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Topic: On Generalized bi-Gamma-Ideals in Gamma-Semigroups
Replies: 1 Last Post: May 23, 2013 5:40 PM
Messages: [ Previous | Next ]
On Generalized bi-Gamma-Ideals in Gamma-Semigroups
Posted: May 23, 2013 7:40 AM
The present day theory of ideals has been standardized in some
respects, and
it is recently being extensively enriched and studied by many
algebraists. This
notion of ideals that was originally formulated by Dedekind for the
ring of
integers of an algebraic number eld, was again generalized by Emmy
in terms of one-sided and two-sided ideals in associative rings.
this theory of ideals has won universal acceptance due to its signi
cance in
characterizing dierent algebraic structures as is evident from the vast
ture available on the topic. Further, Steinfeld[22 and 23] invented
of rings and semigroups in 1953 and in 1956 respectively as a
generalization of
one-sided ideals of rings and semigroups and then in 1952, Good and
[13] jointly announced the arrival of the bi-ideal of semigroups as a
generalization of one-sided ideals of semigroups. Interestingly, the
concept of bi-ideals
of semigroups was given earlier than the concept of quasi-ideals of
and it was subsequently revealed that the bi-ideal of semigroups
not only one-sided ideals of semigroups but also quasi-ideals of
Moreover, in 1962, the concept of the bi-ideal was extended to
rings by Lajos [14].
The notion of the generalized biideal[(or generalized (1,1)-ideal] was
introduced in rings in 1970 by Szasz[3 and 4] and then in semigroups in
by Lajos [15, 16, 17, 18 and 19] as a generalization of bi-ideals of
rings and
semigroups. In fact the notion of gamma-semigroups is a generalization
of the con-
cept of semigroups. For any relevant terminologies and unde ned
concepts on
gamma-semigroups in this paper, readers can see [6, 7 and 10]. The
notion of quasi-
gamma-ideals and bi-gamma-ideals in gamma-semigroups was given by
Chinram[11 and 12] in
2006 and in 2007 respectively. The properties of the bi-ideal and the
alized bi-ideal in semigroups as well as in gamma-semigroups have been
studied by
several authors [1, 2, 8, 9, 20 and 21]. In this paper, we have studied
some gen-
eral classical properties of the generalized bi-gamma-ideal in
gamma-semigroups and also
the prime and irreducible generalized bi-gamma-ideal in
gamma-semigroups. Further, S.
Lajos [17] identi ed a class of semigroups for which some classes of
bi-ideals are distinct from the class of bi-ideals. He also raised the
problem of
characterizing those semigroups whose generalized bi-ideals are
bi-ideals. This
problem was solved by F. Catino[5]. We have investigated it in
In fact the class of generalized bi-gamma-ideals of gamma-semigroups is
a generalization of
the class of generalized bi-ideals in semigroups in the same way as
in gamma-semigroups are a generalization of bi-ideals in semigroups.
For the current
directions of the theory, readers can refer the reference of this paper
and the
references of the reference.
Date Subject Author
5/23/13 On Generalized bi-Gamma-Ideals in Gamma-Semigroups Abul Basar
5/23/13 Re: On Generalized bi-Gamma-Ideals in Gamma-Semigroups Robin Chapman
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Correlation And Prediction
Table of Contents
Correlation And Prediction
Types of Variables
Graphing: Scatter Plot
Patterns of Correlation
More Info on Correlation
Computing the Pearson Correlation Coefficient (r)
Degree of Correlation
Interpreting Correlations
Plots and Computations
Statistical Significance
Correlation Matrix from SPSS
A suggestion...
Multiple Regression
Prediction with Z scores
Manager’s Stress example
Prediction with Raw Scores
Number supervised (X): M=7; SD=2.37; Stress (Y): M=6; SD =2.61 EXAMPLE: Someone Supervising 3 employees
Last Year’s Psych 201 class
This is what you see
This is what you see
Multiple Regression
This is what you would see
This is what you would see
This is what you would see
Stepwise Multiple Regression
Excluded Variables
What you need to know
A suggestion...
Chapter 3 Summary
Author: pcuser
Home Page: http://www.mtholyoke.edu/courses/bpackard/stats/
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Stokes' theorem
July 31st 2008, 05:12 AM #1
May 2008
Stokes' theorem
I got stuck using the Stokes' theorem, the problem is at the bottom of the pic. I found the Curl of F, and also the normal of the Triangle. As you can see, I ended up with an area integer with 3
variables, how do I solve this? did I do it right? http://img2.tapuz.co.il/forums/1_119833460.jpg
I got stuck using the Stokes' theorem, the problem is at the bottom of the pic. I found the Curl of F, and also the normal of the Triangle. As you can see, I ended up with an area integer with 3
variables, how do I solve this? did I do it right? http://img2.tapuz.co.il/forums/1_119833460.jpg
A couple of things:
1. Since you're integrating on the surface (a plane), you can substitute z = 4x + y (which is the equation of the plane). This solves your technical infelicity of three variables in the integrand
2. It looks like you're using a unit vector normal to the surface (plane). That's incorrect. $F \cdot dS eq F \cdot \hat{n} \, dx \, dy$ in general.
In fact, $F \cdot dS = F \cdot n \, dx \, dy$:
$F \cdot dS = F \cdot \left( \frac{\partial z}{\partial x} \, i + \frac{\partial z}{\partial y} \, j - k\right) \, dx \, dy$.
3. Are you OK with the region in the xy-plane you integrate over (and hence the integral terminals) .....? The region is the right angle triangle with vertices at (0, 0), (2, 0) and (2, 2).
A couple of things:
1. Since you're integrating on the surface (a plane), you can substitute z = 4x + y (which is the equation of the plane). This solves your technical infelicity of three variables in the integrand
2. It looks like you're using a unit vector normal to the surface (plane). That's incorrect. $F \cdot dS eq F \cdot \hat{n} \, dx \, dy$ in general.
In fact, $F \cdot dS = F \cdot n \, dx \, dy$:
$F \cdot dS = F \cdot \left( \frac{\partial z}{\partial x} \, i + \frac{\partial z}{\partial y} \, j - k\right) \, dx \, dy$.
3. Are you OK with the region in the xy-plane you integrate over (and hence the integral terminals) .....? The region is the right angle triangle with vertices at (0, 0), (2, 0) and (2, 2).
Ok, I think I fixed it, is that right?
Another thing.
I marked everything in the pic.
Shouldn't I normalize the normal vector?
Did I write the boundaries right?
10x again.
Ok, I think I fixed it, is that right?
Another thing.
I marked everything in the pic.
Shouldn't I normalize the normal vector? Mr F says: No. I've already given the correct formula.
Did I write the boundaries right? Mr F says: I'm not sure why you wanted to switch from xy to uv-coordinates ..... x and y are fine. Using xy-coordinates, the boundary is y = 0 to y = -x + 2 and
x = 0 to x = 2 (or alternatively, x = 0 to x = 2 - y and y = 0 to y = 2).
10x again.
I get 20/3 (and I stayed with xy-coordinates) but I could be wrong since my careless arithmetic errors are legion.
July 31st 2008, 03:38 PM #2
July 31st 2008, 09:37 PM #3
May 2008
July 31st 2008, 10:36 PM #4
July 31st 2008, 10:51 PM #5
May 2008
August 1st 2008, 06:44 AM #6
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Math Forum Discussions
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Topic: Symmetric Group
Replies: 4 Last Post: Mar 1, 2011 12:24 AM
Messages: [ Previous | Next ]
Symmetric Group
Posted: Feb 28, 2011 2:12 PM
I am wondering why the group of all permutations on a set of n
elements is called a "symmetric group". (Sometimes knowing the origin
of a word helps me to better understand it). Neither wikipedia, nor
any of the texts I own (Hungerford's "Intro to Algera", Herstein, or
Dummit and Foote), directly answer this question.
I understand that the subgroup of all symmetries of a regular n-gon
are contained within S_n, but it seems to me that not all the elements
of S_n are in fact symmetries of a regular n-gon. (i.e. for n=4, I
don't think the permutation (1324) is part of the octic subgroup of
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Differentiable but not continuosly differentiable.
December 20th 2012, 11:22 AM #1
Apr 2008
Differentiable but not continuosly differentiable.
Is it true that there are no function which are differentiable (in a complex sense) but not continuously differentiable (in a complex sense)? i.e. If a function f' exist then it is continuous.
This seems to be the only correct conclusion from the theorems in my course but I've never heard it stated explicitly so I just wanted to check I'm not misunderstanding!
Re: Differentiable but not continuosly differentiable.
Re: Differentiable but not continuosly differentiable.
Being "differentiable" for functions of complex variables is much more severe than for real functions because of the increased "dimensionality". To be "differentiable" for functions of a real
variable, we only have to have the limits "from the left" and "from the right" the same. To be "differentiable" for a function of a complex variable we must have all limits from the infinitely
many ways to approach a point in two dimension the same. In fact, one can show that if a function of a complex variable is differentiable in some neighborhood of a point, then it is infinitely
differentiable at that point.
December 20th 2012, 11:35 AM #2
December 20th 2012, 01:54 PM #3
MHF Contributor
Apr 2005
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Options for Graphics
When Mathematica plots a graph for you, it has to make many choices. It has to work out what the scales should be, where the function should be sampled, how the axes should be drawn, and so on. Most
of the time, Mathematica will probably make pretty good choices. However, if you want to get the very best possible pictures for your particular purposes, you may have to help Mathematica in making
some of its choices.
There is a general mechanism for specifying "options" in Mathematica functions. Each option has a definite name. As the last arguments to a function like Plot, you can include a sequence of rules of
the form , to specify the values for various options. Any option for which you do not give an explicit rule is taken to have its "default" value.
Choosing an option for a plot.
A function like Plot has many options that you can set. Usually you will need to use at most a few of them at a time. If you want to optimize a particular plot, you will probably do best to
experiment, trying a sequence of different settings for various options.
Each time you produce a plot, you can specify options for it. "Redrawing and Combining Plots" will also discuss how you can change some of the options, even after you have produced the plot.
Some of the options for Plot. These can also be used in Show.
Some common settings for various options.
When Mathematica makes a plot, it tries to set the and scales to include only the "interesting" parts of the plot. If your function increases very rapidly, or has singularities, the parts where it
gets too large will be cut off. By specifying the option PlotRange, you can control exactly what ranges of and coordinates are included in your plot.
Settings for the option PlotRange.
Mathematica always tries to plot functions as smooth curves. As a result, in places where your function wiggles a lot, Mathematica will use more points. In general, Mathematica tries to adapt its
sampling of your function to the form of the function. There is a limit, however, which you can set, to how finely Mathematica will ever sample a function.
It is important to realize that since Mathematica can only sample your function at a limited number of points, it can always miss features of the function. Mathematica adaptively samples the
functions, increasing the number of samples near interesting features, but it is still possible to miss something. By increasing PlotPoints, you can make Mathematica sample your function at a larger
number of points. Of course, the larger you set PlotPoints to be, the longer it will take Mathematica to plot any function, even a smooth one.
More options for Plot. These cannot be used in Show.
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Is there a name for this property of a topology?
up vote 2 down vote favorite
This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?
For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in I}$ such that $U=\cup_i U_i$.
I suppose this could also be formulated as each nonempty open set having an open cover of proper subsets, or being the colimit of its open subsets.
(Also, apologies if this is something obvious I should have thought of.)
gn.general-topology terminology
Each nonempty open set? – user2734 Mar 19 '10 at 16:41
1 As stated its just not a good question. If the space is perfect, that is every point is a limit point, and points are closed, then a space will have this property. Its not a research level
question. – Charlie Frohman Mar 19 '10 at 17:11
4 @ Ketil Tveiten, re: Charlie Frohman's comment. For what it's worth, I think this is a fine question. I don't think the answer would be obvious to every "research-level" mathematician, although
I'm only a first-year graduate student myself... – Vectornaut Mar 19 '10 at 19:52
5 @Charlie: Are you saying that serious researchers only study T1 spaces?!? – François G. Dorais♦ Mar 19 '10 at 23:25
I agree with Charlie, and here's why: I don't see off hand any good reason for caring what the name of such a space is. I mean, if you have examples of some of these spaces, and some result that
3 says that this precise property is what you need for some application, then by all means, it should have a name, and knowing the conventional name will help you look up the appropriate literature.
But as it is, I'd like some motivation before I'll like the question. – Theo Johnson-Freyd Mar 20 '10 at 1:43
show 3 more comments
3 Answers
active oldest votes
In spaces where singleton points are closed, your property is equivalent to saying that the space has no isolated points. Or in other words, that it is perfect.
Clearly, no space with an isolated point can have your property. Conversely, when singletons are closed, then you can subtract one point from any open set and thereby have a proper open
subset. So if U has at least 2 points x,y, then U = U-{x} union U-{y}, giving an instance with I of size 2.
up vote 10
down vote However, your property does not imply that points are closed, since the space on reals R, where open sets have the form (-infty, a), has your property, but points are not closed in this
Note that in the example space I give, no finite I would suffice, in contrast to the T1 perfect space situation, where I of size 2 suffices. – Joel David Hamkins Mar 20 '10 at 2:26
Sort of going the other way (you gave a condition that implies the OP's condition), the condition implies that any local system, ordered by set inclusion, cannot have a smallest
element. I think this is saying that local base cannot be finite? (This distinguishes your two examples of R with (-infty,a) and Z with (-infty,n). ) – Willie Wong Mar 20 '10 at 14:01
Willie, this is ,clearly, saying that every local base can not be finite. – Alexei Fedotov Mar 20 '10 at 15:01
Thanks for verifying my question (I am a bit rusty here). – Willie Wong Mar 20 '10 at 15:44
add comment
Are you just saying that the topology is an atomless lattice? I'd call it "a space with atomless topology".
up vote 2 down vote
1 Consider the integers Z, with open sets (-infty,n). This is a topology and is an atomless lattice, but it doesn't have the desired property. – Joel David Hamkins Mar 20 '10 at
Hmm, good point. I would like to hear where these spaces come from. – Andrej Bauer Mar 20 '10 at 20:00
add comment
Isn't this just the Base of the topology?
up vote -1 down
1 Huh? Take the set {0,1} with the discrete topology. The subsets { {0}. {1}} form a base of the topology, and they don't satisfy the conditions given in the question. Perhaps you
are thinking of something else? – Willie Wong Mar 20 '10 at 13:39
Stupid me. I missed the word 'Proper'. – Undergrad Mar 20 '10 at 13:59
add comment
Not the answer you're looking for? Browse other questions tagged gn.general-topology terminology or ask your own question.
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Classical type theory
Results 21 - 30 of 88
- Journal of Symbolic Logic , 2004
"... Abstract. In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of
higher-order models with respect to various combinations of Boolean extensionality and three forms of func ..."
Cited by 15 (9 self)
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Abstract. In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of
higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods
(by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes. §1. Motivation. In classical
first-order predicate logic, it is rather simple to assess the deductive power of a calculus: first-order logic has a well-established and intuitive set-theoretic semantics, relative to which
completeness can easily be verified using, for instance, the abstract consistency method (cf. the introductory textbooks [6, 22]). This well understood meta-theory has supported the development of
calculi adapted to special applications—such as automated theorem proving (cf. [16, 47] for an overview). In higher-order logics, the situation is rather different: the intuitive set-theoretic
standard semantics cannot give a sensible notion of completeness, since it does
- AUTOMATED DEDUCTION — CADE-16 INTERNATIONAL CONFERENCE, LNAI 1632 , 1999
"... This paper presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69] , and
acalculusERUE adapting FO RUE-Resolution [Dig79] to classical type theory, i.e., HO logic based on Church ..."
Cited by 14 (8 self)
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This paper presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69] , and
acalculusERUE adapting FO RUE-Resolution [Dig79] to classical type theory, i.e., HO logic based on Church’s simply typed λ-calculus. EP and ERUE extend the extensional HO resolution approach ER
[BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones
provided by ER that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable difference-reducing character the HO paramodulation approach loses its pure
term-rewriting character. On the other hand examples demonstrate that the extensionality rules harmonise quite well with the difference-reducing HO RUE-resolution idea.
- In FroCos , 2005
"... Abstract. We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following Nelson-Oppen schema.
The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enj ..."
Cited by 14 (1 self)
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Abstract. We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following Nelson-Oppen schema. The
procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and allows an abstract version of a ‘Keisler-Shelah like
’ isomorphism theorem. We show that this general decidability transfer result covers as special cases, besides applications which seem to be new, the recent extension of Nelson-Oppen procedure to
non-disjoint signatures [16] and the fusion transfer of decidability of consistency of A-Boxes with respect to T-Boxes axioms in local abstract description systems [9]; in addition, it reduces
decidability of modal and temporal monodic fragments [32] to their extensional and one-variable components. 1
- J. of Formalized Reasoning , 2010
"... Articulate Software The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known and well established infrastructure that supports research, development, and
deployment of Automated Theorem Proving (ATP) systems. The extension of the TPTP from first-order form (F ..."
Cited by 14 (10 self)
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Articulate Software The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known and well established infrastructure that supports research, development, and
deployment of Automated Theorem Proving (ATP) systems. The extension of the TPTP from first-order form (FOF) logic to typed higher-order form (THF) logic has provided a basis for new development and
application of ATP systems for higher-order logic. Key developments have been the specification of the THF language, the addition of higher-order problems to the TPTP, the development of the TPTP THF
infrastructure, several ATP systems for higher-order logic, and the use of higher-order ATP in a range of domains. This paper surveys these developments. 1.
, 1994
"... Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the
suspension notation for lambda terms that is described in a companion paper. This notation obviates ff-conver ..."
Cited by 12 (2 self)
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Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the
suspension notation for lambda terms that is described in a companion paper. This notation obviates ff-conversion in the comparison of terms by using the `nameless' scheme of de Bruijn and also
permits a delaying of substitutions by including a class of terms that encode other terms together with substitutions to be performed on them. The suspension notation contains a mechanism for
`merging' substitutions so that they can be effected in a common structure traversal. The mechanism is cumbersome to implement in its full generality and a simplification to it is considered. In
particular, the old merging operations are eliminated in favor of new ones that capture some of their functionality and that permit a simplified syntax for terms. The resulting notation is refined by
the addition of annotations ...
- JOURNAL OF APPLIED LOGIC , 2008
"... Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as
possible from their weaknesses. Of particular interest is the integration of first-order and higher-order tech ..."
Cited by 11 (7 self)
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Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible
from their weaknesses. Of particular interest is the integration of first-order and higher-order techniques. First-order reasoning systems, on the one hand, have reached considerable strength in some
niches, but in many areas of mathematics they still cannot reliably solve relatively simple problems, for example, when reasoning about sets, relations, or functions. Higher-order reasoning systems,
on the other hand, can solve problems of this kind automatically. But the complexity inherent in their calculi prevents them from solving a whole range of problems. However, while many problems
cannot be solved by any one system alone, they can be solved by a combination of these systems. We present a general agent-based methodology for integrating different reasoning systems. It provides a
generic integration framework which facilitates the cooperation between diverse reasoners, but can also be refined to enable more efficient, specialist integrations. We empirically evaluate its
usefulness, effectiveness and efficiency by case studies involving the integration of first- order and higher-order automated theorem provers, computer algebra systems, and model generators.
- In Kirchner and Kirchner [KK98 , 1998
"... Abstract. In this paper we present an extensional higher-order resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary
for the completeness result – by specialized (goal-directed) inference rules is of practical applicabilit ..."
Cited by 11 (6 self)
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Abstract. In this paper we present an extensional higher-order resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for
the completeness result – by specialized (goal-directed) inference rules is of practical applicability, as an implentation of the calculus in the Leo-System shows. Furthermore, we prove the
long-standing conjecture, that it is sufficient to restrict the order of primitive substitutions to the order of input formulae. 1
- FB Informatik, U. des Saarlandes , 2008
"... Abstract Garg and Abadi recently proved that prominent access control logics can be translated in a sound and complete way into modal logic S4. We have previously outlined how normal multimodal
logics, including monomodal logics K and S4, can be embedded in simple type theory and we have demonstrate ..."
Cited by 11 (9 self)
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Abstract Garg and Abadi recently proved that prominent access control logics can be translated in a sound and complete way into modal logic S4. We have previously outlined how normal multimodal
logics, including monomodal logics K and S4, can be embedded in simple type theory and we have demonstrated that the higher-order theorem prover LEO-II can automate reasoning in and about them. In
this paper we combine these results and describe a sound (and complete) embedding of different access control logics in simple type theory. Employing this framework we show that the off the shelf
theorem prover LEO-II can be applied to automate reasoning in and about prominent access control logics. 1
"... We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We
give examples to demonstrate that these embeddings provide an effective framework for computational inve ..."
Cited by 9 (9 self)
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We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give
examples to demonstrate that these embeddings provide an effective framework for computational investigations of various non-classical logics. We report some experiments using the higher-order
automated theorem prover LEO-II.
- Typed Lambda Calculi and Applications, number 3461 in Lectures , 2005
"... Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on
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Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on
completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular
to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.
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Encyclopaedia Index
Back to start of article
3.4 Sub-group in which two differential equations are used
3.4.1 The k-epsilon (KE-EP) model
1. Introduction
The KE-EP turbulence model proposed by Harlow and Nakayama in 1968 is by far the most widely-used two-equation eddy-viscosity turbulence model, mainly because the EP was long believed to require no
extra terms near walls. The popularity of the model, and its wide use and testing, has thrown light on both its capabilities and its short-comings, which are well-documented in the literature.
PHOENICS provides the standard high-Reynolds-number form of the KE-EP model, as presented by Launder and Spalding [1974], with inclusion of allowance for buoyancy effects.
Also provided are:-
• the modified model of Yap (see Section 3.4.4 below on the Lam-Bremhorst KE-EP model);
• a low-Reynolds extension with an option for the so-called 'Yap'
• correction for separated flows (see Section 3.4.4 below);
• the two-scale split-spectrum extension (see Section 3.5 below);
• the modified model of Chen and Kim (see Section 3.4.2 below); and
• the RNG-derived model (see Section 3.4.3 below).
2. Description of the model
For high turbulent Reynolds numbers, the standard form of the KE-EP model may be summarised as follows, with ,t denoting differentiation with respect to time and ,i denoting differentiation with
respect to distance:
(RHO*KE),t + (RHO*Ui*KE - {RHO*ENUT/PRT(KE)}*KE,i ),i = RHO*(Pk + Gb - EP) (1)
(RHO*EP),t + (RHO*Ui*EP - {RHO*ENUT/PRT(EP)}*EP,i ),i = {RHO*EP/KE}*(C1*Pk + C3*Gb - C2*EP) (2)
ENUT = CMUCD*KE**2/EP (3)
Here KE is the turbulent kinetic energy; EP is the dissipation rate; RHO is the fluid density; ENUT is the turbulent kinematic viscosity.
Energy generation
Pk is the volumetric production rate of KE by shear forces. It is calculated from:
Pk = ENUT * (Ui,j + Uj,i) Ui,j (4)
Gb is the volumetric production rate of KE by gravitational forces interacting with density gradients. It is calculated from:
Gb = - ENUT * gi * {RHO,i}/(RHO * PRT(H1)) (5)
where gi is the gravitational vector and PRT(H1) is the turbulent Prandtl number.
Gb is negative for stably-stratified (dense below light) layers, so that KE is reduced and turbulence damped.
Gb is positive for unstably-stratified (dense above light) layers, in which therefore KE increases at the expense of gravitational potential energy.
With the Boussinesq approximation, in which the variations in density are expressed by way of variations in enthalpy, eqn (5) reduces to:
Gb = ENUT * BETA * gi * {H1,i}/(CP * PRT(H1)) (6)
where H1 is the enthalpy, CP is the specific heat at constant pressure, and BETA is the volumetric coefficient of expansion.
If the energy equation is solved in terms of the temperature TEM1, rather than enthalpy H1, equation (6) is replaced by:
Gb = ENUT * BETA * gi * {TEM1,i}/PRT(TEM1) (7)
The empirical constants
The following constants are normally used:
PRT(KE)=1.0, PRT(EP)=1.314, CMUCD=0.09, C1=1.44, C2=1.92, C3=1.0.
The constant C3 has been found to depend on the flow situation. It should be close to zero for stably-stratified flow, and to 1.0 for unstably-stratified flows. The default in the PHOENICS VR Menu is
to compute C3 from the function proposed by Henkes et al [1991]:
C3 = tanh (v/U) (8)
where v and U are, respectively, the velocity components parallel and perpendicular to the gravity vector. This function arranges that C3e is unity when the main flow direction is aligned with
gravity, and zero when the main flow direction is perpendicular to gravity. The computed value of C3 can be stored by using the command STORE(C3EB) in the Q1 file.
If PHOENICS VR is not used, then a default value of C3=1.0 is employed via the setting GCEB=1.0 in the GROUND subroutine GXKEGB (in the file GXGENK.FOR). A value of C3=0.0 may be effected by simply
not using a COVAL statement for EP. The default value OF GCEB=1.0 may be overwritten by setting, for example:
in the Q1 file. The setting SPEDAT(SET,KEBUOY,GCEB,R,-1.0) arranges that C3 is computed from equation (8) above.
The model presented above is applicable only in regions where the turbulence Reynolds number is high.
Near walls, where the Reynolds number tends to zero, the model requires the application of so-called 'wall functions'(see Section 8 below ) or alternatively, the introduction of a low-Reynolds-
number extension (see Sections 3.3.1 and 3.4.4).
The standard model employs the wall-function approach.
Performance of the model
It should be mentioned that the standard form of the KE-EP model has been found to perform less than satisfactorily in a number of flow situations, including:
• separated flows,
• buoyancy,
• streamline curvature,
• swirl,
• turbulence-driven secondary motions,
• rotation,
• compressibility,
• adverse pressure gradients and
• axi-symmetrical jets.
Nevertheless because the model is so widely used, variants and/or ad-hoc modifications aimed at improving its performance abound in the literature. The most well-known include:
(a) the RNG, Chen-Kim and Yap variants for use in separated flows; and
(b) the ad-hoc Richardson-number modification of Launder et al [1977] for curvature, swirl and rotation.
3. Activation of the model
The standard KE-EP is activated by inserting the PIL command TURMOD(KEMODL) in the Q1 file, which is equivalent to:
SOLVE(KE,EP);ENUT=GRND3;EL1=GRND4;KELIN=0 PATCH(KESOURCE,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP) COVAL(KESOURCE,KE,GRND4,GRND4) COVAL(KESOURCE,EP,GRND4,GRND4);GENK=T TERMS(KE,N,Y,Y,Y,Y,N);TERMS
The sources for KE and EP are calculated and inserted in subroutine GXKESO called from GROUP 13 of GREX. Different linearizations of these sources are selected by the KELIN parameter. The generation
rate used in the source terms can be stored by the command STORE(GENK).
The WALL and CONPOR commands create the required wall-function COVAL settings automatically,
COVAL(WALLN,KE,GRND2,GRND2); COVAL(WALLN,EP,GRND2,GRND2)
COVAL(WALLN,KE,GRND3,GRND3); COVAL(WALLN,EP,GRND3,GRND3)
if the user sets WALLCO=GRND3 in the Q1 file.
Thus, the PHOENICS default is equilibrium (GRND2) wall functions. However, in separated flows the use of non-equilibrium (GRND3) wall functions is recommended, especially if wall heat transfer is
The buoyancy source terms are activated in the KE and EP equations when the user introduces the following PATCH and COVAL statements in the Q1 file:
PATCH(KEBUOY,PHASEM,1,NX,1,NY,1,NZ,1,1) COVAL(KEBUOY,KE,GRND4,GRND4); COVAL(KEBUOY,EP,GRND4,GRND4)
The cartesian components of the gravitational vector, which appears in equation (5), are defined by setting BUOYA, BUOYB and BUOYC in the Q1 file.
These parameters must be set in connexion with the buoyancy source term in the momentum equations. For example, with gravity acting in the negative y-direction in Cartesian coordinates, the following
PIL sequence introduces a buoyancy source in the V1 equation:
BUOYA=0.;BUOYB=-9.81;BUOYC=0.0;BUOYD=0.5*(RHOB+RHOT) PATCH(BUOY,PHASEM,1,NX,1,NY,1,NZ,1,1) COVAL(BUOY,V1,FIXFLU,GRND2)
Here BUOYD is the reference density defined by the user.
With the Boussinesq approximation, and solution of the energy equation via H1, the buoyancy force would be activated as follows:
BUOYA=0.;BUOYB=-9.81;BUOYC=0.0;BUOYD=....;BUOYE=.... PATCH(BUOY,PHASEM,1,NX,1,NY,1,NZ,1,1) COVAL(BUOY,V1,FIXFLU,GRND3)
Here BUOYD = -BETA/CP and BUOYE = -BUOYD*H1,ref and H1,ref is the reference enthalpy.
If the energy equation is solved via the TEM1 variable, then
BUOYD = -BETA and BUOYE = TEM1,ref
where TEM1 is the reference temperature; if the volumetric coefficient of expansion is variable, then BUOYD=GRND10 is appropriate.
The Boussinesq form of the KE-EP buoyancy source terms is used only when the density is set as a constant via the Q1 or the menu, and not when a constant value is set via GREX or GROUND.
For separated flows, the so-called Yap-correction to the EP eqn can be introduced into the KE-EP model, as follows:
TURMOD(KEMODL-YAP) for the high-Re KE-EP model; and TURMOD(KEMODL-LOWRE-YAP) for the low-Re form of the model.
The extension -YAP is equivalent to the following PIL commands:
DISWAL;PATCH(EYAP,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP) COVAL(EYAP,EP,FIXFLU,GRND5) FIINIT(WDIS)=0.1*AMIN1(XULAST,YVLAST,ZWLAST)
TURMOD(KECHEN) selects the modified KE-EP model of Chen and Kim, which is described below in Section 3.4.2.
TURMOD(KERNG) selects the RNG-derived KE-EP model of Yakhot and Orszag, which is described below in Section 3.4.3.
TURMOD(KEMODL-LOWRE) selects the Lam-Bremhorst low-Re extension to KE-EP model, which is described below in Section 3.4.4.
TURMOD(KEMODL-2L) activates the two-layer low-Re model, as an alternative to the conventional low-Re k-e models. The two-layer model employs the high-Re k-e model away from the wall, and a one-
equation in the near-wall region.
4. Sources of further information
F.H.Harlow and P.I.Nakayama, 'Transport of turbulence energy decay rate', LA-3854, Los Alamos Science Lab., U. California, USA, (1968).
R.A.Henkes,F.van der Flugt and C.Hoogendoorn, 'Natural convection in a square cavity with low-Re turbulent fluids', Int.J.Heat Mass Transfer, Vol.34, p1543-1557, (1991).
Press, (1972).
B.E.Launder and D.B.Spalding, 'The numerical computation of turbulent flows', Comp. Meth. in Appl. Mech. & Eng., Vol.3, pp269, (1974).
B.E.Launder, C.H.Priddin and B.R.Sharma, 'The calculation of turbulent boundary layers on spinning and curved surfaces', ASME J Fluids Engng., Vol.99, p321, (1977).
W.Rodi, 'Examples of calculation methods for flow and mixing in stratified fluids', J.Geo.Res., Vol.92, No.C5, p5305, (1987).
See also the Instruction Course lectures on Turbulence Modelling.
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rate of pressure increase
October 15th 2013, 08:17 PM
rate of pressure increase
Okay so this one I thought was going to be more straight forward because I was given an equation that relates that which dependent upon time t.
when a certain gas expands or contracts adiabatically, it obeys the law $PV^1.4 = K$ where P is pressure, V is volume and K is a constant.
At a certain instant the pressure is 40 N/cm^2, the volume is 32cm^3, and the volume is increasing at a rate of 5 cm^3 per second. At what rate is the pressure changing at this instant.
So I thought the given equation was the one I would use. I solve it to find K
$40(32^1.4) = 5120$
then I differentiated the equation
$\frac{d}{dt} [PV^1.4=k] \Rightarrow V^1.4\frac{dP}{dt} + (1.4V^.4)P\frac{dV}{dt} = 0$
I can't figure out how to make exponents with decimal points work in latex.. the exponents are 1.4 and .4 after the differentiation.
anyway, Am I on the right track?
October 15th 2013, 09:13 PM
Re: rate of pressure increase
there appears to be something wrong in the expression. Pl check if it is PV^(1.4) = k
October 15th 2013, 09:23 PM
Re: rate of pressure increase
I don't understand what you mean.
Yes that is the right equation given by the book if that is what you mean.
October 15th 2013, 11:47 PM
Re: rate of pressure increase
I checked the answer and it comes out to $-8.75 g/cm^2/s$
October 16th 2013, 01:31 AM
Re: rate of pressure increase
okay I have solved this and I did it two separate ways
$\frac{d}{dt} [PV^(1.4)=K] \Rightarrow V^(1.4)\frac{dP}{dt} + 1.4V^(0.4)P\frac{dV}{dt} = \frac{dK}{dt}\Rightarrow\frac{dP}{dt} = \frac{-[1.4V^(0.4)(40)(5)}{32^(1.4)}$
$\Rightarrow -8.75$
$\frac{d}{dt} [P = \frac{K}{V^(1.4)} \Rightarrow \frac{dP}{dt}=K[-1.4V^(-2.4)\frac{dV}{dt}]$
$\Rightarrow\frac{dP}{dt} = 5120[-1.4(32^(-2.4)(5)]\Rightarrow -8.75$
Okay so for one I needed the value of K and the other I needed the derivative of K being zero.
I wouldn't have known the g/cm^2/s part though so hopefully I learn that later on when I am required to take some physics.
also, is there a way to use exponents that have decimal points in them in LaTeX? Because, what I had to do makes it look a little bit confusing.
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[SOLVED] Differential Equation Help
September 20th 2008, 05:47 PM #1
[SOLVED] Differential Equation Help
Hi all,
Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having
trouble with it.
Here it is:
"find the equation of R(t) when
$\frac{dE}{dt}= -\gamma_{b}E$where $E(0) = E_0$and $\gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p$,
$\frac{dS}{dt}= -\gamma_{s}S$ where $S(0) = S_0$ and $\gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p$
$\frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS$
where $M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p$ are all constants.
Now I know that
$S(t) = S_0 e^{-\gamma_{s} t}$
$E(t) = E_0 e^{-\gamma_{b} t}$
and then by substituting these values into $\frac{dR}{dt}$ we obtain
$\frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}$
this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
Last edited by Maccaman; September 20th 2008 at 05:48 PM. Reason: Latex Error
Hi all,
Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having
trouble with it.
Here it is:
"find the equation of R(t) when
$\frac{dE}{dt}= -\gamma_{b}E$where $E(0) = E_0$and $\gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p$,
$\frac{dS}{dt}= -\gamma_{s}S$ where $S(0) = S_0$ and $\gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p$
$\frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS$
where $M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p$ are all constants.
Now I know that
$S(t) = S_0 e^{-\gamma_{s} t}$
$E(t) = E_0 e^{-\gamma_{b} t}$
and then by substituting these values into $\frac{dR}{dt}$ we obtain
$\frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}$
this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
Just integrate with respect to t.
Hi all,
Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having
trouble with it.
Here it is:
"find the equation of R(t) when
$\frac{dE}{dt}= -\gamma_{b}E$where $E(0) = E_0$and $\gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p$,
$\frac{dS}{dt}= -\gamma_{s}S$ where $S(0) = S_0$ and $\gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p$
$\frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS$
where $M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p$ are all constants.
Now I know that
$S(t) = S_0 e^{-\gamma_{s} t}$
$E(t) = E_0 e^{-\gamma_{b} t}$
and then by substituting these values into $\frac{dR}{dt}$ we obtain
$\frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}$
this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
Well, as the right hand side doesn't depend on R all you need to do is integrate both sides with respect to t.
ahhhhhh, of course
Is this answer correct? (my integration rules are a bit dodgy)
$R(t) = - \frac{\gamma_{b_1} E_0 e^{-\gamma_b t}}{\gamma_b} - \frac{\gamma_{s_1} M S_0 e^{-\gamma_s t}}{\gamma_s}$
September 20th 2008, 05:49 PM #2
September 20th 2008, 05:50 PM #3
September 20th 2008, 05:53 PM #4
September 20th 2008, 05:56 PM #5
September 20th 2008, 06:35 PM #6
September 20th 2008, 06:48 PM #7
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Prove that f(x)=f(x^2)
January 13th 2013, 08:07 AM
Prove that f(x)=f(x^2)
We have function f:[0,1]->R, which is continuous and f(0)=f(1). Prove that exsists x (0,1) such that f(x) = f(x^2).
I even don't know how to start solving this...
January 13th 2013, 10:11 AM
Re: Prove that f(x)=f(x^2)
Hi happygirl! :)
I'd start with trying to find an example where it works out.
After that we'll see.
So, suppose f(x) = C for some constant C.
Does it have an x in (0,1) such that f(x) = f(x^2)?
Suppose f(x) = |x - (1/2)|.
Does it have an x in (0,1) such that f(x) = f(x^2)?
January 13th 2013, 10:28 AM
Re: Prove that f(x)=f(x^2)
First note that since f(x) is continuous, it has a maximum and a minimum on the closed interval [0,1]. (One or both of them might be equal to f(0) = f(1)). However, we can assume the function is
non constant, otherwise it is trivial and the result would follow immediately. Therefore, f(x) has either an absolute minimum or an absolute maximum on the open interval (0,1), or both.
Let g(x) = f(x) - f(x^2)
Then g(0) = g(1) = 0. You now need to show that there is some other x such that g(x) = 0.
Il show this by contradiction. Lets assume that there is no such x. That means that either f(x) > f(x^2) for all x in (0,1) OR f(x) < f(x^2) for all x in (0,1), due to the intermediate value
Assume f(x) > f(x^2) for all x in (0,1). If f(x) has an absolute maximum, occuring at x = k, then this inequality will by definition fail to hold for x = sqrt(k). Similarly, if f(x) has an
absolute minimum, occuring at x=k, then the inequality will fail to hold for x = k. Therefore, it is not possible that f(x) > f(x^2) for all x in (0,1).
Equivalently, assume f(x) < f(x^2) for all x in (0,1). If f(x) has an absolute maximum, occuring at x = k, then this inequality will by definition fail to hold for x = k. Similarly, if f(x) has
an absolute minimum, occuring at x = k, then the inequality will fail to hold for x = sqrt(k). Therefore, it is not possible that f(x) < f(x^2) for all x in (0,1).
Since neither of those is possible, it has to be the case that f(x) = f(x^2) somewhere in (0,1).
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Re: Cerberus and Quine
Date: 02/23/05
Date: 23 Feb 2005 07:31:08 -0800
Paul Holbach wrote:
> > examachine@gmail.com wrote:
> > While Paul is obviously very right in saying
> > that truth does not have
> > to conform to common sense, I think especially
> > in caring about ordinary
> > language, you have to do ordinary philosophy of
> > language, and respect
> > common sense, because in a strong sense,
> > that is definitely what you
> > want to explain.
> I carefully chose my words when I said that common sense was *not
> always* the key to wisdom.
> I didnīt say "never"!
> > The true philosophy of mathematics
> > has to explain why and how axioms
> > are selected, regardless of this
> > confusing metaphysical talk.
> Apart from the methodology of mathematics and the philosophy of
> mathematics not being one and the same, the "true" philosophy of
> mathematics has to provide substantial and elaborate answers to the
> following crucial questions:
> "The job of the philosopher is to give an account of mathematics and
> its place in our intellectual lives.
> - What is the subject-matter of mathematics (ontology)?
> - What is the relationship between the subject-matter of mathematics
> and the subject-matter of science, which allows such extensive
> application and cross-fertilization?
> - How do we manage to do and know mathematics (epistemology)?
> - How can mathematics be taught?
> - How is mathematical language to be understood (semantics)?
> In short, the philosopher must say something about mathematics,
> something about the application of mathematics, something about
> mathematical language, and something about ourselves.
> A daunting task, even without the job of eliciting first principles."
> ;-)
> [Shapiro, Stewart (2000). /Thinking about mathematics: The
> of mathematics/. Oxford: Oxford University Press. (p. 15f)]
I agree with your explanations, Paul. We merely don't agree that
realism satisfies questions at all these levels. It seems, in
particular, on ontology.
It is a challenge that goes beyond the skills of even the finest
mathematicians and logicians to determine what is "philosophically"
true or significant about mathematics.
Let me point out my position briefly, so it becomes more apparent where
I stand. I would like a metaphysics which is compatible with the modern
view of the mind. This view excludes any kind of dualism and Platonism.
So, these are out of question for me. However, you can still be a
realist on some matters, you don't have to go all solipsist or
fictionalist (agnostic). What I say is that the Quine-Putnam argument
is not true as it stands, but there is indeed some objective truth to
mathematics which aliens would discover. The trouble is that, this
objective truth has many intriguing forms that might deceive a simple
metaphysics. For instance, a theory of infinity will have a necessarily
fictional component, on the other hand it will be an "objective
fiction" which can be analyzed with the scientific method, and it can
even prove use in scientific thought, *as long as it is also a proper
Anyway, the ontology I take is simple: substance monism, and a
mechanical world. I deal in nothing more than that, because I simply
don't believe in angels.
On epistemology, the language of philosophical logic has no first place
for me, but logic as method of philosophy, certainly has. I do not
think that one can access any truth except by experimentation, either
outwardly, or inwardly. I think even the notion of "truth" is an
abstraction we use to make sense of the world, for instance we like to
define it as correspondence to reality. But then, in our conversation,
I see that "reality" has degenerated into a mixture of observations,
story telling, and fantasizing, spanning a wide range of cognitive
activities. I don't find this talk scientific, especially when one is
talking about unscientific concepts like transfinite ordinals.
Therefore, I would like to first consider a positivistic account that
is free of some convictions of Quine: holism (non-explanation),
logicism (mixing knowledge with existence). I think logic is just that,
a way to represent and process your knowledge. But apparently predicate
logic is not the only way to do that, right? Anyway, there are
reductions among many formalisms, but this does not mean that one of
them is somehow special. It's just that whatever formalism "works", the
human keeps. Sometimes, these formalisms can be irrational. I don't
defend "irrationalism", but the situation is this: logic does not even
tell us why its premises are true, right? So, what does this tell you?
How do you know the premises are true? By many layers of cognitive
processing, that are simply not rational in the sense of logicism, or
even proper lambda calculus. If the mind is mechanical you can write a
logical translation of that mechanism in first-order logic, but I don't
think it's going to be any more rational than any "experiment". For the
most part, we ascertain truth to our observations due to statistical
computations that are actually going on in our mind, not scholastic
Eray Ozkural
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Intel Education Designing Effective Projects Choreographing Math
Students learn to graph linear equations and choreograph dance moves to demonstrate them. For example, students modeling the function y=x² hold both arms above their head (similar to the way a
referee in a football game would indicate a touchdown), and they use a graphing calculator to create corresponding figures and graphs. Each dance is comprised of nine equation poses choreographed to
music. Students videotape or photograph their dances, and combine these visual elements with screen shots of the equations and graphs into an electronic presentation.
View how a variety of student-centered assessments are used in the Choreographing Math Unit Plan. These assessments help students and teachers set goals; monitor student progress; provide feedback;
assess thinking, processes, performances, and products; and reflect on learning throughout the learning cycle.
Introducing the Project Ask students to discuss what communicating mathematically means. Engage students in discussion as they offer their ideas and opinions. Discuss how mathematical communication
includes understanding, expressing, and conveying ideas orally, in writing, graphically, and algebraically. Introduce the idea that students will also learn how to communicate their mathematical
understanding through movement by creating a dance comprised of nine equation poses. Pose the Essential Question, How can we communicate through movement? Have students record this information in
math journals and share ideas with the class. Set the stage for the project by modeling a variety of functions with your arms. Play a popular song on the radio and move to the beat of the music. Ask
students if they can identify the functions you model. For example, stretch your arms out at a diagonal to model the equation y=x. Invite students to join you by getting out of their seats and
modeling a few basic functions to the beat of the music. Challenge them to name the equations of the functions they are modeling. Distribute the student handout and go over basic expectations for the
project, including selecting equations, choreographing the dance, choosing music, and collecting visual elements of both the poses and the corresponding graphs and equations. Encourage students to
supply props and costumes. Distribute and discuss common lines using the Common Lines Reference, a sheet of common graphs. Getting to Work Hand out the project rubric and the group task rubric so
students are aware of project expectations. Check for student understanding and answer questions as needed. Students use the group task rubric to self and peer-assess their participation while
working in groups. Allow two days for students, working in small groups, to discover families of linear functions by completing the graphing activity on a graphing calculator. After groups complete
the activity and discuss their findings with the class, have them complete a four-question investigation so you can assess their understanding thus far. Make necessary adjustments to bring all
students to a common point of understanding. Throughout the unit, teach formal lessons to develop students' understanding of linear equations. Pose the Unit Question, How do we represent linear
equations in different ways? Discuss ideas as a class. Then, begin a series of lessons to teach students to identify slope and write equations in standard form, point-slope form, and slope-intercept
form. Have students document understanding of these concepts in their math journals. Collect journals and provide students with feedback. Use the journal entries to reteach concepts as needed. Begin
the dance choreography part of the project by reviewing the student handout. Show part of a sample presentation to demonstrate ways students might represent their functions. Have students reconvene
into small groups. Make sure groups have a recording of their music as they choreograph their moves. Ask students to get their songs approved before bringing in music and starting work on their
presentations. Graphing calculators will be useful for exploring the various functions they may want to model. Have students choose equations and develop corresponding poses for their dances. Have
them experiment with the order of poses and the dance elements between each pose. Instruct students to graph each equation on a separate sheet of graph paper. When all of the groups have their
choreography established, ask each group to turn in an outline of their group’s dance sequence to you. Review each outline and make necessary recommendations and comments to each group. When the
dances are ready, have students begin developing the multimedia slide presentations. Invite other school personnel to help students work on their projects. The dance instructor, physical education
teacher, media center specialist, and video production teacher may be assets. Give students digital cameras to take pictures of their poses as well as their graph and equation sketches. Have students
draft 3- to 5-minute long slideshow presentations. Hand out the slideshow presentation checklist to students and make sure all students understand required expectations. Have students review and
refine their presentations, and practice their delivery with one another. The groups can give feedback to each other using the peer assessment sheet. Performing and Presenting Plan for students to
perform their dances and present their multimedia presentations. Invite other classes, parents, and administrators to watch. If your school has a video production class, allow students to film the
dances and broadcast them into the various classrooms. Revisit the Essential Question, How can we communicate through movement? Have students record their ideas in their math journals and make sure
they provide concrete examples from the unit. Use these entries in final assessment.
Brenda Levert teaches mathematics at the Academy for Academics and Arts in Huntsville, Alabama. Levert's classroom was featured in An Innovation Odyssey, a collection of stories of technology in the
classroom, Story 152: Choreographing Math. A team of teachers expanded the plan into the example you see here.
Mobile apps, reviewed by professional educators for related instructional content.
This unit is aligned to Common Core State Standards for Math.
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Posts by
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5th Grade Math
YOUR MAMA!!!!!!!!!!!!!!!!!
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thank you
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Can some check my personal narrative and give a grade from 1-5? Mario Garcia 10-10-11 Period.3 The Lost Dog I didn t feel like helping my sister wash my dad s car. After our dog left this morning we
were both sad, we didn t feel like doing anything but to just s...
Foreighn Languages
What Language should I learn? Mandarin or French?
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what is the plural form of turkey. i think it is turkie am i right can somebody help me If you look up the word in a hard copy dictionary (as I did), you would find that the plural is turkeys.
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What year was the oldest copy of Ralph Ellison s Going to the Territory published?
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Correlation and its discontents
Posted: October 22, 2011 Filed under: Uncategorized | Tags: maths Leave a comment
Another powerful addition to the business statistics toolbox today: correlation coefficients. These should be used in the context of two golden rules:
1. Draw a graph
2. Keep using your brain (I know it shouldn’t need saying, but honestly, people don’t always manage this one).
Here’s what you need to know: most of the time when people talk about correlation in a business context, they’re talking about linear correlation between two variables – ie how neatly they fit on a
line when plotted against one another. There’s usually an input variable (e.g. advertising spend per month) and an output (e.g. sales per month). The correlation coefficient is somewhere between -1
and 1. The absolute value tells you how well the two variables correlate, with 0 being no connection at all, and 1/-1 being a perfect line so that if you know the input, the output is perfectly
predictable. A positive/negative value tells you whether the line slopes up (positive) or down (negative).
Here are some lovely well-correlated points.
These have a correlation coefficient of 0.99. Which is not very surprising, considering how tidily they line up.
Looking at correlation can be helpful in business, as it measures how well you can predict outputs from inputs. The other essential piece of the puzzle is a regression line: this is the ‘line of best
fit’ between your points. The correlation coefficient tells you how good that ‘best fit’ is. Here’s a regression line (in red) for that suspiciously neat set of points that we just looked at.
If you end up with a clear-cut relationship like this, you’re home and dry: whenever you know ‘input’ then you can predict ‘output’, so for example you can predict your sales from your advertising
spend. You’ll often see linear correlation being assumed by business managers in the form of ‘rules of thumb’ – for example, when breaking into new markets, businesses often guess at a very simple
linear relationship between population and revenue, so if they make £1m in the UK they assume they can make £1m if they go into France as it has a similar population, and £1.3m in Germany as it has
roughly 1/3 more people, all other things being equal.
Now, those golden rules:
1. Draw a graph
Why? First off, because correlation is heavily affected by rogue and outlier data. If you draw a graph, you’ll see the rogue points, and then you can decide whether they are valid or whether you
should exclude them.
For example, if I add one dodgy point into the data above, like this:
then the correlation coefficient plummets from 0.99 to 0.87. You can imagine the mess that you get if your data includes several dodgy points. (Of course, don’t remove data if it really is valid –
otherwise your model will just be wishful thinking, created by removing everything that doesn’t suit you).
Furthermore, this kind of correlation only tells you how well your data fits along a straight line. It could be a terrible fit for a straight line, but you might see immediately on plotting it that
it fits a curve, zigzag, or something else. For a cautionary tale, see Anscombe’s quartet – remember that all four of these datasets have a correlation coefficient of 0.81 and the same line of best
fit, even though when you look at them it is completely obvious that they express different situations.
2. Keep using your brain
Once you’ve plotted your data, you need to keep thinking. For example, is there any ‘signal’ which could be masking the straight line relationship you’re looking for, which you could easily remove?
(see here for an example of an easy to remove signal). Also, how are you going to interpret the relationship if you find one – does one thing actually cause the other, or do they happen to happen at
the same time – for example, does the CMO always gee up the sales team at the same time as a big advertising push? In that case it’s hard to know how much revenue is driven by advertising spend and
how much by the sales team trying extra hard.
This ‘using your brain’ part is extra important when using clever-sounding measures like correlation coefficients – it’s tempting to just assume it must tell you something important because it seems
clever and complicated. As we’ve seen above, there are a few ways in which this can catch you out (rogue data, weird patterns, coincidences) so it’s critical to keep a common sense eye on what you
seem to be finding, to ensure that it really does make sense for your business.
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MathGroup Archive: November 2006 [00300]
[Date Index] [Thread Index] [Author Index]
Re: Question about trig simplify
• To: mathgroup at smc.vnet.net
• Subject: [mg71131] Re: Question about trig simplify
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Thu, 9 Nov 2006 03:37:39 -0500 (EST)
• References: <eisgno$ngg$1@smc.vnet.net>
You could say it is a feauture; not a bug (at least for me).
Anyway Mathematica offers many opportunities for proper/desired
simplifications/factorizations. It just needs to search a little more.
TrigReduce /@ {-2*Sin[a]*Cos[a], 2*Sin[a]*Cos[a]}
{-Sin[2*a], Sin[2*a]}
Execute the following commands for more information
r = {{Cos[a], Sin[a]}, {-Sin[a], Cos[a]}};
r . r
{{Cos[a]^2 - Sin[a]^2, 2*Cos[a]*Sin[a]}, {-2*Cos[a]*Sin[a], Cos[a]^2 -
{{Cos[2*a], Sin[2*a]}, {-Sin[2*a], Cos[2*a]}}
TrigReduce[r . r . r]
{{Cos[3*a], Sin[3*a]}, {-Sin[3*a], Cos[3*a]}}
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A. Vistoli, Z. Reichstein, Birational isomorphisms between twisted group actions, J. Lie Theory, Journal of Lie Theory 16 (2006), Number 4, 791--802.
Abstract: Let X be an algebraic variety with a generically free action of a connected algebraic group G. Given an automorphism f : G -> G, we will denote by X^f the same variety X with the G-action
given by g : x -> f(g). x
V. L. Popov asked if X and X^f are always G-equivariantly birationally isomorphic. We construct examples to show that this is not the case in general. The problem of whether or not such examples can
exist in the case where X is a vector space with a generically free linear action, remains open. On the other hand, we prove that X and X^f are always stably birationally isomorphic, i.e., X \times A
^m and X^f x A^m are G-equivariantly birationally isomorphic for a suitable non-negative integer m.
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CCP4BB Archives
Please excuse my ignorance, but I cannot understand why Rmerge is unreliable for estimation of the resolution?
I mean, from a theoretical point of view, <1/sigma> is indeed a better criterion, but it is not obvious from a practical point of view.
<1/sigma> depends on a method for sigma estimation, and so same data processed by different programs may have different <1/sigma>. Moreover, HKL2000 allows users to adjust sigmas manually. Rmerge estimates sigmas from differences between measurements of same structural factor, and hence is independent of our preferences. But, it also has a very important ability to validate consistency of the merged data. If my crystal changed during the data collection, or something went wrong with the diffractometer, Rmerge will show it immediately, but <1/sigma> will not.
So, please explain why should we stop using Rmerge as a criterion of data resolution?
Sanford-Burnham Medical Research Institute
10901 North Torrey Pines Road
La Jolla, California 92037
On Jun 1, 2012, at 5:07 AM, Ian Tickle wrote:
> On 1 June 2012 03:22, Edward A. Berry <[log in to unmask]> wrote:
>> Leo will probably answer better than I can, but I would say I/SigI counts
>> only
>> the present reflection, so eliminating noise by anisotropic truncation
>> should
>> improve it, raising the average I/SigI in the last shell.
> We always include unmeasured reflections with I/sigma(I) = 0 in the
> calculation of the mean I/sigma(I) (i.e. we divide the sum of
> I/sigma(I) for measureds by the predicted total no of reflections incl
> unmeasureds), since for unmeasureds I is (almost) completely unknown
> and therefore sigma(I) is effectively infinite (or at least finite but
> large since you do have some idea of what range I must fall in). A
> shell with <I/sigma(I)> = 2 and 50% completeness clearly doesn't carry
> the same information content as one with the same <I/sigma(I)> and
> 100% complete; therefore IMO it's very misleading to quote
> <I/sigma(I)> including only the measured reflections. This also means
> we can use a single cut-off criterion (we use mean I/sigma(I) > 1),
> and we don't need another arbitrary cut-off criterion for
> completeness. As many others seem to be doing now, we don't use
> Rmerge, Rpim etc as criteria to estimate resolution, they're just too
> unreliable - Rmerge is indeed dead and buried!
> Actually a mean value of I/sigma(I) of 2 is highly statistically
> significant, i.e. very unlikely to have arisen by chance variations,
> and the significance threshold for the mean must be much closer to 1
> than to 2. Taking an average always increases the statistical
> significance, therefore it's not valid to compare an _average_ value
> of I/sigma(I) = 2 with a _single_ value of I/sigma(I) = 3 (taking 3
> sigma as the threshold of statistical significance of an individual
> measurement): that's a case of "comparing apples with pears". In
> other words in the outer shell you would need a lot of highly
> significant individual values >> 3 to attain an overall average of 2
> since the majority of individual values will be < 1.
>> F/sigF is expected to be better than I/sigI because dx^2 = 2Xdx,
>> dx^2/x^2 = 2dx/x, dI/I = 2* dF/F (or approaches that in the limit . . .)
> That depends on what you mean by 'better': every metric must be
> compared with a criterion appropriate to that metric. So if we are
> comparing I/sigma(I) with a criterion value = 3, then we must compare
> F/sigma(F) with criterion value = 6 ('in the limit' of zero I), in
> which case the comparison is no 'better' (in terms of information
> content) with I than with F: they are entirely equivalent. It's
> meaningless to compare F/sigma(F) with the criterion value appropriate
> to I/sigma(I): again that's "comparing apples and pears"!
> Cheers
> -- Ian
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the first resource for mathematics
The author of the present pleasant paper establishes that if $A$ is a subalgebra of a Banach algebra $B$ and $f:A\to B$ satisfies $\parallel f\left(x+y\right)-f\left(x\right)-f\left(y\right)\parallel
\le \delta$ and $\parallel f\left(xy\right)-xf\left(y\right)-f\left(x\right)y\parallel \le \epsilon$, for all $x,y\in A$ and for some $\delta ,\epsilon \ge 0$, then there exists a unique additive
derivation $d:A\to B$ such that $\parallel f\left(x\right)-d\left(x\right)\parallel \le \delta \phantom{\rule{1.em}{0ex}}\left(x\in A\right)$, and $x\left(f\left(y\right)-d\left(y\right)\right)=0\
phantom{\rule{1.em}{0ex}}\left(x,y\in A\right)$. The result and its proof are still true for a more general case if we consider a normed algebra $A$ and replace $B$ by a Banach $A$-bimodule $X$.
He also proves that if $B$ is a normed algebra with an identity belonging to $A$, then every mapping $f:A\to B$ satisfying $\parallel f\left(xy\right)-xf\left(y\right)-f\left(x\right)y\parallel \le \
epsilon \phantom{\rule{1.em}{0ex}}\left(x,y\in A\right)$ must fulfil $f\left(xy\right)=xf\left(y\right)-f\left(x\right)y\phantom{\rule{1.em}{0ex}}\left(x,y\in A\right)$. This superstability result is
nice, since there is not assumed any (approximately) additive condition on $f$. Some similar results in which one considers generalized derivations can be found in the reviewer’s paper
[“Hyers-Ulam-Rassias stability of generalized derivations”, Int. J. Math. Math. Sci. (to appear)].
39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains and/or ranges
47B47 Commutators, derivations, elementary operators, etc.
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Three Electric Charges Are Arranged On A Straight ... | Chegg.com
Image text transcribed for accessibility: Three electric charges are arranged on a straight line as shown in the figure. Select True or False for the following statements. If Q1 is positive, Q2 is
negative and Q3 is positive, then Q2 MUST feel a net force to the right. If Q1 is positive, Q2 is negative and Q3 is negative, then Q2 MUST feel a net force to the right. If Q1 is positive, Q2 is
negative and Q3 is positive, then Q2 MIGHT feel a net force to the right. If Q1 is negative, Q2 is negative and Q3 is positive, then Q2 MUST feel a net force to the right. Tries 0/6 In the above
figure, R = 1.36 m, Q1 = 1.71times10-6 C, Q2=-Q1 and Q3=-Q1. Calculate the total force on Q2. Give your answer with a positive number for a force directed to the right. Tries 0/20
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Posts by
Total # Posts: 32
How do you find the distance of 2 points?
Eleven members of the team gave each other "high fives". If each player gave each teammate a high five, how many high fives would be given altogether?
business law
IF Byron Cabell forms a corporation named Cabell Inc but he fails to keep corporate reords or seperates business property from personal property and a creditor discovers poor fianancial status of the
corportation can the creditor sue Cabell as an individual? what theory will t...
business law
If Terry mistakes Mr Hall for a sales clerk and He helps her pick out a suit and she gives him money. Then he pockets the money and leaves is the store responsible? does the store have to honor the
A 120-V rms voltage at 2000 Hz is applied to a 6.0-mH inductor, a 2.0-ìF capacitor and a 200-Ù resistor. What is the rms value of the current in this circuit?
A certain ac signal at 60 Hz is applied across a 40-mH inductor and a 200-Ù resistor. What is the power factor of this circuit?
4th grade math
Students are standing in a circle evenly spaced and numbered in order. The 3rd student is directtly opposite the 7th. How many students are in the circle?
The potential difference between the plates of a parallel plate capacitor is 35 V and the electric field between the plates has strength of 750 V/m. If the plate area is 4.0 × 10-2 m2, what is the
capacitance of this capacitor?
An electron, initially at rest is accelerated through a potential difference of 550 V. What is the speed of the electron due to this potential difference?
On the 1st day the movie opened 35 people saw it. On the 2nd day 16 more people came than on the 1st day so that 86 people had seen the movie after the second day. On the 3rd day 67 people came, 16
more than on the 2nd day, If each day 16 more people saw the movie than the day...
It takes Nancy 15 minutes to walk a mile. How many miles can she walk in one and a half hours?
corresponding sides mean: a)the measures are equal b)the side is in a similar location c)the sides are opposite d)one side is shorter than the other
parallel lines never meet because they are a) far apart b)equidistant c)horizontal d)vertical
y = 1/2 x -3. what is the slope? a)-3 b)3 c)1 d)1/2
Mary has cloth in 4 colors (red, blue, green, yellow)and 2 styles (large, small) for the school logo (star). The flag she is making will have 2 different stripes with white in the middle and the
logo. What is the probability that her design will have red and blue stripes and ...
what final scores, between 1 and 30, are not possible for a football team to make?
Compare and contrast the roles the prison system, the family, and the social stratification system play in either contributing to or alleviating the problem of crime
Compare and contrast the roles the prison system, the family, and the social stratification system play
The next dividend payment by Hot Wings, Inc., will be $2.10 per share. The dividends are anticipated to maintain a 5 percent growth rate forever. If the stock currently sells for $48 per share, the
required return is percent. (Do not include the percent sign (%). Round your an...
US history
how is Pickett's charge an example of new tactics?
us history
what control if any should the states have over the use of federal lands within their borders
Oxygen gas reacts with powdered aluminum according to the following reaction: 4Al(s)+3O2(g) yields 2Al2O3(s) What volume of gas (in L), measured at 770 mmHg and 35 degrees C, is required to
completely react with 53.9g of Al?
What mass of lithium in grams is required to react completely with 57.9mL of N2 gas at STP? 6Li(s)+N2 (g)yields 2Li3N (s)
8th Grade Math
I don't know. We were just told to solve them. We were not given a formula.
8th Grade Math
Can someone please help me with the following problems? I do not have the formula for finding the slope of a line. These are not homework they are for a portfolio. Find the slope of each of the
following lines. 1. y=-5-17 2. 10 + y = x 3. y = -2x - 5 4. 4x + y = 0 5. 3x - 4y =...
s+(s-$0.45)+(s-$0.45)=$9.65 Please show me how to do this problem.
Ms Sue, Thank you very much. I hope you can help with another problem. I get mixed up on changing the plus and minus. 4a+17=a-13. I do know that the second a is a 1.
Answer: Z2 Thank you
3Z+6Z+7=29-4 Please shoe how to solve this Problem.
Please show me how to do this problem. 3Z+6Z+7=29-4 Thank You, Rusty
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convert 47.8 kilograms to pounds
You asked:
convert 47.8 kilograms to pounds
105.380961324372 pounds
the mass 105.380961324372 pounds
Say hello to Evi
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Hernán R. Henríquez
Hernán R. Henríquez was born in Santiago, Chile, in 1950. He received the B.E.E. degree from the University of Chile, Santiago, Chile, in 1974, the M.S. degree in mathematics from the University of
Santiago (USACH), Santiago, Chile, in 1975, and the Ph.D. degree in science from the University of Campinas (UNICAMP), Campinas, SP, Brazil, in 1983. He was a Fellow of the Organization of American
States (OAS) and of the National Council for Scientific and Technological Development (CNPq), Brazil. Since 1988, Henríquez has been serving as a Full Professor at the Department of Mathematics at
the University of Santiago, Santiago, Chile. His areas of research include evolution equations, mathematical theory of control, retarded functional differential equations with values in abstract
spaces, the theory of almost periodic functions, and the theory of semigroups of operators. His research work is reflected in approximately 100 scientific articles. In addition, Henríquez is the
author of the following books: Fundamentos de Análisis Funcional and Introducciòn a la Integración Vectorial, both published by Editorial Académica Española. His research work is partially supported
by the National Fund for Science and Technology, Chile.
Biography Updated on 29 December 2013
Scholarly Contributions [Data Provided by
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Finding t intervals
April 14th 2008, 12:40 PM #1
Apr 2008
Finding t intervals
I understand how to work this problem while finding Z sub alpha/2 but I can't figure out how with t sub alpha/2. The mean is unknown, as needed to be a T interval, i think, but Im not sure how to
apply these numbers.
Find the value:
t sub alpha/2 and n= 18 for the 99% confidence interval for the mean.
I understand how to work this problem while finding Z sub alpha/2 but I can't figure out how with t sub alpha/2. The mean is unknown, as needed to be a T interval, i think, but Im not sure how to
apply these numbers.
Find the value:
t sub alpha/2 and n= 18 for the 99% confidence interval for the mean.
df = n - 1 = 17. $t_{\alpha/2} = t_{0.005} = 2.898$ (using http://www.anu.edu.au/nceph/surfstat...e/tables/t.php, for example).
$\bar{x} - t_{\alpha/2} \frac{s}{\sqrt{n}} < \mu < \bar{x} + t_{\alpha/2} \frac{s}{\sqrt{n}}$
where $\bar{x}$ is the sample mean and s is the sample sd.
April 14th 2008, 08:17 PM #2
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Music: a Mathematical Offering
Back to Dave Benson's front page
Music: a Mathematical Offering
The current online version (14 December 2008) is available FREE in pdf format here: music.pdf (10 megabytes, 531 pages).
Please read further down about differences from print version.
I have noticed many people putting old versions of this text online, especially on the usenet group alt.binaries.e-book.technical:
PLEASE, PLEASE don't do this. The text is regularly updated, and your version is almost always out of date, sometimes by several years.
Finally, I get a large volume of email about this book. If I don't answer yours personally, please don't get offended.
Published by Cambridge University Press, Nov 2006, 426 pages. ISBN: 0521853877 (hbk), 0521619998 (pbk).
UK: Hardback £65 / Paperback £26
USA: Hardback $126 / Paperback $48
AUS: Hardback A$225 / Paperback A$80
Can be obtained directly from CUP online (above) or bookshop (tel. 01223-333333);
or from Amazon UK £25.99; Amazon USA $44.10; Amazon FR €29,38;
Amazon DE €32,99; Amazon CA CDN$43.96; Amazon JP ¥5,213.
A review of the book appears in the February 2007 issue (#276) of The Wire, just a few pages away from a review of a concert by Iggy Pop. Must be the first time I've been featured in the same
magazine as Iggy! Could it be a first for him too? You never know.
Also reviewed by American Scientist 95 (4) 2007, Maths Reviews, Nov 2007, The Mathematical Intelligencer, Vol 30 No 1 (2008), 76-77 and The London Mathematical Society, Newsletter 363, Oct 2007
CUP have granted me permission to keep an updated version of the book online in its entirety, even after publication, so the pdf file will continue to be available on this site, and will continue to
be updated periodically. No online version will ever be identical to the published version; each has material not contained in the other. For example, the online version keeps and continues to update
material of a more ephemeral nature such as links to online resources, and some extensive tables not suitable for a published book.
Beware that the online version continues to evolve, so that references to it will always be unstable.
References should be made to the print edition instead.
Current Chapter Headings (online version):
1. Waves and harmonics
1.1 What is sound?
1.2 The human ear
1.3 Limitations of the ear
1.4 Why sine waves?
1.5 Harmonic motion
1.6 Vibrating strings
1.7 Sine waves and frequency spectrum
1.8 Trigonometric identities and beats
1.9 Superposition
1.10 Damped harmonic motion
1.11 Resonance
2. Fourier theory
2.1 Introduction
2.2 Fourier coefficients
2.3 Even and odd functions
2.4 Conditions for convergence
2.5 The Gibbs phenomenon
2.6 Complex coefficients
2.7 Proof of Fejér's theorem
2.8 Bessel functions
2.9 Properties of Bessel functions
2.10 Bessel's equation and power series
2.11 Fourier series for FM synthesis and planetary motion
2.12 Pulse streams
2.13 The Fourier transform
2.14 Proof of the inversion formula
2.15 Spectrum
2.16 The Poisson summation formula
2.17 The Dirac delta function
2.18 Convolution
2.19 Cepstrum
2.20 The Hilbert transform and instantaneous frequency
2.21 Wavelets
3. A mathematician's guide to the orchestra
3.1 Introduction
3.2 The wave equation for strings
3.3 Initial conditions
3.4 The bowed string
3.5 Wind instruments
3.6 The drum
3.7 Eigenvalues of the Laplace operator
3.8 The horn
3.9 Xylophones and tubular bells
3.10 The mbira
3.11 The gong
3.12 The bell
3.13 Acoustics
4. Consonance and dissonance
4.1 Harmonics
4.2 Simple integer ratios
4.3 Historical explanations of consonance
4.4 Critical bandwidth
4.5 Complex tones
4.6 Artificial spectra
4.7 Combination tones
4.8 Musical paradoxes
5. Scales and temperaments: the fivefold way
5.1 Introduction
5.2 Pythagorean scale
5.3 The cycle of fifths
5.4 Cents
5.5 Just intonation
5.6 Major and minor
5.7 The dominant seventh
5.8 Commas and schismas
5.9 Eitz's notation
5.10 Examples of just scales
5.11 Classical harmony
5.12 Meantone scale
5.13 Irregular temparaments
5.14 Equal temperament
5.15 Historical remarks
6. More scales and temperaments
6.1 Harry Partch's 43 tone and other super just scales
6.2 Continued fractions
6.3 Fifty-three tone scale
6.4 Other equal tempered scales
6.5 Thirty-one tone scale
6.6 The scales of Wendy Carlos
6.7 The Bohlen-Pierce scale
6.8 Unison vectors and periodicity blocks
6.9 Septimal harmony
7. Digital music
7.1 Digital signals
7.2 Dithering
7.3 WAV and MP3 files
7.4 MIDI
7.5 Delta functions and sampling
7.6 Nyquist's theorem
7.7 The z-transform
7.8 Digital filters
7.9 The discrete Fourier transform
7.10 The fast Fourier transform
8. Synthesis
8.1 Introduction
8.2 Envelopes and LFOs
8.3 Additive synthesis
8.4 Physical modeling
8.5 The Karplus-Strong algorithm
8.6 Filter analysis for the Karplus-Strong algorithm
8.7 Amplitude and frequency modulation
8.8 The Yamaha DX7 and FM synthesis
8.9 Feedback, or self-modulation
8.10 CSound
8.11 FM synthesis using CSound
8.12 Simple FM instruments
8.13 Further techniques in CSound
8.14 Other methods of synthesis
8.15 The phase vocoder
8.16 Chebychev polynomials
9. Symmetry in music
9.1 Symmetries
9.2 The harp of the Nzakara
9.3 Sets and groups
9.4 Change ringing
9.5 Cayley's theorem
9.6 Clock arithmetic and octave equivalence
9.7 Generators
9.8 Tone rows
9.9 Cartesian products
9.10 Dihedral groups
9.11 Normal subgroups and quotients
9.12 Orbits and cosets
9.13 Burnside's lemma
9.14 Pitch class sets
9.15 Pólya's enumeration theorem
9.16 The Mathieu group M[12]
Appendix A: Answers to Almost All Exercices
Appendix B: Bessel functions
Appendix C: Complex numbers
Appendix D: Dictionary
Appendix E: Equal tempered scales
Appendix F: Frequency and MIDI chart
Appendix G: Getting stuff from the internet
Appendix I: Intervals
Appendix J: Just, equal and meantone scales compared
Appendix L: Logarithms
Appendix M: Music theory
Appendix O: Online papers
Appendix P: Partial derivatives
Appendix R: Recordings
Appendix W: The wave equation
Green's identities
Gauss' formula
Green's functions
Hilbert space
The Fredholm alternative
Solving Laplace's equation
Conservation of energy
Uniqueness of solutions
Eigenvalues are nonnegative and real
Inverting the Laplace operator
Compact operators
The inverse of the Laplace operator is compact
Eigenvalue stripping
Solving the wave equation
Polyhedra and finite groups
An example
© Dave Benson 20011.
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Welcome to Mr. Krupinski's Class
Algebra 1 Objectives
Topics to be integrated thoughout the year.
1. ISAT REVIEW (Basic skills and ISAT type questions)
2. Percents (6b6-6b7)
3. Problem solving using strategies
4. Math writing
5. Mental Math Skills
6. Check/determine that an answer or estimate is correct/reasonable (6.c.4)
7. Use of a graphing calculator
Introduction to Algebra
1. Simplify and evaluate expressions using the order of operations
2. Add, subtract, multiply, and divide three or more integers
3. Simplify expressions by combining like terms
4. Multiply and divide powers.
5. Add and subtract matrices
6. Apply the distributive property
7. Translate phrases into algebraic expressions
8. Solve real world story problems involving integers
Equations and Inequalities
1. Solve one and two-step equations involving rational numbers
2. Solve two-step inequalities involving rational numbers
3. Translate and solve one and two-step equations involving real world problems
4. Solve multi-step equations with variables on the same side.
5. Solve multi-step equations with variables on both sides
6. Solve multi-step equations involving the distributive property
7. Solve literal equations
Graphing Linear Equations
1. Graph linear equations in one variable
2. Graph linear equations in two variables using a table of values
3. Graph linear equations in two variables using intercepts
4. Find the slope of a line numerically and graphically
5. Graph linear equations in two variables using slope/intercept form
6. Graph linear equations using a graphing calculator
7. Solve and graph absolute value equations
Writing Linear Equations
1. Write the equation of a line given the slope and y-intercept
2. Write the equation of a line given the slope and a point
3. Write the equation of a line given two points
4. Convert between linear and standard form
5. Use a graphing calculator to graph linear equations
6. Use linear equations and their graphs to model real-life situations
1. Solve and graph inequalities in one variable
2. Solve and graph compound inequalities
3. Solve and graph absolute value inequalities
4. Solve and graph linear inequalities
5. Use inequalities and their graphs to model real-life situations
1. Solve a system of linear equations by graphing, substitution, using linear combinations, and graphing
2. Find the minimum and maximum values of an objective quantity subject to constraints
Powers and Exponents
1. Use the multiplication properties of exponents to evaluate powers
2. Use negative and zero exponents
3. Use division properties of exponents to evaluate powers
4. Use scientific notation to express large and small numbers
5. Use the compound interest formula and the exponential growth and decay formulas
6. Use models of exponential growth and decay to solve real-life problems
Quadratic Equations
1. Evaluate square roots
2. Use the Pythagorean Theorem
3. Solve quadratic equations by finding square roots and completing the square
4. Graph quadratic equations with and without a graphing calculator
5. Use the Quadratic Formula to solve quadratic equations
6. Graph quadratic inequalities
Polynomial and Factoring
1. Add and subtract polynomials
2. Multiply polynomials using the distributive property and the FOIL method
3. Factor polynomial expressions and equations
4. Factor polynomials by factoring out common factors, using the difference of two squares, and that are perfect square trinomials
5. Find the rational zero of a polynomial function
6. Factor quadratic trinomial
7. Use factoring to solve a quadratic equation
Rationals and Functions
1. Simplify rational expressions
2. Multiply and divide rational expressions
3. Solve rational equations using ratios and proportions
4. Identify relations and functions
5. Solve linear functions
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Racine, WI Algebra Tutor
Find a Racine, WI Algebra Tutor
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Homework Help
Posted by Linda on Monday, October 24, 2011 at 6:27pm.
Little Billy was out trick or treating on Halloween. He had gotten a lot of candy and was peering into his bag, when out from behind a tree came Freddy Krueger. Freddy reached into Billy's bag and
took half of his candy plus one piece. Within a few steps, Billy encountered a Great White Shark, who reached into Billy's sack and took half of his remaining candy plus two pieces. Before Billy
could turn and run, a Ninja Turtle, stole half of the candy that was left plus three pieces. Now Little Billy began to cry because he only had 5 pieces of candy left. How many pieces of candy did
Billy have before Freddy, the Shark, and the Ninja Turtle raided his sack?
• Trick or Treat - DrBob222, Monday, October 24, 2011 at 11:33pm
I would do that this way.
Let X = candy from 3rd party.
(X/2)-3 = 5
Solve for X and I get 16 as follows:
Multiply through by 2
X-6=10 and X = 16 which means the boy had 16 going into the 3rd "robber."
How many into the second?
(X/2)-2 = 16
X = 4=32
X = 36 going into the second.
(X/2)-1 = 36
X = 74 to start.
Check it.
1/2 of 74 = 37 and 37-1=36
1/2 of 36 = 18 and 18-2 = 16
1/2 of 16 = 8 and 8-3=5.
• Trick or Treat - chelsea, Tuesday, October 30, 2012 at 11:21pm
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Integration i cylindrical coordinates
February 2nd 2010, 05:33 PM #1
Feb 2010
Integration i cylindrical coordinates
It has been a while since I've had calculus. I am working on a fluid mechanics problem:
I have reduced an expression and this is what I have:
mu [d/dr (1/r d/dr (r v(angular)))] = 0
How do I solve for v(angular)
Follow Math Help Forum on Facebook and Google+
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Finding absolute value of imaginary number
How do you find the absolute value of 1 plus or minus i. The answer is 1.41
If you were to plot the point $\displaystyle 1 + i$ and draw the length from the origin to that point on an Argand diagram, you'll see that from the origin, you have travelled one unit right and one
unit up. So really, a right-angle triangle has been created, with the two shorter sides = 1 unit in length. How would you find the length of the hypotenuse (in other words, $\displaystyle |1 + i|$)?
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Bethe Ansatz and Open Spin-1/2 XXZ Quantum Spin Chain
Abstract (Summary)
The open spin-1/2 XXZ quantum spin chain with general integrable boundary terms is a fundamental integrable model. Finding a Bethe Ansatz solution for this model has been a subject of intensive
research for many years. Such solutions for other simpler spin chain models have been shown to be essential for calculating various physical quantities, e.g., spectrum, scattering amplitudes, finite
size corrections, anomalous dimensions of certain field operators in gauge field theories, etc. The first part of this dissertation focuses on Bethe Ansatz solutions for open spin chains with
nondiagonal boundary terms. We present such solutions for some special cases where the Hamiltonians contain two free boundary parameters. The functional relation approach is utilized to solve the
models at roots of unity, i.e., for bulk anisotropy values eta = i pi/(p+1) where p is a positive integer. This approach is then used to solve open spin chain with the most general integrable
boundary terms with six boundary parameters, also at roots of unity, with no constraint among the boundary parameters. The second part of the dissertation is entirely on applications of the newly
obtained Bethe Ansatz solutions. We first analyze the ground state and compute the boundary energy (order 1 correction) for all the cases mentioned above. We extend the analysis to study certain
excited states for the two-parameter case. We investigate low-lying excited states with one hole and compute the corresponding Casimir energy (order 1/N correction) and conformal dimensions for these
states. These results are later generalized to many-hole states. Finally, we compute the boundary S-matrix for one-hole excitations and show that the scattering amplitudes found correspond to the
well known results of Ghoshal and Zamolodchikov for the boundary sine-Gordon model provided certain identifications between the lattice parameters (from the spin chain Hamiltonian) and infrared (IR)
parameters (from the boundary sine-Gordon S-matrix) are made.
Bibliographical Information:
Advisor:Changrim Ahn; James Nearing; Rafael I. Nepomechie; Orlando Alvarez
School:University of Miami
School Location:USA - Florida
Source Type:Master's Thesis
Keywords:physics arts sciences
Date of Publication:04/12/2008
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Plancherel formula for non-second-countable (non-unimodular) groups
up vote 3 down vote favorite
The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstrict Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a
square-integrable function on the group from its Fourier transform by taking traces and integrating with respect to the Plancherel measure (alternatively: we can decompose the regular representation
into irreducible representations using the Plancherel measure).
In Representation Theory and Noncommutative Harmonic Analysis by Alexander Kirillov the same theorem is stated but without the restriction “second-countable” (theorem 6.12). It is just a survey
book—there is neither a proof nor an explicit reference. First question: Do you know a reference for this theorem?
Regarding non-unimodular groups: Duflo and Moore proved that the Plancherel formula still works for non-unimodular groups, but you have to introduce some additional unbounded, positive,
“semiinvariant” operators to scale the stuff correctly (see this paper). They require the group to be of type 1 (of course) and second countable.
Second question: Is it known whether this works for non-second-countable groups? Is there any point where second-countability is thought to be crucial?
Third question: Kirillov also mentions generalisations to non-type-1 groups. Then it is not enough to consider irreducible representations, but according to him there is a similar statement. Do you
know what theorem he means and do you know any reference? These are his words:
Another generalization is possible for groups which are not of type I. In this case, the integral on the right hand side of the formula is computed over the larger space $\tilde{G}$ and the
ordinary trace is replaced by the trace in the sense of the corresponding factor.
harmonic-analysis rt.representation-theory reference-request
add comment
1 Answer
active oldest votes
Answer to the first question: Jacques Dixmier, Les C-algèbres et leurs représentations. Section 18.8.1
Comment on the second question: I actually believe a decomposition of von-Neumann algebra into factors is only available for seperable vNas, which should be for the right regular
representation equivalent to the group being second countable.
up vote 1
down vote Comment on the third question: I have no idea what could be meant. But the decomposition into factors will not be unique (this is probably what you mean with not enough to consider
irreducible representations) and I don't even know what kind of traces should be involved. So for me, it seems unreasonable to expect something useful in this context, which has similar
applications as the Plancherel formula.
Thank you. Dixmier requires “separability” (does he mean second-countability? I have seen people using these words synonymously), too. Why don’t you think that your argument regarding
the second question can be applied to the first one, too? – The User May 17 '13 at 8:53
Regarding the third question: I have added a quote by Kirillov. And I have found this paper (ams.org/journals/tran/1962-104-02/S0002-9947-1962-0139959-X/…) regarding decompositions with
respect to this $\tilde{G}$. However, in the formula (theorem 2) there do not appear any traces and there is no relation to any kind of Fourier transform given—the proposed decomposition
is very generic. The paper seems to be too old to be a proper generalisation, too. – The User May 17 '13 at 8:57
No, I am saying the Hilbert space $L^2(G)$ is seperable in the sense that they have a countable orthonormal basis iff $G$ is second countable. Now, how do you define a state on say $C_c^
\infty(G)$ or $C_c(G)$ from a representation $\pi$ if $\pi(\phi)$ is not Hilbert Schmidt, trace class or something analogous. What is the suggested analogon you have in mind? Sure, the
integral decomposition exists, but is not unique and the unitary dual is not a nice space anymore. For type 1 e.g. it will be almost Hausdorff. – plusepsilon.de May 17 '13 at 9:47
1 For your information (I have looked it up): Separability and second-countability are not equivalent for locally compact groups: The compact group $\mathbb{T}^{\omega_1}$ is not
first-countable, but separable. I guess that Dixmier’s definition of the word “separable” is “second-countable” (I have seen that before in harmonic analysis)—thus it is probably the
same condition as Folland’s. – The User May 17 '13 at 14:28
1 Ah okay, first countability is necessary and sufficient for having a metric in a locally compact group. So correction: first countable implies second countable for lc groups if
seperable:( – plusepsilon.de May 17 '13 at 14:49
show 7 more comments
Not the answer you're looking for? Browse other questions tagged harmonic-analysis rt.representation-theory reference-request or ask your own question.
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the encyclopedic entry of Minkowski world
In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other. In classical
physics (Galilean relativity), the only conversion believed necessary was x′ = x − vt, describing how the origin of one observer's coordinate system slides through space with respect to the other's,
at speed v and along the x-axis of each frame. According to special relativity, this is only a good approximation at much smaller speeds than the speed of light, and in general the result is not just
an offsetting of the x coordinates; lengths and times are distorted as well.
If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the
spacetime interval between any two events in Minkowski space. The Lorentz transformations describe only the transformations in which the event at x = 0, t = 0 is left fixed, so they can be considered
as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.
Henri Poincaré named the Lorentz transformations after the Dutch physicist and mathematician Hendrik Lorentz (1853–1928) in 1905. They form the mathematical basis for Albert Einstein's theory of
special relativity. They were derived by Joseph Larmor in 1897, and Lorentz (1899, 1904). In 1905 Einstein derived them under the assumptions of the principle of relativity and the constancy of the
speed of light in any inertial reference frame.
Lorentz transformation for frames in standard configuration
Assume there are two observers O and $Q$, each using their own Cartesian coordinate system to measure space and time intervals. O uses $\left(t, x, y, z\right)$ and Q uses $\left(t", x\text{'}, y\
text{'}, z\text{'}\right)$. Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis overlap, the y-axis is parallel to the y' -axis, as are the z-axis and the z'
-axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate
systems are said to be in standard configuration. A symmetric presentation between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are
in symmetric configuration. The symmetric form highlights that all physical laws should be of such a kind that they remain unchanged under a Lorentz transformation.
The Lorentz transformation for frames in standard configuration can be shown to be:
t' &= gamma left(t - v x/c^{2} right) x' &= gamma left(x - v t right) y' &= y z' &= z end{cases} where
$gamma = 1 / sqrt\left\{1 - v^2/c^2\right\}$
is called the
Lorentz factor
Matrix form
This Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as
begin{bmatrix} c t' x' y' z' end{bmatrix} = begin{bmatrix} gamma&-beta gamma&0&0 -beta gamma&gamma&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c,t x y z end{bmatrix} . More generally for a boost
in an arbitrary direction
$\left(beta_\left\{x\right\}, beta_\left\{y\right\}, beta_\left\{z\right\}\right)$
begin{bmatrix} c,t' x' y' z' end{bmatrix} = begin{bmatrix} gamma&-beta_x,gamma&-beta_y,gamma&-beta_z,gamma -beta_x,gamma&1+(gamma-1)frac{beta_{x}^{2}}{beta^{2}}&(gamma-1)frac{beta_{x}beta_{y}}{beta^
{2}}&(gamma-1)frac{beta_{x}beta_{z}}{beta^{2}} -beta_y,gamma&(gamma-1)frac{beta_{y}beta_{x}}{beta^{2}}&1+(gamma-1)frac{beta_{y}^{2}}{beta^{2}}&(gamma-1)frac{beta_{y}beta_{z}}{beta^{2}} -beta_z,gamma&
(gamma-1)frac{beta_{z}beta_{x}}{beta^{2}}&(gamma-1)frac{beta_{z}beta_{y}}{beta^{2}}&1+(gamma-1)frac{beta_{z}^{2}}{beta^{2}} end{bmatrix} begin{bmatrix} c,t x y z end{bmatrix} , where
$beta = frac\left\{v\right\}\left\{c\right\}=frac\left\{|vec\left\{v\right\}|\right\}\left\{c\right\}$
$gamma = frac\left\{1\right\}\left\{sqrt\left\{1-beta^2\right\}\right\}$
Note that this is only the "boost", i.e. a transformation between two frames in relative motion. But the most general proper Lorentz transformation also contains a rotation of the three axes. This
boost alone is given by a symmetric matrix. But the general Lorentz transformation matrix is not symmetric.
The Lorentz transformation can be cast into another useful form by introducing a parameter $phi$ called the rapidity (an instance of hyperbolic angle) through the equation:
$e^\left\{phi\right\} = gamma\left(1+beta\right) = gamma left\left(1 + frac\left\{v\right\}\left\{c\right\} right\right) = sqrt frac\left\{1 + v/c\right\}\left\{1 - v/c\right\}$
$phi = ln left\left[gamma\left(1+beta\right)right\right] , -phi = ln left\left[gamma\left(1-beta\right)right\right] ,$
Then the Lorentz transformation in standard configuration is:
c t-x = e^{- phi}(c t' - x') c t+x = e^{phi}(c t' + x') y = y' z = z' end{cases}
Hyperbolic trigonometric expressions
It can also be shown that:
$gamma = cosh\left(phi\right) = \left\{ e^\left\{phi\right\} + e^\left\{-phi\right\} over 2 \right\}$
$beta = tanh\left(phi\right) = \left\{ e^\left\{phi\right\} - e^\left\{-phi\right\} over e^\left\{phi\right\} + e^\left\{-phi\right\} \right\}$
and therefore,
$beta gamma = sinh\left(phi\right) = \left\{ e^\left\{phi\right\} - e^\left\{-phi\right\} over 2 \right\}$
Hyperbolic rotation of coordinates
Substituting these expressions into the matrix form of the transformation, we have:
begin{bmatrix} c t' x' y' z' end{bmatrix} = begin{bmatrix} cosh(phi) &-sinh(phi)&0&0 -sinh(phi) & cosh(phi) &0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c t x y z end{bmatrix} .
Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity $phi$ represents the hyperbolic angle of rotation.
General boosts
For a boost in an arbitrary direction with velocity $vec\left\{v\right\}$, it is convenient to decompose the spatial vector $vec\left\{r\right\}$ into components perpendicular and parallel to the
velocity $vec\left\{v\right\}$: $vec\left\{r\right\}=vec\left\{r\right\}_perp+vec\left\{r\right\}_|$. Then only the component $vec\left\{r\right\}_|$ in the direction of $vec\left\{v\right\}$ is
'warped' by the gamma factor:
t' = gamma left(t - frac{vec{r} cdot vec{v}}{c^{2}} right) vec{r'} = vec{r}_perp + gamma (vec{r}_| - vec{v} t) end{cases} where now
$gamma equiv frac\left\{1\right\}\left\{sqrt\left\{1 - vec\left\{v\right\} cdot vec\left\{v\right\}/c^2\right\}\right\}$
. The second of these can be written as:
$vec\left\{r"\right\} = vec\left\{r\right\} + left\left(frac\left\{gamma -1\right\}\left\{v^2\right\} \left(vec\left\{r\right\} cdot vec\left\{v\right\}\right) - gamma t right\right) vec\left\{v\
These equations can be expressed in matrix form as
begin{bmatrix} c t' mathbf{r'} end{bmatrix} = begin{bmatrix} gamma & -gamma mathbf{v}^mathrm{T}/c -frac{gammamathbf{v}}{c} & I+ (gamma-1) frac {mathbf{v} mathbf{v}^mathrm{T}}{v^2} end{bmatrix} begin
{bmatrix} c t mathbf{r} end{bmatrix}text{,} where
is the identity matrix,
is velocity written as a column vector and
is its transpose (a row vector).
Spacetime interval
In a given coordinate system ($x^mu$), if two events $A$ and $B$ are separated by
$\left(Delta t, Delta x, Delta y, Delta z\right) = \left(t_B-t_A, x_B-x_A, y_B-y_A, z_B-z_A\right) ,$
spacetime interval
between them is given by
$s^2 = - c^2\left(Delta t\right)^2 + \left(Delta x\right)^2 + \left(Delta y\right)^2 + \left(Delta z\right)^2 .$
This can be written in another form using the
Minkowski metric
. In this coordinate system,
eta_{munu} = begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} . Then, we can write
s^2 = begin{bmatrix}c Delta t & Delta x & Delta y & Delta z end{bmatrix} begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c Delta t Delta x Delta y Delta z end{bmatrix} or,
using the
Einstein summation convention
$s^2= eta_\left\{munu\right\} x^mu x^nu .$
Now suppose that we make a coordinate transformation $x^mu rightarrow x"^mu$. Then, the interval in this coordinate system is given by
s'^2 = begin{bmatrix}c Delta t' & Delta x' & Delta y' & Delta z' end{bmatrix} begin{bmatrix} -1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&1 end{bmatrix} begin{bmatrix} c Delta t' Delta x' Delta y' Delta z' end
{bmatrix} or
$s"^2= eta_\left\{munu\right\} x\text{'}^mu x\text{'}^nu .$
It is a result of special relativity that the interval is an invariant. That is, $s^2 = s"^2$. It can be shown that this requires the coordinate transformation to be of the form
$x"^mu = x^nu \left\{Lambda^mu\right\}_nu + C^mu .$
is a constant vector and
a constant matrix, where we require that
$eta_\left\{munu\right\}\left\{Lambda^mu\right\}_alpha\left\{Lambda^nu\right\}_beta = eta_\left\{alphabeta\right\} .$
Such a transformation is called a
Poincaré transformation
or an
inhomogeneous Lorentz transformation
. The
represents a space-time translation. When
$C^a , = 0$
, the transformation is called an
homogeneous Lorentz transformation
, or simply a
Lorentz transformation
Taking the determinant of $eta_\left\{munu\right\}\left\{Lambda^mu\right\}_alpha\left\{Lambda^nu\right\}_beta = eta_\left\{alphabeta\right\}$ gives us
$det \left(\left\{Lambda^a\right\}_b\right) = pm 1 .$
Lorentz transformations with
$det \left(\left\{Lambda^mu\right\}_nu\right)=+1$
are called
proper Lorentz transformations
. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with
are called
improper Lorentz transformations
and consist of (discrete) space and time reflections combined with spatial rotations and boosts. They don't form a subgroup, as the product of any two improper Lorentz transformations will be a
proper Lorentz transformation.
The composition of two Poincaré transformations is a Poincaré transformation and the set of all Poincaré transformations with the operation of composition forms a group called the Poincaré group.
Under the Erlangen program, Minkowski space can be viewed as the geometry defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all
Lorentz transformations forms a group, called the Lorentz group.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
Special relativity
One of the most astounding predictions of special relativity was the idea that time is relative. In essence, each observer's frame of reference is associated with a unique clock, the result being
that time passes at different rates for different observers. This was a direct prediction from the Lorentz transformations and is called time dilation. We can also clearly see from the Lorentz
transformations that the concept of simultaneity varies between reference frames. Another startling result is length contraction.
Lorentz transformations can also be used to prove that magnetic and electric fields are simply different aspects of the same force — the electromagnetic force. If we have one charge or a collection
of charges which are all stationary with respect to each other, we can observe the system in a frame in which there is no motion of the charges. In this frame, there is only an electric field. If we
switch to a moving frame, the Lorentz transformation will give rise to a magnetic field. These two fields are unified in the concept of the electromagnetic field.
The correspondence principle
For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle. The correspondence limit is
usually stated mathematically as $v rightarrow 0$, so it is usually said that non relativistic physics is a physics of "instant action at a distance" $c rightarrow infty$.
See also History of Lorentz transformations.
The transformations were first discovered and published by Joseph Larmor in 1897. In 1905, Henri Poincaré named them after the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928) who
had published a first order version of these transformations in 1895 and the final version in 1899 and 1904.
Many physicists, including FitzGerald, Larmor, Lorentz and Woldemar Voigt, had been discussing the physics behind these equations since 1887. Larmor and Lorentz, who believed the luminiferous aether
hypothesis, were seeking the transformations under which Maxwell's equations were invariant when transformed from the ether to a moving frame. In early 1889, Heaviside had shown from Maxwell's
equations that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion relative to the ether. FitzGerald then
conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published his conjecture in Science to explain the baffling outcome
of the 1887 ether-wind experiment of Michelson and Morley. This became known as the FitzGerald-Lorentz explanation of the Michelson-Morley null result, known early on through the writings of Lodge,
Lorentz, Larmor, and FitzGerald. Their explanation was widely accepted as correct before 1905. Larmor gets credit for discovering the basic equations in 1897 and for being first in understanding the
crucial time dilation property inherent in his equations.
Larmor's (1897) and Lorentz's (1899, 1904) final equations are algebraically equivalent to those published and interpreted as a theory of relativity by Albert Einstein (1905) but it was the French
mathematician Henri Poincaré who first recognized that the Lorentz transformations have the properties of a mathematical group. Both Larmor and Lorentz discovered that the transformation preserved
Maxwell's equations. Paul Langevin (1911) said of the transformation:
"It is the great merit of H. A. Lorentz to have seen that the fundamental equations of electromagnetism admit a group of transformations which enables them to have the same form when one passes
from one frame of reference to another; this new transformation has the most profound implications for the transformations of space and time".
The usual treatment (e.g., Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is exposed, for example, in the
second volume of the Course in Theoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say, exerts on
another can not be transmitted instantaneously. Hence, there exists a theoretical maximal speed of information transmission which must be invariant, and it turns out that this speed coincides with
the speed of light in vacuum. The need for locality in physical theories was already noted by Newton (see Koestler's "The Sleepwalkers"), who considered the notion of an action at a distance
"philosophically absurd" and believed that gravity must be transmitted by an agent (interstellar aether) which obeys certain physical laws.
Michelson and Morley in 1887 designed an experiment, which employed an interferometer and a half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected the light
back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, given the results were negative, rather
than validating the aether, based upon the findings aether was not confirmed. This was a major step in science that eventually resulted in Einstein's Special Theory of Relativity.
In a 1964 paper, Erik Christopher Zeeman showed that the causality preserving property, a condition that is weaker in a mathematical sense than the invariance of the speed of light, is enough to
assure that the coordinate transformations are the Lorentz transformations.
From group postulates
Group Postulate Derivation
Following is a classical derivation based on group postulates and isotropy of the space.
Let us consider two inertial frames, K and K', the latter moving with velocity $vec\left\{v\right\}$ with respect to the former. By rotations and shifts we can choose the z and z' axes along the
relative velocity vector and also that the events (t=0,z=0) and (t'=0,z'=0) coincide. Since the velocity boost is along the z (and z') axes nothing happens to the perpendicular coordinates and we can
just omit them for brevity. Now since the transformation we are looking after connects two inertial frames, it has to transform a linear motion in (t,z) into a linear motion in (t',z') coordinates.
Therefore it must be a linear transformation. The general form of a linear transformation is
begin{bmatrix} t' z' end{bmatrix} = begin{bmatrix} gamma & delta beta & alpha end{bmatrix} begin{bmatrix} t z end{bmatrix}, where
$alpha, beta, gamma,$
are some yet unknown functions of the relative velocity
Let us now consider the motion of the origin of the frame K'. In the K' frame it has coordinates (t',z'=0), while in the K frame it has coordinates (t,z=vt). These two points are connected by our
begin{bmatrix} t' 0 end{bmatrix} = begin{bmatrix} gamma & delta beta & alpha end{bmatrix} begin{bmatrix} t vt end{bmatrix}, from which we get
$beta=-valpha ,$.
Analogously, considering the motion of the origin of the frame K, we get
begin{bmatrix} t' -vt' end{bmatrix} = begin{bmatrix} gamma & delta beta & alpha end{bmatrix} begin{bmatrix} t 0 end{bmatrix}, from which we get
$beta=-vgamma ,$.
Combining these two gives
and the transformation matrix has simplified a bit,
begin{bmatrix} t' z' end{bmatrix} = begin{bmatrix} gamma & delta -vgamma & gamma end{bmatrix} begin{bmatrix} t z end{bmatrix},
Now let us consider the inverse transformation. On one hand the inverse transformation is done simply by the inverse matrix,
begin{bmatrix} t z end{bmatrix} = frac{1}{gamma^2+vdeltagamma} begin{bmatrix} gamma & -delta vgamma & gamma end{bmatrix} begin{bmatrix} t' z' end{bmatrix}.
On the other hand the inverse transformation is the one where $v$ is substituted by $-v$,
begin{bmatrix} t z end{bmatrix} = begin{bmatrix} gamma(-v) & delta(-v) vgamma(-v) & gamma(-v) end{bmatrix} begin{bmatrix} t' z' end{bmatrix}, Now the function
can not depend upon the direction of
because it is apparently the factor which defines the relativistic contraction and time dilation. These two (in an isotropic world of ours) cannot depend upon the direction of
. Thus,
and comparing the two matrices, we get
gamma^2+vdeltagamma=1. ,
At last a composition of two coordinate transformations is also a coordinate transformation, thus the product of two of our matrices should also be a matrix of the same form, in particular the
diagonal elements should be equal. Calculating the product of two transformation matrices, one with $v$ the other with $v"$ and comparing the diagonal elements gives
frac{vgamma(v)}{delta(v)}=frac{v'gamma(v')}{delta(v')} Since this holds for any arbitrary
this combination of function must be a universal constant, one and the same for all inertial frames. Let's define this constant as
has a dimension of velocity (we have not yet assumed, that
). Using the equation from the inverse transformation we finally get
and the transformation matrix is given by
begin{bmatrix} t' z' end{bmatrix} = frac{1}{sqrt{1-v^2/c^2}} begin{bmatrix} 1 & -v/c^2 -v & 1 end{bmatrix} begin{bmatrix} t z end{bmatrix}. Apparently
cannot be negative because otherwise there would be a transformation which transforms time into spatial coordinate and vice versa. This is no good (at least in special relativity) since time can only
run in the positive direction while coordinates in both. If then
it is apparently the highest achievable velocity. Theoretically it can be either infinitely large, which gives Galilean transformation and Euclidean world with absolute time, or it can be finite,
which gives Lorentz transformation and Minkowski world of special relativity. The experiment tells us that it is finite,
From Physical Principles
Physical Derivation
The problem is usually restricted to two dimensions by using a velocity along the x axis such that the y and z coordinates do not intervene. It is similar to that of Einstein. More details may be
found in As in the Galilean transformation, the Lorentz transformation is linear : the relative velocity of the reference frames is constant. They are called inertial or Galilean reference frames.
According to relativity no Galilean reference frame is privileged. Another condition is that the speed of light must be independent of the reference frame, in practice of the velocity of the light
Galilean reference frames
In classical kinematics, the total displacement x in the R frame is the sum of the relative displacement x' in frame R' and of the displacement x in frame R. If v is the relative velocity of R'
relative to R, we have v : x = x’+vt or x’=x-vt. This relationship is linear for a constant v, that is when R and R' are Galilean frames of reference. In Einstein's relativity, the main difference
with Galilean relativity is that space is a function of time and vice-versa: t ≠ t’. The most general linear relationship is obtained with four constant coefficients, α, β, γ and v:
$t"=betaleft\left(t+alpha xright\right)$
The Lorentz transformation becomes the Galilan transformation when β = γ = 1 and α = 0.
Speed of light independent of the velocity of the source
Light being independent of the reference frame as was shown by Michelson, we need to have x = ct if x’ = ct’. Replacing x and x' in the preceding equations, one has:
$t"=betaleft\left(1+alpha cright\right)t$
Replacing t’ with the help of the second equation, the first one writes:
$cbetaleft\left(1+alpha cright\right)t=gammaleft\left(c-vright\right)t$
After simplification by t and dividing by cβ, one obtains:
$1+alpha c=frac\left\{gamma\right\}\left\{beta\right\}\left(1-frac\left\{v\right\}\left\{c\right\}\right)$
Principle of relativity
According to the principle of relativity, there is no privileged Galilean frame of reference. One has to find the same Lorentz transformation from frame R to R' or from R' to R. As in the Galilean
transformation, the sign of the transport velocity v has to be changed when passing from one frame to the other.
The following derivation uses only the principle of relativity which is independent of light velocity constancy. The inverse transformation of
$t"=betaleft\left(t+alpha xright\right)$
is :
$x=frac\left\{1\right\}\left\{1-alpha v\right\}left\left(frac\left\{x"\right\}\left\{gamma\right\}-frac\left\{vt\text{'}\right\}\left\{beta\right\}right\right)$
$t=frac\left\{1\right\}\left\{1-alpha v\right\}left\left(frac\left\{t"\right\}\left\{beta\right\}-frac\left\{alpha x\text{'}\right\}\left\{gamma\right\}right\right)$
In accordance with the principle of relativity, the expressions of x and t are:
$t=betaleft\left(t"+alpha x\text{'}right\right)$
They have to be identical to those obtained by inverting the transformation except for the sign of the velocity of transport v:
$x=frac\left\{1\right\}\left\{1+alpha v\right\}left\left(frac\left\{x"\right\}\left\{gamma\right\}+frac\left\{vt\text{'}\right\}\left\{beta\right\}right\right)$
$t=frac\left\{1\right\}\left\{1+alpha v\right\}left\left(frac\left\{t"\right\}\left\{beta\right\}-frac\left\{alpha x\text{'}\right\}\left\{gamma\right\}right\right)$
We thus have the identities, verified for any x’ and t’ :
$x=gammaleft\left(x"+vt\text{'}right\right)=frac\left\{1\right\}\left\{1+alpha v\right\}left\left(frac\left\{x\text{'}\right\}\left\{gamma\right\}+frac\left\{vt\text{'}\right\}\left\{beta\right\}
$t=betaleft\left(t"+alpha x\text{'}right\right)=frac\left\{1\right\}\left\{1+alpha v\right\}left\left(frac\left\{t\text{'}\right\}\left\{beta\right\}-frac\left\{alpha x\text{'}\right\}\left\
Finally we have the equalities :
$beta =gamma=frac\left\{1\right\}\left\{sqrt\left\{1+alpha v\right\}\right\}$
Expression of the Lorentz transformation
Using the relation
$1+alpha c=frac\left\{gamma\right\}\left\{beta\right\}\left(1-frac\left\{v\right\}\left\{c\right\}\right)$
obtained earlier, one has :
$alpha =-frac\left\{v\right\}\left\{c^2\right\}$
and, finally:
$beta =gamma=frac\left\{1\right\}\left\{sqrt\left\{1-frac\left\{v^2\right\}\left\{c^2\right\}\right\}\right\}$
We have now all the coefficients needed and, therefore, the Lorentz transformation :
$mathbf\left\{ x=frac\left\{x" + vt\text{'}\right\}\left\{ sqrt\left[\right]\left\{1 -frac\left\{v^2\right\}\left\{c^2\right\}\right\} \right\}\right\}$
$t=mathbf\left\{frac\left\{t" + frac\left\{vx\text{'}\right\}\left\{c^2\right\}\right\}\left\{ sqrt\left[\right]\left\{1 -frac\left\{v^2\right\}\left\{c^2\right\}\right\}\right\}\right\}$
Lorentz transformation writes, using the Lorentz factor γ:
$mathbf\left\{x"= gammaleft\left(x - vtright\right)\right\}$
$mathbf\left\{t"=gammaleft\left(t - frac\left\{vx\right\}\left\{c^2\right\}right\right)\right\}$
See also
Further reading
External links
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[FOM] [HM] Infinity and the "Noble Lie"
Alexander Zenkin alexzen at com2com.ru
Sun Dec 25 23:52:16 EST 2005
James Landau wrote at the [HM]-list (on 19 Dec 2005 5:58 ):
<begin quote>
I believe there is a way around having to use what Alexander Zenkin
describes as
"Bourbaki formulate this "tenet of the "ZFC religion" as follows:
AXIOM OF INFINITY. There exists an infinite set."
(Sun, 11 Dec 2005 20:11:41 +0300 From: "Alexander Zenkin" <
alexzen at com2com.ru >)
1) Let us start with the positive integers. Without using the (as yet
undefined) word "set" let us define certain binary operations on the
integers, of which one-to-one correspondence is the most important.
We now define "countably infinite set" as "anything which can be
placed in one-to-one correspondence with the positive integers".
2) What about "uncountably infinite sets"? Cantor's diagonal proof is
not necessary, as we can proceed via measure theory. Any countably
infinite set has measure zero. There exist certain sets that have a
measure greater than zero, e.g. any interval on the real number line. We
now define any set with a positive measure as an "uncountably infinite
set." Have I cheated?
<end quote>
Yes, you have cheated: the famous Cantor set ('Cantor's dust') is an
"uncountably infinite set" that has the Lebesgue measure of zero, and
therefore it is a counter-example to your 'definition' of uncountable
sets. Of course, you may ("The essence of mathematics lies in its
freedom") call any set with a positive measure as an "uncountably
infinite set", but I doubt very much that you can prove that the
cardinality of your "uncountably infinite set" is greater than the
cardinality of the set of positive integers, i.e. prove that there is
not a 1-1-correspondence between these sets, not using Cantor's diagonal
James Landau concludes:
<begin quote>
What I think I have accomplished is to defend Cantor and other set
theorists of the charge that they failed to explicitly state certain
axioms. Rather they have assumed or implied certain axioms, including
"the positive integers exist" and "the real number line exists", that
their critics such as Brouwer (cited above) also implicitly accept.
<end quote>
You certainly proved that in modern Axiomatic Set Theory (further -AST)
there are implicit ('assumed or implied') statements that are in reality
necessary conditions of some AST-proofs. But you could not prove that
the explication of such hidden statements is forbidden in AST. Moreover
the history of mathematics itself is a history of explications, more
clear formulations, and formalizations of primarily informal, fuzzy, and
hidden statements some of which later became axioms.
The explication of any hidden statement which is used as a necessary
condition of a mathematical proof can't do harm to mathematics, but
sometimes can help to avoid a trouble in the case when the addition of
the hidden statement to an axiom system of a given area leads to a
So, we have the following situation.
The paradigmatic AST-Theorem on the uncountability of continuum is
proved using algorithmically some consequences of the Cantor axiom ("all
infinite sets are actually infinite"), since if they use the Aristotle
axiom ("the infinite exists potentially"), the AST-Theorem becomes
unprovable (together with all Cantor's 'Study on Transfinitum'). However
modern AST outlaws the Cantor axiom and forbids to mention it at all.
I see no mathematical reasons against the addition of the Cantor
axiom to ZFC-system. By the way, a strict definition of the actual
infinity notion, used in Cantor's axiom formulation, is given first in
the papers shown in my previous message to FOM- and HM-lists.
I (together with Kronecker, Hermit, Poincare, Weyl, Brouwer,
etc., etc., etc.) have a firm belief (together with a strict proof of
the belief) that nobody will be able ever to defend Cantor's theory of
transfinite ordinals and cardinals.
Merry Christmas and Happy New Year !
Alexander Zenkin
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Figure 1 A.1 For V5 - 5 Volts Dc Calculate V Infinity ... | Chegg.com
Information in photo Image text transcribed for accessibility: Figure 1 A.1 For V5 - 5 volts dc calculate V infinity between nodes 3 and 0. Calculate I infinity by short circuiting nodes 3-0.
Calculate Req looking to the left at the nodes 3-0 with V, short circuited and also verify that Req = V infinity / I infinity Draw the Thevenin and Norton equivalent circuits. A.2 For RL = 1K Ohm and
10K Ohm , calculate the current through the resistance. Calculate the value of RL for maximum power transfer. Calculate the power dissipated in RL and R111
Electrical Engineering
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schizophernics - sampling distribution
May 30th 2006, 10:22 AM #1
May 2006
schizophernics - sampling distribution
Paranoid schizophrenics often manifest two different types of problems: depression and trembling. Following is the probability distribution for a discrete random variable, X: the number of these
problems that a paranoid schizophrenic might experience:
X P(X)
0 .35
1 .47
2 .18
a) Generate the sampling distribution for the mean number of these problems that random samples of two paranoid schizophrenics might experience.
b) What are the mean and standard deviation of the sampling distribution of the mean number of these problems that samples of 40 paranoid schizophrenics might experience?
c) What is the probability that the total number of these problems in a random sample of 40 paranoid schizophrenics will exceed 36?
a) 0.83
b) 0.1119
c) 0.2643
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Geometry and the imagination
You are currently browsing the tag archive for the ‘Bowen’ tag.
I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic $3$-manifold $M$ contains a
closed $\pi_1$-injective surface. Equivalently, $\pi_1(M)$ contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing
question in $3$-manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic $3$-manifold groups were LERF, one would be able to deduce
that all hyperbolic $3$-manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic $3$-manifold groups are LERF;
if successful, this would therefore resolve some of the most important outstanding problems in $3$-manifold topology (in fact, I would say: the most important outstanding problems, by a substantial
In fact, the argument appears to work for hyperbolic manifolds of every dimension $\ge 3$, and possibly more generally still. Details on the argument of Markovic-Kahn are scarce (Vlad informs me that
they expect to have a preprint in a few weeks) but the sketch of the argument presented by Kahn is compelling. Roughly speaking, the argument (as summarized by Ian Agol in a comment at Jesse’s blog)
takes the following form:
1. Given $M$, for a sufficiently big constant $R$, one can find “many” immersed, almost totally-geodesic pairs of pants (i.e. thrice-punctured spheres) with geodesic boundary components (i.e.
“cuffs”) of length very close to $2R$. In fact, one can further insist that the complex length of the boundary geodesic is very close to $2R$ (i.e. holonomy transport around this geodesic does
not rotate the normal bundle very much).
2. Conversely, given any geodesic of complex length very close to $2R$, one can find many such pairs of pants that it bounds, and moreover one can find them so that the normal to the geodesic
pointing in to the surface is prescribed.
3. If one takes a sufficiently big collection of such geodesic pairs of pants, one has enough of them in oppositely-aligned pairs along each boundary component, that they can be matched up (by some
version of Hall’s marriage theorem), and furthermore, matched up with a definite prescribed “twist” along the boundary components
4. One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is $\pi_1$-injective
Many aspects of this argument have a lot in common with some previous attempts on the surface subgroup conjecture, including one recent approach by Bowen (note: Bowen’s approach is known to have some
fatal difficulties; the “twist” in 3. above specifically addresses some of them). All of these points deserve some comments.
First, where do the pairs of pants come from? If $P$ is a totally geodesic pair of pants with boundary components of length close to $2R$, the pants $P$ retract onto a geodesic spine, i.e. an
immersed totally geodesic theta graph, whose edges all have length close to $2R$, and which meet at angles very close to $120$ degrees. One can cut this spine up into two pieces, which are obtained
by exponentiating the edges of an infinitesimal (almost)-planar tripod for length $R$.
Given a tripod $T$ in some plane in the tangent space at some point of $M$, one can exponentiate the edges for length $R$ to construct such a half-spine; if $T$ and $T'$ are a pair of tripods for
which the exponentiated endpoints nearly match up, with almost opposite tangent vectors, then the resulting half-spines can be glued up to make a spine, and thickened to make a pair of pants. One key
idea is to use the exponential mixing property of the geodesic flow on a hyperbolic manifold, e.g. as proved by Pollicott. Given some tolerance $\epsilon$, once $R$ is sufficiently large, the mixing
result shows that the set of such pairs of tripods for which such a matching occurs have a definite density in the space of all pairs (and in fact, are more and more equidistributed in this space, in
probability). In fact, one may even insist that two of the pairs of prongs join up to make some specific closed geodesic of length almost $2R$, and vary the pair of third prongs a very small amount
so that they glue up. This takes care of the first two points; this seems quite uncontroversial (exponential mixing comes in, I suspect, to know that one doesn’t need to wiggle the pair of third
prongs much, having paired the first two pairs).
The matching (i.e. the gluing up of opposite pant cuffs) apparently is done by some variant of Hall’s marriage theorem. One needs to know (I think) that for any finite set of cuffs to be glued, the
set of other cuffs that they could potentially be glued to is at least as big in cardinality. This probably needs some thought, but it is plausibly true: given a cuff, it can be glued to any cuff
which is almost oppositely aligned to it, and since there is some tolerance in the angle of gluing — this is where dimension at least $3$ is necessary — and moreover, since oriented cuffs are almost
equidistributed, one can always find “more” cuffs that are opposite, up to a bit of tolerance, to any given subset of cuffs (of course, more details are necessary here). There is an extra wrinkle to
the argument, which is that the gluing must be done with a “twist” of a definite amount, so that cuffs are not glued up in such a way that the perpendicular geodesic arcs joining pairs of cuffs match
(Update 8/8: I think there must necessarily be more details to the matching argument, as very loosely described above. There are at least two additional issues that must be dealt with in order to
perform a matching: a parity issue (since each pants has an odd number of cuffs) and a homology issue (if the argument relativizes, so that one fixes some collection of cuffs in advance and glues up
everything else, one concludes a posteriori that the union of the unglued cuffs is homologically inessential). Probably the parity issue (and more subtle divisibility issues) can be solved by gluing
with real-valued weights, then approximating a real solution by a rational solution, and multiplying through to clear denominators. Maybe the homology issue does not arise, if in fact the argument
doesn’t relativize.) Both these issues suggest that one does not specify in advance a collection of pants to be glued up, but rather wants to glue up a definite number of pants from some subset.)
This issue of a twist is important for the 4th point, which is perhaps the most delicate. In order to know that the resulting surface is $\pi_1$-injective, one must use geometry. A closed (immersed)
surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is $\pi_1$-injective. One way to see this is to observe that a geodesic loop in the surface is almost geodesic
in the manifold; the ambient negative curvature means that the geodesic can be shrunk (by the negative of the gradient of length in the space of loops) to become geodesic in the ambient manifold; if
it is close to being geodesic at the start, it very quickly becomes totally geodesic, without getting much shorter. Any closed geodesic in a hyperbolic manifold is essential.
If one builds a surface by gluing up almost totally geodesic pieces in such a way that there is almost no angle along the gluing, the resulting surface is almost geodesic, and therefore injective.
However, one must be very careful to control the geometry of the pieces that are glued, and this is hard to do if the injectivity radius is very small. A geodesic pair of pants has area $2\pi$ no
matter how long its boundary components are. So if the boundary components have length $2R$, then at the points where they are thinnest, they are only $e^{-R}$ across. If cuffs are glued where the
pants are thinnest, even if the gluing angle is very small, the surfaces themselves might twist through a big angle in a very short time. So one needs to make sure that the thinnest part of one pants
are glued up to a thicker part of the next, which is glued to a thicker part of the next . . . and so on. This is the point of introducing the twist before gluing: the twists accumulate, and before
one has glued $R$ pieces together, one has entered the thick part of some pants, where the injectivity radius is bounded below by some universal constant.
Anyway, this seems like a really spectacular development, with an excellent chance of working out. Some of the ingredients — e.g. the exponential mixing of the geodesic flow — work just as well in
variable negative curvature. In fact, some version of it should work for arbitrary hyperbolic groups (using Mineyev’s flow space). Without knowing more details of the argument, one can’t say how
delicate the last part of the argument is, and how far it generalizes (but readers are invited to speculate . . .)
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The Dancing Doll line sold slightly more than $3.5 million
Author Message
The Dancing Doll line sold slightly more than $3.5 million [#permalink] 29 Apr 2010, 09:14
bupbebeo Difficulty:
Intern 5% (low)
Joined: 16 Apr 2010 Question Stats:
Posts: 36 66%
Followers: 0 (01:47) correct
33% (01:44)
based on 33 sessions
The Dancing Doll line sold slightly more than $3.5 million worth of toys last year, 40% mroe than the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic
Blocks line's sales.
A. the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic Blocks line's sales
B. the Teeny Tiny Trucks did and nearly 3 times what the Basic Blocks' sales were
C. the Teeny Tiny Trucks line sold and nearly 3 times as much as Basic Blocks' sales.
D. the Teeny Tiny Trucks line and nearly 3 times more than Basic Blocks' sales
E. the Teeny Tiny Trucks line and nearly 3 times more than the Basic Blocks line
Re: comparision and parallel [#permalink] 29 Apr 2010, 10:05
bupbebeo bupbebeo wrote:
Intern The Dancing Doll line sold slightly more than $3.5 million worth of toys last year, 40% mroe than the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic
Blocks line's sales.
Joined: 16 Apr 2010
A. the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic Blocks line's sales
Posts: 36 B. the Teeny Tiny Trucks did and nearly 3 times what the Basic Blocks' sales were
C. the Teeny Tiny Trucks line sold and nearly 3 times as much as Basic Blocks' sales.
Followers: 0 D. the Teeny Tiny Trucks line and nearly 3 times more than Basic Blocks' sales
E. the Teeny Tiny Trucks line and nearly 3 times more than the Basic Blocks line
the correct answer is E, but I do not know why. could anyone help me?
Re: comparision and parallel [#permalink] 29 Apr 2010, 11:05
IMO C (Best possible choice)
OA Please !!
D & E are wrong : 3 times as much as is not same as 3 times more than , i.e., say x = 100. 3 times as much as x is 300 and 3 times more than x is incomplete - 3 times of
Intern what, let's assume if 3 times of x more than x then also it is 100 + 300 = 400. Meaning of sentence is changed.
Joined: 06 Mar 2010 B: Idiomatic usage : "as much as" is required, and "of" preposition is missing - 3 times of what
Posts: 20 A: Tiny trucks line did ==> sold
Followers: 0 Sold is more clear than did.
Kudos [?]: 2 [0], given: 1 Ideally the parallel structure should be:
The Dancing Doll line sold :
slightly more than $3.5 million worth of toys last year
40% mroe than the Teeny Tiny Trucks line "sold"
newarly 3 times as much as the Basic Blocks line "sold"
Re: comparision and parallel [#permalink] 29 Apr 2010, 11:07
OA please !!
Joined: 06 Mar 2010
FYI :
Posts: 20
IMO - In My Opinion
Followers: 0 OA - Official Answer
Kudos [?]: 2 [0], given: 1
Re: comparision and parallel [#permalink] 29 Apr 2010, 18:16
bupbebeo the Official answer is E. The book says that we are comparing sold.
Intern Dancing Doll line sold.........the Teeny Tiny Trucks line (sold) and nearly 3 times more than the Basic Blocks line (sold).
Joined: 16 Apr 2010 sold following the Teeny Tiny Trucks line and the Basic Blocks line can be omitted.
Posts: 36 therefore the correct answer is E.
Followers: 0 But I do not think this is a right explanation. Because when we compare how Dancing Doll line sold with how Teeny Tiny Truck & Basic Block line sold, we must use sold or
at least did.
Can anyone help me give the right answer for this problem.
sasen Re: comparision and parallel [#permalink] 30 Apr 2010, 03:09
Manager A. the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic Blocks line's sales
AS MUCH is incorrect,MORE is correct
Joined: 28 Aug 2009 B. the Teeny Tiny Trucks did and nearly 3 times what the Basic Blocks' sales were
MORE is correct
Posts: 196 C. the Teeny Tiny Trucks line sold and nearly 3 times as much as Basic Blocks' sales.
AS MUCH is incorrect,MORE is correct
Followers: 2 D. the Teeny Tiny Trucks line and nearly 3 times more than Basic Blocks' sales
not PARALLEL.
Kudos [?]: 38 [0], given: 1 E. the Teeny Tiny Trucks line and nearly 3 times more than the Basic Blocks line
PARALLEL and MORE used ..CORRECT
Re: comparision and parallel [#permalink] 30 Apr 2010, 06:38
So, after long explanation, what is your choice sasen???
Joined: 16 Apr 2010
Posts: 36
Followers: 0
Manager Re: comparision and parallel [#permalink] 01 May 2010, 12:54
Joined: 29 Dec 2009 i choose c but after the expalanation it cud be E
Posts: 124 coz ven v dunt have a helping verb like here it is sold only... no helping verb ....
to make it compact v can write nouns only ... verb is understood...
Location: india
Followers: 1
Kudos [?]: 13 [0], given: 10
Re: comparision and parallel [#permalink] 02 May 2010, 22:24
calvinhobbes sasen wrote:
Manager A. the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic Blocks line's sales
AS MUCH is incorrect,MORE is correct
Joined: 21 Jan 2010 B. the Teeny Tiny Trucks did and nearly 3 times what the Basic Blocks' sales were
MORE is correct
Posts: 233 C. the Teeny Tiny Trucks line sold and nearly 3 times as much as Basic Blocks' sales.
AS MUCH is incorrect,MORE is correct
Followers: 4 D. the Teeny Tiny Trucks line and nearly 3 times more than Basic Blocks' sales
not PARALLEL.
Kudos [?]: 45 [0], given: 38 E. the Teeny Tiny Trucks line and nearly 3 times more than the Basic Blocks line
PARALLEL and MORE used ..CORRECT
I suppose "3 times as much as sth" is idiomatic as well? no?
Re: The Dancing Doll line sold slightly more than $3.5 million [#permalink] 24 Jan 2013, 05:10
fameatop bupbebeo wrote:
Director The Dancing Doll line sold slightly more than $3.5 million worth of toys last year, 40% mroe than the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic
Blocks line's sales.
Joined: 24 Aug 2009 A. the Teeny Tiny Trucks line did and newarly 3 times as much as the Basic Blocks line's sales
B. the Teeny Tiny Trucks did and nearly 3 times what the Basic Blocks' sales were
Posts: 512 C. the Teeny Tiny Trucks line sold and nearly 3 times as much as Basic Blocks' sales.
D. the Teeny Tiny Trucks line and nearly 3 times more than Basic Blocks' sales
Schools: Harvard, Columbia, E. the Teeny Tiny Trucks line and nearly 3 times more than the Basic Blocks line
Stern, Booth, LSB,
Option A, B, C & D make the same kind of error i.e. Comparing Sales with Line.
Followers: 7
Kudos [?]: 321 [0], given:
241 If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS.
Kudos always maximizes GMATCLUB worth -Game Theory
If you have any question regarding my post, kindly pm me or else I won't be able to reply
gmatclubot Re: The Dancing Doll line sold slightly more than $3.5 million [#permalink] 24 Jan 2013, 05:10
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differentiable map and local isometry
December 20th 2008, 12:29 AM #1
Junior Member
Nov 2008
differentiable map and local isometry
hi dear mathhelpform's citizens ı have a question,ı want to ask you;
if the differential of a differentiable map F preserves,orthonormal basis then F is a(local) isometry.
thanks for your helps.
Why did you put parentheses on "local"? It is true with or without, but it is much easier with the parentheses, and you would have been given a hint to do the global version, so I guess what you
need is the local version and that's what I'll be writing about.
I'll need an additional hypothesis: $F$ is continuously differentiable. Did you forget it?
Note that if $\varphi$ is a linear map, " $\varphi$ preserves orthornormal bases" implies $\|\varphi(x)\|=\|x\|$ for every $x$, (one says that $\varphi$ is orthogonal). This implies $\|\varphi\|=
\max_{xeq 0}\frac{\|\varphi(x)\|}{\|x\|}=1$.
Suppose $F$ is defined on an open convex set $U$. By the previous remark, we have $\|dF_x\|=1$ for every $x\in U$. As a consequence, for any $x,y\in U$, the mean-value theorem gives $\|F(x)-F(y)\
|\leq \max_{z\in[x,y]}\|dF_z\|\times \|x-y\|=\|x-y\|$.
In order to get the reverse inequality, prove that, at any point, $F$ has locally an inverse function (by the inverse function theorem), notice that this inverse function satisfies the same
hypothesis as $F$ and procede like above to find $\|F^{-1}(f(x))-F^{-1}(f(y))\|\leq \|f(x)-f(y)\|$ if $x,y$ are in a (possibly small) open convex set where $F$ is invertible. And this is it.
thanks laurent,god save you! ı know the isometry of f hold the local isometry but my teacher wants everything in order anyway ı appreciate u ı can handle on this from here thanks again.
December 20th 2008, 11:49 PM #2
MHF Contributor
Aug 2008
Paris, France
December 21st 2008, 02:49 AM #3
Junior Member
Nov 2008
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Template Question
07-31-2011 #1
Registered User
Join Date
Jul 2011
Template Question
Hello fellow coders,
I am trying to create a template that behaves differently
if it is give a primative type vs a class. I should be easy
enough but my brain seems to be stuck in neutral.
Here is the general problem:
template< class T, class S>
struct Node
Node* MyNode;
T Value;
template< class T, class S>
class Myclass
T GetValue();
bool operator== (T rhs);
Node node;
// In the case that T is a class, T will look somethink like this.
template< class T, class S>
class MyVector
int size;
float length;
S value;
Here are the requirements.
If T is a a primative (int, long, float, double, char...), S will be a primative of the same type
GetValue will return the value of classInst1.node.value
The operator== will compare classInst1.node.value == classInst2.node.value
If T is a class, S will be a primative.
GetValue will return the value of classInst1.node.value
The operator== will compare classInst1.node.<T>value.value == classInst2.node.<T>value.value
so I could have a program like this:
Myclass<int, int> Foo1;
Myclass<int, int> Foo2;
Myclass<MyVector, float> Bar1;
Myclass<MyVector, float> Bar2;
int myInt = Foo1.GetValue(); //returns Foo1.node.value
MyVector myVect = Bar1.GetValue(); //returns Bar1.node.value
if ( Foo1 == Foo2 ){} //compares Foo1.node.value == Foo2.node.value
if ( Bar1 == Bar2 ){} //compares Bar1.node.value.value == Bar2.node.value.value
Thank in advance for any help.
Last edited by byteherder; 07-31-2011 at 04:21 PM.
You messed up something. Only the 'Node' class is templated, all the others are not. The following is also invalid:
<int, int> Foo1;
<int, int> Foo2;
<MyVector, float> Bar1;
<MyVector, float> Bar2;
You did not specify which class you want to instantiate (Node?):
Node<int, int> Foo1;
Node<int, int> Foo2;
Node<MyVector, float> Bar1;
Node<MyVector, float> Bar2;
Do you really know what you want to do?
I never put signature, but I decided to make an exception.
I'll edit the top post. I want the template to be the same for all of the top code block. Thanks for pointing out my mistake.
Last edited by byteherder; 07-31-2011 at 04:23 PM.
If T is a class, S will be a primative.
What primitive will S be then? Your main() is still broken.
For compile-time type recognition you can use template specialisation. You may not grasp it at once, but here is a hint:
template <typename T>
class is_fundamental {
static const bool result = false;
template <>
class is_fundamental<char> {
static const bool result = true;
template <>
class is_fundamental<int> {
static const bool result = true;
You need to provide specialisation for every primitive. Then you can easily check whether given type is fundamental (unfortunately, this is done at run-time):
if (is_fundamental<int>::result)
// int is a fundamental type
I do not even want to mention that all your classes will also need (partial) specialisation.
Shortly: you need a better (easier that is) way of doing whatever you want to do now.
Last edited by kmdv; 07-31-2011 at 04:42 PM.
I never put signature, but I decided to make an exception.
What primitive will S be then? Your main() is still broken.
For compile-time type recognition you can use template specialisation. You may not grasp it at once, but here is a hint:
template <typename T>
class is_fundamental {
static const bool result = false;
template <>
class is_fundamental<char> {
static const bool result = true;
template <>
class is_fundamental<int> {
static const bool result = true;
You need to provide specialisation for every primitive. Then you can easily check whether given type is fundamental (unfortunately, this is done at run-time):
if (is_fundamental<int>::result)
// int is a fundamental type
I do not even want to mention that all your classes will also need (partial) specialisation.
Shortly: you need a better (easier that is) way of doing whatever you want to do now.
The primative could be any of primative types. I wanted to get away from specifying template specialisation for every single primative type.
Looking a little deeper into partial template specialisation, I am reading about Boost's enable_if or something like that. This is looking more like a metatemplate problem, where the compiler
first has to choose the right template and then fill in that template with the right type.
Has anyone used the the enable_if before?
What you really want is "type traits". Something like what the yasli_vector has I imagine.
My homepage
Advice: Take only as directed - If symptoms persist, please see your debugger
Linus Torvalds: "But it clearly is the only right way. The fact that everybody else does it some other way only means that they are wrong"
Check out boost's type traits. It has a is_fundamental for your purpose.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
In your example, the value of is_fundamental<int>::result is used at run time, but the compiler will evaluate it at compile time. Some compilers will, for example, give a warning about testing a
condition that (in your example) is always true.
However, you have provided the core of a solution ....
Assuming an is_fundamental<> type is specialised as you have shown then the original problem can be addressed through;
template<class T, bool is_fundamental_type = is_fundamental<T>::result > struct MyClass {};
template<class T> struct MyClass<T, false>
// provide all the operations you want for MyClass instantiated for non-fundamental types
template<class T> struct MyClass<T, true>
// provide all the operations you want for MyClass instantiated for fundamental types
Last edited by grumpy; 08-01-2011 at 04:23 AM.
Right 98% of the time, and don't care about the other 3%.
Type traits that was the answer I was looking for, Also, the is_fundamental. Thank you, everyone, for you help.
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the definition of Deduction
deduction (dɪˈdʌkʃən)
1. the act or process of deducting or subtracting
2. something, esp a sum of money, that is or may be deducted
3. a. the process of reasoning typical of mathematics and logic, whose conclusions follow necessarily from their premises
b. an argument of this type
c. the conclusion of such an argument
4. logic
a. a systematic method of deriving conclusions that cannot be false when the premises are true, esp one amenable to formalization and study by the science of logic
b. Compare induction an argument of this type
deduction (dĭ-dŭk'shən) Pronunciation Key
1. The process of reasoning from the general to the specific, in which a conclusion follows necessarily from the premises.
2. A conclusion reached by this process.
Our Living Language : The logical processes known as deduction and induction work in opposite ways. In deduction general principles are applied to specific instances. Thus, using a mathematical
formula to figure the volume of air that can be contained in a gymnasium is applying deduction. Similarly, applying a law of physics to predict the outcome of an experiment is reasoning by
deduction. By contrast, induction makes generalizations based on a number of specific instances. The observation of hundreds of examples in which a certain chemical kills plants might prompt the
inductive conclusion that the chemical is toxic to all plants. Inductive generalizations are often revised as more examples are studied and more facts are known. If certain plants that have not
been tested turn out to be unaffected by the chemical, the conclusion about the chemical's toxicity must be revised or restricted. In this way, an inductive generalization is much like a
The process of reasoning from the general to the specific, in which a conclusion follows necessarily from the premises.
Our Living Language : The logical processes known as deduction and induction work in opposite ways. In deduction general principles are applied to specific instances. Thus, using a mathematical
formula to figure the volume of air that can be contained in a gymnasium is applying deduction. Similarly, applying a law of physics to predict the outcome of an experiment is reasoning by deduction.
By contrast, induction makes generalizations based on a number of specific instances. The observation of hundreds of examples in which a certain chemical kills plants might prompt the inductive
conclusion that the chemical is toxic to all plants. Inductive generalizations are often revised as more examples are studied and more facts are known. If certain plants that have not been tested
turn out to be unaffected by the chemical, the conclusion about the chemical's toxicity must be revised or restricted. In this way, an inductive generalization is much like a hypothesis.
A process of reasoning that moves from the general to the specific. (Compare induction.)
deduction definition
1. n.
a child. (Actually a child is an exemption on the U.S. income tax return. See also expense.) : How many little deductions do you have running around your home?
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156 projects tagged "Windows"
The Maximum Entropy Toolkit provides a set of tools and library for constructing maximum entropy (maxent) models in either Python or C++. It features conditional maximum entropy models, L-BFGS and
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Quantum E6/E7 knot polynomials
up vote 7 down vote favorite
Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8?
I suspect these haven't been worked out, but if anybody knows of a reference containing these or even just discussing them, please let me know.
knot-theory qa.quantum-algebra quantum-topology
It is certainly possible to evaluate these for links which can be obtained as the closure of a braid with at most three strings. – Bruce Westbury Jan 22 '12 at 22:17
I'm pretty sure that the quantum groups Mathematica package which is part of the Knot Atlas package (katlas.org/wiki/Main_Page) can do this. It'll probably be pretty slow. You can ask Scott
Morrison if you want more info. – Noah Snyder Jan 22 '12 at 22:17
I spoke to Scott a couple of days ago. His package cannot compute quantum invariants for F4 or En due to the complexity of some intermediate computations. He noted that one could avoid these issues
by, e.g. writing down the quantum positive roots in terms of the PBW bases... working this out is [currently] a bit beyond my primitive background in QA. However, if anybody knows of a good source
for such things, that would be equally useful. – Ross Elliot Jan 22 '12 at 22:38
Ok, my bad. I'd used it for some more basic quantum En calculations before, but I guess they were all a lot simpler than what you need. – Noah Snyder Jan 23 '12 at 0:15
I would write down the representations of the braid group directly. – Bruce Westbury Jan 23 '12 at 6:59
show 2 more comments
2 Answers
active oldest votes
$(E7, 56)$ belongs to a series for which I computed some skein relations in my phd thesis (see p55-56 of http://web.univ-ubs.fr/lmam/patureau/articles/these.ps.gz with $Y=X^{19}$).
up vote 5 Unfortunately the set of skein relations is not complete but you can use it to compute the quantum invariants of small knots.
down vote
add comment
Dear Bertrand & Ross - could you check the following? I neither speak French nor Math :-) but I computed the same skein relations with magic and trickery (so my results are unproven, of
course). I get 11+6+2*8+2*3=39 basis 3+3 tangles (which I here already split into the D6h symmetry classes) and under the E7 family polynome 10+5+2*7+2*3=35 are independent. The linear rest
should be your skein relations. So, are there exactly 4 (including symmetry-transformed!) of them, i.e. your Figs. 2/3? Since they don't look symmetric under rotation, it's hard to check for
(BTW, for 2+2 tangles, I "know" an even generalized result (of your Fig. 1) for 20 years :-)
up vote Maybe going to 4+4 tangles could solve the problem of having a complete reducing skein set (or at least give an unproven solution) but since that might need hundreds of basis tangles you a)
0 down either have to crowdsource the calculation or b) do it with birdtracks or c) use math (I'm out then :-)
EDIT: Now that I'm back at my PC with access to the literature, 4+4 doesn't look too promising either. Kuperbergs G2 paper says there are 455 crossingless freeways with 8 endpoints. Uck. (And
under E7 instead, a lot of them are inaccessible.) So if one could do the following: 1. Prove that: IF you can reduce 1-gons and 2-gons and two adjacent 3-gons (works for all the E7 family)
AND an adjacent 3- and 4-gon pair AND two 3-gons touching on a corner (five crossings; i.e. simpler diagrams now span the 8-endpoint vector space...or so I hope!) THEN you could reduce any
link diagram, and: 2. the compatibility rules of this reducing set are finite (I don't have much hope for that either!)... the whole E7 family polynomial proof would reduce to skein diagram
manipulation (of gargantuan size, of course). Yup, it's probably better to use math instead :-)
add comment
Not the answer you're looking for? Browse other questions tagged knot-theory qa.quantum-algebra quantum-topology or ask your own question.
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Absolute Value Expression
August 23rd 2009, 11:34 AM #1
Aug 2009
Absolute Value Expression
I am studying for a placement test and this seemingly easy Algebra problem has me confused:
|x-3|+|x+4| when x<-4
This isn't a system of equations, or a graph, but a problem needing to be simplified.
I got the answer -2x-1.
Is that correct?
Yes, it is.
August 23rd 2009, 11:52 AM #2
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My name is Derek H. and I recently graduated from Cornell University's College of Engineering with a degree in Information Science, Systems, and Technology. I have a strong background in Math,
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Advanced Birthday Problem
March 19th 2010, 06:03 PM #1
Mar 2010
In a town of 42,077 people 12 people run for local office.
Of those 12 only 9 are elected.
What is the probability that 2 of the 9 elected have the same birthday?
I don't even know where to start and I think I broke my calculator. And please, I'd love to see the work/equation.
Am I misreading something? I don't see why the majority of the information is being given. Why don't you just need to calculate the probability of matching birthdays in a group of 9 people,
assuming a 365 day year with each day equally likely?
Also, are you looking for the probability of at least one match, or one and only one match?
You're not misreading. The extra information might be there to throw me off and from your reaction it sounds like it is.
I am so horrible at trying to decide what numbers matter in these types of problems.
And the problem is for ONLY 2 of 9 having the same birthday. So...one and only one match.
Do you know combinational mathematics? I don't know your school system
Perhaps this could be something in the right direction? I dont know
P(2 people same)= (365/365)*(364/365)*(363/365)*(362/365)*(361/365)*(360/365)*(359/365)*(358/365)*(8/365)
P(2 people same)=365!/(357!*365^8)*(8/365)
P(2 people same)=(365!*8!)/(357!*8!*365^8)*(8/365)
P(2 people same)=(365 choose 357)*(8!*8)/(365^9) which gives an answer about 0.020288... Even if it's wrong, I hope that you might get any idea or something out of this!
In that case, let's tackle this one step at a time. Hopefully I'll do this right, since I'm not very good at these - if this is wrong hopefully the forum will forgive me. The set of all possible
outcomes is $\{1, 2, ... , 365\}^9$ (using ordered 9-tuples), and each possibility is equally likely. In this situation, you just take the number of ways to have one and only one matching
birthday, and divide that number by the number of possibilities. The number of possibilities is easy: 365^9.
So the majority of the problem is counting the number of vectors in $\{1, 2, ... , 365\}^9$ that have one and only one matching number. First, we will pick the two of the nine that match, and
there are 9C2 ways to do that. Then we assign those two a number: 365 ways to do that. Then, we fill in the remaining 7 numbers, and order matters for the purpose of counting up our
possibilities, so 364 P 7 ways to do that. What you end up with at the end is
$<br /> \frac{<br /> \left( <br /> \begin{array}{c} 9 \\ 2 \end{array} <br /> \right)<br /> P \left( <br /> \begin{array}{c} 365 \\ 8 \end{array} <br /> \right)<br /> }<br /> {<br /> 365^9<br />
}<br />$
Since it's a little ambiguous, the top is 9C2 * 365P8. It's hard to make things particularly intuitive in counting the number of things you have in the numerator. The key, I think, is to make
sure the way you break up "doing the jobs" is one-one and onto the set you are trying to count. Fail at the one-one step, and you end up double counting some possibilities, and if it isn't onto
then you are leaving some out. Basically, if you look at a particular possibility you should be able to reconstruct exactly the things you did to get there. For example, for $(3, 17, 22, 150,
153, 17, 240, 270, 340)$ using my method of counting I
1) Picked the 2nd and 6th place to match (9C2 ways)
2) Filled those both with 17 (365 ways)
3) Filled the 1st position with 3 (364 ways)
4) Filled the 3rd position with 22 (363 ways)
9) Filled the last position with 340 (358 ways)
and this is the only way to do this using the method I outlined.
March 19th 2010, 08:22 PM #2
Senior Member
Oct 2009
March 19th 2010, 09:20 PM #3
Mar 2010
March 20th 2010, 07:20 AM #4
Aug 2009
March 20th 2010, 07:49 AM #5
Senior Member
Oct 2009
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Voltage and Current Sources
Real voltage sources can be represented as ideal voltage sources in series with a resistance r, the ideal voltage source having zero resistance. Real current sources can be represented as ideal
current sources in parallel with a resistance r, the ideal current source having infinite resistance.
Such ideal voltage and current sources are used in modeling real circuits with Thevenin's theorem and Norton's theorem.
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Valley Village Math Tutor
...I help students master algebraic topics such as systems of equations, liner lines and graphing, solving algebraic problems, factoring, etc. Algebra II is one of my specialties & also one of my
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Solve for values of x
June 7th 2010, 03:18 AM #1
Solve for values of x
For what values of x does $(x-\pi)(x+5)(x-3) > 0$ hold?
What's the approach with this, is there an elegant way to solve it, or is a matter of exhausting every combination?
I would solve it by opening the first two followed by the last....
June 7th 2010, 03:27 AM #2
Jun 2010
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Proof That sin(5) is Irrational
Date: 04/24/2001 at 06:57:20
From: Howard
Subject: sin 5
How do you prove that sin(5) is an irrational number?
Date: 04/24/2001 at 09:52:44
From: Doctor Floor
Subject: Re: sin 5
Hi, Howard,
Thanks for writing.
I suppose you mean that 5 is 5 degrees here. If you mean 5 radians,
then the situation is a lot more complicated, and I would not know how
to solve your problem.
We know that 5 degrees is a solution of sin^2(9x) = sin^2(45) = 0.5.
We try to find a formula for sin(9x) in terms of sin(x). In fact we
are only interested in the highest power term. By that we could
rewrite sin^2(9x) = 0.5 into an equation of the form f(sin x) = 0 to
which we will apply the rational root theorem.
First for sin(3x) we write:
sin(3x) = sin(x + 2x)
= sin(x)cos(2x) + cos(x)sin(2x)
= sin(x)(1-2sin^2(x)) + cos(x)*2sin(x)cos(x)
= sin(x) - 2sin^3(x) + 2sin(x)cos^2(x)
= sin(x) - 2sin^3(x) + 2sin(x)*(1 - sin^2(x))
= sin(x) - 2sin^3(x) + 2sin(x) - 2sin^3(x)
= -4sin^3(x) + 3sin(x)
From this we derive:
sin(9x) = -4sin^3(3x) + 3sin(3x)
= -4(-4sin^3(x) + 3sin(x))^3 -4sin^3(x) + 3sin(x)
so that
sin(9x) = -256sin^9(x) + integers times lower powers of sin(x).
Note that the right-hand side does not have a constant term.
This means that sin^2(9x) = 0.5 can be rewritten as:
65536sin^18(x) + integers times lower powers of sin(x) - 0.5 = 0
131072 sin^18(x) + integers times lower powers of sin(x) - 1 = 0
2^17 sin^18(x) + integers times lower powers of sin(x) - 1 = 0
This is a polynomial equation in sin(x) to which we can apply the
rational root theorem, and sin(5) is one of the roots.
For the rational root theorem, see for instance from the Dr. Math
Rational Root Theorem
We find that the rational roots of this equation must be of the form
+- (1/2)^n for n <= 17.
It is easy to verify that sin(5), approximately equal to 0.087, is not
of this form. So sin(5) is a root of the polynomial, but not a
rational root. That means that sin(5) must be irrational.
I hope this helps. If you need more help, just write back.
Best regards,
- Doctor Floor, The Math Forum
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Algebra Tutors
San Francisco, CA 94110
Experienced in English, Writing, ESL, Math - TESL Certified!
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Offering 10+ subjects including algebra 1
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Autonomous quantum error correction technique proposed for quantum memories
August 4th, 2010 in Physics / Quantum Physics
A diagram of the proposed quantum memory with autonomous quantum error control (QEC). Image credit: Kerckhoff, et al. (c)2010 APS.
A diagram of the proposed quantum memory with autonomous quantum error control (QEC). Image credit: Kerckhoff, et al. (c)2010 APS.
(PhysOrg.com) -- While words such as "powerful" and "efficient" are often used to describe the potential of quantum computing, these quantum systems can be very fragile at the same time. Errors in
quantum systems can easily arise due to decoherence - which occurs when a quantum state interacts with its environment - as well as unwanted noise or defective components. In order to protect quantum
systems from these problems, physicists use quantum error correction (QEC) techniques to identify and correct errors without disturbing the system.
In a recent study, physicists have developed a new QEC technique that can be directly embedded into quantum memories. Because the method is implemented "on-chip," it requires no external clocking or
logic. In addition, all control operations are performed autonomously by an embedded feedback loop, which is different than most previous QEC approaches. The researchers hope that the design could be
useful for nanophotonics implementations and quantum-optical engineering.
“Good QEC designs can improve implementation efficiency by reducing the hardware and computational ‘overhead’ that is needed in the implementation of the QEC for a particular quantum memory scheme,”
coauthor Hendra Nurdin of Stanford University and the Australian National University told PhysOrg.com.
In general, it’s much more difficult to design QEC methods than it is to design classical error correction methods because in classical methods, bits can simply be copied for redundancy. However,
qubits cannot be copied in the same way due to the non-cloning theorem. Yet physicists can get around this limitation in a few ways, such as by encoding a single “logical” qubit (representing the
information carried) in the entangled state of three “physical” qubits using a technique called the bit-flip code.
The new autonomous QEC technique is based on the bit-flip code and another similar strategy called the phase-flip code, and can protect the stored information against independent unwanted flips to
any, but not more than one, of the physical qubits. Whereas previous QEC approaches usually involved discrete restoration steps, the new approach involves a continuous syndrome readout to diagnose
and correct errors. In this approach, each physical qubit is strongly coupled to its own optical cavity. If an error occurs so that one of the physical qubits has its state flipped, two feedback
signals are sent to the qubit to flip it back and correct the error. The system is autonomous in that probe signals are continuously providing feedback to the qubits: less than two feedback signals
in the case of no errors, and two feedback signals in the case of an error.
“This QEC design has the potential to be embedded on the same hardware platform as the quantum memory, such as in nanophotonics, and has the potential for reduced hardware overhead requirements
because it does not require external clocking and logic to operate, nor does not it require interfacing to measurement devices,” Nurdin said. “Moreover, since all processing is performed coherently,
no classical computations are required to determine the corrective feedback signals.”
Although the current design is just a proposal, the physicists explain that the circuitry could be realized with available technology, such as solid-state qubits coupled to electromagnetic resonators
and waveguides. In the future, the scientists also plan to find ways to extend the design to QEC feedback networks that can correct a wider variety of qubit errors.
More information: Joseph Kerckhoff, et al. “Designing Quantum Memories with Embedded Control: Photonic Circuits for Autonomous Quantum Error Correction.” Physical Review Letters 105, 040502 (2010).
Copyright 2010 PhysOrg.com.
All rights reserved. This material may not be published, broadcast, rewritten or redistributed in whole or part without the express written permission of PhysOrg.com.
"Autonomous quantum error correction technique proposed for quantum memories." August 4th, 2010. http://phys.org/news200123042.html
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Post a reply
The figure formed by the intersection of a solid with a plane parallel to the base of the solid is congruent to the base if the solid is a?
a. right cylinder
b. right cone
c. rectangular pyramid
d. frustum of a cone
i've read a math book twice to find the answer, i just want to know if i am right - cause i picked A
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Belmont, CA Prealgebra Tutor
Find a Belmont, CA Prealgebra Tutor
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...I strive to break down Algebra 1 into understandable bite-sized chunks that are easy to learn conceptually and procedurally. Algebra 1 is one of my favorite subjects to teach, and the most
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7 Subjects: including prealgebra, geometry, algebra 1, algebra 2
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class Value in scala.Enumeration
Method Summary
override def compare (that : Value) : Int
Result of comparing this with operand that. returns x where x < 0 iff this < that x == 0 iff this == that x > 0 iff this > that
override def equals (other : Any) : Boolean
This method is used to compare the receiver object (this) with the argument object (arg0) for equivalence.
override def hashCode : Int
Returns a hash code value for the object.
abstract def id : Int
the id and bit location of this enumeration value
def mask32 : Int
this enumeration value as an Int bit mask.
def mask64 : Long
this enumeration value as an Long bit mask.
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Network analyst: Accumulation attributes seem to be inconsistent with impedance
07-05-2010 12:19 PM #1
Adam Storeygard
Join Date
Apr 2010
Answers Provided
I'm getting some strange results from a closest facility problem (~40 facilities, ~1100 incidents) in Network analyst. In addition to a Total_Length, which is my impedance measure, I am
calculating lengths for each route within several mutually exclusive and exhaustive categories. In other words, when summed for any edge in my network dataset, they add to Total_Length for that
edge. Let's call them LengthA, LengthB, LengthC etc. I do this by setting them as Accumulation Attributes.
The resulting Routes look fine, except that the Total_Length of each is not the same as the sum of the lengths by category. The difference is sometimes too large to be because e.g. I am assigning
locations to the nearest network location within 5000 meters.
Here's the part that makes me think someone out there will know the answer:
All datasets I am using are in the same feature dataset in a sinusoidal projection. The percent difference between Total_Length and Sum(LengthA, LengthB, LengthC...) is highly correlated (0.85)
with the percent difference between the analogous straight line distances calculated using latitude/longitude in the Haversine formula and using projected (sinusoidal) coordinates.
I can't think of any reason why the Plate Carree ("Geographic") or any projection other than the original sinusoidal would be used for one set of calculations but not the other.
I'm running 9.3 SP1 with an ArcInfo license.
Any ideas would be greatly appreciated.
First are you solving on Length? OR are you solving on say, travel time and accumulating Length to get the Total_Length field?
Second, how was the Length attribute added/defined in the network dataset?
Third, how are the length by category attributes set up? Are they based on the same "Length" attribute of that edge? Or were they set up by some other formula?
Network Analyst itself is not doing any projections to figure out the lengths of the edges. And it will accumulate what ever fields you ask. There is no error in simply summing up attributes to
report out the accumulation of the various attributes. Any differences have to exist in the input data.
If you use the network identify tool (second icon from right on the NA toolbar), it will list out all the attribute values for that edge. Use that to see if the length and the length by category
are the same. Most likely they are not same and that is source of the differences.
Jay Sandhu
Thank you for your response, and sorry for the lack of details. I am solving on Length. The length by category fields were calculated on the the only feature class in the network, prior to
building the network, just working with the table in a python script, assigning the [Shape_Length] field to the length-by-category field for each feature in the corresponding category. Below is a
summary of inputs for the network build. I've removed all but two of the length-by-category fields - they are all analogous.
As you suggest, the discrepancy already appears when using the Network identify tool, so whatever is going wrong is at the network build stage and not the solver. But when using the (simple,
non-network) identify tool on the feature dataset used to build the network, on the analogous fields for a network element that is identical to a line in the feature class, there is no
discrepancy. In fact, of the four lengths identified in this way (network total length, network length-by-category, underlying feature class total length, and underlying feature class
length-by-category), only network total length is different. It can be shorter or longer, depending on the segment (in a way that suggests some sort of implicit reprojection as I mentioned
Name: AfricaRoadsFD_ND
Type: Geodatabase-Based Network Dataset
Edge Sources:
Group 1:
Edge Connectivity:
AfrRoads : Bridge (Any Vertex)
AfrRoads : Missing (Any Vertex)
AfrRoads : Primary (Any Vertex)
AfrRoads : Secondary (Any Vertex)
AfrRoads : Tertiary (Any Vertex)
AfrRoads : Unknown (Any Vertex)
AfrRoads : Urban (Any Vertex)
Usage Type: Cost
Data Type: Double
Units Type: Meters
Use by Default: True
Source Attribute Evaluators:
AfrRoads (From-To): Field - [Shape]
AfrRoads (To-From): Field - [Shape]
Default Attribute Evaluators:
Default Edges: Constant - 0
Default Junctions: Constant - 0
Usage Type: Cost
Data Type: Double
Units Type: Meters
Use by Default: False
Source Attribute Evaluators:
AfrRoads (From-To): Field - [LEN22]
AfrRoads (To-From): Field - [LEN22]
Default Attribute Evaluators:
Default Edges: Constant - 0
Default Junctions: Constant - 0
Usage Type: Cost
Data Type: Double
Units Type: Meters
Use by Default: False
Source Attribute Evaluators:
AfrRoads (From-To): Field - [LEN23]
AfrRoads (To-From): Field - [LEN23]
Default Attribute Evaluators:
Default Edges: Constant - 0
Default Junctions: Constant - 0
Directions Ready: No
-Length Attribute Required
-Street Name Field Required [AfrRoads]
The [Shape_Length] field is the computed length based on the projection of the data.
The [Shape] field used in the evaluator will compute/return the true geodesic length of the feature in the units specified (meters in this case).
So these two will most likely not be the same.
So one way to fix is to continue using the [Shape_Legnth] in all places by changing your settings as:
AfrRoads (From-To): Field - [Shape_Length]
AfrRoads (To-From): Field - [Shape_Length]
Add a field to your street data and calculate it to the true geodesic distance by using the Haversine formula. And then update your Len22, Len23, etc, fields based on this new field. But this
will be more work.
Jay Sandhu
Thanks for this. If the geodesic distances are being calculated for every edge in the network build anyway, is there any way to access them, to put them into a tabular field that can then be
passed into the various categories, either directly or through a spaital join with the original feature dataset from which the network was built? This seems like it would be much better than
adding projection error to the (total) Length field just to make it match the others. Unless there's a better trick for using the Haversine on complex lines, I suspect that splitting into COGO,
with well over a million segments, will crash my machine.
Probably the quickest way would be dump out the network dataset edge attribute information to a table and then add it to ArcMap and do a join.
As long as you have one line source in your network dataset and simple end point connectivity, i.e. the input line features were not split into multiple edges in the network, then you can use the
following VBA to write out a file with the ID and Length. Modify as needed.
Jay Sandhu
Public Sub List_ND_Topology()
Dim pMxDoc As IMxDocument
Set pMxDoc = ThisDocument
Dim pNLayer As INetworkLayer
Set pNLayer = pMxDoc.SelectedLayer
Dim pND As INetworkDataset
Set pND = pNLayer.NetworkDataset
Dim pNQ As INetworkQuery
Set pNQ = pND
Dim pEnumNE As IEnumNetworkElement
Set pEnumNE = pNQ.Elements(esriNETEdge)
Dim pNE As INetworkElement
Set pNE = pEnumNE.Next
Dim pNEdge As INetworkEdge
Set pNEdge = pNE
Dim i As Integer
Open "c:\ND_Length.csv" For Output As #1 'set path as needed
Print #1, """EdgeOID"", ""Length_Attribute"""
Do Until pNE Is Nothing
Print #1, pNE.OID; ","; pNE.AttributeValueByName("Length")
Set pNE = pEnumNE.Next
Close #1
End Sub
Thanks for this. I'll try to implement it when I have a chance.
Quick question, Jay:
What is the difference between Total_Length and Shape_Length?
In this context, the shape_length is the length of each edge in the network and the total_length is he length of the path that the route solver finds. (it may be the sum of the shape_lengths of
each edge if that is what the route was solved on).
Jay Sandhu
07-06-2010 10:45 AM #2
Jay Sandhu
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07-06-2010 11:43 AM #3
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07-06-2010 03:01 PM #5
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Flow Regimes
reynolds number
Re = ρ(Δv[x,z])(Δy) = ρvD = ρvℓ
η η η
• The Reynolds number (Re) is the ratio of inertial resistance to viscous resistance for a flowing fluid.
• The Reynolds number is a non-dimensional (unitless) factor governing resistance due to viscosity (among other things).
• This importance of this dimensionless constant was first determined in 1883 by the British physicist and engineer Osborne Reynolds (1842–1912).
□ Reynolds, Osborne." An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel
channels." Royal Society, Philosophical Transactions, 1883.
□ Reynolds' experiments on flow through pipes. Compare pressure losses due to viscous friction with flow speed through a horizontal pipe. Log-log graph of pressure gradient (ΔP/Δd) vs. flow
speed (v). The slope of the line of best fit on a log-log graph is the power (n) relating the explanatory variable (flow speed) to the response variable (pressure gradient).
Flow Regimes
• For low Reynolds numbers the behavior of a fluid depends mostly on its viscosity and the flow is steady, smooth, viscous, or laminar and n = 1.
• For high Reynolds numbers the momentum of the fluid determines its behavior more than the viscosity and the flow is unsteady, churning, roiling, or turbulent and n = 2.
• For intermediate Reynolds numbers the flow is transitional — partly laminar and partly turbulent.
• The full implications of the Reynolds number were never realized by Reynolds who considered the ratio merely as a criterion for the critical velocity in pipe flow. Lord Rayleigh has shown that it
is a non-dimensional factor which governs all problems on fluid flow frictional resistance, and that similar non-dimensional constants exist for many other natural phenomena.
• It is a practice in engineering design that when a large object such as a ship, airplane, or building is to be made, a scale model is constructed and tested so that the performance of the large
object can be calculated from the test results of the scale model. Lord Rayleigh showed that the scale model tests gave comparable results only when the non-dimensional factor of the model is
equal to that of the large object when working under its design conditions. By equating the non-dimensional factor of the large object to that of the model, the test speed of the model is
obtained. This is known as the corresponding speed and the comparison of the two conditions between the large object and the test results of a scale model at its corresponding speed is known as
the principle of dynamic similarity.
• From NASA "By 1921 more than a score of wind tunnels had been constructed the world over. But all those of substantial size were operating at normal atmospheric pressures. This meant that the
experimental results obtained using scale models in the tunnels were open to question because a special parameter called the Reynolds number did not match those encountered in the actual flights
of full-scale aircraft. In other words, the Reynolds number of 1/20-scale models being tested at operational flight velocities would be too low by a factor of 20. Reynolds' classic experiments
had shown that airflow conditions could be radically different for model and full-scale aircraft. Since the Reynolds number is also proportional to air density, an obvious solution to the problem
of scale effects would be to test 1/20-scale models at a pressure of 20 atmospheres. The Reynolds number would then be the same in the wind tunnel tests and actual full-scale flights."
Fluid Flow Regimes as a Function of Reynolds Number.
Critical Reynolds numbers
object lower upper
circular pipes 2,000 2,500
flat plates 300,000 500,000
Selected Reynolds numbers
Re animal Re aircraft
62,000 seagull 2,000,000,000 boeing 747
50,000 large fish 110,000,000 typical commercial jet
3,900 butterfly 6,300,000 cessna
1,000 honeybee 4,700,000 light plane
300 african frog tadpole 1,600,000 glider
120 housefly 250,000 model airplane
15 chalcid wasp 47,000 paper airplane
0.2 paramecium
0.025 dinoflagellate
0.0035 spermatozoa, sea urchin
0.000,01 bacterium
Re circulatory system Re miscellaneous
3,400 aorta 250,000,000 cumulus cloud formation
3,300 vena cava
500 artery
140 vein
0.7 arteriole
0.01 venule
0.002 capillary
mach number
No two bodies can occupy the same place at the same time. When a solid object and a fluid are in relative motion — like a bird flying through the air or the wind blowing around a mountain — it is
usually the fluid that yields to the solid. Solids are held together by intermolecular forces and atomic bonds. If the cohesive forces between the particles in a solid are considered significant and
long lasting, then the cohesive forces in a liquid are weak and short lived. In a gas they are virtually nonexistent. You might think that fluids are a pushover for a moving solid, but this is not
always the case.
The molecules that make up even the most tenuous of gases won't be able to get out of the way of a solid body moving at a considerable speed. Meteors quite commonly break up on entering the earth's
atmosphere from space. (They also burn up, but that is as much an result of frictional heating as it is of trying to push the air out of the way.) Aircraft are known to have broken up during flight
from the buffeting effects of moving air on a weakened or damaged part. Less commonly and more unfortunately, so too have spacecraft.
In 2003 the Space Shuttle Columbia broke apart in the upper atmosphere during its final descent to landing. A piece of foam isulation about the size of a briefcase fell off the external fuel tank
while the shuttle was taking off. This punched a fist-sized hole into the leading edge of the orbiter's left wing. The hot plasma produced when the shuttle reenters the earth's atmosphere eventually
melted the aluminum frame holding the wing in place. It snapped off and the air rushing by tore the orbiter to pieces. Contact was lost somewhere over northeastern Texas at an altitude of 62,000 m —
on the edge of space where the pressure and density of the atmosphere are roughly one-ten thousandth of their values at sea level. Columbia was scheduled to land sixteen minutes later in Florida.
Flying this distance in a commercial jet would take something like two hours and forty-five minutes — roughly ten times longer. Columbia was destroyed by an exceptionally tenuous gas while it flying
at an exceptionally high speed. At the time of last contact it was traveling at nearly 5600 m/s.
The Mach number (Ma) is a ratio of inertia to compressibility. It is the non-dimensional factor governing resistance due to longitudinal (compressional) wave formation.
Ma^2 = ρv^2ℓ^2 & c^2 = B ⇒ Ma = v
Bℓ^2 ρ c
The ratio of the speed of flow (v) to the speed of sound in a fluid (c) is known as the Mach number.
Mach 0.5 corresponds to a flow speed that's half the speed of sound, Mach 2 to a flow speed that's twice the speed of sound, and so on. Fluid flow can be broken up into two general regimes by Mach
number: those less than Mach 1 are said to be subsonic, while those greater than Mach 1 are said to be supersonic.
A body moving through a fluid at speeds less than the speed of sound in the fluid is preceded by a region of gradually varying density and pressure. At speeds greater than the speed of sound, such a
gradual transition is not possible and a shock wave of nearly discontinuously changing pressure and density is formed. In the case of a supersonic aircraft or a bullet, this shockwave is a double
walled cone that forms with the front and back of the object at its vertices (projections in between like wings and stabilizers are placed at the vertices of intermediate shock waves).
Shock waves can also form whenever a fluid is heated so rapidly that the leading edge of its expansion travels at or above the speed of sound in the fluid. Roughly spherical shock waves form when
bombs, fireworks, and other pyrotechnic devices explode. A bolt of lightning generates a cylindrical shockwave centered on the bolt's path. The sound of a shockwave produced by a supersonic aircraft
is called a sonic boom, while the sound of a shock wave produced by lightning is called thunder.
Mach numbers between 0.8 and 1.5 are said to be transonic.
A transonic flow over an aircraft wing will have pockets of subsonic and supersonic flow mixed together, which leads to a loss of stability.
The effects of the so called sound barrier also tend to be significant and flight can very easily become hard to control.
When the Mach number in a fluid approaches 5, the behavior of the fluid depends more upon the Reynolds number than the Mach number and the flow is said to be hypersonic. A model of an airplane
traveling through any fluid at a certain Mach number will behave much like the real thing traveling through the air at that same Mach number up until one enters the hypersonic regime. Below Mach 5,
the shock wave is separated from the object by a small but significant distance. Objects moving faster than Mach 5 start to interact with this shock front.
Fluid Flow Regimes as a Function of Mach Number.
compressible vs. incompressible flow
froude number
• The Froude number (Fr) is the ratio of the inertial forces to the gravitational forces.
• The Froude number is the non-dimensional factor governing resistance due to surface wave formation (among other things).
• Named after William Froude, a Nineteenth Century English scientist who was one of the first to use a towing tank.
• Hydrostatic jump
• The wake in front of a boat. (Is that what it's called?)
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Geometric Proofs
April 29th 2012, 08:02 AM #1
Apr 2012
United States
Make a two-column proof showing statements and reasons to prove that triangle ABD is similar to triangle ABC.
(10 points)
Look at the figure shown below.
A student made the table below to show the steps to prove that DC is equal to EC.
Statements Justifications
1. Side AC of triangle ACE is equal to side BC of triangle BCD. Given
2. Angle EAC of triangle ACE is equal to angle DBC of triangle BCD. Given
3. Angle ACE = angle ACD + angle DCE From the figure
4. Angle DCB = angle ECB + angle DCE = angle ACD + angle DCE From the figure angle ECB = angle ACD
6. Triangle DBC is congruent to triangle EAC. ASA Postulate
7. DC = EC The corresponding sides of congruent triangles are equal.
A. Provide the missing statement and justification in the proof.
B. Using complete sentences, explain why the proof would not work without the missing step. (8 points)
The figure below shows a kite labeled PQRS.
Write a 2-column, paragraph, or flow-chart proof to show that angle PQR is congruent to angle PSR. (10 points)
I have no idea how to do this!!! -_- please teach me how!! Its for a test!! HELP!!!!
Re: Geometric Proofs
Problem 1 Key statement angles ADB = DBC + ACB Why?
Problem 2 You have to prove that angles ACE=BCD
Problem 3 PQ=PS QR= SR PR is the axis of symetry.DEfinition of kite
April 29th 2012, 12:31 PM #2
Super Member
Nov 2007
Trumbull Ct
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Bapchule Trigonometry Tutors
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WHAT ARE NUMERICALS?EXPLAIN BRIEFLY WITH EXAMPLES. - Homework Help - eNotes.com
Numerical s can be either counting numbers or can be mathematical expression of a concept.
example: 1,2,3,............
or P = F / A or R = V / I
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Intermediate Algebra with P.O.W.E.R. Learning
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Intermediate Algebra with P.O.W.E.R. Learning | 9780073406275
ISBN-13: 9780073406275 See more
Author(s): Messersmith, Sherri; Perez, Lawrence; Feldman, Robert
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Author(s): Messersmith, Sherri; Perez, Lawrence; Feldman, Robert
Publisher: McGraw-Hill Higher Education
Copyright year: © 2014 Pages: 960
Sherri Messersmith’s successful hardcover franchise is expanded with the new softcover P.O.W.E.R. series. The conversational writing style, practical applications, innovative student resources and
student friendly walk through of examples that users of the hard cover books noted and appreciated are also found in the pages of the P.O.W.E.R. series.
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Abu Abd Allah Muhammad ibn Muadh Al-Jayyani
Born: 989 in Cordoba, Spain
Died: after 1079 in possibly Jaén, Andalusia (now Spain)
Previous (Chronologically) Next Main Index
Previous (Alphabetically) Next Biographies index
Little is known of al-Jayyani's life. Even the identification of al-Jayyani the mathematician with al-Jayyani the Spanish scholar who was born in Cordoba in 989 is not absolutely certain. Everything
points to this identification being correct except one (possible) problem.
The Spanish scholar who was born in Cordoba has exactly the same name as the mathematician, and the Spanish scholar is described as an expert in the Qur'an, also being knowledgeable in Arabic
philology, inheritance laws and arithmetic. Al-Jayyani, the mathematician, is described as a judge and a jurist in one of his treatises. The only possible problem to the identification is that
al-Jayyani wrote a treatise on the total solar eclipse which occurred in Jaén on 1 July 1079. The identification means that he was over ninety years old when he wrote this treatise which, although
certainly not impossible, casts a slight doubt.
The only other facts known about al-Jayyani's life are that he lived in Cairo from 1012 to 1017 and that he must have undertaken most of his work in Jaén, the city at the centre of the Moorish
principality of Jayyan. This cannot only be deduced from his name "al-Jayyani" which means "from Jaén", but also from the fact that the astronomical tables that he produced were for the longitude of
Jaén. Certainly he observed the solar eclipse in Jaén in 1079.
Al-Jayyani's work On ratio is almost certainly his most interesting mathematical work. An English translation of this remarkable treatise is given in [2]. In this work al-Jayyani sets out to defend
Euclid's Elements Book V. In [7] Vahabzadeh writes:-
Euclid's definition, in Book V of his "Elements", of the proportionality of four magnitudes gave rise to numerous commentaries. Of these we have selected two [one being al-Jayyani's] whose goal
was not to criticise Euclid's point of view but rather to justify it by trying to make explicit the assumptions underlying Euclid's argument.
Al-Jayyani states that he is writing the treatise On ratio (see for example [1]):-
... to explain what may not be clear in the fifth book of Euclid's writing to such as are not satisfied with it.
There are five magnitudes that, according to al-Jayyani, are used in geometry; number, line, surface, angle, and solid. Neither Euclid nor any other Greek mathematician would have considered "number"
as a geometrical magnitude, but al-Jayyani needs the notion for his definition of ratio which follows the Arabic idea of number. After assuming that every intelligent person has a basic understanding
of ratio, al-Jayyani deduces further properties based on this "commonly understood definition". To justify his approach he writes:-
There is no method to make clear what is already clear in itself.
He then connects this idea of ratio with that given by Euclid. The authors of [1] write:-
Al-Jayyani shows here an understanding comparable with that of Isaac Barrow, who is customarily regarded as the first to have really understood Euclid's Book V.
Another work of great importance is al-Jayyani's The book of unknown arcs of a sphere, the first treatise on spherical trigonometry. The work, which is published together with a Spanish translation
and a commentary in [3], contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle. Proofs are sometimes only
given as sketches. Debarnot, in his review of [3], argues however that Villuendas:-
... in his commentary ... fails to take the originality of the Determination of the magnitudes sufficiently into account.
Al-Jayyani was to have a strong influence on European mathematics. In addition to translations of his works from the Arabic, his work influenced certain European mathematicians. The article [4]
argues that one of Regiomontanus sources was The book of unknown arcs of a sphere. Among the similarities between al-Jayyani's treatise and that of Regiomontanus are the definition of ratios as
numbers, the lack of a tangent function, and a similar method of solving a spherical triangle when all sides are unknown.
However, the author of [4] remarks that there are some marked differences in approach between al-Jayyani and Regiomontanus, such as the proof of the spherical sign law. Although it is certain that
Regiomontanus based his treatise on Arabic works on spherical trigonometry it may well be that al-Jayyani's work was only one of many such sources.
The article [6] describes the treatise Kitab al-asrar fi nata'ij al-Afkar (The book of secrets about the results of thoughts), attributed to al-Jayyani on the basis of internal evidence together with
its date. The work studies hydraulics and water clocks.
Work by al-Jayyani on astronomy was also important. He wrote on the morning and evening twilight, computing the fairly accurate value of 18° for the angle of the sun below the horizon at the start on
morning twilight and at the end of the evening twilight.
In the Tabulae Jahen al-Jayyani gave data to enable the calculation of the time of day, the calendar, the new moon, eclipses and information required for the timing and direction for prayers. As was
common at this time, not only was there astronomical information in the work but also astrological information on horoscopes. Al-Jayyani seems to have considerable respect for al-Khwarizmi's
astronomical data, which he freely used, but he rejects the ideas of al-Khwarizmi on astrology. Much of al-Jayyani's astrology is based on Hindu sources.
Article by: J J O'Connor and E F Robertson
List of References (7 books/articles)
Mathematicians born in the same country
Previous (Chronologically) Next Main Index
Previous (Alphabetically) Next Biographies index
History Topics Societies, honours, etc. Famous curves
Time lines Birthplace maps Chronology Search Form
Glossary index Quotations index Poster index
Mathematicians of the day Anniversaries for the year
JOC/EFR © November 1999 School of Mathematics and Statistics
Copyright information University of St Andrews, Scotland
The URL of this page is:
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The monkey typewriter
by Mithrandir
I bet all of you know the now famous theorem of the infinite monkeys. Starting from this idea someone devised the following problem:
Suppose there is a monkey sitting in front of a terminal consisting of n keys. The monkey will press any of those keys at random and, when all of them will be pressed, she will receive a banana
as a payment for her efforts. We are mainly interested in how long will it take for her to obtain that banana in the following two cases:
□ when a key is pressed it remains pressed
□ when a key is pressed for the second time it will behave as though it was not pressed at all.
Practically, we are interested to see what is the average number of keystrokes for the specific number n of keys.
This is what we’ll be doing here. For the first part of our instalment we will use Numerical Methods to deal with the exact mathematical equation for the average number of keystrokes needed.
The second instalment of this problem will tackle with a Monte-Carlo approach to this problem and (maybe) with reducing the computational cost of Monte-Carlo methods, the main reason behind choosing
this problem.
So, we begin. We won’t go into details about Markov Chains and how one can use the transition matrix of such a chain to determine the average number of jumps in the chain before arriving at an
absorbing state. We will only publish the main concepts behind, no mathematics for now. I think this function (a simple Octave/Matlab script) is pretty self-explanatory.
function t = Markov(P, na)
% computes the average times (as a row vector t) needed
% to reach one state from the na absorbing ones
% existing in a Markov chain, given by the matrix P - the
% transition matrix
n = size(P)(1); % size of the chain
Q = P(1: (n - na), 1: (n - na)); % the transient part of P
N = inv(eye(n - na) - Q); % the expectation matrix
t = N * ones(n - na, 1); % the average number of jumps
Now, we only have to build those P matrices for those two cases. Come to think of it, this is also an easy part after observing that we could define our state by counting the number of keys which are
pressed (thus, the only absorbing state would be the one where all n keys are pressed). The following Octave/Matlab snippet builds the P matrix for the first case.
v = (0:n) / n; % auxiliary vector
P = diag(v) + diag(1 - v(1:n), 1); % the transition matrix
For the second case we have the following, full script this time (including the call and the return value).
function t = f2(n)
% computes the average number of random keystrokes
% (from n available keys) needed to fill the table
% in the second case
v = (0:n) / n; % auxiliary vector
P = diag(v(2: (n + 1)), -1) + diag(1 - v(1:n), 1);
t = Markov(P, 1); % we have only one absorbant state
t = t(1);
Looking at the results, the first case is a lucky one for the monkey: the average number of keystrokes is in $O(1)$, as it is shown by this plot. Unfortunately, the second case has an exponential
growth (as seen from this plot, taken at a logarithmic scale).
As of now, we have a good view on the complexity of the problem, which is one of the reasons which just makes it perfect for being used in Monte Carlo simulation studies along with the fact that it
is not a trivial problem like the one of numerical integration, instead being a problem with different degrees of complexity.
Thank you for your patience, we will come again soon with the second part.
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Congruences, Powers, and Euler's Formula
Hello I can't quite seem how to finish this problem:
The number 3750 satisfies f(3750) = 1000. Find a number a that has the following three properties:
(i) a º 7^3003 (mod 3750).
(ii) 1 < a < 5000.
(iii) a is not divisible by 7.
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Legendre Polynomial
Rodrigues' Formula: The Legendre Polynomials
Generating Function: The generating function of a Legendre Polynomial is
Orthogonality: Legendre Polynomials complete orthogonal set on the interval
By using this orthogonality, a piecewise continuous function
This orthogonal series expansion is also known as a Fourier-Legendre Series expansion or a Generalized Fourier Series expansion.
Even/Odd Functions: Whether a Legendre Polynomial is an even or odd function depends on its degree
Based on
In addition, from
Recurrence Relation: A Legendre Polynomial at one point can be expressed by neighboring Legendre Polynomials at the same point.
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Mplus Discussion >> StdYX in Multilevel Regression
Florian Fiedler posted on Sunday, January 22, 2006 - 8:58 am
Is there a short example how to interpret StdYX of a given parameter? My explanation for Std is like that:
Model: Y=aX+bZ+c
Y ON X + Z;
Std (a): a/Var(Y)
Std (b): b/Var(Y)
So Std is just parameter/Var(latent variable).
But I have no idea what I should devide by the estimates to get StdYX. The user's guide just gives "[...] These coefficients are standardized USING the variances of the cont. latent variables as well
as the variances of the background AND/OR outcome variables."
1) What does USING mean?
2) What does AND/OR mean?
3) Is there anything special about StdYX in multilevel-regression analysis?
Thank you for a short explanation and an equation how to compute StdYX from parameter estimates.
Florian Fiedler.
bmuthen posted on Sunday, January 22, 2006 - 9:56 am
In regression of y on x you simply create Stdyx from the raw slope b as usual,
Stdyx = b * SD(x)/SD(y)
Then the Stdyx value has the regular interpretation: how much of a y SD the y variable changes for a 1 SD change in x.
The text you referred to tries to cover more general cases than standard regression, including latent variables.
Florian Fiedler posted on Sunday, January 22, 2006 - 10:37 am
So far I understand it. But I'm interested in regressions containing two or more observed variables another (2-level) variable is regressed on. And since it's a dependent variable on two levels it's
getting a bit more complicated I guess ...
The model I've got now is the following:
WITHIN ARE X1 X2 X3 X4;
BETWEEN ARE Z1 Z2 Z3;
Y ON X1 + X2 + X3 + X4;
Y ON Z1 + Z2 + Z3;
How is, for instance, StdYX for variable X1 computed and interpreted? Sorry for being impatient, will I get answers on the questions 1) (USING = sum of variances; product of SDs?), 2) and 3)?
Thank you for your work (even on sundays).
Florian Fiedler.
bmuthen posted on Sunday, January 22, 2006 - 7:42 pm
StdYX for X1 in your example uses as the SD(x) and SD(y) of my example, the estimated within-level SDs of x1 and y. Hope that helps.
Linda K. Muthen posted on Monday, January 23, 2006 - 8:48 am
1) What does USING mean?
Multiplying or dividing.
2) What does AND/OR mean?
Observed variable variance when no latent variables in the model - both observed and latent variable variances when latent variables are in the model.
3) Is there anything special about StdYX in multilevel-regression analysis?
Fred Smyth posted on Sunday, February 05, 2006 - 12:02 pm
How can I constrain two parameter estimates to be equal, but with opposite sign?
Fred Smyth
Linda K. Muthen posted on Sunday, February 05, 2006 - 5:09 pm
Use the MODEL CONSTRAINT command and say:
p1 = -p2;
Back to top
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MathGroup Archive: February 1999 [00296]
[Date Index] [Thread Index] [Author Index]
Re: Plot[] problem (plnr) - help !!!
• To: mathgroup at smc.vnet.net
• Subject: [mg15992] Re: [mg15935] Plot[] problem (plnr) - help !!!
• From: "Richard Finley" <rfinley at medicine.umsmed.edu>
• Date: Fri, 19 Feb 1999 03:27:08 -0500
• Sender: owner-wri-mathgroup at wolfram.com
It is not clear to me what you are trying to do?? I presume you mean to
use the assignment operator = instead of the comparator ==?? If
that is true, and I presume you meant to put the Clear[H] before you
define H rather than after?? Even so, you will not be able to plot the
function as is since it has real and imaginary parts. Instead try this:
H[w_]:= Log[10,Sum[a[k] Exp[I 2 Pi w(-k)], {k,0,1}]]
or you could plot the Real and Imaginary parts separately. Regards, RF
>>> Alexander Sirotkin <root at borjch.math.tau.ac.il> 02/17/99 10:34PM >>>
I'm trying to do most trivial and simple thing - define
some function and then Plot[] it. However, I get some very
weird error - " ...is not a machine-size real number at ..."
Here is the code :
H[w] == Log[10,Sum[a[k] Exp[I 2Pi w(-k)], {k,0,1}]]
Plot[H[w], {w, 1, 100}]
I've seen some articles (even at www.wolfram.com) about that
problem, but NONE of them proposed a working solution !!!
Just putting Evaluate[] inside the Plot[] does nothing...
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s for
Results 1 - 10 of 87
- Journal on Satisfiability, Boolean Modeling and Computation , 2006
"... In this paper, we describe and evaluate three different techniques for translating pseudoboolean constraints (linear constraints over boolean variables) into clauses that can be handled by a
standard SAT-solver. We show that by applying a proper mix of translation techniques, a SAT-solver can perfor ..."
Cited by 121 (2 self)
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In this paper, we describe and evaluate three different techniques for translating pseudoboolean constraints (linear constraints over boolean variables) into clauses that can be handled by a standard
SAT-solver. We show that by applying a proper mix of translation techniques, a SAT-solver can perform on a par with the best existing native pseudo-boolean solvers. This is particularly valuable in
those cases where the constraint problem of interest is naturally expressed as a SAT problem, except for a handful of constraints. Translating those constraints to get a pure clausal problem will
take full advantage of the latest improvements in SAT research. A particularly interesting result of this work is the efficiency of sorting networks to express pseudo-boolean constraints. Although
tangential to this presentation, the result gives a suggestion as to how synthesis tools may be modified to produce arithmetic circuits more suitable for SAT based reasoning. Keywords:
pseudo-Boolean, SAT-solver, SAT translation, integer linear programming
- In Proceedings of the 24 th Euromicro Conference
"... In this paper we investigate the Sum Absolute Difference (SAD) operation, an operation frequently used by a number of algorithms for digital motion estimation. For such operation, we propose a
single vector instruction that can be performed (in hardware) on an entire block of data in parallel. We in ..."
Cited by 23 (15 self)
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In this paper we investigate the Sum Absolute Difference (SAD) operation, an operation frequently used by a number of algorithms for digital motion estimation. For such operation, we propose a single
vector instruction that can be performed (in hardware) on an entire block of data in parallel. We investigate possible implementations for such an instruction. Assuming a machine cycle comparable to
the cycle of a two cycle multiply, we show that for a block of 16x1 or 16x16, the SAD operation can be performed in 3 or 4 machine cycles respectively. The proposed implementation operates as
follows: first we determine in parallel which of the operands is the smallest in a pair of operands. Second we compute the absolute value of the difference of each pairs by subtracting the smallest
value from the largest and finally we compute the accumulation. The operations associated with the second and the third step are performed in parallel resulting in a multiply (accumulate) type of
operation. Our approach covers also the Mean Absolute Difference (MAD) operation at the exclusion of a shifting (division) operation.
- In Proceedings of Arith-13 , 1997
"... In recent years computer applications have increased in their computational complexity. The industry-wide usage of performance benchmarks, such as SPECmarks, and the popularity of 3D graphics
applications forces processor designers to pay particular attention to implementation of the floating p ..."
Cited by 19 (1 self)
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In recent years computer applications have increased in their computational complexity. The industry-wide usage of performance benchmarks, such as SPECmarks, and the popularity of 3D graphics
applications forces processor designers to pay particular attention to implementation of the floating point unit, or FPU. This paper presents results of the Stanford subnanosecond arithmetic
processor (SNAP) research effort in the design of hardware for floating point addition, multiplication and division. We show that one cycle FP addition is achievable 32% of the time using a variable
latency algorithm. For multiplication, a binary tree is often inferior to a Wallace-tree designed using an algorithmic layout approach for contemporary feature sizes (0.3m). Further, in most cases
two-bit Booth encoding of the multiplier is preferable to non-Booth encoding for partial product generation. It appears that for division, optimum area-performance is achieved using functional
iteration, ...
- in IEEE Alessandro Volta Memorial Workshop on Low Power Design , 1999
"... Reducing the power dissipation of parallel multipliers is important in the design of digital signal processing systems. In many of these systems, the products of parallel multipliers are rounded
to avoid growth in word size. The power dissipation and area of rounded parallel multipliers can be signi ..."
Cited by 19 (5 self)
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Reducing the power dissipation of parallel multipliers is important in the design of digital signal processing systems. In many of these systems, the products of parallel multipliers are rounded to
avoid growth in word size. The power dissipation and area of rounded parallel multipliers can be significantly reduced by a technique known as truncated multiplication. With this technique, the least
significant columns of the multiplication matrix are not used. Instead, the carries generated by these columns are estimated. This estimate is added with the most significant columns to produce the
rounded product. This paper presents the design and implementation of parallel truncated multipliers. Simulations indicate that truncated parallel multipliers dissipate between 29 and 40 percent less
power than standard parallel multipliers for operand sizes of 16 and 32 bits. 1: Introduction High-speed parallel multipliers are fundamental building blocks in digital signal processing systems [1].
- IEEE TRANSACTIONS ON COMPUTERS , 1995
"... In this paper, a new design technique for column-compression (CC) multipliers is presented. Constraints for column compression with full and half adders are analyzed and, under these
constraints, considerable flexibility for implementation of the CC multiplier, including the allocation of adders, an ..."
Cited by 15 (3 self)
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In this paper, a new design technique for column-compression (CC) multipliers is presented. Constraints for column compression with full and half adders are analyzed and, under these constraints,
considerable flexibility for implementation of the CC multiplier, including the allocation of adders, and choosing the length of the final fast adder, is exploited. Using the example of an 8 8 bit CC
multiplier, we show that architectures obtained from this new design technique are more area efficient, and have shorter interconnections than the classical Dadda CC multiplier. We finally show that
our new technique is also suitable for the design of two's complement multipliers.
- SHA-256 (384, 512),” in Proc. DATE
"... After recalling the basic algorithms published by NIST for implementing the hash functions SHA-256 (384, 512), a basic circuit characterized by a cascade of full adder arrays is given.
Implementation options are discussed and two methods for improving speed are exposed: the delay balancing and the p ..."
Cited by 14 (1 self)
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After recalling the basic algorithms published by NIST for implementing the hash functions SHA-256 (384, 512), a basic circuit characterized by a cascade of full adder arrays is given. Implementation
options are discussed and two methods for improving speed are exposed: the delay balancing and the pipelining. An application of the former is first given, obtaining a circuit that reduces the length
of the critical path by a full adder array. A pipelined version is then given, obtaining a reduction of two full adder arrays in the critical path. The two methods are afterwards combined and the
results obtained through hardware synthesis are exposed, where a comparison between the new circuits is also given. 1.
- Journal of VLSI and Signal Processing-Systems for Signal, Image and Video Technology , 2000
"... . In this paper, we describe the implementation of MorphoSys, a reconfigurable processing system targeted at data-parallel and computation-intensive applications. The MorphoSys architecture
consists of a reconfigurable component (an array of reconfigurable cells) combined with a RISC control process ..."
Cited by 13 (0 self)
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. In this paper, we describe the implementation of MorphoSys, a reconfigurable processing system targeted at data-parallel and computation-intensive applications. The MorphoSys architecture consists
of a reconfigurable component (an array of reconfigurable cells) combined with a RISC control processor and a high bandwidth memory interface. We briefly discuss the system-level model, array
architecture, and control processor. Next, we present the detailed design implementation and the various aspects of physical layout of different sub-blocks of MorphoSys. The physical layout was
constrained for 100 MHz operation, with low power consumption, and was implemented using 0.35 m, four metal layer CMOS (3.3 Volts) technology. We provide simulation results for the MorphoSys
architecture (based on VHDL model) for some typical data-parallel applications (video compression and automatic target recognition). The results indicate that the MorphoSys system can achieve
significantly better performance...
, 1997
"... A signed binary (SB) addition circuit is presented that always produces an even parity representation of the sum word. The novelty of this design is that no extra check bits are generated or
used. The redundancy inherent in a SB representation is further exploited to contain parity information. ..."
Cited by 11 (1 self)
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A signed binary (SB) addition circuit is presented that always produces an even parity representation of the sum word. The novelty of this design is that no extra check bits are generated or used.
The redundancy inherent in a SB representation is further exploited to contain parity information.
, 1998
"... A new IEEE compliant floating-point rounding algorithm for computing the rounded product from a carry-save representation of the product is presented. The new rounding algorithm is compared with
the rounding algorithms of Yu and Zyner [23] and of Quach et al. [18]. For each rounding algorithm, a log ..."
Cited by 10 (2 self)
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A new IEEE compliant floating-point rounding algorithm for computing the rounded product from a carry-save representation of the product is presented. The new rounding algorithm is compared with the
rounding algorithms of Yu and Zyner [23] and of Quach et al. [18]. For each rounding algorithm, a logical description and a block diagram is given and the latency is analyzed. We conclude that the
new rounding algorithm is the fastest rounding algorithm, provided that an injection (which depends only on the rounding mode and the sign) can be added in during the reduction of the partial
products into a carry-save encoded digit string. In double precision the latency of the new rounding algorithm is 12 logic levels compared to 14 logic levels in the algorithm of Quach et al., and 16
logic levels in the algorithm of Yu and Zyner. 1. Introduction Every modern microprocessor includes a floating-point (FP) multiplier that complies with the IEEE 754 Standard [9]. The latency of the
FP multiplier...
- PROCEEDINGS OF THE IEEE , 1997
"... This paper analyzes the effect of this technological progress on the design of nanoelectronic circuits and describes computational paradigms revealing novel features such as distributed storage,
fault tolerance, self-organization, and local processing. In particular, linear threshold networks, the a ..."
Cited by 10 (4 self)
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This paper analyzes the effect of this technological progress on the design of nanoelectronic circuits and describes computational paradigms revealing novel features such as distributed storage,
fault tolerance, self-organization, and local processing. In particular, linear threshold networks, the associative matrix, self-organizing feature maps, and cellular arrays are investigated from the
viewpoint of their potential significance for nanoelectronics. Although these concepts have already been implemented using present technologies, the intention of this paper is to give an impression
of their usefulness to system implementations with quantum-effect devices.
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Latest posts of: JohnMunsch - Java-Gaming.org
> I might have the IQ of a horse's arse but I completely fail to see how this dice roll example would apply to submitting a high score to a server.
He wasn't serious. There are algorithms for fair die roll, coin flip, and card shuffling. Those are probably sufficient to handle what the original poster (way back on page one
The more recent stuff about high scores is a different matter. I don't know a way for a normal algorithm to prove that you did what you say you did when all activity is confined to your machine and
it only has the opportunity to look at the final result of a game playing session.
Maybe there is one, but I certainly don't know it.
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North Wales Geometry Tutor
Find a North Wales Geometry Tutor
...It was amazing to see how excited these kids were to learn about science and now, reflecting on this experience, I see the impact I can have on people directly. My passion for math and science
has given me more than expertise in those subjects, but also a strong analytical mind that is essential...
16 Subjects: including geometry, Spanish, calculus, physics
...Format pages and insert headers and footers. 8. Insert graphics, pictures, and table of contents, and PowerPoint adds easy-to-use interactive features that make the usual slides of boring
bulleted text and charts a relic of the past. Making up for a tedious PowerPoint presentation by being an exceptional speaker is no longer required.
27 Subjects: including geometry, calculus, statistics, algebra 1
...One of my goals is for the student to feel smarter and more confident when I am through with them! By middle school, many students (particularly boys) believe that they aren't capable of
reading well. I work hard to help each student believe in themselves and to understand that they just need a little help to catch up.
28 Subjects: including geometry, reading, English, writing
...I have a Master of Science degree in math, over three years' experience as an actuary, and am a member of MENSA. I am highly committed to students' performances and to improve their
comprehension of all areas of mathematics.I have excelled in courses in Ordinary Differential Equations in both un...
19 Subjects: including geometry, calculus, trigonometry, statistics
I currently attend Kutztown University and I am studying Prek-8 Special Education and 4-8 Math and English Education. I currently tutor at a middle school as well as intern at a middle and
elementary school. In addition to my experiences in school districts, I teach Sunday school.
29 Subjects: including geometry, reading, English, writing
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Non-uniform grid via coordinate transformation in Comsol
Hi all,
I would like to know if there is a way to construct a non-uniform grid using coordinate transformation in Comsol. More specifically, I have a PDE on a rectangle and I would like to transform the
equidistant grid on the x axis
x1_m = m/M, where m = 1,...,M
to the following one
x2_m = A + B * sinh(c1*x1_m + c2*(1 − x1_m)), where A, B, c1 and c2 are scalars.
The reason for this transformation is that choosing adequately c1 and c2 will lead to a finer mesh around B.
Then, I would like to do the same kind of tranform for the y axis. I tried with the distribution option using a Free Quad meshing, but it is linked to a boundary, as I understand.
Thanks in advance.
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Isaak Newton
Isaac Newton, who lived during the 18th century, was one of the greatest mathematicians and physists. He mainly used mathematics as a reasearching tool for astronomy and physics.
Newton was also very interested in theology, like Euler and many other mathematicans before him. He was, however, one of the last mathematicians who did his research in the name of God. And his
discoveries about gravitation and the paths of the different planets around the sun, were an important reason why.
After Newton, the laws of nature became more important than the creator behind them. And through the discoveries of non-Euclidean geometries, scientists started to establish theorems, and then see
how they might be useful. Many people still believe that there is one formula to describe the world. Some of the greatest scientists have spent years trying to find such a formula. Would this formula
be a mathematical way to describe God?
Many of Newton's discoveries are common knowlege today, but were revolutionary in his time. For example, people still believed in the Greek idea that colors consisted of a mixture of light and
shadows. This idea isn't so crazy if you think of how colors grow fainter when the sun goes down. So when Newton came with his idea that light consists of all colors mixed together, people weren't
too happy. Even though anyone could have tested his experiment of breaking the light into its colors with a prism.
One of Newton's most important papers laid the foundation for calculus. Calculus is an extention of Descartes' analytic geometry. For example, with it you can calculate the area of shapes which are
not enclosed with lines and parts of circles.
Even though Newton didn't publish much of his work, it was plenty to give him quite some fame. When he died, he was a rich, honored and knighted gentleman.
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Safety/Relief Valve Quencher Loads: Evaluation for BWR Mark II and III Containments (Generic Letter No. 82-24)
Safety/Relief Valve Quencher Loads: Evaluation for BWR Mark II and III Containments (Generic Letter No. 82-24)
UNITED STATES
NUCLEAR REGULATORY COMMISSION
WASHINGTON, D.C. 20555
November 4, 1982
EVALUATION FOR BWR
MARK II AND III CONTAINMENTS
(Generic Letter No. 82-24)
Enclosed is a copy of NUREG-0802, "Safety/Relief Valve Quencher Loads:
Evaluation for BWR Mark II and III Containments. NUREG-0802 is being issued
to provide acceptance criteria for hydrodynamic loads on piping, equipment,
and containment structures resulting from SRV actuation. The NRC staff finds
that use of these acceptance criteria satisfy the requirements of General
Design Criteria 16 and 29 in Appendix A to 10 CFR Part 50. NUREG-0802,
however, is not a substitute for the regulations, and compliance with the
NUREG is not a requirement. An approach or method different from the
acceptance criteria contained herein will be accepted if the substitute
approach or method provides a basis for determining that the regulations
have been met.
The NRC had issued SRV load acceptance criteria for both Mark II
(NUREG-0487, Supplement No. 1, September 1980) and Mark III (SER for GESSAR,
July 1976). However, the staff, the Mark II Owners Group and GE recognized
that these criteria were very conservative because they were established at
the early stage of quencher development. Since then, extensive quencher test
programs were performed resulting in a sufficient data base to justify
re-evaluation the SRV load criteria. In response to the request by the Mark
II Owners Group and GE, the staff has re-evaluated the SRV loads and
established the new acceptance criteria in NUREG-0802. The staff also finds
the earlier criteria acceptable. The acceptance criteria in NUREG-0487
supplement No. 1 (for Mark II plants) or the acceptance criteria in an
attachment 2 (for Mark III plants) are conservative with respect to the
acceptance criteria proposed in Appendices A and B of NUREG-0802,
respectively and they are acceptable.
The reporting and/or recordkeeping requirements contained in this letter
affect fewer than ten respondents; therefore, OMB clearance is not required
under P.L. 96-511.
Darrell G. Eisenhut, Director
Division of Licensing
Office of Nuclear Reactor Regulation
Attachments 1 & 2
ATTACHMENT 2
ACCEPTANCE CRITERIA
FOR QUENCHER LOADS FOR
THE MARK III CONTAINMENT
I. INTRODUCTION
On September 2, 1975, the General Electric Company submitted topical
reports NEDO-11314-08 (nonproprietary) and NEDE-11314-08 (proprietary)
entitled, "Information Report Mark III Containment Dynamic Loading
Conditions," docketed as Appendix 3-B to the Amendment No. 37 for
GESSAR, Docket No. STN-50-447. As part of this report, a device called
a "quencher" would be used at the discharge end of safety/relief valve
(SRV) lines inside the suppression pool. Tests were performed in a
foreign country to obtain quencher load data that were used to
establish the Mark III data base. A statistical technique using the
test data to predict quencher loads for Mark III containment was also
presented. GE had submitted another topical report NEDE-21078 entitled,
"Test Results Employed by GE for BWR Containment and Vertical Vent
Loads," to substantiate their method to extrapolate the loads obtained
from the tests to the Mark III design.
We reviewed the above topical reports and had identified several areas
of concern. Meetings with GE were held to discuss these concerns. As a
result, GE presented a modified method during the April 2, 1976,
meeting held in Bethesda, Maryland. Subsequent to the meeting, this
modified method and proposed load criteria were reported in Amendment
No. 43, which was received on June 22, 1976. Our evaluation, therefore,
is based on the modified method and the load criteria calculated by
this method.
The statistical method proposed by GE to arrive at design quencher
loads for the Mark III containment consists of a series of steps.
Initially, a multiple linear regression analysis for the first
actuation event is performed with a data base taken from three tests
series: mini-scale (9 points), small scale (70 points) and large scale
(37 points).
Non-linearities are introduced where necessary by using-quadratic
variables and formed straight line segments. The regression
coefficients are estimated from the appropriate data set. The resulting
equation contains a constant term plus corrective terms that take into
account the influence of all key parameters.
In the second step, the subsequent actuation effect is determined by
postulating a direct proportionality between the observed maximum
subsequent actuation pressure and the predicted first actuation
pressure. The proportionality constant is found by considering the
large scale data.
In the third step, the total variance of the predicted future SRV
subsequent actuation is found by noting that the total variance is the
sum of three terms: (1) a term due to the uncertainty in the
first actuation prediction which is calculated from standard (normal
variate) formulas, (2) a term due to the uncertainty in the
proportionality factor as was calculated in the second step above, and
(3) a term due to the variance of the residual maximum subsequent
pressure. It is now assumed that this variance is proportional to the
square of predicted maximum-subsequent actuation pressure. The
proportionality constant is found from the large scale subsequent
actuation data (10 values).
In the fourth step, design values for Mark III are determined from the
estimated (i.e., predicted) values of maximum subsequent actuation
pressure and its standard deviation by employing standard tables of
so-called "tolerance factors." These tables are entered with three
quantities: (1) n, the number of sample data points from which the
estimate of the mean and standard deviations are obtained. GE has set
n x 10, based on 10 maximum subsequent actuation points used in the
third step, (2) the probability value, and (3) the confidence level.
The design value is then simply the predicted value plus the tolerance
factor times the estimated standard deviation.
The approach as outlined above is used to calculate the positive
pressures for a single SRV considering multiple actuations which
represents the most severe SRV operation condition. For the single
actuation case, the calculational procedures are similar with the
method mentioned above with the following exceptions:
1. The calculation which involves subsequent actuations is
eliminated; and,
2. Thirty-seven data points were selected for establishing the
tolerance factor since these data points in the large-scale tests
relate to single value actuation.
For negative pressure calculation, a correlation of peak positive and
negative pressures is developed. The correlation is based on the
principle of conservation of energy and verified by the small-scale and
large-scale test results.
Based on the method outlined above, GE has calculated the SRV quencher
loads for the Mark III and established the load criteria for six cases
of SRV operation. The calculated load criteria based on 95-95%
confidence level are given on Table 1 which is attached.
As a result of our review, we have concluded that the statistical
method proposed by GE and the load criteria shown on Table 1 are
acceptable. This conclusion is based on the following:
1. The method has properly treated all available test data and is
based essentially on the large-scale data with correction terms
that take into account the influence of non-large-scale variables.
Since the large-scale tests were performed in an actual reactor
with a suppression containment conceptually similar with GE
containments extrapolation from the large-scale by statistical
technique, therefore, is appropriate and acceptable.
2. The method has been conducted in a conservative manner. The
primary conservatisms are:
a. The calculation is based on the most severe parameters. For
example, the maximum air volume initially stored in the line,
the maximum initial pool temperature and the highest primary
system pressure were selected to establish quencher load
b. For the cases of multiple valve actuation, the load criteria
are based on the assumption that the maximum pressures
resulting from each valve will occur simultaneously. We
believe that the assumption is conservative since different
lengths of line and SRV pressure set points will result in
the occurrence of maximum pressures at different times and
consequently lower loads.
3. The proposed load criteria, which are provided on the attached
Table 1, are acceptable. The criteria were established by using
95-95% confidence limit. Our consultant, the Brookhaven National
Laboratory, has performed an analysis for the effect of confidence
limit. The result of this analysis indicates that for 95-95%
confidence limit, approximately 1% of the number of RSV actuations
may result in containment loads above the design value. We believe
this low probability is acceptable considering the conservatism of
the method of prediction, i.e., the actual loads should not exceed
the design value.
4. With regard to the subsequent actuation, the load criteria are
based upon a single SRV actuation, G.E. has established this basis
by regrouping the SRV's in each group of pressure set points. As
indicated in Amendment 43, there are three groups of pressure set
points for the 19 SRV's for the 238-732 standard plant, namely,
one SRV at a pressure set point of 1103 psig, 9 SRV's at 1113
psig, and the remaining 9 SRV's at 1123 psig. Only one SRV is now
set at the lowest pressure set point. Based on this pressure set
point arrangement for the 19 SRV's, GE has analyzed the most
severe primary pressure transient, i.e., a turbine trip without
bypass. Results of the analysis shows that initiation of reactor
isolation will activate all or a portion of the 19 SRV's which
will release put the stored energy in the primary system.
Following the initial blowdown, the energy generated in the
primary system consists primarily of decay heat which will cause
the lowest set SRV to reopen and reclose (subsequent actuation).
The time duration between subsequent actuation was calculated to
be a minimum of 62 seconds and increasing with each actuation. The
time duration of each blowdown decreases from 51 seconds for the
initial blowdown and decreases to 3 seconds at the end of the
period of subsequent actuations which is 30 minutes after
initiation of
reactor isolation.
The staff finds the result of the GE analysis reasonable.
Therefore, the assumption of only the lowest set SRV operating in
subsequent actuation is justified and acceptable.
The acceptance of the quencher load criteria is based on the test data
available to us. We realize, however, that the tests lack exact dynamic
or geometric similarity with the quencher system for the Mark III
containment. The test results, therefore, could not be applied
directly. Though the quencher loads for the Mark III appear
conservative in comparison with the test data, some degree of
uncertainty is acknowledged. The uncertainty is primarily due to a
substantial degree of scatter of all test data. We therefore will
require in-plant testing.
It is our position that applicants for Mark III containments using the
criteria specified below:
1. The structures affected by the SRV operation should be designed to
withstand the maximum loads specified in Table 1. For the cases
of multiple valve actuation, the quencher loads from each line
shall be assumed to reach the peak pressure simultaneously and
oscillate in phase.
2. The quencher loads as specified in Item 1 above are for a
particular quencher configuration shown in the topical reports
NEDO-11314-08 and NEDE-11314-08. Since the quencher loads are
sensitive to and dependent upon the parameters of quencher
configuration, the following requirements should be met:
a. the sparger configuration and hole pattern should be
identical with that specified in Section A7.2.2.4 of
b. The value of key parameters should be equal to or less than
that specified below:
Total air volume in each SRV line (ft3) 56.13
Distance from the center of quencher
to the pool surface at high water
level 13'-11"
Maximum pool temperature during
normal plant operation (F) 100
c. The value of those key parameters should be equal to or
larger than that specified below:
Water surface area per quencher (ft2) 295
SRV opening time (sec) 0.020
3. The spatial variation of the quencher loads should be calculated
by the methods shown in Section 2.4 of the topical report
4. The load profile and associated time histories specified in Figure
A5.11 of NEDO-113/4-08 should be used with a quencher load
frequency of 5 to 11 Hz.
5. For the 40 year plant life, the number of fatigue cycles for the
design of the structures affected by the quencher loads should not
be less than that specified in Section A9.0 of NEDO-11314-08.
6. In-plant testing of the quencher should be conducted to verify the
quencher design loads and oscillatory frequency. The in-plant
tests should include the following:
a. single valve actuation;
b. consecutive actuation of the same valve; and,
c. actuation of multiple valves.
Included should be measurements of pressure load, stress, and
strain of affected structures. A prototypical plant should be,
selected for each type of containment structure. For example, the
pressure responses from a concrete containment should not be used
for a free-standing steel containment and vice versa. Tests should
be conducted as soon as operational conditions allow and should be
performed prior to full power operation.
7. Based on the in-plant test results, reanalyses should be performed
to ensure the safety margin for the structures, which include the
containment wall, basemat, drywell walls, submerged structures
inside the suppression pool, quencher supports and components
influenced by S/R loads. If the analysis indicates that the safety
margin for the structures will be reduced because of the
new loads identified from the test, modification or strengthening
of the structures should be made in order to maintain the safety
margin for which the structures were originally designed. The
applicants for the Mark III containment with quenchers for S/R
valves should submit a licensing topical report for approval. This
report should present a test program and identify the feasibility
of modification or strengthening of the structures.
Page Last Reviewed/Updated Tuesday, June 18, 2013
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>Geometry Proofs - CPCTC, Two-Column Proofs, FlowChart Proofs & Proof by Contradiction (with videos, worksheets, games & activities)
Geometry Proofs - CPCTC, Two-Column Proofs, FlowChart Proofs and Proof by Contradiction
Videos, worksheets, games and acivities to help Geometry students learn geometry proofs and how to use CPCTC, Two-Column Proofs, FlowChart Proofs and Proof by Contradiction.
Related Topics:
More Grade 9 and Grade 10 math
CPCTC is an acronym for corresponding parts of congruent triangles are congruent. CPCTC is commonly used at or near the end of a proof which asks the student to show that two angles or two sides are
congruent. It means that once two triangles are proven to be congruent, then the three pairs of sides that correspond must be congruent and the three pairs of angles that correspond must be
How to use the principle that corresponding parts of congruent triangles are congruent, or CPCTC.
This video offers a look at two triangle proofs that involve the CPCTC theorem.
How do we use CPCTC?
Two Column Proofs
Two column proofs are organized into statement and reason columns. Each statement must be justified in the reason column. Before beginning a two column proof, start by working backwards from the
"prove" or "show" statement. The reason column will typically include "given", vocabulary definitions, conjectures, and theorems.
How to organize a two column proof.
This video introduces the structure of 2 column proofs and works through three examples.
A brief lesson and practice on drawing diagrams and completing two column proofs from word problems
Flowchart Proofs
Flowchart proofs are organized with boxes and arrows; each "statement" is inside the box and each "reason" is underneath each box. Each statement in a proof allows another subsequent statement to be
made. In flowchart proofs, this progression is shown through arrows. Flowchart proofs are useful because it allows the reader to see how each statement leads to the conclusion.
How to outline a flowchart proof. Using flow charts to do proofs
Using flowcharts in proofs for Geometry
Proof by Contradiction
CA Geometry: Proof by Contradiction
CA Geometry: More Proofs
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Williamsbridge, New York, NY
Fort Lee, NJ 07024
Math, Science, Technology, and Test Prep Tutor
I'm an experienced certified teacher in NJ, currently pursuing a Doctorate in
Education and Applied
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Offering 10+ subjects including algebra 1, algebra 2 and calculus
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How do you graph the rational function y= x-4/(x^2+2)
for the graph of $y = \frac{x-4}{x^2+2}$ for horizontal asymptote, divide numerator and denominator with highest power $(x^2)$ and take $x\rightarrow \infty$ horizontal asymptote is x-axis (equation
of x-axis is y=0) there is no vertical asymptote. $as\;\; x \rightarrow \infty, \;\;y \rightarrow 0$ $as\;\; x \rightarrow -\infty, \;\;y \rightarrow 0$ also, points (0, -2) and (4,0) lie on the
graph. Please see graph
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Learning Kernel Based Halfspaces with the 0-1 Loss
Shai Shalev-Shwartz, Ohad Shamir and Karthik Sridharan
SIAM Journal on Computing 2011.
We describe and analyze a new algorithm for agnostically learning kernel-based halfspaces with respect to the 0-1 loss function. Unlike most previous formulations which rely on surrogate convex loss
functions (e.g. hinge-loss in SVM and log-loss in logistic regression), we provide nite time/sample guarantees with respect to the more natural 0-1 loss function. The proposed algorithm can learn
kernel-based halfspaces in worst-case time poly(exp(L log(L=))), for any distribution, where L is a Lipschitz constant (which can be thought of as the reciprocal of the margin), and the learned
classier is worse than the optimal halfspace by at most . We also prove a hardness result, showing that under a certain cryptographic assumption, no algorithm can learn kernel-based halfspaces in
time polynomial in L
PDF - Requires Adobe Acrobat Reader or other PDF viewer.
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Topos Theory
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
The notion of geometric theory has many different incarnations. A few are:
The equivalence of these statements involves some serious proofs, including Giraud's theorem characterizing Grothendieck topoi.
Logical definition
In logical terms, a geometric theory fits into a hierarchy of theories that can be interpreted in the internal logic of a hierarchy of types of categories. Since predicates in the internal logic are
represented by subobjects, in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be
well-behaved. For instance:
Note that the axioms of one of these theories are actually of the form
$\varphi \vdash_{\vec{x}} \psi$
where $\varphi$ and $\psi$ are formulas involving only the specified connectives and quantifiers, $\vdash$ means entailment, and $\vec{x}$ is a context. Such an axiom can also be written as
$\forall \vec{x}. (\varphi \Rightarrow \psi)$
so that although $\Rightarrow$ and $\forall$ are not strictly part of any of the above logics, they can be applied “once at top level.” In an axiom of this form for geometric logic, the formulas $\
varphi$ and $\psi$ which must be built out of $\top$, $\wedge$, $\bot$, $\bigvee$, and $\exists$ are sometimes called positive formulas.
The interpretation of arbitrary uses of $\Rightarrow$ and $\forall$ requires a Heyting category. In fact, by the adjoint functor theorem for posets, any geometric category which is well-powered is
automatically a Heyting category, but geometric functors are not necessarily Heyting functors. Likewise, a well-powered geometric category automatically has arbitrary intersections of subobjects as
well, so we can interpret infinitary $\bigwedge$ in its internal logic, but again these are not preserved by geometric functors.
By the usual syntactic constructions (see internal logic and context), any geometric theory $T$ generates a “free geometric category containing a model of that theory,” also known as its syntactic
category $G_T$. This syntactic category $G_T$ has the universal property that for any other geometric category $G'$, geometric functors $G_T \to G'$ are equivalent to $T$-models in $G'$.
• Any finitary algebraic theory is, in particular, a cartesian theory, and hence geometric. This includes monoids, groups, abelian groups, rings, commutative rings, etc.
• The theory of (strict) categories is not finitary-algebraic, but it is cartesian, and hence geometric; this generalises to (finitary) essentially algebraic theories.
• The theory of torsion-free abelian groups is also cartesian, being obtained from the theory of abelian groups by the addition of the sequents $(n\cdot x = 0) \vdash (x = 0)$ for all $n\in \mathbb
• The theory of local rings is a coherent theory, being obtained from the theory of commutative rings by adding the sequent $(0 = 1) \vdash \bot$, for nontriviality, and
$\exists z. ((x+y)z = 1) \vdash (\exists z.(x z = 1) \vee \exists z.(y z = 1))$
saying that if $x+y$ is invertible, then either $x$ or $y$ is so.
• The theory of fields is also coherent, being obtained from the theory of commutative rings by adding $(0 = 1) \vdash \bot$ and also
$\top \vdash (x=0) \vee (\exists y.(x y = 1))$
asserting that every element is either zero or invertible. In the constructive logic that holds internal to the categories in question, this is the notion of a “discrete field;” other classically
equivalent axiomatizations (called “Heyting fields” or “residue fields”) are not coherent.
• The theory of torsion abelian groups is geometric but not coherent; it can be obtained from the theory of abelian groups by adding the sequent
$\top \vdash \bigvee_{n\gt 1} (n \cdot x = 0)$
asserting that for each $x$, either $x=0$ or $x+x=0$ or $x+x+x=0$ or …. Similarly, the theory of fields of finite characteristic is geometric but not coherent.
• The theory of a real number is geometric. This is a propositional theory, having no sorts, and having one relation symbol “$p\lt x\lt q$” for each pair of rational numbers $p\lt q$. Its axioms
□ If $\max(p_1,p_2) \lt \min(q_1,q_2)$, then if $p_1\lt x\lt q_1$ and $p_2\lt x\lt q_2$, then $\max(p_1,p_2)\lt x\lt \min(q_1,q_2)$. Otherwise, if $p_1\lt x\lt q_1$ and $p_2\lt x\lt q_2$ then $
□ If $p\lt p' \lt q' \lt q$, then if $p\lt x\lt q$, then either $p\lt x\lt q'$ or $p' \lt x\lt q$.
□ If $p\lt x\lt q$, then $\bigvee_{p\lt p'\lt q'\lt q} (p' \lt x\lt q')$.
□ Always $\bigvee_{p\lt q} (p\lt x\lt q)$.
The classifying topos of this theory is the topos of sheaves on the real numbers.
• A geometric theory whose classifying topos is a presheaf topos is called a theory of presheaf type.
Other characterizations
In terms of sheaf topoi
Categories of each “logical” type can also be “completed” with respect to a suitable “exactness” property, without changing their internal logic. Any regular category has an completion into an exact
category (see regular and exact completion), any coherent category has a completion into a pretopos, and any geometric category has a completion into an infinitary pretopos.
However, Giraud's theorem says that an infinitary pretopos having a generating set is precisely a Grothendieck topos. Moreover, a functor between Grothendieck topoi is geometric (preserves all the
structure of a geometric category) iff it preserves finite limits and small colimits. By the adjoint functor theorem, this implies that it is the inverse image part of a geometric morphism, i.e. an
adjunction $f^* \dashv f_*$ in which $f^*$ (the “inverse image”) preserves finite limits.
Grothendieck topoi and inverse-image functors are in some sense the “natural home” for models of geometric theories.
Note, though, that geometric morphisms are generally considered as pointing in the direction of the direct image $f_*$, which is the opposite direction to the geometric functor $f^*$. (This is
because when $E$ and $F$ are the toposes of sheaves on sober topological spaces (or locales) $X$ and $Y$ respectively, then continuous maps $X \to Y$ correspond precisely to geometric morphisms $E \
to F$.) Also, of course any Grothendieck topos is an elementary topos (at least, as long as one works in foundations for which Set is an elementary topos), and hence its internal logic also includes
“higher-order” constructions such as function-objects and power-objects. However, these are not preserved by geometric functors, so they (like $\forall$ and $\Rightarrow$) are not part of geometric
logic. (They are, however, preserved by logical functors, a different sort of morphism between topoi.)
In particular, we can apply the “exact completion” operation to the syntactic category $G_T$ of a geometric theory to obtain an infinitary pretopos $Set[T]$. As long as the theory $T$ was itself
small, $Set[T]$ will have a generating set, and therefore will be a Grothendieck topos. The universal property of the syntactic category, combined with that of the exact completion, implies that for
any Grothendieck topos $E$, geometric morphisms $E\to Set[T]$ are equivalent to $T$-models in $E$. This topos $Set[T]$ is called the classifying topos of $T$.
In the other direction, if $C$ is any small site, we can write down a “geometric theory of cover-preserving flat functors $C^{op}\to Set$.” By Diaconescu's theorem classifying geometric morphisms
into sheaf topoi, it follows that $Sh(C)$ is a classifying topos for this theory. Therefore, not only does every geometric theory have a [[classifying topos], every Grothendieck topos is the
classifying topos of some theory. Very different-looking theories can have equivalent classifying topoi; this of course implies that they have all the same models in all Grothendieck topoi (hence a
Grothendieck topos is the “extensional essence” of a geometric theory). We say that two geometric theories with equivalent classifying topoi are Morita equivalent.
Functorial definition
We can approach the same idea by starting instead from the notions of Grothendieck topos and geometric morphism. The following approach is described in B4.2 of the Elephant.
Suppose we consider what sorts of “theories” we can define in terms of Grothendieck topoi, that are preserved by inverse image functors. Any such theory should certainly define a 2-functor $T\colon
GrTop^{op}\to Cat$, where $GrTop$ is the 2-category of Grothendieck topoi and geometric morphisms, so for the moment let’s call any such $T$ a “theory”. The image $T(E)$ of a functor $E$ is supposed
to be “the category of $T$-models,” and a classifying topos for such a 2-functor will be just a representing object for it.
Of course, this notion of theory is far too general; we only want to consider theories that are constructed in some reasonable way. One theory that should certainly be geometric is the theory of
objects, $O$. This 2-functor sends a topos $E$ to itself, considered as a mere category, and an inverse image functor to itself, considered as a mere functor. The theory $O$ can be shown to have a
classifying topos, the object classifier $Set[O]$. Similarly, we have a theory $O_n$ of $n$-tuples of objects that should be geometric.
How can we construct more theories that ought to be geometric? We should start from some finite collection of objects (i.e. a model of $O_n$), “construct some new objects and morphisms,” and then
“impose some axioms on them.” For any theory $T$, let’s call a transformation $T\to O$ a geometric construct. This is supposed to be “an object constructed out of the axioms of $T$ in a natural way.”
More precisely, to every $T$-model in a topos $E$ it assigns an object of $E$, in a way that varies naturally with morphisms of $T$-models and inverse image functors.
Now define a simple functional extension of $T$ to be the inserter of a pair of geometric constructs $T\;\rightrightarrows\; O$. A model of such a theory will consist of a model of $T$, together with
an additional morphism between two objects constructed out of the given $T$-model. By iteratively applying such constructions, we can add in any number of new morphisms between “constructed objects.”
Finally, define a simple geometric quotient of $T$ to be the inverter of a modification between a pair of geometric constructs $T\;\rightrightarrows\; O$. That is, we require that a certain naturally
defined morphism between objects constructed out of $T$-models must be an isomorphism. (Applying equalizers, we see that this also includes the ability to set morphisms equal, i.e. to construct
From this point of view, a geometric theory is a theory $GrTop^{op}\to Cat$ obtained from some $O_n$ by a finite sequence of simple functional extensions and simple geometric quotients. Of course,
once we know that each $O_n$ has a classifying topos, it follows immediately that any geometric theory has a classifying topos, since $GrTop$ has inserters and inverters.
Hybrid definition
The following definition is sort of a “halfway house” between logic and geometry. Start with a first-order signature $\Sigma$ (this is the logical part). Then we have a 2-functor $\Sigma Str\colon
GrTop^{op}\to Cat$ sending a topos $E$ to the category $\Sigma Str(E)$ of $\Sigma$-structures in $E$. A geometric theory over $\Sigma$ is defined to consist of the following.
• For each Grothendieck topos $E$, we have a replete full subcategory $T(E)$ of $\Sigma Str(E)$, such that
• For each geometric morphism $f\colon E\to F$, if $M\in T(F)$ then $f^*M\in T(E)$ (i.e. $T$ is a subfunctor of $\Sigma Str$), and moreover
• For any set-indexed jointly surjective family $(f_i \colon E_i \to E)_{i\in I}$ of geometric morphisms, and any $M \in \Sigma Str(E)$, if $f_i^* M \in T(E_i)$ for all $i$, then $M\in T(E)$.
The equivalence of this definition with the others can be found in
• Olivia Caramello, “A characterization theorem for geometric logic”, arXiv:0912.1404.
It is not sufficient, in the third condition, to restrict to the case when $I$ is a singleton, but it is sufficient to consider the case when $I$ is a singleton together with all families of
coproduct injections $(E_i \to \coprod_i E_i)_{i\in I}$.
By framing this notion in the internal language of a topos $S$ we can talk of geometric theories over $S$, with models in bounded $S$-toposes (the relative version of “Grothendieck topos”). As a
simple example, if we have a sheaf $A$ of rings on a topological space $X$ we can describe left $A$-modules as models of a geometric theory over $Sh(X)$, the topos of sheaves on $X$, and this notion
is definable in $Sh(X)$-toposes.
Similarly to the $Set$-based case, given a geometric theory $T$ over a topos $S$, we can form the $S$-topos $S[T] \to S$ that classifies $T$, for which the category of $T$-models in a bounded $S$
-topos $E$ is naturally equivalent to the category of morphisms of $S$-toposes $E \to S[T]$. In other words
$T Mod(E) \simeq Top_S(E,S[T])$
Morphisms of theories
Since a Grothendieck topos is the “extensional essence” of a geometric theory, it makes sense to define a map of theories $T \to T'$ to be a morphism of $S$-toposes $h: S[T'] \to S[T]$. Equivalently,
of course, this is a $T$-model in $S[T']$. Composition with $h$ induces a functor, forget along $h$, from $T'$-models to $T$-models in any $S$-topos.
Gros categories of models
Define the gros category? $T Mod$ of $T$-models to have as objects pairs $(E,A)$ where $E$ is an $S$-topos and $A$ is a $T$-model in $E$. A map $(f,g): (E,A) \to (F,B)$ is given by a morphism $f: F \
to E$ of $S$-toposes and a homomorphism $g: f^*(A) \to B$ of $T$-models in $F$. The composition of maps should be evident. A map $h: T \to T'$ of geometric theories over $S$ induces a forgetful
functor $T' Mod \to T Mod$ which leaves unchanged the $S$-topos of residence, which has a left adjoint $T Mod \to T' Mod$ which may change the topos. For if $a: E \to S[T]$ is a $T$-model in $E$,
pulling $a$ back along $h$ yields a $T'$-model, not in $E$ but in the pullback. This is a consequence of general facts about finite 2-limits of the 2-category of bounded $S$-toposes.
Standard textbook accounts include
Discussion in the context of computer science is in
Discussion with an eye towards geometric type theory is in
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A Square Of Side A (all Sides Are Equal Length) ... | Chegg.com
A square of side A (all sides are equal length) has charges +Q and -Q alternating from one corner to the next.
One of the -Q charges in the figure is given an outward "kick" that sends it off with an initial speed v_0 while the other three charges are held at rest.
If the moving charge has a mass m, what is its speed when it is infinitely far from the other charges?
Express your answer in terms of the variables Q, m, v_0, a, and appropriate constants.
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Probability question...at least I believe it is.
April 2nd 2012, 08:30 AM #1
Apr 2012
Probability question...at least I believe it is.
I have a question I could never figure out since I would not even know where to begin.
Here it is,
what is the probability that a Mother and Father and their child would all be born on the 7th of the month but of three different months?
This may not be the proper forum for this question, if not I apologize.
To understand me a little better, I was an art student LOL! Philosophy, art, literature ect....Math, grade ten algebra, is all I achieved. Even Bill Nye the science guy could not teach me basic
physics in childs terms. Sad.... I guess we all can't be good at the same thing :0)
Thank-you for your time
and thank-you in advance if you intend to answer!
Re: Probability question...at least I believe it is.
Let (F, M, C) be a triple where each of F, M and C is a number from 1 to 365 (birthdays of mother, father, and child, respectively). Then the total number of such triples is 365^3. The number of
triples where each number corresponds to the 7th of some month is 12 * 11 * 10. Namely, there are 12 variants for mother; for each of those there are 11 variants for father; and for each of the
first two there are 10 variants for child. If every triple (F, M, C) is equally likely, then the probability is the ratio $\frac{12\cdot 11\cdot 10}{365^3}\approx0.002\%$.
Note that this probability corresponds to a given day of the month (7th or any other fixed date <= 28). The probability that all three have birthdays on the same day of the month in different
months without specifying which day it is will be of course greater (by about a factor of 30).
Re: Probability question...at least I believe it is.
Wow emakarov I actually understood the way you put that!!! :0) Hey maybe I could have done better in math! Thank-you!!
Sometimes it may just take a great teacher! So the answer would be .002%?
Lovely thank-you again so much....after reading some of the other questions here I realized very quickly I was in way, way over my head and did not expect anyone to answer. It seemed to simple.
Again I thank-you
April 2nd 2012, 09:22 AM #2
MHF Contributor
Oct 2009
April 2nd 2012, 11:04 AM #3
Apr 2012
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Communicating in the Language of Mathematics
From IAE-Pedia
Information Age Education (IAE) is an Oregon non-profit corporation created by David Moursund in July, 2007. It works to improve the informal and formal education of people of all ages throughout the
world. A number of people have contributed their time and expertise in developing the materials that are made available free in the various IAE publications. Click here to learn how you can help
develop new IAE materials.
Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems. (George Polya, How to Solve It, a New Aspect of Mathematics.)
Extra Short Summary
Communication in math involves making use of the processes of reading, writing, speaking, listening, and thinking as one communicates with one's self, other people, computers, books, and other aids
to the storage, retrieval, and use of the collected mathematical knowledge of the world. Current precollege math education systems have substantial room for improvement in helping students learn to
communicate effectively in the "language" of mathematics.
Short Summary
Math is a huge discipline with great breadth and great depth. The discipline existed before the development of written languages. As you know, reading and writing—including the development of math
notation and math vocabulary—have contributed immensely to the discipline of math.
Many people consider math to be a language. It is not a general purpose language, such as English or Spanish. Rather, it is a discipline-specific language. Each discipline has developed specialized
vocabulary and its own ways of communication that are specific to the discipline. Consider, for example, music notation and a person learning to sight read music. Perhaps you have seen and heard
examples of the language of basketball or the language of football. On televised broadcasts of games one sometimes gets to see diagrams of plays and hear the language used by the coach to communicate
plays and give directions to players.
A summary of arguments supporting the idea that math is a language is available in Logan (2000) and a more recent discussion is available on the Web. Certainly, math is an area in which one can learn
oral and written communication and can learn to think using the vocabulary, symbols, and ideas of the language of math.
Now, add to this concept the word communication in the field called Information and Communication Technology (ICT). Note that in the United States this field is more often called Information
Technology (IT), while the rest of the world tends to use the term ICT. The communication aspects of ICT open up a whole new dimension in every academic discipline.
Logan (2000) argues that the Internet is a language and that computer programming languages (collectively) are language. Clearly the whole field of Computer and Information Science and its
applications in ICT strongly overlap with the discipline of math. Indeed, in a number of colleges, Computer and Information Science is organized within the Department of Mathematics. Thus, it is
appropriate to consider the language aspects of Computer Science and ICT as we consider communication in math.
Here are some of the more important ideas in this document:
□ Although one can spend a lifetime studying math and still learn only a modest part of the discipline, young children can gain a useful level of math knowledge and skill via "oral tradition"
even before they begin to learn to read and write. Oral and tangible, visual communication in and about math is an important part of the discipline.
□ Reading and writing are a major aid to accumulating information and sharing it with people alive today and those of the future. This has proven to be especially important in math, because the
results of successful math research in the past are still valid today.
□ Reading and writing (including drawing pictures and diagrams) are powerful aids to one's brain as it attempts to solve challenging math problems. Reading and writing also help to overcome the
limitations of one's short-term memory.
□ The language of mathematics is designed to facilitate very precise communication. This precise communication is helpful in examining one's own work on a problem, drawing upon the previous
work of others, and in collaborating with others in attempts to solve challenging problems.
□ Our growing understanding of brain science is contributing significantly to our understanding of how one communicates with one's self in gaining increased expertise in solving challenging
problems and accomplishing challenging tasks in math (and in other disciplines).
□ Information and Communication Technology (ICT) has brought new dimensions to communication, and some of these are especially important in math. Printed books and other "hard copy" storage are
static storage media. They store information, but they do not process information. ICT has both storage and processing capabilities. It allows the storage and retrieval of information in an
interactive medium that has some machine intelligence (artificial intelligence). Even an inexpensive handheld, solar-battery 6-function calculator illustrates this basic idea. There is a big
difference between retrieving a book that explains how to solve certain types of equations and making use of a computer program that can solve all of these types of equations.
□ We all understand the idea of a native language speaker of a natural language. Students learning an additional language will often progress better when taught by a native language speaker who
can fluently listen, read, talk, write, and think in the language, and who is skilled in teaching the language. We prefer that this teacher be fluent in a "standard" version of the language
and not have a local accent and vocabulary that would give pause to many native speakers of the same language. The same idea holds in math education. The math educational system in the United
States is significantly hampered because so many of the people teaching math do not have the math knowledge, skills, and math pedagogic knowledge—and, most important, level of fluency—that
would classify them as being math education native language speakers.
The last bulleted item points to a major weakness in our current math educational system. A great many of our children are being taught math by teachers who are ill prepared to deal with the
complexities of being a successful teacher in this discipline. This topic is addressed in more detail throughout the remainder of this document.
Reading & Writing Across the Curriculum
This document is intended mainly for teachers of math teachers and for preservice and inservice teachers of math. Oral and written communication are recognized as being an important part of the core
of a modern education. All preservice teachers learn about the need for students to learn to read and write—both in general, and within each discipline they study. Reading across the curriculum is a
common theme. Sometimes this expression is taken to mean both reading and writing, but often writing does not receive as much emphasis as reading.
Many students find it is quite difficult to reach or exceed contemporary standards in reading and writing in a natural language such as English. Even for those who graduate from high school and go on
to higher education, reading and writing can still be major challenges. Many students find that they have to undertake remedial work in this area when they enter college, trade school, or an
“Contemporary standards” are usually set so that a significant percentage of students do not meet the standards. Failure to meet standards is often then considered to be the fault of the teacher not
teaching well enough or the student being lazy.
I am particularly interested in students learning to read and write in the discipline of mathematics. For example, I am interested in whether typical students learn to read math well enough so they
can use this skill to learn math by reading their textbooks and other books. I am interested in whether students learn to write in the discipline of math well enough so they can communicate
mathematically (in writing) with themselves, their teachers, and others. An immediate complication is that many people learn more easily through discussion, demonstration, or guided practice than by
As I have explored this topic, I have done some comparing and contrasting with reading and writing in other areas, such as music and chess. I have also thought about how Information and Communication
Technology (ICT) is affecting or should be affecting reading and writing across the disciplines. ICT allows experiences akin to personal discussion, demonstration, and guided practice.
While this document focuses on communication in math, it draws on ideas from other curriculum disciplines and lays groundwork for other people to explore the topic of communication in other
disciplines. It also explores current and potential impacts of ICT on communication in math and in other disciplines.
Writing Math to Learn Math
The general idea of writing to learn cuts across the curriculum. There is a substantial amount of research and practitioner knowledge about having students write math in order to learn math.
A 5/19/08 search of "writing to learn" returned abut 90,000 hits. A search on writing to learn math returned about 20,000 hits. Here are some examples of available articles.
Totten, Samuel (2005). Writing to Learn for Preservice Teachers. National Writing Project. Retrieved 5/19/08: http://www.nwp.org/cs/public/print/resource/2231.
The paper is based on a survey study of 104 teacher education programs spread across the United States. The focus is on all preservice teachers learning more about the process of writing across the
curriculum. Here is a paragraph quoted from the paper:
The fact that only four respondents require all preservice candidates to take a separate course in process writing indicates that faculty of many colleges of education do not see the value in
a course that focuses on writing and writing-to-learn strategies. That another four require such a course only for English teachers points to a belief that a separate course cannot be
justified for teachers across the curriculum. Some faculties may still be uninformed about how writing is an integral tool for assisting students to comprehend more deeply and clearly what
they are studying; they may be unaware of the research that underscores the value of incorporating writing-to-learn strategies in every discipline. [Bold added for emphasis.]
Jones, Alexandria—Pseudo name (8/21/08). Writing to Learn Math: Let's Play Math. Retrieved 5/19/08: http://letsplaymath.wordpress.com/2007/08/21/writing-to-learn-math/. Contains a nice assortment of
links to writing to learn math materials.
Options for Writing in Math (n.d.). (Adapted from Marilyn Burns, Writing in Math Class, Math Solutions Publications, 1995.) Retrieved 5/25/08: http://www.springfield.k12.il.us/resources/math/
El-Rahman, Madiha (n.d.) The Effects of Writing-to-Learn Strategy on the Mathematics Achievement of Preparatory Stage Pupils in Egypt. Retrieved 5/19/08: http://math.unipa.it/~grim/
EEl-Rahman26-33.PDF. Here are a couple of examples of activities quoted from the paper:
Pair Share This is a very simple activity to use when the teacher senses that the student does not understand the lesson. He stops and asks them to explain what is giving them trouble. After
the students “Free - write“ for a couple of minutes, they share their writing with their classmates. This can help to remove their confusion (Burchfield and others, 1993).
Journal Writing This is a diary-like series of writing assignments. Each assignment is short and written in prose rather than in the traditional mathematical style. The students can write in
their journals: daily goals, rational for learning any concepts, and the strategies used to solve problems (Bagley, 1992: 660). It can give both the teachers and the students great insight
into a student’s progress (Potter, 1996: 184).
Russek, Bernadette (n.d.). Writing to Learn Mathematics. Retrievedd 5/25/08 from http://wac.colostate.edu/journal/vol9/russek.pdf.
Correlation Between Language Arts and Mathematics
Boaler, Jo & Staples, Megan (2008). Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School. Teachers College Record Volume 110 Number 3, 2008, p. 608-645.
Retrieved 12/8/09 from http://www.tcrecord.org/Content.asp?ContentID=14590.
This research study looked at the teaching of high school math in three different schools over a period of several years. The researchers were particularly interested in the effects of not grouping
students by math ability or previous math performance. Thus, in one of the high schools, all first year high school students began with first year high school algebra and then all proceeded to
geometry. Based on the data from the three high schools that were studied, the argument is that this is a good thing to do.
Here is an interesting tidbit quoted from the Boaler and Staple study:
The correlation between students’ scores on the language arts and mathematics sections of the AYP tests, across the whole state of California, was a staggering 0.932 for 2004. This data point
provides strong indication that the mathematics tests were testing language as much as mathematics. This argument could not be made in reverse as the language tests do not contain
mathematics. [Bold added for emphasis.]
This very high correlation seems to be supportive of Keith Devlin's assertion (in his book, The Math Gene) that anybody who has the brain power to become literate in a natural language can learn
Sparks, Sarah D. (2/17/2011). Studies find language is key to learning math. Education Week. Retrieved 2/19.2011 from http://www.edweek.org/ew/articles/2011/02/17/21math.h30.html?tkn=
RMVFed6tRXiQd1a8OyriYPqFy8qmkJgPYe0F&cmp=clp-sb-ascd. Quoting from the article:
New research shows a lack of language skills can hamstring a student’s ability to understand the most fundamental concepts in mathematics. A series of studies led by Susan Goldin-Meadow, a
psychology professor at the University of Chicago, found that profoundly deaf adults in Nicaragua who had not learned a formal sign language could not accurately describe or understand
numbers greater than three. While hearing adults and those who used formal sign language easily counted and distinguished groups of objects, those who used only self-created “homesigning”
gestures could not consistently extend the proper number of fingers to count more than three objects at a time, nor could they match the number of objects in one set to those in another set.
Quoting from the Susan Goldin-Meadow article:
Does learning language change the way we think about number? The exact quantities to which words like “seven,” “eight,” and “nine” refer seem so basic it is hard to imagine that we might need
the word “seven” to have the concept seven. But evidence from groups who have not been exposed to conven- tional numerical systems suggests that language, particularly the numeral list in a
count routine, may be importantly involved in the ability to represent the exact cardinal values of large sets.
The Mundurukú (1) and Pirahã (2) are Amazonian people in rural Brazil whose languages do not contain words for exact numbers larger than five (the Mundurukú) or any exact number words at all
(the Pirahã).* Adults in these cultures have not been reported to invent ways to communicate about the large numbers for which they do not have words. In addition, these groups do not display
a robust ability to match sets exactly with respect to number, except when a one-to-one correspondence strategy is readily available (e.g., pairing each object in one visible row with a
corresponding object in a second visible row) (3). The absence of a linguistic model for representing exact number (in this case, a count list) could explain the difficulties Pirahã and
Mundurukú adults have representing large exact numbers (2, 4–6). However, their difficulties could just as easily be explained by the absence of culturally supported contexts in which exact
number must be encoded (7, 8). To disentangle these possibilities, we investigated the numerical abilities of individuals who lack a linguistic model for number but who live in the numerate
culture of Nicaragua: “homesigners.”
Specific Problem
The specific problem situation being addressed is that U.S. precollege math education is not as good as most American taxpayers would like it to be. It definitely is not as successful as math
education in a number of other countries. Possible reasons are many, and many people are dedicated to improving the American system.
Considerable literature addresses the problems of math education and how to improve math education. This IAE-pedia contains some of these documents.
Relevant substantial research has been done in many disciplines such as math, music, chess, etc. K. Anders Ericsson is a world leader in this research field. The link is to a short article on
expert’s long-term working memory that summarizes some of the key ideas in teaching and learning. All teachers (and, indeed, all students) can benefit by having some knowledge of this field.
The document you are currently reading focuses specifically on communication in math. This includes looking at some related aspects of ICT, brain science (chunking and expertise especially), and
empowering students and their teachers.
This document is written specifically for use in preservice and inservice math education courses and workshops. The focus is on the idea that the discipline of math includes a language that we call
the language of mathematics. We want students to learn to read, write, speak, listen, and think creatively in the language of mathematics. In essence, the goal is for students to become
mathematicians at the level of the math they have studied. We want them to learn to use effectively their math content knowledge and skill to solve challenging problems and accomplish challenging
tasks that are amenable to effective use of the math they have studied—not to mention their learning the math skills they will need to navigate through all the financial and life-decision hazards and
opportunities they will face.
The goal of this document is to encourage and support discussion and deep thought followed by constructive action. A good use of this document in a preservice or inservice teacher education course
would be to have students read it in advance of a class meeting and form their personal opinions on some of the ideas. During class, students would then share their insights and ideas in small group
and whole class discussion. A follow-up activity might be having students continue the discussion in an online environment, write about this article in their math journals, do research on one
specific idea in the article that came up during the in-class discussions, or develop some instructional materials that could be used to help implement their ideas.
To support the intended use, from time to time this document contains a question suitable for personal reflection or for discussion in a workshop or class. Here is an example:
For reflection and discussion: Drawing upon your knowledge of yourself and other people you know, analyze your levels of expertise in the areas of reading, writing, speaking, listening,
thinking, and problem solving in mathematics. You might find it helpful to use the terms fluency and/or expertise in doing this analysis. Identify your relative strengths and weaknesses.
Think about how our math education system contributed to your relative strengths weaknesses. What does your analysis suggest in terms of possible ways to improve our math education system?
Be tolerant of your first draft; you will gain many ideas when you and your colleagues share your first attempts.
Introduction to Reading and Writing
The development of reading and writing about 5,200 years ago was a major milestone in human history. From then on, there has been an accelerating pace of change in societies of the world brought on
by the accumulation and sharing of data, information, knowledge, and wisdom. Major accelerating inventions in this process are the printing press, electronic communication (telegraph etc.), the
computer, and the Internet and Web. The totality of this accumulation is huge and currently is perhaps doubling every five to ten years. The Web, all by itself, is a virtual library with many times
the content of the largest physical libraries on our planet. It is continuing a rapid pace of growth. More content is added each day than a typical person can read in a lifetime.
The development of general-purpose written languages brought with it a start in the development of a written language for the discipline of mathematics. Over thousands of years, the discipline of
mathematics and its language have grown and matured. Math educators support the idea of students learning to read, write, speak, listen, and think creatively in the language of mathematics. All of
these aspects of communication contribute to representing and solving math problems.
Developing fluency (read, write, speak, listen, understand, and think) in the language of mathematics certainly has some similarities to doing the same thing in a natural language such as English.
However, there are considerable differences, and some are discussed in this article.
An example of a similarity is provided by the challenge a high school student or adult faces trying to learn a foreign language. It is not too hard to memorize a large number of words and phrases.
However, many second language learners of high school age and older find it is hard to learn to think and gain verbal fluency in a foreign language. Research strongly supports the value of starting
to learn a second language at a much younger age and being taught by native language speakers who are skilled teachers.
For reflection and discussion: You know that children are exposed to math as they gain oral communication skills well before they start school. Our school system starts formal math
instruction at the earliest grade levels. Thus, we cannot attribute our lack of success in math education to not starting early enough! Why do you suppose that this large amount of math
instruction over many years produces such poor results for many students?
For reflection and discussion: There are many disciplines that school students are capable of learning. What is there about mathematics that justifies the many years of math coursework
required of all students? Think of possible arguments for having less required coursework in math, thereby freeing up time for more coursework in other areas that might be of more specific
interest and importance to some students. In thinking about and discussing this topic, try to give examples of recent times when it has been beneficial to you to draw upon your knowledge of
high school algebra and geometry to deal with problems outside of a school setting.
Learning to Read and Write in a Natural Language
The undamaged human brain is genetically "wired" for learning oral and visual communication. Children learn to understand, talk, and think in spoken language supplemented by gestures long before they
reach kindergarten. They learn whatever language or languages commonly used in their environment. Thus, children growing up in a bilingual or trilingual home and community environment will become
orally bilingual or trilingual.
When children start on the process of learning to read and write, they already have a substantial level of oral fluency. Young children are displaying a high level of creativity and intelligence as
they communicate orally. Learning to read and write draws heavily on the ability to create meaningful utterances and understand spoken language.
Young students also have a substantial and growing knowledge of the world. This often provides help in discerning the meaning of a sequence of words or a sentence. Students have considerable ability
to extract meaning from context and from pictures. Pictures in story books help students in extracting meaning from the written presentation.
As a child moves through the first few grades of elementary school, the child continues to gain verbal fluency. A combination of informal learning outside of school and the formal schooling adds
thousands of words per year to the child's oral fluency repertoire. This steadily growing oral fluency provides a growing foundation for building fluency in reading and writing.
Being around those whose oral and written language fluency is quite a bit greater than the child’s substantially aids the process. Think of this as role modeling. The child can observe and hear oral
communication being routinely used. As a child attempts to imitate and participate in this oral communication; immediate feedback is provided by proficient speakers of the language.
In many homes, young children are read to frequently. Research strongly supports that this and other adult role modeling in reading and writing makes a major contribution to children's future
linguistic development.
Even with a strong supportive background, most students take many years of instruction and practice to develop a level of reading and writing expertise that meets contemporary standards. Thus, most
colleges and universities require entering freshman to take a year sequence in writing. In the U.S., this course is often called English Composition.
The idea of "contemporary standards" is important. magazines and newspapers are written at or below 10th grade reading level. A great many adults who graduated from high school have considerable
difficulty reading above this level. (This document’s Flesch-Kincaid readability is approximately 10th grade level. The Gunning Fog index suggests high school completion is needed for reading the
Similar observations have been made about average adult performance in other areas. For example, many adults who graduated from high school function in math at about the 6th to 7th grade level.
Such observations point to the major difference between standards that governments and others want to set, and what is readily achievable by our current educational system. The educational leaders in
each academic discipline have created "standards" that they feel students should achieve. While each discipline's standards may appear to be reasonable or desirable when viewed individually, the
collected set of standards far exceed what an ordinary student can achieve, possibly in part because the experts may overestimate the needs of the general public. For example, consider the terms a
capella (music), undecidability (math), and zugzwang (chess)—all of which have ‘real-life’ implications.
Moreover, while students demonstrate they have achieved a standard by passing a particular test, the reality is that forgetting occurs (in many cases, quite rapidly) so that even in a test-based
standards system, relatively few people continue to meet the standards as they become adults.
For reflection and discussion: Why do you think it is so hard to learn to be a good writer, when it is relatively easy to learn to talk in a manner that meets contemporary standards? (Hmm.
Does an average high school graduate meet the oral fluency standards that our schools would like to set?) Next, think about the same question for learning math.
For reflection and discussion:We know that a person's knowledge and skills in an area degradates over time if the knowledge and skills are not being used. A different way of saying this is
that students forget much of what they (supposedly) learn in school. Think about some personal examples. In what ways does our educational system acknowledge that people forget, and attempt
to accommodate the forgetting?
Some Brain Theory: Seven Plus or Minus Two
Written and oral language are aids to thinking. Thinking is sometimes described as "talking silently to oneself." Such thinking allows a person to contemplate various actions and possible outcomes of
the actions—without actually carrying out the actions. You probably know some people who "think out loud." In addition, having s research subject talk aloud while solving a problem is a useful
research technique to gain insight into a person's thinking process during problem solving.
This section provides some general information about the human brain and some roles of language in thinking and problem solving.
Humans have three types of memory:
□ Sensory memory stores data from one’s senses, and for only a short time. For example, visual sensory memory stores an image for less than a second, and auditory sensory memory stores aural
information for less than four seconds.
□ Working memory (short-term memory) can store and actively process a small number of chunks. It retains these chunks for less than 20 seconds.
□ Long-term memory has a large capacity and stores information for a long period of time. Over time, information stored in long-term memory tends to become more and more difficult to
remember—that is, to retrieve—if it is not used very often. However, traces of these stored memories continue to exist, and they can be an aid as one relearns what was learned in the past.
When you work to solve a problem, you bring information and ideas about the problem into your working memory. You consciously manipulate this information and ideas. Research on working memory
indicates that for most people the size of this memory is about 7 ± 2 chunks (Miller, 1956).
This means, for example, that a typical person can read or hear a seven-digit telephone number and remember it long enough to key it into a telephone keypad. The word chunk is very important. For
example, the sequence of four digits 1 4 9 2 can be thought of as four distinct chunks. However, it can also be thought of as one chunk—the year when Columbus discovered America. It can also be
represented as two chunks-14 and 92. The point is that appropriate chunking of ideas and information is a powerful aid to overcoming limitations of short-term memory.
The names of the number words in Chinese are, on average, shorter than the corresponding names of the number words in English. In terms of digit recall, 7 English digits are about the same length as
9 Chinese digits. Native language speakers of Chinese have a greater short term memory digit span than native language speakers of English. The referenced article is about the research work of
Stanislas Dehaene.
Your brain is very good at learning meaningful chunks of information. Think about some of your personal chunks such as constructivism, multiplication, democracy, complex numbers, transfer of
learning, and Mozart. Undoubtedly these chunks have different meanings for me than for you. Moreover, our chunks are of different size. Research indicates that experts in a discipline have more
chunks and much larger chunks (in their discipline) than do novices.
As a personal example, my chunk “multiplication” covers multiplication of positive and negative integers, fractions, decimal fractions, irrational numbers, complex numbers, functions (such as
trigonometric and polynomial), matrices, and so on. My breadth and depth of meaning and understanding were developed through years of undergraduate and graduate work in mathematics. Others might
connect “multiplication” with the pressure of having to learn the multiplication table before a test or recall “go forth and multiply—a paraphrase of various biblical phrases—and related jokes.
Here is another example. You "know" what the number line is. When you think about the number line, your mind probably conjures up some sort of picture, perhaps a line with equally spaced marks on it,
and the marks labeled with digits such as … -4, -3, -2, -1, 0, 1, 2, 3, 4, …. You can think of this as a chunk. My number line mental chunk is not the same as yours. Through years of studying and
using math, my mental math number line chunk has grown to include rational numbers and irrational numbers. It has grown to include irrational numbers that are called transcendental numbers. Moreover,
my number line chunk is closely tied in with chunks about numbers in different bases, different sizes of infinity, some results from the area of math called number theory, complex numbers, and other
components of math. You probably modified your conception of “number line” as soon as you were reminded of these other numbers; I remind myself of these numbers as I’m visualizing a number line.
It is useful to think of a chunk as a label or representation (perhaps a word, phrase, visual image, sound, smell, taste, or touch sensation) and a collection of pointers. A chunk has four important
1. It can be used by short-term memory in a conscious, thinking, problem-solving process.
2. It can be used to retrieve more detailed information from long-term memory.
3. It serves as an anchor for constructing new knowledge and skills. (It lies at the root of constructivism learning theory.)
4. It is a key to higher levels of expertise in a discipline. High-level experts in a discipline have a large repertoire of chunks in that discipline. They think and solve problems making use of
these chunks. Furthermore, as proficiency in a disciple increases, chunks become bigger. Such a larger chunk can include both information about a problem situation and possible actions to
take in attempting to solve the problem.
In terms of communication, chunks and chunking are a critical aspect of how one's working memory communicates with one's long-term memory. A chunk may have a name. As indicated above, the name
"multiplication" allows my short-term memory to access (retrieve) a large "multiplication" chunk in my long-term memory. For another example, the name of one of your friends allows your short-term
memory to access a chunk of information about your friend.
However, seeing your friend in person, seeing a picture of your friend, of smelling a particular smell can also trigger this information retrieval. No words are used in this retrieval process. This
indicates we all have and use an extensive non-verbal language. And, of course, you are familiar with the language of gestures. This language can be very extensive. Think, for example, of American
Sign Language.
There has been substantial research on roles of building and using chunks in gaining a high level of expertise in a discipline. One idea that has emerged is that, in some sense, high-level experts
are able to use such chunks as a kind of an extension of their working memory. That is, within the area of high-level expertise, these experts are able to function as if they had a working memory
that is much larger than “normal.” For instance, driving a route new to you in a city requires concentration. As you become familiar with the route, you are able to mentally construct the route so
that you automatically allow for speeds, lane changes, possible trouble, etc., and you are able to devote much of your consciousness to other matters.
In brief summary, creating, storing, and using chunks of information are essential to building a high level of expertise in an area. Such chunking ties in with oral, written, and nonverbal
communication and thinking, and it is applicable in every academic discipline. Expertise in a discipline is dependent on having an extensive repertoire of large chunks specific to that discipline.
However, there are many chunks that have interdisciplinary use and value. Suppose, for example, that a person develops a high level of expertise in understanding and making use of careful, logical,
rigorous arguments in a discipline such as math. Many of the chunks involved in this type of problem solving in math carry over to other disciplines such as the sciences and law.
For reflection and discussion: Think about some discipline in which you have a reasonably high level of expertise. Identify some chunks in your brain that you use in this discipline. You know
that many other people who have lower expertise in this discipline lack entirely or have much less robust chunks. Also, think about whether you make use of this chunk is other disciplines.
Learning Mathematics
The healthy human brain is genetically wired for learning some math and math-related knowledge and skills. Howard Gardner has identified logical/mathematical and spatial as two types of
intelligences. Spatial intelligence can be quite important in attempting to solve some types of math problems. Learn more about Howard Gardner at http://iae-pedia.org/Howard_Gardner.
Very young infants have a little number sense, such as being able to distinguish between two of an object and three of an object. Recent research suggests that perhaps this is an innate ability of
infants to sense that something is wrong when they are expecting to see two objects and are presented with one or three of the objects. Stanislas Dehaene, who was mentioned earlier in this article,
is a world leader in this type of math-related brain research.
A research experiment might involve showing an infant two objects set on a small stage. A screen comes down in front of the stage and then goes up. There seems to be an innate expectation that the
number of objects will not have changed. Researchers can time the increased eye time fixation of a viewer when a change occurs.
Toddlers who can crawl readily learn to orient themselves in their spatial environment, finding their way around different parts of a house. Such spatial skills are essential to a hunter-gatherer
life style in which people had to forage for food and then find their way back to their clan.
Now, think about a child learning words for numbers. As an example, I have a young grandson who is quite bright. At an early age he could say in order the words one, two, three, … up to about
sixteen. However, his understanding of these words was quite limited. At the time, he had some working understanding of one and two, and perhaps three.
There is a large difference between being able to say words and having an understanding of what they mean. This, of course, is true for both math words and non-math words. The above example suggests
that quantity is a relatively abstract idea that is a challenge to learners.
Attributes such as color, size, shape, numerosity (number, quantity) and so on are all learning challenges. Numerosity and other math-related words and concepts have the added challenge in that our
contemporary standards tend to expect a high level of perfection.
By the time an average child enters the first grade, the child has developed a reasonable level of skill in using the number counting words to be able to determine and say the number of objects in a
small set. The child can do simple additions, such as 2 + 5 through a process of counting. Quite a few children have learned counting on either through their own discovery or through being explicitly
taught by the time they begin the first grade.
This is a major step in learning math and allows relatively young children to do math at a higher level than people growing up in a hunter-gatherer society whose natural language is mathematically
quite limited.
Stanislas Dehaene has shown that the ability to estimate amounts—an innate 'number sense' that human beings have in common with various other species—forms the basis for our mathematical
(abstract reasoning) and arithmetic (calculation) abilities. The latter ability does, however, require a well-developed system of symbols—a language system. Evidence for this duality has been
found not only in scientific experiments but also in anthropological research. One example is the language of the Amazonian Mundurukú tribe, which has words for numbers only up to five. The
Mundurukú are not able to perform precise calculations with larger numbers, but they can approximate and compare larger amounts.
Thus, the average child starting school has a beginning level of understanding of the number line. However, the number line is a quite complex math concept. We expect students to learn about
fractions and decimal fractions. We expect students to learn about both positive and negative numbers. We expect students to learn to perform arithmetic on the various types of numbers on the number
line. If we go back 4,000 years, only the most learned mathematicians of their time could effectively handle the range of math we are expecting grade school students to learn.
For reflection and discussion: One difference between natural language and the language of mathematics is the degree of precision required in communication. In many situations, small errors
in the use of natural language do not destroy the overall correctness or effectiveness of a communication. Explore this idea and its math education implications.
Oral Tradition
This section is a work in progress.
To a very large extent, math is taught using methods that might be described as "oral tradition." Students learn to recognize some math symbols and math words. However, many do not learn to read math
at a level that allows them to learn math by reading math. It is only when students reach the more advanced high school math courses that there is a significant emphasis on learning the math by
reading the math book and other resource materials. Thus, a great many students graduate from high school with a very limited ability to learn math by reading a math book or other math resource
Math education makes extensive use of "word" or "story" problems. Here, a problem that can perhaps be represented and solved mathematically is presented in a natural language statement that may
contain few or perhaps no math symbols and vocabulary. A student must meet the challenge of understanding the problem, translating it into math notation and vocabulary, solving the math problem, and
translating the results back into the context of the original natural language statement of the word problem.
This is an important component of math education, since many of the problems that can be addressed using math are not explicitly stated in math notation. Indeed, we now have very powerful Computer
Algebra Systems that can solve a very wide range of problems that are stated in math notation. Thus, a major challenge in math education is to prepare students to deal with the tasks of determining
when math might be useful in solving a problem and in representing such a problem using math notation.
Comment by Jen Jensen 4/28/09
The following is quoted from an email message sent to the National Council of Supervisors of Mathematics on 4/28/09:
One of the most difficult components to today's problem-based texts such as CMP or Core Plus is the reading component. I have run numerous collaborative coaching cycles with math teachers
over the last two years with the goal of understanding how to implement effective reading strategies in a math classroom. These usually involve reading teachers and their expertise is always
beneficial. I also look back to one of the original goals of public education and see that our charge is to create an educated citizenry that can participate intelligently in the democratic
This being said, I believe that English and Social Studies teachers are critically important in the education of our children and should be paid at the same level of math and science
teachers. To me the issue is not the subject being taught but the quality of the teacher doing the teaching. Highly qualified does not mean quality. We need a process for raising the quality
of our teaching force-such as the coaching model, and then we need to remove those teachers who refuse to participate and improve their practice.
Ken Jensen
Instructional Math Coach
Aurora Public Schools
Native Natural Language Speakers and Native Math Language Speakers
When children grow up in a bilingual or trilingual natural language environment, they grow up bilingual or trilingual. This idea is often incorporated into schools. Some students get to attend a
bilingual elementary school in which the content areas are taught in the student's second or third language. It is highly desirable that the teachers teaching the content areas be native language
speakers of the language(s).
We all understand the idea of a native language speaker of a "standard version" of a natural language. We expect the native language speaker to think in the language, know the culture of the people
who speak the language, and have a native accent. Moreover, we prefer that this person not have a strongly regional accent and vocabulary. We want learners to be learning a relatively standard
version of the language and with a relatively standard accent.
Now, take this idea and carry it over into math education. What might we mean by a "native math language speaker" of mathematics?
First, consider the following quote from George Polya, a world-class math educator and math researcher. In a talk to elementary school teachers, Polya said:
To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher
aims about which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems,
which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the
foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this
or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. (Polya, 1969)
In summary, the term native math language speaker means someone who has a high level of fluency in reading, writing, speaking, listening, thinking, and creative problem solving in the discipline of
mathematics. A native math language speaker knows the culture of mathematicians. In short, a native math language speaker is a mathematician.
In the remainder of this article, I will use the term mathematician interchangeable with a native math language speaker.
However, think about an average 6-year-old, an average 12-year-old, and an average 18-year-old native speaker of a language such as "standard" English. The 6-year-old has substantial oral
communication and thinking capability in the language, but has a relatively limited vocabulary and is just getting started in learning reading and writing. The 12-year-old has a much larger
vocabulary, is still better at oral communication, and has made significant progress in reading and writing. The 18-year-old is still more skilled at reading, writing, speaking, listening, and
thinking in his or her native language.
When applied to math, observation leaves us with the situation in which a person might be a high school graduate but mathematically function like an average 12-year-old. In some cases we have
6-year-olds who mathematically function like average 12-year-olds.
Here is one more piece of the math education puzzle. A person who is certified as a teacher has attended 16 or more years of schooling starting at the first grade or earlier. This person knows a
tremendous amount about general pedagogy (how to teach), only a small fraction of which was learned in teacher education courses. This person also knows a tremendous amount about how to be a math
teacher, and yet may have taken only one or two courses in math pedagogy. Learning on the job is a very important part of becoming a good teacher.
The typical elementary school teacher is a "math teacher" as well as being a teacher of other disciplines. How can we tell if this person is a qualified "math educator?"
Let's take the specific case of an average elementary school teacher who is responsible for teaching math as well as a number of other subjects. We can talk about the number of years of coursework
(precollege and college) experience the teacher has had being explicitly taught in math content, being explicitly taught in math pedagogy, being exposed to general pedagogy, and being exposed to math
This is a work in progress, and the question just asked is a difficult one to address. We might be able to quantify the situation with a statement such as, "This teacher functions at the
level of an 8th grade mathematician, an average 10th grader in math pedagogy, and an average graduate of an elementary teachers' college program in general pedagogy."
The levels of the three different measures of qualification will change as the teacher learns on the job through classroom experience and staff development. We know that, on average, teachers
gain considerably in their overall levels of effectiveness during their first half-dozen years on the job.
I leave this topic for now, still not having given a specific definition of what we mean by an appropriately qualified (well-qualified) math educator for some specific category of students.
For example, a person who is a well-qualified math educator for learning disabled students might not be as well qualified in working with talented and gifted students, and vice versa. I have
suggested (recommended) that this person needs to be a "native language speaker" in math content, math pedagogy, and pedagogy. But I have not specified the grade levels or age levels or other
measures we want to use in each area. I have not specified what might be meant by "standard math," or "standard math education."
Substantial research supports the contention that a major weakness in our precollege math educational system is the relative weakness in the math pedagogy and math problem-solving
capabilities of many teachers of math.
A Personal Story
Being a native math language speaker means that one can "do" mathematics. It means having a high level of expertise in solving math problems, recognizing problem situations in which math is apt to be
useful, having quite a bit of math knowledge and skills, being able to use one's math knowledge and skills, and being comfortable in the culture of mathematicians. It takes many years of time and a
considerable amount of effort to become a native math language speaker (that is, to be come a mathematician).
Here is a personal example. I grew up in a home where both my mother and father taught math in a university. Each was a mathematician. Thus, merely by growing up in this environment, I was given a
large boost in moving toward becoming a native math language speaker. However, much more was required.
I did well in math in elementary and secondary school. I then went on to college and majored in math. By the time I finished a bachelor's degree in math, I had a good start on being a mathematician.
Four years of graduate work, resulting in a doctorate in math, certainly qualified me to be considered a native math language speaker. I was fluent, with a high level of expertise, in reading,
writing, speaking, listening, creatively thinking, and problem solving in math. My four years of graduate work essentially constituted a math immersion program, with all of the teaching being done by
highly-qualified mathematicians with a high level of fluency and competence in math.
However, I was not a mathematics educator. I had very little teaching experience during my graduate work, and I had no specific instruction on how to teach math. I had some math teachers who were
more effective than others. Indeed, some of them satisfy my definition of math educator. Others were clearly much more engrossed in their research. Some were both excellent math educators and
excellent math researchers.
Nor was I a math historian. Sure, I had learned a little math history as I studied math. However, my level of expertise in this component of the discipline of math was minimal.
One Major Math Education Problem
We have now come to one crux of a difficult education problem. Most children do not grow up in a home environment of native math language speakers (that is, mathematicians). Moreover, most students
do not have their elementary school math taught by native math language speakers. Indeed, even in their middle school and high school math courses, many students are not being taught by people who
would be considered to be native math language speakers. It tends to be the students who take the more advanced math courses who are taught by mathematicians.
One way to attack this problem is by departmentalizing the teaching of math at the grade school level, and requiring that teachers of math at every grade level have at least a bachelor's degree in
mathematics. Some of the countries that do well in international math education comparisons do take such an approach.
Another approach is to have math teachers at all grade levels take rigorous and demanding math education workshops and summer courses, year after year after year. Some teachers do this, and indeed
develop into native math language speakers.
Note that the goal is not to make such teachers into research mathematicians. Instead, it is to make them into "expert level" math education mathematicians who specialize in teaching math to specific
groups of students.
A third approach would be to place much greater emphasis in the math curriculum on students learning to read, write, speak, listen, and creatively think and solve challenging problems in math. Such
ideas are often emphasized in both preservice and inservice education for math teachers.
There is a strong parallel between good math communication and the teaching of reading and writing in a natural language such as English. In teaching writing in a natural language, there is a great
need for the teacher to read what a student is writing and provide feedback on the content. How well is the intended "message" being conveyed by what the student has written? Of course, teachers also
provide feedback on spelling and grammar. But feedback on the content is essential. So it is with math. But, especially at the elementary school level, the math feedback is usually focused only on
the correctness of an answer.
The following reference presents research on the mathematics that a third grade teacher needs to know.
Ball, Deborah Loewenberg, Heather C. Hill, and Hyman Bass (Fall 2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?
American Educator. Retrieved 7/13/2010 from http://www.aft.org/pdfs/americaneducator/fall2005/BallF05.pdf.
Compare and Contrast with Music Education
I find it helpful, when thinking about educational strategies in one discipline, to compare and contrast that approach with strategies in other disciplines. I often have selected music and chess as
disciplines to compare and contrast with math. This section looks at the language of music and the next section looks at the language of chess.
Music is one of the nine multiple intelligences identified by Howard Gardner. See http://iae-pedia.org/Howard_Gardner. While people vary in their musical "IQ," most people can gain a relatively high
level of expertise in music if they have appropriate opportunities and interest. Most people gain a useful and enjoyable level of music expertise just through their informal exposures to music and
opportunities to sing. A "professional" high level of expertise takes many thousands of hours of good instruction and practice.
This same situation exists for the multiple intelligence that Howard Gardner labels as logical/mathematical. Both music and math are, at some level, built into our genes. But it takes a great deal of
study and practice to develop this innate capacity to meet contemporary standards of expertise.
Making music includes singing, chanting, humming, whistling, rhythmic clapping, use of drums and other musical instruments, creating music, composing new music, and so on. Music existed long before
the development of reading and writing. Music was part of the environment that children grew up in tens of thousands of years ago. Musical knowledge and skills passed from generation to generation
via children growing up in a musical environment of role models and being expected to participate.
Oral tradition and making and using musical instruments served the discipline of music well for many thousands of years, up until quite recent times. Even after reading and writing were invented,
very few people learned to read and write. Moreover, even for people who did learn to read and write, a decent system of musical notation for the reading and writing of music had not yet been
invented. The invention of a good written language for music took many thousands of years. Quoting from the Wikipedia:
Scholar and music theorist Isidore of Seville, writing in the early 7th century, famously remarked that it was impossible to notate music. By the middle of the 9th century, however, a form of
notation began to develop in monasteries in Europe for Gregorian chant, using symbols known as neumes; the earliest surviving musical notation of this type is in the Musica disciplina of
Aurelian of Réôme, from about 850. There are scattered survivals from the Iberian peninsula [sic] before this time of a type of notation known as Visigothic neumes, but its few surviving
fragments have not yet been deciphered.
The ancestry of modern symbolic music notation originated in the Roman Catholic Church, as monks developed methods to put plainchant (sacred songs) to paper. The earliest of these ancestral systems,
dating from the 8th century, did not originally utilize a staff, and used neum (or neuma or pneuma), a system of dots and strokes that were placed above the text. Although capable of expressing
considerable musical complexity, the dots and strokes could not exactly express pitch or time. This system served mainly as a reminder to one who already knew the tune, rather than a means by which
one who had never heard the tune could sing it exactly at sight.
We are all used to the term "musician" to describe a person who has a high level of knowledge and skill in some parts of the discipline of making music. Music is such a large discipline that no one
can gain a high level of expertise over the entire discipline. A world class opera singer need not know how to play a violin, compose music, or direct an orchestra. However, world class singers,
violinists, composers, and conductors are all immersed in the culture of music and each can communicate effectively in the language of music.
Many musicians have a high level of expertise as music teachers. Indeed, perhaps because music instruction tends to be performed by good music educators, many musicians learn a great deal about how
to teach music as they learn music. In any case, many musicians learn to teach music and come to depend on this knowledge and skill as a source of income.
Our educational system accepts the idea that a person teaching music should be a native language speaker of music (that is, a musician) and a music educator. In telling contrast, American schools
have a large percentage of people teaching math who are ill-prepared in the content and the teaching of math.
Most children grow up in a home environment is which there is singing and other music. They may learn jingles and other music from radio, television, or Internet ads, from computerized games, from
music storage and playback devices, and so on. If parents and other caregivers have a reasonably high level of music interest and fluency, their children will learn a lot of music by being immersed
in such an environment.
However, learning to play a musical instrument, learning to read music, learning to compose music, learning to sing well individually and in a group, and so on, all require many years of informal and
formal instruction plus lots of practice.
Three important characteristics of informal and formal music education are:
1. The music learner is frequently given the opportunity to observe (listen to) the performance of others who have a relatively high level of expertise. Math has this characteristic for very few
children—or adults.
2. Music is a human endeavor that is often performed in a group setting. This setting promotes and facilitates communication and sharing. This sharing—making music together—is an important
aspect of music. There are large intrinsic and extrinsic rewards in such musical sharing. Again, this is seldom true for math.
3. Music provides a learning environment in which the learner can readily tell that he or she is gaining in expertise and can demonstrate to others this increasing expertise. Math exhibits some
of this characteristic, but not nearly as strongly as music.
Each of these characteristics of music education offers us some insights into math education.
The first item suggests that we can improve math education by improving the general math environment. For example, we currently teach math during a specific math period in school, and math may
receive very little attention during the rest of the day. A child's math-related environment at home may be quite limited. Our educational system has accepted the idea of reading and writing across
(throughout) the curriculum. How about making this into "the three R's across the curriculum"?
The second item points to a major challenge in math education. While math, like music, is a human endeavor, it mainly lacks the group learning and performance aspects of music. The intrinsic and
extrinsic reward structure in math is quite different from that in music.
The third item suggests we can improve math education by creating better feedback mechanisms and by helping students gain expertise in sensing and assessing their own progress. Research in math
eduction supports these ideas.
For reflection and discussion. Think about the three ideas listed above in terms of your personal math education experiences. What does your personal analysis suggest to you in terms of
things you can do to improve the math education experiences of students?
Doing and Consuming Music and Math
From an educational point of view, it seems important to distinguish between those who learn to "do" music (perform at some level) and those who merely consume (listen to) music. Both categories of
people existed before the development of electronic technology. We can educate for each of these endeavors, and most people have some level of interest in and expertise in each of them. Participation
in the second category has been greatly influenced by the development of technology for the broadcasting, recording, and playback of sound, and new developments in electronic music composition and
We all "do" math. When my stomach rumbles and I glance at my watch, I do a mental calculation of how long it has been since my last meal and how long before my next meal. When I fill my car's gas
tank, I estimate the cost and the miles per gallon since the last fill up. In a store, I pay some money and get some change, then I check the correctness of the change.
We all consume math, but not in the same way that we consume music. For example, I use my cell phone to make a call. The cell phone system makes quite sophisticated use of computers and a variety of
other equipment. People designing the equipment and the overall phone system made extensive use of math. So, when I use my cell phone, I am making extensive use of math. However, that does not give
me the same sense of feeling contentment that I get by "making use of music" when I am listening to a live or high-quality recording of an expert musical performance.
ICT and Music
The overall discipline of music has been strongly impacted by Information and Communication Technology (ICT). The history of this impact certainly dates back to the development of the telephone (so
music could be transmitted over a wire) and recording devices (so music could be stored, edited, and widely distributed).
We now have relatively inexpensive electronic hardware and software for creating, editing, storing, playback, sharing, and performing music. With such technology, a grade school student can learn to
compose and edit music, and can use a computer as a performance instrument. This is an important idea. When grade school students compose music and use a computer in the editing and playback
process,they can hear what they produce, improve it by editing, and share their music with others. This is in marked contrast to what typically happens in math instruction.
Computer technology allows people to build personal libraries of music. Computer technology can "notate" music—that is produce a written musical score from a live or recorded music performance.
Artificial intelligence music generation systems have generated music "in the style of" various world class composers, music deemed comparable in quality to human-created compositions. These
electronic musical instruments and a wide range of storage and editing tools have greatly changed the music recording and performance industries.
Relating back to math education, one of the key ideas given above is that ICT has brought us a number of new electronic musical instruments to "do" music. In math, ICT has brought us 6-function,
scientific, graphing, and equation-solving calculators, and computers with vastly greater powers. These new math tools are instruments that can be used to "do" math. Thus, math education is faced by
the problem of determining the extent to which it wants to facilitate students learning to use these electronic tools (instruments) and pay less or no attention to other more traditional tools. We
can ask, "What is so special about paper and pencil computational algorithms?"
Here is an idea that might interest you. Solving a complex math problem or producing a proof in math is somewhat akin to composing in music. A musical composer does not have to have a high level of
expertise in playing each of the instruments needed to perform a composition. That is, composing and performing are two different skills. Of course, the composer needs to know the capabilities and
limitations of the various musical instruments and the human limitations in playing the instruments.
In math, for example, a person working to understand and solve a math problem can imagine having a three-dimensional picture of a particular three-dimensional geometric figure, and being able to
readily view this picture from different directions. We have long had computer graphics software that can produce such three-dimensional representations.
Somewhat similarly, a person working to solve a math problem may decide that it would be helpful to fit a mathematical function (perhaps a quadratic, cubic, or higher order polynomial function) to
some data, and then find the places where this function crosses the x axis. Computer programs have long existed that readily accomplish such tasks.
To continue this example, the math problem solver may decide that it would be helpful to perform various statistical computations on some data. Computer programs that can carry out such computations
incorporate a huge amount of collected knowledge from a great many mathematical statisticians.
Perhaps you see a pattern emerging here. Think of a person attempting to solve a complex math problem as a composer, developing a set of instructions that can be carried out to solve the problem. The
math problem-solving composer can draw upon the performance capabilities of calculators and computers. The math problem-solving composer does not need to have a high level of expertise in areas in
which calculators and computers can readily produce very high levels of performance.
Here is another math education idea coming from music. One of the reasons math education is part of the core curriculum is that math is a powerful aid to solving problems in many different
disciplines. What computers and other aspects of ICT have done is make it possible to automate finding the solution to many of these problems. Consider an analogy between having an automated tool
that solves a particular problem, and having a music storage and playback device. A music consumer gets to listen to and have the benefits of recorded music. A person using a computerized tool that
incorporates many mathematical functions gets the advantage of a mathematical performance.
In summary, ICT has strongly affected doing and consuming music. It has equally strong potentials in math. However, our math educational system has made only modest progress in realizing (making use
of) these potentials.
Compare and Contrast with Chess
This section examines the game and language of chess. Extensive research on good chess players has given us good insight into the role of chunking (see above) in learning to be a good chess player.
There is considerable similarity between this chunking used in playing chess (solving the problem of making a good move) and chunking used in solving problems in math.
Chess (just like checkers)is played on an 8-square by 8-square board. The game has a long history and is played by millions of people. At the higher levels of play there are masters grandmasters,
international grandmasters, and world champions. At lower levels, grade school children can learn the game and there are tournaments for players at all levels. Many websites explain the rules and
some of the details of the game.
Chess is a game of skill. This is in marked contrast to card games such as bridge and poker, where the "luck of the draw" makes a big difference in the short run. In a chess game, two players compete
against each other. The game ends in a win for one of the players, or a draw. There is some advantage to being the first player to make a move, but in a chess match between two players, they take
turns in going first. The "luck" that chess players talk about is when their opponent overlooks a possible good move or fails to see that a move s/he is about to make is a bad move.
Chess has an oral and written language. However, chess is not a significant part of the everyday life of most people. Nor is chess one of the nine areas of multiple intelligence identified by Howard
Gardner. See http://iae-pedia.org/Howard_Gardner. Children who grow up in a "chess-playing" family may learn the game when they are quite young and gain from the collected oral history of chess that
is part of the family conversation.
Chess has been extensively studied to help understand problem solving and how humans can improve their problem-solving skills. It has also been extensively studied by people working in the field of
artificial intelligence. How does one go about developing a computer program that is good at playing chess? What can we learn about human intelligence and the education of children through the study
of how humans and computers learn and become better at playing chess?
You don't have to be a skilled chess player to understand the chess section in this Communication in the Language of Mathematics document. Here is enough background to get you going.
Chess Notation
The columns (files) of the 8 x 8 board are lettered a, b, … h, and the rows (ranks) are numbered 1, 2, … 8. In chess, the person playing the White pieces always moves first. The lettering and
numbering notation used to identify the spaces on the board is convenient and natural from the point of view of the person playing the White pieces.
The names of the pieces (in English) are abbreviated as follow: K=King, Q=Queen, R=Rook, B=Bishop, N=Knight, and P=Pawn. This board coordinate system and the piece name abbreviations make it quite
easy to record all of the moves in a game.
For example, here are the first few moves of a game. White always moves first, and White's moves are in the left column. The sequence of moves given below indicates that White’s Bishop captures
Black’s Knight on White’s fourth move.
1. Pe2 to e4 — Pe7 to e5
2. Ng1 to f3 — Nb8 to c6
3. Bf1 to b5 — Pa7 to a6
4. Bb5 x Nc6 —
This notation can be tightened up considerably. Here is a tighter notation that conveys the same information. The notation assumes that the reader knows the legal moves. Thus, the first move of Pe4
means that White's pawn that is at e2 is moved to the e4 location. It is the only pawn that can legally move to that location at this point in the game.
1. Pe4 Pe5
2. Nf3 Nc6
3. Bb5 Pa6
4. BxN
Notice how easy it is to make an exact record of a chess game and to learn to read such a record. Contrast this with musical notation and learning to read music, or the notations used in math. From a
notational point of view, music and math are far more complex than chess. It takes only s few minutes to learn the written language used to store a record of a game of chess.
Communication is More than Just Notation
This section discusses some of the learning that takes place as one develops into a good chess player. If you have not played chess, or not played it much, the ideas will probably pass you by. But,
for each idea, think about whether it might be applicable in learning to being better at math problem solving. Keep asking yourself, "What 'big ideas' in math problem solving do I know that I use
regularly and that I can help my students learn to use?"
Here is a piece of information useful for understanding the example in the following paragraph. In chess, a Knight's move must be either two horizontal and one vertical square or one horizontal and
two vertical squares. This allows a Knight to attack various pieces that cannot, in turn, be attacking back.
Like any well-developed discipline, chess has an extensive vocabulary. Also, as in any discipline-specific vocabulary, many chess terms are adapted from natural language vocabularies. For example,
you might think of a
as an eating utensil. Of course, you have heard of a tuning fork used in music. You have heard of a fork in a road. In the diagram, White has just moved the Knight to d7, actually forming a triple
fork. This particular fork of King, Queen, and Rook is also known in chess as a “family fork.”
You might be able to guess meanings of terms such as open file and Queenside. Other terms such as check, gambit, castle, and fianchetto (Italian for "on the flank") are more challenging.
This special vocabulary and notation are important for communicating about and thinking about chess. However, there is more to such endeavors than just vocabulary and notation. Your brain stores
images that represent emotions, sounds, smells, pictures, and so on. Your brain draws upon these mental images as it works to solve problems and accomplish tasks. Good chess players have stored many
thousands of chess patterns (chess chunks) in their brains. For them, a short look at a chess game in progress provides information needed to retrieve mental chunks of information related to possible
future outcomes of the game in progress.
In this aspect of communication with one's self, there are clear similarities among chess, math, and music. In each discipline, one learns chunks, stores them, and learns to think in terms of these
chunks. Some of the chunks have names, while others are mental patterns that one accesses through other means such as mental pictures, sounds, "gut-level feelings," and so on.
Learning a discipline-specific chunk and how to make effective use of it is a step toward increased expertise in a discipline. However, accumulating a large number of chunks in and by itself does not
make one into a high-level expert in a discipline. It is learning to "see," "sense," "hear," "feel," "recognize," etc., the relationships among chunks, and to make use of appropriate combinations of
these chunks, that is key to having a high level of expertise in a discipline.
The discipline-specificity issue is worth repeating. Substantial research supports the need for discipline-specific knowledge and skill (discipline-specific chunks) in order to have a high level of
expertise in that discipline. That is why, for most people, it takes so many years of effort in order to become a high-level expert in a discipline. But it takes more than just rote memorization of
chunks to achieve this high level of expertise.
Chess Strategy
There are many chess websites available. On the Web one can read about chess, see the rankings of the best players, follow tournaments in progress, play against human opponents, play against a
computer, or try your hand at solving challenging chess problems. These websites can be used just for fun and can also be used to gain increased expertise as a chess player.
Here is a personal story. I learned to play chess when I was relatively young. That is, I learned the legal moves and to play well enough so that it was fun to play with other kids my age.
A number of years later I became interested enough in the game to read a couple of chess books. One was a "how to" book that explained some of the strategies that good players find useful.
Another was a chess history book, looking at some of the great players and games from the past. I found both types of book enjoyable. The "how to" book substantially increased my level of
playing ability.
This is an "aside." Notice that I read these chess books for fun and to learn to be a better chess player. At about the same time I read some books such as The World of Mathematics that had
little to do with the math I was being taught in school. Our educational system places a lot of emphasis on students learning to read well enough so they can learn by reading. In math
education, however, we do not take much advantage of a student's steadily improving expertise in reading. Think back over your own math education. Did you ever read a math book for fun or to
further your math knowledge and skills beyond what was being taught in school?
Here is a simple example. In chess, one of the key ideas is to maintain the mobility of your pieces. That is, to keep as many move options available for your pieces as possible. Another strategy is
to gain control of the center of the board. Among other things, control of the center tends to increase your own level of mobility and decrease your opponent's level of mobility.
Suppose that you had studied a book discussing these two strategies, and that it contained some examples of how to make use of the strategies. You then play a game against an opponent who (up until
now) was your equal, but who had not received formal instruction (from a teacher, book, or opponent) on these two strategies. The chances are quite good that you will now be the superior player. This
little bit of formal instruction gives you a large advantage over an "unschooled" opponent.
This is an important idea. It might well be that, as you play chess, you will discover some of these strategies for yourself. However, there are many strategies that are useful at various points in a
typical game. Many have been discovered and carefully analyzed by world class chess players. The accumulated knowledge in this area is far more than one person could discover (unaided by the previous
work of others) in a lifetime.
Now, consider my opponent who is consistently losing to me because I have been making use of these two strategic concepts. My opponent may carefully analyze these (losing) games and eventually
discover the concepts of mobility and center control. Alternatively, I might mention the two ideas and illustrate them in a game that we have recently completed. In both cases, I am assuming that we
have written down the moves from the games, so that we have a written record that allows us to analyze games we have played in the past.
Chess is both a fun game for children and a discipline of fierce, ego-involved competition. With few exceptions, it takes ten thousand or more hours of study and practice to become an international
grandmaster, assuming inherent talent. Much of this time and effort is spent studying games that have been played by exceptionally good players in the past, and games one has played in the past.
These insights into learning chess strategies provide some useful insights into learning math. There are many different strategies for attempting to solve math problems. Many of these are designed to
aid in communicating with one's self, such as by drawing a diagram, making a table, creating a mental model or image, and so on.
Determining a Chess Player's Strength
Chess is a competitive game. If two players of approximately equal chess-playing strength play against each other a number of times, they will each win about half of the games. If one player is much
stronger than the other, this player will win almost all the time.
Over the years, the discipline of chess has developed a relatively accurate means for determining a player's strength. The method is somewhat like that used in rating teams in competitive sports.
Careful records are kept of how well players do against each other in different tournaments. Even if two players have never played against each other, they will have played against players whose
strength or rank has been determined through tournament play.
As in competitive sports, there are chess tournaments pitting the top players in the world against each other, and there are world championship matches in which two players compete against each
other, with the winner designated as the world champion.
While there are competitions in both math and music, there is essentially nothing like the level of competition one finds in chess. There are many world class mathematicians, and there are many world
class musicians. However, there is no world champion mathematician determined by head-to-head competition.
Computers have brought an additional approach to chess rankings. One can compare (and rank) humans in how well they do playing against various computer chess programs. People throughout the world can
compare themselves in terms of how well they do against a particular chess program, set at a particular level of difficulty.
Without a competition and ranking system, math students have no easy way to compare their own math strengths against each other. Let me share a personal example. For students in college, there is a
national math competition called the Putnam competition. Throughout the United States, on one specific day, entrants spend the day working on 12 problems. This is done on their own campuses, and the
test is carefully proctored.
I was the best math undergraduate at my university. I knew this because of having taken course with the other top students—in some sense, competing with these students in math classes and the tests
given in the classes. I thought of myself as being quite good at math.
I competed in the Putnum contest in both my junior and senior years. In both years, a quarter to a third of all entrants scored better than me! Objectively, I was quite good. Only the better students
would enter the competition, and I was better than about seven of ten other such students. Even so, my ego's feeling of “being quite good” suffered significantly.
One of the goals in the No Child Left Behind Act is to move toward a ranking system in math education that can be used to measure the relative strengths of schools. The people supporting this type of
"competition" believe that it will help to improve the precollege math educational system in this country.
It is not at all obvious that making math into a competitive "sport" will lead to improved math learning and performance for students as a whole. Indeed, it might well do just the opposite. Those who
are not highly talented and highly motivated in math (as well as those who are not basically competitive in what they do) may well choose not to compete. One might well see widespread implementation
of the sentiment: "I’m not very good at math. Why should I compete, when I will always come out in the bottom half?"
Contrast this with a person learning that the knowledge and skills they are gaining in math empower them to do various things they need or want to do. Through study and practice, they get better at
doing those things. This suggests that math education can become more successful through helping students, individually as necessary, to grasp the personal advantages (empowerment) they accrue
through their math studies.
For reflection and discussion: Suppose we had a computer program that could "play the game" of math at different difficulty levels. A student studying math could play against this game to
determine his or her current math ranking. Here, we are assuming that this "game of math" is good enough to be used throughout the world to determine a student's math level of learning,
understanding, and overall "math strength." How do you think this would affect math education? To help your thinking on this question, you might want to read David Moursund's short article
Chesslandia: A Parable.
Rote Memory
It may feel to you that a discussion of rote memory is a far reach from a discussion of communication in math and math education. Here is the way I see it. Much of what a person does when attempting
to solve problems and accomplish tasks in any discipline involves communicating with her or him self. One consciously communicates with data, information, knowledge, and wisdom stored in one's brain.
One carries on a mental conversation. Indeed, you probably know people who verbalize—talk out loud to themselves—during this thinking.
One can memorize with little or no understanding of what is being memorized. One way to think about this is in a stimulus/response setting. A person's brain (or, some other animal's brain) is trained
to respond in a specified manner to a specified stimulus. The stimulus elicits the response, and the responder does not need to have an “understanding” of the meaning of what is stored in the brain
and produces the response.
Of course, we can also have stimulus/response learning in which the response has meaning to the learner. You may be able to respond quite rapidly to the stimulus 8 x 7 =, and produce a response of
56. Upon further reflection, you realize that you have done a "multiplication fact" problem-solving task, producing an answer of 56.You may realize that this answer is a little bigger than 50 and a
lot less than 100. You may realize that a score of 56 on a hundred point test may not lead correspond to a good grade on the test.
Rote memory, with or without understanding, can be used in the storage and retrieval of part of the collected knowledge with a particular area. This can be quite helpful in solving some of the
frequently occurring problems within that area.
Here is an illustration from the game of chess. A chess game begins with White and Black each having 16 playing pieces. It is possible to carefully analyze the board situations that result after all
possible sequences of one move by each player, two moves by each player, and so on. Of course, the number of possible sequences grows exponentially, and soon becomes so huge that no person (indeed,
even all the past and present chess players in the world) can analyze all of them.
However, what has been done is that many of the interesting and potentially good opening sequences of moves have been carefully analyzed by high level chess experts. A huge amount of this collected
chess knowledge is available in books and in other media. Any person who has learned to read chess notation can access this collected chess knowledge.
A chess player gains a considerable advantage by studying these well-analyzed sequences of opening moves and by memorizing a large number of them. Rote memory of the results of work done by others is
a good substitute for "reinventing the wheel." In a game between reasonably highly ranked chess players, the first half dozen or more moves by each player tend to be made quite quickly, using rote
After that, the thinking and chess problem-solving begins. Each player soon encounters a position (a chess problem) that he or she has never encountered before. However, even here having a large
repertoire of memorized chunks is very important. In essence, such chunks correspond to parts of a game position. The good chess player recognizes parts of the problem as being similar to or even
exactly the same as parts of board positions that he or she has carefully analyzed in the past.
Math education can be approached via rote memory. We can have a student memorize facts, definitions, and algorithms. Rote memory is useful in solving frequently occurring problems. Moreover, math
problem solving makes use of chunks much in the same way as chess playing does.
It turns out, however, that math is played on a much larger playing board (many more playing pieces) than is chess. In dealing with the math people encounter in their everyday lives, they quickly
move beyond the point where rote memory suffices. In novel problems, problem solvers quickly move beyond being able to succeed from rote memory and enter the mode of attempting to make effective use
of chunks of information stored in their long-term memories. A large repertoire of such chunks and lots of experience in drawing on such chunks is essential in dealing with challenging math problems.
Artificial Intelligence and Chess Playing
Consider the following type of competition. Well before the competition begins, competitors are given a copy of a very comprehensive dictionary. During the contest, competitors are given a definition
from the dictionary. Their task is to say and spell the word that has been defined.
By dint of considerable study and practice, a person can get very good at this rote memory game. But a computer can become letter perfect in a very short period of time. Computers are very good at
rote memory, and computers can store the spelling and pronunciation of all of the words in a dictionary.
When researchers in artificial intelligence went looking for a game to study in which (at the time) humans were much better than computers, many decided on chess. One can become better at chess by
rote memory, but chess is far more than a rote memory game.
In chess, rote memory is quite helpful at the beginning of a game. It can also be quite helpful near the end of a game in which each player has lost a number of pieces. In that situation, there are
relatively few pieces left on the board. Many such end games have been carefully analyzed by chess experts and computers. The results are available in books and databases.
It is in the mid game—after use of the memorized openings and before use of memorized end games—that intelligence is needed. How does a human chess player analyze possible moves in order to select
the one that improves their situation and/or damages the opponents’ position? One way to gain insight into this is through working with skilled chess players. Get them to "think out loud" as they
analyze chess problems. Of course, there are also many books full of the written analyses of games played by good chess players.
This interaction with expert problem solvers has been used in many different disciplines. It has led to the development of expert systems (computer programs that are good at solving challenging
problems) in many different disciplines.
This is part of the challenge for educators in the information age. Thousands of researchers are working on developing computer programs that make use of computer capabilities (machine intelligence,
artificial intelligence) to solve or help solve problems in various disciplines. Sometimes the artificial intelligence methods parallel human intelligence methods. Often they don't. Rather, they make
use of methods that take advantage of the large memory and great speed of computer systems.
The first chess-playing computer programs were very weak compared to humans. However, over the years, computers got much more capable, and chess-playing programs got much better. By 1997, an IBM
computer named Deep Blue beat Garry Kasparov, the world's human chess champion! A dedicated chess machine called Hydra was programmed to take advantage of a combination of rote memory and a type of
artificial intelligence relevant to chess playing—and became as good at the world's best human chess players. The success of chess-playing computers did not lead to the game of chess gradually going
away. Nowadays there are chess matches that pit computer against computer. There are chess matches that pit human plus computer against human plus computer. Many chess players practice their skills
against computer programs. With all of this, chess remains a game that many people learn to play and enjoy playing against human and computer opponents.
Artificial intelligence has been used in other game-playing computer programs. IBM undertook the challenge of developing a computer program that could play the popular TV game show Jeopardy. In this
game, opponents are given an answer to a quite specific question, and compete to see who can first correctly state the question. In February of 2011, an IBM computer system named Watson defeated two
human champions in this game. See http://i-a-e.org/iae-blog/entry/the-future-of-ibm-s-watson-computer-system.html.
Artificial Intelligence and Solving Math Problems
Artificial intelligence, rote memory, and steadily increasing computer memory size and speed have been applied in mathematics. For many years, there have been high quality Algebra Systems. Such
computer programs can solve a wide range of math problems.
You know that an inexpensive 6-function calculator can add, subtract, multiply, divide, and compute square roots. That is, it can "do" some of the things we are teaching grade school students to do
through rote memory and through use of memorized algorithms. A modern Computer Algebra System (CAS) has this same level of capability up through calculus and linear algebra. That is, in every part of
the math curriculum where rote memory and use of memorized algorithms is useful, artificially intelligent CAS systems can do (typically, faster and more accurately) what we are teaching students to
do by hand.
The math education community needs to think carefully about the steadily growing "intelligence" of computer systems. If a computer system can solve a certain category of math problems, what "by hand
and by brain" knowledge and skills in this specific area do we want students to acquire?
Our math educational system has been struggling with this situation for years. To the extent that math resembles a competitive game, computers are far better than humans at many aspects of it. It
seems evident that this math education quandary will continue to exist for the foreseeable future.
There are no widely agreed on solutions to the computers versus humans in math issue. The Common Core State Standards Math Initiative is down-playing the importance of computers. See http://
www.corestandards.org/Math and http://i-a-e.org/downloads/doc_download/249-common-core-state-standards-for-k-12-education-in-america.html
For reflection and discussion. In chess, the development of computer programs that can outplay even the best of human players has not resulted in the demise of the game. Chess players enjoy
the head-to-head competition with each other and the social aspects of being part of the chess community. Within in certain areas, computers are far better at math than humans. How is this
affecting what we are doing in math education? Are there aspects of math that are very large numbers of people want to learn because they are fun—personally and socially
rewarding—independently of whether computers can do them better than humans? For example, to what extent is Sudoku a math game?
For reflection and discussion. Suppose that our education system decided that all math education above the 8th grade was elective. Any course requiring a higher level of math knowledge and
skill could clearly specify the higher level of math prerequisite that was required. However, students could well graduate from high school and college without taking math courses above the
8th grade level—or, by only taking such higher level math courses when they had a clear personal need to do so. What are your thoughts on how this would affect our overall educational system?
Final Remarks
For me, the ideas that I have discovered and explored while writing this article are quite thought-provoking. While my initial focus was on communication and math education, many of the ideas apply
to learning in every academic discipline. For example, brain research on chunking is applicable in any discipline, including disciplines as diverse as carpentry, dancing, and Texas hold ‘em. Skill in
creating and using personal chunks is an essential component of self-talking and planning in solving challenging problems.
Information and Communication Technology (ICT) brings a new dimension to communication. One way to think about this is that a computer is an artificially intelligent machine that one can communicate
with and make use of in solving problems, accomplishing tasks, and learning. Thus, within each discipline in our educational system, educators are now faced by the challenge of helping their students
communicate effectively with computers and other artificially intelligent aids to solving problems and accomplishing tasks.
This challenge is especially large in math and in other disciplines where computers are especially useful (powerful, capable) in solving or helping to solve problems and accomplish tasks. That is,
many chunks include procedures that one can learn to carry out "by hand" but that computers can carry out faster and (often) more accurately. For each of these, a student could have an option of
learning all details of the chunk, or of learning that such a chunk exists and that a computer can accurately and rapidly carry out the details of the procedure(s) associated with the chunk.
For reflection and discussion: What are your thoughts on education, including having a student learn about (including how to retrieve and how to make use of) a number of computerized chunks?
In my opinion, our math educational system spends far too much time helping students to learn (memorize, often with little understanding) to do things that computers can do faster and more
accurately. This uses up so much of the math education time, that relatively little time is spent on understanding, creative thinking, problem posing, and other activities in which human intelligence
far exceeds computer intelligence.
Some of this memory work is important. Often a person is called upon to make real time decisions (quick decisions) based on using math knowledge and skills. As computerized processing and information
retrieval systems get better, and as more computer intelligence (including math-related computer intelligence) is built into machines, we will need to continually reexamine those aspects of math that
need to be stored in one's head.
For reflection and discussion: Think back over this article. Identify one or two ideas that you found particularly interesting and that you tend to agree with. Find one or two that you fond
uninteresting and/or that you strongly disagree with. Do a compare and contrast, working to increase your insight into communication aspects of improving math education in our information age
Annenberg Media (n.d.). Mathematics illuminated. Retrieved 5/31/08 from http://www.learner.org/channel/courses/mathilluminated/units/1/?pop=yes&pid=2283#.
This is a set of 13 free half-hour videos and accompanying instructional materials. The first of these talks about mathematics as a language and explores prime numbers.
Ball, D.L., Hill, H.C., & Bass, H. (Fall 2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator. Retrieved 7/13/2010
from http://www.aft.org/pdfs/americaneducator/fall2005/BallF05.pdf.
Quoting the first part of the article:
With the release of every new international mathematics assessment, concern about U.S. students’ mathematics achievement has grown. Each mediocre showing by American students makes it plain
that the teaching and learning of mathematics needs improvement. Thus, the country, once more, has begun to turn its worried attention to mathematics education. Unfortunately, past reform
movements have consisted more of effort than effect. We are not likely to succeed this time, either, without accounting for the disappointing outcomes of past efforts and examining the
factors that contribute to success in other countries. Consider what research and experience consistently reveal: Although the typical methods of improving U.S. instructional quality have
been to develop curriculum, and—especially in the last decade—to articulate standards for what students should learn, little improvement is possible without direct attention to the practice
of teaching. Strong standards and quality curriculum are important. But no curriculum teaches itself, and standards do not operate independently of professionals’ use of them. To implement
standards and curriculum effectively, school systems depend upon the work of skilled teachers who understand the subject matter. How well teachers know mathematics is central to their
capacity to use instructional materials wisely, to assess students’ progress, and to make sound judgments about presentation, emphasis, and sequencing.
That the quality of mathematics teaching depends on teachers’ knowledge of the content should not be a surprise. Equally unsurprising is that many U.S. teachers lack sound mathematical
understanding and skill. This is to be expected because most teachers—like most other adults in this country—are graduates of the very system that we seek to improve. Their own opportunities
to learn mathematics have been uneven, and often inadequate, just like those of their non-teaching peers. Studies over the past 15 years consistently reveal that the mathematical knowledge of
many teachers is dismayingly thin.1 Invisible in this research, however, is the fact that the mathematical knowledge of most adult Americans is as weak, and often weaker. We are simply
failing to reach reasonable standards of mathematical proficiency with most of our students, and those students become the next generation of adults, some of them teachers. This is a big
problem, and a challenge to our desire to improve.
Dynarski, et al. (March 2007). Effectiveness of reading and mathematics software products: Findings from the first student cohort. Report to Congress. Retrieved 2/19/08: http://ies.ed.gov/ncee/pdf/
20074005.pdf. Quoting from the report's Executive Summary:
The main findings of the study are: ,
1. Test Scores Were Not Significantly Higher in Classrooms Using Selected Reading and Mathematics Software Products. Test scores in treatment classrooms that were randomly assigned to use
products did not differ from test scores in control classrooms by statistically significant margins.
2. Effects Were Correlated With Some Classroom and School Characteristics. For reading products, effects on overall test scores were correlated with the student-teacher ratio in first grade
classrooms and with the amount of time that products were used in fourth grade classrooms. For math products, effects were uncorrelated with classroom and school characteristics.
Ericcson, K.A. (n.d.). Superior memory of experts and long-term working memory (LTWM): An updated and extracted version of Ericsson (in press). Retrieved 3/4/08: http://www.psy.fsu.edu/faculty/
Fenwick, C. (n.d.). UCL's University Preparatory Course in Science and Engineering. Retrieved 12/6/07: http://www.ucl.ac.uk/~uczlcfe/main.html. Quoting from this website:
The Mathematics Course : Surprising as it may sound the learning of mathematics is not just about learning to 'get the right answer'. It is also (amongst other things) about being able to
think mathematically and read mathematically, and then being able to show how you develop your ability, reading and thinking. Consequently, as part of the coursework you will need not only to
be able to do the mathematics set but also be able to describe exactly the process by which you went about doing such mathematics.
Hence, throughout the course you will need to demonstrate your developing mathematical thinking, technical and reading ability by:
solving specific mathematical problems
adopting the approach of reading mathematics. This will be done by interpreting technical text, mathematical expressions, solutions to mathematical problems, diagrams, etc...
studying and learning how you go about working on, solving and hence, learning mathematics
Specifically, the course aims to help you develop the following abilities :
1. the ability to solve appropriate mathematical problems
2. the ability to construct appropriate mathematical proofs
3. the ability to read mathematically by interpreting/describing mathematical text, expressions, solutions and/or proofs as appropriate, and demonstrate this through written and/or oral work
4. think mathematically by identifying mathematical patterns and use these to extend given mathematics
5. the ability to critically analyze and discuss issues in mathematics, as well as your learning of mathematics
6. the ability to work individually and in groups on the topic of mathematics
7. the ability to improve &/or extend any aspect of 1) - 6) above.
Kadiec, A., & Friedman, W. (2007). Important, but not for me: Kansas and Missouri students and parents talk about Math, Science and Technology Education. Public Agenda. Retrieved 9/20/07: http://
Logan, R.K. (2000). The sixth language: Learning and living in the Internet age. Toronto, Canada: Stoddard. (See also.)
Mazur, E. (n.d.). Interview by Marty Abrahamson. Retrieved 2/24/08: phttp://www.bedu.com/Newsletterarticle/mazurperspective.html. Mazur is a physics professor and highly acclaimed teacher at Harvard.
Quoting from the interview:
1. Mazur assigns reading and expects his students to email him questions about what they do not understand.
2. Mazur assigns reading and gives an online quiz to see what they do not understand.
3. Mazur make sues of "clickers" (hand held student response units) in class to get feedback from students.
MA: … Despite the fact that it is possible to accomplish all of these objectives and many more with a single question, do you think that it is useful to have a specific primary objective when
designing and planning the delivery of a question?
EM: Oh, yes ! I often actually use students' questions. I actually use this now with a teaching technique called "Just-in-Time Teaching" …. Basically, the students read before class and then
they tell me in an e-mail what they find difficult or confusing. I use that to prepare my lecture. In other words rather than lecture on what I find difficult, I will take some of their
confusion and bounce it straight back at them.
Miller, G.A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Originally published in The Psychological Review, 1956, vol. 63, pp. 81-97.
Retrieved 2/29/08: http://www.musanim.com/miller1956/.
Moursund, D.G. (June 2006). Computational thinking and math maturity: Improving math education in K-8 schools. Retrieved 3/4/08: http://uoregon.edu/~moursund/Books/ElMath/ElMath.html.
Polya, G. (circa 1969). The goals of mathematical education. Mathematically sane. Retrieved 9/16/07: http://mathematicallysane.com/analysis/polya.asp.
Rohrer, D., & Pashler, H. (2007). Increasing retention without increasing study time. Current Directions in Psychological Science. vol. 16—no. 4. Retrieved 5/19/08: http://www.pashler.com/Articles/
RohrerPashler2007CDPS.pdf. Quoting the Abstract:
Because people forget much of what they learn, students could benefit from learning strategies that provide long-lasting knowledge. Yet surprisingly little is known about how long-term
retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple
sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of
the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions – the spacing effect – depends
jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning.
[Bold added for emphasis.]
Sylwester, R. (March 2008). How children learn a language: Part 2 – Knowing what to say and how to sayit. Brain Connection. Retrieved 3/19/08: http://www.brainconnection.com/content/267_1. Quoting
from the article:
Last month’s column described how mirror neurons provide us with a mental template of the active motor neurons of someone who is speaking. The person’s comments create an analogous template
of the content of the speaker’s thoughts. So if a person says cat, it activates the mirror neurons our brain uses to say cat, but also the neurons that process our memories and images of cat
as a concept.
When I begin to write an article, I have a general sense but no set outline of what I hope to write. I explore the concept on my keyboard, and the article gradually begins to emerge. As in
conversation, the focus may shift from the original idea. At one point, though, everything becomes clearer, and then considerable rewriting sharpens the text. This often also occurs in a
conversation or meeting, when a consensus suddenly occurs, and the issue is then quickly resolved.
What’s odd is that when things are most confusing, I’ll often suddenly wake up from sleep with the mental clarity that had eluded me while writing during the day. I have no explanation for
this, except that my thoughts about current tasks seem to continue at a subconscious level, whether awake or asleep. We’ve all experienced this when we can’t recall a familiar name. We go on
with other thoughts, and then hours later the name suddenly pops up in our mind.
This suggests that while thought and language are perhaps two sides of a single coin, thought can occur without language—and alas, a lot of language occurs without thought.
Waters, R. (3/4/08). World-wise web? Financial Times. Retrieved 4/5/08: http://www.ft.com/cms/s/0/4fba0434-e98c-11dc-8365-0000779fd2ac.html?nclick_check=1.
This article looks at possible futures of the Web. It focuses specifically on increasing linguistic "intelligence" of the Web. Web 3.0 will have a much better ability to "read" the content of
websites, extract meaning, and link this meaning to that stored in other Websites.
Wiggins, G. (1990). The case for authentic assessment: Practical assessment, research & evaluation, 2(2). Retrieved 9/16/07: http://PAREonline.net/getvn.asp?v=2&n=2.
Links to Other IAE Resources
This is a collection of IAE publications related to the IAE document you are currently reading. It is not updated very often, so important recent IAE documents may be missing from the list.
This component of the IAE-pedia documents is a work in progress. If there are few entries in the next four subsections, that is because the links have not yet been added.
Note added 1/7/8/2014: The following article probably fits well into the discussion.
http://www.smartbrief.com/01/02/14/ideas-help-students-learn-academic-language-1#.UssP9fY512A, The basic idea is that there is an academic language in each academic discipline. Learning to
communicate in a discipline's language (read, write, speak, listen) is a critical part of developing a high level of expertise in the discipline. The same idea holds for non-academic disciplines.
IAE Blog
In some sense, all teachers are ethnographers.
IAE Newsletter
IAE-pedia (IAE's Wiki)
I-A-E Books and Miscellaneous Other
David Moursund' Learning and Leading with Technology Editorials
The original version of this document was written by David Moursund. Editing and a number of revisions were provided by Dick Ricketts.
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Recent Items
Technology Overview
Dr. Herman has been involved in a variety of aspects of using instructional technology in mathematics and physics classes. He has also attended all of the ICTCM (International Conference on
Technology in Collegiate Mathematics) since 1993. The type of involvement includes, but is not limited to
• Programming - He taught Fortran and Pascal and has created over 20 years applications using BASIC, Turbo Pascal, Visual Basic and has worked with many mathematics packages, including MuMath,
MathCAD, Matlab, and Maple.
• Mathematics on Pocket PCs - He has worked on funded projects to explore the use of handheld devices in the classroom. This has lead to investigations of the software that can be used to do
Mathematics on Pocket PCs.
• Multimedia Presentations - He has created presentations in Toolbook and PowerPoint and conducted workshops for faculty in using multimedia, including using the Internet and creating web pages
with MS Front Page and other editors. Some samples are at WebCalc.
• Mathematics and Physics Worksheets - He has made use of MathCAD, Matlab, and Maple to create simple presentations for the classroom.
• Lab Manuals - With others he has developed sets of labs for calculus, differential equations and introductory physics classes.
• Online Courses - He had co-authored an Online College Algebra Course and has worked with many faculty in developing online course in other fields using tools like WebCT.
• Funded Projects - He has worked with others to obtain funding for several projects, which are at the forefront of technology use in the classroom. These include the funded projects: MCP Project
(Multimedia Instruction in Mathematics, Chemistry, and Physics), iLumina Digital Library (a part of the National Science Digital Library), and the Numina Project (an exploration of the use of
handheld and pocket PCs in science and mathematics classrooms). More recently (2005) the Laboratory for Research on Mobile Learning Environments has been funded to study the use of mobile devices
(Pocket PCs and Tablet PCs) in chemistry and mathematics classes.
• Department Contributions - He has been chair of the department Technology Committee, worked to have graphing calculators required in many lower level classes, and helped to implement a Long Range
Computing Plan. He has been the Web Master for the Mathematics and Statistics Department, and has helped graduate students with LaTeX issues and created a thesis format for their use.
• Mathematics Taxonomy The Core Taxonomy for Mathematical Sciences Education is an approved taxonomy for use in classifying digital resources in mathematics education.
Classroom Software Written
• GraphData 2002. - Graphing Tool for Jornada Handheld for graphing and analysis of Pocket Excel Data, used in Chemistry Labs, developed in the Fall and used in 2002. Revised for Pocket PC and has
been in use for the past year in many Chemistry Labs
• Menten-Michaelis Reaction, for Organic Chemistry, 2000. on Chemical Kinetics for Biochemistry class, jointly with G. Lugo and C. Halkides. Wrote in Summer, 2000. Classroom tested in Fall 2000-03;
• Geometric Optics, 1998. Used in physics labs for three years.
• LRC Circuit Laboratory, 1997. The LRC circuit software that I had written a couple of years ago. This past year I was asked for, and granted, permission for this to be put on a CD of classroom
tested software for At Home Science, Inc. http://www.athomescience.com/ .
• Over 116 items as one of the authors have been put into the iLumina Digital Library, which is part of the NSDL (NSF supported National Science Digital Library). Recently this material has been
harvested at NSDL (National Science Digital Library) and is now part of the electronic resources accessible through Randall Library.
• I had done some work for Dr. Turrisi on automating Existential Graphs for interested people at NSA. I wrote a prototype for doing some testing of validity of statements and associated truth
tables. Both a desktop and pocket PC version were developed. The presentation, of which I wrote most is included in the supporting material.
Links to Some Materials
Some material can be found in the iLumina Digital Library by doing a search on "herman".
The information contained on this Web page and related pages
does not necessarily represent UNC Wilmington official information.
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The post Fixing lists defined a (commonly used) type of vectors, whose lengths are determined statically, by type. In Vec n a, the length is n, and the elements have type a, where n is a type-encoded
unary number, built up from zero and successor (Z and S).
infixr 5 :<
data Vec ∷ * → * → * where
ZVec ∷ Vec Z a
(:<) ∷ a → Vec n a → Vec (S n) a
It was fairly easy to define foldr for a Foldable instance, fmap for Functor, and (⊛) for Applicative. Completing the Applicative instance is tricky, however. Unlike foldr, fmap, and (⊛), pure
doesn’t have a vector structure to crawl over. It must create just the right structure anyway. I left this challenge as a question to amuse readers. In this post, I give a few solutions, including my
current favorite.
You can find the code for this post and the two previous ones in a code repository.
In the post Memoizing polymorphic functions via unmemoization, I toyed with the idea of lists as tries. I don’t think [a] is a trie, simply because [a] is a sum type (being either nil or a cons),
while tries are built out of the identity, product, and composition functors. In contrast, Stream is a trie, being built solely with the identity and product functors. Moreover, Stream is not just
any old trie, it is the trie that corresponds to Peano (unary natural) numbers, i.e., Stream a ≅ N → a, where
data N = Zero | Succ N
data Stream a = Cons a (Stream a)
If we didn’t already know the Stream type, we would derive it systematically from N, using standard isomorphisms.
Stream is a trie (over unary numbers), thanks to it having no choice points, i.e., no sums in its construction. However, streams are infinite-only, which is not always what we want. In contrast,
lists can be finite, but are not a trie in any sense I understand. In this post, I look at how to fix lists, so they can be finite and yet be a trie, thanks to having no choice points (sums)?
You can find the code for this post and the previous one in a code repository.
• 2011-01-30: Added spoilers warning.
• 2011-01-30: Pointer to code repository.
I have another paper draft for submission to ICFP 2009. This one is called Beautiful differentiation, The paper is a culmination of the several posts I’ve written on derivatives and automatic
differentiation (AD). I’m happy with how the derivation keeps getting simpler. Now I’ve boiled extremely general higher-order AD down to a Functor and Applicative morphism.
I’d love to get some readings and feedback. I’m a bit over the page the limit, so I’ll have to do some trimming before submitting.
The abstract:
Automatic differentiation (AD) is a precise, efficient, and convenient method for computing derivatives of functions. Its implementation can be quite simple even when extended to compute all of
the higher-order derivatives as well. The higher-dimensional case has also been tackled, though with extra complexity. This paper develops an implementation of higher-dimensional, higher-order
differentiation in the extremely general and elegant setting of calculus on manifolds and derives that implementation from a simple and precise specification.
In order to motivate and discover the implementation, the paper poses the question “What does AD mean, independently of implementation?” An answer arises in the form of naturality of sampling a
function and its derivative. Automatic differentiation flows out of this naturality condition, together with the chain rule. Graduating from first-order to higher-order AD corresponds to sampling
all derivatives instead of just one. Next, the notion of a derivative is generalized via the notions of vector space and linear maps. The specification of AD adapts to this elegant and very
general setting, which even simplifies the development.
You can get the paper and see current errata here.
The submission deadline is March 2, so comments before then are most helpful to me.
Enjoy, and thanks!
I’ve just finished a draft of a paper called Denotational design with type class morphisms, for submission to ICFP 2009. The paper is on a theme I’ve explored in several posts, which is
semantics-based design, guided by type class morphisms.
I’d love to get some readings and feedback. Pointers to related work would be particularly appreciated, as well as what’s unclear and what could be cut. It’s an entire page over the limit, so I’ll
have to do some trimming before submitting.
The abstract:
Type classes provide a mechanism for varied implementations of standard interfaces. Many of these interfaces are founded in mathematical tradition and so have regularity not only of types but
also of properties (laws) that must hold. Types and properties give strong guidance to the library implementor, while leaving freedom as well. Some of the remaining freedom is in how the
implementation works, and some is in what it accomplishes.
To give additional guidance to the what, without impinging on the how, this paper proposes a principle of type class morphisms (TCMs), which further refines the compositional style of
denotational semantics. The TCM idea is simply that the instance’s meaning is the meaning’s instance. This principle determines the meaning of each type class instance, and hence defines
correctness of implementation. In some cases, it also provides a systematic guide to implementation, and in some cases, valuable design feedback.
The paper is illustrated with several examples of type, meanings, and morphisms.
You can get the paper and see current errata here.
The submission deadline is March 2, so comments before then are most helpful to me.
Enjoy, and thanks!
The post Sequences, streams, and segments offered an answer to the the question of what’s missing in the following box:
┃ │infinite│finite ┃
┃discrete │Stream │Sequence┃
┃continuous │Function│??? ┃
I presented a simple type of function segments, whose representation contains a length (duration) and a function. This type implements most of the usual classes: Monoid, Functor, Zip, and
Applicative, as well Comonad, but not Monad. It also implements a new type class, Segment, which generalizes the list functions length, take, and drop.
The function type is simple and useful in itself. I believe it can also serve as a semantic foundation for functional reactive programming (FRP), as I’ll explain in another post. However, the type
has a serious performance problem that makes it impractical for some purposes, including as implementation of FRP.
Fortunately, we can solve the performance problem by adding a simple layer on top of function segments, to get what I’ll call “signals”. With this new layer, we have an efficient replacement for
function segments that implements exactly the same interface with exactly the same semantics. Pleasantly, the class instances are defined fairly simply in terms of the corresponding instances on
function segments.
You can download the code for this post.
• 2008-12-06: dup [] = [] near the end (was [mempty]).
• 2008-12-09: Fixed take and drop default definitions (thanks to sclv) and added point-free variant.
• 2008-12-18: Fixed appl, thanks to sclv.
• 2011-08-18: Eliminated accidental emoticon in the definition of dup, thanks to anonymous.
What kind of thing is a movie? Or a song? Or a trajectory from point A to point B? If you’re a computer programmer/programmee, you might say that such things are sequences of values (frames, audio
samples, or spatial locations). I’d suggest that these discrete sequences are representations of something more essential, namely a flow of continuously time-varying values. Continuous models,
whether in time or space, are often more compact, precise, adaptive, and composable than their discrete counterparts.
Functional programming offers great support for sequences of variable length. Lazy functional programming adds infinite sequences, often called streams, which allows for more elegant and modular
Functional programming also has functions as first class values, and when the function’s domain is (conceptually) continuous, we get a continuous counterpart to infinite streams.
Streams, sequences, and functions are three corners of a square. Streams are discrete and infinite, sequences are discrete and finite, and functions-on-reals are continuous and infinite. The missing
corner is continuous and finite, and that corner is the topic of this post.
┃ │infinite│finite ┃
┃discrete │Stream │Sequence┃
┃continuous │Function│??? ┃
You can download the code for this post.
• 2008-12-01: Added Segment.hs link.
• 2008-12-01: Added Monoid instance for function segments.
• 2008-12-01: Renamed constructor “DF” to “FS” (for “function segment”)
• 2008-12-05: Tweaked the inequality in mappend on (t :-># a).
While working on Eros, I encountered a function programming pattern I hadn’t known. I was struck by the simplicity and power of this pattern, and I wondered why I hadn’t run into it before. I call
this idea “semantic editor combinators”, because it’s a composable way to create transformations on rich values. I’m writing this post in order to share this simple idea, which is perhaps “almost
obvious”, but not quite, due to two interfering habits:
• thinking of function composition as binary instead of unary, and
• seeing the functions first and second as about arrows, and therefore esoteric.
What I enjoy most about these (semantic) editor combinators is that their use is type-directed and so doesn’t require much imagination. When I have the type of a complex value, and I want to edit
some piece buried inside, I just read off the path in the containing type, on the way to the buried value.
I started writing this post last year and put it aside. Recent threads on the Reactive mailing list (including a dandy explanation by Peter Verswyvelen) and on David Sankel’s blog reminded me of my
unfinished post, so I picked it up again.
• 2008-11-29: added type of v6 example. Tweaked inO2 alignment.
Fixing lists
Paper: Beautiful differentiation
Denotational design with type class morphisms
Sequences, segments, and signals
Sequences, streams, and segments
Semantic editor combinators
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Golden rectangle.
January 19th 2010, 02:44 PM #1
Jan 2010
Golden rectangle.
Hello all,
I'm wondering if someone here could kindly give me a hand to solve a small problem.
I'm trying to draft a golden spiral but I keep finding myself snagged mid-way through the drawing.
I'll explain my procedure first, leading into the issue i'm having.
I start by creating the largest square of the sequence using inches, I pick any size to fit within the page i'm working on... so anywhere from 4"x4" - 9x9.
Once i've done that I divide the length in half and draw my diagonal line from the mid-point of the length to one of the corners on the adjacent side. Then while using a compass pivoting on the
mid-point, I create my arc starting at the corner traveling down to level with the length of the square.
Now, to my understanding i've created the golden rectangle.
So, I then proceed on to creating the second, smaller square by using the measurement of the length I had just created (the golden mean of the first square)
Now i'm left with two squares, one big and one small and one small golden rectangle. Using the same method as in creating the second square, I create the third square within the smaller
rectangle, which then leaves me with an even smaller rectangle.
Repeating this pattern once or twice more I hit a point where the rectangle i'm left with is too narrow to continue on ( it's proportions are outside of being a golden rectangle ) ...For example.
The next square in the sequence I have to create may be 3"x3".. the rectangle would be 3"x8". leaving me with a 5 unit remainder and if I was to treat the 3"x3" square as the 1,1 of the fibonacci
sequence then I would create another 3"x3" square however, that creates a 6"x3" rectangle, still leaving me with a 3"x2" square...
*the measurements I used are just for a rough example.
-So, what the heck am i'm doing wrong? lol. What i'm hoping to get is a 9 square sequence for this spiral.
I would also like to mention that i've gone over my measurements many times now and all of my boxes are completely square.... my only guess is that my arc may be too wide but I don't understand
how that would be if I was accurate with the measurement of the first square....
Any help will be greatly appreciated
Hello, artofstoo!
I'm trying to draft a golden spiral but I keep finding myself snagged
mid-way through the drawing.
I'll explain my procedure first, leading into the issue i'm having.
I start by creating the largest square of the sequence using inches.
I pick any size to fit within the page i'm working on, anywhere from 4"x4" - 9x9.
Once i've done that I divide the length in half and draw my diagonal line from the midpoint
of the side to one of the corners on the adjacent side.
Then while using a compass pivoting on the midpoint, I create my arc
starting at the corner traveling down to level with the length of the square.
Now, to my understanding i've created the golden rectangle.
So, I then proceed on to creating the second, smaller square by using
the measurement of the length I had just created (the golden mean of the first square)
Now i'm left with two squares, one big and one small and one small golden rectangle.
Using the same method as in creating the second square, I create the third square
within the smaller rectangle, which then leaves me with an even smaller rectangle.
Repeating this pattern once or twice more I hit a point where the rectangle i'm left with
is too narrow to continue on. (It's proportions are outside of being a Golden Rectangle).
Your construction is correct.
D 1 C
* - - - - - *
| /| *
| / | *
1 | / | *
| / |
| / | *
* - - * - - * - - * - -
A M B P
We have square $ABCD\!:\;\;AB = BC = CD = DA = 1$
$M$ is the midpoint of $AB\!:\;\;MB = \tfrac{1}{2}$
In right triangle $CBM\!:\!:\;MC^{\:\!2} \:=\:1^2 + \left(\tfrac{1}{2}\right)^2 \:=\:\frac{5}{4} \quad\Rightarrow\quad MC = \tfrac{\sqrt{5}}{2}$
With center $M$ and radius $MC$, draw arc $CP$, cutting $AB$ extended at $P.$
Therefore: . $MP \:=\:\tfrac{1}{2} + \tfrac{\sqrt{5}}{2} \;=\;\frac{1+\sqrt{5}}{2}$ . . . the Golden Mean.
There should be no integers after the initial square.
$\text{We have a }\text{1-by-}\phi\text{ rectangle }APQD\;\text{ . . . a Golden Rectangle.}$
: - - - - - φ - - - - - :
D C Q
- *-------------*---------*
: | | |
: | | |
1 | | |
: | R *---------* S
: | | |
- *-------------*---------*
A B P
Cut off square $ABCD$
and the remaining rectangle $CQPB$ is a Golden Rectangle.
Cut off square $CQSR$
and the remaining rectangle $RSPB$ is a Golden Rectangle.
In theory, we can repeat this process forever.
. . The ratio of the sides will always be $1\!:\!\phi.$
Thanks for the response Soroban, Fancy ASCII illustration too btw haha.
The problem i am having though is when I do repeat the process.
Lets say I cut off R,S,B,P to get:
: - - - - - φ - - - - - :
D C Q
- *-------------*---------*
: | | |
: | | |
1 | | t |
: | R *---------* S
: | | | |
- *-------------*---------*
A B u P
Now I have R,T,B,U .. The problem i'm getting usually happens while squaring off this section on either this step or on the following step ( R,T,"V,W"? ) ...somehow I keep finding myself left
with more space then what I should have.
I assume, obviously, that my measurements have to be off in order for this to happen. What I don't get is how they could be off in the first place unless the "negative space" ,i'll call it, is
the summation of some earlier lines being short a millimeter or two.
I'll try to illustrate what i'm being left with -
^ ^
^ ^
C* *Q
| |
R T |
| | |
X*----------*Y |
| | |
| | |
V*----------*W |
| | |
| | |
<<*------------ *---------*-------------------*
A B U P
Alright, So this illustration shows the rectangle zoomed in to the R,T,B,U sector.
now, lets say I had just created the cut from T-U and I shifted my attention to R,T,B,U...the next line I need to create is V,W, so I take the length of B,U and bring it over to B,R to get B,V
and I create my square V,W,B,U.... Now, notice R,T,V,W... it's an usual shape.
-The cut for V,W should actually be where I placed X,Y... giving me the horizontal rectangle R,T,X,Y which actually looks a lot closer to the dimensions of a golden rectangle even in this
However, I keep getting that V,W line...which I should call "checkmate" lol because I can't follow it up with another cut. When I do I end up creating an adjacent square to V,W,B,U and of the
same dimension, leaving me with the "negative space" I was explaining earlier, similar to the shape of R,T,X,Y...
...it's definitely not an infinite spiral i'm drawing lol. If anything B,U,V,W and X,Y,V,W should both = 1 leaving me with nothing correct?
So whats potentially causing this problem?
my lines are accurate to 1/16 of an inch.
Last edited by artofstoo; January 19th 2010 at 11:33 PM.
yes code reformatting!.
-i'll have to edit that later.
January 19th 2010, 07:19 PM #2
Super Member
May 2006
Lexington, MA (USA)
January 19th 2010, 11:20 PM #3
Jan 2010
January 19th 2010, 11:40 PM #4
Jan 2010
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Computational Statistics
Computational statistics, or statistical computing, is the interface between statistics and computer science. It is the area of computational science (or scientific computing) specific to the
mathematical science of statistics. This area is also developing rapidly, leading to calls that a broader concept of computing should be taught as part of general statistical education. The terms
'computational statistics' and 'statistical computing' are often used interchangeably, although Carlo Lauro (a former president of the International Association for Statistical Computing) proposed
making a distinction, defining 'statistical computing' as "the application of computer science to statistics", and 'computational statistics' as "aiming at the design of algorithm for implementing
statistical methods on computers, including the ones unthinkable before the computer age (e.g. bootstrap, simulation), as well as to cope with analytically intractable problems" The term
'Computational statistics' may also be used to refer to computationally intensive statistical methods including resampling methods, Markov chain Monte Carlo methods, local regression, kernel density
estimation, artificial neural networks and generalized additive models.
Faculty in this Research Group
Michael Trosset, Professor of Statistics
Education: Ph.D., Statistics, University of California Berkeley, 1983
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Power & Heat
Reducing power by design
Reducing the power in a circuit means reducing voltage or current. Hence the current move away from 5V devices to 3V3 or even 1V8 and lower. Modern micro controller systems often have a 3V3 interface
with a 1V2 core, requiring two supply voltages but dissipating less power.
That is also why FETs are much preferred nowadays over transistors for switching large currents. Where as a transistor can have a saturation voltage of between 0.9V and 2V when it is fully on a FET
can have a very low “On resistance”. Typically a FET’s On resistance is in the order of 0.1R (ohms) so for 1A flowing down it it will only dissipate 1 * 1 * 0.1 = 100mW. Where as a transistor with
0.9V saturation voltage switching the same current will dissipate 0.9 * 1 = 900mW so in this case there is nine times less power being dissipated or wasted than in a transistor. Some FETs can push
this ON resistance down even further to values like 0.01R or lower.
One mistake beginners often make is to think if they are switching say a 50W load they are going to dissipate 50W in the switch, this is not the case. You only dissipate the power in the switch given
by the current down it and either the series resistance or the saturation voltage across it. So with a good FET you can switch 50W of power and only dissipate a few milliwatts in the switch.
Power supply regulation can be done using a series regulator and such three pin regulators are very common. Suppose you have a 5V regulator being supplied by a 12V supply. This means that the
regulator has to “drop” 12 - 5 = 7V across it. If the circuit is taking 500mA that is a power of 3.5W, all that is wasted in heat and has to be removed from the regulator, a large heat sink being
necessary. However, using switching regulators does not require such a voltage drop and their efficiency is around 90% so the heat dissipated in them is much smaller and is related only to the load
current, so for the same example the actual circuit is taking 5V at 500mA so the power is 2.5W. A switch mode power supply is 90% efficient so only 10% of this power is dissipated giving 0.1 * 2.5 =
250mW of heat in the power regulator. Compare this with the 3.5W of a conventional series regulator.
House keeping
Pull up resistors are usually a higher value than pull down resistors so if you have the choice use them in stead. The savings are not so dramatic but when you have a lot it all adds up. Also instead
of using a 1K pull up how about using a 10K, that is using only one tenth of the power.
LEDs are normally run at 20mA but sometimes this can be too bright, 10mA is often more than bright enough for a non multiplexing LED.
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Note: This page documents a possible idea for developing new patch types and tweaking existing ones. It doesn’t currently form the basis for any implementation.
As we add new patch types, the number of commute rules we will have to implement could grow quadratically, although in some cases we might just say that patches never commute - e.g. syntax tree edits
and hunk changes. We need a way of figuring out these rules. This proposal has one idea for doing that.
Let’s start with an example of hunk patches within a single file. These are currently represented as a line number plus a sequence of existing lines to remove and a sequence of lines to put in their
place. For simplicity lets ignore the surrounding file system and just think about the file itself.
We can model the hunk patch as a Haskell type (playing fast and loose with the precise syntax):
Hunk (n :: Int) (old :: [String]) (new :: [String])
In fact, we don’t really care whether the contents are Strings or not. Let’s abstract that to the type variable line:
Hunk (n :: Int) (old :: [line]) (new :: [line])
The underlying assumption here is that a file is modelled as a list of lines; we can index into the file at position n, chop out old and insert new.
Let’s call the initial contents of the file oldcontents:
oldcontents :: [line]
If the hunk patch applies to the file, then we actually know that oldcontents must contain old at position n:
oldcontents = prefix ++ old ++ suffix
length prefix = n
From this we can get the new contents:
newcontents = prefix ++ new ++ suffix
How does this help us? Let’s now consider two patches in sequence:
patch1 = Hunk n1 old1 new1
patch2 = Hunk n2 old2 new2
The overall goal is to figure out how we can commute the sequence patch1; patch2 into an alternate sequence patch2'; patch1'
The file contents then goes through three states - initial, intermediate (after patch1) and final (after patch2):
initial = prefix1 ++ old1 ++ suffix1 length prefix1 = n1 intermediate = prefix1 ++ new1 ++ suffix1 intermediate = prefix2 ++ old2 ++ suffix2 length prefix2 = n2 final = prefix2 ++ new2 ++ suffix2
We have two equations for intermediate. Consider how the two possible breakdowns might line up, and in particular where old2 is found within prefix1, new1 and suffix1. I’ll break it down into three
1. old2 is entirely contained within prefix1, i.e. length prefix2 + length old2 <= length prefix1
2. old2 is entirely contained within suffix1, i.e. length prefix2 >= length prefix1 + length new1
3. otherwise old2 overlaps with new1
Appealing to intuition, we declare that case 3 implies that patch1 and patch2 don’t commute.
I’ll consider only case 1 for now.
Given condition 2, we can say that
prefix1 = prefix2 ++ old2 ++ gap
and so our set of equations becomes
initial = prefix2 ++ old2 ++ gap ++ old1 ++ suffix1
length prefix2 + length old2 + length gap = n1
intermediate = prefix2 ++ old2 ++ gap ++ new1 ++ suffix1
intermediate = prefix2 ++ old2 ++ suffix2
length prefix2 = n2
final = prefix2 ++ new2 ++ suffix2
from this we can derive
suffix2 = gap ++ new1 ++ suffix1
and so our set of equations becomes
initial = prefix2 ++ old2 ++ gap ++ old1 ++ suffix1
length prefix2 + length old2 + length gap = n1
intermediate = prefix2 ++ old2 ++ gap ++ new1 ++ suffix1
length prefix2 = n2
final = prefix2 ++ new2 ++ gap ++ new1 ++ suffix1
Again appealing to intuition, after a commute, the intermediate state between patch2' and patch1' should be
intermediate' = prefix2 ++ new2 ++ gap ++ old1 ++ suffix1
from this, we can derive
patch2' = Hunk (length prefix2) old2 new2
patch1' = Hunk (length prefix2 + length new2 + length gap) old1 new1
Using what we know about the lengths, we get
patch2' = Hunk n2 old2 new2
patch1' = Hunk (n1 + length new2 - length old2) old1 new1
and this is subject to the condition that
n2 + length old2 <= n1
Using a similar argument for case 2 gives us that
patch2' = Hunk (n2 - length new1 + length old1) old2 new2
patch1' = Hunk n1 old1 new1
n2 >= n1 + length new1
(TODO: check this, I didn’t actually go through the derivation :-)
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Find all the value of `x` satixfying the condition in as many way as possible:
`root(3)(25x(2x^2+9))>= 4x+3/x`
`... - Homework Help - eNotes.com
Find all the value of `x` satixfying the condition in as many way as possible:
`root(3)(25x(2x^2+9))>= 4x+3/x`
` `
` `
We are asked to find all values for x satisfying `root(3)(25x(2x^2+9))>=4x+3/x`
(1) Looking at the graph of the left side and the right side it appears that the condition holds for all x<0. The graph of the left side in black, the right side in red:
But you might see a point of equality in the first quadrant.
(2) `root(3)(25x(2x^2+9))>=4x+3/x` Cube both sides -- this does not change the inequality
Multiply by `x^3` -- note that we have two cases:
(a) x>0 ; the inequality remains the same.
This function is nonnegative for all x>0 -- however it is zero at `x=+-sqrt(3)` .
Thus in the original inequality, there is an equality at `x=sqrt(3)`
(b) x<0; the inequality changes.
`14x^6-81x^4+108x^2+24>=0` is true for all x<0.
The solutions to `root(3)(25x(2x^2+9))>=4x+3/x` are x<0 and `x=sqrt(3)` . (There is equality at `x=+-sqrt(3)` )
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Conservation Laws
What Pete and Ich were describing can be summed up in the conservation of the
for any isolated system. In one nice package the conservation of four-momenum gives you the classical conservation of momentum, conservation of energy, and conservation of mass.
The relevant quote from the Wikipedia link:
"Note that the mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame counts as system mass. As an example, two
particles with the four-momentums {-5 Gev, 4 Gev/c, 0, 0} and {-5 Gev, -4 Gev/c, 0, 0} each have (rest) mass 3 Gev/c2 separately, but their total mass (the system mass) is 10 Gev/c2. If these
particles were to collide and stick, the mass of the composite object would be 10 Gev/c2."
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[Help-gsl] Does anybody know how to use FFT to compute numerical integra
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[Help-gsl] Does anybody know how to use FFT to compute numerical integra
From: Michael
Subject: [Help-gsl] Does anybody know how to use FFT to compute numerical integration?
Date: Wed, 27 Jun 2007 20:41:16 -0700
The FFT used for numerical integral has the problem of non-adaptive sample
points, which is inefficient.
However I have to use it in my 2D integration problem, for the following
My 2D integration is:
Integrate( F(v) * FourierTransform[g(t)] (v), v from -infinity to
+infinity ).
g(t)'s evaluation is costly. So I plan to fix the parameter for g(t), and
compute the Fourier Transform of g(t) only once, and store in memroy. And
then in calibration loop, I only vary the parameters for F(v). And each time
for each different set of parameters of F(v), I compute the dot-product of
F(v) sample points and the FT[g(t)] sample points to obtain approximation to
the integral.
My question is: how to improve the accuracy of FFT-based integration? I know
it's inefficient, but is there any remedy at least?
Moreover, is there a better adaptive quadature based "smart" integration
method that can help me deal with the above situation efficiently? I am
thinking of doing a cache for the Fourier Transform of g(t), which is
FT[g(t)](v), since adaptive quadature based integration may sample different
point of FT[g(t)](v) each time... but perhaps the overhead introduced in the
cache may outweigh the smart adaptive integration itself...
Any suggestions? Thanks a lot!
[Prev in Thread] Current Thread [Next in Thread]
• [Help-gsl] Does anybody know how to use FFT to compute numerical integration?, Michael <=
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[SciPy-dev] Ideas for scipy.sparse?
Viral Shah vshah@interactivesupercomputing....
Mon Apr 14 13:36:31 CDT 2008
Hi Brian, and others,
I have been following this thread for the past few days, and its been
quite interesting. Some of my comments interspersed below:
> So, I am currently implementing a distributed memory array package
> for python:
> http://projects.scipy.org/ipython/ipython/browser/ipythondistarray
> The goal is to have distributed/parallel arrays that look and feel
> just like numpy arrays. Here is an example:
> import ipythondistarray as ipda
> a = ipda.random.rand((10,100,100), dist=(None,'b','c'))
> b = ipda.random.rand((10,100,100), dist=(None,'b','c'))
> c = 0.5*ipda.sin(a) + 0.5*ipda.cos(b)
> print c.sum(), c.mean(), c.std(), c.var()
> This works today on multiple processors. Don't get too excited
> though, there is still _tons_ of work to be done....
This looks really exciting. I looked at the project, and it seems like
you are using MPI as the basic infrastructure - but I also see
references to GASNet. My first question looking at this was, why not
just use Global Arrays for the basic array infrastructure ? It has a
bunch of stuff that would let you get quite far quickly:
I believe, looking at the project organization, that you plan to add
fft, sparse arrays, the works.
I must mention that I work with Star-P (www.interactivesupercomputing.com
), and we have plugged in our parallel runtime to overload parts of
numpy and scipy. We have sparse support for M (matlab), but not for
python yet.
I am guessing that your framework allows one to run several instances
of python scripts as well on each node ? That way, one could solve
several ODEs across nodes. You'd need some dynamic scheduling for
workload balancing, perhaps.
Have you thought about integrating this with the stackless python
project, for a nice programming model that abstracts away from MPI and
allows high performance parallel programs to be written entirely in
python ?
> Here is the main issue: I am running into a need for sparse arrays.
> There are two places I am running into this:
> 1) I want to implement sparse distributed arrays and need sparse local
> arrays for this.
> 2) There are other places in the implementation where sparse arrays
> are needed.
> Obviously, my first though was scipy.sparse. I am _really_ excited
> about the massive improvements that have been happening in this area
> recently. Here are the problems I am running into:
Not having followed the numpy/scipy debate for too long, I must say
that I was a little surprised to find sparse array support in scipy
rather than numpy.
> 1) I need N-dimensional sparse arrays. Some of the storage formats in
> scipy.sparse (dok, coo, maybe lil) could be generalized to
> N-dimensions, but some work would have to be done.
From the discussion, it seemed to me that these formats are written
in python itself. In general, I like the idea of writing data
structures in the same language, and as much in higher level languages.
Some people pointed out that this tends to be slow when called in for
loops. I couldn't figure out what the cause of this was. Is it that
for loops in python are generally slow, or is it that indexing
individual elements in sparse data structures is slow or are python's
data structures slow, or some combination of all three ?
I can say something about Matlab's sparse matrices, which I am
familiar with. Its not that Matlab's for loops are the cause of
slowness (they may be slow), but that indexing single elements in a
sparse matrix is inherently slow. Every instance of A(i, j) has to
perform a binary search. Insertion is even more expensive because it
requires a memmove for CSC/CSR formats. Thus you almost never work
with for loops and single element indexing for sparse matrices in
Matlab. That said, one could design a data structure that partially
solves some of these problems.
I thought initially that dok may solve this for some kinds of sparse
matrix problems, but it seems that its too slow for large problems.
Perhaps a little benchmark must be setup.
For N-d are you thinking of something along the lines of Tammy Kolda's
tensor toolbox for Matlab ? That could be accomplished with just
distributed dense vectors - I'm not sure what the performance would be
like. Its the same as using a distributed coo format.
> 3) That we begin to move their implementation to using Cython (as an
> aside, cython does play very well with templated C++ code). This
> could provide a much nicer way of tying into the c++ code than using
> swig.
Not being intimately familiar with cython or swig, I did take a look
at the cython homepage and it sounded like a good fit. It does seem
that some people have strong oppositions to it. Support for templated
C++ code would be nice to have though !
More information about the Scipy-dev mailing list
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Re: On Formal IS-A definition
From: Keith H Duggar <duggar_at_alum.mit.edu> Date: Sat, 22 May 2010 12:53:46 -0700 (PDT) Message-ID: <ad1c2142-724e-4d85-b43d-9662ddee91ec_at_q13g2000vbm.googlegroups.com>
On May 17, 10:57 pm, David BL <davi..._at_iinet.net.au> wrote:
> On May 10, 5:54 pm, David BL <davi..._at_iinet.net.au> wrote: > > On May 10, 1:34 pm, Keith H Duggar <dug..._at_alum.mit.edu> wrote: > > > On May 9, 9:29 am, David BL <davi..._at_iinet.net.au>
wrote: > > > > On May 9, 11:38 am, Bob Badour <bbad..._at_pei.sympatico.ca> wrote: > > > > > > My set of three variables and a dog fully complies with ZFC. > > > > > Here is a quote from (http://
en.wikipedia.org/wiki/Zermelo > > > > %E2%80%93Fraenkel_set_theory) > > > > > "ZFC has a single primitive ontological notion, that of a hereditary > > > > well-founded set, and a single ontological
assumption, namely that all > > > > individuals in the universe of discourse are such sets. Thus, ZFC is a > > > > set theory without urelements (elements of sets which are not > > > > themselves
sets)." > > > > > and this (fromhttp://en.wikipedia.org/wiki/Hereditary_set) > > > > > "In set theory, a hereditary set (or pure set) is a set all of whose > > > > elements are hereditary sets. That
is, all elements of the set are > > > > themselves sets, as are all elements of the elements, and so on." > > > > > I wonder whether Bob enjoys putting a leash on a set and taking it for > > > > a
walk. > > > > I wonder if you know what a variable is? Or more specifically I > > > wonder if you can prove that a variable is not a set? Well, that > > > is a rhetorical question really because I
already know that a > > > variable /is/ a set. Or rather, because the word "is" is vacuous > > > most of the time, a variable can be represented by a set. Since > > > you enjoy wikipedia so much
(since when did wikipedia become an > > > authoritative source?) try reading this (thoughtfully): > > > > http://en.wikipedia.org/wiki/Variable_(mathematics) > > > > and see if you can figure out how
it is that variables can be > > > represented by sets. Hint, a variable is a /symbol/. > > > You are using "variable" in the sense that a logician would use it. > > This discussion actually began
with variables accessed by programs > > that support imperative assignment statements. Let's be sure we > > don't confuse these. > > > In any case you are still wrong. I believe you are suggesting
one > > can > > > 1) Have a symbol x > > > 2) Form a set {x} > > > 3) Have a logic formula where symbol x is a variable, such as > > > for all x, x+0 = x > > > 4) Deduce that variables can appear in
sets. > > > I accept 1), 2) and 3) but not 4). You make the mistake of thinking > > that symbols represent variables outside the context of the formula > > they appear in - even when the variable is
bound. If that were true > > that would be remarkably bad! > > I've been reading the following Stanford articles: > > http://plato.stanford.edu/entries/types-tokens/ > > (note well section 8 on
occurrences), and > > http://plato.stanford.edu/entries/logic-classical/ > > I'm going to eat my words. I see now that I was wrong. I was > associating the term "variable" with occurrences of
symbols, whereas > the Stanford articles go to the trouble to distinguish between a > variable and an occurrence of a variable. > > Therefore assuming this terminology it is indeed valid to have a
set > of variables.
Good. We've come full circle jerk in yet another extravagant DBL pose fest. You should study how your profound ignorance and lame attraction to fallacies (context shifts, strawmen, etc) required
nearly 30 posts across multiple days and posters to correct.
> According to Section 4 (Semantics) in the SEP article on
> classical logic, an interpretation M = <d,I> assigns denotations to
> constants. E.g. For constant c, I(c) is an element of d, whereas a
> variable-assignment function s on M is required to assign a denotation
> to a free variable. The denotation of variable v is s(v) which is an
> element of d, not the variable itself. It seems that although one can
> have sets of variables, it is rather difficult to denote them!
"not the variable itself" and "it is rather difficult to denote them!" are just more nonsense meaningless drivel.
> > > DBL knows that in formal semantics /variables/ are /interpreted/
> > Wrong (assuming "interpreted" means mapped by an interpretation
> > function). Only function symbols and predicate symbols are
> > interpreted.
> I was correct there.
No, you were and remain wrong because the context was not FOL and never has been! These ignorant context shifts you keep trying to impose on the discussion are plain stupid.
The context of my statements was and continues to be mathematics and formal languages in general and formal semantics in general. In that broader context variable assignment functions are simply a
"part" of an interpretation. Read section 1 of the following:
Do you understand now? An "interpretation" is the totality of the "added information". Or as wikipedia concisely puts it
"an interpretation is an assignment of meaning to the symbols of a language." Across a variety of formal languages and semantics this is extra information is formalized as a /relation/. Sometimes
that relation is thought of in parts (for various reasons) such as the "denotation assigment function" and the "variable assignment function" etc.
But of course, you are near totally ignorant of this broader context. As evidenced by this post
you were even ignorant of the field of formal semantics until I told you about it a month ago. Now, after a month, you think you are qualified to pronounce yourself right and your teacher wrong??
This has got to be one of the clearest examples of an idiotic vociferous ignorant poser we've seen in a long while.
That what the rest of formal semantics calls a "model" (which is exactly why it is commonly represented by the letter M even in FOL) is often called the "interpretation" in classic first order
interpretation, is completely irrelevant to the more general context of model theory applied to mathematics and formal languages as a whole.
As already demonstrated you were nearly ignorant of all this even just days ago. Had you been aware of variable assignment functions you would have understood that my general point made in a more
general context, applied equally well to FOL because a function (the variable assignment function) is a relation!
> > > DBL knows that an interpretation is formally a /relation/ mapping
> > > variables (and all other symbols) to elements of the /domain of
> > > interpretation/ also sometimes called a "universe"
> > Wrong. FOL variables are not mapped to anything. They are *only*
> > used to express quantification in logic.
Obviously the above is flat wrong because free variables are not used to express quantification. This is part of the Dense Bullshit and Lies (DBL) that is so time consuming to respond to.
Also, we see yet another example of you trying to impose a context shift (from languages in general to FOL only). A dishonest "tactic" that is so blatantly easy to spot for those trained to do so and
yet so annoying and time consuming to repeatedly correct.
> > A sentence (i.e. formula where all variables are bound) is interpreted
> > according to the semantics of existential or universal quantification
> > on the bound variables that are assumed to range over the universe of
> > discourse.
> I will qualify that. An interpretation function I doesn't map a
> variable to anything. Rather a variable assignment function s defined
> on an interpretation M is used to assign denotations to free
> variables.
> Keith's comment was incorrect. A variable assignment function s is
> not part of an interpretation M, and therefore it is incorrect to say
> that an interpretation M assigns a denotation to a free variable.
Wrong. See above. In the general context of formal semantics and model theory the variable assignment functions discussed in FOL are just one part of what formal semantics calls "interpretation".
The problem was and remains that DBL is nearly completely ignorant of formal semantics. He's never sat in a class for it, never worked through examples of interpretation, never heard a professor warn
you of some common ambiguities and overloaded terminology and to explain the history behind them. In other words, DBL is ignorant of the whole and worse is vociferously arrogant in that ignorance.
KHD Received on Sat May 22 2010 - 14:53:46 CDT
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Comparison for formal local cohomology
up vote 3 down vote favorite
Let $(R, \mathfrak{m})$ be a local ring and $X = Spec(R)$. Let $Y = V(I)$ be a closed subscheme of $X$, defined by an ideal $I \subset R$, and let $P \in X$ (in fact, $P \in Y$) be the closed point.
Let $(\hat{X}, \mathcal{O}_{\hat{X}})$ be the formal completion of $X$ along $Y$ ($\hat{X} = Y$ as a topological space, and the sheaf of rings $\mathcal{O}_{\hat{X}}$ is $\varprojlim \mathcal{O}_X/\
Can we write the cohomology of $\mathcal{O}_{\hat{X}}$ with support in the closed point $P$ in terms of local cohomology on $R$? Specifically, do we have isomorphisms $H^j_P(\hat{X}, \mathcal{O}_{\
hat{X}}) \simeq \varprojlim H^j_{\mathfrak{m}}(R/I^n)$?
Without the "support at P" subscript, this is Corollaire 4.1.7 of EGA III, Grothendieck's comparison theorem. So essentially what I am asking is: is there an analogue of the comparison theorem for
local cohomology?
One natural strategy for deducing a local comparison theorem from the global one would be to use the long exact sequence relating $H^j_P(\hat{X}, \mathcal{O}_{\hat{X}})$ to $H^j(\hat{X}, \mathcal{O}_
{\hat{X}})$ and $H^j(\hat{X} \setminus P, \mathcal{O}_{\hat{X}})$, but this runs into trouble because the inclusion $X \setminus P \hookrightarrow X$ won't be proper, so the global comparison theorem
will fail for $X \setminus P$.
SGA 2 would seem to be the natural place to look for results of this type, but its Expose IX (where formal completions are treated) doesn't address local cohomology, as far as I can tell.
ag.algebraic-geometry local-cohomology
The topological space of the formal scheme $\widehat{X}$ consists of the single point $P$ (so $\widehat{X} - P$ is empty). The functor of global sections on $\widehat{X}$ with supports at $P$ is
the same as the one without mention of the supports, so it is an exact functor (even coincides with the identity functor, so to speak). Hence, $H^j_P(\widehat{X},\cdot)$ vanishes for $j > 0$ and
is the identity functor for $j = 0$. Is there some motivation to guide the way to an interesting reformulation, or is this idle curiosity? – user29720 Feb 27 '13 at 6:10
1 @kreck: I think the poster is completing along $I = 0$, so the topological space of $\widehat{X}$ is the same as that of $Spec(R/I)$. @Nick: The obstruction to the isomorphism you want lies in the
$\lim^1$ of the projective system $\{ H^{j-1}_m(R/I^n) \}$. I don't see a reason why these should vanish in general. Have you tried looking at homogeneous ideals (so it becomes a question in
projective geometry)? – anon Feb 28 '13 at 2:33
@anon: Ah, so I was misreading; thanks for the correction. – user29720 Feb 28 '13 at 12:33
1 Actually, now it seems to me that this is always true for any noetherian ring $R$: each $H^i_m(R/I^n)$ is an artinian $R$-module, so the ML condition in the comment above is trivially verified (as
any projective system of artinian $R$-modules is automatically ML). – anon Feb 28 '13 at 15:27
I agree with anon's comment. Proposition 2.2 in the reference in my answer below has several parts and the Gorenstein assumption may be needed for the proof of other parts of the proposition, but
I don't see it used in the proof of the statement you want. – Mahdi Majidi-Zolbanin Feb 28 '13 at 19:25
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1 Answer
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Dear Nick: If your local ring $R$ is Gorenstein, then $H^i_p(\hat{X},\mathcal{O}_{\hat{X}})\cong\varprojlim H^i_{\mathfrak{m}}(R/I^n)$ for all $i$, as you want. This is proved, for
instance, in Proposition 2.2, page 334 of A. Ogus' Local cohomological dimension of algebraic varieties, Annals of Mathematics, Vol. 98, No. 2 (1973). He attributes it to Peskine
Szpiro. You should also check Proposition A8 and Corollary A9 on pages 362 and 363 of Ogus' paper, as well as Propositions 2.1 and 2.2 on pages 106-107 of Peskine-Szpiro's Dimension
projective finie et cohomologie locale.
up vote 3
down vote I would like to add that as anon commented above, the proof given by Ogus is achieved by showing that when $R$ is Gorenstein the suggested (by anon) $\mathrm{lim}^1$ vanishes.
Edit. Following anon's comment, I don't think Gorenstein assumption is needed for this.
Excellent. Thank you very much for this. – Nick Switala Mar 1 '13 at 1:18
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Management Accounting
Posted by please help !!! on Friday, May 25, 2007 at 8:27pm.
Deer Valley Lodge, a ski resort in the Wasatch Mountains of Utah, has plans to eventually add five new chair lifts. Suppose that one lift costs $2 million and preparing the slope and installing the
lift costs another $1.3 million. The lift will allow 300 additional skiers in the slopes, but there are only 40 days a year when the extra capacity will be needed. (Assume that Deer park will sell at
300 lift tickets on those 40 days) Running that new lift will cost $500 a day for the entire 200 days the lodge is open. Assume that the lift tickets at Deer Valley cost $55 a day and the added cosh
expenses for each skier-day are $5. The new lift has an economic life of 20 year.
1. What if the before-tax required rate of return for Deer Valley is 14%. Compute the before tax NPV of the new lift and advise the managers of Deer Valley about adding the lift to be profitable
2. What if the after-tax required rate of return for Deer Valley is 8% the income tax rate is 40%, and the MACRS recovery period is 10 years. Compute the after-tax NPV of the new lift and advise the
managers of Deer Valley about adding the lift will be profitable investment.
3. What subjective factors would afect the investment decision?
An Excel spreadsheet is very helpful for these kinds of questions. Plug in what you know. The capital cost is $3.3 million. The net increase revenue stream is 40*300*(55-5) = $0.6 million for 20
years. Discount the first year by 1.14, the second year by 1.14^2, and so on. Add discounted revenues to get the NPV of all future revenues. Compare this to $3.3 million to see if the lift is a
profitable investment.
2) Taxes complicate the problem, but do not change the basic nature of the problem. Instead of a 14% discount rate, use an 8% rate. Second, convert the revenue stream to and after-tax stream. Using
straight line depreciation, the Lodge would get a $0.33 deduction for the first 10 years. So year 1 after tax revenue is (1-.4)*(0.6-.33)=$.162 million. Deflate this by 1.08. Repeat for 9 more years,
at which time the depreciation deduction goes away. But keep going for another 10 years.
Assume that the after-tax required rate of return for Deer Valley is 8%, the income tax rate is 40%, and the MACRS recovery period is 10 years. Compute the after-tax NPV of the new lift and advise
the managers of Deer Valley about whether adding the lift will be a profitable investment
• Management Accounting - ask, Sunday, February 7, 2010 at 10:54pm
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