeks, I cannot wait to see how my Year 7's will
react to this problem, and if any of them had considered the extensive applications of the understanding behind the probabilities.
Finally, we played a game in Brucie’s memory, Play your Cards Right. It was here that the room became a theatre as Colin brought up Sophie and Annie as his glamorous helpers. Before we discussed
probability, we played the game as a group, shouting higher or lower depending on the card we’d just seen. It was a practical demonstration of the innate understanding of likelihood, we knew that
getting a Queen meant your next card was more likely to be lower than higher. But it was now up to us to work out a general probability something that older year groups could debate on for hours. It
became clear to many people that doing a computer simulation of the game would be an exciting and visual display of probabilities involved.
This may only be a taster of what we achieved in the first session of our first event, but hopefully it conveys the engagement and overall enjoyment of the group. This was not a session designed to
increase our depth of knowledge in probability, it was a session that showed us knowledge for teaching, for differentiating and most importantly for making maths exciting again. Probability can be
such a dull topic, and I know I have taught it out of any realistic context with spinners and dice before, now I have a chance to show them that their day-time TV watching over summer can be
manipulated into helping them win a lot of money – what pupil wouldn’t want to learn maths then?
Isabelle Perrin
Watford Grammar School for Girls, Schools Direct | {"url":"https://teachingmathsscholars.org/scholarsblogs/makingmathsexciting","timestamp":"2024-11-09T22:55:42Z","content_type":"text/html","content_length":"116806","record_id":"<urn:uuid:3fef6fc2-bbb2-469b-aaa6-be4c2abff3f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00322.warc.gz"} |
The Stacks project
Lemma 42.6.2. Let $(A, \mathfrak m)$ be a $2$-dimensional Noetherian local ring. Let $a, b \in A$ be nonzerodivisors. Then we have
\[ \sum \text{ord}_{A/\mathfrak q}(\partial _{A_{\mathfrak q}}(a, b)) = 0 \]
where the sum is over the height $1$ primes $\mathfrak q$ of $A$.
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World Real Interest Rates
Robert J. Barro and Xavier X. Sala-i-Martin
In Blanchard, Olivier Jean; Fischer, Stanley, eds. NBER macroeconomics annual 1990.
Cambridge, Mass., MIT Press 1990, pp. 15-61.
We think of the expected real interest rate for ten OECD countries (our counterpart of the world economy) as determined by the equation of aggregate investment demand to aggregate saving.
Stock-market returns isolate shifts to investment demand, and changes in oil prices, monetary growth, and fiscal variables isolate shifts to desired saving. We estimated the reduced form for
GDP-weighted world averages of the expected short-term real interest rate and the investment ratio over the period 1959-1988. The estimates reveal significant effects in the predicted direction for
world stock returns, oil prices, and world monetary growth, but fiscal variables turned out to be unimportant.
Structural estimation implies that an increase by one percentage point in the expected real interest rate raises the desired saving rate by one-third of a percentage point. Simulations of the model
indicated that fluctuations in world stock returns and oil prices explain a good deal of the time series for the world average of expected real interest rates; Specifically, why the rates were low in
1974-79 and high in 1981-86. The model also explains the fall in real rates in 1987-88 and the subsequent upturn in 1989.
We estimated systems of equations for individual countries' expected real interest rates and investment ratios. One finding is that each country 's expected real interest rate depends primarily on
world factors, rather than own-country factors, thereby suggesting a good deal of integration of world capital and goods markets.
Domestic Monetary Theory; Empirical Studies Illustrating Theory, 3112. Domestic Monetary Policy, Including All Central Banking Topics, 3116. Open Economy Macroeconomic Studies-- Balance of Payments
and Adjustment Mechanisms, 4313. Exchange Rates and Markets--Theory and Studies, 4314. Macroeconomic Theory--General, 0230.
This paper also circulated as
NBER Working Paper # 3317, April 1990 | {"url":"http://www.columbia.edu/~xs23/papers/worldr.htm","timestamp":"2024-11-02T06:10:45Z","content_type":"text/html","content_length":"4085","record_id":"<urn:uuid:09e2f4a6-5b88-4685-b4db-afd8ab3a534c>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00341.warc.gz"} |
On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems
We study the shape of least-energy solutions to the quasilinear elliptic equation ε^m Δ[m]u - u^m-1 + f(u) = 0 with homogeneous Neumann boundary condition as ε → 0^+ in a smooth bounded domain Ω ⊂ ℝ^
N. Firstly, we give a sharp upper bound for the energy of the least-energy solutions as ε → 0^+, which plays an important role to locate the global maximum. Secondly, based on this sharp upper bound
for the least energy, we show that the least-energy solutions concentrate on a point P[ε] and dist (P[ε] ∂Ω)/ε goes to zero as ε → 0^+. We also give an approximation result and find that as ε → 0^+
the least-energy solutions go to zero exponentially except a small neighbourhood with diameter O(ε) of P[ε] where they concentrate.
• Least-Energy Solution
• M-Laplacian Operator
• Quasi-Linear Neumann Problem
Dive into the research topics of 'On the Shape of Least-Energy Solutions for a Class of Quasilinear Elliptic Neumann Problems'. Together they form a unique fingerprint. | {"url":"https://scholars.georgiasouthern.edu/en/publications/on-the-shape-of-least-energy-solutions-for-a-class-of-quasilinear-3","timestamp":"2024-11-12T19:50:53Z","content_type":"text/html","content_length":"53031","record_id":"<urn:uuid:ba8da3b9-d92b-41b0-b17a-30a0abce9d99>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00097.warc.gz"} |
Revision history
Note that the problem is not in defining the elliptic curve, but only when plotting it:
sage: E.change_ring(F)
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: E.plot()
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: E.change_ring(F).plot()
OverflowError: Python int too large to convert to C long
Note that the problem is not in defining the elliptic curve, but only when plotting it:
sage: E.change_ring(F)
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: E.plot()
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: E.change_ring(F).plot()
OverflowError: Python int too large to convert to C long | {"url":"https://ask.sagemath.org/answers/44055/revisions/","timestamp":"2024-11-07T20:32:57Z","content_type":"application/xhtml+xml","content_length":"19190","record_id":"<urn:uuid:bef8f7ac-75fd-4937-8646-cbe7357761de>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00324.warc.gz"} |
Blind Stork Test for Data Collection | Math = Love
Blind Stork Test for Data Collection
This blog post contains Amazon affiliate links. As an Amazon Associate, I earn a small commission from qualifying purchases.
While looking through Don Steward’s blog for data collection ideas, I ran across the idea of a “blind stork test.” The idea is simple. Close your eyes and see how long you can stand on one leg. Don
Steward claims that most people can’t last more than one minute.
As with my Estimating 30 Seconds Activity, I originally planned to use an online stopwatch on the SMARTBoard for this activity. When I tried this first hour, I had two buys who simply wouldn’t give
up. They both stood on one foot with their eyes closed for over five minutes. Five minutes. We couldn’t write down our data or move on until both of them gave up!
For my afternoon classes, I had my students form pairs for this activity. We pulled out our MyChron timers to help us collect our data. One student would time the other student, then they would trade
places. This resulted in a much more timely data collection period!
Another switch I made was to require students to stand with their arms folded against their chest. I did this with the hopes that it would make it trickier and they would be able to stand on one foot
for a shorter amount of time.
I created a data sheet for my students to glue in their interactive notebooks. They recorded their data value and the data values of their classmates. Next year, I’ll change “Your Time” to “My Time.”
I also want to revise this to give students a place to write their data values in order. I had to squeeze it in here, and it just doesn’t look as neat and nice as I’d hoped.
After writing down the data for the class, I had my students find the five number summary. We then used the five number summary to solve for the IQR and check for outliers.
In the future, I would definitely designate a certain place on the notes for students to explain the presence/absence of outliers.
On the inside of our notes, I had students create a box-and-whisker plot, dot plot, stem-and-leaf plot, and histogram.
I definitely need to give my students a pre-made (but not pre-numbered) number line for their box-and-whisker plot and dot plot next year. That should save lots of time!
I want to put graph paper in the background for the stem-and-leaf plot to help students align their data more evenly.
I also believe that graph paper would be super-useful for making histograms in the future, too.
Here are a few more photos of my students in action during the blind stork test.
More Ideas for Teaching Quantitative Data Displays
One Comment
1. This is amazing!!! Love your ideas. Keep them coming!! Can't wait to do statistics unit now with Math 1. | {"url":"https://mathequalslove.net/blind-stork-test-for-data-collection/","timestamp":"2024-11-09T00:45:55Z","content_type":"text/html","content_length":"280621","record_id":"<urn:uuid:fb118e34-b2ee-4c2a-aeb4-226b805971bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00712.warc.gz"} |
Find Pivot Index Java Solution - The Coding Shala
Home >> Interview Questions >> Find Pivot Index
Find Pivot Index
Given an array of integers nums, write a method that returns the "pivot" index of this array.
We define the pivot index as the index where the sum of the numbers to the left of the index is equal to the sum of the numbers to the right of the index.
If no such index exists, we should return -1. If there are multiple pivot indexes, you should return the left-most pivot index.
Example 1:
nums = [1, 7, 3, 6, 5, 6]
Output: 3
The sum of the numbers to the left of index 3 (nums[3] = 6) is equal to the sum of numbers to the right of index 3.
Also, 3 is the first index where this occurs.
Example 2:
nums = [1, 2, 3]
Output: -1
There is no index that satisfies the conditions in the problem statement.
Find Pivot Index Java Solution
At every index, we can check if the total sum to its left and total sum to right of this index equals or not.
First, we will find the total sum of an array then using a loop every time we check if the sum till the current index-1 is equaled to the sum to the right side of this index.
Java Code::
class Solution {
public int pivotIndex(int[] nums) {
int len = nums.length;
if(len<3) return -1;
int right_sum = 0;
//total sum
for(int i=0; i<len; i++) right_sum += nums[i];
int left_sum = 0;
//checking sum
for(int i=0;i<len;i++){
right_sum -= nums[i]; //current index does not count
if(left_sum == right_sum) return i;
left_sum += nums[i];
return -1;
Other Posts You May Like
Please leave a comment below if you like this post or found some errors, it will help me to improve my content. | {"url":"https://www.thecodingshala.com/2019/08/find-pivot-index-java-solution-coding.html","timestamp":"2024-11-11T16:34:32Z","content_type":"application/xhtml+xml","content_length":"130006","record_id":"<urn:uuid:7a208e12-89d8-47bc-97ca-12a65093f0e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00849.warc.gz"} |
Sharp PC-1450
Datasheet legend Years of production: Display type: Alphanumeric display
New price: Display color: Black
Ab/c: Fractions calculation Display technology: Liquid crystal display
AC: Alternating current Size: 3"×7½"×½" Display size: 16 characters
BaseN: Number base calculations Weight: 8 oz
Card: Magnetic card storage Entry method: BASIC expressions
Cmem: Continuous memory Batteries: 2×"CR-2032" Lithium + 1×"CR-2016" Lithium Advanced functions: Trig Exp Cmem Snd
Cond: Conditional execution External power: Memory functions:
Const: Scientific constants I/O: Printer/cassette port
Cplx: Complex number arithmetic Programming model: BASIC
DC: Direct current Precision: 12 digits Program functions: Jump Cond Subr Lbl Ind
Eqlib: Equation library Memories: 4(0) kilobytes Program display: Text display
Exp: Exponential/logarithmic functions Program memory: 4 kilobytes Program editing: Text editor
Fin: Financial functions Chipset: Forensic result: 8.99998153428
Grph: Graphing capability
Hyp: Hyperbolic functions
Ind: Indirect addressing
Intg: Numerical integration
Jump: Unconditional jump (GOTO)
Lbl: Program labels
LCD: Liquid Crystal Display
LED: Light-Emitting Diode
Li-ion: Lithium-ion rechargeable battery
Lreg: Linear regression (2-variable statistics)
mA: Milliamperes of current
Mtrx: Matrix support
NiCd: Nickel-Cadmium rechargeable battery
NiMH: Nickel-metal-hydrite rechargeable battery
Prnt: Printer
RTC: Real-time clock
Sdev: Standard deviation (1-variable statistics)
Solv: Equation solver
Subr: Subroutine call capability
Symb: Symbolic computing
Tape: Magnetic tape storage
Trig: Trigonometric functions
Units: Unit conversions
VAC: Volts AC
VDC: Volts DC
Sharp PC-1450
calculators. The other day I had a very successful hunting trip: I returned home with an old scientific calculator (an APF Mark 50), a Casio FX-795P, and this rare Sharp PC-1450, all three machines
in excellent working condition.
The PC-1450 is a BASIC programmable calculator. It shares a highly useful feature with the TI-74: it is a calculator mode, in which its buttons function much like the buttons on an ordinary
scientific calculator. In other words, you can invoke functions that operate on the contents of the display register, as opposed to having to key in BASIC expressions in immediate mode. (As to why
the designers of many newer calculators abandoned this calculator mode altogether, opting for the more cumbersome "formula display" mode of operation, I really have no idea.)
I have no manual for this beast, but I was able to discover a few things about its built-in BASIC interpreter anyway. For one thing, the PC-1450 BASIC has PEEK and POKE keywords, indicating the
possibility of accessing the hardware or doing machine language programming. For another, the BASIC is fairly advanced and complete, it even contains the DATA statement, which I found is missing from
the BASIC implementation on many a handheld device.
As usual, I wrote a Gamma function program to test the capabilities of this machine. In it, I made use of the DATA statement to simplify the program a little. Here's the result:
10:DATA 76.18009172,9.5E-9
20:DATA -86.50532032,-9.4E-9
30:DATA 24.01409824,8E-10
40:DATA -1.231739572,-4.5E-10
50:DATA 1.208650973E-3,8.7E-13
60:DATA -5.395239384E-6,-9.5E-16
100:INPUT "X=",X
120:IF X>=0 THEN 160
150: GOTO 120
170:FOR I=1 TO 6
180:READ C,D
200:NEXT I
210:G=LN (SQR (2*π)*G/X)
220:G=G-X-5.5+LN (X+5.5)*(X+.5)
230:PRINT EXP(G)/T | {"url":"https://rskey.org/CMS/?view=article&id=7&manufacturer=Sharp&model=PC-1450","timestamp":"2024-11-03T01:00:05Z","content_type":"application/xhtml+xml","content_length":"27738","record_id":"<urn:uuid:245ef490-9ee5-4cab-b051-cf882700d289>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00035.warc.gz"} |
What is a sampling distribution? | Hire Someone To Do Exam
What is a sampling distribution? Where would a sampling distribution be presented, or how could one define it? If sampling is based on the distribution of the sampling condition of the signal, you
could say it may be a multivariate process, as explained above. So, the sampling distribution suggests that the different degrees might take into account the different qualities of the signal, but
there is a more complicated dependence than this expression can encompass. Note, however, this is based on a method called FIT from the work of Staudinger, which starts by making the signal a mixture
of two functions: mixture Mixture = { r x, s } Where m is the distribution of the mixture, and r is a function that should be understood as a subset. An example of a mixture can be found here. {x,s}
A typical FIT from Staudinger would be to create symbols for the samples, called ϕ = mixture mixture = { r x, s , n x, y x, y } Where m is the number of the samples, and n is a dimension of the
signal. The number x is a number less than the number n x: = < mixture mixture = r x , s > mixture mixture = n n, y > k = ϕ >> n >> k = ϕ ω = hire someone to do medical assignment ϕ a value of the
function r from a normal distribution is generated by normalising the value r to a number n′ = -n*x, where n is a number. For example, one could write ϕ(mixture) = (mixture) q α = (ϕ) ∑ i:: i ≤ i0 ∑
j ∑ i ∑ n j ∑ k = 1 2.5 A similar result could be obtained from a likelihood as L(mixture) = ϕ(mixture) μ = ϕ(mixture) μ = ϕ(mixture) μ = ϕ(mixture) ∑ i:: i ≤ i0 ∑ j1 ∑ i ∑ k1 ∑ i ∑ n1 ∑ n j1 ∑ i ∑
n2 ∑ j2 ∑ n j2 ∑ i ∑ k2 ∑ i ∑ click for more info ∑ i ∑ n j2 ∑ i ∑ k3 ∑ i ∑ n4 ∑ j3 ∑ i ∑ j ∑ i ∑ k4 What is a sampling distribution? A sampling distribution (e.g..pdf) consists of small sample of
data with the more length as those in the data set. Prove If more information distribution whose r. M is written as a function of the number of bits in alphabet We define a sampler as follows: 1. The
frequency with which the data is sampled over a few “size” of the sample array 2. The expected number of samples per 100,000 period in 3. The expected number of samples per 100,000 period of time
look at more info the data come back in (random object with 100,000 elements in order so that each sample is roughly 100,000 elements.) The requirement that each sample be has exactly 100,000
elements makes this sampler a bit-clock. What is medical assignment hep sampling click for info Let us consider the case of a data-degenerate distribution: (1) In a local unit of measure, the r. M is
a distribution on the probability measure of a sample, as if a random variable with r = 1 is an r. M (2) In a local unit of distance zero, the r.
Law Will Take Its Own Course Meaning
M is a distribution on the probability measure of the samples not in the initial distribution, as if the random variable has r.M = 1 (3) In a local unit of magnitude, the r. M is a read review on the
probability measure of a sample, as if a random variable with r = 0 is an r. M (4) In a local unit of angular momentum, the view M is a distribution on the probability measure of a sample, as if a
random variable with r = 1 has r = 0.M = 0. Its tail is the number, that is, \begin{align} np_0 = 0.52428 \; p_1 = 0What is a sampling distribution? Frequencies in which the fraction of the sample
over a given fraction is rounded towards zero or one are find more fractional quantities. Fractional quantity is the fraction of a sampling distribution and its characteristic function with respect
to which fractions of the sample have sample normalization. Variance in the fractional quantity is the variance of the distribution fraction of the sample. Fractional quantities are values of a
function which accounts for the number of samples of a sample. This function is defined by $$R(x)=|x/ x^2|. \label{multisamples}$$ In practice, the sample mean may be divided by a binomial
distribution in samples of 1 to 5. It may be positive if the samples are equally distributed. The difference is then the variance of the fractional sample divided by the known mean. Further, each
sampling sample has its own sample variance and sample normalization. This is the portion of the sample that it represents. This type of distribution has particular advantages provided by certain
high order statistics methods. They are described in detail in Research Note 8 at https://researchnotes.org/association/
Pay Someone To Take Clep Test
The first key, of course, is the problem of information propagation in high order moments. As was suggested in Chapter 2, he cited a few recent papers which attempt to address this issue. The
statistics methods which have been used are not restricted to high order moments, for example Gaussian moments. In the next chapter we will refer to these methods as the multisamples method in
reference to distributions. ### The multisamples–from-the-sample-distribution We consider the four groups of samples in which the fraction of the sample is 1: B … \ | {"url":"https://paytodoexam.com/what-is-a-sampling-distribution","timestamp":"2024-11-10T09:28:04Z","content_type":"text/html","content_length":"192733","record_id":"<urn:uuid:daec6dea-c3c4-4afa-9612-0e382b9703a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00589.warc.gz"} |
An example problem on wind load calculation according to NSCP 2010 ;)
A 20-meter-high square-plan five-storey building with flat roof and 4m-high floors, located in Makati CBD, has sides of 10 meters length each, and a large open front door on the first floor that is
2m x 2m in dimension. Assuming that G = 0.85 and that torsion is negligible,
1. Show how this maybe is an open, partially enclosed, or enclosed building.
2. Determine the internal pressure coefficients.
3. Determine the external pressure coefficients for the design of main girders and shear walls.
4. Determine the base reactions due to along-wind loads acting on the front wall of the building.
1. The building satisfies all definitions of a partially enclosed building (NSCP 2010 Section 207.2).
2. The internal pressure coefficients for a partially enclosed building (GCpi) are +/- 0.55 (NSCP Figure 207-5).
3. The external pressure coefficients on MWFRS (from NSCP 2010 Figure 207-6) are as follows: - windward wall, Cp = 0.8 - leeward wall, Cp = -0.5 since L = 10m, B = 10m, and L/B = 1 - side walls, Cp =
-0.7 - whole roof, Cp = -1.04 or -0.18 since h = 20m, h/L = 2, L <= h/2 = 10m, and Roof Area = 100 sq.m > 93 sq.m
4. The base reactions can be calculated after we calculate the design wind force at each level. However, taking x = along-wind direction, y = across-wind direction, z = vertical direction, we already
can deduce that Vy = 0, and Mx = 0. Additionally, Mz is given as zero. We only need to estimate Vx, Vz, and My.
To calculate the design wind force at each level, we need to multiply net design wind pressures at each level with tributary areas. To get net design wind pressures, we calculate pressures on both
windward and leeward faces. On each face, we need to calculate the net of external and internal pressures. To get external and internal pressures, we need first to calculate the velocity pressures at
each level. To calculate by hand, it is easiest to do this in table form but with a computer, a spreadsheet makes it much easier:
│Assume: Exposure Terrain Category B Case 2, Iw = 1.0, Kd = 0.85, Kzt = 1.0, V = 200 kph │
│ │Windward wall pz (kPa) │Leeward wall pz (kPa) │
│z (m) │Kz │qz (kPa) │with +Gcpi │with -Gcpi │with +Gcpi │with -Gcpi │
│20 │0.88 │1.42 │0.18 │1.75 │-1.38 │0.18 │
│16 │0.82 │1.32 │0.12 │1.68 │-1.38 │0.18 │
│12 │0.76 │1.22 │0.05 │1.61 │-1.38 │0.18 │
│8 │0.67 │1.08 │-0.05 │1.52 │-1.38 │0.18 │
│4 │0.57 │0.92 │-0.16 │1.41 │-1.38 │0.18 │
│0 │0.57 │0.92 │-0.16 │1.41 │-1.38 │0.18 │
│Net along wind pressures pz (kPa)│ │Net along wind loads Fz (kN) │Base bending moment contribution My,z (kNm) │
├────────────────┬────────────────┤Afz (sqm)├──────────────┬──────────────┼──────────────────────┬──────────────────────┤
│with +Gcpi │with -Gcpi │ │with +Gcpi │with -Gcpi │with +Gcpi │with -Gcpi │
│1.56 │1.57 │20 │31 │31 │620 │620 │
│1.5 │1.5 │40 │60 │60 │960 │960 │
│1.43 │1.43 │40 │57 │57 │684 │684 │
│1.33 │1.34 │40 │53 │54 │424 │432 │
│1.22 │1.23 │40 │49 │49 │196 │196 │
│1.22 │1.23 │20 │24 │25 │0 │0 │
│ │ │Vx (kN) =│274 │276 │ │ │
├────────────────┼────────────────┼─────────┴──────────────┴──────────────┤2884 │2892 │
│ │ │My (kNm) = │ │ │
│ │Roof loads 1, p (kPa)│Roof loads 2, p (kPa)│Vz = Roof loads 1 (kN)│Vz = Roof loads 2 (kN) │
│Af,roof (sqm)├──────────┬──────────┼──────────┬──────────┼──────────┬───────────┼───────────┬───────────┤
│ │with +Gcpi│with -Gcpi│with +Gcpi│with -Gcpi│with +Gcpi│with -Gcpi │with +Gcpi │with -Gcpi │
│100 │-2.04 │-0.47 │-1 │0.56 │-204 │-47 │-100 │56 │
│Vz (kN) = │-204 │56 │
Spot any errors? Sound out in the comments. :)
28 comments:
Sir, diba yung roof pressure coefficient value must be multiplied by 0.80 to account for its area? So -0.18 should be -0.144? :)
The NSCP 2010 appears to say that the 0.8 reduction factor for area should be applied only to -1.3 and not to -0.18. The NSCP 2001 did not specify a -0.18 coefficient but it did say that the
reduction factor applies only to the -1.3 coefficient.
Thanks, sir!
Hi, Ronjie. How would you evaluate the pressure coeff. for structures
with irregular polygon shapes, say a structure with a plan shape of the
letter "W" or "k". Most of the pressure coefficients in the code are
for uniform section such as Circle, rectangle, square, etc.
have my own approach on this (mostly based on engineering judgement),
but I would like to here it first on a wind load aficionado, such as
John, thanks for your question. You are right, codes usually do not provide for such rather uncommonly used building shapes. You can of course use engineering judgment based on the coefficients
for shapes given in the code, which are based on wind tunnel tests on those specific "regular" shapes. That said, the more accurate and most rational method is to use the wind tunnel approach
(i.e. Method 3 in NSCP 2010, or ASCE7). Unless it is a shape that has been tested before and is published in literature, you might not be accounting for unexpected wind flow patterns that could
increase wind pressures at specific locations, or you might not be realizing cost savings by using a more accurate approach. If you are dealing with tall buildings, it is easy to justify the
relative low cost of doing a wind tunnel test.
Good Day Sir!
We're currently designing a building for a project and I hope it's okay to ask some concerns regarding the NSCP code.
The one that would be designed here is the MWFRS part of the bldg. The building is a low rise type and the calculations will be done using the Method 1 based from NSCP 2010. It is said that
Method 1 is applicable to both gable type and flat roof type buildings. The one indicated on Figure 2 of NSCP is a gable type building where there are loads A,B,C,D,E,F,G,and H applied.
a.) What will happen to the loads B and D for flat roof type of buildings? According to the table provided by NSCP there are negative horizontal pressures on these zones but since the eave height
is equal to the height of the structure, i think B and D should be zero. |
b.) Will there be no pressure applied on the rear side of the building?
Adolfo, thanks for asking. I can't help but notice two things...
1. "WE... designing a building"? You have a license already? ;)
2. "NSCP code"... National Structural Code of the Philippines Code? :D
As to your two questions, reading the NSCP, I think it's quite clear what the answers are. As I have emphasized in class, just make sure to consider at least 4 wind directions. Good luck!
Sir, pare-pareho lang po ba talaga ung values ng Leeward wall pz sa kahit anong elevation?
Thank you po in advance. :)
NSCP Figure 207-6 answers your question. :)
Oh. Thank you so much Sir! :D
Hi ronjie, do you know where Table 207-11 is located in NSCP 2010?
Thanks, I think I do. How about you?
I have trouble finding it yesterday, it seems that the code has some typo graphical errors in sec. 207.7.1 which directs you to table 207-11. Table 207-11 does not exist but the intended content
of table 207-11 is available in table 207-5.
Additional info. Eqn 207-38 & 207-39 is equal to 0.016/h and 0.23/h respectively. however Eqn 207-40 is equal to 0.007/n, in which h = ht of structure, and n = natural freq of structure. are Eqns
207-38 & 207-39 correct? or h should be replaced by n?
Yeah, thanks for pointing that out. I've started to use the latest NSCP only recently and I've also spotted a lot of typographical errors already. You are right, Table 207-11 should be Table
207-5, and note also that alpha-overbar and a small letter "L" in Times New Roman are mentioned in 207.7.1 but in Table 207-5, it's actually A-overbar and a small letter "L" in script font in
there. There are also a number of instances where h or h <= 18 m or h >= 18 m appears to have been omitted.
As to your question on Equations 207-38, -39, and -40, note that 207-38 and -39 are formulas for one thing (Beta_s), while 207-40 is a formula for another thing (Beta_a). I think that is quite
clear, so my answer to your question is: no, h should not be replaced by n.
Good day sir. do you have a sample problem like above to get the roof wind load for sawtooth type of buildings?
I don't have one, but I think the NSCP's figures for sawtooth roofed buildings is pretty straightforward. Thanks for commenting!
Good day sir! Im a bit confused. Can you show me how the wind pressure
Hi, can I show you how the wind pressure.....?
Good day sir! i'd like to ask if the qz/velocity pressure in the nscp acts perpendicular to the roof surface or perpendicular to the vertical projection of the roof..
You don't actually apply q_z on a roof surface, you apply q_h (which is q_z with z = h). The figures show or indicate that the NSCP wind pressures are either positive (pressing action) or
negative (suction) -- which means they're always perpendicular to the surface.
thank you sir..so, how about if its a truss analysis? where should i start my computation sir? with reference to the Nscp..im at a lost in wind load computations.
For vertical truss structures, wind forces are always applied horizontally. For roof trusses, they are applied perpendicular to the surface receiving the wind pressures.
in reference with NSCP sir,wat is the equation to use for the wind load acting on a truss alone? And also In staad, Can i consider a single truss only considering loads(dead,live,wind) in
tributary area?!
The answer to your first question is in the NSCP. For any structural analysis whether using STAAD or another software, of course you can consider a single truss considering only the tributary
area but make sure to use the appropriate loads for that area.
What did you need this for? A homework or a project?
for a project sir. can i use fig. 207-1 in the computation of wind load and apply it to the truss considering only the tributary area?
for a project sir. im considering loads acting on a truss alone and not on d whole structure..in what part of the nscp will i refer to directly? if wimd velocity is 350kph. can i use ds formula
p=Qh[(GCp) - (GCpi)]
to directly act normally to the roof?
or formula (207-15):
Good day sir. I'd like to ask you about how to use the reduction factor to -1.3. | {"url":"http://engg.ronjie.com/2013/08/an-example-problem-on-wind-load.html?showComment=1376994209843","timestamp":"2024-11-03T13:25:59Z","content_type":"application/xhtml+xml","content_length":"134088","record_id":"<urn:uuid:45a69797-cad0-4121-95ee-9792dcb864dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00887.warc.gz"} |
What does it mean to evaluate a function?
What does it imply to judge a perform?
To judge a perform is to: Substitute (substitute) its variable with a given quantity or expression.
Which is an instance of an argument?
Right here is an instance of an argument: If you wish to discover a good job, you need to work laborious. You do wish to discover a good job. So you need to work laborious.
What’s a ultimate analysis?
A ultimate evaluation is the final exercise college students should full in a course. A ultimate evaluation could also be an examination, a culminating exercise or a mixture of the 2. A ultimate
evaluation refers back to the process that evaluates college students’ progress in a course. …
How do you write a ultimate analysis?
1. What do you consider the general design ?
2. Are you pleased with the supplies you selected ?
3. Is the color scheme precisely what you anticipated ?
4. Did the venture take too lengthy to make ?
How have you learnt if it’s a perform?
The vertical line check can be utilized to find out whether or not a graph represents a perform. If we are able to draw any vertical line that intersects a graph greater than as soon as, then the
graph doesn’t outline a perform as a result of that x worth has multiple output. A perform has just one output worth for every enter worth.
How do you begin a analysis?
How one can Write an Analysis Essay
1. Select your matter. As with all essay, this is among the first steps .
2. Write a thesis assertion. It is a key factor of your essay because it units out the general function of the analysis.
3. Decide the standards used to evaluate the product.
4. Search for supporting proof.
5. Draft your essay.
6. Assessment, revise & rewrite.
What are key arguments?
n (Philosophy) an argument designed to make specific the situations underneath which a sure sort of information is feasible, esp.
How do you resolve an expression with two variables?
Divide either side of the equation to “resolve for x.” After you have the x time period (or whichever variable you’re utilizing) on one aspect of the equation, divide either side of the equation to
get the variable alone. For instance: 4x = 8 – 2y. (4x)/4 = (8/4) – (2y/4) | {"url":"https://wanderluce.com/blog/what-does-it-mean-to-evaluate-a-function/","timestamp":"2024-11-09T00:57:49Z","content_type":"text/html","content_length":"117712","record_id":"<urn:uuid:cc70b21f-e550-4bda-be86-ea4549665fc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00083.warc.gz"} |
Long Run Production Function
Last updated on 17/04/2020
Production in the short run in which the functional relationship between input and output is explained assuming labor to be the only variable input, keeping capital constant.
In the long run production function, the relationship between input and output is explained under the condition when both, labor and capital, are variable inputs.
In the long run, the supply of both the inputs, labor and capital, is assumed to be elastic (changes frequently). Therefore, organizations can hire larger quantities of both the inputs. If larger
quantities of both the inputs are employed, the level of production increases. In the long run, the functional relationship between changing scale of inputs and output is explained under laws of
returns to scale. The laws of returns to scale can be explained with the help of isoquant technique.
Isoquant Curve
The relationships between changing input and output is studied in the laws of returns to scale, which is based on production function and isoquant curve. The term isoquant has been derived from a
Greek work iso, which means equal. Isoquant curve is the locus of points showing different combinations of capital and labor, which can be employed to produce same output.
It is also known as equal product curve or production indifference curve. Isoquant curve is almost similar to indifference curve. However, there are two dissimilarities between isoquant curve and
indifference curve. Firstly, in the graphical representation, indifference curve takes into account two consumer goods, while isoquant curve uses two producer goods. Secondly, indifference curve
measures the level of satisfaction, while isoquant curve measures output.
Some of the popular definitions of isoquant curve are as follows:
According to Ferguson, “An isoquant is a curve showing all possible combinations of inputs physically capable of producing a given level of output.”
According to Peterson, “An isoquant curve may be defined as a curve showing the possible combinations of two variable factors that can be used to produce the same total product”
From the aforementioned definitions, it can be concluded that the isoquant curve is generated by plotting different combinations of inputs on a graph. An isoquant curve provides the best combination
of inputs at which the output is maximum.
Following are the assumptions of isoquant curve:
• Assumes that there are only two inputs, labor and capital, to produce a product
• Assumes that capital, labor, and good are divisible in nature
• Assumes that capital and labor are able to substitute each other at diminishing rates because they are not perfect substitutes
• Assumes that technology of production is known
On the basis of these assumptions, isoquant curve can be drawn with the help of different combinations of capital and labor. The combinations are made such that it does not affect the output.
Figure-1 represents an isoquant curve for four combinations of capital and labor:
In Figure-1, IQ1 is the output for four combinations of capital and labor. Figure 1 shows that all along the curve for IQ1 the quantity of output is same that is 200 with the changing combinations of
capital and labor. The four combinations on the IQ1 curve are represented by points A, B, C, and D.
Some of the properties of the isoquant curve are as follows:
1. Negative Slope
Implies that the slope of isoquant curve is negative. This is because when capital (K) is increased, the quantity of labor (L) is reduced or vice versa, to keep the same level of output.
2. Convex to Origin
Shows the substitution of inputs and diminishing marginal rate of technical substitution (which is discussed later) in economic region. This implies that marginal significance of one input (capital)
in terms of another input (labor) diminishes along with the isoquant curve.
3. Non-intersecting and Non-tangential
Implies that two isoquant curves (as shown in Figure-1) cannot cut each other.
4. Upper isoquant have high output
Implies that upper curve of the isoquant curve produces more output than the curve beneath. This is because of the larger combination of input result in a larger output as compared to the curve that
s beneath it. For example, in Figure-5 the value of capital at point B is greater than the capital at point C. Therefore, the output of curve Q2 is greater than the output of Q1.
Forms of Isoquants
The shape of an isoquant depends on the degree to which one input can be substituted by the other. Convex isoquant represents that there is a continuous substitution of one input variable by the
other input variable at a diminishing rate.
However, in economics, there are other forms of isoquants, which are as follows:
1. Linear Isoquant
Refers to a straight line isoquant. Linear isoquant represents a perfect substitutability between the inputs, capital and labor, of the production function. It implies that a product can be produced
by using either capital or labor or using both, if capital and labor are perfect substitutes of each other. Therefore, in a linear isoquant, MRTS between inputs remains constant.
The algebraic form of production function in case of linear isoquant is as follows:
Q = aK + BL
Here, Q is the weighted sum of K and L.
Slope of curve can be calculated with the help of following formula:
MP[K] = ∆Q/∆K = a
MP[L] = ∆Q/∆L = b
MRTS = MP[L]/MP[K]
MRTS = -b/a
However, linear isoquant does not have existence in the real world.
Figure-2 shows a linear isoquant
2. L-shaped Isoquant
Refers to an isoquant in which the combination between capital and labor are in a fixed proportion. The graphical representation of fixed factor proportion isoquant is L in shape. The L-shaped
isoquant represents that there is no substitution between labor and capital and they are assumed to be complementary goods.
It represents that only one combination of labor and capital is possible to produce a product with affixed proportion of inputs. For increasing the production, an organization needs to increase both
inputs proportionately.
Figure-3 shows an L-shaped isoquant
In Figure-3, it can be seen OK1 units of capital and OL1 units of labor are required for the production of Q1. On the other hand, to increase the production from Q1 to Q2, an organization needs to
increase inputs from K1 to K2 and L1 to L2 both.
This relationship between capital and labor can be expressed as follows:
Q = f (K, L) = min (aK, bL)
Where, min = Q equals to lower of the two terms, aK and bL
For example, in case aK > bL, then Q = bL and in case aK < bL then, Q = aK.
L-shaped isoquant is applied in many production activities and techniques where labor and capital is in fixed proportion. For example, in the process of driving a car, only one machine and one labor
is required, which is a fixed combination.
3. Kinked Isoquant
Refers to an isoquant that represents different combinations of labor and capital. These combinations can be used in different processes of production, but in fixed proportion. According to L-shaped
isoquant, there would be only one combination between capital and labor in a fixed proportion. However, in real life, there can be several ways to perform production with different combinations of
capital and labor.
For example, there are two machines in which one is large in size and can perform all the processes involved in production, while the other machine is small in size and can perform only one function
of production process. In both the machines, combination of capital employed and labor used is different.
Let us understand kinked isoquant with the help of another example. For example, to produce 100 units of product X, an organization has used four different techniques of production with fixed-factor | {"url":"https://indiafreenotes.com/long-run-production-function/","timestamp":"2024-11-03T00:32:15Z","content_type":"text/html","content_length":"230493","record_id":"<urn:uuid:2e5b6bff-8823-4cda-b137-f880af39590b>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00272.warc.gz"} |
Dividing Whole Numbers Worksheets Answers
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Divide Whole Numbers By Larger Powers Of Ten And Answer In A Decimal Number Grade 5 Math
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Dividing Whole Numbers 6th Grade Math Worksheets
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Division Worksheets K5 Learning
Grade 5 division worksheets Divide 3 or 4 digit numbers by 1 digit numbers mentally Division with remainder 1 100 1 1 000 Dividing by whole tens or hundreds with remainders Long division with 1 digit
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Experts’ Guide – A Comprehensive List of Integer Functions and Methods in Java
Overview of Integer Functions and Methods in Java
In Java, the Integer class provides numerous functions and methods for working with integer values. Understanding these functions and methods is crucial for Java developers. Whether you are dealing
with arithmetic operations, conversions, comparisons, bit manipulations, or number properties, the Integer class has you covered.
Constructors and Initialization
When working with integers in Java, you can create Integer objects using different constructors and initialization methods:
Integer() – Creating a New Integer Object with Default Value
The Integer() constructor creates a new Integer object with a default value of zero. This is useful when you need to initialize an integer variable with a default value.
Integer(int value) – Creating a New Integer Object with Specified Value
The Integer(int value) constructor allows you to create a new Integer object with a specified value. This is useful when you already know the value that you want to assign to the integer variable.
parseInt(String s) – Parsing a String to an int Value
The parseInt(String s) method is a static method of the Integer class that allows you to parse a String representation of an integer and convert it into an int value. This is useful when you need to
extract integer values from user input or manipulate integer values stored as strings.
Conversion and Comparison
When working with Integer objects, you may need to perform conversions or comparisons:
intValue() – Converting an Integer Object to int Primitive
The intValue() method is used to convert an Integer object to an int primitive. This is particularly useful when you need to perform calculations or operations that require an int value rather than
an Integer object.
compareTo(Integer anotherInteger) – Comparing Two Integer Objects
The compareTo(Integer anotherInteger) method compares two Integer objects. It returns a negative value if the calling object is smaller, a positive value if the calling object is larger, and zero if
the two objects are equal.
equals(Object obj) – Checking if an Integer Object is Equal to Another Object
The equals(Object obj) method checks if an Integer object is equal to another object. It returns true if the objects are equal and false otherwise. This method is often used to compare Integer
objects for equality.
toString() – Converting an Integer Object to a String
The toString() method converts an Integer object to a String. This is useful when you need to represent the integer value as a string, such as when displaying it in user interfaces or concatenating
it with other strings.
Arithmetic Operations
The Integer class also provides methods for performing various arithmetic operations on integer values:
add(int a, int b) – Adding Two int Values
The add(int a, int b) method adds two int values and returns the result as an int. This method is useful when you need to perform addition operations on integers.
subtract(int a, int b) – Subtracting One int Value from Another
The subtract(int a, int b) method subtracts one int value from another and returns the result as an int. It is helpful when you need to perform subtraction operations on integers.
multiply(int a, int b) – Multiplying Two int Values
The multiply(int a, int b) method multiplies two int values and returns the result as an int. Use this method when you need to perform multiplication operations on integers.
divide(int dividend, int divisor) – Dividing One int Value by Another
The divide(int dividend, int divisor) method divides one int value by another and returns the quotient as an int. It is useful when you need to perform division operations on integers.
modulo(int dividend, int divisor) – Calculating the Remainder of the Division
The modulo(int dividend, int divisor) method calculates the remainder of the division between two int values and returns the result as an int. This method comes in handy when you need to find the
remainder after division.
Bit Manipulation
The Integer class provides methods for manipulating the individual bits of an integer:
bitCount(int n) – Counting the Number of One-Bits in an int Value
The bitCount(int n) method counts the number of one-bits in the binary representation of an int value and returns the result as an int. It is useful when you need to determine the number of set bits
in an integer.
bitShiftLeft(int n, int positions) – Shifting the Bits of an int Value to the Left
The bitShiftLeft(int n, int positions) method shifts the bits of an int value to the left by a specified number of positions and returns the result as an int. This method is helpful for performing
left shift operations on integers.
bitShiftRight(int n, int positions) – Shifting the Bits of an int Value to the Right
The bitShiftRight(int n, int positions) method shifts the bits of an int value to the right by a specified number of positions and returns the result as an int. Use this method for performing right
shift operations on integers.
bitwiseAnd(int a, int b) – Performing a Logical AND Operation on Two int Values
The bitwiseAnd(int a, int b) method performs a logical AND operation on two int values and returns the result as an int. It is useful when you need to perform bitwise AND operations on integers.
bitwiseOr(int a, int b) – Performing a Logical OR Operation on Two int Values
The bitwiseOr(int a, int b) method performs a logical OR operation on two int values and returns the result as an int. This method is helpful for performing bitwise OR operations on integers.
bitwiseXor(int a, int b) – Performing a Logical XOR Operation on Two int Values
The bitwiseXor(int a, int b) method performs a logical XOR operation on two int values and returns the result as an int. Use this method for performing bitwise XOR operations on integers.
Number Properties
The Integer class provides methods for checking various number properties:
isEven(int n) – Checking if an int Value is Even
The isEven(int n) method checks if an int value is even and returns true if it is, and false otherwise. This method comes in handy when you need to identify even numbers.
isPositive(int n) – Checking if an int Value is Positive
The isPositive(int n) method checks if an int value is positive and returns true if it is, and false otherwise. Use this method when you need to determine if a number is positive.
isNegative(int n) – Checking if an int Value is Negative
The isNegative(int n) method checks if an int value is negative and returns true if it is, and false otherwise. This method is useful when you need to identify negative numbers.
isPrime(int n) – Checking if an int Value is a Prime Number
The isPrime(int n) method checks if an int value is a prime number and returns true if it is, and false otherwise. It is a helpful tool for identifying prime numbers.
abs(int n) – Calculating the Absolute Value of an int Value
The abs(int n) method calculates the absolute value of an int value and returns the result as an int. It is useful when you need to obtain the positive magnitude of a number, regardless of its
original sign.
The Java Integer class provides an extensive set of functions and methods for working with integer values. Whether you need to perform arithmetic operations, conversions, comparisons, bit
manipulations, or check number properties, the Integer class has you covered.
By understanding and utilizing the functions and methods covered in this blog post, you can enhance your Java programming skills and tackle various programming tasks more efficiently. So, why not
dive deeper into the documentation and explore the full potential of the Integer class in your future Java projects? Happy coding! | {"url":"https://skillapp.co/blog/experts-guide-a-comprehensive-list-of-integer-functions-and-methods-in-java/","timestamp":"2024-11-03T06:33:39Z","content_type":"text/html","content_length":"114587","record_id":"<urn:uuid:9b16cf9f-fc76-490f-bd70-ebec776f4a58>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00124.warc.gz"} |
Section: New Results
Online Matrix Completion Through Nuclear Norm Regularisation [14]
It is the main goal of this paper to propose a novel method to perform matrix completion on-line. Motivated by a wide variety of applications, ranging from the design of recommender systems to sensor
network localization through seismic data reconstruction, we consider the matrix completion problem when entries of the matrix of interest are observed gradually. Precisely, we place ourselves in the
situation where the predictive rule should be refined incrementally, rather than recomputed from scratch each time the sample of observed entries increases. The extension of existing matrix
completion methods to the sequential prediction context is indeed a major issue in the Big Data era, and yet little addressed in the literature. The algorithm promoted in this article builds upon the
Soft Impute approach introduced in Mazumder et al. (2010). The major novelty essentially arises from the use of a randomised technique for both computing and updating the Singular Value Decomposition
(SVD) involved in the algorithm. Though of disarming simplicity, the method proposed turns out to be very efficient, while requiring reduced computations. Several numerical experiments based on real
datasets illustrating its performance are displayed, together with preliminary results giving it a theoretical basis.
Synthèse en espace et temps du rayonnement acoustique d'une paroi sous excitation turbulente par synthèse spectrale 2D+T et formulation vibro-acoustique directe [33]
Une méthode directe pour simuler les vibrations et le rayonnement acoustique d'une paroi soumise à un écoulement subsonique est proposée. Tout d'abord, en adoptant l'hypothèse d'un écoulement
homogène et stationnaire, on montre qu'une méthode de synthèse spectrale en espace et temps (2D+t) est suffisante pour obtenir explicitement une réalisation d'un champ de pression pariétale
excitatrice p(x,y,t) dont les propriétés inter-spectrales sont prescrites par un modèle empirique de Chase. Cette pression turbulente p(x,y,t) est obtenue explicitement et permet de résoudre le
problème vibro-acoustique de la paroi dans une formulation directe. La méthode proposée fournit ainsi une solution complète du problème dans le domaine spatio-temporel : pression excitatrice,
déplacement en flexion et pression acoustique rayonnée par la paroi. Une caractéristique de la méthode proposée est un cout de calcul qui s'avère similaire aux formulations inter-spectrales
majoritairement utilisées dans la littérature. En particulier, la synthèse permet de prendre en compte l'intégralité des échelles spatio-temporelles du problème : échelles turbulentes, vibratoires et
acoustiques. A titre d'exemple, la pression aux oreilles d'un auditeur suite à l'excitation turbulente de la paroi est synthétisée.
Bandits attack function optimization [27]
We consider function optimization as a sequential decision making problem under the budget constraint. Such constraint limits the number of objective function evaluations allowed during the
optimization. We consider an algorithm inspired by a continuous version of a multi-armed bandit problem which attacks this optimization problem by solving the tradeoff between exploration (initial
quasi-uniform search of the domain) and exploitation (local optimization around the potentially global maxima). We introduce the so-called Simultaneous Optimistic Optimization (SOO), a deterministic
algorithm that works by domain partitioning. The benefit of such an approach are the guarantees on the returned solution and the numerical eficiency of the algorithm. We present this machine learning
rooted approach to optimization, and provide the empirical assessment of SOO on the CEC'2014 competition on single objective real-parameter numerical optimization testsuite.
Optimistic planning in Markov decision processes using a generative model [30]
We consider the problem of online planning in a Markov decision process with discounted rewards for any given initial state. We consider the PAC sample com-plexity problem of computing, with
probability 1−δ, an -optimal action using the smallest possible number of calls to the generative model (which provides reward and next-state samples). We design an algorithm, called StOP (for
Stochastic-Optimistic Planning), based on the "optimism in the face of uncertainty" princi-ple. StOP can be used in the general setting, requires only a generative model, and enjoys a complexity
bound that only depends on the local structure of the MDP.
Near-Optimal Rates for Limited-Delay Universal Lossy Source Coding [3]
We consider the problem of limited-delay lossy coding of individual sequences. Here, the goal is to design (fixed-rate) compression schemes to minimize the normalized expected distortion redundancy
relative to a reference class of coding schemes, measured as the difference between the average distortion of the algorithm and that of the best coding scheme in the reference class. In compressing a
sequence of length $T$, the best schemes available in the literature achieve an $O\left({T}^{-1/3}$) normalized distortion redundancy relative to finite reference classes of limited delay and limited
memory, and the same redundancy is achievable, up to logarithmic factors, when the reference class is the set of scalar quantizers. It has also been shown that the distortion redundancy is at least
of order ${T}^{-1/2}$ in the latter case, and the lower bound can easily be extended to sufficiently powerful (possibly finite) reference coding schemes. In this paper, we narrow the gap between the
upper and lower bounds, and give a compression scheme whose normalized distortion redundancy is $O\left(ln\left(T\right)/{T}^{1/2}\right)$ relative to any finite class of reference schemes, only a
logarithmic factor larger than the lower bound. The method is based on the recently introduced shrinking dartboard prediction algorithm, a variant of exponentially weighted average prediction. The
algorithm is also extended to the problem of joint source-channel coding over a (known) stochastic noisy channel and to the case when side information is also available to the decoder (the Wyner–Ziv
setting). The same improvements are obtained for these settings as in the case of a noiseless channel. Our method is also applied to the problem of zero-delay scalar quantization, where $O\left(ln\
left(T\right)/{T}^{1/2}\right)$ normalized distortion redundancy is achieved relative to the (infinite) class of scalar quantizers of a given rate, almost achieving the known lower bound of order $1/
{T}^{-1/2}$. The computationally efficient algorithms known for scalar quantization and the Wyner–Ziv setting carry over to our (improved) coding schemes presented in this paper.
Online Markov Decision Processes Under Bandit Feedback [4]
Software systems are composed of many interacting elements. A natural way to abstract over software systems is to model them as graphs. In this paper we consider software dependency graphs of
object-oriented software and we study one topological property: the degree distribution. Based on the analysis of ten software systems written in Java, we show that there exists completely different
systems that have the same degree distribution. Then, we propose a generative model of software dependency graphs which synthesizes graphs whose degree distribution is close to the empirical ones
observed in real software systems. This model gives us novel insights on the potential fundamental rules of software evolution.
A Generative Model of Software Dependency Graphs to Better Understand Software Evolution [37]
Software systems are composed of many interacting elements. A natural way to abstract over software systems is to model them as graphs. In this paper we consider software dependency graphs of
object-oriented software and we study one topological property: the degree distribution. Based on the analysis of ten software systems written in Java, we show that there exists completely different
systems that have the same degree distribution. Then, we propose a generative model of software dependency graphs which synthesizes graphs whose degree distribution is close to the empirical ones
observed in real software systems. This model gives us novel insights on the potential fundamental rules of software evolution.
Preference-Based Rank Elicitation using Statistical Models: The Case of Mallows [8]
We address the problem of rank elicitation as-suming that the underlying data generating pro-cess is characterized by a probability distribu-tion on the set of all rankings (total orders) of a given
set of items. Instead of asking for complete rankings, however, our learner is only allowed to query pairwise preferences. Using information of that kind, the goal of the learner is to reliably
predict properties of the distribution, such as the most probable top-item, the most probable rank-ing, or the distribution itself. More specifically, learning is done in an online manner, and the
goal is to minimize sample complexity while guaran-teeing a certain level of confidence.
Preference-based reinforcement learning: evolutionary direct policy search using a preference-based racing algorithm [1]
We introduce a novel approach to preference-based reinforcement learn-ing, namely a preference-based variant of a direct policy search method based on evolutionary optimization. The core of our
approach is a preference-based racing algorithm that selects the best among a given set of candidate policies with high probability. To this end, the algorithm operates on a suitable ordinal
preference structure and only uses pairwise comparisons between sample rollouts of the policies. Embedding the racing algorithm in a rank-based evolutionary search procedure, we show that
approxima-tions of the so-called Smith set of optimal policies can be produced with certain theoretical guarantees. Apart from a formal performance and complexity analysis, we present first
experimental studies showing that our approach performs well in practice.
Biclique Coverings, Rectifier Networks and the Cost of ε-Removal [16]
We relate two complexity notions of bipartite graphs: the minimal weight biclique covering number Cov(G) and the minimal rec-tifier network size Rect(G) of a bipartite graph G. We show that there
exist graphs with Cov(G) ≥ Rect(G) 3/2−ǫ . As a corollary, we estab-lish that there exist nondeterministic finite automata (NFAs) with ε-transitions, having n transitions total such that the smallest
equivalent ε-free NFA has Ω(n 3/2−ǫ) transitions. We also formulate a version of previous bounds for the weighted set cover problem and discuss its con-nections to giving upper bounds for the
possible blow-up.
Efficient Eigen-updating for Spectral Graph Clustering [2]
Partitioning a graph into groups of vertices such that those within each group are more densely connected than vertices assigned to different groups, known as graph clustering, is often used to gain
insight into the organisation of large scale networks and for visualisation purposes. Whereas a large number of dedicated techniques have been recently proposed for static graphs, the design of
on-line graph clustering methods tailored for evolving networks is a challenging problem, and much less documented in the literature. Motivated by the broad variety of applications concerned, ranging
from the study of biological networks to the analysis of networks of scientific references through the exploration of communications networks such as the World Wide Web, it is the main purpose of
this paper to introduce a novel, computationally efficient, approach to graph clustering in the evolutionary context. Namely, the method promoted in this article can be viewed as an incremental
eigenvalue solution for the spectral clustering method described by Ng. et al. (2001). The incremental eigenvalue solution is a general technique for finding the approximate eigenvectors of a
symmetric matrix given a change. As well as outlining the approach in detail, we present a theoretical bound on the quality of the approximate eigenvectors using perturbation theory. We then derive a
novel spectral clustering algorithm called Incremental Approximate Spectral Clustering (IASC). The IASC algorithm is simple to implement and its efficacy is demonstrated on both synthetic and real
datasets modelling the evolution of a HIV epidemic, a citation network and the purchase history graph of an e-commerce website.
From Bandits to Monte-Carlo Tree Search: The Optimistic Principle Applied to Optimization and Planning [36]
This work covers several aspects of the optimism in the face of uncertainty principle applied to large scale optimization problems under finite numerical budget. The initial motivation for the
research reported here originated from the empirical success of the so-called Monte-Carlo Tree Search method popularized in computer-go and further extended to many other games as well as
optimization and planning problems. Our objective is to contribute to the development of theoretical foundations of the field by characterizing the complexity of the underlying optimization problems
and designing efficient algorithms with performance guarantees. The main idea presented here is that it is possible to decompose a complex decision making problem (such as an optimization problem in
a large search space) into a sequence of elementary decisions, where each decision of the sequence is solved using a (stochastic) multi-armed bandit (simple mathematical model for decision making in
stochastic environments). This so-called hierarchical bandit approach (where the reward observed by a bandit in the hierarchy is itself the return of another bandit at a deeper level) possesses the
nice feature of starting the exploration by a quasi-uniform sampling of the space and then focusing progressively on the most promising area, at different scales, according to the evaluations
observed so far, and eventually performing a local search around the global optima of the function. The performance of the method is assessed in terms of the optimality of the returned solution as a
function of the number of function evaluations. Our main contribution to the field of function optimization is a class of hierarchical optimistic algorithms designed for general search spaces (such
as metric spaces, trees, graphs, Euclidean spaces, ...) with different algorithmic instantiations depending on whether the evaluations are noisy or noiseless and whether some measure of the
”smoothness” of the function is known or unknown. The performance of the algorithms depend on the local behavior of the function around its global optima expressed in terms of the quantity of
near-optimal states measured with some metric. If this local smoothness of the function is known then one can design very efficient optimization algorithms (with convergence rate independent of the
space dimension), and when it is not known, we can build adaptive techniques that can, in some cases, perform almost as well as when it is known. | {"url":"https://radar.inria.fr/report/2014/sequel/uid86.html","timestamp":"2024-11-08T11:13:43Z","content_type":"text/html","content_length":"56261","record_id":"<urn:uuid:a49c6425-44cc-4818-9ee4-419382007ffa>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00644.warc.gz"} |
Applied Math Seminar | Peter Kloeden, Random ordinary differential equations and their numerical approximation | Applied Mathematics
Thursday, August 22, 2019 2:30 pm - 2:30 pm EDT (GMT -04:00)
MC 6460
Peter Kloeden | Universitat Tubingen, Germany
Random ordinary differential equations and their numerical approximation
Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. They have been used for many years
in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion or a
Poisson process, but when it is a diffusion process then there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE
and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors.
Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the
driving process. It will be shown how, nevertheless, various kinds of Taylor-like expansions of the solutions of RODES can be obtained when the stochastic process has Hölder continuous or even
measurable sample paths and then used to derive pathwise convergent numerical schemes of arbitrarily high order. The use of bounded noise and an application in biology will be considered. | {"url":"https://uwaterloo.ca/applied-mathematics/events/applied-math-seminar-peter-kloeden-random-ordinary","timestamp":"2024-11-11T04:43:21Z","content_type":"text/html","content_length":"112370","record_id":"<urn:uuid:05f210ce-f1d5-4016-859c-e4890988384d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00491.warc.gz"} |
NCERT Solutions for Class 9 Maths Exercise 10.3 Circles
NCERT Solutions for Class 9 Maths Exercise 10.3 Circles in Hindi and English Medium free to download in PDF. We have updated the solutions related to class 9 Maths Exercise 10.3 for new academic
session 2024-25. Video solutions of exercise 10.3 is also given in Hindi and English for better understanding.
About Class 9 Maths Exercise 10.3
Before proceeding exercise 10.3 which has only three questions, we have to understand theorem 10.3, 10.4 and 10.5 also. Theorem 10.3 says that if a perpendicular is drawn from the center of the
circle to a chord. This perpendicular bisects the chord.
The term bisects means dividing into two equal halves. Students should know that the term BI at many places 2. Similarly, Theorem 10.4 is the converse of theorem 10.3. It says that the line drawn
through the center of the circle to bisect a given chord must be perpendicular to the chord.
How to prepare Exercise 10.3 in Class 9
Students must have notice that right through the study of geometry most of theorems have their converse too. For example – In higher classes you could study the Pythagoras theorem and its converse
In this exercise we shall also learn that we need minimum three points to drawn a circle. It is to be noted that these three point must not be collinear. The word collinear means passing through the
same straight line or line on the same straight line.
Questions of Exercise 10.3 Class 9 Maths
We should also learn how to draw a circle which passes through three non – collinear points. By drawing perpendicular bisectors through them. Based on the above discussions Theorem 10.5 states that
only one circle can be drawn which passes through three given non – collinear points.
In addition to the above, we have to learn how to complete the circle given an arc of a circle. To draw a circle we need to know the center of the circle and the radius.
Solving 9th Maths Exercise 10.3 using Theorems
By using theorem 10.5 we can find both of them and draw the required circle. It will not be out of place to mention that student should carry the geometry box. In particular – the compass, a
sharpened pencil and an eraser.
Because many times it has been observe that students tried to resort to these small items even in examination hall. Exercise 10.3 has just three question and you should not mind doing all the things.
Last Edited: November 4, 2023 | {"url":"https://www.tiwariacademy.com/ncert-solutions/class-9/maths/chapter-10/exercise-10-3-old/","timestamp":"2024-11-11T07:04:08Z","content_type":"text/html","content_length":"230397","record_id":"<urn:uuid:04818202-ae5a-40bc-a04a-a669aa37f87c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00859.warc.gz"} |
Invited Talk: A domain theory for quasi-Borel spaces and statistical probabilistic programming
EasyChair Smart Slide Docs/Log in
Title:Invited Talk: A domain theory for quasi-Borel spaces and statistical probabilistic programming
Authors:Ohad Kammar
Tags:axiomatic domain theory, commutative monads, essentially algebraic theories, Grothendieck quasi-topos, Jung-Tix problem,
probabilistic powerdomains, quasi-Borel spaces, s-finite distributions and synthetic measure theory
I will describe ongoing work investigating a convenient category of pre-domains and a probabilistic powerdomain construction MoreRelated
suitable for statistical probabilistic programming semantics, as used in statistical modelling and machine learning.
Specifically, we provide (1) a cartesian closed category; (2) whose objects are (pre)-domains; and (3) a commutative monad for Invited Talk: A domain theory for quasi-Borel spaces and 1
probabilistic choice and Bayesian conditioning. Jones and Plotkin have shown that conditions (2)--(3) hold when one restricts statistical probabilistic programming
attention to continuous domains, and Jung and Tix have proposed to search for a suitable category of continuous domains 2
possessing all three properties (1)--(3), a question that remains open to date. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
I propose an alternative direction, considering spaces with separate, but compatible, measure-theoretic and domain-theoretic
structures. On the domain-theoretic side, we require posets with suprema of countably increasing chains (omega-cpos). On the 4
measure-theoretic side, we require a quasi-Borel space (qbs) structure, a recently introduced algebraic structure suitable for
modelling higher-order probability theory. There are three equivalent characterisations of this category given by: imposing an
order-theoretic separatedness condition on countable-preserving omega-cpo-valued presheaves; internal omega-cpos in the
quasi-topos of quasi-Borel spaces; and an essentially algebraic presentation. The category of these omega-qbses validates Fiore
and Plotkin's axiomatic domain theory, yielding semantics for recursive types. To conclude, I will describe a commutative
powerdomain construction given by factorising the Lebesgue integral from the space of random elements to the space of
sigma-linear integration operators.
Title:Invited Talk: A domain theory for quasi-Borel spaces and statistical probabilistic programming
Authors:Ohad Kammar
Tags:axiomatic domain theory, commutative monads, essentially algebraic theories, Grothendieck quasi-topos, Jung-Tix problem,
probabilistic powerdomains, quasi-Borel spaces, s-finite distributions and synthetic measure theory
I will describe ongoing work investigating a convenient category of pre-domains and a probabilistic powerdomain construction MoreRelated
suitable for statistical probabilistic programming semantics, as used in statistical modelling and machine learning.
Specifically, we provide (1) a cartesian closed category; (2) whose objects are (pre)-domains; and (3) a commutative monad for Invited Talk: A domain theory for quasi-Borel spaces and 1
probabilistic choice and Bayesian conditioning. Jones and Plotkin have shown that conditions (2)--(3) hold when one restricts statistical probabilistic programming
attention to continuous domains, and Jung and Tix have proposed to search for a suitable category of continuous domains 2
possessing all three properties (1)--(3), a question that remains open to date. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
I propose an alternative direction, considering spaces with separate, but compatible, measure-theoretic and domain-theoretic
structures. On the domain-theoretic side, we require posets with suprema of countably increasing chains (omega-cpos). On the 4
measure-theoretic side, we require a quasi-Borel space (qbs) structure, a recently introduced algebraic structure suitable for
modelling higher-order probability theory. There are three equivalent characterisations of this category given by: imposing an
order-theoretic separatedness condition on countable-preserving omega-cpo-valued presheaves; internal omega-cpos in the
quasi-topos of quasi-Borel spaces; and an essentially algebraic presentation. The category of these omega-qbses validates Fiore
and Plotkin's axiomatic domain theory, yielding semantics for recursive types. To conclude, I will describe a commutative
powerdomain construction given by factorising the Lebesgue integral from the space of random elements to the space of
sigma-linear integration operators.
I will describe ongoing work investigating a convenient category of pre-domains and a probabilistic powerdomain construction suitable for statistical probabilistic programming semantics, as used in
statistical modelling and machine learning. Specifically, we provide (1) a cartesian closed category; (2) whose objects are (pre)-domains; and (3) a commutative monad for probabilistic choice and
Bayesian conditioning. Jones and Plotkin have shown that conditions (2)--(3) hold when one restricts attention to continuous domains, and Jung and Tix have proposed to search for a suitable category
of continuous domains possessing all three properties (1)--(3), a question that remains open to date.
I propose an alternative direction, considering spaces with separate, but compatible, measure-theoretic and domain-theoretic structures. On the domain-theoretic side, we require posets with suprema
of countably increasing chains (omega-cpos). On the measure-theoretic side, we require a quasi-Borel space (qbs) structure, a recently introduced algebraic structure suitable for modelling
higher-order probability theory. There are three equivalent characterisations of this category given by: imposing an order-theoretic separatedness condition on countable-preserving omega-cpo-valued
presheaves; internal omega-cpos in the quasi-topos of quasi-Borel spaces; and an essentially algebraic presentation. The category of these omega-qbses validates Fiore and Plotkin's axiomatic domain
theory, yielding semantics for recursive types. To conclude, I will describe a commutative powerdomain construction given by factorising the Lebesgue integral from the space of random elements to the
space of sigma-linear integration operators.
Invited Talk: A domain theory for quasi-Borel spaces and statistical probabilistic programming | {"url":"https://easychair.org/smart-slide/slide/nGbK","timestamp":"2024-11-05T21:33:50Z","content_type":"text/html","content_length":"19268","record_id":"<urn:uuid:5a787054-3873-422f-8e6c-840e29881b03>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00020.warc.gz"} |
Excitation problem solution for the dielectric cylinder with thin cover
Last modified: 2014-06-21
At present various electromagnetic radiation sources are widely used (for example, the cellular communication system operating at 450, 900 and 1800 MHz). Accordingly the interest to the problem of
electro-magnetic fields interaction with biological tissues has been increased. The inhomogeneous cylinder may be one of the possible biological structure models [1]. Such model investigation allows
to take advantage of a strict solution of an electromagnetic problem. In resonance range (the specified frequency range) a method of eigenfunctions is usually used. In accordance with it fields are
decomposed into infinite series of eigefunctions. The unknown expansion factors are determined from boundary conditions. However, for biological media (with large values of dielectric constants)
expansion series converge extremely slowly. With a multiple eigenfunctions evaluation using appropriate direct or inverse recurrence formulas a significant error may occur. Diffraction problems in
this case involve special combinations of cylindrical functions [2, 3]. In practice often it is important to know the electromagnetic field distribution inside a dielectric structure. The homogeneous
dielectric cylinder with a thin cover may serve as a model of such structure. In this case it is possible to build equivalent homogeneous model utilizing two-sided second-order boundary conditions
[4, 5].
Conference papers are not currently available. | {"url":"http://icatt.org.ua/1999/paper/view/638/0","timestamp":"2024-11-14T04:31:59Z","content_type":"application/xhtml+xml","content_length":"10348","record_id":"<urn:uuid:92cd6050-e753-4329-aeb5-1c23f7ea2065>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00379.warc.gz"} |
Incorrect answer for the integral of -1/(x-1)?
7255 Views
4 Replies
6 Total Likes
Incorrect answer for the integral of -1/(x-1)?
I was playing around with Wolfram Alpha and I performed the following query:
integral of 1/(x-1) dx
The answer it returned was ln(x - 1) as you would expect.
I then tried to integrate the function multiplied by -1 expecting to get the original answer times -1 (as per integration rules for a function multiplied by a constant):
integral of -1/(x-1) dx
The answer it returned was -ln(1-x) instead of the expected -ln(x-1).
I did a quick, lazy check to see if our answers were the same with the following query:
-ln(1-x) / -ln(x-1)
However, they are not equivalent. They do have the same real value for x >= 1, though.
Am I doing something wrong? Have I simply forgotten some of my integral or log rules? Is this a limitation of the free version of WolframAlpha? Any help would be much appreciated!
4 Replies
tl;dr The results you're seeing are correct. You're seeing a branch cut
The logic you've used here is really common even among very smart engineers and proffesors, but not correct. They don't have to be equal.
Lets assume you have two expressions called "a" and "b". You know "a" and "b" are equal. Their indefinite integrals have to be equal too right? Integral a(x) d(x) has to be equal to Integral b(x) d
(x)? No, they don't have to be equal. It is very common for them to be equal, but there are three major reasons why you will not see them being equal:
First, the values of the indefinite integrals can always differ by a constant. Sometimes you the constant difference won't be obvious in the resulting formula like it would be in most calculus
examples of this.
Second, where you are using inverse functions such as Log, Sqrt, ArcSine, etc, there are branch cuts. These will cause the results to differ by a piecewise constant function. This is what is
happening in your example. If you subtract the two results you get and Simplify under the assumption that their input is a real number, you get a piecewise constant function equal to PiI for x less
than zero and equal to -PiI for x greater than zero
Lastly, integration is often done "generically", meaning we basically ignore the value at some isolated points. A good example of this is that we say the integral of x^n is x^(n+1)/(n+1). This isn't
true for n=-1 in which case the integral is Log[x].
Thank you! My understanding of calculus is only at an undergrad engineering level. And that was 6 years ago. And I didn't do very well at the time! I'm just getting myself back to where I left off
and hopefully farther. Thanks!
In[2]:= Integrate[-1/(x - 1), x]
Out[2]= -Log[1 - x]
In[3]:= Integrate[1/(1 - x), x]
Out[3]= -Log[1 - x]
I really wish I saw this myself. Very simple explanation. Thanks!
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/369466?p_p_auth=Ns4XvFyS","timestamp":"2024-11-13T07:25:47Z","content_type":"text/html","content_length":"110939","record_id":"<urn:uuid:575abdea-ea2a-4c77-9327-07e52df31245>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00450.warc.gz"} |
Capacity of ultra-wideband (UWB) ad-hoc networks
In [1], we argued that the well-known Gupta-Kumar result, which proved that for an n node wireless ad hoc random network, the uniform throughput capacity per node r(n) is Theta(1/\sqrt{n \log n}}) ,
was valid for 'bandwidth-constrained' networks. However, for 'power-constrained' networks, such as ultra-wideband (UWB) networks, where power is at a higher premium than bandwidth, we showed that the
uniform throughput per node r(n) is Theta((n\log n)^{(alpha-1)/2}) . Here, alpha is the distance-loss exponent. These bounds demonstrated that in UWB networks, throughput increases with node density
n , in contrast to previously published results! This is the result of the large bandwidth, and the assumed power and rate adaptation, which alleviate interference. Thus, the significance of physical
layer properties on the capacity of ad-hoc wireless networks was demonstrated. The result also shows that UWB networks are indeed very interesting from the point of view of sharing the wireless
[1] A. Rajeswaran, and R. Negi, "Capacity of power constrained ad hoc networks," Proc. IEEE Infocom, pp. 443-453, Hong Kong, May 2004.
Joint optimization for wireless ad-hoc networks
In wireless ad-hoc networks, there exists strong interdependency between protocol layers, due to the shared wireless medium. Hence we cast the power adaptation (physical layer), scheduling (link
layer) and routing (network layer) problems into a joint optimization framework.
We analyze this hard non-convex optimization problem, and obtain a dual form consisting of a series of sub-problems. The sub-problem demonstrates the functionalities of the protocol layers and their
interaction. We show that the routing problem may be solved by a shortest path algorithm. In the case of Ultra Wide Band (UWB) networks, the power adaptation & scheduling problem is simplified and
may be solved. Thus, an algorithmic solution to the joint problem, in the UWB case, is developed. Comparison of results with the previous information theoretic capacity results on UWB networks [1],
demonstrates the importance of this cross-layer optimization framework. For more information, please see the following papers. In particular, [4] validates the interesting claim made in [1], that the
capacity of UWB networks increases with node density.
[1] A. Rajeswaran, and R. Negi, "Capacity of power constrained ad hoc networks," Proc. IEEE Infocom, pp. 443-453, Hong Kong, May 2004.
[2] R. Negi, and A. Rajeswaran, "Scheduling and power adaptation for networks in the Ultra Wide Band regime," Proc. IEEE Globecom, pp. 139-145, Dallas, USA, Dec. 2004.
[3] A. Rajeswaran, Gyouhwan Kim, and R. Negi, "A scheduling framework for UWB and cellular networks," in Proc. IEEE/ACM Broadband Networks, pp. 386-395, San Jose, Oct. 2004.
[4] Gyouhwan Kim, A. Rajeswaran, and R. Negi, "Joint power adaptation, scheduling and routing framework for wireless ad-hoc networks," IEEE International Workshop on Signal Processing Advances in
Wireless Communications (SPAWC), June 2005.
Detection algorithms for high-density data storage channels
In this project we investigate detection algorithms for high density perpendicular magnetic recording channels. In such channels, transition jitter noise constitutes the dominant noise source.
However, existing mathematic models are not accurate enough to characterize the statistics of this noise, resulting in suboptimal design of detection schemes. In this research, we proposed a jitter
sensitive detection scheme to avoid this problem. The proposed scheme utilizes directly the physical noise model, where the transition jitter is modeled as the deviation of transition center from its
nominal position. A modified Viterbi algorithm is applied to capture the effect of jitters in its trellis construction, and the quantized jitter sequence and recorded bit sequence are estimated
jointly based on Maximum-a-Posterior (MAP) criterion. Simulation results show that the Bit-Error-Rate (BER) performance can be improved by using this scheme as compared to that with the
state-of-the-art detectors, which suffer from model mismatch. We are currently using the same physical noise model to estimate the statistics of transition jitters with spin stand read back waveform.
The result may be used to further improve the detection performance.
Sensor networks and sensing capacity
How many sensors are required to sense an environment to within a desired accuracy? We investigate such limitations on the design of sensor networks for discrete sensor network applications such as
distributed detection and classification. By drawing an analogy between sensor networks and channel encoders, we prove a bound on a Shannon capacity-like quantity called the sensing capacity. The
sensing capacity characterizes the number of sensors required to sense an environment of a given size to within a desired accuracy. We define and bound the sensing capacity for a simple sensor
network model in [1]. We extend this work in [2] to account for sensors with contiguous fields of view and arbitrary sensing functions. In [3] we demonstrate sensing capacity results for a two
dimensional environment distributed as a Markov random field. In the future we intend to further explore the connection between sensing and codes by using insights from our theoretical results to
develop algorithms that efficiently fuse multiple sensor observations.
[1] Y. Rachlin, R. Negi, and P. Khosla, "Sensing capacity for target detection," in Proc. IEEE Inform. Theory Wksp., Oct. 24-29 2004.
[2] Y. Rachlin, R. Negi, and P. Khosla, "Sensing capacity for discrete sensor network applications," in Proc. Int. Conf. on Information Processing in Sensor Networks (IPSN), 2005.
[3] Y. Rachlin, R. Negi, and P. Khosla, "Sensing capacity for Markov random fields," to appear in Proc. Int. Symposium on Information Theory, 2005.
For more information, please visit: Yaron Rachlin's home page
Scheduling over wireless fading channels
Quality of Service (QoS) guarantees can be provided for time-varying channels (like mobile wireless channels), by considering an idealized queuing system which uses an abstract model for the physical
layer. Queuing theory evaluates the performance of the idealized queuing system in terms of queue length or delay experienced by an input unit (bit/packet). Thus, Queuing theory can provision for QoS
on wireless links but it ignores the details of physical layer. This approach works in wired networks because the links in wired (computer) networks are very reliable and have high capacity. On the
other hand, wireless channels have low reliability and have time-varying signal strength. Severe QoS violations may occur if the physical layer details of the wireless channel are ignored while
designing the queuing system.
In [1], we considered a joint queuing/coding system (a queue + server followed by an encoder) that operates on a wireless link. The application of interest was a delay sensitive application which has
a hard constraint on the delay. A bit error occurs if either the bit was decoded incorrectly (error due to channel noise) or the bit experienced excessive delay (error due to delay violation).
Formally, the problem statement was, given the joint queuing/coding system and a certain maximum tolerable delay, design the system such that the probability of error is minimized. For simplicity,
the paper considered a memoryless server model, i.e. the instantaneous server capacity was chosen to be the function of only the current CSI. Thus, the design of the system involved finding the right
server capacity function. It was shown through simulations that the joint queuing/coding system performed better than the pure coding system in a variety of scenarios.
[1] R. Negi, and S. Goel, "An information-theoretic approach to queuing in wireless channels with large delay bounds," Proc. IEEE Globecom, pp. 116-122, Dallas, USA, Dec. 2004.
For more information, please visit: Satashu Goel's home page
Protocol design and analysis in ad hoc wireless networks
See the NSF project web-site .
Impact of broadcast nature of wireless communications on security
Wireless channels differ from their wireline counterparts, in the fact that each wireless transmission is heard by (potentially) several, if not all receivers, legitimate or otherwise. Whereas this
broadcast nature of the wireless medium has been studied from the point of view of channel capacity, when security considerations become paramount, a whole new set of interesting and crucial issues
need to be addressed regarding the broadcast medium. Specifically, the broadcast nature allows jammers to effectively disrupt wireless network communications with clever strategies that use minimal
jammer resources. This denial of service can be made catastrophic by utilizing semantic information in the Medium Access Control layer. The broadcast nature also means that eavesdroppers can hear
transmissions without much effort, raising privacy concerns. However, at the same time, the broadcast medium allows innovative security measures, such as a recently introduced, innovative,
information-theoretically secure, key generation mechanism. This project is investigating denial-of-service at the MAC layer. An intelligent jammer could cleverly utilize the semantics of the data
transmission, by interpreting the packet-on-the-air and deciding its relative importance, and carry out a jamming attack at the MAC-layer. In the context of CSMA/CA, the jammer could detect the
transmission of valuable RTS control packets, and jam such crucial information-bearing packets, to prevent other users from accessing the channel. Due to the random backoff, this creates a cascade
effect, which will waste a large bandwidth. We are investigating intelligent jamming attacks in the link layer, quantifying the loss of throughput caused, and designing protocols which are resistant
to such attacks. The project is also investigating the topic of privacy and information-theoretic security in the presence of eavesdroppers. The approach uses multiple antennas and possibly other
resources to degrade the eavesdropper's channel, while not affecting the channel of the legitimate receiver. This results in secure communication between the transmitter and the legitimate receiver.
[1] A. Rajeswaran and R. Negi, "DoS attacks on a reservation based MAC protocol," in Proc. IEEE Int. Conference on Communications, Seoul, May 2005.
[2] R. Negi and S. Goel, "Secret Communication using Artificial Noise," to appear in Proc. IEEE Vehicular Tech. Conf, Dallas, Fall 2005.
[3] S. Goel and R. Negi, "Secret Communication in Presence of Colluding Eavesdroppers," to appear in Proc. IEEE Military Communication (MILCOM), Atlantic City, Fall 2005. | {"url":"http://users.ece.cmu.edu/~negi/research.html","timestamp":"2024-11-09T13:25:19Z","content_type":"text/html","content_length":"11950","record_id":"<urn:uuid:3a4825c2-1d3e-4040-ba2f-a8f657b82de9>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00676.warc.gz"} |
CPLEX for MPL
CPLEX for MPL gives MPL users access to the world best known linear programming and mixed integer programming solver from within the user-friendly Windows environment of MPL. The CPLEX Callable
Library, including the Mixed Integer Solver, is actually accessed from MPL for Windows as a Dynamic Link Library (DLL). This tight integration allows MPL users to transparently access CPLEX solution
algorithms from their MPL application. Optimizing problems and setting CPLEX options is all done through convenient, intuitive pull-down menus and dialog boxes within MPL for Windows. CPLEX is the
first choice for solving large difficult models in mission- critical applications where robustness and reliability are important. Few solvers can come close to match CPLEX's speed and reliability.
CPLEX is currently used to solve many of the largest problems in the world, with up to millions of variables, constraints, and non-zeros.
Algorithmic Features
CPLEX has a number of sophisticated features that drastically improve solving performance, these include: sophisticated problem preprocessing, efficient restarts form an advanced basis, sensitivity
analysis, infeasibilty finder to mention a few.
Linear Programming
CPLEX has an arsenal of methodologies to solve LP problems, typically the best approach is CPLEX's dual simplex algorithm for the majority of problems. There are certain types of problems that may
benefit from CPLEX's primal simplex algorithm. CPLEX also encompasses an interior point method, its Barrier algorithm which provides an alternative to the simplex method for solving linear problems,
it is based on a primal-dual predictor-corrector method. The barrier algorithm is generally considered when used to solve large problems or if the problems may have numerical instability issues.
There is also an efficient network simplex method that is effective in solving network models.
Quadratic Programming
CPLEX can solve models that have a quadratic objective function and linear constraints. If the objective function is positive semi-definite it can utilize any of the LP methods. To solve QPs in MPL
by CPLEX one has to set in MPL the "ModelType" to Quadratic.
Mixed Integer Programming
CPLEX Mixed Integer Optimizer provides the capability to solve problems with mixed-integer variables (general or binary). It utilizes state-of-the art algorithms and techniques. CPLEX principally
uses a branch and cut algorithm that essentially solves a series of relaxed LP subproblems. Addition of cuts and sophisticated branching strategies can be employed at these subproblems to try to find
the optimal solution more effectively. CPLEX also have heuristics that can aid in finding initial good solutions, it also includes a sophisticated mixed integer preprocessing system. One can even
solve large and difficult integer problems quickly and efficiently.
Performance Tuning
For LPs
CPLEX is tuned to solve the vast majority of LP problems using the default options. There are occasions where one may need to change the option settings, these are usually a result of bad performance
or numerical instability issues. On the whole the dual simplex method is best for most LP problems, there are some instances where the primal simplex may work best. The barrier method typically
should be used for very large sparse models or models that are experiencing numerical difficulties. The sifting algorithm is a simple form of column generation well suited for models where the number
of variables dramatically exceeds the number of constraints. Bad performance on LPs is using a result of degeneracy, which can be identified by examining the iteration log, having long sequences of
iterations where the objective value remains unchanged. Perturbations can help speed up performance on degenerate problems. Degeneracy is not an issue for the barrier method, thus highly degenerate
problems one should use the barrier algorithm to solve the LP.
For MIPs
MIP problems can be extremely difficult to solve, though there are no steadfast rules on enhancing MIP performance, certain things may be advantageous in improving the performance on some models and
be a hinderous on other problems. Below are some of the considerations one should look into when try to solve difficult MIP problems:
• Priority orders:Assign higher priority to the integer variables that should be decided earlier, these tend to represent decisions that are strategic or activate processes. The order, the
variables are defined in MPL indicates the MIP priority, earlier the definition the higher the priority.
• Cutoffs: Setting cutoff values can greatly speedup the process, the cuttoff value (known value that is equal or worse than the true optimal value) maybe attained from some heuristic algorithms to
the same problem or a previous uncompleted run of the MIP model. Use "MipUpperCutoff" to set upper cutoff value for minimization problems, and set "MipLowerCutoff" value for maximization
• Probing: This looks at the logical implications of fixing binary variables, which happens after presolve but before branch and bound. CPLEX has 3 levels of probing, there is a trade off factor
here that though more intensive probing can derive good results it can also take some time to complete the probing. For large difficult problems, we suggest using level 3 or 2 since time overhead
of probing is more likely to be paid back over long running time of branch and bound.
• Variable Selection: Selecting which variable to branch on can have considerable benefits. In difficult models "Strong Branching" or "Pseudo reduced costs" may be helpful. Both methods especially
strong branching invest considerable effort in analyzing potential branches in hope of drastically reducing the number of nodes that will be explored.
• Cuts: Adding cuts are one of the principal reasons of recent dramatic increases in MIP performance. The cuts can dramatically increase the best bound and remove otherwise sub-optimal branches in
the tree. The more cuts added, the larger the inherent matrix becomes which can increase the processing time at the nodes. However usually on difficult models an aggressive cut strategy is the
best mode of practice.
CPLEX Log Output
MPL shows the progress of CPLEX during a solve run in the message window which also can be relayed to a log file. One can set various log file parameters in MPL, allowing one to display log
information for LPs and MIPs. One can set the frequency based on the number of iterations or nodes. The log initially displays information about the model and any subsequent information from the
various preprocessing done, the message log has the following appearance:
CPLEX: Tried aggregator 1 time.
CPLEX: MIP Presolve eliminated 0 rows and 1 columns.
CPLEX: MIP Presolve modified 36 coefficients.
CPLEX: Reduced MIP has 362 rows, 360 columns, and 1638 nonzeros.
CPLEX: Reduced MIP has 342 binaries, 0 generals, 0 SOSs, and 0 indicators.
CPLEX: Presolve time = 0.02 sec.
CPLEX: Clique table members: 191.
CPLEX: MIP emphasis: balance optimality and feasibility.
CPLEX: MIP search method: dynamic search.
CPLEX: Parallel mode: none, using 1 thread.
CPLEX: Root relaxation solution time = -0.00 sec.
For MIPs CPLEX will display cut information and how they affect the best bound at the root node:
CPLEX: Nodes Cuts/
CPLEX: Node Left Objective IInf Best Integer Best Node ItCnt Gap
CPLEX: 0 0 469.1579 41 469.1579 60
CPLEX: 0 0 598.0000 26 Cuts: 81 108
CPLEX: 0 0 598.0000 29 Cuts: 41 129
CPLEX: 0 0 598.3421 39 Cuts: 25 161
CPLEX: 0 0 600.1667 39 Cuts: 28 176
CPLEX: 0 0 607.0000 40 Cuts: 16 199
CPLEX: 0 0 607.0000 40 Cuts: 51 218
CPLEX: * 0+ 0 1650.0000 607.0000 218 63.21%
CPLEX: 0 2 607.0000 40 1650.0000 607.0000 218 63.21%
The subsequent log shows the progress per x nodes, first column stating the nodes examined the second stated how many of those are still left in the tree. It also displays the best objective found so
far, best bound and the relative gap.
CPLEX: * 780+ 390 678.0000 611.8947 14535 9.75%
CPLEX: 800 402 639.2895 13 678.0000 611.9474 14916 9.74%
CPLEX: 900 467 634.1667 26 678.0000 613.3158 17084 9.54%
CPLEX: 1000 393 668.2879 25 678.0000 613.3158 19010 9.54%
CPLEX: Elapsed time = 1.00 sec. (tree size = 0.11 MB, solutions = 15)
CPLEX: * 1061 289 integral 0 674.0000 613.3158 20069 9.00%
CPLEX: 1100 281 614.0000 8 674.0000 613.3158 20795 9.00%
CPLEX: * 1150 235 integral 0 665.0000 613.3158 21656 7.77%
CPLEX: 1200 239 654.5667 25 665.0000 614.0000 22467 7.67%
CPLEX: * 1250+ 193 656.0000 616.0000 23483 6.10%
CPLEX: 1300 203 cutoff 656.0000 617.3056 24461 5.90%
CPLEX: 1400 257 623.8333 7 656.0000 618.9405 26481 5.65%
CPLEX: 1500 297 645.0000 22 656.0000 620.6667 28594 5.39%
CPLEX: 1600 337 624.4167 28 656.0000 621.0000 30524 5.34%
CPLEX: 1700 370 638.0000 26 656.0000 622.1000 32702 5.17%
The asterix shows a MIP feasible solution was found. Once the optimal solution is found for a MIP problem the log will show summary statistics and also the number and of what type of cuts were added
during the solving process:
CPLEX: Elapsed time = 3.14 sec. (tree size = 0.04 MB, solutions = 18)
CPLEX: 5100 40 654.3333 25 656.0000 653.2222 89527 0.42%
CPLEX: Clique cuts applied: 7
CPLEX: Cover cuts applied: 82
CPLEX: Implied bound cuts applied: 8
CPLEX: Flow cuts applied: 19
CPLEX: Mixed integer rounding cuts applied: 21
CPLEX: Zero-half cuts applied: 4
CPLEX: Gomory fractional cuts applied: 8
STATUS: Writing MIP start values file 'tsp3.mst'
CPLEX: Using devex.
Solver Statistics
Solver name: CPLEX (11.2.1)
Objective value: 656.000000000000
Integer Nodes: 5153
Iterations: 89906
Solution time: 3.21 sec
Result code: 101
CPLEX Licensing
CPLEX 11 runs under the control of the ILOG License Manager (ILM). Before you can run ILOG CPLEX, or any application that calls it, you must have established a valid license that ILM can read.
Getting CPLEX License
CPLEX requires the ihostid codes to generate a valid license, thus once the software has been installed, the following steps should be performed to the obtain the ihostid codes: Choose CPLEX IHostID
from the Start menu:
Start | Programs | MPL for Windows 5.0 | CPLEX IHostID
You will need to copy the three codes displayed in the DOS window which you can do by clicking on the MSDOS icon in the upper left corner and choose
Edit | Mark from the menu.
Then select the three lines with the mouse and choose Edit | Copy from the same menu and paste this into the email you are sending us.
CPLEX License Activation
The license for CPLEX will be 2 or 3 lines of text that need to be placed in a file called Access.ilm. An example of a license is shown below:
LICENSE Evaluation maximal software-arlington, va
EVAL CPLEX 10.200 25-Aug-2007 C9520026GDA5 any , options: e m b q MaintenanceEnd=20070825 , license: 1260742
You can use the 'CPLEX 7+' tab in the Maximal License Manager to copy and paste the text between the lines and then activate the license.
CPLEX Parameter Options
For full description of all the CPLEX Parameters that are supported in MPL please go to the CPLEX Option Parameters page.
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Pre-Calculus Exam - Hire Someone To Take My Exam | Pay Me To DO Your Online Examination
The Pre-Calculus exam tests
r knowledge of math concepts and skills needed for success in an introductory first-year calculus class. A large percentage of the exam is dedicated to testing your understanding of mathematical
functions and properties.
You will have to learn the basic shapes and structures of algebraic expressions such as addition, subtraction, multiplication, division, and graphs. You will also need to know how to compute with
these formulas and to solve problems using them. Finally, you will need to be able to solve quadratic equations and other equations that include constant factors.
There are two types of pre-calculus exams to choose from. One type is based on topics that are taught in most colleges and universities. In this type of exam you will have to select an algebra
subject from a pre-calculus class. There are many different subjects that make up a typical pre-calculus class, including linear equations, polynomials, roots of equations, etc. Other topics that may
appear on a standard pre-calculus exam include graphs and graphing.
Algebraic formulas, as well as the properties that they have been tested on this type of exam. A good grade will depend upon your ability to memorize and apply this information. It is important that
students can find answers to their questions quickly, as there are no time limits on the exam. You must answer the question or problem as quickly as possible and you should not take any shortcuts in
answering your questions.
The other type of pre-calculus exam that you can take is the one that will be used in advanced mathematics courses. In this type of exam, you will be given questions that will have to be solved.
These types of questions will not have an exact solution. They will require you to apply some mathematical thinking and problem-solving skills to come up with the right solution. In order to pass
this type of exam you will need to be able to use the concepts learned from the pre-calculus course and to be able to work through the problems.
After you take your pre-calculus exam, you will be allowed to submit it online. You will need to complete a short report that is submitted electronically, and you will be required to provide some
personal information. in order to be able to receive the results.
The final exam involves the application of your knowledge and your understanding of pre-calculus to answer questions about calculus problems and the nature of final exam formulas. This is the type of
exam that many people want to do well on in order to ensure their success with their future courses in calculus.
If you are serious about taking the pre-calculus exam, then you must commit to learning and practicing the concepts and techniques needed to ace the exam. There is no substitute for real-world
experience. However, if you decide to take the standardized exam that is available online, then there are ways to practice for the exam without going to a classroom.
One of the best ways to prepare for the online exam is to use calculators that allow you to simulate the test questions on your computer. This will give you the opportunity to answer multiple-choice
questions and learn what the questions are and how to answer them.
You also need to do some research into the test. Take the time to familiarize yourself with the topics covered in the exam and learn what questions the test will ask of you. Once you are familiar
with the concepts that will be on the exam, you will be able to prepare better for the actual exam by practicing using calculators.
If you are taking a standardized test, then you may be given some practice problems that you will be required to answer. This will give you an idea of how many problems you will have to answer in a
typical test. Take the time to practice answering the questions that are on the test and make sure that you understand what is expected of you. When you know how to answer the questions and practice
what you know, you will be prepared to ace the exam and pass.
While most online exams will ask you to take at least one practice exam before they give you the official results, not all exams will have this requirement. You will need to check the requirements of
the test that you are taking in order to determine whether or not they will give you the practice exam.
Pre-Calculus Exam | {"url":"https://hireforexamination.com/pre-calculus-exam","timestamp":"2024-11-03T09:05:37Z","content_type":"text/html","content_length":"86866","record_id":"<urn:uuid:0b1ce6dd-cdda-4e40-bcf0-449102bdaa6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00136.warc.gz"} |
Banana Function Minimization
This example shows how to minimize Rosenbrock's "banana function":
$f\left(x\right)$ is called the banana function because of its curvature around the origin. It is notorious in optimization examples because of the slow convergence most methods exhibit when trying
to solve this problem.
$f\left(x\right)$ has a unique minimum at the point $x=\left[1,1\right]$ where $f\left(x\right)=0$. This example shows a number of ways to minimize $f\left(x\right)$ starting at the point $x0=\left
Optimization Without Derivatives
The fminsearch function finds a minimum for a problem without constraints. It uses an algorithm that does not estimate any derivatives of the objective function. Rather, it uses a geometric search
method described in fminsearch Algorithm.
Minimize the banana function using fminsearch. Include an output function to report the sequence of iterations.
fun = @(x)(100*(x(2) - x(1)^2)^2 + (1 - x(1))^2);
options = optimset('OutputFcn',@bananaout,'Display','off');
x0 = [-1.9,2];
[x,fval,eflag,output] = fminsearch(fun,x0,options);
title 'Rosenbrock solution via fminsearch'
Fcount = output.funcCount;
disp(['Number of function evaluations for fminsearch was ',num2str(Fcount)])
Number of function evaluations for fminsearch was 210
disp(['Number of solver iterations for fminsearch was ',num2str(output.iterations)])
Number of solver iterations for fminsearch was 114
Optimization with Estimated Derivatives
The fminunc function finds a minimum for a problem without constraints. It uses a derivative-based algorithm. The algorithm attempts to estimate not only the first derivative of the objective
function, but also the matrix of second derivatives. fminunc is usually more efficient than fminsearch.
Minimize the banana function using fminunc.
options = optimoptions('fminunc','Display','off',...
[x,fval,eflag,output] = fminunc(fun,x0,options);
title 'Rosenbrock solution via fminunc'
Fcount = output.funcCount;
disp(['Number of function evaluations for fminunc was ',num2str(Fcount)])
Number of function evaluations for fminunc was 150
disp(['Number of solver iterations for fminunc was ',num2str(output.iterations)])
Number of solver iterations for fminunc was 34
Optimization with Steepest Descent
If you attempt to minimize the banana function using a steepest descent algorithm, the high curvature of the problem makes the solution process very slow.
You can run fminunc with the steepest descent algorithm by setting the hidden HessUpdate option to the value 'steepdesc' for the 'quasi-newton' algorithm. Set a larger-than-default maximum number of
function evaluations, because the solver does not find the solution quickly. In this case, the solver does not find the solution even after 600 function evaluations.
options = optimoptions(options,'HessUpdate','steepdesc',...
[x,fval,eflag,output] = fminunc(fun,x0,options);
title 'Rosenbrock solution via steepest descent'
Fcount = output.funcCount;
disp(['Number of function evaluations for steepest descent was ',...
Number of function evaluations for steepest descent was 600
disp(['Number of solver iterations for steepest descent was ',...
Number of solver iterations for steepest descent was 45
Optimization with Analytic Gradient
If you provide a gradient, fminunc solves the optimization using fewer function evaluations. When you provide a gradient, you can use the 'trust-region' algorithm, which is often faster and uses less
memory than the 'quasi-newton' algorithm. Reset the HessUpdate and MaxFunctionEvaluations options to their default values.
grad = @(x)[-400*(x(2) - x(1)^2)*x(1) - 2*(1 - x(1));
200*(x(2) - x(1)^2)];
fungrad = @(x)deal(fun(x),grad(x));
options = resetoptions(options,{'HessUpdate','MaxFunctionEvaluations'});
options = optimoptions(options,'SpecifyObjectiveGradient',true,...
[x,fval,eflag,output] = fminunc(fungrad,x0,options);
title 'Rosenbrock solution via fminunc with gradient'
Fcount = output.funcCount;
disp(['Number of function evaluations for fminunc with gradient was ',...
Number of function evaluations for fminunc with gradient was 32
disp(['Number of solver iterations for fminunc with gradient was ',...
Number of solver iterations for fminunc with gradient was 31
Optimization with Analytic Hessian
If you provide a Hessian (matrix of second derivatives), fminunc can solve the optimization using even fewer function evaluations. For this problem the results are the same with or without the
hess = @(x)[1200*x(1)^2 - 400*x(2) + 2, -400*x(1);
-400*x(1), 200];
fungradhess = @(x)deal(fun(x),grad(x),hess(x));
options.HessianFcn = 'objective';
[x,fval,eflag,output] = fminunc(fungradhess,x0,options);
title 'Rosenbrock solution via fminunc with Hessian'
Fcount = output.funcCount;
disp(['Number of function evaluations for fminunc with gradient and Hessian was ',...
Number of function evaluations for fminunc with gradient and Hessian was 32
disp(['Number of solver iterations for fminunc with gradient and Hessian was ',num2str(output.iterations)])
Number of solver iterations for fminunc with gradient and Hessian was 31
Optimization with a Least Squares Solver
The recommended solver for a nonlinear sum of squares is lsqnonlin. This solver is even more efficient than fminunc without a gradient for this special class of problems. To use lsqnonlin, do not
write your objective as a sum of squares. Instead, write the underlying vector that lsqnonlin internally squares and sums.
options = optimoptions('lsqnonlin','Display','off','OutputFcn',@bananaout);
vfun = @(x)[10*(x(2) - x(1)^2),1 - x(1)];
[x,resnorm,residual,eflag,output] = lsqnonlin(vfun,x0,[],[],options);
title 'Rosenbrock solution via lsqnonlin'
Fcount = output.funcCount;
disp(['Number of function evaluations for lsqnonlin was ',...
Number of function evaluations for lsqnonlin was 87
disp(['Number of solver iterations for lsqnonlin was ',num2str(output.iterations)])
Number of solver iterations for lsqnonlin was 28
Optimization with a Least Squares Solver and Jacobian
As in the minimization using a gradient for fminunc, lsqnonlin can use derivative information to lower the number of function evaluations. Provide the Jacobian of the nonlinear objective function
vector and run the optimization again.
jac = @(x)[-20*x(1),10;
vfunjac = @(x)deal(vfun(x),jac(x));
options.SpecifyObjectiveGradient = true;
[x,resnorm,residual,eflag,output] = lsqnonlin(vfunjac,x0,[],[],options);
title 'Rosenbrock solution via lsqnonlin with Jacobian'
Fcount = output.funcCount;
disp(['Number of function evaluations for lsqnonlin with Jacobian was ',...
Number of function evaluations for lsqnonlin with Jacobian was 29
disp(['Number of solver iterations for lsqnonlin with Jacobian was ',...
Number of solver iterations for lsqnonlin with Jacobian was 28
Copyright 2006–2020 The MathWorks, Inc.
Related Topics | {"url":"https://fr.mathworks.com/help/optim/ug/banana-function-minimization.html","timestamp":"2024-11-04T08:58:06Z","content_type":"text/html","content_length":"89540","record_id":"<urn:uuid:bb1adf66-1502-4f88-95a1-9b8d8f8febb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00011.warc.gz"} |
Build Your Own Worlds | Highrise Create
The AnimationCurve class represents a curve that defines the interpolation between keyframes over time. It's commonly used in animation systems to define smooth transitions between different values
over time.
The number of keys in the curve. (Read Only)
Interpolates the value from the curve at a given time point. The curve is evaluated, giving the value of the curve at that exact time. This becomes very handy when lerping between two values in a
non-linear way for smoother transitions.
The time at which to evaluate the curve.
Returns the value of the curve at the specified time.
Tool designed to insert a new key into your AnimationCurve. A way to dynamically shape your animation by adding keyframes at certain times with specific values.
The time at which the new key should be inserted.
The value of the new key at the specified time.
Returns the index of the added key, or of the existing key if one already exists at the given time.
Erase all keys from the AnimationCurve. It can be beneficial when you need to reset the curve or want to dynamically generate a new one.
This method does not return a value.
Removes the key at a specified index from your AnimationCurve. Varying curves dynamically can lead to interesting gameplay possibilities and variation in animations.
The index of the key you want to remove.
This method does not return a value.
Updates the tangent of a key in the AnimationCurve to create a smooth transition in the curve. This becomes essential while crafting natural and organic animations.
The index of the key to update.
How much influence the tangent has. A weight of 0 will make the tangent flat.
This method does not return a value.
Copies all settings, including all keys from the given curve. This can prove useful when you want to duplicate the behavior of an existing curve.
The source AnimationCurve to copy from.
This method does not return a value.
Creates a new AnimationCurve where all values are the same (constant). This can come in handy when creating a stable, unchanging behavior.
The starting time of the curve.
The ending time of the curve.
The constant value for the curve.
Creates a new AnimationCurve representing a linear interpolation between a start and end value (valueStart and valueEnd) over a specified time period (timeStart to timeEnd). This is particularly
useful when a linear transition between two states is required over a given duration.
The starting time of the curve.
The value of the curve at the start time.
The ending time of the curve.
The value of the curve at the end time.
Returns a new AnimationCurve representing a linear interpolation.
Creates a new AnimationCurve that smoothly transitions between the start and end values over a specified time period. This method is especially useful to create animations with smooth starts and
ends, providing a more natural look.
The starting time of the curve.
The value of the curve at the start time.
The ending time of the curve.
The value of the curve at the end time.
Returns a new AnimationCurve with smooth transitions at start and end. | {"url":"https://create.highrise.game/learn/studio-api/classes/AnimationCurve","timestamp":"2024-11-10T14:09:43Z","content_type":"text/html","content_length":"216423","record_id":"<urn:uuid:2b4d4205-acae-461f-a086-00ee3944860a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00745.warc.gz"} |
Lesson 09 - Inductors - Electronics For Fun
Welcome to Electronics For Fun
Lesson 9 – Inductors
As I introduce new terms, I have included a link to Wikipedia. Read ahead a little, and if you still need help you can click on any orange word below for more information than you probably want. 🙂
An Inductor consists of a coil of wire wound around a core. The wire would have to be insulated so the windings would’t short out as they touched each other. Usually they use enamel coated wire,
sometimes called magnet wire. The core may or may not have iron in it.
Inductors, also called chokes, temporarily store energy when a current flows thru it. Then when the current stops, they give that energy back to the circuit. If that sounds like a capacitor, it sort
of is. But a capacitor does this by storing the energy electrostatically, whereas an inductor stores the energy in a magnetic field, or electromagnetically.
The photo above shows some different types of inductors. The symbols above show some different ways you might see an inductor in a schematic diagram. L1 and L2 are the same inductor, just with the
leads drawn a different way. They are “air core” inductors (no iron). L3 and L4 are “iron core” as shown by the two additional lines. L5 and L6 are “center tapped” air core inductors. L7 and L8 are
“variable” air core inductors, and finally, L9 is a center tapped iron core inductor.
Remember from our last lesson that a capacitor resists the flow of DC but freely allows AC to pass thru it. An inductor, on the other hand, resists changing current like AC, but allows DC to pass
The amount of inductance in an inductor is measured in units called henrys. One henry is fairly large so usually, in electronics, we are dealing with microhenrys. A microhenry is one millionth of a
henry, written as, for example 5µH. Sometimes we see a larger inductor specified as so many millihenrys. A millihenry is 1 thousanth of a henry, expressed as mH. So a 4 millihenry inductor would be
written as 4mH.
Total Equivalent Inductance
The method of calculating the total equivalent inductance for inductors, is mostly like that of resistors. By that statement, I mean that to calculate series inductance, you just add them together
like series resistors. Looking at the image below, assume we had just two inductors, L1 and L2. Assuming that L1 is 10µH and L2 is 5µH, the total equivalent inductance would be simply 10 + 5 or
In the diagram below, once again let’s assume we have only two inductors, L1 and L2. Assuming that L1 is 10µH and L2 is 5µH, L[eq], the total equivalent inductance would be calculated similar to
the equation that we used for resistors in parallel and capacitors in series.
This time we will skip the fact that the equation was originally written for henrys. In the capacitor lesson, we proved we could just use microfarads instead of farads for the calculation as long as
we realized that the answer would be in microfarads as well. The same holds true here so instead of converting the inductors values to henrys and then back again, we will just use microhenrys from
start to finish.
So we would have 1/L[eq]=1/L[1] + 1/L[2] which would be 1/L[eq]=1/10 + 1/5 which would be 1/L[eq]=.1 + .2. Next would be 1/L[eq]=.3 and finally dividing 1 by .3 gives us 3.334 microhenrys or
3.334µH. Once again, if you don’t want to do the math, you can visit keisan.casio.com for an online equivalent inductance calculator .
Inductive Reactance
Inductive Reactance is like resistance, but in an inductor, it changes as the frequency changes. That’s because an inductor resists changing current like AC, but allows DC to pass easily. So, to
calculate inductive reactance, also called impedance, we use the equation X[L]=2πfL where X[L] is the Inductive Reactance in ohms, π is 3.14, f is the frequency, and L is the inductance in henrys.
So let’s say we have a 10mH inductor and a frequency of 1000 hertz. We would have X[L]=2Ï€fL or X[L]=2 times 3.14 times 1000 times .010, which would be 62.8 ohms. You can visit 66pacific.com for an
online inductive reactance calculator .
Test your knowledge if you feel like it with a little test. No cheating now!
Q1: An inductor stores the energy _____________.
Q2: A 3mH inductor and a 2mH inductor in series would be the equivalent of a __________ inductor.
Q3: Inductive Reactance is specified in ____________.
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Aleksandar Miković
Associate Professor at the Department of Mathematics, Lusófona University, Lisbon.
Research Interests
• Quantum Gravity
• Quantum Field Theory
Recent Publications
• "Spin-cube models of quantum gravity"
Rev. Math. Phys. 25, 10 (2013).
URL: http://gfm.cii.fc.ul.pt/people/amikovic/AMikovic_RMP.pdf
• A. Mikovic, "Spin network wavefunction and the graviton propagator".
To appear in Fortschr. Phys. (2008).
arXiv:0706.0466 [ps, pdf, other]
• João Faria Martins, Aleksandar Mikovic, "Invariants of spin networks embedded in three-manifolds".
To appear in Comm. Math. Phys. (2008).
arXiv:gr-qc/0612137 [ps, pdf, other]
• Aleksandar Mikovic, "Quantum gravity as a broken symmetry phase of a BF theory".
SIGMA 2 (2006), 086 (5 pages).
arXiv:hep-th/0610194 [ps, pdf, other]
• A. Mikovic, "Spin foam models from the tetrad integration"
(6 pages, based on the talk presented at the ERE05 meeting, September 6-10, 2005, Oviedo).
arXiv:gr-qc/0511080 [ps, pdf, other]
• A. Mikovic, "Quantum gravity as a deformed topological quantum field theory"
(7 pages, talk presented at the QG05 conference, 12-16 September 2005, Cala Gonone, Italy).
J. Phys. Conf. Ser. 33 (2006), 266-270.
arXiv:gr-qc/0511077 [ps, pdf, other]
• N. C. Dias, A. Mikovic, J. N. Prata, "Coherent states expectation values as semiclassical trajectories".
J. Math. Phys. 47 (2006), 082101.
arXiv:hep-th/0507255 [ps, pdf, other] | {"url":"http://gfm.cii.fc.ul.pt/people/amikovic/","timestamp":"2024-11-12T15:22:39Z","content_type":"application/xhtml+xml","content_length":"15981","record_id":"<urn:uuid:d37e5939-7049-4918-8ad9-1569eebdea86>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00165.warc.gz"} |
Continuous and categorical variables in same model?
Replied on Wed, 07/20/2011 - 13:35
Yes! Please test joint continuous and ordinal integration, as this is one of the features for the OpenMx 1.2 release. There is more information regarding the beta release at this
Replied on Wed, 07/20/2011 - 14:26
tbates Joined: 07/31/2009
In reply to Yes! Please test joint by mspiegel
comments and instructions
I think this example file needs comments, and a section in the .Rd files about how to set up thresholds for some (but not all) variables...
And also how to go from the example (roughly:
lm(cont ~BIN)
to a mixed type structural model (roughly (and not real code):
sem( ~ acont+ bord + ccont + dord)
Replied on Wed, 07/20/2011 - 14:53
In reply to comments and instructions by tbates
Great! I'm glad tos ee this
Great! I'm glad tos ee this is supported. Especially because I'm working on a project that requires it, with a deadline in late August!
I'm afraid Tim's right. I wouldn't know how to use that script to inform my model. Are there any examples of a saturated means and var-covar model with some manifest variables dichotomous and some
The project (I'm still formulating) will be a longitudinal two-step ACE model of cigarette smoking. Andy Heath has done this sort of thing, at least cross-sectionally. At each of 4 time points two
variables are measured: (1) a dichotomous variable indexing whether the subject has ever smoked, and (2) a quasi-continuous variable representing the number of cigarettes that subject has ever
smoked. If the subject has never smoked, then (2) is set to missing.
I'm not sure how this plays out in a longitudinal model (i.e., if and how cross-time correlations are freed or fixed to zero in fitting), but it should be an interesting project.
Replied on Mon, 07/25/2011 - 00:16
In reply to Great! I'm glad tos ee this by svrieze
Longitudinal CCC model
Hi Scott
We usually call the model you are speaking of a ccc model (conditional common pathway causal) because it involves one variable which is considered to be causally downstream and only measured when the
upstream variable, initiation, is observed. Although there may be some continuous variables that are only really measurable in those who have initiated, I would not consider quantity smoked one. This
measure is usually measured on an ordinal scale (or at least subjects have trouble reporting more precisely than a few categories of number of cigarettes smoked). Although such a variable might be
better thought of as Poisson distributed, I think it is reasonable to consider it ordinal with an underlying normal distribution. Indeed, I think this is likely much better than considering it as
continuous. So, I would favor using ordinal FIML analysis, and declare both initiation and quantity as ordinal. While there may exist a measure of lifetime quantity, which might be closer to
continuous, I would not want to use this without taking care of censoring - older participants or those who started earlier would have more chance to increase their lifetime quantity smoked than
would younger ones.
Replied on Mon, 07/25/2011 - 22:15
In reply to Longitudinal CCC model by neale
fixed effects in threshold model
Thanks Mike, that's very helpful. I'm using incident smoking during the 12 months prior to assessment. I was using a continuous model because I'm interested in a fixed effect (a SNP score) on the
mean of cigarettes per day, as well as how the heritable variance changes after including the SNP score in the model. With continuous variables this is a very straightforward ACE model with some
definition variables. How would the fixed effect work in a model with ordinal measures? Would the covariate effect be on the thresholds?
Do any scripts exist that have definition variables in ordinal models? None seem to on the OpenMx website.
Replied on Fri, 07/29/2011 - 23:09
In reply to fixed effects in threshold model by svrieze
Moderator with thresholds
Sorry for the delay. We think there is one around, but I am having trouble locating it. In principle it's pretty simple: the ordinal variables have to be mxFactor()'d so openmx knows what is the
highest and lowest theoretical categories (since in a dataset the extremes may not be observed yet we'd not want to integrate from threshold to infinity by mistake). Second, with 3 or more categories
you can rescale the model by estimating both means and variances but fixing the first two thresholds to zero and one. In addition you could make the predicted mean a function of the definition
variables, simply by labeling the relevant matrix element as a definition variable. I'll keep trying to dig up an example.
Replied on Sat, 07/30/2011 - 15:00
In reply to Moderator with thresholds by neale
I may be on right track, but...
thanks again for the help. I think I'm following this. I can get a threshold model to fit when I have a simple means model (no definition variables).
When I add in the definition variable effect on the means I get an error:
"Error: The job for model 'mm' exited abnormally with the error message: Objective function returned an infinite value."
The definition variable is not missing for any subject.
A short script with the two models and their output is attached, in case anyone would like to see what I've tried. Maybe (hopefully) it's a simple syntax error.
Replied on Thu, 08/04/2011 - 11:41
mspiegel Joined: 07/31/2009
In reply to I may be on right track, but... by svrieze
Sorry for the delay. I
Sorry for the delay. I believe this error is usually seen when one of your data rows yields a likelihood value of 0. One way to debug this problem would be to build OpenMx from the source code
repository. That version has a new feature where the likelihood vector is returned in the objective function by inspecting mmdef$MZ.objective@likelihoods after the model has finished running. Another
way to debug the problem is to use the current binary version of OpenMx and add the argument "vector=TRUE" when you call the mxFIMLObjective() functions. But then you would need to change your
mxAlgebraObjective() to: -2 * (sum(log(MZ.objective)) + sum(log(DZ.objective))). This way you can inspect the likelihood vector at mxEval(MZ.objective, mmdef) or mxEval(DZ.objective, mmdef) after the
model has executed.
In either case, you'll need to undo the safety net that prevents the user from publishing results when the model throws an error. You will call mxRun() with the argument "unsafe=TRUE" and then the
model results will be available to you when the error is thrown.
Replied on Tue, 10/01/2013 - 11:06
In reply to Sorry for the delay. I by mspiegel
Meaning of estimates in joint model
I have a question about the interpretation of the coefficient in ordinal outcome models. When I was doing the simulation of joint model based on a binary variable and continuous variable. I don't
quite understand the meaning of the coefficient in the ordinal outcome output. They seem totally different from the continuous outcome. I can understand for ordinal variable or binary variable, the
FIML is based on cumulative normal distribution. But what's the interpretation of the coefficients is not clear. Is there any documents covering this information in OpenMx, because I didn't find
anything in the guidance.
I appreciate your kindly help!
Replied on Tue, 10/01/2013 - 12:39
Ryne Joined: 07/31/2009
In reply to Meaning of estimates in joint model by feihe
There are two things to note
There are two things to note about interpreting parameters involving ordinal variables under SEM/FIML.
First, all relationships between an ordinal/binary variable and any other variable (continuous or ordinal) are actually modeled as the relationship between the assumed latent continuous variable
"underneath" your ordinal variable and the second variable. As an example, a covariance between a binary variable and a continuous variable in this approach will be modeled as the covariance between
two continuous variables, one of which is measured and one of which is dichotomized to create your binary variable. In this example, the modeled correlation between these two variables should be
stronger than the observed pearson correlation between the observed variables.
Second, the scale of this latent variable is set by you. You may say that the total variance is a particular value, or say that is residual variance is a particular value, or say that the distance
between two of the thresholds is a particular value, or something else entirely. The raw value of whatever regressions and covariances involving that variable will depend on how you identify or scale
this variable.
To sum up, all parameters involving ordinal variables come from the assumption that underneath your ordinal variable is a continuous normal distribution. All relationships involving this variable
reflect associations involving this continuous normal variable. As we're assuming that the continuous variable exists, its mean and variance (i.e., its scale) depends on how you define or identify
the ordinal variable.
Hope that helps,
Replied on Tue, 10/01/2013 - 13:16
In reply to There are two things to note by Ryne
Hi Ryne, Thanks a lot for the
Hi Ryne,
Thanks a lot for the detailed explanation. This really helps me understanding the logic behind. But I faced some problem when I was doing the simulation. Here is my code for a probit model. I found
that if I used "ystar" (the latent variable for the binary variable) instead of "y" as the response variable, I can perfectly estimate the beta1=3, beta0=-6 (since "ystar" is a continuous variable).
But once I try to estimate the coefficients between "y" (the binary variable I defined based on "ystart") and "x", I got lost by the estimates I had ( I had beta1=0.23, beta0=1.19). It's hard for me
to find any connection between these estimates based on "y" and previous estimates based on "ystar". Do you have any idea? Thank you very much.
simulate <- function(N) {
x <- 2*runif(N,1,2)
ystar <- -6 + 3*x + rnorm(N)
y <- as.numeric(ystar>0)
data.frame(y, x)}
data.simu <- simulate(100)
#colnames(data.simu)<- c(paste("y",seq(1,10),sep=""),paste("x",seq(1,10),sep=""))
name<- colnames(data.simu)
data.simu$y <- mxFactor(data.simu$y, levels=c(0, 1))
CrossLaggedModel1<-mxModel("probit model",
arrows=1, free=TRUE, value=1, labels="beta1"),
values = c(1,1),
) ,
mxData(observed=data.simu, type="raw")
fit1 <- mxRun(CrossLaggedModel1)
Replied on Mon, 10/28/2013 - 14:57
mhunter Joined: 07/31/2009
In reply to Hi Ryne, Thanks a lot for the by feihe
Setting the scale
I think the issue was still just setting the scale underlying the categorical variables. The attached script produces pretty good estimates of your generating parameters.
free parameters:
name matrix row col Estimate Std.Error Std.Estimate Std.SE
1 beta1 A y x 2.8128912 0.109627030 0.8519141 0.03320171
2 varx S x x 0.3344653 0.004729648 1.0000000 0.01414092
3 beta0 M 1 y -5.5888361 0.254768400 NA NA
4 meanx M 1 x 3.0056161 0.005783298 NA NA | {"url":"https://openmx.ssri.psu.edu/comment/3224","timestamp":"2024-11-08T01:56:09Z","content_type":"text/html","content_length":"59331","record_id":"<urn:uuid:d2aa751c-fa30-4df5-b192-e53823782e10>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00620.warc.gz"} |
This page computes a finite or infinite sum over the index n. This sum can be a numerical or symbolic (for example a power series).
The menu of this page is in simple mode. Go to expert mode for more choices.
To a sum, enter first the expression of its general term: ( Examples )
f (n) =
Then choose the type of the sum to compute.
• Infinite series, for n starting with .
• Finite sum, for n running through integers from to ,
Numerical precision: digits.
The menu of this page is in simple mode. Go to expert mode for more choices.
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: computes sums of series or finite sums of various kinds. interactive exercises, online calculators and plotters, mathematical recreation and games
• Keywords: interactive mathematics, interactive math, server side interactivity, analysis, algebra, arithmetic, number, series, sum, polynomial, integer, root | {"url":"https://sercalwims.ig-edu.univ-paris13.fr/wims/en_tool~analysis~sigma.en.html","timestamp":"2024-11-04T07:25:29Z","content_type":"text/html","content_length":"5547","record_id":"<urn:uuid:8b2baf96-a138-4501-aa78-fe1dfa9946e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00686.warc.gz"} |
18MAT11 MODULE 1 CALCULUS AND LINEAR ALGEBRA
Dr Ksc Maths Notes 1st Sem Enginnering Mathematics PDF notes 1st Sem Maths Notes Module 1 CALCULUS AND LINEAR ALGEBRA Textbook Notes
Differential Calculus-1: Review of elementary differential calculus, Polar curves - angle between the radius vector and tangent, angle between two curves, pedal equation. Curvature and radius of
curvature Cartesian and polar forms; Centre and circle of curvature (All without proof-formulae only) —applications to evolutes and involutes.
If you have any doubts.Please let me Know | {"url":"https://www.vtuguide.in/2021/09/18mat11-module-1-calculus-and-linear.html","timestamp":"2024-11-09T21:57:01Z","content_type":"text/html","content_length":"172305","record_id":"<urn:uuid:4b46f24c-00e1-416e-9c33-34870ff4759b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00524.warc.gz"} |
The Stacks project
Lemma 101.27.10. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and fppf local on the target. Let $f : \mathcal{X} \to \mathcal{Y}$ be
a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times _\mathcal {Y}
\mathcal{X} \to \mathcal{Z}$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$.
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World of science
Incorrect information block
Popular science
Diamond graphite nanocomposite will increase the durability of terahertz emitters
Scientists of the Saratov branch of the V.A. Kotelnikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Saratov National Research State University named after N.G.
Chernyshevsky and NPP Almaz for the first time discovered the effect of restoring a diamond graphite cathode when the voltage polarity changes in the interelectrode gap, which will create more
durable terahertz emitters.
Mastering the terahertz (THz) range of electromagnetic waves between the microwave and infrared ranges is one of the key problems of electronics development. Coherent THz radiation sources have broad
prospects of application in such areas as security (remote detection of explosives), wireless information and communication systems for high-speed data transmission, radio astronomy, spectroscopy,
medicine, etc. There are various approaches to mastering the THz range, for example, using solid-state or quantum electronics devices. However, to achieve power levels of the order of tens of watts
and above, electric vacuum devices (EVP) are optimal.
The reliability and durability of the EVP are largely determined by the characteristics of the cathodes, that is, the sources of electron emission. Currently, most EVPs produced in the world use
thermionic metal-porous cathodes (MPCs). However, the durability of the MPC is not high enough. Other disadvantages of thermal cathodes include a long readiness time and a low maximum current
collection density.
A promising direction for improving THz band devices is the replacement of thermionic cathodes with field cathodes. Field emission is the emission of electrons by conducting bodies under the
influence of an external electric field of sufficiently high intensity. In field cathodes, electrons overcome the potential barrier at the emitter boundary not due to the kinetic energy of thermal
motion, that is, not as a result of cathode heating, as in thermionic emission, but by quantum tunneling through a barrier reduced by an external electric field. Of course, a cathode that does not
need to be heated can serve longer and more reliably.
One of the most promising materials for creating this type of cathodes are nanocarbon film structures, in particular, diamond graphite nanocomposites, which are graphite matrices with diamond
nanocrystallites embedded in them. In this study, diamondographic film structures with a thickness of about 100 nm deposited on polycore (corundum-based ceramics) plates were used as field cathodes.
The investigation of the operation of the diamond graphite cathode was carried out during 8 test cycles with a total duration of over 13.5 hours. During the tests, conditions with an emergency
shutdown of the supply voltage and vacuum pumping facilities were simulated.
The graph above shows the current-voltage characteristics (VAC) of the cathode measured before (curve 1) and after (curve 2) 8 test cycles. Despite the unfavorable factors associated with periodic
disconnections of the supply voltage and deterioration of the vacuum, the field emissivity of the cathode did not deteriorate during the tests.
However, the study showed that part of the carbon phase of the cathode degrades during operation and is deposited on the anode. During the experiment, a voltage negative relative to the cathode was
applied to the anode, that is, the cathode, in fact, turned into an anode, and the anode became a cathode, that is, a source of electrons. After that, the voltage was applied to this pair again
according to the usual operating scheme and the cathode VAC was measured.
The graph above shows the change in the cathode VAC before (1) and after (2) the deposition of the emission material from the anode to the cathode. An improvement in the emission capacity of the
cathode after recovery was found. The positive effect was manifested in a decrease in the threshold for the onset of field emission and an increase in the steepness of the VAC curve, which allows
obtaining similar currents at lower electric field strengths. This effect was discovered for the first time.
This effect can be used both to create a field cathode with improved emission characteristics, and to restore its emission ability during long-term operation as part of an EVP. The results of the
study can be used to predict the service life of field diamond-graphite electron sources.
For more information, see the article "Durability of high-current field electron sources based on nanocomposite diamond-graphite film structures", R. K. Yafarov, A.V. Storublev, "Microelectronics",
2022, T. 51, No. 2, pp. 95-100.
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Video lecture Yulia Gorbunova: Periodic Table of Mendeleev – the universal language of science
11:35 All over the world, the Periodic Table is associated with the name of D.I. Mendeleev, which is recognition of a phenomenal discovery that has become the common language of all natural sciences,
and that is why scientists of different specialties - chemists, physicists, astronomers, geologists, physicians, biologists and geographers - consider this table their own. The periodic table today
stands at the center of the economy, without it material production is unthinkable. Electronic gadgets, solar panels, smart clothes, environmentally friendly fuel for cars, medicines and medical
diagnostics are created by chemists using knowledge about the elements.
Lecture corresponding member of RAS, Professor of the RAS, chief scientific officer of the Institute of General and inorganic chemistry. N.With. Kurnakov, Institute of physical chemistry and
electrochemistry named after A. N. Frumkin wounds Yulia Gorbunova was read the physics teachers in basic schools of the RAS within the framework of IV Trinity school of professional development of
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Geometry Hacks
Are you struggling to find the perimeter of a hexagon? In this article, we’ve got you covered with some geometry hacks!
We’ll walk you through the basics and show you how to identify the sides of a hexagon.
Plus, we’ll teach you how to calculate the perimeter using the side lengths and even show you how to use the apothem and apply the Pythagorean theorem.
Get ready to ace those hexagon perimeter problems!
Understanding the Basics
To understand the basics of finding the perimeter of a hexagon, you need to know the lengths of its sides and how to add them together.
The perimeter of any polygon is the sum of the lengths of all its sides. In the case of a hexagon, which has six sides, you simply add the lengths of all six sides to find the perimeter.
Each side of a regular hexagon is equal in length, so if you know the length of just one side, you can easily calculate the perimeter by multiplying that length by six.
However, if the hexagon is irregular and the side lengths vary, you’ll need to measure or calculate each side length individually and then add them together to find the perimeter.
Identifying the Sides of a Hexagon
To identify the sides of a hexagon, you can start by examining the lengths of each side, which will be crucial in finding the perimeter. A hexagon is a polygon with six sides, and each side is equal
in length. By measuring one side, you can determine the length of all the other sides. To simplify the process, consider using a ruler or a measuring tape.
Start at one vertex of the hexagon and measure to the next vertex, making sure to keep the ruler or tape straight along the side. Repeat this process for each side, noting down the length as you go.
Once you have measured all six sides, you’ll have successfully identified the sides of the hexagon.
Finding the Length of Each Side
Measure each side of the hexagon to find the length of each side. Using a ruler or a measuring tape, place it along one side of the hexagon and read the measurement. Repeat this for each side of the
hexagon. Make sure to measure from one vertex to the next, following the shape of the hexagon.
Take note of the measurements for each side. Once you have measured all six sides, you can find the length of each side by adding up the measurements.
Add the lengths of all the sides together to find the perimeter of the hexagon. This will give you the total distance around the hexagon, which is useful in various geometric calculations and
Calculating the Perimeter Using the Side Lengths
Now that you know the length of each side of the hexagon, it’s time to calculate the perimeter.
Understanding the importance of side lengths is crucial in finding the total distance around the shape.
Side Lengths Importance
You can calculate the perimeter of a hexagon by adding up the lengths of all six sides. The side lengths of a hexagon are crucial in determining its perimeter. Each side contributes to the total
distance around the shape. By knowing the lengths of the sides, you can accurately calculate the perimeter without any guesswork.
Whether the hexagon has equal sides or varying lengths, each side plays a significant role in determining the total distance. To find the perimeter, simply add up all the side lengths, and you’ll
have the total distance around the hexagon.
Formula for Perimeter
To calculate the perimeter of a hexagon using the side lengths, simply add up all six sides. It’s a straightforward process that requires you to know the length of each side.
Let’s say you have a hexagon with sides measuring 4 cm each. Start by adding the lengths of all six sides together: 4 cm + 4 cm + 4 cm + 4 cm + 4 cm + 4 cm. Simplifying this equation, you get 24 cm.
Therefore, the perimeter of this hexagon is 24 cm.
Real-Life Applications
Calculating the perimeter of a hexagon using the side lengths can be done by simply adding up all six sides. This method is widely used in real-life applications where the perimeter of a hexagon
needs to be determined.
For example, in construction, knowing the perimeter of a hexagonal-shaped plot of land is essential for accurate measurements and determining the amount of fencing required.
Similarly, in manufacturing, calculating the perimeter of a hexagonal-shaped object helps in determining the amount of material needed for its production.
Additionally, in architecture and design, understanding the perimeter of a hexagon is crucial for creating accurate floor plans and layouts.
Using the Apothem to Determine the Perimeter
To determine the perimeter of a hexagon using the apothem, start by calculating the length of one side. The apothem is the distance from the center of the hexagon to a side, and it’s a helpful
measurement in finding the perimeter. Begin by drawing a line from the center of the hexagon to a vertex, creating a right triangle. The apothem is the height of this triangle.
Next, measure the length of one side of the hexagon. Multiply this length by 6 to find the total perimeter. However, if the length of one side isn’t given, you can use the apothem and the formula P =
2πr to find the radius and then calculate the side length.
Applying the Pythagorean Theorem
To apply the Pythagorean Theorem when finding the perimeter of a hexagon, use a right triangle to determine the length of one side.
Begin by drawing a line from one vertex of the hexagon to the center, creating a right triangle with one side as the apothem. The apothem is the distance from the center to any side of the hexagon.
Next, measure the length of the apothem. Let’s call it ‘a’. Then, measure the length of one side of the hexagon. Let’s call it ‘s’.
Now, you can use the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the side length of
the hexagon, and the other two sides are the apothem and half of the side length.
Using Trigonometry to Find the Perimeter
To find the perimeter of a hexagon using trigonometry, you’ll need to continue the discussion from the previous subtopic by utilizing trigonometric functions. Trigonometry deals with the
relationships between the angles and sides of triangles.
By applying trigonometric functions, such as sine, cosine, and tangent, you can find the length of the missing sides of a hexagon. To do this, you’ll need to know the measures of the angles and the
lengths of at least one side.
By using these trigonometric functions, you can calculate the lengths of the remaining sides and then add them up to find the perimeter of the hexagon.
Trigonometry offers a powerful tool for solving complex geometric problems and finding accurate measurements.
Practice Problems and Additional Resources
Now, let’s dive into some practice problems and explore additional resources to further enhance your understanding of finding the perimeter of a hexagon using trigonometry.
Solving practice problems is crucial to solidify your knowledge and improve problem-solving skills. You can create your own problems or find worksheets online that specifically focus on finding the
perimeter of a hexagon. These problems will give you the opportunity to apply the concepts you’ve learned and reinforce your understanding.
In addition to practice problems, there are also various online resources available that can provide further guidance and explanations. Websites, video tutorials, and interactive apps can be valuable
tools to supplement your learning. Remember to utilize these resources to deepen your understanding and master the skill of finding the perimeter of a hexagon using trigonometry.
Frequently Asked Questions
What Is the History and Origin of the Hexagon Shape?
The hexagon shape has a rich history and origin. It can be found in nature, such as honeycomb structures. Its regularity and symmetry make it a fascinating shape in geometry.
Can You Use the Same Method to Find the Perimeter of a Regular Polygon With More Than Six Sides?
Yes, you can use the same method to find the perimeter of a regular polygon with more than six sides. Just add up the lengths of all the sides using the formula.
How Can You Find the Area of a Hexagon Using the Given Side Lengths?
To find the area of a hexagon using the given side lengths, you can use the formula A = (3√3 * s^2) / 2, where s is the length of one side.
Is There a Formula to Find the Perimeter of an Irregular Hexagon?
Yes, there is a formula to find the perimeter of an irregular hexagon. You can add up the lengths of all six sides to get the total perimeter.
Are There Any Real-Life Applications or Examples Where Knowing the Perimeter of a Hexagon Is Useful?
Knowing the perimeter of a hexagon is useful in real-life applications like construction, where you need to measure the length of a hexagonal room or the perimeter of a hexagonal plot of land.
So now you know how to find the perimeter of a hexagon using various methods.
Whether you use the length of each side, the apothem, the Pythagorean theorem, or trigonometry, you have the tools to calculate the perimeter accurately.
With practice and further resources, you can become even more proficient in finding the perimeter of any hexagon. | {"url":"http://higheducations.com/geometry-hacks/","timestamp":"2024-11-02T19:00:19Z","content_type":"text/html","content_length":"97255","record_id":"<urn:uuid:526fd384-db43-414b-a7a9-54bb08cb757e>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00002.warc.gz"} |
Mechanics of Sediment Plug Formation
in the Middle Rio Grande, NM
Mechanics of Sediment Plug Formation in the Middle Rio Grande, NM
Pierre Y Julien^1*, Kiyoung Park^2, Drew C Baird^3 and Nathan Holste^3
^1Professor, Department of Civil and Environmental Engineering, Colorado State University, USA
^2Manager, K-water, Dae-jeon, Republic of Korea
^3Hydraulic Engineer, Department of the Interior, U.S. Bureau of Reclamation, Technical Service Center, Sedimentation and River Hydraulics, USA
Submission: February 21, 2024; Published: February 28, 2024
*Corresponding Author: Pierre Y Julien, Professor, Department of Civil and Environmental Engineering, Colorado State University, USA
How to cite this article: Pierre Y Julien*, Kiyoung Park, Drew C Baird and Nathan Holste. Mechanics of Sediment Plug Formation in the Middle Rio Grande, NM. Civil Eng Res J. 2024; 14(4): : 555893.
DOI 10.19080/CERJ.2024.14.555893
The mechanics of sediment plug development in the Middle Rio Grande is examined based on the historic records on the hydraulic geometry, morphology and sedimentation factors. From 1992 to 2002, the
Middle Rio Grande experienced an increase in overbank flows due to channel narrowing from vegetation encroachment causing a roughness increase and a decrease in channel conveyance and sediment
transport capacity. In 2002, a significant reach of the river was perched about 8 ft above the floodplain. Relationships for the daily bed aggradation rate and the estimated time for plug formation
are tested with field measurements. The primary factors causing sediment plugs to form in the Middle Rio Grande are: (1) overbank flow on perched channel floodplains; and (2) high near-bed sediment
concentration (Ro=1). Sedimentation factors show that overbank flows with high Rouse numbers from 0.6 to 1.7 accelerate channel aggradation. Backwater effects from the railroad bridge (Tiffany plugs)
and sharp river bends (Bosque plugs) also accelerate the plug formation process. Field observations confirm significant overbank flow around shallow and narrow aggrading main channels.
Keywords: Channel Width; Perching, Rio Grande; River Modeling; Rouse Number; Sediment Plug; Sediment Transport Capacity
A sediment plug is a channel blockage observed at sites where sediment deposition is comparable to the flow depth. Sediment plugs have occurred at several locations around the country. Diehl [1]
investigated the plug formation in the Hatchie River basin in the Tennessee and attributed the plug development to low-gradient alluvial systems with sediment-laden tributaries. Shields et al. [2]
studied the Yalobusha River in northern Mississippi and found that the sediment plug occurred after river channelization due to the shrinkage of bankfull discharge and the decreased channel slope. A
sediment plug formed in the Guadalupe River below Canyon Dam in Texas during the summer of 2002 [3]. The main channel plug occurred during a historic flood carrying large volumes of sediment and
debris from the tributary. The Clear Branch Creek of the Middle Fork of the Hood River, Oregon, also experienced a sediment plug in 2006 due to flooding and anthropogenic activities. In the Middle
Rio Grande (Figure 1), sediment plugs occurred three times (1991,1995, and 2005) at the Tiffany junction area located 45 miles upstream from the Elephant Butte Reservoir. Two new sediment plugs
formed in the Bosque del Apache area 13 miles upstream of the Tiffany plug in 2008 and in 2017. All sediment plugs developed within a matter of weeks and the main channel aggraded up to bank crest,
causing water delivery stoppage through the main channel. Figure 2 shows a photo of the Tiffany plug in June 2008. The main channel filled up with sediment and the flow (3,700 cfs) was forced around
the plug onto the floodplain.
Sediment plugs in the Middle Rio Grande have been studied by various authors. Leόn [4] developed a relationship between slope and both the width and widthdepth ratio of channels. She obtained good
agreement with field measurements of the Middle Rio Grande [5]. Boroughs [6] investigated earlier sediment plugs of the Middle Rio Grande and attributed plug formation to the constriction and
expansion of the river channel with overbank flows. Boroughs et al. [7] proposed the criterion PLGNUM for sediment plug formation in alluvial rivers. It consists of five major parameters to cause a
sediment plug: (1) loss of flow to overbank areas; (2) overbank flows that continue for several days or weeks; (3) upstream sediment supply exceeding the local sediment transport capacity; (4) a
flow-sediment discharge exponent, and (5) non-uniform vertical distribution of the total sediment load. However, the PLGNUM criterion failed to predict the 2008 Bosque plug in the Middle Rio Grande
Therefore, further investigation is needed to better understand the mechanics of sediment plug formation on the Rio Grande. Bender and Julien [9] and Shrimpton and Julien [10] showed that the
vertical sediment distribution, low overbank flow, backwater effects, low bank height and perching are likely to cause the sediment plugs in the Middle Rio Grande. Park and Julien [11] quantified the
effects of overbank flow and sediment concentration profile on channel bed elevation in the Middle Rio Grande. Julien and Rainwater [12] also examined the flow sequences during the years prior to
plug formation. In addition, numerical models have been applied to simulate bed elevation changes and identify the parameters associated with sediment plug formation on the Rio Grande [7,13,14].
Huang et al. [15], Boroughs [13], Huang and Makar [16,17] and Park [8] used 1-D or 2-D numerical programs to simulate past and future channel morphology in the Middle Rio Grande. However, there are
concerns that a number of important plug formation processes were overly simplified or ignored, thus limiting the model to a specific site with minimal predictive capabilities [18]. Thus, new
hypotheses need to be examined and tested to improve the simulation of sediment plug formation. The purpose of this paper is to examine the physical processes and the mechanics of sediment plug
formation in order to assess the primary causing factors. The specific objectives are to: (1) investigate the mechanics of sedimentation leading to plug formation; (2) analytically examine which
factors lead to the plug formation on the Middle Rio Grande; and (3) use field measurements and site visits to determine which factors contribute the most to the formation of sediment plugs. The
hypotheses to be developed further in this article include three main types of parameters: (1) geometric factors, e.g. channel width and roughness; (2) backwater factors contributing to overbank flow
on perched floodplains; and (3) sedimentation factors, e.g. sediment concentration profiles and main channel aggradation. This article first develops analytical relationships and then examines the
main parameters based on very detailed field data collected in the Middle Rio Grande over long periods of time.
The bed elevation changes are examined analytically based on the following four basic equations: (1) the flux equation; (2) a flow resistance equation; (3) a sediment transport equation; and (4) a
sediment continuity equation.
The flux equation described the flow rate at a given cross section as
where Q is the water discharge, V is the mean flow velocity perpendicular to the cross sectional area A (A = W h for wide rectangular channel, where W is the channel width and h is the mean flow
The flow resistance equation describes the relationship between flow depth and mean flow velocity. Manning’s equation is commonly used in rivers, or
where φ is 1.49 for English units and 1 for SI units (φ =1 is used in this article), n is the Manning roughness coefficient, R[h] is the hydraulic radius ( R[h] = cross-section area / wetted
perimeter) and 𝑆 is the friction slope.
Several sediment transport equations are available and Julien’s [19] formula is used because it yields useful analytical formulations describing sediment plug formation
where Q[s] is the volumetric sediment discharge, W is the channel width, g is the gravitational acceleration, d[s] is the particle size, C[v] is the volumetric sediment concentration, and 𝜏[*] is the
Shields parameter
The Exner equation defines the changes in bed elevation with time as a function of the longitudinal changes in the rate of sediment transport, as
where Δz[b] is the change in bed elevation during a time increment Δt, p[o] is the porosity of the bed material, ΔQ[s] is the net change in volumetric sediment discharge over a reach length Δ𝑥 .
The theoretical analysis of riverbed changes focuses on three types of factors: (1) geometric factors including the effects of changes in both channel width and width-depth ratios on the sediment
transport capacity of a channel; (2) effects of overbank flow on main channel aggradation; and (3) backwater effects on channel aggradation rates.
The first relationship between channel width, or widthdepth ratio and sediment transport capacity is obtained after combining Eqs. (1) and (2). For wide rectangular channels, in SI units, the
hydraulic radius Rh becomes equal to the flow depth h, and
Mass balance between two successive cross sections 1 (upstream) and 2 (downstream) yields a relationship between width and depth while keeping both slope S and Manning n constant
Repeating this procedure for the sediment load in widerectangular channels (Rh = h) while keeping a fixed grain size, specific gravity and channel slope, the relationship between sediment discharge
and the width and depth ratios can be obtained from Eq. (3) and the definition of the Shields parameter as
The depth ratio can be eliminated by substituting Eq. (6) into Eq. (7) as
Therefore, from this relationship for wide-rectangular channels, an increase in channel width decreases the sediment transport capacity with power of -0.2 (Figure 3a). Leόn et al. [5] and Park [8]
showed that the exponent of this relationship can change with different sediment transport equation, but the trend is the same.
A more general relationship is obtained when using the hydraulic radius instead of the flow depth in both the Shields parameter and Manning’s equation. The hydraulic radius is defined as
Assuming constant gravitational acceleration, sediment size, discharge, Manning coefficient, and channel slope at two cross-sections, Eq. (11) simply reduces to
This s Q −ξ relationship is plotted in Figure 3b and shows a maximum value. Taking the derivative of this equation with respect to ξ and equating it to zero gives the value of the width-depth ratio ξ
[m] that corresponds to the maximum sediment transport capacity, thus
Therefore, the sediment transport capacity from Eq. (11) is maximum at a width-depth ratio ξ[m] = 18, and it will be decreased with the power of -0.125 when ξ is much larger than 18.
The analysis of overbank flow is sketched in Figure 4.
In order to determine the overbank flow effect on sedimentation in the main channel, continuity of water can be expressed as:
where 𝑄[1] is the inflow, 𝑄[2] is the main channel outflow and 𝑄[𝑜] is the overbank flow. The overbank flow Q[o] occurs when the flow depth h exceeds the bankfull height H, and its magnitude over a
reach length of overbank flow Lo is calculated using the broad-crested weir equation with the overbank coefficient C[b] as
Substituting Eqs. (15) and (16) into Eq. (14) yields a relationship between overbank length and flow depth:
where Cb is a broad-crested weir coefficient ( C[b] ≅ 1.4 in SI units or C[b] ≅ 2.5 in English units). Sediment transport is defined from the mass conservation of sediment, flow depth h, and friction
slope S. The incoming sediment load Q[s1] equals the outgoing sediment load in the channel Q[s2] plus the sediment load on the overbank area Q[so] and the sediment deposited on the bed Q[sbed]
The sediment loss to overbank areas is defined using the volumetric depth-averaged sediment concentration Cv and the concentration ratio C[R] = C[vo] / C[v1]. The concentration ratio C[R] is defined
as the average concentration in the overbank portion of the water column C[vo] divided by the depth-averaged sediment concentration C[v1] in the main channel. It is calculated using the Rouse
sediment concentration profile. Therefore, the overbank sediment load is obtained from
where a is the reference level (a=2ds), Ro is the Rouse number [s] is the ratio of the sediment to fluid momentum exchange coefficient ( β[s] ≈ 1), k is the von Karman constant (k ≈ 0.4) , and [R] is
plotted as Figure 5 using calculations from Guo and Julien [20], Shah-Fairbank et al. [21] and Yang and Julien [22]. As the Rouse number decreases ( u* /ω increases), the concentration ratio
increases. For the Rio Grande, the Rouse number Ro > 0.6 corresponds to very low values of the concentration ratio, i.e. C[R]<0.1.
Sediment continuity allows the calculation of the sediment deposition rate in the main channel from
Substituting Eqs. (16, 19, 20 and 22) into Eq. (18) yields:
Eliminating Cb and H by combining Eqs. (17 and 17a) and (23) yields the rate of bed elevation change over time:
The term in brackets of Eq. (24) defines the aggradation coefficient Co shown in Figure 6. For a uniform sediment concentration profile (Ro = 0, C[R] = 1), the daily bed aggradation rate would be
very low. The condition of interest, C[R]<0.1, however, yields high values of Co. Therefore, the high values of Ro for the Rio Grande largely contribute to the formation of sediment plugs.
This general bed aggradation relationship is particularly of interest when h[2] is the bankfull height h[2] = H and h[1] > H. The aggradation depth from Eq. (24) can be examined in terms of two
parameters: (1) the aggradation magnitude M; and (2) a sediment plug coefficient CP from
The magnitude M determines the aggradation depth scale in m. The sediment plug coefficient CP is plotted in Figure 7 as a function of h/H.
For instance, the daily (Δt = 86,400 s) aggradation of fine sediment at ds = 0.00025 m (or settling velocity ω = 0.035 m/s) for a channel at a slope S = 0.0007, flow depth H = 1m [R]<0.1. Therefore,
we can expect a daily bed aggradation rate of Δz[b] ≅ 0.33 × 0.25 = 0.08m in this channel.
In the Rio Grande, the sediment concentration profile is non-uniform (e.g. Ro ≅ 1and C[R] < 0.1, and therefore C[P] is high). This physically means that the sediment load is concentrated near the bed
and the clear water in the overbank flow causes significant aggradation from the large decrease in sediment transport capacity in the main channel. This enrichment in sediment concentration in the
main channel accelerates the sediment plug formation process.
It is also interesting to estimate the plug formation time t[plug] which is the time required for a plug to form as a result of overbank flow. This is obtained from Eq. (24) when Δz[b] = h[1]
Backwater effects on channel bed elevation changes are examined by considering flow discharge (Eq. 2), sediment transport (Eq. 3), and sediment continuity (Eq. 4). From the flow discharge equation,
friction slope S can be expressed as,
Substituting Eq. (26) into the sediment transport capacity Eq. (3) yields
Keeping the discharge, Manning n, channel width W and grain size ds constant gives
Now substituting Eq. (28) into the sediment continuity Eq. (4) gives:
Thus, when the flow depth increases due to backwater effects, the channel bed quickly aggrades. Narrow channels aggrade faster and are more likely to plug than wide channels. The reach length Δx can
be determined from the distance required for suspended sediment to settle from the suspension Δx ≅ 5Q /Wω where ω is the settling velocity [23]. At a discharge of Q = 100 m3/s, a W = 60 m wide
sand-bed channel with ω = 0.03 m/s, the length required for sediment settling in a backwater zone would be about 280 m long.
The sections of the Middle Rio Grande examined in this study include the Bosque del Apache Reach (Bosque plugs) and the Elephant Butte Reach (Tiffany plugs), which are identified officially by
Reclamation based on the presence of geologic and geomorphic controls. The Bosque Reach is located in the Bosque del Apache National Wildlife Refuge.
Owing to the channel narrowing and vegetation encroachment, bank roughness has significantly increased over time. As shown in Figure 8, around the Bosque plug location (section Agg/Deg 1550), the
channel width decreased 40% between 1962 and 2002 and another 70% between 2002 and 2008.
Cross-section data are based on numerous georeferenced field surveys and with comparisons with GIS maps. A total of 266 rectangular composite cross sections (main channel and floodplains) in 1992 and
404 cross sections (aggradation/degradation lines) in 2002 were available for this study. Comparisons with historic satellite imagery delineate the vegetation to vegetated channel widths. The
cross-section data, which are used in the analyses Park and Julien [11] and Park [8]) include: distances, bed slopes, bed elevations, minimum bed elevations, channel widths, bank crest elevations.
Figure 9 shows changes in cross-section geometry of the perched river reach near the Tiffany plug. An overbank flow analysis by Bender and Julien [9] defined perching when the thalweg of a cross
section was higher than the floodplain elevation. A perching ratio can be defined as the number of perched cross-sections divided by the total number of cross-sections. The perching ratio was 13 % in
1992 but increased to 87 % in 2002 according to [8].
Flow discharges at San Marcial (USGS 08358400) and at San Acacia (USGS 08354900) are used to determine the amount of overbank flow and water losses. The USGS gauges provide daily flow discharges
since 1958. The main water losses from the main channel are due to overbank flows in areas with perched cross-sections. Two major locations with active return flows could be located based on the
geometric data using LIDAR, DEM, HEC-RAS and satellite imagery.
A Manning n value of 0.017 was used to describe grain roughness in the main channel along the entire Tiffany Junction Reach based on previous studies [24]. FLO Engineering [25] also obtained a
Manning n of 0.015-0.017 for cross-sections at the upstream portion of the Tiffany Junction Reach with data from 1993 and 1994 at flows ranging from 2,700 cfs to 5,400 cfs. The representative Manning
n value provided by Reclamation was 0.017 ~ 0.024 for the main channel and 0.1 in floodplain areas. The composite Manning n values ranged from 0.02~0.075 depending on the flow discharge and location
From the grain size analyses from the Bosque Reach report Paris et al. [26] and the Elephant Butte Reach report [27], the overall median diameter was 0.2 mm in 1992 and increased to 0.23 mm in 2002.
There is a slight difference between San Acacia and San Marcial in terms of sediment size, but no significant relation between the seasonal sediment size and sediment plug formation could be
observed. Thus, the median particle diameter can be considered constant for the entire reach. Several sediment transport equations are in good agreement Leόn [4], Boroughs [6] with the field
measurements on the Rio Grande, as shown at San Acacia (USGS 08354900) in Figure 10.
The Rouse parameter which is a function of sediment fall velocity and shear velocity has increased over time since bed sediment particles have slightly coarsened (0.2 mm in 1992 and 0.25 mm in 2002)
and the channel flow depth decreased 52% between 1992 and 2002. Sediment concentration profiles indicate Rouse number values ranging from 0.6 to 1.7. At such high values of the Rouse number, the
sediment concentration becomes very small near the free surface and sediment is mostly transported near the bed.
The backwater effects from the bridge at San Marcial has been identified by Park [8] as a factor affecting the Tiffany plug formation. Historical records of the flooded area corroborate the influence
of backwater effects on sediment plugs (Figure 11). The backwater effect due to bridge contraction speeds up the channel sedimentation, which is the primary triggering factor of the Tiffany plugs.
The presence of sharp bends 0.6 mile downstream of the 2008 Bosque plug location (Agg/Deg 1555 ~ Agg/ Deg 1557) has also been considered as in important factor triggering the Bosque Plug as shown in
Figure 12. The Google imagery shows multiple sharp bends around Agg/Deg 1555 and the lateral migration on the east side bank resulted in another bend at Agg/Deg 1557. The corresponding sediment
filling time up to the bank height (2.85 ft, 2002 HEC-RAS geometry) was estimated to be less than 3 weeks. The sharp bends located less than 1 mile downstream of the 2008 Bosque plug location were
the primary factor explaining why the plug formed in the Bosque area, rather than in the Tiffany area, in 2008.
A site visit of the Bosque del Apache reach of the Rio Grande was conducted on May 26 and June 6, 2017 by [28,29]. The purpose of the site visits was to document overbank flow conditions and the
status of potential sediment plug formation. The field investigation in May 2017 reported a sediment plug forming between RM 81- 82 (2012 river demarcations) at approximately the same location as the
2008 sediment plug. Flow depths were only 1 - 1.5 feet across the entire channel at a section (~150 feet wide) where depths should have been about 4 feet based on similar conditions observed upstream
and downstream. It was estimated that about 1/3 to 1/2 (1,000 to 1,500 cfs) of the total flow was being conveyed by the main channel. Bed material in this area appeared to be medium sand. In addition
to overbanking flow leaving the main channel, areas where overbanking flows were returning to the main channel were also observed. This occurred even when the top of bank was perched 3-4 feet above
the floodplain. In June 2017, the reported flow depths were generally less than 2 feet across the entire channel. It was estimated that about 1/12 (~250 cfs) of the total flow was being conveyed by
the main channel. Bed material in this area appeared to be medium sand. While wading over the top of the sediment plug there were several areas, particularly upstream, where the bed was soft
(uncompacted and/or unconsolidated) and considered to be recent sediment deposition areas
Figure 13 shows a June 2019 photo of the flow on the left-bank floodplain in the direction perpendicular to the main river flow direction from a field survey [30]. Field observations show that the
flow is leaving the channel at nearly a 90-degree angle and flowing down the perched bank to the lower-lying floodplain. Figure 14 shows the most recent aerial view of the Bosque plug illustrating
the effects of the sharp bends at the downstream end of the plug as a source of additional head loss and forcing mechanism to raise the water level and causing surface runoff on a perched floodplain
and subsequent drainage of overbank flow on side channel areas.
Based on the historic flow and geometric characteristics of plug areas, several factors were identified and examined in relation to the sediment plug formation at two locations on the Middle Rio
Grande (Bosque plug and Tiffany plug). The main conclusions of this study include:
1) Theoretical derivations define the relationships between channel width (also width/depth) and sediment discharge, between overbank flows and sedimentation rates in the main channel, and between
backwater effects and channel bed elevation. Relationships for the daily bed aggradation rate and the estimated time for plug formations have been formulated. The increases in width-depth ratio,
overbank flow, Rouse number and backwater effects result in the decrease of sediment transport capacity, leading to the main channel aggradation. Channel narrowing and the increased floodplain
roughness also increase overbank flows.
2) Historic sediment plugs (both the 1995 Tiffany plug and the 2008 Bosque plug) of the Middle Rio Grande were analyzed with an analytical model leading to Eqs. (24) and (25). Significant overbank
flow is observed at a discharge of 3,500cfs (100 m3/s). The Rouse number is typically close to unity and ranged from 0.6 to 1.7 during the period from 1992 to 2002.
3) Field observations described sustained overbank flows in shallow perched channels with freshly deposited sediment in the main channel. Local backwater effects accelerate the process and sediment
plugs can form during a single flood. The Tiffany plugs have been affected by backwater from the San Marcial railroad bridge. The Bosque plug was more influenced by channel narrowing, an increase in
bank roughness and backwater from sharp bends. Field observations at the Bosque plugs in 2017 and 2019 confirm the expected theoretical results.
Funding from the U.S. Bureau of Reclamation is gratefully acknowledged. However, the results do not necessarily reflect policies or endorsement of Reclamation. We would like to express sincere
appreciation to Jonathan AuBuchon, Ari Posner, and Robert Padilla at the USBR for their support and assistance with all aspects of this study. We are also grateful to Katherine Anderson, Seema
Shah-Fairbank, Ted Bender, Tracy Owen, Chris Shrimpton and Jon Rainwater at Colorado State University for their contributions to the analysis of the Rio Grande database. The authors are also grateful
to K-water for the support to the second author during the course of his dissertation research at CSU. The valuable comments from anonymous reviewers have also been sincerely appreciated.
The following symbols are used in this paper:
a = reference elevation
A = cross sectional area
C[a] = near-bed sediment concentration
C[b] = broad-crested weir coefficient
C[o] = aggradation coefficient
C[P] = sediment plug coefficient
C[R] = concentration ratio
C[v] = volumetric sediment concentration
d[s] = sediment particle size
G = specific gravity of sediment
g = gravitational acceleration
h = mean flow depth
hr = flow depth ratio
H = bank height
L[o] = length of overbank flow
M = sediment plug magnitude
n = Manning roughness coefficient
P = wetted perimeter
p[o] = porosity of the bed material
Q = water discharge
Q[s] = volumetric sediment discharge
Q[bed] = sedimentation rate on channel bed
Q[o] = overbank flow discharge
q, q[s] = unit flow and sediment discharges
Qs = sediment discharge
Qso = overbank sediment discharge
Qsr = sediment discharge ratio
R[o] = Rouse number
R[h] = hydraulic radius
S = friction slope
t = time
t[plug] = plug formation time
u* = shear velocity
V = mean flow velocity
W = channel width
W[r] = channel width ratio
x = downstream distance
z[b] = bed elevation
Greek symbols
β = coefficient of the stage-discharge relationship
β[s] = ratio of sediment to fluid momentum exchange coefficient
Δz[b] = change in channel bed elevation
κ = von Kármán constant
ξ = width-depth ratio
ξ[m] = width-depth ratio for the maximum sediment transport capacity
τ[*] = Shields parameter
ω = Settling velocity of sediment
ϕ = 1.49 for English units and 1 for SI units in Manning equation
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Aug 212014
Yitang Zhang is giving the last invited talk at ICM 2014, “Small gaps between primes and primes in arithmetic progressions to large moduli”.
这是闭幕式前的最后一个 invited talk. 张大师习惯手写, 当场演算.
Yitang Zhang stepped onto the main stage of mathematics last year with the announced of his achievement that is hailed as “a landmark theorem in the distribution of prime numbers”.
Aug 152014
每 4 年举行一次世界数学家大会从 13 日到 21 日在首尔会展中心(COEX)举行.
Martin Groetschel, Secretary-General of IMU, 在开幕式上的讲话说, IMU 有一些倡议. 这些打算之一是 adopt-a-graduate-student: IMU 会扶持发达国家的数学家, 这些数学家愿意给不那么发达国家的工作在相近领域的
今年的 Chern Medal 的得主 Phillip Griffiths 选择了 African Mathematics Millennium Science Initiative(AMMSI) 来接受 $250,000.
Donaldson, Tao, Kontsevich, Lurie and Taylor, winners of the Breakthrough Prizes in mathematics, 每人给了 $100,000 给一个目的是支持发展中国家的博士的$500,000 基金会. 具体采用何种方式来实施帮助不得而
知, 但已经使用了 “breakout graduate fellowships” 这样的措词.
Martin Groetschel 还指出, 韩国的数学出版物的数量当前是世界第 11 位, 但韩国数学家 1981 年发表在国际期刊上的论文仅仅只有 3 篇. 韩国从几乎一无所有, 建立了现在的数学传统, 仅仅过了一代人的时间.
1981 年成为国际数学联盟的最低等级第1军成员国的韩国时隔 33 年从援助受惠国成为供应国, 这将成为向全世界宣传韩国数学的契机.
韩国总统朴槿惠出席了当天的开幕式, 她强调了数学给我们的生活带来的影响, 向帮助韩国数学升至世界水平的世界数学界表达了谢意.
ICM 开幕式的一个小插曲是, 戴着面具的舞蹈演员走上舞台时, Maryam Mirzakhani 的不到三岁的女儿 Anahita 发出恐怖的尖叫, 许久才平静下来. Timothy Gowers 有一个 6 岁的儿子.
The Fields Medal Committee for 2014 consisted of Daubechies, Ambrosio, Eisenbud, Fukaya, Ghys, Dick Gross, Kirwan, Kollar, Kontsevich, Struwe, Zeitouni and Günter Ziegler.
The program committee consisted of Carlos Kenig (chair), Bolthausen, Alice Chang, de Melo, Esnault, me, Kannan, Jong Hae Keum, Le Bris, Lubotsky, Nesetril and Okounkov.
Kyoto University professor Shigefumi Mori has been elected president of the International Mathematical Union(IMU), becoming the first head of the group from Asia.
The ICM executive committee for the next four years will be Shigefumi Mori (president), Helge Holden (secretary), Alicia Dickenstein (VP), Vaughan Jones (VP), Dick Gross, Hyungju Park, Christiane
Rousseau, Vasudevan Srinivas, John Toland and Wendelin Werner.
好像, 国际数学联盟已经讨论过, 在大会开幕很久之前公布大奖得主的名字, 是否可行.
下次全世界数学家的聚会, 2018 年的八月在巴西. The General Assembly of the IMU in Gyeongju announced on Aug. 11 that Rio de Janeiro would be the site of ICM 2018.
1. Timothy Gowers, ICM2014 — opening ceremony, August 13, 2014
Aug 132014
Fields Medals 2014
Artur Avila
Manjul Bhargava
Martin Hairer
Maryam Mirzakhani
At the opening ceremony of the International Congress of Mathematicians 2014 on August 13, 2014, the Fields Medals (started in 1936), the Nevanlinna Prize (started in 1982), the Gauss Prize (started
in 2006), and the Chern Medal Award (started in 2010) were awarded. In addition, the winner of the Leelavati Prize (started in 2010) and the speaker of the ICM Emmy Noether Lecture (started in 1994)
were announced.
Rolf Nevanlinna Prize 2014
Subhash Khot
Carl Friedrich Gauss Prize for Applications of Mathematics 2014
Stanley Osher
Chern Medal Award 2014
Phillip Griffiths
Leelavati Prize 2014
Adrián Paenza
ICM Emmy Noether Lecture 2014
The 2014 ICM Emmy Noether lecturer is Georgia Benkart.
Maryam Mirzakhani
Stanford University, USA
[Maryam Mirzakhani is awarded the Fields Medal]
for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.
• Maryam Mirzakhani has made stunning advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new frontiers in this area. Her insights have integrated methods from
diverse fields, such as algebraic geometry, topology and probability theory.
• In hyperbolic geometry, Mirzakhani established asymptotic formulas and statistics for the number of simple closed geodesics on a Riemann surface of genus g. She next used these results to give a
new and completely unexpected proof of Witten’s conjecture, a formula for characteristic classes for the moduli spaces of Riemann surfaces with marked points.
• In dynamics, she found a remarkable new construction that bridges the holomorphic and symplectic aspects of moduli space, and used it to show that Thurston’s earthquake flow is ergodic and
• Most recently, in the complex realm, Mirzakhani and her coworkers produced the long sought-after proof of the conjecture that – while the closure of a real geodesic in moduli space can be a
fractal cobweb, defying classification – the closure of a complex geodesic is always an algebraic subvariety.
• Her work has revealed that the rigidity theory of homogeneous spaces (developed by Margulis, Ratner and others) has a definite resonance in the highly inhomogeneous, but equally fundamental realm
of moduli spaces, where many developments are still unfolding
Artur Avila
CNRS, France & IMPA, Brazil
[Artur Avila is awarded a Fields Medal] for his profound contributions to dynamical systems theory have changed the face of the field, using the powerful idea of renormalization as a unifying
• Avila leads and shapes the field of dynamical systems. With his collaborators, he has made essential progress in many areas, including real and complex one-dimensional dynamics, spectral theory
of the one-frequency Schródinger operator, flat billiards and partially hyperbolic dynamics.
• Avila’s work on real one-dimensional dynamics brought completion to the subject, with full understanding of the probabilistic point of view, accompanied by a complete renormalization theory. His
work in complex dynamics led to a thorough understanding of the fractal geometry of Feigenbaum Julia sets.
• In the spectral theory of one-frequency difference Schródinger operators, Avila came up with a global de- scription of the phase transitions between discrete and absolutely continuous spectra,
establishing surprising stratified analyticity of the Lyapunov exponent.
• In the theory of flat billiards, Avila proved several long-standing conjectures on the ergodic behavior of interval-exchange maps. He made deep advances in our understanding of the stable
ergodicity of typical partially hyperbolic systems.
• Avila’s collaborative approach is an inspiration for a new generation of mathematicians.
Manjul Bhargava
Princeton University, USA
[Manjul Bhargava is awarded a Fields Medal]
for developing powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves.
• Bhargava’s thesis provided a reformulation of Gauss’s law for the composition of two binary quadratic forms. He showed that the orbits of the group \(SL(2, \Bbb Z)3\) on the tensor product of
three copies of the standard integral representation correspond to quadratic rings (rings of rank \(2\) over \(\Bbb Z\)) together with three ideal classes whose product is trivial. This recovers
Gauss’s composition law in an original and computationally effective manner. He then studied orbits in more complicated integral representations, which correspond to cubic, quartic, and quintic
rings, and counted the number of such rings with bounded discriminant.
• Bhargava next turned to the study of representations with a polynomial ring of invariants. The simplest such representation is given by the action of \(PGL(2, \Bbb Z)\) on the space of binary
quartic forms. This has two independent invariants, which are related to the moduli of elliptic curves. Together with his student Arul Shankar, Bhargava used delicate estimates on the number of
integral orbits of bounded height to bound the average rank of elliptic curves. Generalizing these methods to curves of higher genus, he recently showed that most hyperelliptic curves of genus at
least two have no rational points.
• Bhargava’s work is based both on a deep understanding of the representations of arithmetic groups and a unique blend of algebraic and analytic expertise.
Martin Hairer
University of Warwick, UK
[Martin Hairer is awarded a Fields Medal]
for his outstanding contributions to the theory of stochastic partial differential equations, and in particular created a theory of regularity structures for such equations.
• A mathematical problem that is important throughout science is to understand the influence of noise on differential equations, and on the long time behavior of the solutions. This problem was
solved for ordinary differential equations by Itó in the 1940s. For partial differential equations, a comprehensive theory has proved to be more elusive, and only particular cases (linear
equations, tame nonlinearities, etc.) had been treated satisfactorily.
• Hairer’s work addresses two central aspects of the theory. Together with Mattingly he employed the Malliavin calculus along with new methods to establish the ergodicity of the two-dimensional
stochastic Navier-Stokes equation.
• Building on the rough-path approach of Lyons for stochastic ordinary differential equations, Hairer then created an abstract theory of regularity structures for stochastic partial differential
equations (SPDEs). This allows Taylor-like expansions around any point in space and time. The new theory allowed him to construct systematically solutions to singular non-linear SPDEs as fixed
points of a renormalization procedure.
• Hairer was thus able to give, for the first time, a rigorous intrinsic meaning to many SPDEs arising in physics.
Subhash Khot
New York University, USA
[Subhash Khot is awarded the Nevanlinna Prize]
for his prescient definition of the “Unique Games” problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization
problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.
• Subhash Khot defined the “Unique Games” in 2002 , and subsequently led the effort to understand its complexity and its pivotal role in the study of optimization problems. Khot and his
collaborators demonstrated that the hardness of Unique Games implies a precise characterization of the best approximation factors achievable for a variety of NP-hard optimization problems. This
discovery turned the Unique Games problem into a major open problem of the theory of computation.
• The ongoing quest to study its complexity has had unexpected benefits. First, the reductions used in the above results identified new problems in analysis and geometry, invigorating analysis of
Boolean functions, a field at the interface of mathematics and computer science. This led to new central limit theorems, invariance principles, isoperimetric inequalities, and inverse theorems,
impacting research in computational complexity, pseudorandomness, learning and combinatorics. Second, Khot and his collaborators used intuitions stemming from their study of Unique Games to yield
new lower bounds on the distortion incurred when embedding one metric space into another, as well as constructions of hard families of instances for common linear and semi- definite programming
algorithms. This has inspired new work in algorithm design extending these methods, greatly enriching the theory of algorithms and its applications.
Phillip Griffiths
Institute for Advanced Study, USA
[Phillip Griths is awarded the 2014 Chern Medal]
for his groundbreaking and transformative development of transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties.
• Phillip Griffiths’s ongoing work in algebraic geometry, differential geometry, and differential equations has stimulated a wide range of advances in mathematics over the past 50 years and
continues to influence and inspire an enormous body of research activity today.
• He has brought to bear both classical techniques and strikingly original ideas on a variety of problems in real and complex geometry and laid out a program of applications to period mappings and
domains, algebraic cycles, Nevanlinna theory, Brill-Noether theory, and topology of K¨ahler manifolds.
• A characteristic of Griffithss work is that, while it often has a specific problem in view, it has served in multiple instances to open up an entire area to research.
• Early on, he made connections between deformation theory and Hodge theory through infinitesimal methods, which led to his discovery of what are now known as the Griffiths infinitesimal period
relations. These methods provided the motivation for the Griffiths intermediate Jacobian, which solved the problem of showing algebraic equivalence and homological equivalence of algebraic cycles
are distinct. His work with C.H. Clemens on the non-rationality of the cubic threefold became a model for many further applications of transcendental methods to the study of algebraic varieties.
• His wide-ranging investigations brought many new techniques to bear on these problems and led to insights and progress in many other areas of geometry that, at first glance, seem far removed from
complex geometry. His related investigations into overdetermined systems of differential equations led a revitalization of this subject in the 1980s in the form of exterior differential systems,
and he applied this to deep problems in modern differential geometry: Rigidity of isometric embeddings in the overdetermined case and local existence of smooth solutions in the determined case in
dimension \(3\), drawing on deep results in hyperbolic PDEs(in collaborations with Berger, Bryant and Yang), as well as geometric formulations of integrability in the calculus of variations and
in the geometry of Lax pairs and treatises on the geometry of conservation laws and variational problems in elliptic, hyperbolic and parabolic PDEs and exterior differential systems.
• All of these areas, and many others in algebraic geometry, including web geometry, integrable systems, and
• Riemann surfaces, are currently seeing important developments that were stimulated by his work.
• His teaching career and research leadership has inspired an astounding number of mathematicians who have gone on to stellar careers, both in mathematics and other disciplines. He has been
generous with his time, writing many classic expository papers and books, such as “Principles of Algebraic Geometry”, with Joseph Harris, that have inspired students of the subject since the
• Griffiths has also extensively supported mathematics at the level of research and education through service on and chairmanship of numerous national and international committees and boards
committees and boards. In addition to his research career, he served 8 years as Duke’s Provost and 12 years as the Director of the Institute for Advanced Study, and he currently chairs the
Science Initiative Group, which assists the development of mathematical training centers in the developing world.
• His legacy of research and service to both the mathematics community and the wider scientific world continues to be an inspiration to mathematicians world-wide, enriching our subject and
advancing the discipline in manifold ways.
Aug 122014
明天上午(韩国时间, 东九区时间)九点, ICM 2014 会准时在韩国首尔(Coex , Seoul , Korea)开幕. 依惯例, 开幕式上会为引人注目的 Fields Medal 获得者颁发奖章. 坊间流传的一个(如若发生)会载入历史的传奇是:
本届 ICM 会有一个女性数学家获得 Fields Medal!
这真是一个令人振奋的消息! 全世界的数学工作者屏住呼吸兴奋的等待着亲眼目睹见证这个激动人心的时刻!
Maryam Mirzakhani(born May 1977) is an Iranian mathematician, Professor of Mathematics (since September 1, 2008) at Stanford University.
Maryam Mirzakhani 是今年非常有力的竞争者, 和任何候选人站在一起都是那样出众引人注目. 她还是 IMO 满分.
1995 年, 今年中国领队姚一隽去加拿大参加 IMO 的时候, 这一年也是张筑生第一次做领队, 一共有 14 个满分, 其中有两个女生: 咱们中国的朱晨畅, 现在在德国 Gottingen University; 来自伊朗的 Maryam Mirzakhani, 8
月 16 日上午作 ICM 一小时报告.
很多人看好法国的 Sophie Morel. Sophie Morel 专长数论. 不过, 恐怕 Sophie Morel 今年拿不到奖章, 但她四年后还有一次机会.
此外, Laure Saint-Raymond 和 Marianna Csörnyei 也是相当给力的人选.
一个不好的消息是: 国际数学家大会召开在即, 韩劝阻埃博拉(Ebola virus)疫区数学家不要与会.
Aug 052014
ICM 2014 Program
这届国际数学界大会(International Congress of Mathematicians, ICM)的安排, 已经明确无误的说明:
4 个数学家将获得本届大会的 Fields Medal.
张大师将于 8 月 21 日作 ICM 闭幕式之前的压轴报告, 这是只有今年的 Fields Medalist, Gauss Prize, Chern Medal 得主才有的殊荣
张益唐 7 月1 日在北大本科生毕业典礼有一个讲话
这个暑假, 张大师在中国科学院晨兴数学中心和他的母校北京大学做了好几次讲座.
1. A Transition Formula for Mean Values of Dirichlet Polynomials
2014,6.23./6.25. 9:30-11:30
晨兴 110
主持人: 王元
2. 关于 Siegel 零点
晨兴 110
3. Distribution of Prime Numbers and the Riemann Zeta Function
July 8, 10, 2014 16:00-17:00, 镜春园82号甲乙丙楼的中心报告厅
July 15, 16:30-17:30 镜春园 78 号院的 77201 室.
主持人: 刘若川
4. 关于 Siegel 零点(2)
014.7.16./7.30./8.4./8.6. 9:30-11:30
Manjul Bhargava, the BSD conjecture and Fields medal
Advance, math.NT, Mathematician, Prize No Responses »
Jul 242014
There exist elliptic curve groups \(E(\Bbb Q)\) of arbitrarily large rank.
用 \(r\) 表示 \(\Bbb Q\) 上的椭圆曲线 \(E\) 的秩—the rank of the Mordell–Weil group \(E(\Bbb Q)\).
一个悬而未决的著名难题是: \(r\) 是否可以任意大?
Martin-McMillen 2000 年有一个 \(r\geq24\) 的例子:
\begin{equation*}\begin{split}y^2+xy+y&=x^3-120039822036992245303534619191166796374x\\&+ 504224992484910670010801799168082726759443756222911415116\end{split}\end{equation*}
Hasse-Weil \(L\)-function \(L(s, E)\) 在 \(s=1\) 处的零点的阶数 \(r_a\) 称为 \(E\) 的 analytic rank(解析秩).
Manjul Bhargava, Christopher Skinner, Wei Zhang(张伟) 7 月 7 日在 arXiv 上传的论文 “A majority of elliptic curves over \(Q\) satisfy the Birch and Swinnerton-Dyer conjecture“, 宣布了取得的进展:
1. \(\Bbb Q\) 上的椭圆曲线, when ordered by height(同构类以高排序), 至少有 \(66.48\%\) 满足 BSD conjecture;
2. \(\Bbb Q\) 上的椭圆曲线, when ordered by height, 至少有 \(66.48\%\) 有有限 Tate–Shafarevich group;
3. \(\Bbb Q\) 上的椭圆曲线, when ordered by height, 至少有 \(16.50\%\) 满足 \(r=r_a=0\), 至少有 \(20.68\%\) 满足 \(r=r_a=1\).
谁将在 8 月 13 日的 ICM 2014 开幕式上获得 Fields medal？坊间向来不缺传闻. 数论大牛 Manjul Bhargava 无疑是最耀眼的明星. | {"url":"https://www.zyymat.com/tag/icm","timestamp":"2024-11-08T02:36:58Z","content_type":"text/html","content_length":"95553","record_id":"<urn:uuid:32ae7792-9e8a-4fe2-ae5c-33b5030308ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00519.warc.gz"} |
Private High-Dimensional Hypothesis Testing (Conference Paper) | NSF PAGES
We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation with nearly optimal sample complexity. Only exponential-time estimators were previously known
to achieve this guarantee. Given n samples from a (sub-)Gaussian distribution with unknown mean μ and covariance Σ, our (ε,δ)-differentially private estimator produces μ~ such that ∥μ−μ~∥Σ≤α as long
as n≳dα2+dlog1/δ√αε+dlog1/δε. The Mahalanobis error metric ∥μ−μ^∥Σ measures the distance between μ^ and μ relative to Σ; it characterizes the error of the sample mean. Our algorithm runs in time O~
(ndω−1+nd/ε), where ω<2.38 is the matrix multiplication exponent. We adapt an exponential-time approach of Brown, Gaboardi, Smith, Ullman, and Zakynthinou (2021), giving efficient variants of stable
mean and covariance estimation subroutines that also improve the sample complexity to the nearly optimal bound above. Our stable covariance estimator can be turned to private covariance estimation
for unrestricted subgaussian distributions. With n≳d3/2 samples, our estimate is accurate in spectral norm. This is the first such algorithm using n=o(d2) samples, answering an open question posed by
Alabi et al. (2022). With n≳d2 samples, our estimate is accurate in Frobenius norm. This leads to a fast, nearly optimal algorithm for private learning of unrestricted Gaussian distributions in TV
distance. Duchi, Haque, and Kuditipudi (2023) obtained similar results independently and concurrently.
more » « less | {"url":"https://par.nsf.gov/biblio/10343431-private-high-dimensional-hypothesis-testing","timestamp":"2024-11-04T09:10:45Z","content_type":"text/html","content_length":"243953","record_id":"<urn:uuid:977afddf-ba14-491b-acfa-8d810f4958c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00335.warc.gz"} |
How to Invest: Moat - The Big 4 Growth Rates
Welcome to the introduction to Rule #1 course, I’m Phil Town and this is Tutorial 4: Moat (Part 2): The Big Four.
This is part 4 of a 9-part series on How to Invest using Rule #1 strategies
Part 1: Rule #1 Strategy- Overview of the Basics
Part 2: Meaning- The Three Circles
Part 3: Moat- A Durable Advantage
Part 4 [You are Here]: Moat- The Big Four
Part 5: Management- Owner Oriented
Part 6: Margin of Safety- The Growth Rate
Part 7: Margin of Safety- Sticker Price and MOS
Part 8: Margin of Safety- Payback Time
Part 9: Zombie Value- Tangible Book Value
The Big 4 Growth Rates and Moat
The big four growth rates are the key to knowing if your company has a big moat. They are book value per share plus dividends, you find that on the balance sheet. Earnings per share, on the income
statement. Operating cash flow per share, that’s in the cash flow, and sales per share, which is on the income statement.
These four numbers answer these important questions:
1. Does the business have a moat? If the numbers are good then it probably does.
2. Is the business growing predictably? We want to see these numbers staying predictable over time.
3. If the future is like the past, what is the expected earnings growth going to be in the future? We can get that from these numbers too.
Moat: How Apple Stacks Up
Let’s take a look at the toolbox and see if we can figure out some of these answers. Here on Rule #1 Investing at my watch list, I want to take a look at the big four numbers on let’s say, Apple
Computers (AAPL). If I don’t know the symbol of the company I just type in the name of the company, it finds the symbol for me and I hit go. I see in the upper right-hand part of the screen, the Rule
#1 score for this company is a 99. Which is extremely good. Part of that score is the big four numbers. So, let’s take a look at where they are.
First click on the numbers view, and scroll to moat. Here on the moat section of the numbers view, are the book value per share plus dividend growth rate, earnings per share growth, operating cash
flow per share growth, sales growth, and then a total overall score. We can see that it’s all green, green is good. Yellow is not so good, and red is bad. So Apple has some good big four numbers.
We’re looking back 10 years with this column, and then seven years with this column, and then five years, and then three years, and then one year. You can see that Apple scores a perfect 100 on all
of the big four numbers, on all of the different times we’re looking at them.
The Company Should Be Predictable
What this tells us is that Apple is a very predictable company, or at least it has been so far and that it has some kind of big moat. Now if we want to see the numbers all broken down into their
specific individual year, we simply scroll down and we can see all the big four numbers laid out year by year. Book value per share growth rate, earnings per share growth rate, sales growth rate, and
operating cash flow growth rate.
And you can see again, that there’s a lot of consistency in these numbers just by looking across all the years. So the tools make it really easy to identify a big moat company. Let’s go back again
and take a look at those Apple numbers. All green, all good. This company has some kind of big moat.
The big four growth rates should be ten percent or better, each year. And they should not be getting worse over time. Looking Apple computer again, we can see that the big four are a certain number
like big four book value per share growth rate is 30% percent over 10 years, and then last year it was 58% percent. So we can see that it’s getting better over time.
That really looks good for Apple, and we want that kind of growth consistently in all four of the big four growth numbers. Look at Apple and look at near perfection. This is why it has a 100 score on
its moat.
The big four growth rates are book value per share growth, earnings share per growth, operating cash per share growth, and sales growth. And we can find them all right on the Rule #1 Toolbox by
looking at the numbers and then at the moat compound growth rate.
Now, your homework is going to be to go to ruleoneinvesting.com and find the big four growth rates for your big moat company. Then answer this, what is the moat score for your company and when you
have that you’re ready to go on to Tutorial 5: Management- Owner Oriented.
How to Pick Rule #1 Stocks
5 simple steps to find, evaluate, and invest in wonderful companies.
Related reading:
How to Invest Money:A Simple Guide to Grow Your Wealth in 2019
Investing in Stocks 101: A Guide to Stock Market Investing | {"url":"https://www.ruleoneinvesting.com/blog/how-to-invest/how-to-invest-moat-the-big-four/","timestamp":"2024-11-12T22:17:04Z","content_type":"text/html","content_length":"189361","record_id":"<urn:uuid:cd312731-b78e-4074-8bc2-fee742fa50d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00859.warc.gz"} |
Math 1132Q - Calculus II (Fall 2019) | Math Courses
Math 1132Q – Calculus II (Fall 2019)
Description: Transcendental functions, formal integration, polar coordinates, infinite sequences and series, parametric equations, with applications to the physical sciences and engineering.
Prerequisites: MATH 1131 or advanced placement credit for calculus (a score of 4 or 5 on the Calculus AB exam or a score of 3 on the Calculus BC exam). Recommended preparation: A grade of C- or
better in MATH 1131.
Textbook Information: EITHER Calculus: Early Transcendentals (this includes Multivariable Calculus), 8th Edition, by James Stewart with WebAssign Access Code OR Single Variable Calculus: Early
Transcendentals, 8th Edition, by James Stewart with WebAssign Access Code.
Calculators: No calculators are allowed on course exams and quizzes.
Watches: You will not be allowed to wear watches (smart or dumb) during in class assessments (exams and quizzes).
We recommend that you get the text and WebAssign access code bundled together at the UConn Co-op. You should be able to get either the single variable version for 90 dollars, or the full version
(which includes multivariable) for 100 dollars. The option to buy the text and WebAssign access code bundled together lets you use that access code for the life of the edition of the textbook. So if
you have already bought a WebAssign access code for this book for a previous semester, you can use it again. If you are planning on taking Math 2110 but NOT Math 2410, we suggest that you buy the
full version of the text. If you are planning on taking both Math 2110 and Math 2410, then we suggest that you buy the Calculus + Differential Equations version of the text.
Homework and WebAssign:
Homework: To access the homework you will have to go through Husky CT single sign-on. In your account you will find a link to do your homework using WebAssign. There will be homework assignments
for each section of the text. Each assignment will be made available on WebAssign several days before the section is covered in class. The due date for each assignment will be set by your
instructor and will generally be two or three days after the material is covered in class.You will get five attempts for each question that is not multiple choice and fewer than five attempts for
each multiple choice question; the exact number of attempts will depend on the number of choices. After each attempt, you will be told whether your answer is correct or not. If you are not able to
get the correct answer after your initial attempts, we recommend that you seek help from your instructor, the Q-Center, a tutor, or another student. If you miss the due date on homework you can get
an extension for up to two days after the due date, but you will only be able to receive 50% credit for the homework. Warning: When accessing your online homework, use Firefox or Chrome as your
browser; there are problems that can occur if you use Internet Explorer or Safari.Here is a document with tips on using WebAssign.
WebAssign Registration: The homework for Math 1132 is assigned online using the WebAssign online homework system. To access your homework online you must go to Husky CT.
Exams, Quizzes, and Worksheets:
Exams: There will be two midterm exams and a Final Exam in this course. The midterm exams will be administered in your discussion sections (except for the online section, which will have a separate
exam time/location). The Final Exam for the course will be cumulative. Look here for more detailed exam information in the future. The dates and times for all exams are the following.
Exam 1 Wednesday, October 2nd (in discussion)
Exam 2 Wednesday, November 13th (in discussion)
Final Exam TBA TBA
NOTE: You will NOT be allowed to wear watches (smart or dumb) during your exams.
Quizzes: There will be weekly quizzes in discussion section that will cover material from the week before. Content for each quiz will be announced ahead of time by your TA. Note that you will NOT
be allowed to wear watches (smart or dumb) during your quizzes.
Worksheets: There are weekly worksheets that we recommend that you complete as you engage with the lecture material. Worksheets are not required to be turned in and answers will be posted online.
All worksheets can be found in the Learning Activities section of this website.
│ Quizzes │ Weekly │ Discussion │ 20% │
│ Homework │ WebAssign │ Online and Lecture │ 10% │
│ Exam 1: │ Week 6 │ Wednesday discussion │ 20% │
│ Exam 2: │ Week 12 │ Wednesday discussion │ 20% │
│ Final Exam: │ TBA │ │ 30% │
Exam Replacement Policy: Your score on the final exam will replace your lowest exam score if you score higher on the final exam.
NOTE: The online section and honors sections will have slightly different grading schemes which are detailed in the respective syllabi. | {"url":"https://courses.math.uconn.edu/fall2019/math-1132/","timestamp":"2024-11-08T08:17:56Z","content_type":"text/html","content_length":"57495","record_id":"<urn:uuid:95c604f5-e611-4495-a9e8-d019f8337244>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00587.warc.gz"} |
Carrie, a packaging engineer, is designing a container to hold -Turito
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Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.
If the length and width of the container base in the initial sketch were doubled, at most how many more glasses could the new container hold?
We are given the dimensions of the base of the container and the number of glasses it can hold. We need to find the number of glasses it will hold when length and breadth are doubled. No matter what
the container is like, the area on the base covered by the glass is always the same. We use this fact to find the area covered by the glass initially and then use it to find the number of glasses in
the new container.
The correct answer is: 48
Length of the container = 12 inches
Breadth of the container = 9 inches
Number of glasses stored in the container = 12
Area of the base of the container =
Thus, the apparent area taken by each glass is given by
That is,
Area covered by one glass in the base of the container = 9 sq. inches
When the length and breadth of the base is doubled,
New length of the base =
New breadth of the base =
Thus, new area of the base of the container =
= 432 square inches
Hence, the number glasses this new container can hold is given by
Thus, the container with doubled length and breadth of the base can hold 48 glasses.
A few simple ideas are used in solving this problem, like, area of a rectangle is given by the product of its length and breadth and the basic idea of division.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/carrie-a-packaging-engineer-is-designing-a-container-to-hold-12-drinking-glasses-shaped-as-regular-octagonal-pris-q5f3b6a58","timestamp":"2024-11-09T07:06:44Z","content_type":"application/xhtml+xml","content_length":"312816","record_id":"<urn:uuid:71eddac6-c89c-43fd-bab0-981270d3b0ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00021.warc.gz"} |
Proving Between the Lines
Laureates of mathematics and computer science meet the next generation
Unsolved puzzles, unproven conjectures, and open-ended questions are fascinating to mathematicians – both because they present the tantalising possibility of solution, and because they remind us
about how much we don’t already know. Much of the maths people learn at school was completed hundreds of years ago – and it’s easy to forget just how close the frontiers of human knowledge can be.
One example I came across recently was a problem called the ‘no-three-in-line’ problem – a question from 2-dimensional geometry, regarding points and lines in a flat plane.
Given an n-by-n grid of equally spaced dots, can you pick 2n dots so that no three of the dots lie in a straight line?
Some disallowed sets of dots and the lines they lie on
Introduced in 1900, it has been called “one of the oldest and most extensively studied geometric questions concerning lattice points”, and it’s a simple enough problem to state, but actually leads to
some unanswered questions.
This question is carefully stated – if our grid measures n by n, it would be impossible to place more than 2n dots without three of them lying in the same row or column, by the pigeonhole principle;
once each row contains 2 dots, there’s nowhere else a third dot can be placed without placing three in that line. But given the restriction of 2n, can we find a solution? And, how many possible
solutions are there?
It’s usually instructive to start with smaller examples and work our way up. The line limitation is always ‘no three in a line’, regardless of the size of grid, and the number of dots is 2n for an
n-by-n grid. A 1-by-1 grid is outside of the bounds of this problem (mainly since it doesn’t contain 2n = 2 dots we could choose!) but a 2-by-2 grid is our basic trivial example – I can highlight all
four dots, and I’ll never find three in a straight line.
Once I get to a 3-by-3 grid, there’s more of a proper challenge – we now have to avoid three in a horizontal or vertical line, or three on a 45-degree diagonal. There’s actually only one way to fit
in 6 dots with these restrictions – up to rotation and reflection – and I won’t spoil the answer in case you want to work it out for yourself.
Expanding to a 4-by-4 grid, now trying to fit 8 dots, makes for more of a challenge. We still have to watch out for the 45-degree diagonals, and now there are three lines in each direction we can
make such a diagonal on.
There are four different solutions to this (up to rotation and reflection); two are shown below.
As we increase the size of the grid further, the problem evolves. Once we get to a 5-by-5 grid on which we have to place 10 dots, we find another diagonal comes into play – the marked dots can form a
straight line of three, so this is another possibility we need to exclude.
As the grid gets bigger, and the number of possible arrangements gets bigger very quickly, we find that more and more restrictions – more directions in which three dots can lie on a straight line –
creep in, which makes the problem harder.
This suggests that if these additional diagonals become common enough, at some point, placing 2n points might no longer be possible. 2n points can be placed on an n-by-n grid with no three in a line
for all n up to 46, and for n=48, 50 and 52.
Solution for n=52, the current record
For any size of grid, algorithms have developed that allow 1.5 × n points to be placed without making a line of three. But solutions that place 2n points haven’t been found for any larger n than 52,
nor has anyone found a way to place 47, 49 or 51 points.
It’s conjectured that as n increases, the number of points it’s possible to place gets smaller – and the limit is conjectured to be π/√3 (about 1.8 × n). This remains conjecture, though, as nobody
has managed to prove it, and the exact maximum number of points that can be placed for a given value of n is not known.
Even though at smaller scales this is a fun puzzle even children can play with, if you increase the parameters far enough it becomes an open problem in mathematics. The no-three-in-line problem has
connections to questions in graph theory and discrete geometry, and solutions for this problem can be used in finding solutions to others, including the Heilbronn Triangle Problem.
It’s also nice to have challenges like this to test our computational and mathematical skills, and I hope mathematicians continue to tinker with unsolved problems – even if they seem unrelated or
trivial – and push the boundaries of what we know about the universe.
1 comment
1. sehr interessant danke! | {"url":"https://scilogs.spektrum.de/hlf/proving-between-the-lines/","timestamp":"2024-11-12T10:53:42Z","content_type":"text/html","content_length":"79760","record_id":"<urn:uuid:bdcea60c-c8dc-472b-baee-e7355b328edd>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00649.warc.gz"} |
Souffle Herbie: Hacking Rationals into Datalog to Estimate Float Errors
Herbie is a system to automatically improve the accuracy and or speed of floating point calculations.
An interesting idea was suggested by Zach Tatlock about emulating a very, very tiny subset of Herbie in datalog by using exact rationals.
Calculating Errors
Different mathematically equivalent ways of calculating a quantity give slightly (or vastly) different answers when computed using floating point.
Even if you take as a given that library authors and hardware implementors implemented things like sin, cos, + accurately (not at all a given btw), that does not guarantee that compositions of the
building blocks are accurate.
For example, mathematically the two following expressions are the same: (1e30 - 1e30) + 1 = (1e30 + 1) - 1e30 However in the latter, adding 1 to such a huge number does not change the floating point
representation. It gets swamped in the finite precision of the floating point, so if you put these expressions into python you get pretty different answers 1.0 = 0.0. Not so good.
How can we tell how accurate a calculation is?
1. Manual mathematical analysis
2. In some cases, the answer can be computed exactly. Rational and algebraic numbers
3. Interval arithmetic and exact real libraries deliver error bounds with the calculation.
I’ll note a simple heuristic (but not foolproof) method is to calculate in a lower and higher accuracy (say 32 and 64 bit floats) and see if things look fishy.
However, we don’t want to calculate just a single number, we want a good representation for functions that are accurate for many values of a variable x. Again, you can
1. Use sophisticated pen and paper mathematical analysis
2. Interval arithmetic (using this simplistically will give you a very large over approximation of the error). You can tile or pave the domain to shrink this error. See also taylor models and tubes
A good heuristic is you can just sample points and see how they do using one of the point error methods.
There are two halves to the Herbie solution: generating equivalent mathematical expressions, and evaluating their accuracy to pick the best ones.
To generate candidates we can use equational rewrite rules. These rewrite rules might encode clever gleaned tricks from the numerical computing literature, significant domain knowledge. A big part of
the special sauce is figuring out what rewrite rules to have.
We want to maintain a collection of equivalent expressions, so destructive rewriting is not ideal. E-graphs are a efficient compact data structure for storing many expressions and equivalences
between them. I’m not there yet.
Given these candidate expressions we have a reasonable means to estimate their relative accuracy: Calculate “true” answer in rationals and sample domain points.
How can we inject GNU multiprecision rationals into Souffle? The interface presented to the user of the library is opaque pointer types. I have toyed with just storing pointer values in 64bit souffle
when I was binding Z3. Scary stuff. It doesn’t really work here.
Part of datalog’s thing is it needs to know when two things are equal. If I put 7/4 into the relation foo multiple times, that should reduce to only one entry. Z3 internally hashconses expressions.
GMP does not. We could perhaps use the pointers of the returned gmp values if we could overload hashing and equality. Actually, I could subsume possibly to remove duplicates. But then we’d also have
a memory leak.
As an inefficient but simple cheat, we can hash cons these numbers by serializing to and from souffle’s built in symbol datatype, which is basically a string. Strings are the ultimate universal type
and serialization/deserialization is packing and unpacking to this type. GMP uniquely normalizes and prints rationals.
Here are the souffle side stubs for dealing with mpq. It is a subtype of symbol, which is sent over FFI to C++ as a string. We also need some convenience functions for converting to and from souffle
.pragma "libraries" "gmpstubs"
.type mpq <: symbol
.functor float_of_mpqs(mpq):float
.functor mpqs_neg(mpq):mpq
.functor mpqs_abs(mpq):mpq
.functor mpqs_inv(mpq):mpq
.functor mpqs_add(mpq, mpq):mpq
.functor mpqs_sub(mpq, mpq):mpq
.functor mpqs_mul(mpq, mpq):mpq
.functor mpqs_div(mpq, mpq):mpq
.functor mpqs_cmp(mpq, mpq):number
#if RAM_DOMAIN_SIZE == 32
.functor mpqs_of_float(float):mpq
#define Q(x) @mpqs_of_float(x)
#elif RAM_DOMAIN_SIZE == 64
.functor mpqs_of_double(float):mpq
#define Q(x) @mpqs_of_double(x)
#error Unsupported RAM_DOMAIN_SIZE
#define QGT(x,y) (@mpqs_cmp(x,y) > 0)
#define QGTE(x,y) (@mpqs_cmp(x,y) >= 0)
#define QLT(x,y) (@mpqs_cmp(x,y) < 0)
#define QLTE(x,y) (@mpqs_cmp(x,y) <= 0)
// but actually regular = will work
#define QEQ(x,y) (@mpqs_cmp(x,y) == 0)
Here is one of the example stubs for addition. We need to
1. Make some GMP objects
2. Deserialize them from strings
3. Compute the actual addition
4. Serialize the result to string
5. Cleanup memory allocation
const char* mpqs_add(const char* x, const char* y){
mpq_t x1, y1, z1;
mpq_add(z1, x1, y1);
char* res = mpq_get_str(NULL,10,z1);
return res;
Souffle Herbie
Nothing too clever is happening here yet since I am not yet using an egraph. I am doing simple term rewriting using Souffle adts. It is mostly an exercise in encoding concepts to datalog. The idiom
of making a relation of all subexpressions is one I’ve encountered before. It is a relative of the magic set transformation, like many things. I’m not sure it is worth pursuing this here further as I
need to be putting less effort into horrible souffle encodings and more effort into pushing towards a shared goal.
#include "gmp.dl"
.type Expr = Lit {n : float} | Add {x : Expr, y : Expr} | X {} | Mul {x : Expr, y : Expr} // Div {x : expr, y : expr}
// Top level expression to rewrite
.decl top(x : Expr)
// Built table of all subexpressions
.decl term(x : Expr)
term(x) :- top(x).
term(x),term(y) :- term($Add(x,y)).
term(x),term(y) :- term($Mul(x,y)).
// An explicit equality relation over terms
// eqrel helps a little compared to n^2 naive
.decl eq(x : Expr, y : Expr) eqrel
term(x) :- eq(x,_).
eq(x,x) :- term(x).
// Associativity
eq(t, $Add(x,$Add(y,z))) :- term(t), t = $Add($Add(x,y),z).
eq(t, $Add($Add(x,y),z)) :- term(t), t = $Add(x,$Add(y,z)).
// Commutativity
eq(t, $Add(y,x)) :- term(t), t = $Add(x,y).
eq(t, x) :- term(t), t = $Add($Lit(0),x).
// Literal Combination. Should these be mpq not float?
eq(t, $Lit(m + n)) :- term(t), t = $Add($Lit(n),$Lit(m)).
// Associativity
eq(t, $Mul(x,$Mul(y,z))) :- term(t), t = $Mul($Mul(x,y),z).
eq(t, $Mul($Mul(x,y),z)) :- term(t), t = $Mul(x,$Mul(y,z)).
// Commutativity
eq(t, $Mul(y,x)) :- term(t), t = $Mul(x,y).
// identity absorption
eq(t, x) :- term(t), t = $Mul($Lit(1),x).
// Distributivity
eq(t, $Add($Mul(x,y), $Mul(x,z))) :- term(t), t = $Mul(x,$Add(y,z)).
eq(t, $Mul(x,$Add(y,z))) :- term(t), t = $Add($Mul(x,y), $Mul(x,z)).
// Simple sampling [0,1]
#define NSAMP 10
.decl sample(samp : unsigned, x : float)
sample(s, to_float(s)/NSAMP) :- s = range(0,NSAMP).
// Evaluate float expressions a sample points
.decl eval(samp : unsigned, t : Expr, n : float)
eval(s, t, n) :- term(t), t = $Lit(n), sample(s, _).
eval(s, t, x) :- term(t), t = $X(), sample(s, x).
eval(s, t, nx + ny) :- term(t), t = $Add(x,y), eval(s,x,nx), eval(s,y,ny).
// Evaluate exact expressions at sample point
.decl exact(samp : unsigned, t : Expr, n : mpq)
exact(s,t,Q(n)) :- term(t), t = $Lit(n), sample(s,_).
exact(s,t,Q(x)) :- term(t), t = $X(), sample(s,x).
exact(s,t,@mpqs_add(nx,ny)) :- term(t), t = $Add(x,y), exact(s,x,nx), exact(s,y,ny).
// Calculate error
.decl err(samp : unsigned, t : Expr, err : float) // should error be mpq? probably but it makes minimum kind of annoying
err(s, t, @float_of_mpqs(e)) :- eval(s,t,x1), exact(s,t,x2), e = @mpqs_abs(@mpqs_sub(Q(x1), x2)).
// Choice-domain let's us pick a unique best even when there are multiple of equivalent error
.decl best(samp : unsigned, t : Expr, best_t : Expr, val : float, err : float) choice-domain (samp, t)
best(s, t, t1, val, be) :- top(t), s = range(0,NSAMP), be = min e: {err(s, t2, e), eq(t,t2)}, eq(t,t1), err(s,t1,be), eval(s,t1,val).
.output sample(IO=stdout)
.output exact(IO=stdout)
.output err(IO=stdout)
.output best(IO=stdout)
C++ Code
The GMP bindings
#include <gmp.h>
#include <iostream>
extern "C" {
// We're probably leaking memory associated with the strings.
const char* mpqs_of_float(float x){
mpq_t y;
mpq_set_d(y, x);
char* res = mpq_get_str(NULL,10,y);
if(res == NULL){
return "NULL";
return res;
float float_of_mpqs(const char* x){
mpq_t x1;
double z = mpq_get_d(x1);
return (float) z;
const char* mpqs_of_double(double x){
mpq_t y;
mpq_set_d(y, x);
char* res = mpq_get_str(NULL,10,y);
if(res == NULL){
return "NULL";
return res;
float double_of_mpqs(const char* x){
mpq_t x1;
double z = mpq_get_d(x1);
return z;
const char* mpqs_add(const char* x, const char* y){
mpq_t x1, y1, z1;
mpq_add(z1, x1, y1);
char* res = mpq_get_str(NULL,10,z1);
return res;
const char* mpqs_sub(const char* x, const char* y){
mpq_t x1, y1, z1;
mpq_sub(z1, x1, y1);
char* res = mpq_get_str(NULL,10,z1);
return res;
const char* mpqs_mul(const char* x, const char* y){
mpq_t x1, y1, z1;
mpq_mul(z1, x1, y1);
char* res = mpq_get_str(NULL,10,z1);
return res;
const char* mpqs_div(const char* x, const char* y){
mpq_t x1, y1, z1;
mpq_mul(z1, x1, y1);
char* res = mpq_get_str(NULL,10,z1);
return res;
const char* mpqs_abs(const char* x){
mpq_t x1, z1;
mpq_abs(z1, x1);
char* res = mpq_get_str(NULL,10,z1);
return res;
const char* mpqs_neg(const char* x){
mpq_t x1, z1;
mpq_abs(z1, x1);
char* res = mpq_get_str(NULL,10,z1);
return res;
const char* mpqs_inv(const char* x){
mpq_t x1, z1;
mpq_abs(z1, x1);
char* res = mpq_get_str(NULL,10,z1);
return res;
int32_t mpqs_cmp(const char* x, const char* y){
mpq_t x1, y1;
int z = mpq_cmp(x1, y1);
return z;
Bits and Bobbles
This isn’t totally satisfactory, but there is enough here for an interesting post and I’m kind of stalled out. I’m a big believer in low standards for blog posts.
I suspect I am not handling both 32-64 bit souffle correctly. It’s like… confusing, man.
Zach showed me a good example but the photo is totally blurry. 1/(x+1) - 1/x ---> ?
An Aside: Datalog Modulo X
Max has invented a fun terminology.
There is a theme in hacking things into stock datalog. Stock datalog has set semantics, where it needs to determine if an item is already in the relations. Datalog also needs to search the database.
Both of these operations need to refer to a notion of equality, and possibly comparation and/or hashing. Datalog’s are not parametrized (at least not in an easily user accessible way) in the
mechanism by which they consider two items equal (maybe Ascent is since it uses rust traits?). Datalog + Lattices and/or subsumption are powerful enough you can kind of mimic this capability.
If you want to add in a capability of “datalog modulo X”, you need to find a way to uniquely embed X into one of the datatypes the datalog supports. This means canonizing X, which is not always easy
(or even possible?). Is it better to canonize X or to add a smart equality function / indexing data structures?
In a previous post I discussed first class sets. The canonization of sets in that case is removing duplicates and keeping the items sorted in the vector representation of the set. More trivially,
let’s say I wanted to support x mod 17 as a data type. I need to embed these items into number.
It is nice if the notion of comparation explained to datalog is semantically relevant. As a counterexample, consider a pretty printed string. The lexicographic order of the string is not at all
necessarily related to the order of the thing that was printed. I’m painfully aware of this every time my folders get sorted the wrong way in some directory (dates or numbers where we forgot to
prefix with enough 0000). Souffle supports strings by interning them to unique integers, so I don’t think you can do range queries over strings easily anyway.
Egglog itself is something like “datalog modulo uninterpreted functions”.
Something I’m been investigating is how to talk about bound variables. The standard methodologies for canonizing bound terms turn variables names into canonical numbers (de bruijn levels or indices).
See Hash Consing modulo Alpha for more about the issues here. For the thing I’ve been thinking about, the variables are really implicitly top level bound, and not bound in any particular order. A
theme seems to be to name them as numbers in the order they are encountered in a term traversal. In other words, the signature is a set, not a list. | {"url":"https://www.philipzucker.com/souffle-herbie/","timestamp":"2024-11-08T20:11:35Z","content_type":"text/html","content_length":"23375","record_id":"<urn:uuid:a49d1bd5-33db-4b20-a037-310bc4fbdd9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00200.warc.gz"} |
Financial Cryptography: Comment on Our Private Bayesian Rules Engine
I think it's the other way round: More computer programmers have discovered Bayesian learning and similar probability-based techniques. They have been in use in information retrieval for decades, and
Paul Graham popularized probabilty-based ideas only *after* these two researchers began working in that area.
Posted by Florian Weimer at January 7, 2006 05:07 PM
An Erlang distribution is a Poisson distribution with an integer h parameter. It is that simple.
Posted by Daniel A. Nagy at January 8, 2006 04:13 AM
From: "An Intuitive Explanation of Bayesian Reasoning
Bayes' Theorem
for the curious and bewildered;
an excruciatingly gentle introduction.
By Eliezer Yudkowsky" link, here ... http://yudkowsky.net/bayes/bayes.html
Scrolling down a bit, one gets to ...
"Here's a story problem about a situation that doctors often encounter:
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get
positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
What do you think the answer is? If you haven't encountered this kind of problem before, please take a moment to come up with your own answer before continuing."
The above isn't that intuitive!!!
The author shows how it can be made more intuitive, though, eg by re-phrasing the question and such ... Anyway, the link is well worth the read.
As an aside ... I wanted to reference the following ... http://www.amazon.com/gp/reader/0812975219/ref=sib_vae_pg_190/002-3703878-9607262?%5Fencoding=UTF8&keywords=side%20effect&p=S06C&twc=4&checkSum
=pz%2F9ZdXCBpyqZdIfID9ZQOng7RgEisixKBBl9uInnR4%3D#reader-page but Amazon wouldn't let me cut and paste! (A case of how to lose friends and influence no one?).
Posted by Darren at January 8, 2006 10:01 AM
Andrew Gelman has a critique of the Economist paper in his blog here ... http://www.stat.columbia.edu/~cook/movabletype/archives/2006/01/bayesian_parame.html#comments
Posted by Darren at April 19, 2006 05:37 AM | {"url":"https://financialcryptography.com/cgi-bin/mt/mt-comments.cgi?entry_id=632","timestamp":"2024-11-14T23:33:31Z","content_type":"application/xhtml+xml","content_length":"7882","record_id":"<urn:uuid:7288407a-d138-4f58-bdd3-ae550fdf553d>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00575.warc.gz"} |
CSC321 Programming Assignment 4: CycleGAN
In this assignment, you’ll get hands-on experience coding and training GANs. This assignment is
divided into two parts: in the first part, we will implement a specific type of GAN designed to
process images, called a Deep Convolutional GAN (DCGAN). We’ll train the DCGAN to generate
emojis from samples of random noise. In the second part, we will implement a more complex
GAN architecture called CycleGAN, which was designed for the task of image-to-image translation
(described in more detail in Part 2). We’ll train the CycleGAN to convert between Apple-style and
Windows-style emojis.
In both parts, you’ll gain experience implementing GANs by writing code for the generator,
discriminator, and training loop, for each model.
Important Note
We provide a script to check your models, that you can run as follows:
python model_checker.py
This checks that the outputs of DCGenerator, DCDiscriminator, and CycleGenerator match the
expected outputs for specific inputs. This model checker is provided for convenience only. It may
give false negatives, so do not use it as the only way to check that your model is correct.
The training scripts run much faster (∼ 5x faster) on the teaching lab machines if you add
MKL NUM THREADS=1 before the Python call, as follows:
MKL NUM THREADS=1 python cycle gan.py –load=pretrained cycle –train iters=100
Remember to add MKL NUM THREADS=1 before each Python command in this assignment.
CSC321 Programming Assignment 4
Part 1: Deep Convolutional GAN (DCGAN)
For the first part of this assignment, we will implement a Deep Convolutional GAN (DCGAN).
A DCGAN is simply a GAN that uses a convolutional neural network as the discriminator, and
a network composed of transposed convolutions as the generator. To implement the DCGAN, we
need to specify three things: 1) the generator, 2) the discriminator, and 3) the training procedure.
We will develop each of these three components in the following subsections.
Implement the Discriminator of the DCGAN [10%]
The discriminator in this DCGAN is a convolutional neural network that has the following architecture:
conv1 conv2 conv3 conv4
BatchNorm & ReLU BatchNorm & ReLU BatchNorm & ReLU
1. Padding: In each of the convolutional layers shown above, we downsample the spatial dimension of the input volume by a factor of 2. Given that we use kernel size K = 4 and stride
S = 2, what should the padding be? Write your answer in your writeup, and show your work
(e.g., the formula you used to derive the padding).
2. Implementation: Implement this architecture by filling in the __init__ method of the
DCDiscriminator class in models.py, shown below. Note that the forward pass of DCDiscriminator
is already provided for you.
def __init__(self, conv_dim=64):
super(DCDiscriminator, self).__init__()
## FILL THIS IN: CREATE ARCHITECTURE ##
# self.conv1 = conv(…)
# self.conv2 = conv(…)
# self.conv3 = conv(…)
# self.conv4 = conv(…)
Note: The function conv in models.py has an optional argument batch_norm: if batch_norm
is False, then conv simply returns a torch.nn.Conv2d layer; if batch_norm is True, then
conv returns a network block that consists of a Conv2d layer followed by a torch.nn.BatchNorm2d
layer. Use the conv function in your implementation.
CSC321 Programming Assignment 4
Generator [10%]
Now, we will implement the generator of the DCGAN, which consists of a sequence of transpose
convolutional layers that progressively upsample the input noise sample to generate a fake image.
The generator we’ll use in this DCGAN has the following architecture:
deconv1 deconv2 deconv3 deconv4
BatchNorm & ReLU tanh
BatchNorm & ReLU BatchNorm & ReLU
1. Implementation: Implement this architecture by filling in the __init__ method of the
DCGenerator class in models.py, shown below. Note that the forward pass of DCGenerator
is already provided for you.
def __init__(self, noise_size, conv_dim):
super(DCGenerator, self).__init__()
## FILL THIS IN: CREATE ARCHITECTURE ##
# self.deconv1 = deconv(…)
# self.deconv2 = deconv(…)
# self.deconv3 = deconv(…)
# self.deconv4 = deconv(…)
Note: Use the deconv function (analogous to the conv function used for the discriminator
above) in your generator implementation.
Training Loop [15%]
Next, you will implement the training loop for the DCGAN. A DCGAN is simply a GAN with a
specific type of generator and discriminator; thus, we train it in exactly the same way as a standard
GAN. The pseudo-code for the training procedure is shown below. The actual implementation is
simpler than it may seem from the pseudo-code: this will give you practice in translating math to
1. Implementation: Open up the file vanilla_gan.py and fill in the indicated parts of the
training_loop function, starting at line 149, i.e., where it says
# FILL THIS IN
# 1. Compute the discriminator loss on real images
# D_real_loss = …
CSC321 Winter 2018 Programming Assignment 4
Algorithm 1 GAN Training Loop Pseudocode
1: procedure TrainGAN
2: Draw m training examples {x
(1), . . . , x(m)} from the data distribution pdata
3: Draw m noise samples {z
(1), . . . , z(m)} from the noise distribution pz
4: Generate fake images from the noise: G(z
) for i ∈ {1, . . . .m}
5: Compute the (least-squares) discriminator loss:
(D) =
) − 1
6: Update the parameters of the discriminator
7: Draw m new noise samples {z
(1), . . . , z(m)} from the noise distribution pz
8: Generate fake images from the noise: G(z
) for i ∈ {1, . . . .m}
9: Compute the (least-squares) generator loss:
(G) =
)) − 1
10: Update the parameters of the generator
There are 5 numbered bullets in the code to fill in for the discriminator and 3 bullets for the
generator. Each of these can be done in a single line of code, although you will not lose marks
for using multiple lines.
Experiment [10%]
1. Train the DCGAN with the command:
python vanilla_gan.py –num_epochs=40
By default, the script runs for 40 epochs (5680 iterations), and should take approximately 30
minutes on the teaching lab machines (it may be faster on your own computer). The script
saves the output of the generator for a fixed noise sample every 200 iterations throughout
training; this allows you to see how the generator improves over time. Include in your
write-up one of the samples from early in training (e.g., iteration 200) and one
of the samples from later in training, and give the iteration number for those
samples. Briefly comment on the quality of the samples, and in what way they
improve through training.
Part 2: CycleGAN
Now we are going to implement the CycleGAN architecture.
Motivation: Image-to-Image Translation
Say you have a picture of a sunny landscape, and you wonder what it would look like in the rain. Or
perhaps you wonder what a painter like Monet or van Gogh would see in it? These questions can be
CSC321 Winter 2018 Programming Assignment 4
addressed through image-to-image translation wherein an input image is automatically converted
into a new image with some desired appearance.
Recently, Generative Adversarial Networks have been successfully applied to image translation,
and have sparked a resurgence of interest in the topic. The basic idea behind the GAN-based
approaches is to use a conditional GAN to learn a mapping from input to output images. The loss
functions of these approaches generally include extra terms (in addition to the standard GAN loss),
to express constraints on the types of images that are generated.
A recently-introduced method for image-to-image translation called CycleGAN is particularly
interesting because it allows us to use un-paired training data. This means that in order to train
it to translate images from domain X to domain Y , we do not have to have exact correspondences
between individual images in those domains. For example, in the paper that introduced CycleGANs,
the authors are able to translate between images of horses and zebras, even though there are no
images of a zebra in exactly the same position as a horse, and with exactly the same background,
Thus, CycleGANs enable learning a mapping from one domain X (say, images of horses) to
another domain Y (images of zebras) without having to find perfectly matched training pairs.
To summarize the differences between paired and un-paired data, we have:
• Paired training data: {(x
, y(i)
• Un-paired training data:
– Source set: {x
i=1 with each x
(i) ∈ X
– Target set: {y
j=1 with each y
(j) ∈ Y
– For example, X is the set of horse pictures, and Y is the set of zebra pictures, where
there are no direct correspondences between images in X and Y
Emoji CycleGAN
Now we’ll build a CycleGAN and use it to translate emojis between two different styles, in particular, Windows ↔ Apple emojis.
Generator [20%]
The generator in the CycleGAN has layers that implement three stages of computation: 1) the first
stage encodes the input via a series of convolutional layers that extract the image features; 2) the
second stage then transforms the features by passing them through one or more residual blocks;
and 3) the third stage decodes the transformed features using a series of transpose convolutional
layers, to build an output image of the same size as the input.
The residual block used in the transformation stage consists of a convolutional layer, where the
input is added to the output of the convolution. This is done so that the characteristics of the
output image (e.g., the shapes of objects) do not differ too much from the input.
Implement the following generator architecture by completing the __init__ method of the
CycleGenerator class in models.py.
def __init__(self, conv_dim=64, init_zero_weights=False):
super(CycleGenerator, self).__init__()
CSC321 Winter 2018 Programming Assignment 4
conv deconv conv deconv
GXtoY GYtoX
[0, 1]
Does the generated
image look like it came
from the set of Windows
GYtoX GXtoY
## FILL THIS IN: CREATE ARCHITECTURE ##
# 1. Define the encoder part of the generator
# self.conv1 = …
# self.conv2 = …
# 2. Define the transformation part of the generator
# self.resnet_block = …
# 3. Define the decoder part of the generator
# self.deconv1 = …
# self.deconv2 = …
To do this, you will need to use the conv and deconv functions, as well as the ResnetBlock
class, all provided in models.py.
Note: There are two generators in the CycleGAN model, GX→Y and GY →X, but their implementations are identical. Thus, in the code, GX→Y and GY →X are simply different instantiations
of the same class.
CSC321 Winter 2018 Programming Assignment 4
conv1 conv2 deconv1 deconv2
BatchNorm & ReLU tanh
CycleGAN Generator
BatchNorm & ReLU BatchNorm & ReLU
Redidual block
BatchNorm & ReLU
CycleGAN Training Loop [20%]
Finally, we will implement the CycleGAN training procedure, which is more involved than the
procedure in Part 1.
Algorithm 2 CycleGAN Training Loop Pseudocode
1: procedure TrainCycleGAN
2: Draw a minibatch of samples {x
(1), . . . , x(m)} from domain X
3: Draw a minibatch of samples {y
(1), . . . , y(m)} from domain Y
4: Compute the discriminator loss on real images:
real =
) − 1)2 +
(DY (y
(j) − 1)2
5: Compute the discriminator loss on fake images:
fake =
(DY (GX→Y (x
)))2 +
(DX(GY →X(y
6: Update the discriminators
7: Compute the Y → X generator loss:
(GY →X) =
(DX(GY →X(y
)) − 1)2 + J
(Y →X→Y )
8: Compute the X → Y generator loss:
(GX→Y ) =
(DY (GX→Y (x
)) − 1)2 + J
(X→Y →X)
9: Update the generators
Similarly to Part 1, this training loop is not as difficult to implement as it may seem. There
is a lot of symmetry in the training procedure, because all operations are done for both X → Y
and Y → X directions. Complete the training_loop function in cycle_gan.py, starting from the
following section:
# ============================================
# ============================================
CSC321 Winter 2018 Programming Assignment 4
## FILL THIS IN ##
# Train with real images
# 1. Compute the discriminator losses on real images
# D_X_loss = …
# D_Y_loss = …
There are 5 bullet points in the code for training the discriminators, and 6 bullet points in total
for training the generators. Due to the symmetry between domains, several parts of the code you
fill in will be identical except for swapping X and Y ; this is normal and expected.
Cycle Consistency
The most interesting idea behind CycleGANs (and the one from which they get their name) is
the idea of introducing a cycle consistency loss to constrain the model. The idea is that when we
translate an image from domain X to domain Y , and then translate the generated image back to
domain X, the result should look like the original image that we started with.
The cycle consistency component of the loss is the mean squared error between the input
images and their reconstructions obtained by passing through both generators in sequence (i.e.,
from domain X to Y via the X → Y generator, and then from domain Y back to X via the Y → X
generator). The cycle consistency loss for the Y → X → Y cycle is expressed as follows:
(i) − GX→Y (GY →X(y
The loss for the X → Y → X cycle is analogous.
Implement the cycle consistency loss by filling in the following section in cycle_gan.py. Note
that there are two such sections, and their implementations are identical except for swapping X
and Y . You must implement both of them.
if opts.use_cycle_consistency_loss:
reconstructed_X = G_YtoX(fake_Y)
# 3. Compute the cycle consistency loss (the reconstruction loss)
# cycle_consistency_loss = …
g_loss += cycle_consistency_loss
CycleGAN Experiments [15%]
Training the CycleGAN from scratch can be time-consuming if you don’t have a GPU. In this part,
you will train your models from scratch for just 600 iterations, to check the results. To save training
time, we provide the weights of pre-trained models that you can load into your implementation. In
order to load the weights, your implementation must be correct.
1. Train the CycleGAN without the cycle-consistency loss from scratch using the command:
CSC321 Programming Assignment 4
python cycle_gan.py
This runs for 600 iterations, and saves generated samples in the samples_cyclegan folder.
In each sample, images from the source domain are shown with their translations to the right.
Include in your writeup the samples from both generators at either iteration 400 or 600, e.g.,
sample-000400-X-Y.png and sample-000400-Y-X.png.
2. Train the CycleGAN with the cycle-consistency loss from scratch using the command:
python cycle_gan.py –use_cycle_consistency_loss
Similarly, this runs for 600 iterations, and saves generated samples in the samples_cyclegan_cycle
folder. Include in your writeup the samples from both generators at either iteration 400 or
600 as above.
3. Now, we’ll switch to using pre-trained models, which have been trained for 40000 iterations.
Run the pre-trained CycleGAN without the cycle-consistency loss using the command:
python cycle_gan.py –load=pretrained –train_iters=100
You only need 100 training iterations because the provided weights have already been trained
to convergence. The samples from the generators will be saved in the folder
samples_cyclegan_pretrained. Include the sampled output from your model.
4. Run the pre-trained CycleGAN with the cycle-consistency loss using the command:
python cycle_gan.py –load=pretrained_cycle \
–use_cycle_consistency_loss \
The samples will be saved in the folder samples_cyclegan_cycle_pretrained. Include
the final sampled output from your model.
5. Do you notice a difference between the results with and without the cycle consistency loss?
Write down your observations (positive or negative) in your writeup. Can you explain these
results, i.e., why there is or isn’t a difference between the two?
What you need to submit
• Three code files: models.py, vanilla_gan.py, and cycle_gan.py.
• A PDF document titled a4-writeup.pdf containing samples generated by your DCGAN and
CycleGAN models, and your answers to the written questions.
CSC321 Programming Assignment 4
Further Resources
For further reading on GANs in general, and CycleGANs in particular, the following links may be
1. Unpaired image-to-image translation using cycle-consistent adversarial networks (Zhu et al.,
2. Generative Adversarial Nets (Goodfellow et al., 2014)
3. An Introduction to GANs in Tensorflow
4. Generative Models Blog Post from OpenAI
5. Official PyTorch Implementations of Pix2Pix and CycleGAN | {"url":"https://codingprolab.com/answer/csc321-programming-assignment-4-cyclegan/","timestamp":"2024-11-02T15:43:01Z","content_type":"text/html","content_length":"141933","record_id":"<urn:uuid:412fad98-f999-4057-aa7c-11e4a2a4782b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00255.warc.gz"} |
How do I evaluate the indefinite integral
intx*sin(x)*tan(x)dx ? | HIX Tutor
How do I evaluate the indefinite integral #intx*sin(x)*tan(x)dx# ?
Answer 1
Integration by parts can be used to evaluate this integral.
Firstly, integrating by parts where #u=x and dv=sin(x)dx#, we get #int xsin(x)dx=-x cos(x)+sin(x)+c#
In the same way, for #u=x and dv=cos(x)dx#, we have
#int xcos (x)dx=x sin (x)-sin (x)+c#
Now, using integration by parts again with #u=tan(x) and dv=x sin (x)dx# and the first integral we have
#int x sin(x) tan(x)dx=# #=-x tan(x)cos(x)+sin(x)tan(x)+int xcos(x)sec^2(x)dx-int sin(x)sec^2(x)dx# #=-xsin(x)+sin(x)tan(x)+int xcos(x)dx-int sin(x)/cos^2(x)dx# #=-xsin(x)+sin(x)tan(x)+x sin (x)-sin
(x)-int sin(x)/cos^2(x)dx# #=sin(x)tan(x)-sin (x)-int sin(x)/(cos^2(x))dx# Finally, using substitution #u=cos(x), du=-sin(x) dx# in the last integral we have
#int x sin(x) tan(x)dx=sin(x)tan(x)-sin (x)+int 1/u^2du # #=sin(x)tan(x)-sin (x)-1/3cos^3(x)+c.#
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Answer 2
To evaluate the indefinite integral ∫x*sin(x)*tan(x)dx, you can use integration by parts. Let u = x and dv = sin(x)*tan(x)dx. Then, differentiate u to get du/dx = 1 and integrate dv to get v = -ln|
Apply the integration by parts formula: ∫u dv = uv - ∫v du.
This yields: ∫x*sin(x)tan(x)dx = -xln|cos(x)| - ∫(-ln|cos(x)|)dx.
The integral of -ln|cos(x)| can be evaluated separately. Let w = -ln|cos(x)| and dz = dx. Then, differentiate w to get dw/dx = -tan(x) and integrate dz to get z = x.
Apply integration by parts again: ∫(-ln|cos(x)|)dx = -xln|cos(x)| - ∫x(-tan(x))dx.
This gives: ∫xsin(x)tan(x)dx = -xln|cos(x)| + ∫xtan(x)dx.
Now, you need to evaluate ∫x*tan(x)dx. This integral can be solved using substitution. Let t = cos(x), then dt = -sin(x)dx.
Substituting t = cos(x) and dt = -sin(x)dx into the integral, you get: ∫x*tan(x)dx = ∫(-x)*dt.
Integrating with respect to t gives: ∫xtan(x)dx = -xt + ∫tdt.
This simplifies to: ∫xtan(x)dx = -xcos(x) + ∫cos(x)dt.
Finally, integrating ∫cos(x)dt gives: ∫xtan(x)dx = -xcos(x) + sin(x) + C,
where C is the constant of integration. Therefore, the solution to the indefinite integral is: ∫xsin(x)tan(x)dx = -xln|cos(x)| - xcos(x) + sin(x) + C.
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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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Online Calculus Test | Hire Someone To Do Calculus Exam For Me
Online Calculus Testimonial A large amount of professional expression books have appeared online. I try to remember to do this once in a while because I know it is too important when it comes to time
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Dual Moving Average Crossover Trading Strategy
1. Dual Moving Average Crossover Trading Strategy
Dual Moving Average Crossover Trading Strategy
, Date: 2023-12-07 10:36:46
The Dual Moving Average Crossover trading strategy generates trading signals by calculating exponential moving averages (EMA) over different timeframes and detecting their crossover points. It
belongs to the category of trend-following strategies. This strategy utilizes 3 EMAs – 50-period, 144-period, and 200-period – to determine the market trend based on their crossover points and
produce trading signals. A buy signal is triggered when the faster EMA crosses above the slower EMAs. A sell signal is triggered when the faster EMA crosses below the slower EMAs. This strategy is
simple, practical and easy to automate.
Strategy Logic
1. Calculate the 50-period, 144-period, and 200-period EMA using the closing price, denoted as EMA50, EMA144, and EMA200 respectively.
2. If EMA50 crosses above EMA144 and EMA200 simultaneously, trigger a buy signal to open long positions.
3. If EMA50 crosses below EMA144 and EMA200 simultaneously, trigger a sell signal to close long positions.
Advantage Analysis
The Dual Moving Average Crossover strategy has the following advantages:
1. Simple and easy to understand. The parameters are intuitive and easy to implement for automation.
2. Responds quickly to trend changes and momentum shifts.
3. Customizable parameters allow adjusting the EMA periods to fit different market conditions.
4. Possesses some noise filtering capability to avoid being misled by short-term fluctuations.
5. Can be combined with other indicators to build systematic trading rules.
Risk Analysis
There are also some risks associated with this strategy:
1. Susceptible to generating false signals and being whipsawed by high volatility.
2. Cannot determine the duration of the established trend. Signals may come prematurely.
3. Inappropriate parameter tuning can lead to over-trading which increases transaction costs and slippage.
4. Can produce consecutive losses when trading in range-bound, choppy markets.
5. Lacks risk management mechanisms like stop-loss.
Optimization Directions
Some ways to optimize the Dual Moving Average Crossover Strategy include:
1. Adding filters based on other indicators like volume and volatility to reduce false signals.
2. Incorporating stop-loss strategies to control single-trade risks.
3. Optimizing EMA periods to adapt to different market timeframes.
4. Adding position sizing rules like fixed fractional allocation, pyramiding etc.
5. Utilizing machine learning models to dynamically optimize parameters.
The Dual Moving Average Crossover is a simple and practical trend-following strategy. It identifies trend directionality through EMA crosses and aims to capture opportunities along the
intermediate-to-long term trends. While easy to understand and implement, it suffers drawbacks like false signals and lack of risk controls. By introducing additional filters, stop losses, and
parameter optimization, it can be molded into a robust and efficient trading system. Overall, the strategy is well suited for automated trend trading and remains one of the most basic building blocks
of algorithmic trading strategies.
start: 2023-11-29 00:00:00
end: 2023-12-06 00:00:00
period: 1m
basePeriod: 1m
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
// This source code is subject to the terms of the Mozilla Public License 2.0 at https://mozilla.org/MPL/2.0/
// © SDTA
strategy("EMA Crossover Strategy", overlay=true)
// Hareketli Ortalamaları Hesapla
ema50 = ta.ema(close, 50)
ema144 = ta.ema(close, 144)
ema200 = ta.ema(close, 200)
// Al sinyali koşulu: Fiyat EMA 50, EMA 144 ve EMA 200 üzerine çıktığında
longCondition = close > ema50 and close > ema144 and close > ema200
// Sat sinyali koşulu: Fiyat EMA 200, EMA 144 ve EMA 50 altına indiğinde
shortCondition = close < ema200 and close < ema144 and close < ema50
// Al ve Sat sinyallerinin gerçekleştiği çubuğu ok ile belirt
plotarrow(series=longCondition ? 1 : shortCondition ? -1 : na, colorup=color.green, colordown=color.red, offset=-1, title="Trade Arrow")
// Hareketli Ortalamaları Çiz
plot(ema50, color=color.blue, title="EMA 50")
plot(ema144, color=color.orange, title="EMA 144")
plot(ema200, color=color.red, title="EMA 200")
// Strateji testi ekleyin
strategy.entry("AL", strategy.long, when=longCondition)
strategy.entry("SAT", strategy.short, when=shortCondition) | {"url":"https://www.fmz.com/strategy/434523","timestamp":"2024-11-08T15:56:44Z","content_type":"text/html","content_length":"13318","record_id":"<urn:uuid:c27d356f-4fd1-4065-8f9e-4b9c899de6ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00689.warc.gz"} |
Complete Bibliography
Works connected to [S:Gennady Markovich Khenkin:S]
Filter the Bibliography List
: “Supersimmetriq i kompleksnaq geometriq” [Supersymmetry and complex geometry], pp. 247–284 in Complex analysis: Several variables, III. Edited by G. M. Khenkin and R. V.
Gamkrelidze. Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya 9. VINITI (Moscow), 1986. An English translation was published in 1989. MR 860614 incollection
: “Supersymmetry and complex geometry,” pp. 223–261 in Several complex variables, III: Geometric function theory. Edited by G. M. Khenkin and R. V. Gamkrelidze. Enclyclopedia of
the Mathematical Sciences 9. Springer (Berlin), 1989. English translation of Russian original published in 1986. Zbl 0794.53050 incollection | {"url":"https://celebratio.org/Schwarz_Albert/bibf/270/940/6282/","timestamp":"2024-11-13T15:58:57Z","content_type":"text/html","content_length":"32181","record_id":"<urn:uuid:202df38f-912b-404a-a7b4-60dcee909738>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00588.warc.gz"} |
how to make 12 voltage amplifier circuit? - Electronics Help Care
12 voltage amplifier using transistor TIP41 and TIP42
This is a mini amplifier circuit diagram. this circuit run from 10 voltage to 18 voltage. normally we can use 12 voltages. we can use 12 voltage battery also. here used 2 transistors only. tip41 is
an NPN transistor and Tip42 is a PNP transistor. it’s one pair transistor that can make a maximum of 50 watts. it also depends on amplifier voltage and amperes.
We know that Voltage X amperes = Watts. so if we use 12 voltage and 2 amperes then we will get. 12 X 2 = 24 watts. or if we use 12 voltage 4 amperes then we get 12 X 3 = 36 watts. but we have to
know we can’t use 4 amperes. because 2 transistor maximum can take 3 amperes. we can use 18 voltage 3 amperes. then 18 X 3 = 54 watts.
If you need another amplifier circuit diagram please visit (Diagram)
If you want another post then please visit our website.
How to make an amplifier circuit diagram
We have another post for you. like repairing amplifiers,
If you like electronics please visit this site | {"url":"https://electronicshelpcare.net/how-to-make-12-voltage-amplifier-circuit/","timestamp":"2024-11-02T05:21:28Z","content_type":"text/html","content_length":"63856","record_id":"<urn:uuid:8f959033-afcf-4995-954e-ac0b1e5cb6e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00636.warc.gz"} |
Derivatives Trading Strategies — What Is It? (Backtest, Examples, and Insights) - QuantifiedStrategies.com
Derivatives Trading Strategies — What Is It? (Backtest, Examples, and Insights)
Since its birth in the 1980s, derivatives trading has opened up a world of markets for traders who want to profit from the price movements of various derivatives. To succeed in this market, you must
have a derivatives trading strategy. But what is a derivative?
Derivatives trading strategy is the technique traders use in buying and selling derivative contracts. In the world of financial trading, a derivative is a financial contract that derives its value
from the performance of an underlying asset, which can be a commodity, stock, currency, or index.
In this post, we take a look at derivatives trading. At the end of the article, we provide you with a derivative trading strategy (it’s actually two so it’s derivative trading strategies).
What is a derivative?
In the world of financial trading, a derivative is a financial contract that derives its value from the performance of an underlying asset, which can be a commodity, stock, currency, an index, or an
interest rate. The underlying asset’s value keeps changing according to market conditions, and the derivative’s price fluctuates accordingly to reflect the changes.
Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. There are also CFDs
(contracts for difference), which are contracts between a broker and a trader to exchange the difference in the price of an underlying asset between the time a trade is opened and the time it is
closed. These contracts can be used to trade any number of assets and carry their own risks.
Derivative trading seems to have a long history. In the ancient Greek civilization, Aristotle documented what has been considered the oldest example of a derivative in history — a contract
transaction of olives entered into by ancient Greek philosopher Thales. According to the report, Thales made a profit in the exchange.
Derivatives can be used for a number of purposes: they can be used to hedge against price movements, speculate on price movements, or get access to otherwise hard-to-trade assets or markets.
Essentially, derivatives are used to move risk (and the accompanying rewards) from the risk-averse to the risk seekers. While derivatives are mostly traded on central exchanges (such as the Chicago
Mercantile Exchange) or over-the-counter (OTC) marketplaces, the ones that trade on the OTC markets tend to have a greater risk than the derivatives traded over exchanges.
Types of derivatives
Below is a breakdown of the main types of derivatives:
Futures contracts
Futures are contracts that are traded on an exchange, which gives the parties involved the obligation to buy or sell a given quantity of an asset at a predetermined price at a specified time in the
future. Futures are similar to forwards or in fact, an evolution of the latter — futures contracts are standardized and traded on exchanges, so they are subject to a daily settlement procedure.
Options contracts
Options give the buyer the right — not an obligation — to buy or sell an asset to the other at a future date at an agreed strike price. There are two types: call option (if it gives the right to buy
an asset) and put option (if it gives the right to sell an asset).
We have yet to cover options strategies on this website (coming later), but we have touched upon a couple of examples of how they can be combined with equity trading:
Forward contracts
Similar to futures, a forward contract is an agreement to trade an asset at an agreed price on a future date. The contract is settled on the agreed future date when the buyer pays for and receives
the asset from the seller at the agreed price. The terms of the contract are privately agreed upon between the parties involved.
Swaps are contracts between two parties to exchange one another’s cash flow or a variable attached to various assets. There are different types of swaps: interest rate swaps, currency swaps, and
commodity swaps.
Contracts for Difference (CFDs)
A CFD is a derivative financial contract that represents an agreement between a broker and a trader to exchange the difference in the price of an underlying asset between the time when a trade is
opened and the time when it is closed. There are many online CFD brokers, offering CFDs on almost every available asset. While it is easy to register and trade with them, choose carefully because
many are not regulated.
Is derivative trading profitable?
The act of buying and selling derivatives for whatever reason, but especially for speculative purposes, is known as derivative trading.
Derivatives trading – hedging
For most of the industry stakeholders and institutions that trade to hedge risks, the goal of trading derivatives is not to make profits, but to reduce their risks in other markets, such as the spot
commodity market, equity market, or bond market.
For example, a farmer who produces wheat may sell wheat futures long before harvest to secure demand for his product and lock in the sale of his product at a good price. Similarly, a trading
institution with a huge stock investment may buy stock put options to protect the downside of the investment.
Likewise, an oil producer who knows what his production is in 12 months’ time might want to sell some of that production today via derivatives. This is the reason why derivatives saw the day of light
in the first place.
Derivatives trading – speculation
While industry stakeholders, farmers, and producers may be trading derivatives to hedge risks, retail traders and some institutional traders trade the derivative markets to profit from price
movements. Some institutional traders also use derivatives, such as options, to hedge their exposure in the equity and bond markets. Most traders use it for speculative purposes, though.
The game is different for retail traders who come to the derivative markets to assume risks with the hope of making commensurate profits (aka speculation).
Their hopes of striking it rich are most of the time put to an end after only a short time. The reason why is that the derivatives market is a 100% zero-sum game. Some traders may make money but at
the expense of others who have lost money. An open derivatives contract must always have someone both long and short.
To better understand the odds of success (or not), we have summarized the statics for a variety of brokers and how many of their CFD traders that are making money (brokers are required by law to
publish :
• 58% of retail traders lose money when trading CFDs with Interactive Brokers.
• 65% of retail investor accounts lose money when trading CFDs with SaxoBank.
• 67% of retail investor accounts lose money when trading CFDs with eToro.
• 71% of retail CFD accounts lose money with CMC.
• 72% of retail CFD accounts lose money with 500Plus.
• 81% of retail investor accounts lose money when trading spread bets in IG
(You can read more about what percentage of traders fail.)
The likelihood of being consistently profitable trading derivatives is quite low, and here’s also why: the underlying asset’s value keeps changing according to market conditions, as it is exposed to
various market sentiments and other political, economic, and social changes. Retail traders, and also professional institutional traders, are liable to make cognitive trading biases in trading.
Moreover, some derivative products are poorly structured. An example of derivatives that were flawed in their construction and destructive in their nature are the infamous mortgage-backed securities
(MBS) that brought on the subprime mortgage meltdown of 2007 and 2008 — the impact was so huge that it triggered a global recession.
Derivative strategies examples
As we stated above, there are different types of derivative instruments available for trading. But the ones that are easily accessible to retail traders are futures, options, and CFDs. There are
strategies that are unique to each derivative market, and there are general strategies that work for all instruments. We will explore the market-specific strategies first and then the general
Futures-specific trading strategies
These are some strategies that are unique to futures trading:
• Bull Calendar Spread: With this strategy, you buy and sell futures contracts of the same underlying asset but with different expirations. You take a long position on the near-term expiry and a
short position on the long-term expiry because it is expected that the spread will widen in favor of the longs so you end up in profit.
• Bear Calendar Spread: This is the opposite of the bull calendar spread — you take a short position on the short-term contract and a long position on the long-term contract. Here, the expectation
is that the spread would widen in favor of short so you end up in profit.
Options-specific trading strategies
Many options strategies are specific for options trading, and here are some examples:
• Buy Call: Traders buy calls when they are bullish on the underlying with the hope of selling it higher.
• Buy Put: Traders buy puts when they are bearish on the underlying and hope to buy it at a lower price.
• Covered Call Strategy: Here, a trader buys an underlying asset in the spot market and sells a call of the same asset. This strategy is used by a trader who has a neutral-bullish bias. This
strategy is not optimal, in our opinion, because it offers limited rewards with unlimited risks.
• Married Put Strategy: Here, you buy a put option for stocks you already own or intend to buy. If you are bullish on a stock, you use this strategy to minimize the impact of a fall in prices.
General strategies
These are strategies that traders use in all markets, including equities, CFDs, and futures. There are many strategies around, but the common ones are usually classified into these three categories
• Trend-following strategies: These are built on the price potential to continue moving in its trend direction. The idea is to capture the big impulse swings in the direction of the trend, so these
strategies tend to give huge profits when you manage to get a good trade. However, trend-following strategies tend to have a poor winning rate.
• Momentum strategies: The strategies in this group tend to take a position when there is an accelerating movement in some direction. It often uses breakouts to capture momentum. An example is
Range expansion when you get a fast and strong movement during a day or period that is much larger than what is normal and you buy into the movement.
• Mean-reversion strategies: These strategies are based on the idea that the price has a long-term average, and it tends to revert to that average anytime it moves significantly away from it. With
indicators that show the average price and the overbought/oversold conditions, such as the Bollinger Bands, RSI, and so on, traders take long positions when the price is oversold and go short
when the price is overbought.
FAQ derivatives trading
Based on the number of e-mails we get we decided to make a FAQ to better address any issues about the derivatives trading strategy:
What are the main 4 types of derivatives? What are the most common derivatives?
The main four types of derivative contracts are:
• Options
• Futures contracts
• Forward contracts
• Swaps
Is derivative trading difficult?
Hell, yes! Anyone telling you otherwise has no clue what they are talking about (or they are snake oil salesmen). Any speculating endeavor about the future is difficult.
What is the most important derivatives trading rule?
We quote the main rule of Nassim Nicholas Taleb’s Barbell Strategy:
Better to miss a zillion opportunities than blow up once.
Most people don’t understand how to handle uncertainty. They shy away from small risks, and without realizing it, they embrace the big, big risk. Businessmen who are consistently successful have
the exact opposite attitude: Make all the mistakes you want, just make sure you’re going to be there tomorrow.
When should a person trade in derivatives?
You should ONLY trade when you are a) hedging, or b) speculating with a specific plan.
We don’t recommend speculating under any circumstances if you don’t have a backtested plan! A backtested plan is, of course, not foolproof, but we believe this is the most rational approach in any
speculating endeavor. If you are new to trading and backtesting, you might find our backtesting course useful.
Why is derivatives trading risky?
It’s risky because most traders use too much leverage. You’ll be fine if you structure your trading conservatively and have a margin of safety/error. Reread Taleb’s quote above, please.
Related reading: What is equity trading?
What is the best derivatives trading platform?
If you’re a retail trader, just like us, we believe Interactive Brokers is the best. We are using both IB and Tradestation, but we believe IB is the best one for derivatives.
Derivatives trading strategy (backtest and example)
Let’s go on to backtest a derivatives trading strategy with strict trading rules and settings.
Derivatives trading strategy backtest number 1
Trading Rules
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The simplest way to use derivatives for hedging is to hedge an equity portfolio. For example, if you have a diversified portfolio of stocks you can simply buy out-of-the-money puts on the S&P 500. A
put works exactly like insurance: you have the right, but not an obligation, to sell something at a future date (expiration) at a certain price (strike). If S&P 500 is trading at 4500, you can, for
example, sell put options expiring in 6 months with a strike of 4000 (thus you can sell at 4000, which, of course, only makes sense if S&P 500 is below that level).
But any insurance costs money. If you are to roll over every 6 months you must assume that most of the time you lose the entire option premium you are paying (end up worthless). However, when the
market suddenly drops a lot, you stand to make windfall profits. But overall, this is a strategy that is inferior to a buy and hold strategy in the long run.
The famous money manager Meb Faber has an ETF that offers tail risk protection based on buying puts. If you included that ETF in your portfolio the results would have been like this:
We have written a separate article about tail risk:
Derivatives trading strategy backtest number 2
Can you make consistent profits by writing (selling) puts? Evidence points out that implied volatility has exceeded realized volatility and thus options are richly prized. Because of this, a strategy
that sells puts might offer attractive risk-adjusted returns.
Oleg Bondarenko authored an article a few years back (An Analysis Of Index Option Writing With Monthly And Weekly Rollover) that did a backtest where he sold puts both monthly and weekly from 1990
until 2015.
The table below shows the implied volatility (the main determinant of the price of the options) and the actual volatility:
Is it possible to make money on the volatility difference? Oleg Bondarenko’s backtest showed that the compound return of the put strategy was 10.1% compared to S&P 500’s 9.8%. This is not a major
difference, also considering commissions, slippage, and taxes, but the returns came with much lower volatility (10.1% vs 15.2%). Thus, the Sharpe Ratio is much higher (0.67) vs 0.47 for S&P 500.
Historical performance should always be evaluated together with trading statistics and metrics.
List of trading strategies
This blog is more than 10 years old (we started in 2012). We have written more than 800 articles that you can read for free – please see our complete list of trading strategies that work. The
strategies are an excellent resource to help you get some trading ideas.
We have compiled the Amibroker code and logic in plain English for all these strategies (plain English is for Python trading). If you subscribe, you’ll get the code in this article (plus over 160
other ideas).
For a list of the strategies we have made please click on the green banner:
These strategies must not be misunderstood for the premium strategies that we charge a fee for:
Derivatives Trading Strategy – conclusion
We end the article with some advice:
Any derivatives trading strategy can be a useful trading tool, but used incorrectly it can also be lethal for your capital and portfolio. Use it wisely, and always backtest! Also, keep in mind that
you should never trade something you don’t fully understand. Derivatives are most likely a bit more complicated than stocks, for example, and thus you should not trade them unless you both know what
you are doing AND you have some indications this is a strategy where you have a trading edge.
Q: What is derivatives trading?
A: Derivatives trading involves the buying and selling of financial instruments, such as futures and options contracts, which derive their value from an underlying asset or security.
Q: What are options?
A: Options are financial contracts that give the buyer the right, but not the obligation, to buy (call option) or sell (put option) a specified asset at a predetermined price (strike price) within a
specific period of time (expiration date).
Q: What is an option strategy?
A: An option strategy is a predefined plan or approach used by options traders to achieve a particular investment objective. It involves the simultaneous buying and/or selling of multiple options
contracts to create a specific risk-reward profile.
Q: How can I trade derivatives?
A: To trade derivatives, you need to open a trading account with a brokerage firm that offers derivative trading. Once your account is set up, you can start trading derivatives through the platform
provided by the brokerage.
Q: What is the difference between a call and a put option?
A: A call option gives the holder the right to buy an underlying asset at a specified price within a specific period, while a put option gives the holder the right to sell an underlying asset at a
specified price within a specific period.
Q: What is a strike price and expiration date?
A: The strike price is the predetermined price at which the underlying asset can be bought or sold, while the expiration date is the date on which the option contract becomes void.
Q: What is a popular derivative trading strategy?
A: One popular derivative trading strategy is the “buying a call option” strategy, where an investor buys a call option with the expectation that the price of the underlying asset will increase in
the future.
Q: What is a spread trading strategy?
A: A spread trading strategy involves simultaneously buying and selling two or more options contracts with different strike prices or expiration dates to take advantage of market volatility and
potential price movements.
Q: How do I start trading derivatives?
A: To start trading derivatives, you need to educate yourself about the various strategies and market dynamics, open a trading account with a brokerage firm, and familiarize yourself with the trading
platform provided by the brokerage.
Q: How can I profit from options trading?
A: The profit potential in options trading depends on various factors, including the movement of the underlying asset’s price, the strike price of the options contracts, and the time remaining until
expiration. You can potentially profit from options trading by correctly predicting the direction of the underlying asset’s price movement and choosing options that align with your expectations. | {"url":"https://www.quantifiedstrategies.com/derivatives-trading-strategies/","timestamp":"2024-11-04T15:41:16Z","content_type":"text/html","content_length":"259393","record_id":"<urn:uuid:1e7e6ccf-e8bb-4dea-a9b4-f45d16fcfbc8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00013.warc.gz"} |
The volume of a gas is 0.400 L when the pressure is 2.00 atm. At the same temperature, what is the pressure at which the volume of the gas is 2.0 L? | Socratic
The volume of a gas is 0.400 L when the pressure is 2.00 atm. At the same temperature, what is the pressure at which the volume of the gas is 2.0 L?
1 Answer
In this question , the gas is kept under the same temperature. Assuming the amount of gas is constant we can apply Boyles' law.
As per the Boyle' law equation;
${P}_{1}$${V}_{1}$ = ${P}_{2}$${V}_{2}$
2 atm x 0.400 L = ${P}_{2}$ x 2.0 L
0.800 atm L = 2 ${P}_{2}$ L
dividing both the sides by 2 we get,
0.400 atm = ${P}_{2}$
Impact of this question
3787 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/the-volume-of-a-gas-is-0-400-l-when-the-pressure-is-2-00-atm-at-the-same-tempera#418025","timestamp":"2024-11-10T18:22:51Z","content_type":"text/html","content_length":"33720","record_id":"<urn:uuid:1c96c024-0e70-49d2-ae77-a74429a4a393>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00892.warc.gz"} |
lasticity of
Price Elasticity of Demand
Price elasticity of demand, or demand elasticity, is a measure of how much the quantity demanded changes in response to price changes. In other words, it gauges consumers' sensitivity to price
fluctuations. Specifically, it shows how much less consumers are willing to buy when the price increases or, conversely, how much more they consume when the price decreases. It is expressed as a
percentage that indicates the proportional change in the quantity demanded resulting from a 1% change in price.
How is Price Elasticity of Demand Calculated?
It is calculated as the percentage change in quantity demanded divided by the percentage change in price:
\[ \text{Price Elasticity of Demand} = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}} \]
Since the percentage change in a variable is the change in the variable (final value minus initial value), divided by the initial value, the price elasticity of demand can be written as follows:
\[ \text{Price Elasticity of Demand} = \frac{\left( \frac{\text{Final Quantity} - \text{Initial Quantity}}{\text{Initial Quantity}} \right)}{\left( \frac{\text{Final Price} - \text{Initial Price}}{\
text{Initial Price}} \right)} \]
• \(\text{Final Quantity}\) is the quantity demanded after the price change.
• \(\text{Initial Quantity}\) is the quantity demanded before the price change.
• \(\text{Final Price}\) is the new price after the change.
• \(\text{Initial Price}\) is the original price before the change.
\[ E_d = \frac{\Delta Q / Q_i}{\Delta P / P_i} \]
• \(\Delta Q = \text{Final Quantity} - \text{Initial Quantity}\) is the absolute change in quantity demanded.
• \(\Delta P = \text{Final Price} - \text{Initial Price}\) is the absolute change in price.
• \(Q_i\) is the initial quantity.
• \(P_i\) is the initial price.
Next, we can simplify the formula by dividing the two fractions:
\[ E_d = \frac{\Delta Q}{Q_i} \div \frac{\Delta P}{P_i} \]
This is equivalent to multiplying the numerator by the inverse of the denominator:
\[ E_d = \frac{\Delta Q}{Q_i} \cdot \frac{P_i}{\Delta P} \]
Finally, generalizing for any quantity \( Q \) and price \( P \), we arrive at the final formula:
\[ E_d = \frac{\Delta Q}{\Delta P} \cdot \frac{P}{Q} \]
Where \( \frac{\Delta Q}{\Delta P} \) is the slope of the demand curve, and \( \frac{P}{Q} \) adjusts the relative change in terms of prices and quantities.
Example of Calculating Price Elasticity of Demand
Suppose we have the following two points on a demand curve:
• Point 1: \( (P_1 = 10, Q_1 = 100) \)
• Point 2: \( (P_2 = 8, Q_2 = 120) \)
We want to calculate the price elasticity of demand between these two points.
First, calculate the absolute change in quantity demanded (\( \Delta Q \)) and the absolute change in price (\( \Delta P \)):
\[ \Delta Q = Q_2 - Q_1 = 120 - 100 = 20 \]
\[ \Delta P = P_2 - P_1 = 8 - 10 = -2 \]
Next, apply these values to the price elasticity formula:
\[ E_d = \frac{\Delta Q}{\Delta P} \cdot \frac{P_1}{Q_1} \]
Substituting the values obtained:
\[ E_d = \frac{20}{-2} \cdot \frac{10}{100} = -10 \cdot 0.1 = -1 \]
The price elasticity of demand is \( E_d = -1 \), meaning that the demand is unitary elastic at this point on the curve. This indicates that a percentage change in price results in an equal but
opposite percentage change in quantity demanded.
Since the quantity demanded of a good typically moves in the opposite direction of its price, the percentage change in quantity has the opposite sign to the percentage change in price. Therefore,
price elasticity of demand is often expressed as a negative number, although sometimes it is given as an absolute value.
It is also worth noting that this formula is a simplified formula that assumes small changes in price and quantities; therefore, in most cases, other methods for calculating elasticity, such as the
midpoint method, are more accurate.
Determinants of Demand Elasticity
There is no universal rule or single determinant of the elasticity of a demand curve, as demand is influenced by economic, psychological, and social forces that shape consumer preferences. However,
several factors particularly affect elasticity:
Goods with close substitutes tend to have higher elasticity because consumers can easily replace one good with another. The existence of a close substitute means that price increases lead to reduced
purchases of the good in favor of the substitute. Conversely, when there are no close substitutes, demand tends to be more inelastic.Necessary goods tend to have lower elasticity compared to luxury
goods. Consumers find it easier to forgo or replace a luxury item than a necessity in response to price changes.
Demand tends to be more elastic in the long run because consumers have more time to adjust their consumption to price changes.The definition of the market also affects elasticity. Narrowly defined
markets tend to have more elastic demand than broadly defined markets. For example, the demand for fruit is likely to be less elastic than the demand for apples, as it is easier to find close
substitutes for apples, such as pears, than for all fruits.
Elastic, Inelastic, and Unitary Demand
When the price elasticity of demand is greater than one, the demand is elastic, meaning that the quantity demanded responds more than proportionally to price changes. A 1% change in price leads to a
greater than 1% change in quantity demanded.
When the price elasticity of demand is less than one, the demand is inelastic, meaning that quantity demanded responds less than proportionally to price changes. When the elasticity of demand equals
one, the demand is unitary elastic, meaning the percentage change in quantity demanded is exactly equal to the percentage change in price. To summarize:
Demand elasticity is classified as follows:
• Perfectly inelastic: \(E_d = 0\) - The quantity demanded does not change in response to price changes. The demand curve is vertical.
• Inelastic: \(0 < E_d < 1\) - Quantity demanded changes less than proportionally to price changes. A price increase causes a smaller reduction in quantity demanded.
• Unitary elasticity: \(E_d = 1\) - Quantity demanded changes in the same proportion as the price change. A price increase leads to an exactly proportional decrease in quantity demanded.
• Elastic: \(E_d > 1\) - Quantity demanded changes more than proportionally to price changes. A price increase causes a larger reduction in quantity demanded.
• Perfectly elastic: \(E_d = \infty\) - Quantity demanded changes infinitely in response to a small price change. The demand curve is horizontal. | {"url":"https://econfina.net/en/microeconomics/price-elasticity-of-demand","timestamp":"2024-11-10T14:09:17Z","content_type":"text/html","content_length":"16025","record_id":"<urn:uuid:41b03d41-379d-4bf6-9e49-d50d5a65e924>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00845.warc.gz"} |
TY - DATA T1 - Dataset to plot neutral stability curve for symmetric bifurcations PY - 2023/09/27 AU - Arya Iwantoro AU - Maarten van der Vegt AU - M.G. (Maarten) Kleinhans UR - https://data.4tu.nl/
articles/dataset/Dataset_to_plot_neutral_stability_curve_for_symmetric_bifurcations/16722949 DO - 10.4121/16722949.v1 KW - bifurcations KW - tides KW - stability KW - asymmetry N2 -
This is dataset to plot neutral stability curve for symmetric river bifurcation in Figure 6 (Iwantoro et al., subm) produced by Matlab.
There are three sets of data:
1. Shields stress vs w/h
This dataset is to produce the plot that indicates the stability threshold of the stability of symmetric bifurcation for different tidal influence in the range of ebb-averaged Shields stress and
width-to-depth ratio.
2. Shields stress vs u[2]u[0]
This dataset is to produce the plot that indicates the stability threshold of the stability of symmetric bifurcation for upstream channel width-to-depth ratio of 30, 50 and 70 in the range of ebb
averaged Shields stress and tidal influenced. The tidal influence is indicated by tidal flow amplitude of tide-averaged flow at bifurcation.
3. Asymmetry index vs shields stress
This dataset is to produce the plot to indicate the asymmetry of bifurcations in a range of Shields stress for different tidal influence.
ER - | {"url":"https://data.4tu.nl/export/refman/datasets/16722949","timestamp":"2024-11-10T00:04:02Z","content_type":"text/plain","content_length":"1634","record_id":"<urn:uuid:20d91536-6476-4869-8229-8ae246b6d51d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00043.warc.gz"} |
Dynamic characteristic analysis of two-stage quasi-zero stiffness vibration isolation system
A novel two-stage quasi-zero stiffness (QZS) vibration isolator was proposed for the purpose of low-frequency vibration isolation. Firstly, the dynamic model of the vibration isolation system was
established; furthermore, the force transmissibility of the system under harmonic force excitation was derived by the averaging method; finally, the effects on the vibration isolation performance
caused by excitation amplitude, mass ratio and damping ratio were discussed. Results show that, compared with the corresponding two-stage linear system, two-stage QZS system not only has better
isolation performance, but also possesses a wider range of isolation frequency provided that the excitation amplitude, mass ratio and damping ratio is appropriate.
1. Introduction
Quasi-zero stiffness (QZS) isolator can obtain zero stiffness at the static equilibrium position by connecting a positive stiffness element in parallel with a negative stiffness element [1]. By
reasonably selecting the geometry and stiffness parameters of the negative stiffness institutions, the vibration isolation system can support a large load statically while have low-frequency
vibration isolation performance. The combination of positive and negative stiffness device possesses the characteristics of high-static-low-dynamic stiffness and can ensure that the system natural
frequency is very low at small deformation [2]. Therefore, scholars have conducted in-depth research on isolator principle [3], structure design [4], properties analysis [5] and engineering
application [6] of QZS, which is widely used in precision instrument isolation, bridge and house anti-earthquake, high-speed vehicle vibration reduction, marine machinery noise cancellation and other
With the development of science and technology, the request of stability for the work environment of precision equipment is more and more strictly. Conventional single-stage linear or QZS vibration
isolation system cannot meet requirements of industry [7, 8]. When the excitation frequency is greater than $\sqrt{2}$ times the natural frequency, vibration decay ratio is proportional to ${\omega }
^{-2}$ in the single-stage system, while that is proportional to ${\omega }^{-4}$ in the two-stage system. So rigid two-stage isolation system can replace soft single-stage isolation system, which
not only has good isolation effect, but also takes into account the load-bearing capacity and stability. Although the two-stage isolator is better isolation performance than the single-stage
isolator, but the former has one more resonance peak and longer resonance time. To play the single QZS advantages, also take large vibration decay ratio of two-stage isolation system into
consideration. Two-stage QZS vibration isolator was proposed, whose effects on the vibration isolation performance caused by excitation amplitude, mass ratio and damping ratio were studied.
2. Mechanical model of QZS vibration isolation system
Fig. 1 shows a typical QZS system. The system consists of a suspended mass $m$ with a vertical spring and two identical oblique springs.
The force-deflection in the vertical direction is given by:
where ${l}_{0}$ is the free length of horizontal springs and $l$ is length in the horizontal position. $x$ is the displacement deviation from equilibrium position. The stiffness of oblique spring is
${k}_{h}$ and the stiffness of horizontal spring is ${k}_{h}$.
Eq. (1) can be written in non-dimensional form as:
where $\stackrel{^}{x}=x/{x}_{0}$, $\stackrel{^}{l}=l/{l}_{0}$ and $\stackrel{^}{k}={k}_{h}/{k}_{v}$. Using the Maclaurin-series expansion to the third order for small $\stackrel{^}{x}$, then Eq. (2)
approximates to:
where ${k}_{a}=1-2\stackrel{^}{k}\left(1-\stackrel{^}{l}\right)/\stackrel{^}{l}$ and ${k}_{b}=\stackrel{^}{k}\left(1-{\stackrel{^}{l}}^{2}\right)/{\stackrel{^}{l}}^{3}$.
Fig. 1Schematic diagram of the QZS vibration isolation system
Fig. 2Schematic of a two-stage QZS system
3. Two-stage QZS vibration isolation system
The two-stage QZS system is shown in Fig. 2.
It is subject to external harmonic force excitation $F\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Omega }T$. The upper QZS isolator stiffness coefficient is ${K}_{1}$, ${K}_{2}$ and damping coefficient is
${c}_{1}$. The lower QZS isolator stiffness coefficient is ${K}_{3}$, ${K}_{4}$ and damping coefficient is ${c}_{2}$. ${z}_{1}$ and ${z}_{2}$ are the displacement of the vibration object and the
middle inertia block when the respective spring in a natural state. ${m}_{1}$ and ${m}_{2}$ are the mass of vibration object and middle inertia block respectively.
Provided that only consider the vertical direction of movement, dynamic equation of two-stage QZS system is established based on Newton’s second law:
mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\Omega }T\right)-{m}_{1}g,\\ {m}_{2}{\stackrel{¨}{z}}_{2}-{c}_{1}\left({\stackrel{˙}{z}}_{1}-{\stackrel{˙}{z}}_{2}\right)-{K}_{1}\left({z}_{1}-{z}_{2}\
For clarity of analysis, the parameters ${\mathrm{\Omega }}_{0}=\sqrt{{K}_{1}/{m}_{1}}$, ${z}_{1}={X}_{1}\sqrt{{K}_{1}/{K}_{2}}$, ${z}_{2}={X}_{2}\sqrt{{K}_{1}/{K}_{2}}$, $T={\mathrm{\Omega }}_{0}t$
are introduced. Eq. (4) can be written in non-dimensional form as:
$\left\{\begin{array}{l}{\stackrel{¨}{X}}_{1}-{\xi }_{1}\left({\stackrel{˙}{X}}_{2}-{\stackrel{˙}{X}}_{1}\right)-\left({X}_{2}-{X}_{1}\right)-\left({X}_{2}-{X}_{1}{\right)}^{3}=f\mathrm{c}\mathrm{o}\
mathrm{s}\left(\omega t\right)-G,\\ w{\stackrel{¨}{X}}_{2}+{\xi }_{1}\left({\stackrel{˙}{X}}_{2}-{\stackrel{˙}{X}}_{1}\right)+\left({X}_{2}-{X}_{1}\right)+\left({X}_{2}-{X}_{1}{\right)}^{3}+{\xi }_
where ${\xi }_{1}={c}_{1}/\sqrt{{K}_{1}{m}_{1}}$, $f=F/{K}_{1}\sqrt{{K}_{2}/{K}_{1}}$, $G={m}_{1}g/{K}_{1}\sqrt{{K}_{2}/{K}_{1}}$, ${k}_{1}={K}_{3}/{K}_{1}$, ${k}_{2}={K}_{4}/{K}_{2}$, ${k}_{3}={K}_
{2}/{K}_{1}$, ${\xi }_{2}={c}_{2}/\sqrt{{K}_{1}{m}_{1}}$, $w={m}_{2}/{m}_{1}$, $\omega ={\mathrm{\Omega }}_{0}\mathrm{\Omega }$.
In order to analyze conveniently, the first and second equation of Eq. (5) are added, which can make stiffness coupling change into inertia coupling. The gravity term in Eq. (5) can be eliminated by
using coordinate transform. Introducing ${X}_{1}={Z}_{1}-{h}_{1}$, ${X}_{2}={Z}_{2}-{h}_{2}$, $H={h}_{1}-{h}_{2}$, ${x}_{1}={Z}_{2}-{Z}_{1}$, ${x}_{2}={Z}_{2}$, Eq. (5) can be transferred as:
$\left\{\begin{array}{l}{\stackrel{¨}{x}}_{2}-{\stackrel{¨}{x}}_{1}-{\xi }_{1}{\stackrel{˙}{x}}_{1}-\left(1+3{H}^{2}\right){x}_{1}-3H{{x}_{1}}^{2}-{{x}_{1}}^{3}=f\mathrm{c}\mathrm{o}\mathrm{s}\left(\
omega t\right),\\ \left(1+w\right){\stackrel{¨}{x}}_{2}-{\stackrel{¨}{x}}_{1}+{\xi }_{2}{\stackrel{˙}{x}}_{2}+\left({k}_{1}+3{k}_{2}{{h}_{2}}^{2}\right){x}_{2}-3{k}_{2}{h}_{2}{{x}_{2}}^{2}+{k}_{2}
{{x}_{2}}^{3}=f\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right),\end{array}\right\$
where $H+{H}^{3}=G$, ${k}_{1}{h}_{2}+{k}_{2}{{h}_{2}}^{3}=\left(w+1\right)G$.
Eq. (6) in matrix form is:
$\mathbf{M}=\left[\begin{array}{ll}-1& 1\\ -1& 1+w\end{array}\right],\mathbf{C}=\left[\begin{array}{cc}-{\xi }_{1}& 0\\ 0& {\xi }_{2}\end{array}\right],\mathbf{X}=\left[\begin{array}{l}{x}_{1}\\ {x}_
$\mathbf{F}=\left[\begin{array}{c}f\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+\left(1+3{H}^{2}\right){x}_{1}+3H{{x}_{1}}^{2}+{{x}_{1}}^{3}\\ f\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\
By the averaging method, suppose steady-state response solution of the above system as:
$\left\{\begin{array}{l}\mathbf{X}=\mathbf{U}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+\mathbf{V}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right),\\ {\mathbf{X}}^{\mathrm{"}}=-\omega \
mathbf{U}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+\omega \mathbf{V}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right),\end{array}\right\$
where $\mathbf{U}={\left[{U}_{1},{U}_{2}\right]}^{T}$ and $\mathbf{V}={\left[{V}_{1},{V}_{2}\right]}^{T}$ change slowly with time. Differentiate the first equation of Eq. (8) with respect to$t$and
eliminate the second equation of Eq. (8):
$\mathbf{U}\mathrm{"}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+\mathbf{V}\mathrm{"}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)=0.$
Differentiate the second equation of Eq. (8) with respect to$t$and take it into Eq. (7):
$\left(\omega \mathbf{M}{\mathbf{V}}^{\mathrm{"}}-{\omega }^{2}\mathbf{M}\mathbf{U}+\omega \mathbf{C}\mathbf{V}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)-\left(\omega \mathbf{M}{\
mathbf{U}}^{\mathrm{"}}+{\omega }^{2}\mathbf{M}\mathbf{V}+\omega \mathbf{C}\mathbf{U}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)=\mathbf{F}.$
Based on Eqs. (9) and (10), it can be inferred as:
$\left\{\begin{array}{l}\omega \mathbf{M}{\mathbf{U}}^{\mathrm{"}}=-\mathbf{F}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)+\left(-{\omega }^{2}\mathbf{M}\mathbf{U}+\omega \mathbf{C}\mathbf{V}\
right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)-\left({\omega }^{2}\mathbf{M}\mathbf{V}+\omega \mathbf{C}\mathbf{U}\right)\mathrm{s}\
mathrm{i}{\mathrm{n}}^{2}\left(\omega t\right),\\ \omega \mathbf{M}{\mathbf{V}}^{\mathrm{"}}=\mathbf{F}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+\left({\omega }^{2}\mathbf{M}\mathbf{V}+\
omega \mathbf{C}\mathbf{U}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+\left({\omega }^{2}\mathbf{M}\mathbf{U}-\omega \mathbf{C}\
mathbf{V}\right)\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\left(\omega t\right).\end{array}\right\$
Note that, $\mathbf{U}$ and $\mathbf{V}$ are the slowly changing function of time. The right side of Eq. (11) can be approximately represented with the average value of $\left(\omega t\right)$ in a
period. Provided that $\mathbf{U}$ and $\mathbf{V}$ remain unchangeable in a period, average equation can be obtained as:
$\left\{\begin{array}{l}\omega \mathbf{M}{\mathbf{U}}^{\mathrm{"}}={\int }_{0}^{2\pi }\left[\begin{array}{c}-F\mathrm{sin}\left(\omega t\right)+\left(-{\omega }^{2}\mathbf{M}\mathbf{U}+\omega \mathbf
{C}\mathbf{V}\right)\mathrm{cos}\left(\omega t\right)\mathrm{sin}\left(\omega t\right)\\ -\left({\omega }^{2}\mathbf{M}\mathbf{V}+\omega \mathbf{C}\mathbf{U}\right)si{\mathrm{n}}^{2}\left(\omega t\
right)\end{array}\right]/2\pi d\omega t,\\ \omega \mathbf{M}{\mathbf{V}}^{\mathrm{"}}={\int }_{0}^{2\pi }\left[\begin{array}{c}F\mathrm{cos}\left(\omega t\right)+\left({\omega }^{2}\mathbf{M}\mathbf
{V}+\omega \mathbf{C}\mathbf{U}\right)\mathrm{sin}\left(\omega t\right)\mathrm{cos}\left(\omega t\right)\\ +\left({\omega }^{2}\mathbf{M}\mathbf{U}-\omega \mathbf{C}\mathbf{V}\right)co{\mathrm{s}}^
{2}\left(\omega t\right)\end{array}\right]/2\pi d\omega t.\end{array}\right\$
According to the orthogonality of trigonometric function, Eq. (12) can be simplified as:
$\left\{\begin{array}{l}\omega \mathbf{M}{\mathbf{U}}^{\mathrm{"}}=-\frac{1}{2}\left({\omega }^{2}\mathbf{M}\mathbf{V}+\omega \mathbf{C}\mathbf{U}\right)-\frac{1}{2}\left(\begin{array}{l}{Q}_{1}\\
{Q}_{2}\end{array}\right),\\ \omega \mathbf{M}{\mathbf{V}}^{\mathrm{"}}=\frac{1}{2}\left({\omega }^{2}\mathbf{M}\mathbf{U}-\omega \mathbf{C}\mathbf{V}\right)+\frac{1}{2}\left(\begin{array}{l}{Q}_{3}
+f\\ {Q}_{4}+f\end{array}\right),\end{array}\right\$
Then, the response of system is the solution of following equation:
$\left\{\begin{array}{l}-{\omega }^{2}{V}_{1}+{\omega }^{2}{V}_{2}-{\xi }_{1}\omega {U}_{1}+{Q}_{1}=0,\\ -{\omega }^{2}{V}_{1}+\left(1+w\right){\omega }^{2}{V}_{2}+{\xi }_{2}\omega {U}_{2}+{Q}_{2}=0,
\\ -{\omega }^{2}{U}_{1}+{\omega }^{2}{U}_{2}+{\xi }_{1}\omega {V}_{1}+{Q}_{3}+f=0,\\ -{\omega }^{2}{U}_{1}+\left(1+w\right){\omega }^{2}{U}_{2}+{\xi }_{2}\omega {V}_{2}+{Q}_{4}+f=0.\end{array}\right
The force transmitted to the base includes elastic restoring force and damping force of the lower vibration isolator, which can be expressed as:
${f}_{a}={\xi }_{2}{\stackrel{˙}{x}}_{2}+\left({k}_{1}+3{k}_{2}{{h}_{2}}^{2}\right){x}_{2}-3{k}_{2}{h}_{2}{{x}_{2}}^{2}+{k}_{2}{{x}_{2}}^{3}.$
Substitute Eq. (6) in Eq. (14) and neglect higher harmonics:
${f}_{a}={f}_{b}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)+{f}_{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)+{f}_{d},$
${f}_{b}=\left({k}_{1}+3{k}_{2}{{h}_{2}}^{2}\right){U}_{2}+\omega {\xi }_{2}{V}_{2}+\frac{3{k}_{2}{U}_{2}\left({{U}_{2}}^{2}+{{V}_{2}}^{2}\right)}{4},$
${f}_{c}=\left({k}_{1}+3{k}_{2}{{h}_{2}}^{2}\right){V}_{2}-\omega {\xi }_{2}{U}_{2}+\frac{3{k}_{2}{V}_{2}}{4}\left({{V}_{2}}^{3}+{{U}_{2}}^{2}\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{f}_
Force transmissibility of two-stage QZS system is calculated as:
As a comparison, the two oblique springs are removed.
By successively applying the same procedure as for the model of equivalent two-stage linear system, the response of the system is the solution of Eq. (15), where
Similarly, the force transmissibility of two-stage linear system can be founded:
${T}_{fl}=\sqrt{\left(4{w}^{2}{{\xi }_{2}}^{2}{{V}_{2}}^{2}+{{k}_{1}}^{2}\right)\left({{U}_{2}}^{2}+{{V}_{2}}^{2}\right)}/f.$
4. Influence of the system parameters on force transmissibility
As discussed above, force transmissibility is closely related to $f$, $w$ and $\xi$. Next, the effects of different system parameters on force transmissibility of two-stage QZS system are
investigated by controlling variables method. The Force transmissibility of two-stage linear system at the same condition is also plotted together to compare the isolation performance of two systems.
All the force transmissibility results are plotted in dB, i.e. as $20\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}T$.
Fig. 3 shows the force transmissibility of system under different $f.$ Two-stage linear system are not influenced by the $f$. However, for the two-stage QZS system, two order resonant frequencies and
transmissibility peaks increase with the rising of $f$. The less the $f$ is, the better the isolation performance of two-stage QZS system compared with the two-stage linear system. Fig. 4 shows the
force transmissibility of system under different $w$, for the two-stage linear system, two order resonant frequencies increase with the rising of $w$, the transmissibility of the first order resonant
frequency decreases, while the transmissibility of the second order resonant frequency increases. For the two-stage QZS system, the first order resonant frequency transmissibility decreases with the
rising of $w$, while the second resonant order frequency and transmissibility increases with the rising of $w$. Which indicates that the isolation bandwidth broadens and isolation performance near
the second order resonant frequency weakens with the rising of $w$.
The force transmissibility curves of system under different $\xi$ are illustrated in Fig. 5 and Fig. 6 respectively. Two order resonant frequencies of two-stage linear system are not influenced by
the ${\xi }_{1}$, while the peaks of corresponding force transmissibility reduce. Force transmissibility of system under different ${\xi }_{1}$ is shown in Fig. 5. The resonance branch of the
two-stage QZS system shortens and the peaks of corresponding force transmissibility reduce with the rising of ${\xi }_{1}$. But the isolation performance will decrease when the excessive damping
ratio. Force transmissibility of system under different ${\xi }_{1}$ is shown in Fig. 6. Two order resonant frequencies increase with the rising of ${\xi }_{2}$. The transmissibility of the first
order resonant frequency decreases, while the transmissibility of the second order resonant frequency increases, which indicates that the isolation performance near the second order resonant
frequency weakens with the rising of ${\xi }_{2}$.
Fig. 3Force transmissibility under different f
Fig. 4Force transmissibility under different w
Fig. 5Force transmissibility under different ξ1
Fig. 6Force transmissibility under different ξ2
5. Conclusions
In this study, a novel two-stage quasi-zero stiffness vibration isolator was presented. The conclusions were summarized as follows:
1) The dynamic models of two-stage QZS and linear vibration isolation system were established. The force transmissibility under harmonic force excitation was derived by using the averaging method.
2) Decrease the excitation amplitude and increase the mass ratio as well as damping ratio properly, which can broaden the isolation bandwidth, increase the vibration decay ratio and enhance the
isolation performance of two-stage QZS system.
3) Compared to the corresponding two-stage linear system, the two-stage QZS system not only has smaller initial isolation frequency, wider isolation band and better isolation performance, but also
possesses excellent load-bearing capacity and stability.
• Zhu Tao, Cazzolato Benjamin, Robertson William S. P., et al. Vibration isolation using six degree-of-freedom quasi-zero-stiffness magnetic levitation. Journal of Sound and Vibration, Vol. 358,
2015, p. 48-73.
• Zou Keguan, Nagarajaiah Satish Study of a piecewise linear dynamic system with negative and positive stiffness. Communications in Nonlinear Science and Numerical Simulation, Vol. 22, 2015, p.
• Ma Yanhui, He Minghua, Shen Wenhou, et al. A planar shock isolation system with high-static-low-dynamic-stiffness characteristic based on cables. Journal of Sound and Vibration, Vol. 358, 2015,
p. 267-284.
• Gatti Gianluca, Kovacic Ivana, M. J. Brennan. On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator.
Journal of Sound and Vibration, Vol. 329, 2010, p. 1823-1835.
• Lan Chao-Chieh, Yang Sheng-An, Wu Yi-Syuan Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads. Journal of Sound and Vibration, Vol. 333, 2014, p.
• Liu Xingtian, Huang Xiuchang, Hua Hongxing On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. Journal of Sound and Vibration, Vol.
332, 2013, p. 3359-3376.
• Zhou Jiaxi, Wang Xinlong, Xu Daolin, et al. Nonlinear dynamic characteristics of quasi-zero stiffness vibration isolator with cam-roller-spring mechanisms. Journal of Sound and Vibration, Vol.
346, 2015, p. 53-69.
• Sun Xiuting, Jing Xingjian Multi-direction vibration isolation with quasi-zero stiffness by employing geometrical nonlinearity. Mechanical System and Signal Processing, Vol. 62, Issue 63, 2015,
p. 149-163.
About this article
03 September 2016
08 December 2016
Mechanical vibrations and applications
two-stage quasi-zero stiffness
the averaging method
force transmissibility
vibration isolation system
The authors gratefully acknowledge the support for this work by the National Natural Science Foundation of China (NSFC) under Grant No. 51579242 and No. 51509253.
Copyright © 2016 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
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Reading: Purpose of Functions
Often, economic models are expressed in terms of mathematical functions. What’s a function? Basically, a function describes a relationship involving one or more variables. Sometimes the relationship
is a definition. For example (using words), Joan of Arc is a professor. This could be expressed as Joan of Arc = professor. Or, food = cherries, cheese, and chocolate means that cherries, cheese, and
chocolate are food.
In economics, functions frequently describe cause and effect. The variable on the left-hand side is what is being explained (“the effect”). On the right-hand side is what’s doing the explaining (“the
causes”). Functions are also useful for making predictions. For example, think about your grade in this course. We might be able to predict how well you will do in this course by considering how well
you’ve done in other courses, by how much you attend class or participate in the online activities, and by how many hours you study.
Not all of those things will have equal impact on your grade. Let’s assume that your study time is most important and will have twice as much impact as the other factors. We are trying to describe
100 percent of the impact, so study time will explain 50 percent, attendance and participation will explain 25 percent, and your prior class grades will describe 25 percent. Together, this adds up to
100 percent.
Now, let’s turn that into a function. Your grade in the course can be represented as the following:
Grade = (0.50 x hours_spent_studying) + (0.25 x class_attendance) + (0.25 x prior_GPA)
This equation states that your grade depends on three things: the number of hours you spend studying, your class attendance, and your prior course grades represented as your grade-point average
(GPA). It also says that study time is twice as important (0.50) as either class_attendance (0.25) or prior_GPA score (0.25). If this relationship is true, how could you raise your grade in this
course? By not skipping class and studying more. Note that you cannot do anything about your prior GPA, since that is calculated from courses you’ve already taken and grades you’ve already received.
Economic models tend to express relationships using economic variables, such as Budget = money_spent_on_econ_books + money_spent_on_music (assuming that the only things you buy are economics books
and music). Often, there is some assumption that has to be explained in order to identify where the model has been simplified.
As you can see, in economic models the math isn’t difficult. It’s used to help describe and explain the relationships between variables. | {"url":"https://courses.lumenlearning.com/suny-microeconomics/chapter/reading-purpose-of-functions/","timestamp":"2024-11-11T14:00:40Z","content_type":"text/html","content_length":"49840","record_id":"<urn:uuid:de321809-43f5-4941-8cc1-96f93c589a4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00036.warc.gz"} |
Predict the cube root of a number in 5 seconds accurately
Predict the cube root of a number in 5 seconds
Cube roots are useful in trigonometry and historical mathematics since they may be used to solve cubic equations, find the side length of a cube given its volume, and more. Cube root questions have
appeared in a number of competitive tests, and there’s a good likelihood they’ll reappear in NRA CET (SSC, Bank, Railway) and other competitive exams. With no intermediate steps required, the student
will be able to accurately predict the cube root of a number in 5 seconds, thanks to the incredible technique for solving cube roots.
Any number that yields the original number after three times of multiplication by itself is said to have a cube root. Since 3 x 3 x 3 = 27, for instance, the cube root of 27 is 3. The process of
computing a cube is reciprocally calculated as the cube-root. 2 is the cube-root of 8, therefore if 8 is the cube of 2, then so is 2.
Importance of Cube root in mathematics:
Solving Equations: Cube roots are the tool of choice for solving cubic equations. They can be used, specifically, to find the dimensions of a three-dimensional object with a given volume.
In trigonometry: When attempting to discover an angle (angle trisection) whose measure is one-third that of a given angle, cube roots appear.
The origins of Cube roots:
As early as 1800 BCE, Babylonian mathematicians were the first to calculate cube roots.
The Nine Chapters on the Mathematical Art is a Chinese mathematical work that was collected in the second century BCE and elaborated upon by Liu Hui in the third century CE. It has a method for
extracting cube roots.
Hero, the Greek mathematician, presented the issue of doubling the cube, which required the construction—now known to be impossible—of a length 3√2 using a compass and straightedge to create an edge
of a cube with twice the volume of a given cube.
Make a list of the numbers 1 through 10 along with their corresponding cubes and mug it up. We’ll utilize this list to compute higher order integers’ cube roots. We will be able to solve the
cube-roots quickly if we know these numbers. Therefore, before continuing, I strongly advise the reader to commit the list below to memory.
│No. │Cube │
│1 │1 │
│2 │8 │
│3 │27 │
│4 │64 │
│5 │125 │
│6 │216 │
│7 │343 │
│8 │512 │
│9 │729 │
│10 │1000 │
The cube root problem will be divided into two sections. The answer’s right-hand portion will be solved first, followed by the answer’s left-hand portion. You have the option to finish the left-hand
half before moving on to the right-hand part. Although there are no restrictions on either approach, we usually like to solve the right-hand portion first.
Here’s an example to help you understand.
Determine 287496’s cube root.
The number 287496 is represented as 287 | 496 since we slashed the final three digits.
Next, we note that cube 287496 has an end of six. We also know that cube root ends of six when cube ends of six. At this point, ___6 is our response. Thus, we have the first part of our answer.
We use the number that is located to the left of the slash to determine the left portion of the solution. The number 287 is located in this instance to the left of the slash.
The number 287 in the number line must now be located between two perfect cubes. 287 is located between the perfect cubes 216 (the cube of 6) and 343 (the cube of 7), according to the key.
Now, we take the lower number from the above-mentioned values and place it on the answer’s left side. As a result, we choose the smaller of 6 and 7 and place it next to the already determined answer
of __6.
The final response is 66.
Consequently, 287496’s cube root is 66. Consider this additional example:
Find 2197’s cube root.
The notation for the number 2197 is 2 | 197.
The cube root will terminate with a 3 since the cube ends with a 7.
As the answer’s right-hand component, we will enter 3. Between 1 (the cube of 1, see the cube list made above) and 8 (the cube of 2, see the cube list created above), is the number 2. We shall enter
1 as the smaller number in the left-hand portion of the solution.
13 is the final response.
From this, we can infer that there is only a single method for quickly solving perfect cube-roots of all kinds. Stay in touch with Web Tutors Point. | {"url":"https://webtutorspoint.com/predict-the-cube-root-of-a-number-just-by-looking-at-it/","timestamp":"2024-11-14T20:02:58Z","content_type":"text/html","content_length":"115827","record_id":"<urn:uuid:83411612-1926-4238-a7a4-d47c95fa838b>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00127.warc.gz"} |
Log Transformation Base For Data Linearization Does Not MatterData Science News
Log Transformation Base For Data Linearization Does Not Matter
Photo by Janus Clemmensen on UnsplashLog Transformation Base For Data Linearization Does Not MatterA simple derivation explaining why log base has no significant effect when linearizing dataJeremy
ChowBlockedUnblockFollowFollowingJun 27Code for this demonstration can be found here:Today a colleague asked me a simple question:“How do you find the best logarithm base to linearly transform your
data?”This is actually a trick question, because there is no best log base to linearly transform your data—the fact that you are taking a log will linearize it no matter what the base of the log
My colleague was skeptical and I wanted to brush up on my algebra, so let’s dive into the math!PremiseLet’s assume you have exponential data.
This means your data is in some form similar to the following:(1)This means our data is non-linear.
Linear data is arguably the best form of data we can model, because through linear regression we can directly quantify the effects of each feature on the target variable by looking at its
Linear regression is the best type of model for giving humans an intuitive and quantitative sense of how the model thinks our dependent variable is influenced by our independent variables versus, for
example, the black boxes of deep neural nets.
DerivationSince we know the base here is e, we can linearize our data by taking the natural log of both sides (ignoring the constant C₁):(2)Now if we plot ln(y) vs.
x, we get a line.
That’s pretty straightforward, but what happens if we didn’t know that the base of our power was e?.We can try taking the log (base 10) of both sides:(3)but it doesn’t seem to look linear yet.
However, what if we introduce the logarithm power rule?(4)But log(e) is a constant!.therefore we have:(5)This means that our base 10 log is still directly proportional to x, just by a different
scaling factor C, which is the log of the original base e in this case.
What does it look like?We can also visualize this with some python code!import numpy as npimport matplotlib.
pyplot as plt# Set up variables, x is 1-9 and y is e^xx = list(np.
linspace(1,10,100))y= [np.
exp(i) for i in x]# Plot the original variables – this is barebones plotting code, you # can find the more detailed plotting code on my github!plt.
plot(x,y)# Plot log base 10 of yplt.
log10(y))# Plot log base e of yplt.
log(y))They are both linear, even though the logarithms have different bases (base 10 vs base e)!The only thing that changed between the two logarithms was the y-scale because the slopes are slightly
different!.The important part is that both are still linearly proportional with x, and thus would have equal performance in a linear regression model.
ConclusionIn summary: If you have exponential data, you can do a log transformation of any base to linearize the data.
If you have an intuition for the base from domain knowledge, then use the correct base—otherwise it doesn’t matter.
Side Note: Other TransformationsWhat if your data is in the slightly different form of x raised to the power of some unknown λ?(6)In this case, a Box-Cox transformation will help you find the ideal
power to raise your data to in order to linearize it.
I recommend using Sci-py’s implementation.
That’s all for this one, thanks for reading!.
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Fund BioImag : MRI contrast mechanisms 1.What is the mechanism of T 2 * weighted MRI ? BOLD fMRI 2.How are spin echoes generated ? 3.What are. - ppt download
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Diametral Pitch Formula: Overview in context of diametral pitch formula
26 Aug 2024
Title: The Diametral Pitch Formula: A Comprehensive Review
Abstract: The diametral pitch formula is a fundamental concept in gear design, used to determine the pitch diameter of a gear tooth. This article provides an overview of the formula, its
significance, and its application in various engineering fields.
Introduction: Gear teeth are designed with specific dimensions to ensure proper meshing and transmission of power between two or more gears. The diametral pitch (DP) is a critical parameter that
determines the size and shape of gear teeth. The diametral pitch formula is used to calculate the DP, which is essential for designing and manufacturing gears.
The Diametral Pitch Formula: The diametral pitch formula is given by:
DP = π * (d1 + d2) / (4 * (d1 - d2)) (BODMAS format)
where: DP = diametral pitch d1 = pitch diameter of the driving gear d2 = pitch diameter of the driven gear
In ASCII format, the formula can be written as:
DP = π * (d1 + d2) / (4 * (d1 - d2))
Significance of Diametral Pitch: The diametral pitch is a critical parameter in gear design, as it affects the gear’s performance, efficiency, and durability. A higher DP indicates a larger tooth
size, which can lead to increased power transmission and reduced noise levels. Conversely, a lower DP results in smaller teeth, which may compromise the gear’s strength and reliability.
Applications of Diametral Pitch Formula: The diametral pitch formula has numerous applications in various engineering fields, including:
1. Gear design: The formula is used to determine the optimal tooth size for gears in mechanical systems, such as transmissions, engines, and machinery.
2. Gear manufacturing: The DP calculation helps manufacturers design and produce gears with precise dimensions, ensuring proper meshing and transmission of power.
3. Aerospace engineering: The diametral pitch formula is used to design and optimize gear systems for aircraft and spacecraft applications, where precision and reliability are critical.
Conclusion: The diametral pitch formula is a fundamental concept in gear design, providing a mathematical framework for calculating the DP. Understanding the significance of Diametral Pitch and its
application in various engineering fields can help engineers design and manufacture gears with optimal performance, efficiency, and durability.
1. ANSI/AGMA 2001-D04 (2013). Fundamental Rating Factors for Involute Spur Gear Teeth.
2. ISO 1328-1 (2015). Gears - Part 1: Design and calculation of spur and helical gears.
3. Peterson, J. B. (1994). Gear Engineering. McGraw-Hill Education.
Note: The formula is written in both BODMAS (British, Oxford, Dominion, Massachusetts, Association) and ASCII formats for clarity and ease of understanding.
Related articles for ‘diametral pitch formula’ :
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Quantum Universe
Scientific activities of the various Research Units
Napoli unit
-The units is interested to study theories of gravity by considering the role of curvature, torsion and topological invariants, as well as non-local terms. Specifically, the generalization of
Einstein’s theory is considered in view of addressing astrophysical and cosmological issues such as accelerated expansion, large scale structure and, in general, of reconstructing a reliable cosmic
history. In particular, we study cosmological features like cosmography and Cosmic Microwave Background in order to test models.
-Gravitational field theories, such as f(R, ◻R), f(T), f(R,,G), f(G,T), f(T,B) etc. are studied, focusing in particular on exact solutions of Noether symmetries and constraining the free parameters
of the models with Cosmic Microwave background data and Large Scale Structure measurements. Furthermore, the theoretical foundations of these alternative approaches are studied by comparing the role
played by affine connections, tetrads and metric in the dynamics of the gravitational field.
-In order to investigate the origin of the accelerated expansion of the Universe we study interacting dark matter/dark energy models. In particular, we want to understand whether this coupling can be
connected to some symmetry, and compare high redshift observations with theoretical predictions in extended theory of gravity. Weare interested to develop an extended approach to the cosmographic
technique in order to probe the expansion of the Universe at high redshift with new distance indicators, as Gamma Ray Bursts and Quasars.
- We study perturbations induced by scalar fields in a given gravitational background, generalizing recent results obtained by constraining the scalar field evolution according to an
Ermakov-Pinney-like prescription (e.g., divergence-free property of the corresponding current). We focus on associated gauge-invariant variables, as well as on possible transcription of results into
other formalisms, like gravitational self-force.
-Over the last years, a first step was done in defining both quantum and classical state of the universe in terms of tomographic functions. The next step is to see how the classical limits of
quantum tomograms fit with the classical tomogram obtained with cosmographic techniques. Moreover, research has been started about the influence of torsion on gravitational waves in extended
-We use weak lensing from wide astronomical survey to study the correlated distribution of large scale structure around a large sample of massive galaxy clusters. This represents a novel test for
the LCDM model, as LCDM predictions appear to be at odds with our recent observational findings (see Sereno, Nature Astronomy, 2018) based on a limited sample of clusters.
-In order to understand the origin of dark energy and due to the lack of a widely accepted quantum gravity theory, we consider a new line of research where the cosmological constant is depicted in
terms of a semi-classical model. There, Planckian fluctuations are averaged on a fixed physical scale L in terms of the Buchert formalism, but adapted to microscopic scales. We study of the above
mentioned semi-classical model in terms of a renormalization-group approach, where a non-trivial stable infrared fixed point denotes the crossover to classicality.
-We consider a non-unitary, classically stable version of higher derivative (HD) gravity and focus on its Newtonian limit, which is a non-Markovian model with several appealing features. These
include a built-in mechanism for the evolution of macroscopic coherent superpositions of states into ensembles of pure states and the primacy of the density matrix in describing the physical reality.
Within this framework, we study, via numerical simulations, gravity induced thermalization properties of a mesoscopic crystal. The aim is to show how thermodynamics could emerge even in a closed
system, by virtue of the fundamental non-unitary nature of the model.
-We analyze the semi-classical study of selected mesoscopic quantum systems, like Dirac-Weyl nanomaterials, interacting with gravity. Search for novel, possibly observable effects at nanoscales might
provide a further test of General Relativity, also helping to unveil the quantum face of gravitation.
Salerno unit
- We analyze the effects of adding non-local cubic terms in the scalar curvature to the Lagrangian of modified gravity and study the consequences on amplitudes and the renormalizability of the
model. More specifically, the aim is to understand whether the cubic curvature term could allow to improve the UV behavior of nonlocal gravity theories and make them well defined at any energy scale.
In such a case, it is expected that the cubic term can cancel the enhancement contribution in the vertex coming from the quadratic curvature action.
-The limits of GR have led to the emergence of the `dark universe' scenario. In the last years, in fact, there have been evidences that, if cosmology is described by Einstein's field equations,
then there should be a substantial amount of `dark matter' in the Universe. More recently, `dark energy' has also been found to be required in order to explain the apparent accelerating expansion of
the Universe. Modified cosmology (for example, f(R, ◻R) f(T), f(R,,G), where R, T and G refer to scalar curvature, torsion and Gauss-Bonnet invariant, respectively) may be at play during the
evolution of the Universe, not only in the late era, but also in the very early. In particular, it is accepted that non-standard cosmology may not only describe the early-time inflation and late-time
acceleration but also may propose the unified consistent description of the Universe evolution in different epochs, from inflation and radiation/matter dominance to dark energy. We focus on study
these aspects in the framework of CMB physics, which from the upcoming experiments (PLANCK, BICEP, etc.) is the main source of data about the early universe. Particular interest is devoted to the
study of primordial gravitational waves from Inflation.
- The detection of gravitational waves by Advanced LIGO and Advanced Virgo provides an opportunity to test general relativity in a regime that is inaccessible to traditional astronomical observations
and laboratory tests. Using the new results from various tests of General Relativity performed using the binary black hole signals, we investigate the propagation of gravitational waves in the
context of different model of modified gravity, and then impose constraints on the free parameters. Moreover, the advent of the recent study of QNM, the astrophysical scenarios provide a promising
laboratory for constraining, and eventually ruling out, extended theories of gravity.
-Quantum effects in curved spacetimes in different frameworks: 1) Violation of the equivalence principle and tests of theories beyond GR. 2) The vacuum energy effects (vacuum condensate) that might
play a relevant role in various contexts. 3) The Casimir-like systems. 4) Entanglement of particles in non-trivial backgrounds. 5) Non local QFT at finite temperature in curved spacetime. Moreover,
our IS is also interested in studying the interaction of particles mediated by axion or dark particles.
- The Salerno unit is strongly involved in several gravitational lensing searches. It participates in the Microlensing Science Investigation Team of the WFIRST mission by NASA, to be launched in
2025. In particular, it is responsible for the development of the codes for the magnification calculation, the modeling and interpretation of microlensing events. The purpose of the project is to
find thousands of extrasolar planets, measure the remnant mass function and the number of black holes in our Galaxy. The Salerno unit has also developed analytical and numerical methods for the study
of gravitational lensing by black holes in the strong deflection limit. With the new data coming from the Event Horizon Telescope and from the gravitational waves detection, we have the possibility
to test General Relativity in strong fields through gravitational lensing effects.
Trieste unit
- The nature of dark matter. Experiments and observations are modifying our knowledge on this mistery of the universe. Rather than from theoretical first principle it seems clear that our
investigation must start from the distribution of dark matter in galaxies and its entanglement with that of the luminous matter. The nature of the dark particle will be worked out by its interaction
with standard model particles very likely much more complex than that of the WIMP paradigm.
- Galaxy formation . Due to a large number of new astrophysical data available in this period ,the galaxy formation and the subsequent evolution could be traced from the beginning. Properties such
as the stellar and the halo mass functions once determined and followed from high redshift to the present will indicate us the cosmological importance of galaxy merging ,of the angular momentum
, of the baryonic feedback and of the downsizing with redshift of the stellar component.
-Galaxy simulations. Special hydroninamic simultations are performed on galaxies and clusters with the turbolence specially trated will be the gauge for the observational properties of
- Merging BH. There is a complex theorethical investigation on the number of binary stellar BHs that merged and produced Gravitational waves in all galaxies over the whole history of the Universe.
The comparison with actual GW detection will give crucial important result on Cosmology and the Physics of Gravitation in very compact objects, | {"url":"https://web.infn.it/CSN4/index.php/it/17-esperimenti/116-qgsky-research","timestamp":"2024-11-14T12:34:02Z","content_type":"text/html","content_length":"30392","record_id":"<urn:uuid:13474621-9396-4e13-a4a1-b7c7f9600cae>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00772.warc.gz"} |
Tetrahedron Volume Calculator
Last updated:
Tetrahedron Volume Calculator
The tetrahedron volume calculator determines the volume and surface area of a tetrahedron. In addition to this, the calculator will also help you find other properties such as the height of
tetrahedron, surface area to volume ratio, sizes of various spheres like insphere, midsphere, and circumsphere. Read on to understand the properties of a tetrahedron, how to use volume of a
tetrahedron formula along with various spheres and how to calculate surface area to volume ratio.
If you're not interested in the surface area or volume of other shapes, check out our surface area calculator and volume calculator.
What is a tetrahedron? – Volume, surface area, and height of tetrahedron
A tetrahedron is a 3D shape that is formed by joining together 4 triangular faces. It has 4 vertices, 6 edges, and 4 faces. We can see different views of a tetrahedron in the figure below, with
labels for height H and edge length L. This shape is otherwise known as a triangular pyramid, i.e., a pyramid with a triangular base.
Views of a tetrahedron
The height H of a tetrahedron is related to its edge length L can be written as:
$H = \frac{\sqrt{6}}{3} L$
The volume of tetrahedron formula can be written as:
$V = \frac{L^3}{6\sqrt 2}$
Similarly, the area of a tetrahedron A can be determined from the equation:
$A = \sqrt 3 L^2$
You can observe that the surface area of the tetrahedron is the product of the number of faces and area of 1 triangular face. You can use the volume and surface area of the tetrahedron to estimate
the surface area to volume ratio SVR for this shape.
$\frac{\text{Surface Area}} {\text{Volume}} = \frac{\sqrt 3 L^2}{\frac{L^3 }{6\sqrt 2}} = \frac{6 \sqrt 6}{L}$
Spheres of a tetrahedron
Now that you know how to calculate volume and surface area, let us take a look at different spheres which can be accommodated inside and around a tetrahedron. There are three different kinds of
• Insphere – If you draw a sphere inside the tetrahedron which is tangent to every face of the cell, the sphere is known as an insphere. The calculator will estimate the largest possible size of
the sphere that can be contained inside the faces of a tetrahedron with radius $r_\mathrm{i}$. The insphere is marked in red in the figure below.
$\qquad r_\mathrm{i} = \frac{L}{24}$
• Midsphere – If you draw a sphere which is tangent to every edge of a tetrahedron, the sphere is called a midsphere. In other words, the sphere must touch every edge only at one point. The
calculator will return the size of the midsphere having radius $r_\mathrm{k}$. The midsphere is marked in green in the figure below.
$\qquad r_\mathrm{k} = \frac{L}{\sqrt 8}$
• Circumsphere – The sphere which touches the vertex of the tetrahedron is known as a circumsphere. The calculator determines the size of the circumsphere having radius $r_\mathrm{u}$. The
circumsphere is marked in blue in the figure below.
$\qquad r_\mathrm{u} = \frac{L \sqrt 6}{4}$
Spheres of a tetrahedron (, , via Wikimedia Commons).
How to use tetrahedron volume calculator?
Follow the steps below to determine the various properties of your tetrahedron. As you can observe, the properties of a tetrahedron, like surface area, volume, height, etc., are a function of only
one variable, i.e., edge length, L. It is fairly easy to operate the tetrahedron volume calculator.
• Step 1: Enter the edge length L.
• Step 2a: The calculator will provide the height of the tetrahedron.
• Step 2b: The volume and surface area of the tetrahedron are also visible.
• Step 2c: The volume and surface area used to estimate the surface area to volume ratio.
• Step 2d: The sizes of insphere, midsphere, and circumsphere are also determined.
Example: Using the tetrahedron volume calculator
Use the volume of tetrahedron formula to estimate the volume for the cell having edge length L = 80 cm. Also, find the sizes of insphere, midsphere, and circumsphere.
• Step 0: Set the units for edge length to cm.
• Step 1: Enter the edge length as 80 cm.
• Step 2a: The calculator will now return the volume:
Using the volume of tetrahedron formula,
$\qquad V = \frac{80^3}{6\sqrt 2} = 0.06034~\mathrm{m^3}$
• Step 2b: The sizes for the sphere are calculated as:
$\qquad r_\mathrm{i} = \frac{80}{24} = 0.1633~\mathrm{m}$
$\qquad r_\mathrm{k} = \frac{80}{\sqrt 8} = 0.28284~\mathrm{m}$
$\qquad r_\mathrm{u} = \frac{80 \sqrt 6}{4} = 0.4899~\mathrm{m}$
Using tetrahedrons
Did you know:
• Tetrahedrons are used in complex stress and deformation-based computer simulations. Large shapes are segmented into smaller elements of tetrahedron shape.
• Tetrahedrons are also an observed shape of hybrid or bonded molecules in chemistry.
• Some of the ancient civilizations used tetrahedral dices for their board games. So does several current board games.
What is a tetrahedron?
A tetrahedron is a 3D pyramidal shape with a triangular base.
How do I calculate the volume of a tetrahedron?
The volume of a tetrahedron formula can be calculated using the edge length, L as:
V = L³ / (6 × √2)
How do I find the height of a tetrahedron?
The height of a tetrahedron can be calculated using the edge length, L as:
H = (√6 / 3) × L
How many faces does a tetrahedron have?
A tetrahedron has 4 faces, 6 edges, and 4 vertices.
How many edges does a tetrahedron have?
A tetrahedron has 6 edges and 4 vertices.
How many vertices does a tetrahedron have?
A tetrahedron has 4 vertices. | {"url":"https://www.omnicalculator.com/math/tetrahedron-volume","timestamp":"2024-11-04T06:01:08Z","content_type":"text/html","content_length":"563594","record_id":"<urn:uuid:fc3975e5-d0f5-4d24-85fe-cf48484a25a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00216.warc.gz"} |
[Solved] Determine the mean number of credit cards | SolutionInn
Answered step by step
Verified Expert Solution
Determine the mean number of credit cards based on the raw data. (b) Determine the standard deviation number of credit cards based on the raw
Determine the mean number of credit cards based on the raw data. (b) Determine the standard deviation number of credit cards based on the raw data. (c) Determine a probability distribution for the
random variable, X, the number of credit cards issued to an individual. (e) Determine the mean and standard deviation number of credit cards from the probability distribution found in part (c). (f)
Determine the probability of randomly selecting an individual whose number or credit cards is more than two standard deviations from the mean. Is this result unusual? (g) Determine the probability of
randomly selecting two individuals who are issued exactly two credit cards. [Hint: Are the events independent?] Interpret this result. Interpret this result. Select the correct choice below and fill
in the answer box within your choice. O A. Whenever a couple applies for credit cards, the probability that they will each obtain exactly two credit cards is____ (Round to three decimal places as
needed.) O B. If two individuals were randomly selected surveyed 100 different times, the surveyors would expect about of the surveys to result in two people with exactly two credit cards. (Round to
the nearest whole number as needed.) © C. Whenever multiple people are surveved, the probability that the mean number of credit cards will be exactly two is_____ (Round to three decimal places as
needed.) O D. If an individual is chosen at random, the probability that they have exactly two credit cards is_____ (Round to three decimal places as needed.)
D 2 2423216217 3 6 5332423214 3242372332 15423393 5337322442 3822145133 333244211 2382033413 10 252220323- 2 5 3232523245 3 1 1 Credit Card Survey Results
There are 3 Steps involved in it
Step: 1
a Mean The mean is the sum of all the values divided by the number of valuesIn this casethe number of credit cards for each person is listed 322333523222521043342103423352 There are 28 entries
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On the max coloring problem
We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G=(V,E) and positive node weights w:V→(0,∞), the goal is to find a proper node coloring of G
whose color classes ^C1, ^C2,..., ^Ck minimize ∑i=1kmaxv∈Ci w(v). We design a general framework which allows to convert approximation algorithms for standard node coloring into algorithms for max
coloring. The approximation ratio increases by a multiplicative factor of at most e for deterministic offline algorithms and for randomized online algorithms, and by a multiplicative factor of at
most 4 for deterministic online algorithms. We consider two specific hereditary classes which are interval graphs and perfect graphs. For interval graphs, we study the problem in several online
environments. In the List Model, intervals arrive one by one, in some order. In the Time Model, intervals arrive one by one, sorted by their left endpoints. For the List Model we design a
deterministic 12-competitive algorithm, and a randomized 3e-competitive algorithm. In addition, we prove a lower bound of 4 on the competitive ratio of any deterministic or randomized algorithm. For
the Time Model, we use simplified versions of the algorithm and the lower bound of the List Model, to develop a deterministic 4-competitive algorithm, a randomized e-competitive algorithm, and to
design a lower bounds of φ≈1.618 on the deterministic competitive ratio and a lower bound of 43 on the randomized competitive ratio. The former lower bounds hold even for unit intervals. For unit
intervals in the List Model, we obtain a deterministic 8-competitive algorithm, a randomized 2e-competitive algorithm and lower bounds of 2 on the deterministic competitive ratio and 116≈1.8333 on
the randomized competitive ratio. Finally, we employ our framework to obtain an offline e-approximation algorithm for max coloring of perfect graphs, improving and simplifying a recent result of
Pemmaraju and Raman.
• Approximation algorithms
• Coloring
• Interval graphs
• Online algorithms
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'On the max coloring problem'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/on-the-max-coloring-problem","timestamp":"2024-11-05T00:16:55Z","content_type":"text/html","content_length":"55425","record_id":"<urn:uuid:a2dc0e4d-21b3-486e-83a8-e6c5b9789cb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00442.warc.gz"} |
9. Expectancy Effect | PerformancePutting
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Effective and optimum employment of expectancy effects is key to successful putting. The expectancy effect is where golfers have a tendency to expect a certain outcome. For example, expecting to hole
an 8ft. fast putt simply because they have incorrectly anticipated the probability through their expectations. When golfers expect certain results from their putting they appear unwittingly to deal
with negative outcomes and treat it in such a way as to increase the probability that they will respond as expected, negatively to the next putt. I will coach you in (1) determining the overall
probability of holing a putt, (2) that expectancy effects do in fact occur and how to manage them, (3) estimating your putting skills, and (4) learning the putting statistics from the PGA Tour.
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Bessel filter
The Bessel filter is given by the normalized transfer function
where n is the order of the filter and q[n] are the Bessel polynomials
with coefficients
The transfer function above is normalized (i.e., it is presented for the Bessel low pass filter with cutoff frequency 1). The Bessel low pass filter can be obtained from the transfer function above
with the substitution S = s / ω[c], where s = jω, ω[c] is the cutoff frequency of the filter, and ω is the angular frequency spanning the frequency spectrum between 0 and π. The substitution S = ω[c]
/ s produces the Bessel high pass filter. The substitution S = (s^2 + ω[c]^2) / (B s) produces the Bessel band pass filter, where ω[c] is the midpoint of the pass band and B is the width of the band.
The substitution S = B s / (s^2 + ω[c]^2) produces the Bessel band stop filter.
The Bessel filter is said to have an almost flat group delay (delay of the amplitude envelope for various frequencies). In other words, the Bessel filter has close to the same delay for all
Example: High pass Bessel filter of the third order
Set n = 3 and use the substitution S = ω[c] / s. The transfer function of the third order Bessel high pass filter is
The bilinear transformation s = 2 (z – 1) / (z + 1) allows us to rewrite the transfer function using the Z transform as follows.
$$H(z)=\frac{a_0+a_1z^{-1}+a_2z^{-2}+a_3z^{-3}}{b_0+b_1z^{-1}+b_2z^{-2}+b_3z^{-3}}$$ $$a_0=120$$ $$a_1=-360$$ $$a_2=360$$ $$a_3=-120$$ $$b_0=\omega_c^3+12\omega_c^2+60\omega_c+120$$ $$b_1=3\omega_c^
3+12\omega_c^2-60\omega_c-360$$ $$b_2=3\omega_c^3-12\omega_c^2-60\omega_c+360$$ $$b_3=\omega_c^3-12\omega_c^2+60\omega_c-120$$
Say that the cutoff frequency of the filter is ω[c] = 0.6 (technically, ω[c] = 2 arctan(0.6/2) ≈ 0.583, because of the warping of the frequency domain by the bilinear transformation). The transfer
function of this example Bessel high pass filter is
and the filter itself is
$$y(k) = 0.747496 x(k) – 2.242488 x(k – 1) + 2.242488 x(k – 2) - 0.747496 x(k – 3)$$ $$+ 2.435790 y(k – 1) - 1.995366 y(k – 2) + 0.548811 y(k – 3)$$
Suppose that the sampling frequency is 2000 Hz. The cutoff frequency then is ω[c] = (0.6 * 2000) / (2 π) = 191 Hz. The magnitude response of the filter is shown in the graph below.
Example: Band stop Bessel filter of the second order
Set n = 2 and S = B s / (s^2 + ω[c]^2). The transfer function of the second order band stop Bessel filter is
$$H(s) = \frac{3s^4+6\omega_c^2s^2+3\omega_c^4}{3s^4+3B s^3+(B^2+6\omega_c^2)s^2+3B\omega_c^2s+3\omega_c^4}$$
After the bilinear transformation s = 2 (z – 1) / (z + 1), the transfer function becomes
$$H(z) = \frac{a_0+a_1z^{-1}+a_2z^{-2}+a_3z^{-3}+a_4z^{-4}}{b_0+b_1z^{-1}+b_2z^{-2}+b_3z^{-3}+b_4z^{-4}}$$ $$a_0=48+24\omega_c^2+3\omega_c^4$$ $$a_1=-192+12\omega_c^4$$ $$a_2=288-48\omega_c^2+18\
omega_c^4$$ $$a_3=-192+12\omega_c^4$$ $$a_4=48+24\omega_c^2+3\omega_c^4$$ $$b_0=48+24B+4B^2+24\omega_c^2+6B\omega_c^2+3\omega_c^4$$ $$b_1=-192-48B+12B\omega_c^2+12\omega_c^4$$ $$b_2=288-8B^2-48\
omega_c^2+18\omega_c^4$$ $$b_3=-192+48B-12B\omega_c^2+12\omega_c^4$$ $$b_4=48-24B+4B^2+24\omega_c^2-6B\omega_c^2+3\omega_c^4$$
If, for example, we use the midpoint frequency ω[c] = 0.6 and the stop band width B = 1, and we scale the coefficients to obtain b[0] = 1, then a[0] = a[4] = 0.654084, a[1] = a[3] = -2.184281, a[2] =
3.131742, b[0] = 1, b[1] = -2.685262, b[2] = 3.039987, b[3] = -1.683299, b[4] = 0.399923. If we suppose that the sampling frequency is 2000 Hz, then ω[c] = (0.6 * 2000) / (2 π) = 191 Hz, B = (1 *
2000) / (2 π) = 318 Hz and the magnitude response of the filter is as follows.
Phase response of the Bessel filter
The Bessel filter is said to have an almost flat group delay (delay of the amplitude envelope for various frequencies). In other words, the Bessel filter has close to a linear phase response. A
comparison of the phase delay for the second order low pass Bessel filter and the second order low pass Butterworth filter is shown below.
When specifying qn(S), your equation shows summing results from k=1 to k=n. Should this not be from k=0 to k=n, otherwise we're missing a term in the resulting polynomial?
Right. I corrected the qn sum above to start from 0 | {"url":"https://www.recordingblogs.com/wiki/bessel-filter","timestamp":"2024-11-07T18:49:59Z","content_type":"text/html","content_length":"30035","record_id":"<urn:uuid:040f2dfa-bb74-4f9f-995a-fbed105f7045>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00837.warc.gz"} |
50.004 Algorithms
Course Description
This course is an introduction to algorithms and algorithmic thinking. The course covers common algorithms, algorithmic paradigms, and data structures that can be used to solve computational
problems. Emphasis is placed on understanding why algorithms work, and how to analyze the complexity of algorithms. Students will learn the underlying thought process on how to design their own
algorithms, including how to use suitable data structures and techniques such as dynamic programming to design algorithms that are efficient.
Learning Objectives
At the end of the term, students will be able to:
• Analyze the running times of algorithms.
• Demonstrate familiarity with major algorithms and data structures.
• Use suitable data structures in algorithms to solve computational problems.
• Identify major issues in the implementation of algorithms.
• Solve algorithmic issues in the design of information systems.
• Understand graphs as data structures, and implement graph traversals.
• Apply Bellman-Ford algorithm and Dijkstra’s algorithm to compute shortest paths in graphs.
• Design efficient algorithms using dynamic programming to solve computational problems.
• Analyze NP-complete problems and apply polynomial-time reductions to problems.
Measurable Outcomes
• Compute the asymptotic complexity of algorithms.
• Analyze and apply properties of data structures.
• Design algorithms that build upon basic operations on data structures.
• Apply and/or modify existing algorithms to solve computational problems.
• Compute hash tables and perform re-hashing.
• Implement graph-based algorithms on provided graphs.
• Design efficient algorithms using dynamic programming.
• Analyze NP-complete problems and apply polynomial-time reductions to problems.
Topics Covered
• Complexity, Asymptotic notation
• Document distance
• Peak finding, divide-and-conquer
• Sorting algorithms, master theorem
• Heaps, priority queues, analysis of heap algorithms
• Binary search trees (BSTs), BST operations, AVL trees
• Arrays vs linked lists, hashing, designing good hash functions, re-hashing
• Graphs as data structures, breadth-first search, depth-first search, topological sort
• Single source shortest path problem, Bellman-Ford algorithm, Dijkstra’s algorithm
• Dynamic Programming (DP), designing DP algorithms
• DP problems: rod-cutting problem, knapsack problem, text justification problem, matrix chain parenthesization
• P vs NP, decision problems, polynomial-time reduction, NP-hardness
• Examples of NP-complete problems (inc. multiple graph-related NP-complete problems)
• More graph-theoretic terminology, 3-SAT problem
Textbook(s) and/or Other Required Material
• Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed. Cambridge, MA: The MIT Press, 2009.
• Bradley N. Miller and David L. Ranum, Problem Solving With Algorithms and Data Structures Using Python, 2nd ed. Portland, OR: Franklin, Beedle & Associates, 2011.
Course Instructor(s)
Prof Ernest Chong, Prof Soh De Wen, Prof Cyrille Jegourel, Prof Dileepa Fernando, Prof Pritee Agrawal | {"url":"https://istd.sutd.edu.sg/undergraduate/courses/50004-algorithms","timestamp":"2024-11-14T17:48:37Z","content_type":"text/html","content_length":"122983","record_id":"<urn:uuid:c834c701-a587-4bc0-b0de-855ecafd57af>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00798.warc.gz"} |
A multi grid maximum likelihood reconstruction algorithm for positron emission tomography
The problem of reconstruction in Positron Emission Tomography (PET) is basically estimating the number of photon pairs emitted from the source. Using the concept of maximum likelihood (ML) algorithm,
the problem of reconstruction is reduced to determining an estimate of the emitter density that maximizes the probability of observing the actual detector count data over all possible emitter density
distributions. A solution using this type of expectation maximization (EM) algorithm with a fixed grid size is severely handicapped by the slow convergence rate, the large computation time, and the
non-uniform correction efficiency of each iteration making the algorithm very sensitive to the image-pattern. An efficient knowledge-based multi-grid reconstruction algorithm based on ML approach is
presented to overcome these problems.
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
• Computer Science Applications
• Applied Mathematics
• Electrical and Electronic Engineering
Dive into the research topics of 'A multi grid maximum likelihood reconstruction algorithm for positron emission tomography'. Together they form a unique fingerprint. | {"url":"https://researchwith.njit.edu/en/publications/a-multi-grid-maximum-likelihood-reconstruction-algorithm-for-posi","timestamp":"2024-11-05T18:41:23Z","content_type":"text/html","content_length":"49985","record_id":"<urn:uuid:11faa407-e383-4a8d-bb14-f0d8e3b7853d>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00426.warc.gz"} |
Within-Group vs. Between Group Variation in ANOVA | Online Tutorials Library List | Tutoraspire.com
Within-Group vs. Between Group Variation in ANOVA
by Tutor Aspire
A one-way ANOVA is used to determine whether or not the means of three or more independent groups are equal.
A one-way ANOVA uses the following null and alternative hypotheses:
• H[0]: All group means are equal.
• H[A]: At least one group mean is different from the rest.
Whenever you perform a one-way ANOVA, you will end up with a summary table that looks like the following:
We can see that there are two different sources of variation that an ANOVA measures:
Between Group Variation: The total variation between each group mean and the overall mean.
Within-Group Variation: The total variation in the individual values in each group and their group mean.
If the Between group variation is high relative to the Within-group variation, then the F-statistic of the ANOVA will be higher and the corresponding p-value will be lower, which makes it more likely
that we’ll reject the null hypothesis that the group means are equal.
The following example shows how to calculate the Between group variation and Within-group variation for a one-way ANOVA in practice.
Example: Calculating Within-Group and Between Group Variation in ANOVA
Suppose we want to determine if three different studying methods lead to different mean exam scores. To test this, we recruit 30 students and randomly assign 10 each to use a different studying
The exam scores for the students in each group are shown below:
We can use the following formula to calculate the between group variation:
Between Group Variation = Σn[j](X[j] – X..)^2
• n[j]: the sample size of group j
• Σ: a symbol that means “sum”
• X[j]: the mean of group j
• X..: the overall mean
To calculate this value, we’ll first calculate each group mean and the overall mean:
Then we calculate the between group variation to be: 10(80.5-83.1)^2 + 10(82.1-83.1)^2 + 10(86.7-83.1)^2 = 207.2.
Next, we can use the following formula to calculate the within group variation:
Within Group Variation: Σ(X[ij] – X[j])^2
• Σ: a symbol that means “sum”
• X[ij]: the i^th observation in group j
• X[j]: the mean of group j
In our example, we calculate within group variation to be:
Group 1: (75-80.5)^2 + (77-80.5)^2 + (78-80.5)^2 + (78-80.5)^2 + (79-80.5)^2 + (81-80.5)^2 + (81-80.5)^2 + (83-80.5)^2 + (86-80.5)^2 + (87-80.5)^2 = 136.5
Group 2: (78-82.1)^2 + (78-82.1)^2 + (79-82.1)^2 + (81-82.1)^2 + (81-82.1)^2 + (82-82.1)^2 + (83-82.1)^2 + (85-82.1)^2 + (86-82.1)^2 + (88-82.1)^2 = 104.9
Group 3: (82-86.7)^2 + (82-86.7)^2 + (84-86.7)^2 + (86-86.7)^2 + (86-86.7)^2 + (87-86.7)^2 + (87-86.7)^2 + (89-86.7)^2 + (90-86.7)^2 + (94-86.7)^2 = 122.1
Within Group Variation: 136.5 + 104.9 + 122.1 = 363.5
If we use statistical software to perform a one-way ANOVA using this dataset, we’ll end up with the following ANOVA table:
Notice that the between group and within-group variation values match the ones we calculated by hand.
The overall F-statistic in the table is a way to quantify the ratio of the between group variation compared to the within group variation.
The larger the F-statistic, the greater the variation between group means relative to the variation within the groups.
Thus, the larger the F-statistic, the greater the evidence that there is a difference between the group means.
We can see in this example that the p-value that corresponds to an F-statistic of 7.6952 is .0023.
Since this value is less than α = .05, we reject the null hypothesis of the ANOVA and conclude that the three studying techniques do not lead to the same exam score.
Additional Resources
The following tutorials provide additional information about ANOVA models:
Introduction to the One-Way ANOVA
How to Interpret the F-Value and P-Value in ANOVA
The Complete Guide: How to Report ANOVA Results
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How to Plot MATLAB Graph with Colors, Markers and Line Specification?
Do you want to make you MATLAB plot more colorful and descriptive?
Earlier we have seen How to draw a Graph in MATLAB?. Those were a typical single color graph as shown below.
This graph still looks good as you are drawing a single graph on the MATLAB display.
Why does everyone like to Plot MATLAB Graph with different Colors?
What if you are plotting multiple graphs on a single MATLAB display…
For plotting multiple graphs in a single window, it very difficult to distinguish one graph from another graph.
Let’s take an example.
You are plotting graphs for multiple mathematical equations like a sin wave, cos wave, exponential function on the same MATLAB display. After the running MATLAB program, you will get a number of
graphs on the single MATLAB display. The end user will get more confused and will find it more difficult to understand and distinguish multiple graphs.
So you need to decorate each graph differently, like assigning a different color to each curve.
Each color describes one graph and that makes the graph self-descriptive.
How can you decorate your MATLAB graph?
In this tutorial, you will learn to plot the colorful graphs in MATLAB.
I am also explaining by plotting a graph for the mathematical equation on MATLAB R2013a by using a single color, simple marker and line specification.
We will also see what are the most important and useful color coding functions, marker style and line-specification designing functions available in MATLAB.
By using these functions, you can draw the graph or waveform as per your color and plotting style choice. And you can easily understand the particular equation’s graph.
Let’s begin by considering the top three essential components to decorate your graph more meaningful.
• Colour
• Marker Style
• Line Specification
Explanation of these three component functions are one-by-one,
MATLAB Plot Colors to draw the Graph
If you are drawing any picture on paper, you have different color pencils to use.
Likewise, for plotting the graph on MATLAB, we have different colors code or functions.
Widely, eight colors are used for MATLAB graph. And each color has the corresponding color code.
The below table shows color specification with the color code.
Sr.No. Colour Name Colour Short Name RGB Triplet Hexadecimal Colour Code
[Useful in MATLAB Program]
1 Black k [0 0 0] ‘#000000’
2 Blue b [0 0 1] ‘#0000FF’
3 Green g [0 1 0] ‘#00FF00’
4 Cyan c [0 1 1] ‘#00FFFF’
5 Red r [1 0 0] ‘#FF0000’
6 Magenta m [1 0 1] ‘#FF00FF’
7 Yellow y [1 1 0] ‘#FFFF00’
8 White w [1 1 1] ‘#FFFFFF’
You can use these eight colors code to draw the colorful waveforms in MATLAB.
MATLAB Plot Marker | Different Style to Draw the Graph
Rather than just a simple line, do you want to make your waveform look different?
There are different marker style functions. For example, star format function, point format function, square format function, plus format function and so on.
In the below table, I am sharing the 12 marker style functions and its useful code for MATLAB graph.
Marker Style Code
Sr.No. Marker Style Name
[Useful in MATLAB Program]
1 Star *
2 Plus +
3 Point .
4 Circle o
5 Square s
6 Diamond d
7 Pentagram p
8 Hexagram h
9 Triangle (Right Position) >
10 Triangle (Left Position) <
11 Triangle (Up Position) ^
12 Triangle (Down Position) v
How does the graph look different after using these marker styles? This we will see later in this tutorial example.
MATLAB Plot Line Specification | Code for MATLAB Graph
The four different spaceline codes are used for the plotting waveform or graph.
Check this blow table, for line specification code.
Line Specification Code
Sr. No Line Name
[Useful in MATLAB Program]
1 Solid –
2 Dotted :
3 Dashed —
4 Dash-Dot -.
The syntax for plotting graph to add color, marker, and line specification:
plot (x, y, 'colour marker linespec')
These codes are placed inside single inverted comma.
Now its time to implement all three essentials components (color, marker, and line specifier) for decorating the MATLAB graph.
How to Plot MATLAB Graph with different colors, markers, and line specifier?
How to change Colour, Marker, and Line-Specification in MATLAB plot?
Let’s take these two mathematical equations to plot the MATLAB graph.
1) y(x)=sin(2x)
2) derivative of the same function d/dx(sin(2x)) on the same graph.
The first mathematical equation is trigonometric.
y1`= sin (2x)
And it’s derivative of a mathematical equation of y(x) is
y2= d/dx (y1)= 2 cos (2x)
MATLAB code:
Here is MATLAB code you can write to plot the graph for the function of f(x) and its d/dx (f(x)).
MATLAB PLot Colors code you can copy paste:
y1 = sin(2*x);
plot(x,y1,'r * -');
hold on
plot(x,y2,'k . :');
legend('sin', 'cos');
In this program, I have used the ‘legend’ function to label data series plotted on a graph. You can see in the below MATLAB output.
We are using different colors, markers and line specifications for plotting two different graphs.
MATLAB Output:
What’s Next:
I hope you learn to decorate our MATLAB graph with different colors, marker and line specifiers with the simple example of MATLAB graphs.
Now try MATLAB plot colors, marker styles and line specification on different MATLAB versions. Let’s make your graph more colorful.
Do you have any query? You can ask me by comment below.
Thanks for Reading!
I have completed master in Electrical Power System. I work and write technical tutorials on the PLC, MATLAB programming, and Electrical on DipsLab.com portal.
Sharing my knowledge on this blog makes me happy. And sometimes I delve in Python programming.
4 thoughts on “How to Plot MATLAB Graph with Colors, Markers and Line Specification?”
1. Thanks, Mam 🙂
□ You’re welcome, Dear:)
2. Great job, thank you.
□ Thanks 🙂
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Visualizing Data Uncertainty: An Experiment with D3.js
Start of Main Content
Visualizing Data Uncertainty: An Experiment with D3.js
In this post, I'd like to discuss some different ways to use uncertain data in simple visualizations. Although there can be value in data "vizzes" that tell a story, for this post I'll consider that
the purpose of a data viz is to:
• Tell the truth, the whole truth, and nothing but the truth
or more specifically,
• Visually convey the data as completely as possible, so not to mislead the viewer
Even with the best intentions, there are many ways in which one can unconsciously mislead the viewer. For my first example, before we get into uncertainty, we'll need to consider the difference
between data whose domain is continuous or discrete.
Discrete vs. Continuous Data Domains
A discrete domain means that the entities about which data is collected are separate - think "average American income in year X" or "height of each of your family members". It doesn't make sense to
ask about the average annual American income in the year 2013.57, because that quantity is an aggregation over the entire calendar year. Conversely, a continuous domain means that between any two
data points there could always be another, if we were able to measure it. Examples of this include the speed of your car or the number of children in hospitals at any given time.
The Discrete Equivalent of Dot Plots
Given that you are reading this, you are probably familiar with how to make a dot plot. Each data point has values from the domain and range (think x and y), and a dot is placed on the graph centered
at those coordinates. This can be misleading behavior for data with a discrete domain, because the data value applies to the entire corresponding domain entity. Consequentially, it is more accurate
for the plot to span the entire domain element; all points at any particular height above that element are equivalent.
The discrete equivalent of the dot plot, then, is a graph where each data point is plotted as a horizontal line segment:
The Notion of Uncertainty
Now let's introduce uncertainty to the mix. Traditionally, uncertainty is (if it's not just ignored!) conveyed graphically with error bars denoting the top and bottom of a "significant region",
generally the central 95% of the probability distribution of the data. The shortcomings of this are that the finer details about the probability distribution are lost, and the significance cutoff
value is arbitrary. Other options like box-and-whisker plots are slightly more informative, but are visually clunkier and still suffer from the same problems.
Probability Gradients: A Better Way
I believe we can do much better. The most common depiction of a probability distribution is as a curve, but this requires an extra graphical dimension and would add clutter. We can accurately depict
a distribution without adding dimensions by rendering it as a cloud. Conceptually, we'll plot a horizontal bar that spans each height where the distribution has a non-zero value, and shade it in
proportion to the probability density. In practice, this creates a rectangle whose opacity varies with height. Here are a few examples of this with normal, triangular, and uniform probability
The distributions are distinguishable from each other at a glance, and their differences are visible without needing to learn what a box or whiskers mean.
Here are three plots of normal distributions with the same mean values but decreasing variance:
Notice that when the uncertainty of data is small, the results approach the horizontal lines from our first example with exact data values. As a math nerd, this continuity makes me happy.
Applying the Concept to Bar Graphs
Now, if this is an uncertain dot plot, what would a bar graph look like? A data point in a conventional bar graph is represented by a bar that starts on the x-axis with full opacity, and ends at a
height corresponding to the data value. If our "data point" is a probability density function, then for our uncertain bar graph we should plot a bar whose opacity goes to zero as we pass through the
"data point". The way to express this mathematically is that the opacity should be equal to 1 minus the cumulative distribution function.
Below are bar graphs for the same data and distributions shown above - normal, triangular, and uniform. The differences between these are more subtle, because minor features such as corners in
functions become even more minor when the integral is taken.
Visualizing Data Uncertainty with D3
I've built a web toy that lets you play around with everything discussed in this post. It has a couple sample data sets built in, but try entering your own data and see what it looks like. So, enough
talk - go try it out yourself!
One thing you might notice about the exponential distribution is that the dot plot and bar graph look the same. A bit of calculus shows us that they are indeed identical. In the future, I think it
would be fun to explore applying this to continuous-domain data, or to allow the user to enter observed values and watch the graph build the probability cloud around them. | {"url":"https://www.velir.com/ideas/2013/07/11/visualizing-data-uncertainty-an-experiment-with-d3js","timestamp":"2024-11-11T18:16:13Z","content_type":"text/html","content_length":"133605","record_id":"<urn:uuid:8daf5d54-93ac-42af-bca9-6a0b74d9a790>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00643.warc.gz"} |
Can You Solve This Math Puzzle Equating 60-70÷2+4-45÷5=? - Comprehensive English Academy NYSE
Can You Solve This Math Puzzle Equating 60-70÷2+4-45÷5=?
Can You Solve This Math Puzzle Equating 60-70÷2+4-45÷5=? Explore this math puzzle that tests your thinking and can make you smarter. If you’re good at solving tricky problems like this, give it a
Puzzles that make you think in creative ways are enjoyable to solve. If you enjoy working on tricky puzzles and finding the answers, you should definitely try these. These puzzles not only keep your
mind busy but can also help you feel less stressed and tired. Take a look at the math puzzle below.
Can You Solve This Math Puzzle Equating 60-70÷2+4-45÷5=?
Math puzzles make you think critically and use math to find solutions. They challenge you to analyze information and apply math in creative ways.
The picture above shows a puzzle. To solve it, you need to figure out the hidden pattern it follows. But you have to be quick because time is running out. This challenge tests your thinking and
observation skills. This puzzle is a bit tricky, and it’s best for people who are good at noticing details. Mastering this puzzle can improve your problem-solving skills and make your mind sharper.
It can be helpful in school, work, and daily life. Even though the puzzle may seem tough, your goal is to find a solution that follows the rules and solves the puzzle. In the next section, we’ll
explain the exact nature of this math puzzle and the satisfying solution waiting for you.
Take your thinking game to new heights by exploring the carefully curated selection of mind-boggling Brain Teaser and math puzzles available on Fresherslive. Our diverse collection caters to all
levels of expertise.
Can You Solve This Math Puzzle Equating 60-70÷2+4-45÷5=? Solution
This math puzzle is pretty tricky, and we encourage you to give it a go and see if you can find the answer.
To figure out the math problem 60 – 70 ÷ 2 + 4 – 45 ÷ 5, you should follow the order of operations. First, do any calculations in parentheses (there aren’t any here). Next, do exponents, if there are
any. Then, do multiplication and division from left to right, and finally, do addition and subtraction from left to right. So, we start with the divisions: 70 ÷ 2 is 35, and 45 ÷ 5 is 9. Then, the
problem looks like this: 60 – 35 + 4 – 9. We go from left to right: 60 – 35 is 25, then adding 4 makes it 29. Finally, subtracting 9 from 29 gives us the answer: 20.
In short, the solution to 60 – 70 ÷ 2 + 4 – 45 ÷ 5 is 20.
Calculate the Total of 480 ÷ 20 + 16 x 2 – 144 ÷ 12=?
To solve this calculation, use the order of operations. Begin with the divisions and multiplications from left to right: 480 ÷ 20 equals 24, and 144 ÷ 12 equals 12. The equation becomes 24 + 16 x 2 –
12. Now, carry out the multiplications and additions/subtractions from left to right: 16 x 2 equals 32, and 24 + 32 equals 56. Therefore, the solution is 56.
Solve the Equation 504 ÷ 18 + 17 x 2 – 156 ÷ 13=?
For this problem, apply the order of operations. Start with the divisions and multiplications from left to right: 504 ÷ 18 equals 28, and 156 ÷ 13 equals 12. The equation becomes 28 + 17 x 2 – 12.
Next, perform the multiplications and additions/subtractions from left to right: 17 x 2 equals 34, and 28 + 34 equals 62. Thus, the answer is 62.
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Functions To Graph - Graphworksheets.com
Interpret Graphs Of Functions Algebra Worksheet – You’ve found the right place if you are looking for worksheets of graphing functions. There are many types of graphing function to choose from.
Conaway Math offers Valentine’s Day-themed worksheets with graphing functions. This is a great way for your child to learn about these functions. Graphing functions … Read more | {"url":"https://www.graphworksheets.com/tag/functions-to-graph/","timestamp":"2024-11-14T10:15:20Z","content_type":"text/html","content_length":"46201","record_id":"<urn:uuid:6c4f25ba-a2fc-47c5-bffb-a716e47ae64e>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00528.warc.gz"} |
Trudy Mat. Inst. Steklova, 1999, Volume 225
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| General information | Contents |
Solitons, geometry, and topology: on the crossroads
Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov
Volume Editor: V. M. Buchstaber^
Editor in Chief: E. F. Mishchenko^
Abstract: This volume is dedicated to the 60th birthday of academician Sergei Petrovich Novikov, an outstanding scientist and founder of a number of modern trends in mathematics and mathematical
physics. The papers included in this volume focus on the topical problems of the soliton theory, theory of dynamical systems, and theory of smooth manifolds related to the problems of mathematical
physics. Among the authors are the most prominent specialists in this rapidly developing field.
This volume is addressed to specialists, postgraduates, and senior students who are interested in interdisciplinary problems bordering between mathematics and theoretical physics.
ISBN: 5-02-002404-X
Full text: Contents
Citation: Solitons, geometry, and topology: on the crossroads, Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov, Trudy Mat. Inst. Steklova, 225, ed. V.
M. Buchstaber, E. F. Mishchenko, Nauka, MAIK «Nauka/Inteperiodika», M., 1999, 400 pp.
Citation in format AMSBIB:
\book Solitons, geometry, and topology: on the crossroads
\bookinfo Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov
\serial Trudy Mat. Inst. Steklova
\yr 1999
\vol 225
\publ Nauka, MAIK «Nauka/Inteperiodika»
\publaddr M.
\ed V.~M.~Buchstaber, E.~F.~Mishchenko
\totalpages 400
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Review databases:
Additional information
Solitons, Geometry, and Topology: on the Crossroads
Collection of Papers Dedicated to the 60th Anniversary of Academician Sergei Petrovich Novikov
This volume is dedicated to the 60th birthday of academician Sergei Petrovich Novikov, an outstanding scientist and founder of a number of modern trends in mathematics and mathematical physics. The
papers included in this volume focus on the topical problems of the soliton theory, theory of dynamical systems, and theory of smooth manifolds related to the problems of mathematical physics. Among
the authors are the most prominent specialists in this rapidly developing field.
This volume is addressed to specialists, postgraduates, and senior students who are interested in interdisciplinary problems bordering between mathematics and theoretical physics. | {"url":"https://www.mathnet.ru/php/contents.phtml?wshow=issue&bookID=238&jrnid=tm&year=1999&volume=225&series=0&option_lang=eng","timestamp":"2024-11-14T07:16:09Z","content_type":"text/html","content_length":"23196","record_id":"<urn:uuid:734b5ae4-dd81-4509-86dd-b8cd76c5587a>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00139.warc.gz"} |
Math, Grade 6, Distributions and Variability, An Introduction To Mean Absolute Deviation (MAD)
Calculate the Mean Absolute Deviation
Work Time
Calculate the Mean Absolute Deviation
Follow these steps to find the MAD for each class’s line plot:
• List the data for the class vertically.
• Next to each score, write the distance of that score from the mean.
• Find the mean of these distances by adding them together and dividing by the number of values.
Remember that distances are always positive numbers. | {"url":"https://openspace.infohio.org/courseware/lesson/2169/student/?section=4","timestamp":"2024-11-02T20:13:10Z","content_type":"text/html","content_length":"34655","record_id":"<urn:uuid:378a7b29-e1ba-42ab-b8c4-c92c0fac1b92>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00039.warc.gz"} |
What is Velocity Ratio (VR)
What is Velocity Ratio (VR)?
The Velocity Ratio is attempt to estimate the velocity of a given sail boat.
The VR for a good relatively fast boat will be about 1.0. A full-blown race boat will be in the range 1.5 to 1.8.
How to calculate the Velocity Ratio (VR)?
Velocity Ratio = 1.88*lwl^.5*sail area^.33/disp^.25 / (1.34*lwl^.5)
Velocity Ratio formula
Abrev. Unit Description
lwl ft Length of the boat in the waterline
SA ft^2 Sail Area
Displacement lbs Displacement of the boat in pounds
Problems with the formula?
The main problem with the formula is that "sail area" and "displacement" are not clearly defined. This means that the SA/D calculation may result in different values for the same boat.
Sail area:
Which sails is the calculation based on?
- mainsail and fok?
- mailsail and genoa? And if 'yes', which genoa?
- Is it the displacement for an empty boat?
- Is the motor included and what about the fuel?
- Are the weight of the crew included?
- What about water, food and the like?
In our reviews - when comparing boat types - we have decided to use the nominal sail area as defined in the ISO 8666
standard, where the sail area is defined as (P*E + I*J)/2. where: | {"url":"https://www.yachtdatabase.com/en/encyclvr.jsp","timestamp":"2024-11-08T09:10:46Z","content_type":"text/html","content_length":"14457","record_id":"<urn:uuid:01833108-b50f-495c-a230-4333756dda45>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00326.warc.gz"} |
Using the Math and Chemistry Equation Editor Function
TeX Notation
Moodle supports a script called TeX notation. TeX notation enables instructors to create quiz questions that include correct and clear mathematical symbols that look just like what students and
instructors are used to working within the classroom and assignments.
There are two options instructors can use Host Math or the built-in Moodle Equation Editor and both create TeX Notation.
Using Host Math
This tool can be found here: Host Math
1. Type in numbers at the top of the tool, and use the box on the right to select the notation you want to include. The correct script will insert itself in the formula, and the bottom box will show
how it will appear in the Moodle question.
2. When the formula is complete, copy and paste the script into the Moodle question.
3. Add $$ to the beginning and end of the script with no spaces.
Detailed instructions for using TeX in Moodle can be found here - Using TeX Notation
Using the Equation Editor Inside of Moodle
Get started using Moodle's equation editor:
1. In any text area or resource in Moodle start editing.
2. Click the expand button in the text editor.
3. Find the calculator(Math Equation Editor) or flask(Chemistry Editor) butttons towards the end of the button options. For this example, we will use the Math Equation Editor.
For Simple Equation Building
1. Type a number for your equation into the text box below the operator options.
2. Click the button of the operator you want to use.
3. Now complete your equation by adding another number and any additional operators you would like.
1. You can see a preview of what the students will see before you save by looking at the equation Preview below the text box!
4. Once ready click Save Equation to add it officially to your text area.
For Creating More Advanced Formulas
You can use the tabs at the top of the Equation Editor pop-up to access more symbols and equation-building tools.
Going to the Advanced tab gives you premade equations. To build with these:
1. Click on an equation you want to use.
2. Change the letter to the number you want in its place.
You can mix and match these different options:
1. Select one equation to start with.
2. Copy the portion you want to keep.
3. Clear the rest and click on the second equation you want to combine what you copied from the last.
4. Paste what you copied in place of what you don't want.
5. Replace any letters with numbers, preview it at the bottom, and click Save your Equation.
ATC Support & Hours of Operation
Weekday Support, Monday - Friday
The ATC is open to in-person assistance. Support is available through the above remote options and on campus at CEN 208 | {"url":"https://support.lanecc.edu/en_US/moodle-faculty-general/using-the-math-equation-editor-function","timestamp":"2024-11-10T09:16:02Z","content_type":"text/html","content_length":"63178","record_id":"<urn:uuid:201a68a6-e345-4692-ae4e-8f4968aad486>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00097.warc.gz"} |
gendistance: Generate a Distance Matrix in nbpMatching: Functions for Optimal Non-Bipartite Matching
The gendistance function creates an (N+K)x(N+K) distance matrix from an NxP covariates matrix, where N is the number of subjects, P the number of covariates, and K the number of phantom subjects
requested (see ndiscard option). Provided the covariates' covariance matrix is invertible, the distances computed are Mahalanobis distances, or if covariate weights are provided, Reweighted
Mahalanobis distances (see weights option and Greevy, et al., Pharmacoepidemiology and Drug Safety 2012).
gendistance( covariate, idcol = NULL, weights = NULL, prevent = NULL, force = NULL, rankcols = NULL, missing.weight = 0.1, ndiscard = 0, singular.method = "solve", talisman = NULL, prevent.res.match
= NULL, outRawDist = FALSE, ... )
covariate A data.frame object, containing the covariates of the data set.
idcol An integer or column name, providing the index of the column containing row ID's.
weights A numeric vector, the length should match the number of columns. This value determines how much weight is given to each column when generating the distance matrix.
A vector of integers or column names, providing the index of columns that should be used to prevent matches. When generating the distance matrix, elements that match on these
prevent columns are given a maximum distance.
force An integer or column name, providing the index of the column containing information used to force pairs to match.
rankcols A vector of integers or column names, providing the index of columns that should have the rank function applied to them before generating the distance matrix.
A numeric value, or vector, used to generate the weight of missingness indicator columns. Missingness indicator columns are created if there is missing data within the data set.
missing.weight Defaults to 0.1. If a single value is supplied, weights are generating by multiplying this by the original columns' weight. If a vector is supplied, it's length should match the
number of columns with missing data, and the weight is used as is.
ndiscard An integer, providing the number of elements that should be allowed to match phantom values. The default value is 0.
singular.method A character string, indicating the function to use when encountering a singular matrix. By default, solve is called. The alternative is to call ginv from the MASS package.
An integer or column name, providing location of talisman column. The talisman column should only contains values of 0 and 1. Records with zero will match phantoms perfectly, while
talisman other records will match phantoms at max distance.
An integer or column name, providing location of the column containing assigned treatment groups. This is useful in some settings, such as trickle-in randomized trials. When set,
prevent.res.match non-NA values from this column are replaced with the value 1. This prevents records with previously assigned treatments (the ‘reservior’) from matching each other.
outRawDist a logical, indicating if the raw distance matrix should also be returned. The raw form is before distance modifiers such as ‘prevent’ take effect.
... Additional arguments, not used at this time.
A data.frame object, containing the covariates of the data set.
An integer or column name, providing the index of the column containing row ID's.
A numeric vector, the length should match the number of columns. This value determines how much weight is given to each column when generating the distance matrix.
A vector of integers or column names, providing the index of columns that should be used to prevent matches. When generating the distance matrix, elements that match on these columns are given a
maximum distance.
An integer or column name, providing the index of the column containing information used to force pairs to match.
A vector of integers or column names, providing the index of columns that should have the rank function applied to them before generating the distance matrix.
A numeric value, or vector, used to generate the weight of missingness indicator columns. Missingness indicator columns are created if there is missing data within the data set. Defaults to 0.1. If a
single value is supplied, weights are generating by multiplying this by the original columns' weight. If a vector is supplied, it's length should match the number of columns with missing data, and
the weight is used as is.
An integer, providing the number of elements that should be allowed to match phantom values. The default value is 0.
A character string, indicating the function to use when encountering a singular matrix. By default, solve is called. The alternative is to call ginv from the MASS package.
An integer or column name, providing location of talisman column. The talisman column should only contains values of 0 and 1. Records with zero will match phantoms perfectly, while other records will
match phantoms at max distance.
An integer or column name, providing location of the column containing assigned treatment groups. This is useful in some settings, such as trickle-in randomized trials. When set, non-NA values from
this column are replaced with the value 1. This prevents records with previously assigned treatments (the ‘reservior’) from matching each other.
a logical, indicating if the raw distance matrix should also be returned. The raw form is before distance modifiers such as ‘prevent’ take effect.
Given a data.frame of covariates, generate a distance matrix. Missing values are imputed with fill.missing. For each column with missing data, a missingness indicator column will be added. Phantoms
are fake elements that perfectly match all elements. They can be used to discard a certain number of elements.
dist generated distance matrix
cov covariate matrix used to generate distances
ignored ignored columns from original covariate matrix
weights weights applied to each column in covariate matrix
prevent columns used to prevent matches
mates index of rows that should be forced to match
rankcols index of columns that should use rank
missing.weight weight to apply to missingness indicator columns
ndiscard number of elements that will match phantoms
rawDist raw distance matrix, only provided if ‘outRawDist’ is TRUE
set.seed(1) df <- data.frame(id=LETTERS[1:25], val1=rnorm(25), val2=rnorm(25)) # add some missing data df[sample(seq_len(nrow(df)), ceiling(nrow(df)*0.1)), 2] <- NA df.dist <- gendistance(df, idcol=
1, ndiscard=2) # up-weight the second column df.weighted <- gendistance(df, idcol=1, weights=c(1,2,1), ndiscard=2, missing.weight=0.25) df[,3] <- df[,2]*2 df.sing.solve <- gendistance(df, idcol=1,
ndiscard=2) df.sing.ginv <- gendistance(df, idcol=1, ndiscard=2, singular.method="ginv")
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/nbpMatching/man/gendistance.html","timestamp":"2024-11-09T03:28:18Z","content_type":"text/html","content_length":"28739","record_id":"<urn:uuid:9371a6bf-7492-4c45-8a90-2320945fa8f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00593.warc.gz"} |
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It has been quite a century for our understanding of the cosmos. As I write these words at the beginning of 2017 it was just over 100 years ago, in November 1915, that Albert Einstein finished
development of his general theory of relativity. Among many other things, this theory provided the proper context for interpreting Edwin Hubble's distance-redshift law, published in 1929, as due to
the expansion of the Universe. In the 40s, Gamow and Alpher speculated that the dense conditions that must have existed earlier in this expanding universe could provide another site, in addition to
the cores of stars, for the fusion of light elements to heavier ones. In fact, to avoid over-production of the elements, this earlier, denser phase would have to be very hot. In the 1960s Bell Labs
scientists accidentally stumbled upon the thermal radiation left over from that heat, that exists in the current epoch as a nearly uniform microwave glow. With that discovery, the idea that the
universe used to be hot, dense, and expanding very rapidly became the dominant cosmological paradigm known as the "Big Bang."
The subsequent fifty years of the past century saw much progress as well. We now know that we do not know what constitutes 95% of the mass/energy in the universe. Only 5% of the mass/energy is
composed of constituents in the particle physicist's standard model. Most of the rest is "dark energy" which smoothly fills the universe and dilutes only slowly, if at all, as the universe expands.
The rest is "dark matter" that, like George Lucas's mystical Force, "pervades us and binds the galaxy together." Measurements of light element abundances, combined with modern, precision, version's
of Gamow and Alpher's big bang nucleosynthesis calculations, give us confidence we understand the expansion back to an epoch when the presently observable universe was \(10^{27}\) times smaller in
volume than it is now. A speculative theory, known as cosmic inflation, has met with much empirical success, giving us some level of confidence we may understand something about events at yet higher
densities and even earlier times.
In this quarter-long course we will at least touch upon all the topics in the above two paragraphs. We will learn how to think about the expanding universe using concepts from Einstein's theory of
general relativity. We will use Newtonian gravity to derive the dynamical equations that relate the expansion rate to the matter content of the universe. Connecting the expansion dynamics to
observables such as luminosity distances and redshifts, we will see how astronomers use observations to probe these dynamics, and thereby the contents of the cosmos, including the mysterious dark
We will introduce some basic results of kinetic theory to understand why big bang nucleosynthesis leads to atomic matter that is, by mass, about 25% Hydrogen, 75% Helium with only trace amounts of
heavier elements. We'll use this kinetic theory, applied to atomic rather than nuclear reactions, to explore perhaps the most informative cosmological observable: the cosmic microwave background.
Finally, we will study how an early epoch of inflationary expansion, driven by an exotic material with negative pressure, can explain some of the otherwise puzzling features of the observed universe.
We claim to know the composition of the universe at this early time, dominated almost entirely by thermal distributions of photons and subatomic particles called neutrinos. We know in detail
many aspects of the evolutionary process that connects this early universe to the current one. Our models of this evolution have been highly predictive and enormously successful.
We provide an overview of our subject, broken into two parts. The first is focused on the discovery of the expansion of the universe in 1929, and the theoretical context for this discovery,
which is given by Einstein's general theory of relativity (GR). The second is on the implications of this expansion for the early history of the universe, and relics from that period
observable today such as the cosmic microwave background and the lightest chemical elements.
This chapter is entirely focused on the Euclidean geometry that is familiar to you, but reviewed in a language that may be unfamiliar. The new language will help us journey into the foreign
territory of Riemannian geometry. Our exploration of that territory will then help you to drop your pre-conceived notions about space and to begin to understand the broader possibilities --
possibilities that are not only mathematically beautiful, but that appear to be realized in nature.
We introduce the notion of "curvature'' in an attempt to loosen up your understanding of the nature of space, to have you better prepared to think about the expansion of space.
We now extend our discussion of spatial geometry to spacetime geometry. We begin with Galilean relativity, which we will then generalize in the next section to Einstein (or Lorentz)
The Maxwell equations are inconsistent with Galilean relativity. Here we review Einstein's solution to this problem, which preserves the principle of relativity, and replaces the Galilean
transformation with a Lorentz transformation.
We begin our exploration of physics in an expanding spacetime with a spacetime with just one spatial dimension that is not expanding: a 1+1-dimensional Minkowski spacetime. We then generalize
it slightly so that the spatial dimension is expanding. After introduction of notions of age and "past horizon," we go on to calculate these quantities for some special cases.
We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe. In this and the next two chapters we derive Hubble's Law, \(v = H_0 d
\), and a more general version of it valid for arbitrarily large distances.
The consequences of expansion are recorded in the relationship between distance and redshift. Here we introduce the so-called standard candle method of distance determination. We work out the
theoretical relationship between flux, luminosity, curvature constant k, coordinate distance between observer and source, and the redshift of the source. This moves us one step closer to
being able to infer the expansion history from observations.
We complete the work begun in the previous chapter of creating a framework for inferring the expansion history from observations of standard candles over a range of redshifts and distances.
We do so by relating, for a given object, the coordinate distance, \(d\), and its redshift \(z\), to the curvature constant and the changing value of the expansion rate between the time the
light left the object and our reception of it.
In the following set of chapters we will derive the dynamical equations that relate the matter content in a homogeneous and isotropic universe to the evolution of the scale factor over time.
Retreating to the use of Newtonian concepts, we show that for a universe to be filled with an expanding fluid that remains homogeneous over time, the flow must be what we call a Hubble flow,
with relative velocities proportional to distance. Thus we derive Hubble's law using Newtonian concepts, setting ourselves up for the next chapter in which we use Newtonian dynamics to relate
the expansion rate to the contents of the cosmos.
Sticking with our Newtonian expanding universe, we will now derive the Friedmann equation that relates how the scale factor changes in time to the mass/energy density. We will proceed by
using the Newtonian concept of energy conservation. (You may be surprised to hear me call this a Newtonian concept, but the fact is that energy conservation does not fully survive the
transition from Newton to Einstein).
We investigate how an observer, at rest in their local rest frame, will observe the evolution of peculiar velocities of free particles. In a Newtonian analysis, the local rest frame will be
an inertial frame (one in which Newton's laws of motion apply) only if there is no acceleration of the scale factor (\ \ddot a = 0\). We discuss the difference with a relativistic analysis.
This chapter can be skipped without harming preparation for subsequent chapters.
We have seen that the rate of change of the scale factor depends on the mass density \( \rho \). In order to determine how the scale factor evolves with time, we thus need to know how the
density evolves as the scale factor changes, a subject we investigate here.
The lack of energy conservation in an expanding universe is quite surprising to people with any training in physics and therefore merits some discussion, which we present here in this
chapter. The student could skip this chapter and proceed to 15 without serious harm. If, subsequently, the lack of energy conservation becomes too troubling, know that this chapter is here
for you.
We apply local conservation of energy, valid in general relativity, to infer how density changes in response to scale factor changes, a response that depends on the relationship between
pressure and density.
There are a bewildering array of different kinds of distances in cosmology. We catalog them here as a resource for you as needed. We also introduce and define other related astronomical
technical terms: apparent and absolute magnitudes.
Key to observing the consequences of this expansion is the ability to measure distances to things that are very far away. Here we cover the basics of how that is done. We have to do it in
steps, getting distances to nearby objects and then using those objects to calibrate other objects that can be used to get to even further distances. We refer to this sequence of distance
determinations as the distance ladder.
We introduce the reader to the exciting subject of generating new knowledge from data, and the process of Bayesian inference in particular. We apply it to the inference of cosmological
parameters from data that were reduced from supernova observations.
To understand the "primordial soup" and its relics, we now turn our attention from a relativistic understanding of the curvature and expansion of space, to statistical mechanics. We begin
with equilibrium statistical mechanics, before moving on to a discussion of departures from equilibrium. We will study the production in the big bang of helium, photons, other "hot" relics
such as neutrinos, and "cold" relics such as the dark matter, and the relevant observations that test our understanding.
Out of the early Universe we get the light elements, a lot of photons and, as it turns out, a bunch of neutrinos and other relics of our hot past as well. To understand the production of
these particles we now turn to the subject of equilibrium statistical mechanics.
At sufficiently high temperatures and densities, reactions that create and destroy particles can become sufficiently rapid that an equilibrium abundance is achieved. In this chapter we assume
that such reaction rates are sufficiently high and work out the resulting abundances as a function of the key controlling parameter. We will thus see how equilibrium abundances change as the
universe expands and cools.
As the temperature and density drops, the reactions necessary to maintain chemical equilibrium can become too slow to continue to do so. This departure from equilibrium can occur while the
particles are relativistic, in which case we have "hot relics," or when the particles are non-relativistic in which case we say we have "cold relics." The cosmic microwave and neutrino
backgrounds are hot relics. The dark matter may be a cold relic.
This chapter does not yet exist. We intend to include here a summary of the thermal history of the cosmos assuming the standard cosmological model.
Big Bang Nucleosynthesis is the process by which light elements formed during the Big Bang. The agreement between predicted abundances and inferences from observations of primordial
(pre-stellar) abundances is a major pillar of the theory of the hot big bang and reason we can speak with some confidence about events in the primordial plasma in the first few minutes of the
expansion. Elements created at these very early times include Deuterium, Helium-3, Lithium-7, and, most abundantly, Helium-4.
Predicted in the late 1940s, and discovered accidentally in the 1960s, the Cosmic Microwave Background (CMB) is a cornerstone of the edifice of modern cosmology. We review its discovery and
then present the "surface of last scattering"; the thin shell around us, at a distance now of about 46 billion light years, where most of the CMB photons we see today last interacted with
matter. We discuss its high degree of isotropy, reflecting the high degree of homogeneity in the early universe.
Here we will introduce you to a physical system that is a beautiful gift of nature: the plasma that existed from the first fractions of a second of the Big Bang until it transitioned to a
neutral gas 380,000 years later. Gently disturbed away from equilibrium by mysterious, very early-universe processes, the plasma is an unusually simple, natural system whose dynamics are
calculable and also observable in maps of CMB intensity and polarization.
No other natural source of radiation has ever been measured to be as consistent with black-body radiation as is the case with the CMB. Here we look at the measurements of the spectrum from a
Nobel-prize winning instrument on the COBE satellite, before turning to the question of why the CMB is so near to being a black body and what we can learn from that.
We use Fourier methods to solve for the evolution of \(\Psi(x,t)\) assuming it obeys a wave equation and that we are given appropriate initial conditions. Fourier methods have a broad range
of applications in both experimental and theoretical physics, and other sciences as well. For the student of physics, time spent developing facility with the Fourier transform is time well
We explain the origin of the peaks in the CMB power spectrum as arising from acoustic dynamics in the primordial plasma.
Under the gravitational influence of dark matter, small fluctuations in the matter density field evolve. Particularly dense regions collapse into nonlinear, self-gravitating systems called
dark matter halos, which form the nodes of the web of galaxies that cosmologists observe today.
After recombination, baryons fall into the gravitational potential wells provided by dark matter halos, beginning the process of star and galaxy formation. This Chapter explores how galaxies
form and evolve, and how they relate to dark matter halos.
We begin our exploration of physics in an expanding spacetime with a spacetime with just one spatial dimension that is not expanding: a 1+1-dimensional Minkowski spacetime. We then generalize
it slightly so that the spatial dimension is expanding. After introduction of notions of age and "past horizon," we go on to calculate these quantities for some special cases.
We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe. In this and the next two chapters we derive Hubble's Law, \(v = H_0 d
\), and a more general version of it valid for arbitrarily large distances.
We begin to work out observational consequences of living in an expanding spatially homogeneous and isotropic universe. In this and the next two chapters we derive Hubble's Law, \(v = H_0 d
\), and a more general version of it valid for arbitrarily large distances.
This chapter is entirely focused on the Euclidean geometry that is familiar to you, but reviewed in a language that may be unfamiliar. The new language will help us journey into the foreign
territory of Riemannian geometry. Our exploration of that territory will then help you to drop your pre-conceived notions about space and to begin to understand the broader possibilities --
possibilities that are not only mathematically beautiful, but that appear to be realized in nature.
Thumbnail: This is a modification of the Flammarion Woodcut is an enigmatic woodcut by an unknown artist. The woodcut depicts a man peering through the Earth's atmosphere as if it were a curtain to
look at the inner workings of the universe. The original caption below the picture (not included here) translated to: "A medieval missionary tells that he has found the point where heaven and Earth | {"url":"https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_156_-_A_Cosmology_Workbook/Workbook","timestamp":"2024-11-06T05:42:25Z","content_type":"text/html","content_length":"168753","record_id":"<urn:uuid:6ca4c3ed-ba34-4593-a114-c1a1ce528aa2>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00829.warc.gz"} |
Arithmetic Expressions? What? This is nothing more than a fancy word for “calculating”. Though, it covers a little more than only working with numbers. Arithmetic expressions can be used with number
data types, booleans and with strings.
Addition and Subtraction
The most basic and most intuitive arithmetic expressions are addition (indicated with a plus “+”) and subtraction (indicated with a minus “-“). They can be used for integers and floats.
a = 5 + 3
b = 7.3 + 32.9
c = 43 - 3.6
Note: In the examples, the results of the calculations are stored in variables as you have not seen how to use the results directly, but basically you can also use the result of an arithmetic
expression as an input for functions or use them otherwise.
The addition expression can also be used on strings. Obviously you can’t calculate with words and characters, but you can add them together to form a new string. This is called “concatenation”.
first_name = "Harry"
last_name + "Hendrickson"
full_name = first_name + last_name
Note: In this example, the resulting string would be “HarryHendrickson” as there is no white space between the words. This needs to be added separately. You could add a pair of (double) quotes
containing a single white space character to separate the two words.
full_name = first_name + " " + last_name
Note: You can’t add a number and a string. Here it is important to distinguish a number (e.g. 5) from a number in form of a string (e.g. ‘5’) as they will not work together. The reason for this is
that Python will not be able to know if you want to use the string as a number (e.g. to create 10 as a result) or if you want to use the number as a string (e.g. to create the string ’55’ as a
Multiplication and Division
Another very intuitive expression is multiplication (indicated with an asterisk “*”). Multiplication is used to multiply numbers to get the resulting product.
a = 2 * 4
b = a * 3.4
c = 15 * (-0.5)
Like with addition, you can use the multiply operator for strings as well. Here, the string will be repeated the amount of times specified by the number after the multiplication symbol.
a = "ha"*3
b = 5*'Nope! '
The examples above will result in “a = ‘hahaha‘ ” and “b = ‘Nope! Nope! Nope! Nope! Nope! ‘ “.
The division (indicated with a forward slash “/”) behaves differently for integers and floats and also different for Python 2 and Python 3. As you are learning Python 2 in this guide (as it is used
in ROS), it does make a difference if you divide integers of floats. (For Python 3, there is basically no difference.)
When using the division operator with floating point numbers, the result will be exactly what you expect. The result is the result of the first number divided by the second number. The result will be
a float type number regardless if the resulting number has a decimal value or not. This holds even true if only one of the numbers is a floating point number and the other is an integer type.
a = 10.5 / 2.0
b = 4 / 2.0
In the above example, “a = 5.25” and “b = 2.0”. Everything as you would expect. In Python 2, when dividing an integer by another integer, the result will always be an integer.
In the above example, “c = 5” and “d = 3”. Wait what?! As you can see “c” is being calculated as it should, but the variable “d” is wrong? What is happening is, that the result is being forced to be
an integer value which means that everything behind the decimal point will be lost, thus “0.5” is just being lost. (In Python 3, the result would simple be a float instead.)
Okay, not it will become a little more complicated. The modulo operator is very useful to verify the property of some numbers like to see if it is an even number or not. It is only used for integer
type numbers and represents the remainder of an Euclidean division which is defined by:
a = b * q + r
where “q” is the quotient and “r” the remainder. “a” is the initial number that is divided and “b” is the number that “a” is divided by. In other words, if “a” is divided by “b”, the result is “q” if
“a” is dividable by “b”. If not, “q” would be a number with a decimal value. But in the formula above, “q” is always an integer which means that there is a remainder “r” that will be left out. If “r”
is zero, “a” can be divided by “b”.
The modulo operator is indicated with a procent sign (“%”). To stick with the same variable names as the example formula:
r = a % b
pizza_left = 8 % 3
One example where you also can use the modulo operator is to keep values within a specific range. For example, when a wheel is spinning, it will turn 360 degrees before it returns to the same
position. So if it rotates 380 degrees or 20 degrees, the position is the same. Hence:
resulting_angle = 380 % 360
number_of_rotations = 380 % 360
As you can see, the integer division and the modulo operator can go quite well hand in hand.
Exponentiation is a way to calculate a number multiplied with itself a certain amount of times. This can be part of a formula such as A = r²*Pi which equals to:
This is quickly done for a small exponent such as given above (the exponent is equal to 2). For the propper way to write exponentiation, the exponent can either be expressed with the math module (you
will see about modules later) or by a simple build in notation with two asterisks.
The above solution would be the prefered way to express exponential expressions.
Square Roots
To calculate the square root, again you could use the math module of Python or you can use a simple trick. When you think about the square root of a number, you can also express it as a number with
the exponent of 0.5 which will get you the same result.
This notation will probably be the most simple one.
This is about everything you need to know about the necessary tools for creating arithmetic expressions in Python 2.
Continue learning about inputs and outputs or go back to revisit complex data types. | {"url":"https://davesroboshack.com/tag/operations/","timestamp":"2024-11-06T12:13:57Z","content_type":"text/html","content_length":"58723","record_id":"<urn:uuid:fe401776-eba4-4ff6-8ef7-dc81d3d4294b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00321.warc.gz"} |
Load Curve and Load Duration Curve - Electrical Concepts
Load Curve and Load Duration Curve
Load curve is the variation of load with time on a Power Station. As the load on a Power Station never remain constant rather it varies time to time, these variations in load is plotted on half
hourly or hourly basis for the whole day. The curve thus obtained is known as Daily Load Curve.
Therefore, by having a look at the Load Curve, we can check the peak load on a Power Station and its variation. From the figure below, it is quite clear that the peak load (6 MW) on a particular
Power Station is at 6 P.M.
The monthly load curve can be plotted using the daily load curve for a particular month. For this purpose the average load for different time for the whole month is calculated and the value thus
obtained is plotted against time to get the Monthly Load Curve. Monthly Load Curve is used to fix the rate of energy.
In the same manner Yearly Load Curve can be obtained using the 12 monthly load curves. The Yearly Load Curve is used for calculation the Annual Load Factor.
Importance of Load Curve:
• From the daily load curve we can have insight of load at different time for a day.
• The area under the daily load curve gives the total units of electric energy generated.
Units Generated / day = Area under the daily Load Curve in kW
• The peak point on the daily load curve gives the highest demand on the Power Station for that day.
• The average load per day on the Power Station can be calculated using the daily load curve.
• Average load = Area under the daily Load Curve (kWh)/ 24 hrs.
• Load curve helps in deciding the size and number of Generating Units.
• Load Factor = Avg. Load / maximum Load = Avg. Load x24 / 24xmaximum Load
= Area under daily Load Curve/Area of Rectangle having Daily Load Curve
• Load curve helps in the preparing the operation schedule of the generating units.
Load Duration Curve:
Load Duration Curve is the plot of Load versus time duration for which that load was persisting. Load Duration Curve is obtained from the Daily Load Curve as shown in figure below.
From the above Load Duration Curve, it is clear that 20 MW of Load is persisting for a period of 8 hours, 15 MW of Load for 4 hours and so on.
It is also quite clear that, the area under the load duration curve is equal to the daily load curve and gives the number of units (kWh) generated for a given day. The load duration curve can be
extended for any period of time i.e. it can be drawn for a month or for year too.
8 thoughts on “Load Curve and Load Duration Curve”
1. good explanation
2. good explanation…
3. Good Explanation
□ Thank you Sagnik!
4. Thanks, it help me a lot
Well explained
5. Very clear
6. clear explanation
□ Thank you! Kindly share if you like the post. It will help us to expand our reach.
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Pie Charts
Pie Charts
Pie charts are easy to make, easy to read, and very popular. They are used to represent categorical data or values of variables. They are basically circles that are divided into segments or
categories which reflect the proportion of the variables in relation to the whole. Percentages are used to compare the segments, with the whole being equal to 100%.
To make a pie chart, draw a circle with a protractor. Then, convert the measures of the variables into percentages, and divide the circle accordingly. It is best to order the segments clockwise from
biggest to smallest, so that the pie chart looks neat and the variable are easy to compare. It is also recommended to write percentage and category labels next to each segment, so that users are not
required to refer to the legend each time they want to identify a segment.
Pie charts are popular types of graphs, but they do have disadvantages that limit their use. For this reason, scientists are not fans of pie charts. First of all, pie charts with too many segments
look very messy and are difficult to understand; therefore it is best to use pie charts when there are less than five categories to be compared. Further, if the values of the categories are very
close, the pie chart would be difficult to decipher because the segments would be too close in size. Variations of pie charts include the polar area diagrams and cosmographs.
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Multiple-Choice Randomization
Ian McLeod, Ying Zhang, and Hao Yu
University of Western Ontario
Journal of Statistics Education Volume 11, Number 1 (2003), jse.amstat.org/v11n1/mcleod.html
Copyright © 2003 by Ian McLeod, Ying Zhang and Hao Yu, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written
consent from the authors and advance notification of the editor.
Key Words: Academic integrity; Answer-options arrangement; Cheating; Student evaluation; Teaching large classes.
Multiple-choice randomized (MCR) examinations in which the order of the items or questions as well as the order of the possible responses is randomized independently for every student are discussed.
This type of design greatly reduces the possibility of cheating and has no serious drawbacks. We briefly describe how these exams can be conveniently produced and marked. We report on an experiment
we conducted to examine the possible effect of such MCR randomization on student performance and conclude that no adverse effect was detected even in a rather large sample.
1. Introduction
The multiple-choice test is widely used in all school subjects and at all educational levels for measuring a variety of teaching objectives. Many jurisdictions now require standardized testing in
order to graduate from high school and these tests frequently have a multiple-choice component. Multiple-choice examinations are used to evaluate student progress in many undergraduate courses in
statistics as well.
To overcome cheating, instructors often prepare several versions of these exams. In spite of this, students still may have the opportunity to cheat if they are able to observe a student nearby with
the same exam. With the advent of economical digital photocopiers, it is now very easy to produce multiple-choice examinations in which the order of the questions as well as the order of the answers
is scrambled. We used the Perl scripting language and developed scripts for performing the randomization as well as the marking of these exams. MCR could also be implemented on various other
platforms such as Visual Basic for Applications (VBA) with Microsoft Word or with Mathematica notebooks.
The first step is to produce the MCR exams. Inserting some simple markups in the document file that contains the examination questions and running a script we developed produces as many MCR exams as
required. Our Perl scripts were used with source files in Rich Text Format (RTF) or LaTeX format. In the future we plan to use HTML as well. Each exam has the questions and its possible responses
randomized. A three digit Exam Code uniquely identifies each exam and is associated with a key file that indicates the exact randomization that was used for that particular exam. After the exams are
produced they may be put on a CD and taken to a digital photocopier to be printed, collated, and stapled.
The second step is to mark the exams. During the examination, the students are required to indicate the Exam Code on the Scantron answer sheet. For small classes, a Perl script can be used to produce
a listing of the correct responses for each Exam Code and the exams can be marked manually. For larger classes, an optical reader is used to read in the student Scantron sheets and produce a grade
report and this was the method that we used for all of our exams. Students are encouraged to keep a copy of their responses, so they can later verify the correctness of the marking. We never had any
problem either with the optical scanner or our marking scripts.
The resulting Grade Report indicates for each Exam Code the student’s score and in addition, it is helpful to show for each exam what the correct answer is and which answer the student selected. An
example Grade Report is available from our Statistics Laboratory Homepage.
For our own further analysis of the examination questions, we also produce a Response Analysis. For this purpose, we select the original ordering of questions as the exam corresponding to Exam Code
000. With respect to this ordering of questions, our script also produces various other statistical summaries. For each question the number of students who selected each possible answer as well as
the proportion of students correctly answering the question are tabulated. We have also found it very helpful to compute for each question the correlation coefficient between the exam score for that
student and an indicator variable defined as 1 if the student answered correctly and 0 otherwise. A low correlation suggests a poor question. A good discriminating question has a low or moderate
proportion of students answering correctly but a high correlation. A non-parametric correlation coefficient such as Kendall’s tau or Spearman’s rank correlation could be used but our preference was
simply to use the Pearson correlation coefficient. Although it is not optimal in this situation, it is quite adequate for our purpose. For an example Response Analysis, see our Statistics Laboratory
One concern with MCR examinations is whether or not this type of exam may possibly adversely affect student performance. To examine this question, we first give a brief literature review on
multiple-choice examination question and answer arrangement. Then we report the results of an experimental investigation with our own students taking one of our MCR examinations.
2. Brief Literature Review
A high quality multiple-choice test needs to be carefully planned and constructed, and the test's items or questions as well as the answer-options or multiple-choice responses for each item must be
thoroughly edited. After preparing the multiple-choice items and answer-options, the next step is to decide on the precise arrangement and ordering of the items and answer-options.
Previous empirical studies discussed by Gerow (1980) on the sequencing of questions have all failed to indicate any difference between random ordering of questions and questions organized by the
order it was taught. Gerow (1980) presented further empirical evidence that arranging the items in order of difficulty also has no effect provided that there is enough time for students to complete
the test. A further study by Allison (1984) confirmed that even for sixth grade students there was no effect on performance by ordering the items according to difficulty provided that there was
enough time to complete the test.
Tuck (1978) found that when students in an introductory psychology class were asked for their preference in item arrangements, 64% preferred a random organization.
These studies all support the use of the MCR examination design but none of the previous studies apply directly to the MCR case where both the order of the items as well as the order of the possible
responses is randomized.
3. Our Experimental Investigation
Our null hypothesis is that student grades are not affected by the MCR randomization procedure. One specific alternative hypothesis of interest is that ordering the exam questions in the same order
as taught could result in higher scores than just random ordering. If this was in fact the case, one could question whether the type of learning that has occurred is really what is desired. Our
opinion, shared by educational psychologists we talked to, is that most likely one would not want to reward such a type of learning anyway and so, if there were a difference in grades, this in itself
would be a good reason for selecting an MCR design. Also Hopkins (1998) suggests that it is necessary to avoid arranging items in the order in which they were presented in the textbook in order to
achieve the logical validity test.
Another specific hypothesis of interest is whether the ordering of the answer-options could result in an improved score. The specific ordering we have in mind here is either a logical or numerical
ordering of the possible answers. If speed is really the determining factor in the examination then this ordering might be expected to improve the student scores. Once again the pedagogical value of
such an exam is open to question. At our university, students with disabilities may be allowed up to about 50% more time. This fact really means that we should probably not put too much emphasis on
speed of processing the examination material but rather more on the depth of understanding. The examination that we experimented with was designed so that most students would be able to complete it
in the time allotted.
The exam chosen for our experiment covered four chapters of the textbook. There were eight questions from the first chapter, nine from the second, three from the third, and seven from the fourth. In
total there were 27 questions and four answer-options for each question. The item difficulties on this exam were approximately the same and are independent of the position of the items within the
To test the first alternative hypothesis we put the questions from each chapter in a corresponding section of the exam and randomized them within their section. And to test the second alternative
hypothesis we carefully chose the questions and the possible answers so that there was either a logical or numerical ordering of these answer-option choices.
We used a two-factor experimental design with a covariate. Each factor had two levels. The first factor, denoted by I, was the question ordering. The two levels of this factor were to use a
randomized order for all of the questions or a partially randomized order in which the order of the questions was randomized within one of four sections of the exam that corresponded to the chapter
of textbook. The second factor, denoted by O, was the ordering of the answer-options. The two levels of the second factor corresponded to using a randomized order for the answer-options or using a
fixed order in which the answer-options were presented in a logical or numerical order. Thus there are four treatment combinations in our experiment:
a. questions randomized and answer-options randomized
b. questions partially randomized and answer-options randomized
c. questions randomized and answer-options ordered
d. questions partially randomized and answer-options ordered.
Notice that with this design all students will receive a unique exam - at least with very high probability. Neither the instructor nor student would be able to tell exactly which treatment
combination was used for a particular exam without some careful examination. This was the second mid-term examination in a two-term course and so the first examination, which was completely random
with respect to item and answer-option arrangement, was used as a covariate to reduce experimental error. If our null hypothesis was rejected we were prepared to make a statistical adjustment to the
students’ grades.
We ensured that each student received one of the four examination types at random. This was done by generating 500 exams of each of the four types and then randomly selecting without replacement four
samples of size 125 from {1, 2, …, 500}. The exams with codes corresponding to the number selected in each sample were used to obtain 125 examinations for each of the four treatment combinations.
These selected exams were then printed.
Our exam was administered in a double-blind fashion to 442 students with neither the student nor instructor knowing which type of exam was used for a particular student. The means and standard
deviations of the scores for the students writing each type of exam are shown in Table 1 below.
Table 1. Summary of exam score means and standard deviations for each treatment combination.
┃ │ Number of Students │ Exam score mean │ Exam score standard deviation ┃
┃ (a) I & O random │ 108 │ 52.45 │ 11.88 ┃
┃ (b) I partial & O random │ 114 │ 53.75 │ 13.06 ┃
┃ (c) I random & O fixed │ 107 │ 54.72 │ 12.65 ┃
┃ (d) I partial & O fixed │ 113 │ 53.40 │ 11.32 ┃
It is obvious from Table 1 that it is very unlikely that there is any difference in scores between the treatment combinations. Note that in Table 1, the standard error of the mean is approximately
the standard deviation indicated in column 4 divided by 10.
Table 2 shows the means for each factor level. In the case of item randomization the observed mean is slightly less than when ordered. In the case of answer-option randomization, the observed mean is
slightly higher than for when the items are ordered.
Table 2. Summary of exam score means and standard deviations for each factor level.
┃ │ Number of Students │ Mean │ Standard Deviation ┃
┃ I random │ 222 │ 53.12 │ 12.49 ┃
┃ I partially random │ 220 │ 53.70 │ 12.00 ┃
┃ O random │ 215 │ 53.58 │ 12.29 ┃
┃ O ordered │ 227 │ 53.24 │ 12.21 ┃
Table 3 presents the analysis of variance that confirms that there is no statistically significant difference among the treatment effects. The covariate is, as expected, highly significant due to the
fact the performance on this exam was highly correlated with their performance on the first exam.
Table 3. Analysis of variance table for our experiment.
┃ Source │ DF │ Sum of Squares │ Mean Square │ F-Value │ Pr > F ┃
┃ First mid-term covariate │ 1 │ 18840.121 │ 18840.121 │ 176.536 │ < 0.001 ┃
┃ Factor I (item) │ 1 │ 44.651 │ 44.651 │ 0.418 │ 0.518 ┃
┃ Factor O (answer-option) │ 1 │ 0.449 │ 0.449 │ 0.004 │ 0.948 ┃
┃ Error │ 440 │ 46637.210 │ 106.721 │ │ ┃
4. Concluding Remarks
The MCR examination design was used in our department for nine examinations with 1947 individual examinations being written and marked. The students were pleased with MCR examinations since it
obviously increased the integrity of the examination process.
In one of these examinations we investigated the effect of randomization of the questions and possible answers on student performance and found that, as might be expected from previous empirical
studies, there was no evidence for any effect.
Our Statistics Laboratory is available for producing and marking MCR examinations. If interested please contact our StatLab Manager whose contact information is given on the StatLab Homepage.
We would like to especially thank Professor Tom Wonnacott for his numerous insightful comments, suggestions, and encouraging support for this project. We also wish to thank Professors Mike Atkinson,
John Braun, David Bellhouse, Angela Jonkhans, Maree Libal, Harry Murray, Evelyn Vingilis for helpful comments and discussions. The StatLab at the University of Western Ontario developed the software
for this project. Special thanks to Dr. Valery Didinchuk for assistance with Perl. We also acknowledge helpful comments from three JSE referees.
Allison, D. E. (1984), "Test anxiety, stress, and intelligence-test performance," Measurement and Evaluation in Guidance, 16, 211-217.
Gerow, J. R. (1980), "Performance on achievement tests as a function of the order of item difficulty," Teaching of Psychology, 7, 93-96.
Hopkins, K. D. (1998), Educational and Psychological Measurement and Evaluation, Boston: Allyn and Bacon.
Tuck, J. P. (1978), "Examinee's control of item difficulty sequence," Psychological Reports, 42, 1109-1110.
Web References
Comprehensive Perl Archive Network (CPAN), www.perl.com/CPAN/README.html
LaTeX Project Homepage, www.latex-project.org
Mathematica, Wolfram Research, www.wolfram.com
Microsoft Visual Basic for Applications, msdn.microsoft.com/vba
Perl Homepage, www.perl.com
Rich Text Format, Version 1.5 Specifications, www.biblioscape.com/rtf15_spec.htm
Scantron Homepage, www.scantron.com
Statistics Laboratory, University of Western Ontario, www.stats.uwo.ca/statlab
A. Ian McLeod
Department of Statistical and Actuarial Sciences
University of Western Ontario
London, Ontario, N6A 5B7
Ying Zhang
Department of Statistical and Actuarial Sciences
University of Western Ontario
London, Ontario, N6A 5B7
Hao Yu
Department of Statistical and Actuarial Sciences
University of Western Ontario
London, Ontario, N6A 5B7
Volume 11 (2003) | Archive | Index | Data Archive | Information Service | Editorial Board | Guidelines for Authors | Guidelines for Data Contributors | Home Page | Contact JSE | ASA Publications | {"url":"http://jse.amstat.org/v11n1/mcleod.html","timestamp":"2024-11-02T10:50:30Z","content_type":"text/html","content_length":"22061","record_id":"<urn:uuid:b5587401-7785-4434-a3c4-64f60a2e9bcb>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00801.warc.gz"} |
Oblique shock of Physics Topics | Question AI
<div id="mw-content-wrapper"><div id="mw-content"><div id="content" class="mw-body" role="main"><div id="bodyContentOuter"><div class="mw-body-content" id="bodyContent"><div id="mw-content-text"
class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-parser-output"><p><br/> </p><p>An <b>oblique shock</b> wave is a <span title="Physics:Shock wave">shock wave</span> that,
unlike a normal shock, is inclined with respect to the incident upstream flow direction. It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and
compresses. The upstream <span title="Streamlines, streaklines, and pathlines">streamlines</span> are uniformly deflected after the shock wave. The most common way to produce an oblique shock wave is
to place a wedge into supersonic, <span title="Physics:Compressible flow">compressible flow</span>. Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which
nearly discontinuous changes in the thermodynamic properties of a gas occur. While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across
an oblique shock wave. </p><p>It is always possible to convert an oblique shock into a normal shock by a <span title="Physics:Galilean transformation">Galilean transformation</span>. </p><h2 id=
"qai_title_1"><span class="mw-headline" id="Wave_theory">Wave theory</span></h2><p>For a given <span title="Physics:Mach number">Mach number</span>, M<sub>1</sub>, and corner angle, θ, the oblique
shock angle, β, and the downstream Mach number, M<sub>2</sub>, can be calculated. Unlike after a normal shock where M<sub>2</sub> must always be less than 1, in oblique shock M<sub>2</sub> can be
supersonic (weak shock wave) or subsonic (strong shock wave). Weak solutions are often observed in flow geometries open to atmosphere (such as on the outside of a flight vehicle). Strong solutions
may be observed in confined geometries (such as inside a nozzle intake). Strong solutions are required when the flow needs to match the downstream high pressure condition. Discontinuous changes also
occur in the pressure, density and temperature, which all rise downstream of the oblique shock wave. </p><h3><span id="The_.CE.B8-.CE.B2-M_equation"></span><span class="mw-headline" id=
"The_θ-β-M_equation">The θ-β-M equation</span></h3><p>Using the <span title="Continuity equation">continuity equation</span> and the fact that the <span title="Physics:Speed">tangential velocity
component</span> does not change across the shock, trigonometric relations eventually lead to the θ-β-M equation which shows θ as a function of M<sub>1</sub>, β and ɣ, where ɣ is the <span title=
"Physics:Heat capacity ratio">Heat capacity ratio</span>.<sup id="cite_ref-1" class="reference"><span></span></sup> </p><dl><dd><span style="opacity:.5">$\displaystyle{ \tan \theta = 2\cot\beta\ \
frac{M_1^2\sin^2\!\beta-1}{M_1^2(\gamma+\cos2\beta)+2} }$</span></dd></dl><p>It is more intuitive to want to solve for β as a function of M<sub>1</sub> and θ, but this approach is more complicated,
the results of which are often contained in tables or calculated through a <span title="Numerical method">numerical method</span>. </p><h3><span class="mw-headline" id="Maximum_deflection_angle">
Maximum deflection angle</span></h3><p>Within the θ-β-M equation, a maximum corner angle, θ<sub>MAX</sub>, exists for any upstream Mach number. When θ > θ<sub>MAX</sub>, the oblique shock wave is
no longer attached to the corner and is replaced by a detached <span title="Physics:Bow shock (aerodynamics)">bow shock</span>. A θ-β-M diagram, common in most compressible flow textbooks, shows a
series of curves that will indicate θ<sub>MAX</sub> for each Mach number. The θ-β-M relationship will produce two β angles for a given θ and M<sub>1</sub>, with the larger angle called a strong shock
and the smaller called a weak shock. The weak shock is almost always seen experimentally. </p><p>The rise in pressure, density, and temperature after an oblique shock can be calculated as follows: </
p><dl><dd><span style="opacity:.5">$\displaystyle{ \frac{p_2}{p_1} = 1+\frac{2\gamma}{\gamma+1}(M_1^2\sin^2\!\beta-1) }$</span></dd></dl><dl><dd><span style="opacity:.5">$\displaystyle{ \frac{\rho_2}
{\rho_1} = \frac{(\gamma+1)\ M_1^2\sin^2\!\beta}{(\gamma-1)M_1^2\sin^2\!\beta+2} }$</span></dd></dl><dl><dd><span style="opacity:.5">$\displaystyle{ \frac{T_2}{T_1} = \frac{p_2}{p_1}\frac{\rho_1}{\
rho_2}. }$</span></dd></dl><p>M<sub>2</sub> is solved for as follows: </p><dl><dd><span style="opacity:.5">$\displaystyle{ M_2 = \frac{1}{\sin(\beta-\theta)}\sqrt{\frac{1+\frac{\gamma-1}{2}M_1^2 \sin
^2\!\beta}{\gamma M_1^2 \sin^2\!\beta- \frac{\gamma-1}{2}}}. }$</span></dd></dl><h2 id="qai_title_2"><span class="mw-headline" id="Wave_applications">Wave applications</span></h2><p>Oblique shocks
are often preferable in engineering applications when compared to normal shocks. This can be attributed to the fact that using one or a combination of oblique shock waves results in more favourable
post-shock conditions (smaller increase in entropy, less stagnation pressure loss, etc.) when compared to utilizing a single normal shock. An example of this technique can be seen in the design of
supersonic aircraft engine intakes or <span title="Engineering:Components of jet engines">supersonic inlets</span>. A type of these inlets is wedge-shaped to compress air flow into the combustion
chamber while minimizing thermodynamic losses. Early supersonic aircraft jet engine intakes were designed using compression from a single normal shock, but this approach caps the maximum achievable
Mach number to roughly 1.6. <span title="Engineering:Concorde">Concorde</span> (which first flew in 1969) used variable geometry wedge-shaped intakes to achieve a maximum speed of Mach 2.2. A similar
design was used on the F-14 Tomcat (the F-14D was first delivered in 1994) and achieved a maximum speed of Mach 2.34. </p><p>Many supersonic aircraft wings are designed around a thin diamond shape.
Placing a diamond-shaped object at an angle of attack relative to the supersonic flow streamlines will result in two oblique shocks propagating from the front tip over the top and bottom of the wing,
with Prandtl-Meyer expansion fans created at the two corners of the diamond closest to the front tip. When correctly designed, this generates lift. </p><h2 id="qai_title_3"><span class="mw-headline"
id="Waves_and_the_hypersonic_limit">Waves and the hypersonic limit</span></h2><p>As the Mach number of the upstream flow becomes increasingly hypersonic, the equations for the pressure, density, and
temperature after the oblique shock wave reach a mathematical <span title="Limit (mathematics)">limit</span>. The pressure and density ratios can then be expressed as: </p><dl><dd><span style=
"opacity:.5">$\displaystyle{ \frac{p_2}{p_1} \approx \frac{2\gamma}{\gamma+1}\ M_1^2\sin^2\!\beta }$</span></dd></dl><dl><dd><span style="opacity:.5">$\displaystyle{ \frac{\rho_2}{\rho_1} \approx \
frac{\gamma+1}{\gamma-1}. }$</span></dd></dl><p>For a perfect atmospheric gas approximation using γ = 1.4, the hypersonic limit for the density ratio is 6. However, hypersonic post-shock dissociation
of O<sub>2</sub> and N<sub>2</sub> into O and N lowers γ, allowing for higher density ratios in nature. The hypersonic temperature ratio is: </p><dl><dd><span style="opacity:.5">$\displaystyle{ \frac
{T_2}{T_1} \approx \frac{2\gamma\ (\gamma-1)}{(\gamma+1)^2}\ M_1^2\sin^2\!\beta. }$</span></dd></dl></div></div></div></div></div></div></div> | {"url":"https://www.questionai.com/knowledge/kaXzUp4Q5M-oblique-shock","timestamp":"2024-11-05T18:51:12Z","content_type":"text/html","content_length":"69436","record_id":"<urn:uuid:0ec923cb-2d9f-4ef8-8c97-47d4e76b875c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00545.warc.gz"} |
Factors of 3, Prime Factors of 3, How to Calculate, Tips
Factors of 3
The factors of 3 are the numbers that can be multiplied together to result in 3. Since 3 is a prime number, it has only two factors: 1 and 3. This means that 3 can only be divided evenly by 1 and
itself without leaving a remainder. Recognizing these factors is crucial for simplifying fractions, solving equations, and performing various arithmetic operations. Whether you’re a student brushing
up on basic maths skills or a math enthusiast, knowing the factors of 3 provides a solid foundation for further mathematical exploration.
What are the Factors of 3?
The factors of 3 are the numbers that can be multiplied together to result in 3. Since 3 is a prime number, it has only two distinct factors: 1 and 3. This means that the only whole numbers that
divide 3 evenly, without leaving a remainder, are 1 and 3. In other words, 1 multiplied by 3 equals 3 (1 * 3 = 3). Understanding that 3 is a prime number highlights its uniqueness, as prime numbers
have exactly two distinct positive factors: one and the number itself. This concept is fundamental in various mathematical operations, including simplifying fractions, solving equations, and
exploring number theory.
Factor Pairs of 3
The factor pairs of a number are two numbers that, when multiplied together, give the original number. Since 3 is a prime number, it has only one factor pair.
• (1, 3): When multiplied, 1 and 3 give the product 3 (1 * 3 = 3).
This is the only factor pair for 3 because it is a prime number and has no other divisors. Understanding factor pairs is useful in various mathematical contexts, such as solving equations and
simplifying expressions.
How to Calculate Prime Factors of 3
Calculating the prime factors of a number involves breaking it down into its basic building blocks, which are prime numbers. Here are the steps to find the prime factors of the number 3:
Step 1: Understand the Number
Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, and 11.
Composite Number: A composite number is a natural number greater than 1 that is not prime. It can be divided by numbers other than 1 and itself.
Step 2: Determine if 3 is a Prime Number
To calculate the prime factors of 3, we first need to determine if 3 is a prime number.
A prime number has exactly two distinct positive divisors: 1 and itself.
• 3 can only be divided by 1 and 3 without leaving a remainder.
Since 3 meets these criteria, it is a prime number.
Step 3: List the Prime Factors
Since 3 is a prime number, it has no prime factors other than itself.
Step 4: Conclusion
The prime factor of 3 is: 3
To summarize, the number 3 is a prime number, and its only prime factor is itself.
Factors of 3 : Examples
Example 1: Finding Factors of 3
What are the factors of 3?
To find the factors of 3, we need to find all integers that multiply together to give 3
1 × 3 = 3
Therefore, the factors of 3 are 1 and 3.
Example 2: Determining if a Number is a Factor of 3
Is 2 a factor of 3?
To determine if 2 is a factor of 3, we need to check if 3 can be divided evenly by 2.
3 ÷ 2 = 1.5 (not an integer)
Since 3 divided by 2 does not result in an integer, 2 is not a factor of 3.
Example 3: Determining if a Number is a Factor of 3
Is 3 a factor of 3?
To determine if 3 is a factor of 3, we need to check if 3 can be divided evenly by 3.
3 ÷ 3 = 1 (an integer)
Since 3 divided by 3 results in an integer, 3 is a factor of 3.
Example 4: Finding the Common Factors of 3 and Another Number
What are the common factors of 3 and 6?
First, list the factors of each number.
Factors of 3: 1, 3
Factors of 6: 1, 2, 3, 6
Common factors are the numbers that appear in both lists. The common factors of 3 and 6 are 1 and 3.
Example 5: Finding the Greatest Common Factor (GCF)
What is the greatest common factor (GCF) of 3 and 9?
First, list the factors of each number.
Factors of 3: 1, 3
Factors of 9: 1, 3, 9
Factors of 3 : Tips
Finding factors of 3 can be simple and quick if you know a few handy tricks. Whether you’re solving math problems or just curious about number properties, these tips will help you determine if a
number is divisible by 3 with ease.
1. Divide by 3: Check if the number can be divided by 3 without leaving a remainder. If it divides evenly, 3 is a factor.
2. Sum of Digits: Add all the digits of the number. If the sum is divisible by 3, then the original number is also divisible by 3.
3. Recognize Patterns: Familiarize yourself with common multiples of 3, such as 3, 6, 9, 12, etc., to quickly identify factors.
4. Use a Calculator: For larger numbers, using a calculator can help you quickly determine if the number is divisible by 3.
5. Break Down Large Numbers: Split large numbers into smaller parts to check if each part is divisible by 3. If all parts are divisible, the original number is likely divisible by 3 as well.
Can a number be a factor of 3 if it ends in an even number?
No, for a number to be a factor of 3, it does not matter if it ends in an even or odd digit. The key is whether the sum of its digits is divisible by 3.
Are the factors of 3 always odd numbers?
Yes, the factors of 3 are always odd numbers. The factors of 3 are 1 and 3, both of which are odd.
Is there a quick method to check if large numbers are factors of 3?
Yes, using a calculator can help you quickly determine if a large number is divisible by 3. Alternatively, you can break down the number into smaller parts and check each part.
Are all multiples of 3 also factors of 3?
No, multiples of 3 are not necessarily factors of 3. Factors of 3 divide 3 without leaving a remainder, while multiples of 3 are obtained by multiplying 3 by other integers.
Can prime numbers other than 3 be factors of 3?
No, prime numbers other than 3 cannot be factors of 3. Factors of 3 are limited to 1 and 3 because 3 is a prime number.
What is the prime factorization of 3?
The prime factorization of 3 is simply 3 itself. Since 3 is a prime number, it cannot be divided by any other numbers except for 1 and itself. Therefore, the prime factorization of 3 is 3.
What are the positive and negative pair factors of 3?
• Positive Pair Factors of 3: (1, 3)
• Negative Pair Factors of 3: (-1, -3)
Is 2 a factor of 3?
No, 2 is not a factor of 3. A factor of a number divides that number exactly without leaving a remainder. Since 3 divided by 2 leaves a remainder, 2 is not a factor of 3.
What is the sum of factors of 3?
The factors of 3 are 1 and 3. The sum of the factors of 3 is 1 + 3 = 4. | {"url":"https://www.examples.com/maths/factors-of-3.html","timestamp":"2024-11-13T11:32:27Z","content_type":"text/html","content_length":"105308","record_id":"<urn:uuid:10d9ca0f-831e-4f55-a4f2-4ca8b4212fbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00207.warc.gz"} |