content stringlengths 86 994k | meta stringlengths 288 619 |
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(a) Laws of Red-Shifts
(Stationary Universe) dk r
(Expanding '' ) = kr + l r^2 + ...
(k = 5.37 × 10^-10; l = 2.54 × 10^-19; r = distance)
(b) Laws of Nebular Distribution
(Stationary Universe) log[10] N[m] = 0.6 m[c] + C[s]
(Expanding ,, ) = 0.6(m[c] - dC[v]) + C[e]
C[s] = - 4.437; C[e] = - 4.391; N[m] = number of nebulae over entire sky, brighter than m[c]; m[c] = apparent magnitude corrected for local obscuration and for energy effects of red-shifts; dm[c].
The term C[v] is represented by the equation
in which x = r / R[0] where R[0] is the present radius of spatial curvature, and r is the distance corresponding to apparent magnitude (m[c] - d
(c) Derived Constants in an Expanding Universe
Present Radius of Spatial Curvature = R[0] = 4.7 × 10^8 (light-years).
Cosmological Constant = 4.5 × 10^-18 (years)^-2.
Present Mean Density = 6 × 10^-27 gm./cm^3.
The table neatly summarizes the explorations in the observable region of space, the immense region within which reliable data can be gathered with the largest telescope in operation. The results are
a definite step in the observational approach to cosmology. The field is entered, the problems formulated, and the solutions restricted to a fairly narrow range. But the essential clue, the
interpretation of red-shifts, must still be unravelled. The former sense of certainty has faded and the clue stands forth as a problem for investigation.
Larger telescopes may resolve the question, or theory may be revised to account for the new data. But with regard to relativistic cosmology in its present form, and the observations now available,
the conclusion can be stated quite simply. Two pictures of the universe are sharply drawn. Observations, at the moment, seem to favour one picture, but they do not rule out the other. We seem to
face, as once before in the days of Copernicus, a choice between a small, finite universe, and a universe indefinitely large plus a new principle of nature. | {"url":"http://ned.ipac.caltech.edu/level5/Sept04/Hubble/Hubble4.html","timestamp":"2014-04-19T01:51:16Z","content_type":null,"content_length":"5460","record_id":"<urn:uuid:5fe944e7-2630-4f15-8f9e-5ed8612ba9f0>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00423-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ortho Pictures
This work portrays several ways in which a pair (triplet) of completely independent 2D data sets, such as regular 2D color pictures or Height fields, are merged into one 3D model, creating an
Ortho-picture. The Ortho-picture will look like the first input set from one view and will identify with the second set from an orthogonal view. While techniques to reconstruct 3D geometry from
several 2D data sets of the same 2D models are well known, herein we strive to merge pairs of completely independent 2D input data sets. The end result of this effort is a regular 3D model that is
automatically synthesized from a pair of two completely independent pictures or 3D objects.
Given the following two pictures:
One can synthesize the following output from two orthogonal views, using zero-dimensional blob primitives:
Output as an animated gif (click on the animation to see a better resolution, DIVX, AVI animation):
One can synthesize similar 3D geometry using long strips instead blobs:
With strips ordered in random, or alternatively along a single diagonal plane:
Colors are possible too while manufacturing a colored version is still an open question:
Output as an animated gif for three (!) images (click on the animation to see a better resolution AVI animation): | {"url":"http://www.cs.technion.ac.il/~gershon/V3dShaded/","timestamp":"2014-04-19T22:16:55Z","content_type":null,"content_length":"2473","record_id":"<urn:uuid:6672314a-d804-4fd3-b67b-0ca228dba24c>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00166-ip-10-147-4-33.ec2.internal.warc.gz"} |
Enumerating 0-1 finite boxes without null rays.
up vote 2 down vote favorite
Here rays are called lines. Call $M(a_1,a_2)$ the number for matrices of length $a_1$ and height $a_2$, made of $0$ and $1$, having neither null vector nor null co-vector. In other words any line
(row or column) contains at least one $1$.
QUESTION 1 : How to count them?
QUESTION 2 : Same as above in higher dimensions.
It looks like a standard problem:
For example $M(1,p) =1$ , $M(2,p) = 3^p -2$ , $M(3,p) = 7^p -3.3^p +3$ , $M(4,p) = 15^p -4.7^p + 6.3^p -4$ The closed formula being clear but I have no clean proof for it.
Question for higher dimensions; for a cube $M(a_1,a_2,a_3)$ in $0-1$ ($2^{a_1a_2a_3}$ of them) with no null lines (in any of the three directions ).
For example $M(2,2,2) = 35$ (by hand).
Is there a closed formula for M(a_1,a_2,...a_k) in particular k = 3,4?
It seems that things change a lot when passing from k=2 to k=3, Even for k=3 and constraining last dimension to 2 : $M(a_1,a_2,2)=? $
co.combinatorics enumerative-combinatorics
1 I think something is wrong with your examples since clearly $M(a_1,a_2)=M(a_2,a_1)$ but your formulas give $M(3,4)=2161$ and $M(4,3)=1421$. Anyways, here is $M(a,a)$: oeis.org/A048291 – Casteels
Mar 29 '13 at 19:26
$M(4,p)=15^p−4\cdot 7^p+6\cdot 3^p−4$. The general 2-D case a simple inclusion-exclusion. Higher dimensions are harder. – Brendan McKay Mar 30 '13 at 6:04
@Casteels : OK you are right there was a typo the 4 and the 6 where exchanged in front of the powers ^p. – Jérôme JEAN-CHARLES Mar 30 '13 at 18:21
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2 Answers
active oldest votes
Define $$ N(m,r,s) = \begin{cases} 0 & \text{if } r+s\gt m, \text{ otherwise} \\\\ 3^m-2 & \text{if } r=s=0 \\\\ 3^{m-s}-1 & \text{if } r=0,s\gt 0, \\\\ 3^{m-r}-1 & \text{if } r\gt 0,s=0, \
\\\ 3^{m-r-s} & \text{if } r,s\gt 0. \end{cases} $$ Then $$ M(m,n,2) = \sum_{r,s,\ge 0} (-1)^{r+s} \binom{m}{r,s,m-r-s} N(m,r,s)^n. $$
For $M(n,n,2)$ I get 1, 35, 12757, 35420099, 780742441861, 145246791109197875, ...
Not in OEIS (and shouldn't be put there without checking).
Proof: $N(m,r,s)$ is the number of $2\times m$ binary arrays such that none of the $m+2$ lines sum to 0, and moreover a specified set of $r$ positions in the top row are 0 and a disjoint
up vote 1 specified set of $s$ positions in the bottom row are 0. Now make an array by placing $n$ such arrays vertically. There are $2m$ horizontal lines which we still have to make non-zero. The
down vote multinomial counts the ways to choose $r$ positions in the top row and $s$ in the bottom row (which must be disjoint positions or one of the 2-element lines is zero). The formula is then
just inclusion-exclusion.
The asymptotic number is $$ N(n,n,2) \sim 3^{n^2}, $$ which is easy to prove.
Another case: $M(n,2,2)$ is 1, 35, 313, 2339, 16681, 117395, 823033, 5763779, 40351561, 282471155, 1977318553, 13841270819, ... The sum above only has 4 non-zero terms so there is an
explicit formula. Maybe it is $M(n,2,2) = 7^n - 2^{n+2} + 2$.
To get your formula for $M(n,2,2)$ via inclusion-exclusion, stack $n$ levels of $2 \times 2$ matrices, then use $M(2,2)=7$ to overcount the number of ways to fill each level. This
includes $4 \cdot 2^n$ cases where one of the four $n$-lines is all $0$s, and that overcounts the $2$ cases when two lines are all $0$s. The same inclusion/exclusion idea gives $M(n,3,2)
= 25^n -6(8^n+3^n) +6(3\cdot2^n +1)$, and the asymptotics $M(m,n,p) \sim M(m,n)^p$. – Zack Wolske Mar 30 '13 at 10:51
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To extrapolate on Brendan McKay's comment for question one, here is a recurrence for $M(m,n)$. Though not exactly an inclusion exclusion argument (it's not hard to turn it into one), this
essentially counts the matrices with no null rows two ways.
For matrices of size $m \times n$, there are $(2^n-1)^m$ with at least one $1$ in each of the $m$ rows, since each row can be anything but the all $0$ vector.
On the other hand, given $k$ columns, there are $M(m,k)$ matrices with at least one $1$ in each row, and in each of those $k$ columns, and all $0$s in the other columns. If a matrix in the
up vote 1 first count fails to have at least one $1$ in each column, then it has some number of all $0$ columns - at least $1$, and at most $n-1$. This gives $$ M(m,n) = (2^n-1)^m - \sum_{k=1}^{n-1}\
down vote binom{n}{k}M(m,k) $$ Putting the summation on the other side gives the counts of matrices with no null rows.
Attempting to generalize this to higher dimensions didn't get past the very small numbers before breaking into a plethora of cases - even $M(m,3,2)$ and $M(m,3,3)$ are difficult. Maybe you
can see a better way to extend it than I can.
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Boece - Book 4
Bo4 p1 18 namely a ryght gret cause of my sorwe: that so 18
Bo4 p1 38 " that were a greet merveille and an abaysschinge 38
Bo4 p1 42 hous of so mochel a fadir and an ordeynour of 42
Bo4 p1 47 I have concluded a litel herebyforn ben kept 46
Bo4 m1 13 and he, imaked a knyght of the clere 82
Bo4 p2 31 yif thou see a wyght that wolde geten that 144
Bo4 p2 36 " And yif thou se a wyght, " quod sche, " that 150
Bo4 p2 42 who seith, in so moche as a man is myghty to 156
Bo4 p2 43 doon a thing, in so mochel men halt hym 156
Bo4 p2 65 " This is a verray consequence, " quod I. 178
Bo4 p2 105 " Thanne, " quod sche, " yif that a wight be 216
Bo4 p2 189 a merveile to seien, that schrewes, whiche 300
Bo4 p2 196 myghtest seyn of the careyne of a man, that it 308
Bo4 p2 197 were a deed man, but thou ne myghtest nat 308
Bo4 p2 198 symply callen it a man; so graunte I wel forsothe 310
Bo4 p2 216 and proevid a litil herebyforn that evel is 328
Bo4 p2 223 I have diffinysched a litil herbyforn that no thing 334
Bo4 p2 250 desired ben referred to good, ryght as to a 362
Bo4 p3 9 mede of that; as thus, yif a man renneth in 412
Bo4 p3 16 right as a comune mede, which mede ne 420
Bo4 p3 44 yaf the a litel herebyforn, and gadre it togidre 448
Bo4 p3 80 lerned a litil herebyforn that alle thing that 484
Bo4 p3 102 betidith it that, yif thou seest a wyght that be 506
Bo4 p3 104 that he be a man. For if he be ardaunt in avaryce, 508
Bo4 p3 105 and that he be a ravynour by violence of foreyne 508
Bo4 p3 109 likne hym to the hownd; and if he be a 512
Bo4 p3 114 bereth the corage of a lyoun; and yif he be 518
Bo4 p3 124 bounte and prowesse, he forletith to ben a man; 528
Bo4 p3 126 God, he is torned into a beeste. 530
Bo4 m3 10 hem is coverid his face with forme of a 540
Bo4 m3 11 boor; the tother is chaungid into a lyoun 540
Bo4 m3 14 chaunged into a wolf, and howleth whan he 544
Bo4 m3 16 the hows as a tigre of Inde. But al be it so 546
Bo4 m3 43 todrawen a man to hem more myghtely than 572
Bo4 p4 16 a gret partie of the peyne to schrewes scholde 594
Bo4 p4 42 this lif, that is long to abyde, nameliche to a corage 620
Bo4 p4 45 is ofte destroyed by a sodeyn ende, or 622
Bo4 p4 50 lengest is a schrewe. The whiche wikkide 628
Bo4 p4 64 an hard thing to accorde hym to a conclusioun, 642
Bo4 p4 68 nis nat spedful to a necessarie conclusioun; 646
Bo4 p4 110 he is a wrecche, that ther be yit another 686
Bo4 p4 142 concluded a lytel herebyforn. But I preye the 718
Bo4 p4 152 ben excercised by a purgynge mekenesse; but 728
Bo4 p4 212 wyltow seyn of this: yif that a man hadde al 788
Bo4 p4 234 " Yif thou were thanne iset a juge or a 810
Bo4 p4 234 " Yif thou were thanne iset a juge or a 810
Bo4 p4 262 schrewes it were a more covenable thing that the 838
Bo4 p4 290 overmochel a fool, and for to haten 866
Bo4 m4 16 ryghtful. Wiltow thanne yelden a covenable 892
Bo4 p5 27 of this so wrongful a confusioun; for I wolde 920
Bo4 p5 44 ne knowe nat the cause of so gret a disposicioun, 938
Bo4 m5 18 strokes. (That is to seyn, that ther is a maner 960
Bo4 p6 9 And thanne sche, a litelwhat smylinge, 988
Bo4 p6 21 manere ne noon ende, but if that a wyght 1000
Bo4 p6 22 constreynede tho doutes by a ryght lifly and 1000
Bo4 p6 32 as the knowynge of thise thinges is a maner 1010
Bo4 p6 37 deliteth the, thou most suffren and forberen a 1016
Bo4 p6 83 ryght as a werkman that aperceyveth in his 1062
Bo4 p6 117 tornen aboute a same centre or aboute a poynt, 1096
Bo4 p6 117 tornen aboute a same centre or aboute a poynt, 1096
Bo4 p6 120 and is, as it were, a centre or a poynt to the 1098
Bo4 p6 120 and is, as it were, a centre or a poynt to the 1098
Bo4 p6 122 and thilke that is utterest, compased by a largere 1100
Bo4 p6 155 men by a bond of causes nat able to ben 1134
Bo4 p6 179 a worse confusioun than that gode men 1158
Bo4 p6 196 seith, may a man speken and determinen of 1174
Bo4 p6 201 (as who seith, but it is lik a mervayle or 1180
Bo4 p6 225 to comprehende and to telle) a fewe thingis of 1204
Bo4 p6 237 opynioun it is a confusioun. But I suppose that 1216
Bo4 p6 251 wolde deme that it were a felonie that he 1230
Bo4 p6 253 nat suffre that swich a man be moeved with any 1232
Bo4 p6 254 bodily maladye. But so as seyde a philosophre, 1232
Bo4 p6 294 schrewes scheweth a gret argument to good 1272
Bo4 p7 15 " Forsothe this is a ful verray resoun, " quod 1430
Bo4 p7 17 destyne that thou taughtest me a litel herebyforn 1432
Bo4 p7 21 whiche thow seydest a litel herebyforn that 1436
Bo4 p7 30 a litil to the wordis of the peple, 1444
Bo4 p7 93 plawntest a ful egre bataile in thy corage ayeins 1508
Bo4 m7 48 torned hym into a bole, and Hercules brak of 1568
Bo4 m7 71 erthly lust is overcomyn, a man is makid 1592
Bo4 p7 77 stronge man ne semeth nat to abaissen or disdaignen 1492
Bo4 p6 262 abated. And God yeveth and departeth to other 1240
Bo4 p1 38 " that were a greet merveille and an abaysschinge 38
Bo4 p6 155 men by a bond of causes nat able to ben 1134
Bo4 p7 22 thei ne were nat able to ben wened to the 1436
Bo4 p6 117 tornen aboute a same centre or aboute a poynt, 1096
Bo4 p6 117 tornen aboute a same centre or aboute a poynt, 1096
Bo4 p6 121 tothere cerklis that tornen abouten hym; 1100
Bo4 p3 42 mede and ryght greet aboven alle medes. Remembre 446
Bo4 m6 10 abowte the sovereyn heighte of the world 1364
Bo4 p2 200 nat graunten absolutly and symply that thei 312
Bo4 p4 42 this lif, that is long to abyde, nameliche to a corage 620
Bo4 p6 373 abydest som swetnesse of songe. Tak thanne this 1352
Bo4 p4 80 lasse wrecches, that abyen the tormentz 656
Bo4 p1 30 folk, and it abyeth the tormentz in stede of 30
Bo4 m6 22 the sterres. This accordaunce atempryth by evenelyke 1376
Bo4 p4 64 an hard thing to accorde hym to a conclusioun, 642
Bo4 p6 240 accorden hem togidre of hym; but he is so 1218
Bo4 p4 143 that thow telle me, yif thow accordest to leten 718
Bo4 p3 48 and thilke folk that ben blisful it accordeth and 452
Bo4 p4 60 that it accordeth moche to the thinges that 638
Bo4 p4 217 wene that he were blynd? Ne also ne accordeth 792
Bo4 p4 229 " It accordeth, " quod I. 804
Bo4 p4 233 " It accordeth wel, " quod I. 808
Bo4 p4 272 sholde be torned into the habyte of accusacioun. 848
Bo4 p4 273 (That is to seyn, thei scholden accuse 848
Bo4 p4 263 accusours or advocattes, nat wrooth but pytous 838
Bo4 p4 285 to hir juges and to hir accusours. For whiche it 860
Bo4 m7 45 strondes (that is to seyn, that Achaleous coude 1566 | {"url":"http://machias.edu/faculty/necastro/chaucer/concordance/bo/bo4.txt.WebConcordance/c1.htm","timestamp":"2014-04-19T12:00:34Z","content_type":null,"content_length":"29492","record_id":"<urn:uuid:7463f168-fe23-4472-8f74-bd755153acc4>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00313-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fully Reflexive Intensional Type Analysis
Results 1 - 10 of 22
, 2002
"... Intensional polymorphism, the ability to dispatch to di#erent routines based on types at run time, enables a variety of advanced implementation techniques for polymorphic languages, including
tag-free garbage collection, unboxed function arguments, polymorphic marshalling, and flattened data structu ..."
Cited by 142 (38 self)
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Intensional polymorphism, the ability to dispatch to di#erent routines based on types at run time, enables a variety of advanced implementation techniques for polymorphic languages, including
tag-free garbage collection, unboxed function arguments, polymorphic marshalling, and flattened data structures. To date, languages that support intensional polymorphism have required a type-passing
(as opposed to type-erasure) interpretation where types are constructed and passed to polymorphic functions at run time. Unfortunately, type-passing su#ers from a number of drawbacks: it requires
duplication of run-time constructs at the term and type levels, it prevents abstraction, and it severely complicates polymorphic closure conversion.
, 2003
"... introduc e a notion of guarded rec ursive (g.r.) datatype c#w struc tors, generalizing the notion ofrec# rsive datatypes in func tional programming languages suc h as ML and Haskell. We address
both theoret ic#t and prac# ic## issues resulted from this generalization. On one hand, we design a type s ..."
Cited by 129 (10 self)
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introduc e a notion of guarded rec ursive (g.r.) datatype c#w struc tors, generalizing the notion ofrec# rsive datatypes in func tional programming languages suc h as ML and Haskell. We address both
theoret ic#t and prac# ic## issues resulted from this generalization. On one hand, we design a type system to formalize the notion of g.r. datatypec onstruc - tors and then prove the soundness of the
type system. On the other hand, we present some signific ant applic ations (e.g., implementing objec ts, implementing stagedc omputation, etc# ) of g.r. datatype c# nstruc#S rs, arguing that g.r.
datatypec onstruc torsc an have far-reac hingc onsequenc es in programming. The mainc ontribution of the paper lies in the rec#I0 ition and then the formalization of a programming notion that is of
both theoretic# l interest and prac tic# l use.
- In Seventeenth IEEE Symposium on Logic in Computer Science , 2002
"... Proof-Carrying Code (PCC) is a general framework for verifying the safety properties of machine-language programs. PCC proofs are usually written in a logic extended with language-specific
typing rules. In Foundational Proof-Carrying Code (FPCC), on the other hand, proofs are constructed and verifie ..."
Cited by 94 (19 self)
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Proof-Carrying Code (PCC) is a general framework for verifying the safety properties of machine-language programs. PCC proofs are usually written in a logic extended with language-specific typing
rules. In Foundational Proof-Carrying Code (FPCC), on the other hand, proofs are constructed and verified using strictly the foundations of mathematical logic, with no type-specific axioms. FPCC is
more flexible and secure because it is not tied to any particular type system and it has a smaller trusted base. Foundational proofs, however, are much harder to construct. Previous efforts on FPCC
all required building sophisticated semantic models for types. In this paper, we present a syntactic approach to FPCC that avoids the difficulties of previous work. Under our new scheme, the
foundational proof for a typed machine program simply consists of the typing derivation plus the formalized syntactic soundness proof for the underlying type system. We give a translation from a
typed assembly language into FPCC and demonstrate the advantages of our new system via an implementation in the Coq proof assistant. 1.
- In ACM Symposium on Principles of Programming Languages , 2002
"... A certified binary is a value together with a proof that the value satisfies a given specification. Existing compilers that generate certified code have focused on simple memory and control-flow
safety rather than more advanced properties. In this paper, we present a general framework for explicitly ..."
Cited by 84 (12 self)
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A certified binary is a value together with a proof that the value satisfies a given specification. Existing compilers that generate certified code have focused on simple memory and control-flow
safety rather than more advanced properties. In this paper, we present a general framework for explicitly representing complex propositions and proofs in typed intermediate and assembly languages.
The new framework allows us to reason about certified programs that involve effects while still maintaining decidable typechecking. We show how to integrate an entire proof system (the calculus of
inductive constructions) into a compiler intermediate language and how the intermediate language can undergo complex transformations (CPS and closure conversion) while preserving proofs represented
in the type system. Our work provides a foundation for the process of automatically generating certified binaries in a type-theoretic framework. 1
, 2007
"... We introduce System FC, which extends System F with support for non-syntactic type equality. There are two main extensions: (i) explicit witnesses for type equalities, and (ii) open,
non-parametric type functions, given meaning by toplevel equality axioms. Unlike System F, FC is expressive enough to ..."
Cited by 75 (25 self)
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We introduce System FC, which extends System F with support for non-syntactic type equality. There are two main extensions: (i) explicit witnesses for type equalities, and (ii) open, non-parametric
type functions, given meaning by toplevel equality axioms. Unlike System F, FC is expressive enough to serve as a target for several different source-language features, including Haskell’s newtype,
generalised algebraic data types, associated types, functional dependencies, and perhaps more besides.
- SCIENCE OF COMPUTER PROGRAMMING , 2004
"... A polytypic function is a function that can be instantiated on many data types to obtain data type specific functionality. Examples of polytypic functions are the functions that can be derived
in Haskell, such as show , read , and ` '. More advanced examples are functions for digital searching, patt ..."
Cited by 59 (21 self)
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A polytypic function is a function that can be instantiated on many data types to obtain data type specific functionality. Examples of polytypic functions are the functions that can be derived in
Haskell, such as show , read , and ` '. More advanced examples are functions for digital searching, pattern matching, unification, rewriting, and structure editing. For each of these problems, we not
only have to define polytypic functionality, but also a type-indexed data type: a data type that is constructed in a generic way from an argument data type. For example, in the case of digital
searching we have to define a search tree type by induction on the structure of the type of search keys. This paper shows how to define type-indexed data types, discusses several examples of
type-indexed data types, and shows how to specialize type-indexed data types. The approach has been implemented in Generic Haskell, a generic programming extension of the functional language Haskell.
- In the International Conference on Functional Programming (ICFP ’02 , 2002
"... Multi-stage programming languages provide a convenient notation for explicitly staging programs. Staging a definitional interpreter for a domain specific language is one way of deriving an
implementation that is both readable and efficient. In an untyped setting, staging an interpreter "removes a co ..."
Cited by 53 (11 self)
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Multi-stage programming languages provide a convenient notation for explicitly staging programs. Staging a definitional interpreter for a domain specific language is one way of deriving an
implementation that is both readable and efficient. In an untyped setting, staging an interpreter "removes a complete layer of interpretive overhead", just like partial evaluation. In a typed setting
however, Hindley-Milner type systems do not allow us to exploit typing information in the language being interpreted. In practice, this can have a slowdown cost factor of three or more times.
"... Existential types are the standard formalisation of abstract types. While this formulation is sufficient in entirely statically typed languages, it proves to be too weak for languages enriched
with forms of dynamic typing: in the presence of operations performing type analysis, the abstraction barri ..."
Cited by 44 (11 self)
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Existential types are the standard formalisation of abstract types. While this formulation is sufficient in entirely statically typed languages, it proves to be too weak for languages enriched with
forms of dynamic typing: in the presence of operations performing type analysis, the abstraction barrier erected by the static typing rules for existential types is no longer impassable, because
parametricity is violated. We present a light-weight calculus for polymorphic languages with abstract types that addresses this shortcoming. It features a variation of existential types that retains
most of the simplicity of standard existentials. It relies on modified scoping rules and explicit coercions between the quantified variable and its witness type.
"... We present parametric higher-order abstract syntax (PHOAS), a new approach to formalizing the syntax of programming languages in computer proof assistants based on type theory. Like higherorder
abstract syntax (HOAS), PHOAS uses the meta language’s binding constructs to represent the object language ..."
Cited by 20 (2 self)
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We present parametric higher-order abstract syntax (PHOAS), a new approach to formalizing the syntax of programming languages in computer proof assistants based on type theory. Like higherorder
abstract syntax (HOAS), PHOAS uses the meta language’s binding constructs to represent the object language’s binding constructs. Unlike HOAS, PHOAS types are definable in generalpurpose type theories
that support traditional functional programming, like Coq’s Calculus of Inductive Constructions. We walk through how Coq can be used to develop certified, executable program transformations over
several statically-typed functional programming languages formalized with PHOAS; that is, each transformation has a machine-checked proof of type preservation and semantic preservation. Our examples
include CPS translation and closure conversion for simply-typed lambda calculus, CPS translation for System F, and translation from a language with ML-style pattern matching to a simpler language
with no variable-arity binding constructs. By avoiding the syntactic hassle associated with first-order representation techniques, we achieve a very high degree of proof automation. Categories and
Subject Descriptors F.3.1 [Logics and meanings
, 2004
"... Two different ways of defining ad-hoc polymorphic operations commonly occur in programming languages. With the first form polymorphic operations are defined inductively on the structure of types
while with the second form polymorphic operations are defined for specific sets of types. In intensional ..."
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Two different ways of defining ad-hoc polymorphic operations commonly occur in programming languages. With the first form polymorphic operations are defined inductively on the structure of types
while with the second form polymorphic operations are defined for specific sets of types. In intensional type analysis operations are defined by induction on the structure of types. Therefore no new
cases are necessary for user-defined types, because these types are equivalent to their underlying structure. However, intensional type analysis is “closed ” to extension, as the behavior of the
operations cannot be differentiated for the new types, thus destroying the distinctions that these types are designed to express. Haskell type classes on the other hand define polymorphic operations
for sets of types. Operations defined by class instances are considered “open”—the programmer can add instances for new types without modifying existing code. However, the operations must be extended
with specialized code for each new type, and it may be tedious or even impossible to add extensions that apply to a large universe of new types. Both approaches have their benefits, so it is
important to let programmers decide which is most appropriate for their needs. In this paper, we define a language that supports both forms of ad-hoc polymorphism, using the same basic constructs. | {"url":"http://citeseerx.ist.psu.edu/showciting?doi=10.1.1.27.295","timestamp":"2014-04-17T23:26:57Z","content_type":null,"content_length":"39406","record_id":"<urn:uuid:1e4227a8-330d-4bb0-8d13-7917645d3fb4>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00551-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Transformation Groups, Vol. ?, No. ?, ??, pp. 1--?? c
#Birkh˜auser Boston (??)
BEN WEBSTER #
Department of Mathematics
University of California
Berkeley, CA 94720
MILEN YAKIMOV ##
Department of Mathematics
University of California
Santa Barbara, CA 93106
Abstract. We describe a partition of the double flag variety G/B +
× G/B - of a complex
semisimple algebraic group G analogous to the Deodhar partition on the flag variety G/B + .
This partition is a refinement of the stratification into orbits both for B +
× B - and for the
diagonal action of G, just as Deodhar's partition refines the orbits of B + and B - .
We give a coordinate system on each stratum, and show that all strata are coisotropic
subvarieties. Also, we discuss possible connections to the positive and cluster geometry of | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/489/3909779.html","timestamp":"2014-04-18T18:27:00Z","content_type":null,"content_length":"7949","record_id":"<urn:uuid:12437ad2-f256-4e08-8f82-2ee97dc959fa>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00088-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solution and Stability of a General Mixed Type Cubic and Quartic Functional Equation
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 673810, 8 pages
Research Article
Solution and Stability of a General Mixed Type Cubic and Quartic Functional Equation
^1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
^2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
^3College of Mathematics and Information Science, Hebei Normal University, and Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, China
^4Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan
Received 23 August 2013; Accepted 5 September 2013
Academic Editor: Jinlu Li
Copyright © 2013 Xiaopeng Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
We consider the following mixed type cubic and quartic functional equation = where is a fixed integer. We establish the general solution of the functional equation when the integer , and then, by
using the fixed point alternative, we investigate the generalized Hyers-Ulam-Rassias stability for this functional equation when the integer .
1. Introduction
In 1940, Ulam [1] asked the fundamental question for the stability for the group homomorphisms.
Let be a group, and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there is a homomorphism with for all ?
In other words, under what conditions, does there exist a homomorphism near an approximately homomorphism? In the next year, Hyers [2] gave the first affirmative answer to the question of Ulam for
Cauchy equation in the Banach spaces. Then, Rassias [3] generalized Hyers’ result by considering an unbounded Cauchy difference, and this stability phenomenon is known as generalized
Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. During the last three decades, the stability problem for several functional equations has been extensively investigated by many
mathematicians; see, for example, [4–9] and the references therein. We also refer the readers to the books [10–13].
In [14], Jun and Kim introduced the following functional equation It is easy to see that the function satisfies the functional equation (3). Thus, it is natural that (3) is called a cubic functional
equation and every solution of the cubic functional equation is said to be a cubic function. In [14], Jun and Kim established the general solution and the generalized Hyers-Ulam-Rassias stability for
(3). They proved that a function between real vector spaces is a solution of the functional equation (3) if and only if there exists a function such that for all and is symmetric for each fixed one
variable and is additive for fixed two variables.
In [15], Lee et al. considered the following quartic functional equation Since the function satisfies the functional equation (4), the functional equation (4) is called a quartic functional equation
and every solution of the quartic functional equation is said to be a quartic function. In [15], the authors solved the functional equation (4) and proved the stability for it. Actually, they
obtained that a function between real vector spaces satisfies the functional equation (4) if and only if there exists a symmetric biquadratic function such that for all . A function between real
vector spaces is said to be quadratic if it satisfies the following functional equation for all , and a function is said to be bi-quadratic if is quadratic for each fixed one variable (see [8]). The
following mixed type cubic and quartic functional equation was introduced by Eshaghi Gordji et al. [16] It can be verified that the function satisfies the functional equation (6). In [16], the
authors obtained the general solution and the generalized Hyers-Ulam-Rassias stability of (6) in quasi-Banach space. The literature on the stability of the mixed type functional equations is very
rich; see [17–22].
In the present paper, we extend (6) and consider the following functional equation where is a fixed integer. One can see that the functional equation (6) is a special case of (7) when we take the
integer .
In 2003, Radu [23] noticed that the fixed point theorem, which was established by Diaz and Margolis [24], plays an important part in solving the stability problem of functional equations.
Subsequently, this method has been successfully used by many mathematicians to investigate the stability of several functional equations; see, for example, [21, 25, 26] and the references therein.
In this paper, we first establish the general solution of functional equation (7) when the integer and is a mapping between vector spaces. Then, by using the fixed point method, we prove the
generalized Hyers-Ulam-Rassias stability of the functional equation (7) when the integer and is a mapping from the normed space to the Banach space.
2. Solution of the Functional Equation (7)
Recall form [14, 15] that every solution of the cubic functional equation (3) and the quartic functional equation (4) is said to be a cubic function and a quartic function, respectively. In this
section, we investigate the general solution of the mixed type cubic and quartic functional equation (7). Throughout this section, let and be two real vector spaces, and we always assume that the
integer in the functional equation (7) is different from 0, −1, and 1. Before proving our main theorem, we first give the following two lemmas.
Lemma 1. If an odd mapping satisfies (7), then is cubic.
Proof. Note that, in view of the oddness of , we have and for all . Letting in (7), we get for all . Applying (8) to (7), we obtain for all . Replacing by in (9), we get for all . Now, if we replace
by in (9) and use (8), we see that for all . Interchanging with in (11) and using the oddness of , we get the relation for all . Then, subtracting (12) from (10), one has for all . Now, replacing by
in (13) and noting that is odd, we have for all . Adding (13) to (14) gives for all . Replacing by in (9) and using (8), we get Applying (9) and (16) to (15) yields that for all . Thus, the mapping
is cubic. This completes the proof.
Lemma 2. If an even mapping satisfies (7) for all , then is quartic.
Proof. In view of the evenness of , we have for all . Putting in (7), we get . Then, let in (7), we obtain for all . Combing (7) and (18) implies the following equation for all . Replacing by in (19)
and note that is even, we get for all . Replacing by in (19) and using (18), we get for all . Interchanging the roles of and in (21), we obtain for all . If we subtract (22) from (20), we obtain for
all . Replacing by in (23), we get for all . If we add (23) to (24), we have for all . Replacing by in (19) and using (18), we get for all . Applying (19) and (26) to (25), we obtain that for all .
Therefore, the mapping is quartic and the proof is complete.
Now, we are ready to find out the general solution of (7).
Theorem 3. A mapping satisfies (7) for all if and only if there exist a symmetric multiadditive mapping and a symmetric bi-quadratic mapping such that for all .
Proof. First, we assume that there exist a symmetric multi-additive mapping and a symmetric bi-quadratic mapping such that for all . We need to show that the function satisfies (7). By a simple
computation, one can obtain that the function satisfies (7). Thus, to show that the function satisfies (7), we only need to show that the function also satisfies (7); namely, for all . Since is a
bi-quadratic mapping, it can be verified that for all integer and all . Then, (28) becomes To establish (28), it suffices to show (30). Note that if (30) holds for some integer , then so does . Thus,
in the following, we will show that (30) holds for all positive integers with and all . To do this, we use induction on . Fix any . In the case when , we have So (30) is true for . Here, we have used
the bi-quadratic property of the function and (29). Now, assume that (30) is true for all positive integers that are less than or equal to some integer . Then, Applying (31) to (32), one can obtain
that This means that (30) is true for , and we have showed that the function satisfies (7). Therefore, the mapping satisfies (7).
Conversely, we decompose into the odd part and the even part by putting for all . Then, for all . It is easy to show that the mappings and satisfy (7). Hence, it follows from Lemmas 1 and 2 that the
function is cubic and is quartic, respectively. Therefore, there exist a symmetric multi-additive mapping such that for all (see [14, Theorem2.1]) and a symmetric bi-quadratic mapping such that for
all (see [15, Theorem2.1]). Hence, we get for all . The proof is complete.
3. Generalized Hyers-Ulam-Rassias Stability of the Functional Equation (7)
In this section, we will investigate the stability of the functional equation (7) by using the fixed point alternative. Throughout this section, let be a real normed space and be a real Banach space,
and we always assume that the integer used in the section is greater than or equal to 2. For convenience, we use the following abbreviation for a given function for all .
Let us recall the following result by Diaz and Margolis.
Proposition 4 (see [24]). Let be a complete generalized metric space (i.e., one for which may assume infinite value), and let be a strictly contractive mapping with Lipschitz constant ; that is,
Then, for each fixed element , either for all nonnegative integers or there exists a non-negative integer such that(a) for all ;(b)the sequence converges to a fixed point of ;(c) is the unique fixed
point of in the set ;(d) for all .
Lemma 5. Let be an odd function for which there exists a function such that for all . If there exists a constant such that for all , then there exists a unique cubic mapping such that for all , where
Proof. It follows from (39) that for all . Letting in (38) and replacing by , we have for all . By (39), we have for all . This, together with (42), implies that for all . Let be the set of all odd
mappings . We introduce the generalized metric on : It is easy to show that is complete. Now, we define a function by for all and all . Note that, for all , Hence, we obtain that for all ; that is,
is a strictly contractive mapping of with Lipschitz constant . It follows from (43) that . Therefore, according to Proposition 4, the sequence converges to a fixed point of ; that is, and for all .
Also is the unique fixed point of in the set and which yields the inequality (40). It follows from the definition of , (38), and (41) that for all ; that is, the mapping satisfies (7). Since is odd,
is odd. Therefore, Lemma 1 guarantees that is cubic. Finally, it remains to prove the uniqueness of . Let be another cubic function satisfying (40). Since and is cubic, we get and for all ; that is,
is a fixed point of . Since is the unique fixed point of in , it follows that .
Lemma 6. Let be an odd function for which there exists a function such that for all . If there exists a constant such that for all , then there exists a unique cubic mapping such that for all , where
Proof. It follows from (52) that for all . Letting in (51) and replacing by , we have for all . We introduce the same definitions for and as in the proof of Lemma 5 (by replacing by ) such that
becomes a generalized complete metric space. Let be the mapping defined by for all and all . One can show that for all . It follows from (55) that . Due to Proposition 4, the sequence converges to a
fixed point of ; that is, and for all . Also, which yields the inequality (53). The rest of the proof is similar to the proof of Lemma 5, and we omit the details.
Similarly, we can prove the following two lemmas on even functions.
Lemma 7. Let be an even function with for which there exists a function such that for all . If there exists a constant such that for all , then there exists a unique quartic mapping such that for all
, where .
Lemma 8. Let be an even function with for which there exists a function such that for all . If there exists a constant such that for all , then there exists a unique quartic mapping such that for all
, where .
Now, we are ready to give our main theorems in this section.
Theorem 9. Let be a function with for which there exists a function such that for all . If there exists a constant such that for all , then there exist a unique cubic mapping and a unique quartic
mapping such that for all , where
Proof. Let and denote the odd and the even part of , respectively. Then, it can be verified from (65) that for all . Moreover, by (66), it is easy to compute that for all . Thus, by applying Lemmas 5
and 7, one can obtain that there exist a unique cubic mapping and a unique quartic mapping such that for all , where and . Moreover, combining (71) and (72) yields the inequality (67). The proof is
Theorem 10. Let be a function with for which there exists a function such that for all . If there exists a constant such that for all , then there exist a unique cubic mapping and a unique quartic
mapping such that for all , where
Proof. Similar to the proof of Theorem 9, the result follows from Lemmas 6 and 8.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper. | {"url":"http://www.hindawi.com/journals/jfs/2013/673810/","timestamp":"2014-04-16T05:18:06Z","content_type":null,"content_length":"722143","record_id":"<urn:uuid:bb1625a1-2b96-4481-92b1-1c48f441465f>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
Function approximation
February 16th 2010, 09:12 PM
Function approximation
Use the first three terms of the series expansion as an approximation for the function to be integrated. Estimate the value of the definite integral.
$\int^{0.2}_{0.1} \sqrt[3]{1-x^2} dx$
series expansion:
so i have:
$\int^{0.2}_{0.1}(1-\frac{1}{3}x^2-\frac{1}{9}x^4) dx$
answer is 0.9921556
Where is my error?
February 17th 2010, 06:10 AM
Use the first three terms of the series expansion as an approximation for the function to be integrated. Estimate the value of the definite integral.
$\int^{0.2}_{0.1} \sqrt[3]{1-x^2} dx$
series expansion:
so i have:
$\int^{0.2}_{0.1}(1-\frac{1}{3}x^2-\frac{1}{9}x^4) dx$
answer is 0.9921556
Where is my error?
I don't see an error here.
February 17th 2010, 03:40 PM
ok, then the other explanation is the answer is inaccurate? | {"url":"http://mathhelpforum.com/calculus/129197-function-approximation-print.html","timestamp":"2014-04-20T21:39:24Z","content_type":null,"content_length":"6605","record_id":"<urn:uuid:1f8b113b-a6fd-4d23-bfd5-b5be2e19af08>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00465-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Pro-affine varieties over a local field
up vote 2 down vote favorite
Let $K$ be a (perfect) local field, and let $S = \lim (\mathrm{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the affine
varieties $\mathrm{Spec} A_i$ form a projective system. Note that $S$ is an affine $K$-scheme which is not of finite type in general.
I have some elementary questions about such schemes.
Q1.Is $S$ regular if and only if $\mathrm{Spec} A_i$ is regular for all $i=0,\ldots$?
Q1*.Is $S$ normal (resp. irreducible or reduced) if and only if $\mathrm{Spec} A_i$ is normal (resp. irreducible or reduced) for all $i=0,\ldots$?
Q2. How can I determine the dimension of $S$ from the dimension of $A_i$. (Assume the schemes to be integral for this question.)
Unfortunately, I have little feeling for such pro-affine varieties at the moment.
A last (and bit vague) question:
Q3. Is there a moduli space of smooth connected pro-affine varieties of given dimension?
ag.algebraic-geometry limits arithmetic-geometry
It looks like any affine scheme of countable type over $K$ is a pro-affine $K$-variety in your sense. Am I mistaken? – S. Carnahan♦ Apr 26 '12 at 3:59
I think you're right. You just write $A$ as a direct limit of its finitely generated sub-$K$-algebras. – Harry Apr 26 '12 at 7:16
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The nature of the ground field $K$ doesn't seem to be very relevant. In general $S$ won't even be noetherian. The only positive result I know is the following: if $S_i$ is regular for all
up vote 1 sufficiently large $i$, and the map $S_j\to S_i$ is etale for all sufficiently large $i,j$, then $S$ is noetherian and regular (and $dim S=dim S_i$ for all sufficiently large $i$.
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csc A + tan C = 5x = 7/6 + AB/6 AB = sqrt(13) from the Pythagorean formula, so: 5x = (7 + sqrt(13))/6 x = (7 + sqrt(13))/30
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i got 7+sqrt.30/30 but im feeling that im wrong
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how do you know it though?
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Your post posted aat the same time as mine. After reading mine, do you see how to do it now?
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Which step of mine are you stuck at?
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i see it now but is that the way to check your work?
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What I wrote is a little condensed, but the work is basically all there. I could expand it a little, but I have to know where you begin not to follow it. That's where I'll expand my explanation
for you. np.
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The first part of the first line is given. The second part of the first line comes from the definition of csc and tan.
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Do approximately the same polynomials have approximately the same roots?
up vote 2 down vote favorite
"If $U$ is an open subset of the complex plane, then matrices $X\in\textrm{M}(n,\mathbb C)$ all of whose eigenvalues belong to $U$ make up an open subset of $\textrm{M}(n,\mathbb C)$." Trying to
prove this by using the argument principle, I was led to the following question: do approximately the same polynomials have approximately the same roots? More precisely, fix a complex polynomial $p
(z)=a_0+a_1 z+\ldots+a_n z^n$ with $a_n\neq 0$. Let us say that a polynomial $q(z)=b_0+b_1 z+\ldots+b_n z^n$ is in the $\varepsilon$-neighborhood of $p(z)$ if $|b_i-a_i|<\varepsilon$ for all $0\leq i
\leq n$. Now suppose all roots of $p(z)$ lie in, say, the open unit disk. Does $p(z)$ have an $\varepsilon$-neighborhood consisting of polynomials with the same property?
polynomials matrices complex-analysis
8 Yes, the roots of a polynomial depend continously on the coefficients; see en.wikipedia.org/wiki/Properties_of_polynomial_roots . This is better asked on math.stackexchange.com IMO. – quid Aug 3
'12 at 15:39
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closed as too localized by quid, Dmitri Pavlov, Chris Godsil, Qiaochu Yuan, Felipe Voloch Aug 4 '12 at 18:53
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3 Answers
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I think so, yes. Suppose all the roots lie inside the unit disc. By compactness $|p(z)|$ takes a positive minimum $\delta$ on the contour (the unit circle). For $q(z)$ in the
epsilon-neighbourhood, $q(z)-p(z)$ is a polynomial with epsilon coefficients, so we can take epsilon small enough so that on the contour $|q(z)-p(z)|<\delta<|p(z)|$. Then appeal to
up vote 3 down Rouché's theorem to learn that $p(z)$ and $q(z)$ have the same number of zeros in the region.
vote accepted
1 How is this related to the question? – Qfwfq Aug 3 '12 at 20:47
It simply solves the more-precisely-formulated question above, which generalizes at once to give a proof of the proposition in quotation marks. – Murat Güngör Aug 4 '12 at 11:18
add comment
The question about matrices is analyzed to death in Kato's "perturbation theory of linear operators", chapters 1 and 2. The question in the title than is answered (in the affirmative by
up vote 5 associating to each polynomial its companion matrix. Note that the roots depend analytically on the coefficients when they are all distinct, but only $C^0$ when there are multiple roots.
down vote
add comment
For the headline question, no. See, for example, Wilkinson's polynomial: http://en.wikipedia.org/wiki/Wilkinson's_polynomial.
Edit: From the response to this answer, it seems that some expansion is needed. The OP asks two questions. In the the title, and repeated in the text, the question is: Do approximately the
up vote same polynomials have approximately the same roots? The answer to this question is no, as Wilkinson shows with his monic degree-20 polynomial, which has well separated roots of moderate size,
4 down namely the integers 1 to 20. A decrement of $2^{-23}$ in the coefficient of the degree-19 term (which is $-210$)---a proportional change of less than 1 in 1.7 billion---induces substantial
vote changes in the roots: for example, the root pair $16.5\pm 0.5$ becomes the root pair $16.73024\pm 2.81262\mathrm i$ (to 5 d.p.). The second question asked by the OP is whether the
coefficients-to-roots map is continuous, which was already answered (affirmatively) by others.
10 This is a beautiful example, but perturbing the coefficients still perturbs the roots. It's just that the derivative can be far larger than one would naively expect. – Henry Cohn Aug 3
'12 at 16:30
The OP does not ask two questions, the same question is phrased formally and informally. – quid Aug 5 '12 at 12:17
add comment
Not the answer you're looking for? Browse other questions tagged polynomials matrices complex-analysis or ask your own question. | {"url":"http://mathoverflow.net/questions/103874/do-approximately-the-same-polynomials-have-approximately-the-same-roots/103875","timestamp":"2014-04-18T18:29:18Z","content_type":null,"content_length":"58588","record_id":"<urn:uuid:4fd2fee3-f85a-4024-9e5d-4f0109884d5a>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00196-ip-10-147-4-33.ec2.internal.warc.gz"} |
Paul Wilmott's Blog: Science in Finance IV: The feedback effect
For every buyer there is a seller and vice versa. So at a first glance derivatives is a zero-sum game, someone wins and someone loses, and the amounts are identical. Therefore there can be no impact
on the rest of us or on the economy if two adults want to bet large sums on money on the outcome of what may just be the roll of a dice. Well, it isn’t that simple for at least two reasons.
First, many of those trading derivatives are hedging with the underlying and this can affect the behaviour of the underlying: hedging positive gamma can decrease volatility and hedging negative gamma
can increase volatility. When hedging positive gamma (i.e. replicating negative gamma) as the price rises you have to sell more of the underlying, and when the price falls you buy back, thus reducing
volatility if your trades are in sufficient size to impact on the market. But hedging negative gamma is not so nice, you buy when the price rises and sell when it falls, exacerbating the moves and
increasing volatility. The behaviour of stocks on which there are convertible bonds is often cited as a benign example, with the rather more dramatic '87 crash, replicating a put i.e. hedging
negative gamma, as the evil version. (See Wilmott, P. and Schonbucher, P 2000 The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61 232—272, also PS’s dissertation.) You will
probably find some reluctance for people to sell certain derivatives if they are not permitted to dynamically hedge. (Not that it works particularly well anyway, but that is what people do, and that
is what most pricing theory is based on. Static hedging with other derivatives is better, and does not cause such (in)stability problems.)
(We’ve had newspaper headlines about damage done by excessive risk taking, whether by single, roguish, individuals or by larger institutions such as hedge funds, or banks and corporates investing in
products they don’t fully understand. I expect it won’t be long before the attempt to reduce risk is the cause of similar headlines!)
Second, with the leverage available with derivatives it is possible, and actually rather simple, for people to trade so much as to get themselves into a pickle when things go wrong. This has many
consequences. For example a trader loses his bank so much money that the bank collapses or is taken over, job losses ensue and possibly the man in the street loses his savings. Is wealth conserved
during this process, as would be the case in a zero-sum game? I think not.
Of course, we don’t know what proportion of derivatives trades are being used for hedging, speculation with leverage, etc. and how many are being dynamically hedged. But while derivatives trading is
such a large business and while pricing theory is underpinned by dynamic hedging then we can say that the game of derivatives is not zero sum. Of course, this should spur on the implementation of
mathematical models for feedback…which may in turn help banks and regulators to ensure that the press that derivatives are currently getting is not as bad as it could be. | {"url":"http://www.wilmott.com/blogs/paul/index.cfm/2008/1/29/Science-in-Finance-IV-The-feedback-effect","timestamp":"2014-04-18T05:32:07Z","content_type":null,"content_length":"23215","record_id":"<urn:uuid:393c79d2-3765-433e-9524-e3d01e3b2d15>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00076-ip-10-147-4-33.ec2.internal.warc.gz"} |
How many thousands are in ten million.
Hi Echoe.
Here's an easier question for you: how many tens are there in ten million? Answer: one million. The reason is that you divide the two numbers. ten million divided by ten is one million. If you write
it in fraction form, this is
and you can cross off a zero on the top and a zero on the bottom. This leaves just a 1 on the bottom and 10 000 00 (or 1 million) on the top. Anything divided by 1 is just the original number, so the
answer is 1 million. If you have 100 on the bottom (what is ten million divided by 100?) then you can cross off two zeros on the bottom and two zeros on the top. That gives you 10 000 0 over 1, which
is one hundred thousand.
Just extend this procedure to solve your own question.
Hope this helps,
Stephen La Rocque. | {"url":"http://mathcentral.uregina.ca/QQ/database/QQ.02.06/echoe1.html","timestamp":"2014-04-19T12:11:57Z","content_type":null,"content_length":"5924","record_id":"<urn:uuid:aaab9179-10a4-4033-97d3-f9dbead63133>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00648-ip-10-147-4-33.ec2.internal.warc.gz"} |
Embedding a Latin square with transversal into a projective space
Pretorius, Lou M. and Swanepoel, Konrad (2011) Embedding a Latin square with transversal into a projective space. Journal of Combinatorial Theory, Series A, 118 (5). pp. 1674-1683. ISSN 0097-3165
Full text not available from this repository.
A Latin square of side n defines in a natural way a finite geometry on 3. n points, with three lines of size n and n2 lines of size 3. A Latin square of side n with a transversal similarly defines a
finite geometry on 3n+1 points, with three lines of size n, n2-n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn
points, with k lines of size n and n2 lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser.
A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.
Actions (login required)
Record administration - authorised staff only | {"url":"http://eprints.lse.ac.uk/33718/","timestamp":"2014-04-20T11:01:06Z","content_type":null,"content_length":"19932","record_id":"<urn:uuid:fbfe69ac-a999-488a-b53c-a5311a5c301e>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00522-ip-10-147-4-33.ec2.internal.warc.gz"} |
The next problem is shown below:
This time the -3 is inside parentheses. Instead of carrying down the negative sign, each 3 is made negative.
1 * -3 * -3 * -3 * -3 * -3 * -3
As you can see, the multiplication simplifies to 729. Note that aside from the result of the previous example being negative, the result here is the same. Pay careful attention to finding exponents
when negative signs are involved, as this is a common source of error.
The next pages explain the zero exponent in addition to some tips and tricks. | {"url":"http://algebrahelp.com/lessons/simplifying/numberexp/pg3.htm","timestamp":"2014-04-19T02:17:36Z","content_type":null,"content_length":"5991","record_id":"<urn:uuid:4f8502d4-00b6-4a70-b7f0-9b95e8f43218>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00158-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Is MIT Prof. Lewin wrong about Kirchhoff's law?
I'm interested in the other version of KVL. Is it "sum of voltage drop = sum of emf"? Anyway, how do we define the term "voltage" in the case of varying magnetic field?
Voltage can still be defined when there are time varying fields. You can research the well known vector potential A. The scalar potential (which is voltage) and the vector potential can be combined
into a 4-vector in relativity theory, and a complete field description (including time varying fields) can be provided by scalar and vector potentials. But, this goes a little beyond the thread
The "other" version of KVL could also be called the original version. I say this because Kirchoff's original experiments were with batteries and resistors. The experiments revealed that the sum of
EMFs from batteries in a loop equals the sum of the currents times the resistances in the loop. The word potential does not even come up, but it's clear from a modern perspective that resistance
times current is a potential drop. Maxwell quotes this version of KVL and gives credit to Kirchoff in his famous Treatise on Electricity and Magnetism. Actually, he mentions both Kirchoff's voltage
law and his current law, which nobody argues about at all.
For some reason, this other simplified version of KVL (sum of potentials equals zero) has cropped up in the literature. I'm not sure why, but it is very common to see it in text books. So, it's not
too surprising to see Prof. Lewin quoting this as the definition. Again, this is all just semantics, but it is certainly instructive to study and understand the original intent of KVL.
Now it should be said that the original experiments were with EMFs from batteries and not from time changing magnetic flux, but the concept is basically the same. Non-conservative EMFs can be grouped
together and used with a very straightforward definition of KVL which says that the sum of EMFs around a loop equals the sum of potential drops around the loop (or some variation on that). Kirchoff
and Maxwell define it without reference to potential at all, which can avoid the confusion of what potential means. However, modern theory uses the concept of potential, so it's perhaps better not to
avoid it.
To help answer your question, I've attached a copy of Kraus' description of the classical definition of KVL. (Electromagnetics by John D. Kraus, 3rd ed. 1984). Note the footnote at the bottom which
mentions time varying fields. As I mentioned above, this is the version of KVL I carry around in my head and use in my professional work. I really don't know why anyone would be interested in the
other version of KVL that we commonly see, but who am I to judge? | {"url":"http://www.physicsforums.com/showpost.php?p=3020528&postcount=23","timestamp":"2014-04-17T15:30:47Z","content_type":null,"content_length":"11269","record_id":"<urn:uuid:22915917-0caf-405e-9e47-867e69042667>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00431-ip-10-147-4-33.ec2.internal.warc.gz"} |
Critical points
Date: 19 Dec 1994
From: Bruce Wise
Subject: calculus question
This is a problem that I have in my calculus book:
Find all critical points (if any) in the following problem.
k(t)=1/the square root of (t-squared +1).
Date: 19 Dec 1994
From: Dr. Sydney
Subject: Re: calculus question
Hello there!
Thanks for writing to Dr. Math. The critical points of a function are the
points where the derivative is 0 (at these points, functions either reach a
local maximum, local minimum, or "saddle point."). You can see why
this is true...the derivative is the slope of the line tangent to the function,
right? So, when the derivative is 0 the function has "flattened out." Take
for instance the graph y = x^2. We know what that looks like--kind of a U
shape, right? Well, the place where the tangent line is horizontal is where
the graph has a local minimum--at the point (0,0). That is precisely the
point where the derivative is 0. Neat, huh?
Anyway, to find the critical points of your function, we just need to figure
out where the derivative is 0.
I find it easier to differentiate problems like this when I write them with
exponents, so I'd write your problem as follows:
k(t) = (t^2 + 1)^(-1/2)
So, to differentiate, we just use power rule and chain rule to get:
k'(t) = -t(t^2 + 1)^(-3/2)
(If your wondering how exactly I got that, you can write back and I'll
explain in more detail)
Now, all we want to know is for what values of t is k'(t) = 0?
So, when is -t
---- = 0?
(t^2 + 1)^3/2
The denominator is always going to be positive, no matter what t you choose,
so we need not worry about it being 0 (phew!).
So, the derivative will be 0 when the numerator is 0. And, the numerator is
0 when t = 0, right?
So, the derivative is 0 when t = 0.
For a complete answer, you usually want to say what k(t) is at the critical
point. k(0) = 1 in this problem, so we would say that k has a critical
point at the point (0,1).
Hope this helps! Write back if you have any more questions. | {"url":"http://mathforum.org/library/drmath/view/53395.html","timestamp":"2014-04-19T10:02:49Z","content_type":null,"content_length":"6927","record_id":"<urn:uuid:7990a334-1e15-40bd-b95e-317da11eaa74>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reading Comprehension Part 3: Deductive Reasoning
By Michael Maloney
Printed in Practical Homeschooling #71, 2006.
stated facts. They use the information given to answer questions which can be directly answered by applying the appropriate fact. With inferences, students have to draw a relationship between two or
more objects or events with information that is either given or assumed in the material.
In this article I will outline how to use deductive reasoning in order to understand information. Deductive reasoning is what many teachers consider "higher order thinking skills." The student has to
use at least two facts, rules, or laws and to draw a valid conclusion based on the information given.
Deductive reasoning is sometimes presented so that it appears to be more complicated than it actually is. The fact that deductions are sometimes called logical syllogisms and taught as part of
university philosophy courses in logic allows them to be considered more complex than is necessary to understand them.
Rules about Deductions
Like almost any other thought process, logical deductions become much easier to understand if the student is equipped with certain strategies. First of all, deductions can always be simplified into
three distinct parts: the rule, a given fact, and a conclusion. The first part of the deduction, called the rule, is the most critical component. The form in which the rule is given predetermines the
conclusion which the student can draw from any fact.
Deduction rules have two critical components. They can be either universal or particular. Universal deduction rules apply to all members of the set or class to which they refer. Particular rules
apply to only some of the members or examples of the set.
An example of a universal statement about all members of a class or set would be: All snakes crawl.
An example of a particular statement which applies to only some members of a class or set is: Some snakes are poisonous.
Universal or Particular?
Unless students can differentiate between a universal rule and a particular rule, they are likely to make errors in their conclusions about the information they have read. If the rule is particular,
in that it only refers to some of the members of a class, the conclusion of any deduction must always reflect that by saying "maybe" or "perhaps."
A deduction with a universal rule, a fact, and a conclusion would be:
Universal Rule: All snakes crawl.
Fact: An adder is a snake.
Conclusion: So, an adder crawls.
We know that the conclusion is correct because the rule applies to all snakes, including adders.
A deduction with a particular rule, a fact, and a conclusion would be:
Particular Rule: Some snakes are poisonous.
Fact: An adder is a snake.
Conclusion: So, maybe an adder is poisonous.
We cannot conclude that an adder is poisonous on the basis of this information alone, because only some snakes are poisonous and the adder may or may not fall into that group.
Positive or Negative?
The Deduction Rule can also either be positive or negative. Whether it is positive or negative predetermines whether the conclusion is positive or negative.
Deductions with a positive rule are given in examples 1 and 2:
Universal Rule: All snakes crawl
Particular Rule: Some snakes are poisonous.
Deductions with negative rules would be stated like this:
Universal Rule: No constrictors are poisonous.
Particular Rule: Some snakes are not poisonous.
If the rule is a negative statement, the conclusion must also always be a negative statement.
Universal Negative Rule: No constrictors are poisonous.
Fact: A python is a constrictor.
Conclusion: No python is poisonous (or, A python is not poisonous.)
Given the nature of Deduction Rules we can create a simple matrix for dealing with deductions:
│ │ Is the Deduction Rule │
│ ├───────────────┬─────────────────────┤
│ │ Universal? │ Particular │
│ ├───────────────┴─────────────────────┤
│ │ then use: │
│ Positive? │ All are │ Maybe some are │
│ Negative? │ None is, are │ Maybe most are not │
Once students can determine the nature of the rule of a deduction, the rest is relatively easy. They add the fact as the middle of the deduction (e.g. A python is a snake) and then use the matrix to
draw the conclusion.
Universal Positive. All snakes crawl. A python is a snake. So a python crawls (or, So all pythons crawl.)
Particular Positive. Some snakes are poisonous. A python is a snake. So maybe a python is poisonous.
Universal Negative. No constrictors are poisonous. A python is a constrictor. So no pythons are poisonous.
Particular Negative. Some snakes are not poisonous. A python is a snake. So maybe a python is not poisonous.
A Rule About Conclusions
Sometimes the most difficult part of forming the conclusion is in deciding which nouns and or adjectives from the rule and the fact to put in the conclusion.
The student might conclude:
Some snakes are not poisonous. A python is a snake. So a python is not a snake.
To avoid having students chose the incorrect noun to complete the deduction, we can teach them a rule. The noun that is in the rule and in the fact cannot be in the conclusion.
Some snakes are not poisonous. A python is a snake. So maybe a python is not poisonous.
The noun "snake" is in the rule and in the fact, so it cannot appear in the conclusion. The student should choose the remaining noun (python) and adjective (poisonous) to make up a sentence for the
Finding Deductions in Passages
In most cases, the information in textbooks will not be stated directly as deductions for the student. The student will have to sift through the material to find statements that could be the rule
component of a deduction. Key words such as "all, most, some, no, none" followed by a statement of fact are indicators of a possible deduction. The author might write, "Some people think that all
snakes are dangerous." The author may then continue on to describe particular snakes. The student is left to conclude which ones may or may not be dangerous. Being able to analyze the deduction makes
coming to the correct conclusion much easier.
In order to help make students more familiar with deductive reasoning, the following strategy is suggested.
• Make up some rule statements
• Use both positive and negative rule statements
• Use both universal and particular rule statements.
• Add a fact statement.
• Use the same fact statement with different types of rule statements.
• Ask the student to make the conclusion statement.
• Look through your student's textbooks for rule statements.
• Find the fact statements that follow.
• See if the author draws a conclusion using the rule and the fact statement.
• See if the student can analyze the deduction using the matrix to determine if it is valid.
Teaching deductive reasoning is a lot like teaching other skills. If you can show the student a pattern or system that works reliably, the learning becomes more systematic and easier for both teacher
and learner. Teaching deductive thinking means teaching the students how to find the rule and the fact that make up a deduction. Then the student can see if the conclusion follows the pattern and is
valid or if it breaks the pattern and is invalid.
Was this article helpful to you?
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Elementary Lgebra 4 Th Ed Tussy
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MT4111 SYMBOLIC COMPUTATION
The overall aim of the course is to have students using Maple as a tool in their other courses and have them naturally turn to such a package when solving mathematical problems. The course aims to
illustrate the following points:
- A symbolic computation package allows one to conduct mathematical experiments.
- A symbolic computation package allows one to collect data about a problem being studied. This is similar to the way other scientists work.
- It is easier to try several different approaches to a problem and see which works.
- The machine is stupid. Intelligence comes from the user. The user thinks, the user interprets, the computer calculates.
By the end of the course students are expected to be able to
- use the Maple on-line help facilities;
- use standard Maple functions such as diff, int, solve, taylor etc.;
- use the basic data structures of Maple, namely sets, sequences, arrays, vectors and matrices;
- write Maple programs with loops, conditional etc.;
- define functions using procedures;
- understand how Maple is effective in problem solving;
- use Maple to produce data and look for patterns in the data i.e. be able to conjecture theorems;
- understand the strengths and limitations of a symbolic computation package;
- use Maple to solve problems for other honours courses.
- Rational and real arithmetic in symbolic computation.
- Symbolic differentiation and integration. What "simplifying" an expression means to a symbolic computation program.
- Writing Maple programs, procedures in Maple. Recursive definitions.
- The linear algebra package in Maple.
- The geometry package in Maple.
- Plotting with Maple.
To try to allow students to concentrate on the computing and the important points, duplicated course notes are given to each student. On-line help is available on the computers.
Maple manuals are available in computer room adjacent to microcomputer lab.
The university has a site licence for Maple, and students may put it up on their own machines.
2 hour examination = 70% , Project = 30%
The students decide on their own project. They are advised to choose an area of mathematics (another course) that they like and investigate how Maple can be used in that area. Students are advised to
talk with other members of staff about their projects.
MT3501 or MT3503 or MT3504
Academic year 2012/13 in semester 2 at 9
Dr J M Mitchell (Module coordinator), Dr M Neunhoeffer, Dr C M Roney-Dougal
Click here for access to past examination papers via iSaint.
Click here to see the University Course Catalogue entry.
Revised: PMH (September 2012) | {"url":"http://www.mcs.st-and.ac.uk/ug/hon4/MT4111.shtml","timestamp":"2014-04-20T18:39:08Z","content_type":null,"content_length":"5235","record_id":"<urn:uuid:9843d733-519c-4867-9d01-98e801e06587>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00520-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trig shortcuts for quick nav solutions
Jan 1, 2003
In navigation, trigonometry is a highly useful discipline; estimating set and drift of tides, courses to steer to make good a target, true winds from apparent winds, even tide heights, are all
exercises in plane trigonometry.
The problem, of course, is that few people have either the ability or the desire to memorize a trig table. Fortunately, there is no reason to, at least for those cases where what one desires is a
quick, "dirty" answer. With some practice, one can easily determine answers to acceptable precision (say, ±10%) using mental arithmetic and the three trig tricks presented below. These tricks may
seem overly quantitative to some, or imprecise to others. However, the advantages while sailing or racing are many.
All three of the shortcuts are based on the mathematical properties of the trig functions, principally sine and cosine. Rule one allows easy calculation of the sine; rule two easy calculation of the
cosine; and rule three eliminates ever having to calculate either sine or cosine for more than six angles. In all three cases, my goal is to be precise to better than 10%, which is certainly
acceptable for mental arithmetic. If more precision is needed, the quick answer provided here will probably satisfy any pressing requirements while you spin the maneuvering board or crunch the
calculator. Those readers familiar with trig may cringe at some of the shortcuts, but all answers will be within the magic 10% error, our goal.Rule one:
The sine of an angle equals the angle measured in radians, up to angles of 50°. Radians are an alternative, and more natural, way of measuring angles. In any circle, an arc as long as the circle's
radius maps out one radian, which equals approximately 57°. To find the sine of an angle, divide the angle by the number of degrees per radian, or 60. (Use 60 rather than 57 because the division is
easiermost navigators seem pretty proficient at dividing by 60 anyway.) For example, by this method the sine of 15° is 15/60 = 0.25. The actual value is 0.26, for an error of 4%. This method is in
error by more than 10% for angles of 50° to 90°, but this isn't a problem either: remember that sin 60° = 0.9, and the sine of anything higher than 70° is 1. While this list may seem arbitrary, rule
three will show that one only needs to know the sine of only six angles to figure out the rest.Rule two:
The cosine of an angle is the same as the sine of the angle's complement. (Brings you back to that high school geometry class, doesn't it?) An angle and its complement sum to 90°. Thus Cos 20° = Sin
70°. This rule is good for all angles, and is exact. Another (though less convenient) rule for the coSine is below. It is significantly less simple than the corresponding rule for the Sine, and isn't
really needed once you've mastered rules one and three.Rule three:
Interpolate whenever possible, preferably by using angles easily divisible by 60. For example, to determine Sin 22°, I would go through the following process: Sin 15/60 = 0.25, Sin 30/60 = 0.5, so
Sin 22° would be in the middle, or 0.375. The actual value is 0.3746. Interpolation is a learned skill, but one in which most navigators are well practiced.
We'll close with two examples of the rules in action. Case one: On course 000°, speed 6 knots, determine speed over the ground with tide setting 135°, drifting 2 knots. Speed along the course will be
reduced by the drift multiplied by the coSine of the angle between the course and drift. First, determine the coSine of the angle: Cos 45° = Sin 90°- 45° = Sin 45° = 45°/60 = 0.75. So, our speed made
good is (6 - 2 x 0.75) knots = 4.5 knots. The exact result is 4.6 knots, for an error of 6%.
Case two: How much northing and easting will a vessel on course 033°, speed 5 knots make in one hour? The northing is equal to the total distance traveled multiplied by the Sine of 33°. Approximate
Sin 33° as one fifth of the way between Sin 30° and Sin 45°: Sin 30° = 30°/60 = 0.50 Sin 45° = 45°/60 = 0.75 thus, Sin 33° = 0.55.
Northing is then 5 miles x 0.55 = 2.75 miles. Easting is total distance traveled multiplied by the coSine of 33°. Cos 33° = Sin 57°, which is close enough to Sin 60° = 0.9 (rule three) for me. Thus,
easting is 5 miles x 0.9 = 4.5 miles. Actual values are 2.7 and 4.2 miles, so we're within 1% and 7%.
These rules allow you to perform first-level navigation calculations while you're in the cockpit, the head, or eating dinner. Equally important, they provide a rapid way of getting an approximate
answer to a calculation as a check of more complicated procedures, or answers provided by those black boxes on so many navigation tables. Do not rely only on these mental calculations, though. As
always, the prudent navigator uses more than one tool to guide his or her boat through the water. With practice, these calculations become automatic, as will the savings in time and effort their use
will provide.
Larry McKenna is an assistant professor of geology at the University of Kansas and a navigation enthusiast. | {"url":"http://www.oceannavigator.com/January-February-2003/Trig-shortcuts-for-quick-nav-solutions/","timestamp":"2014-04-19T01:52:26Z","content_type":null,"content_length":"20414","record_id":"<urn:uuid:214a69e3-98e6-4dbd-9aba-1a791b0254f5>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00035-ip-10-147-4-33.ec2.internal.warc.gz"} |
Michael Makkai
Michael Makkai
Michael Makkai (Hungarian original Mihály Makkai) is a Canadian mathematician born in Hungary; he studied in Budapest and Warsaw. He is the inventor of anafunctors and FOLDS, which are both ways of
doing category theory and higher category theory without saying anything evil or requiring the axiom of choice. Makkai is a pioneer in the category-theoretic approach to logic (which is mostly the
approach that you see here on the $n$-Lab).
• Models, logics, and higher-dimensional categories. A tribute to the work of Mihály Makkai. Proceedings of the meeting held at the Université de Montréal, June 18–20, 2009. Edited by Bradd Hart,
Thomas G. Kucera, Anand Pillay, Philip J. Scott and Robert A. G. Seely. CRM Proceedings & Lecture Notes 53, Amer. Math. Soc. 2011. x+426 pp.
• math genealogy entry
Revised on May 31, 2012 17:28:41 by
Zoran Škoda | {"url":"http://www.ncatlab.org/nlab/show/Michael+Makkai","timestamp":"2014-04-16T13:42:48Z","content_type":null,"content_length":"13888","record_id":"<urn:uuid:b5b30054-b617-4d7b-938c-d453dcec4d1b>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00166-ip-10-147-4-33.ec2.internal.warc.gz"} |
Isomonodromy and integrability
Seminar Room 1, Newton Institute
The property of having no movable critical points for an ordinary differential equation was linked with integrable systems via theta functions in the 19th century, and more recently, since the 1970s,
with integrable partial differential equations via similarity reduction. A geometric integration of these features will be explored in the first part of the talk, after work by H. Flaschka (1980),
which suggests a deformation of the spectral curve. This provides the segue to the second part of the talk, concerning a joint project with F.W. Nijhoff. The isomonodromy equations for spectral data
(e.g., the Baker function) are studied as systems of ODEs, following R. Garnier (1912). Special functions, specifically the Kleinian sigma function, are implemented in the equations, to seek the
Gauss-Manin-connection counterpart of the Legendre equation, by which R. Fuchs (1906) had connected the isomonodromy property and the absence of movable critical points. Work by Nijhoff et al. on
discrete and Schwarzian equations would be related to this higher-genus Legendre version of the isomonodromy condition.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | {"url":"http://www.newton.ac.uk/programmes/DIS/seminars/2009063009001.html","timestamp":"2014-04-19T22:44:55Z","content_type":null,"content_length":"6951","record_id":"<urn:uuid:cc329517-d6a0-4631-8152-c9229ea4a065>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00535-ip-10-147-4-33.ec2.internal.warc.gz"} |
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A probability question related to extremal combinatorics
up vote 18 down vote favorite
$k$ people play the following game: person $i$ independently picks a subset $S_i$ of $\{ 1,2,\ldots,n \}$ according to some distribution $p$ on the $2^n$ subsets; each person uses the same
distribution $p$. If some $S_i$ is contained in $\cup_{j \neq i} S_j$, they all lose; else, they all win. What distribution $p$ maximizes the probability of winning?
I am actually only interested in the case where $n/k$ is an integer, in which case I would conjecture that the optimal distribution is for each person to pick a random subset with $n/k$ elements. I
can prove this only for $k=2$, in which case it follows straightforwardly from Sperner's theorem.
Edit: JBL points out in the comments that its also easy to confirm the $n=k$ case of the conjecture in the previous paragraph.
pr.probability co.combinatorics
1 I'm not sure what "replaced" means in this context; they are not picking balls out of a common bin. Each person independently picks a subset of $1,2,\ldots,n$. – alex Jul 14 '10 at 23:44
1 Regardless of n/k it seems to me that the most likely optimum is where each randomly picks a set of size $\lfloor n/2 \rfloor$. If everyone is always picking sets of the same size, then they lose
only when two of them pick the same set, so it makes sense to have as many sets available as possible. – Michael Albert Jul 15 '10 at 1:17
3 @Michael Albert, that's not true: if we're picking pairs and we get {1, 2} and {1, 3} and {2, 3}, we lose. – JBL Jul 15 '10 at 1:19
3 (In fact, for $n = k$ I think it shouldn't be hard to show that a uniform distribution over singletons is optimal, since any positive probability on a non-singleton is completely wasted, no?) –
JBL Jul 15 '10 at 1:57
The conjecture is false for $k=3$. If $n=6$ it's better to choose random singletons (the probability to lose is $4/9$) than random doubletons (the probability to lose is, counting conditionally on
1 the number of elements in the intersection of doubletons of the first two players, $\frac1{15} + \frac8{15}\times \frac{9}{15} + \frac{6}{15}\times \frac{6}{15} = \frac{41}{75}$, unless I messed
up). In fact, for $n$ big enough, when $k=3$ it is better to pick random singletons than random doubletons. – zhoraster Oct 17 '10 at 16:07
show 12 more comments
1 Answer
active oldest votes
Here's a quick thought about the case k=3. Let's suppose we go for the uniform distribution on sets of size cn. Then with very high probability the union of the first two sets has size
about (2c-c^2)n, so that almost all the time the conditional probability that the third set is contained in the union is about $\binom{(2c-c^2)n}{cn}/\binom n{cn}.$
This isn't necessarily a very good heuristic, since if the first two sets are more disjoint, then it becomes much more likely that the third will be contained in the union, so the fact
that the probability is small doesn't mean that one can disregard the possibility. But if one does the calculation more carefully, it doesn't seem obvious that the optimum will be at c=1/
up vote 4
down vote Actually, for comparison let's look at the probability that the first two sets are disjoint and that their union contains the third. This is $\binom{(1-c)n}{cn}/\binom n{cn}$ multiplied
by $\binom {2cn}{cn}/\binom n{cn}$. Again, we get an unpleasant enough number that I would be surprised if it were optimized at c=1/3.
So this isn't exactly an answer. It's just a suggestion that it ought to be instructive to think about which layer is best if you want to take the uniform distribution on some layer.
Thanks; maybe I was too hasty in conjecturing an answer. I'll write some maple code to evaluate the probability explicitly for various $n,k$. – alex Jul 16 '10 at 1:46
add comment
Not the answer you're looking for? Browse other questions tagged pr.probability co.combinatorics or ask your own question. | {"url":"http://mathoverflow.net/questions/31925/a-probability-question-related-to-extremal-combinatorics","timestamp":"2014-04-19T12:05:38Z","content_type":null,"content_length":"59947","record_id":"<urn:uuid:8b70af79-b963-4c2c-bc7a-6689258a28dc>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00033-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fachbereich Mathematik
4 search hits
The Quantum Zero Space Charge Model for Semiconductors (1999)
Andreas Unterreiter
The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain \(\Omega\subset I\!\!R^d,d=1,2\) or \( d=3\) is the minimizer of the total energy \({\mathcal E}_
{\epsilon\lambda}\); \({\mathcal E}_{\epsilon\lambda}\) involves the squares of the scaled Planck's constant \(\epsilon\) and the scaled minimal Debye length \(\lambda\). In applications one
frequently has \(\lambda^2\ll 1\). In these cases the zero-space-charge approximation is rigorously justified. As \(\lambda \to 0 \), the particle densities converge to the minimizer of a
limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal
energies one gets an differential-algebraic system for the limiting \((\lambda=0)\) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit \(\epsilon
\to 0\) exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small \(\epsilon\), a reasonable approximation of the quantum model if and only if the quantum
gap vanishes. The simultaneous limit \(\epsilon =\lambda \to 0\) is analyzed.
A Singular-Perturbed Two-Phase Stefan Problem Due to Slow Diffusion (1999)
Jens Struckmeier Andreas Unterreiter
The asymptotic behaviour of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit the model equations reduce to a one-phase
Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by
numerical experiments using a front-tracking method.
On a Recursive Approximation of Singularity Perturbed Parabolic Equations (extended version) (1999)
Jens Struckmeier Andreas Unterreiter
The asymptotic analysis of IBVPs for the singularly perturbed parabolic PDE ... in the limit epsilon to zero motivate investigations of certain recursively defined approximative series
("ping-pong expansions"). The recursion formulae rely on operators assigning to a boundary condition at the left or the right boundary a solution of the parabolic PDE. Sufficient conditions for
uniform convergence of ping-pong expansions are derived and a detailed analysis for the model problem ... is given.
On an Integral Equation Model for Slender Bodies in Low Reynolds-Number Flows (1999)
Thomas Götz Andreas Unterreiter
The interation of particular slender bodies with low Reynolds-number flows is in the limit 'slenderness to 0' described by a linear Fredholm integral equation of the second kind. The integral
operator of this equation has a denumerable set of polynomial eigenfunctions whose corresponding eigenvalues are non-positive and of logarithmic growth. A theorem similiar to a classical result
of Plemelj-Privalov for integral operators with Cauchy kernels is proven. In contrast to Cauchy kernel operators, the integral operator maps no Hölder space into itself. A spectral analysis of
the integral operator restricted to an appropriate class of analytic functions is performed. The spectral properties of this restricted integral operator suggest a collocation-like method to
solve the integral equation numerically. For this numerical scheme, convergence is proven and several computations are presented. | {"url":"https://kluedo.ub.uni-kl.de/solrsearch/index/search/searchtype/collection/id/15997/start/0/rows/10/yearfq/1999/author_facetfq/Andreas+Unterreiter","timestamp":"2014-04-18T09:34:39Z","content_type":null,"content_length":"26422","record_id":"<urn:uuid:e162e38d-be4c-4a72-9d65-2454652ae3f0>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00019-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kicker physics
I was working with a physics student the other day and came across a football trajectory problem which was very timely because Superbowl Sunday fast approaching and the Pittsburgh Steelers are still
in the mix. Yeah!
Most trajectory problems in physics problems meant to be worked “by hand” have to make some assumptions so that the equations can be solved without using a computer. The biggest and most common
assumption is that the effect of the air resistance on the object is negligible. “Negligible” means that we consider (or at least are willing to temporarily pretend) that the effect is small enough
that if we neglect accounting for it in our calculations our answer could still be useful in some way. For example, if:
a) we know our kicker can usually kick with an certain initial speed v, and if
b) we know how to use basic trajectory physics calculations to tell us the maximum possible range R for a kick at a given angle from the horizontal (let’s call it theta), and if
c) we know how to use calculus to optimize our choice of initial conditions for various purposes, such as maximizing range, or height above the cross bar, and if
d) we are willing to accept an answer whose mileage may vary depending on how the ball is kicked (a tight spiral is best, but end over end is next best) due to the variety of magnitudes of air
resistance, and if
e) we are willing to recognize that our theoretically calculated range is in practice greatly reduced due to air resistance forces (drag) by an additive quantity which is some factor times the square
of the result of taking the ball’s forward speed minus the wind’s forward speed, with both speeds being measured with respect to the ground, and
f) point e) applies to a greater extent as for initial velocity and final range increase because drag forces and the resulting decelerations factor in more heavily as the speed and its square
increase (therefore drag typically especially decelerates the ball during the initial fastest moments of the trajectory)
Note that in the coordinate system I am defining for these problems, a headwind (kicking into the direction the wind is coming from) will be considered a negative wind speed while a tailwind (kicking
with the wind) could be considered a positive wind speed. That way, the difference between the ball’s (also defined as positive in the forward-going direction) forward speed and the wind’s forward
speed increases to a value higher than the ball’s speed relative to the ground in a headwind, and decreases to a value less than the ball’s speed relative to the ground if kicked in a tailwind.
We can also see that in a very unusual special situation where the tailwind is roughly constant, in the same direction of the kick and equal to the ball’s initial horizontal ground speed, the
assumption of negligible air resistance is much more reasonable, especially for fairly low angles of kick (drag forces would still be acting in the vertical equation of motion, even in this
situation). Although highly unlikely this is theoretically possible. In fact, to get an idea of how much wind would be reasonable to hypothesize, I searched the web for "windiest NFL game" and found
a hilarious video of some field goal attempts gone wild during an NFL game that was played in winds gusting to 45mph, which is fairly close to a typical kickoff’s initial horizontal velocity. Such
weather could happen again, and if the wind’s direction and speed happens to line up with the kicker’s targeted direction, we can expect negligible drag predictions to actually become fairly
realistic. In such ideal conditions, for example one might expect a kick with an initial velocity of 30m/s to fly around 30 yards further (between 80 and 85 yards) before returning to earth than the
identically launched kick would travel under calm conditions. I would guess that wind conditions were favorable when the records for longest field goal (63 yards for the NFL, 62 yards CFL) and
longest punts (98 yards NFL, 108 CFL) were set. Note that with the longer field (110 yards) and the rules for recording a punt, the difference in best punt likely has more to do with how the
statistics for punting are kept than the actual distance the ball traveled -- there may have also been some lucky bounces involved).
However for typical football weather, we would simply expect our answer to over-estimate the range because in realty the football simply does not have enough mass to avoid letting the small forces
due to air resistance (also known as drag) apply a negative horizontal acceleration (or a slowing effect) to the ball’s trajectory, especially for long punts, field goal attempts or hail mary passes.
Let’s take a brief side trip to examine why the football’s relatively low mass means neglecting air resistance throws off our answer more than if the football were heavier. We will compare and
contrast the flight of a shot put and a football with identical initial speeds and launch angles in air and in a vacuum. In a vacuum, with no air resistance one would expect both objects to travel
identical trajectories, but in air, we would expect both objects, especially the football, to slow down and fall short of that ideal. Let’s apply Newton’s second law of motion, F = m*a (where F is
the force acting on an object, m = the mass of that object, and a = the acceleration, or rate of change of its speed, of an object) to examine this more closely. I will assume that the drag force on
the shot put and the football are about equal when the football is thrown in a tight spiral because they have roughly the same cross-sectional area when viewed from the front, and aerodynamicists say
that drag forces on an object moving through a fluid are typically proportional to cross-sectional area, fluid density, and the square of v_fluid, where v_fluid represents the velocity of the object
with respect to the fluid. There is also another factor called the drag coefficient that accounts for the effect on drag of an object’s shape but for the purposes of this discussion we will consider
the drag coefficients for a tightly spiraling football and a shot put to be close enough not to worry about. To simplify comparing the acceleration due to air’s drag on a football with the
acceleration due to air’s drag on a men’s Olympic shotput we can solve F = ma for a (by dividing both sides by m) to write the same law in a different form:
a = F/m.
This expression for acceleration shows that the resulting accelerations of the objects due to drag will increase in direct proportion to the drag forces acting on the objects. In our case we are
assuming that the drag forces on the two objects are roughly equal, so this is of not much concern. The equation also shows that the resulting accelerations of the objects due to drag are inversely
proportional to the masses of the objects. In other words, if the weight of a football ranges around 14-15 oz | 0.9 lb |0.4 kg, and the weight of a (men’s Olympic size) shot put is 16 lb | 7.26 kg
(women’s is 8.8 lb or 4 kg), then the roughly 16 times more massive shotput will experience roughly one sixteenth as much acceleration due to drag as the football. I chose not to use the word
deceleration here despite recognizing that these drag forces, and therefore the accelerations due to drag, always oppose the motion of the objects relative to the fluid, because in the case of a tail
wind, “drag” forces actually impart a positive acceleration to the object; more on that later. The main point of all this is that we can conclude that neglecting drag always gives erroneous results
when calculating trajectories of object’s moving through fluids, but the magnitude of the error is inversely proportional to the object’s mass. We would expect drag to shorten a football’s flight
trajectory somewhere between ten and twenty times more than a men’s Olympic shot put thrown with the same initial velocity.
Having given a little justification for the common practice of assuming negligible air resistance, in my next blog I will discuss how to determine the general formula for the expected range when air
resistance is neglected. I'm planning to follow that up with another blog outlining the procedure for determining the optimal launch angle is (assuming negligible air resistance). Later, if anyone
expresses interest, I’m prepared to discuss the general procedure necessary to take wind and drag into account; practical basic computing techniques will be involved in developing a simple physics
simulation. If there is any interest I may even eventually write a companion entry touching on ordinary differential equations and their digital corollary, difference equations, and how they apply to
this kind of simulation.
If you enjoyed this post and would like to see more on this or similar topics, please let me know by sending me a message or leave a comment for all to share. | {"url":"http://www.wyzant.com/resources/blogs/8824/kicker_physics","timestamp":"2014-04-19T02:30:44Z","content_type":null,"content_length":"48212","record_id":"<urn:uuid:c6031860-59ac-4044-bc2c-fbaeb342dffb>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00659-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Tutor] Need help with rewriting script to use Decimal module
[Tutor] Need help with rewriting script to use Decimal module
Terry Carroll carroll at tjc.com
Thu Jan 4 17:20:41 CET 2007
On Wed, 3 Jan 2007, Dick Moores wrote:
> Terry, I just noticed that farey(0.36, 10) returns (1, 3), a pretty
> big miss, IMO. The correct fraction with smallest error and maximum
> denominator of 10 is 3/8, which I'm proud to say my klunky frac.py
> (<http://www.rcblue.com/Python/fracForWeb.py>) produces.
Dick, good catch. Consider adding a comment to the Cookbook page pointing
out that the method sometimes does not produce the incorrect answer,
including for that input.
I note that the web page for the recipe says only that its produces "a
rational," not necessarily the closest rational, but the description in
the book says it's the closest:
You have a number v (of almost any type) and need to find a rational
number (in reduced form) that is as close to v as possible but with a
denominator no larger than a prescribed value. (Recipe 18.13)
More information about the Tutor mailing list | {"url":"https://mail.python.org/pipermail/tutor/2007-January/051817.html","timestamp":"2014-04-16T08:43:06Z","content_type":null,"content_length":"3710","record_id":"<urn:uuid:093fd540-73de-4c14-81e5-8f9eb0e55bad>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00421-ip-10-147-4-33.ec2.internal.warc.gz"} |
Libraries in R
Next: Installing R libraries locally Up: R, S, and Splus Previous: If/else syntax issues
The comprehensive R archive (CRAN) has a bunch of very useful libraries. You can install a library from within R using the install.packages() function. Among the useful libraries I use are:
• maps: allows you to make maps of the world, the US, and smaller areas
• mapproj: allows you to do cartographic projections
• fields: lots of useful functions for spatial data, including kriging and thin-plate splines
• nlme: linear and non-linear mixed effects models
• mgcv: provides a gam() function that allows you to do multivariate non-parametric regression (i.e., non-additive modelling) using multiple penalty optimization - this is a much more flexible
model that the gam() in Splus that does additive modelling
• ts: time series stuff
If your favorite Splus command is not available in R, it's likely that the command or an equivalent version of it is available in an R library that hasn't been loaded. A little rooting around at CRAN
will usually do the trick.
To load a library automatically when you begin your R session, include the line
in your .Rprofile file.
Next: Installing R libraries locally Up: R, S, and Splus Previous: If/else syntax issues Chris Paciorek 2012-01-21 | {"url":"http://www.stat.berkeley.edu/~paciorek/computingTips/Libraries_in_R.html","timestamp":"2014-04-16T23:07:15Z","content_type":null,"content_length":"4394","record_id":"<urn:uuid:5568ef03-ed7a-4461-b1fa-6718cf0409e4>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00297-ip-10-147-4-33.ec2.internal.warc.gz"} |
Momentum and the pressure of an ideal gas. Easy Question.
My book (Halliday, 6th ed: Section 20-4), uses the momentum of the individual molecules in a gas to derive the pressure of the gas. They imagine the molecules hitting a wall. I'm a little rusty on my
memory of conservation of momentum, so this equation is confusing me a bit:
(delta)p[x] = (-mv[x]) - (mv[x]) = -2mv[x]
They then say that the molecule delivers +2mv[x] of momentum to the wall.
Are they approximating the wall as much more massive than the molecule, so the molecule's speed is unchanged (velocity opposite)? So to get from +mv[x] to -mv[x] you would take
mv[x] -mv[x] -mv[x] = -mv[x]
to get the change in momentum?
I'm thinking I'm probably right, but I just feel a little queezy about it. Like there is something I'm not getting. I guess what I don't understand is the sentence: "the momentum (delta)p[x]
delivered to the wall by the molecule during the collision is +2mv[x]."
Could someone possibly start from scratch and show how all this comes from the conservation of momentum?
I hope I made sense.
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Maths Challange
Challenge #04 - Training at the Academy
Your Answers ----------------------------------------
(12/12) Eric Ferraro : The answer to the IQ challenge #5 can range in any multiple of 4. Logic tells you that if he devides the remainder into groups of 4s then there has to be an answer with a
multiple of 4 students.
Shika : This defeats the purpose of the story I cooked up. Next!
(10/12) Klave : well, first off...you are right...my first assertion was wrong...it couldn't be even just because the list was identical...and I found another way to look at it...Firstly...they were
already divided into
4 squads...which is something i don't keep my mind to so I'm writing it...and a student or a few is left behind so ranging from 1-3 students...and it didn't have to be identifical if I was to raise
the arguement that you said "could be identical if the students were re-labeled"...and where am i going in this...back to the drawing board...
back to finding out the number of students...the total number of students was not only even it could also be odd...
the list could be made identical if and only if all the students gang up on one...meaning all groups of four gang up on one student...and there you have it my answer...baaah!!! my first answer was
probably more
interesting...maybe not...i'm guessing this... Oh yeah...if there were two or three being ganged up by many...then the list couldn't be identical anymore as they will have to spar each other.I think
I'm still wrong though...Well...here it goes
Shika : You're right. You're still wrong. Sorry, but your answer is not exhaustive and lack of justification. :)
(09/12) Zulhilmi Zainuddin : I think the answer will be 4. reason: The number of the genins should be even, through my understanding. We divide them into 2 teams, first team fight with the second
team. They can be any number (team 1 can be 55, team 2 can be 45). Because they can't team up with genins who they have already fought, generally they will be team up with they own group, then if
possible with the other groups.
I don't know how to elaborate more but this is my equation: a+b/4 = y.
(a=team1, b=team2, y=total team). I think you can get it from my equation, where the 4 people left are those who have fought with each other, so they can't be teammates. In case my assumption is
wrong, that the number is not even (I hope not), then possible
answer will only be 1,2,3.
Shika : Your assumption is wrong.And you are basically suggesting all possible remainders, which defeats the purpose of the question.
(07/12) Lynn Kang : The answer is 1.
Shika : Incorrect. Looks like I'll be forced to reveal the answer soon.
(07/12) KingR@ : The correct answer is very simple......your mama!!! LOL, I am done.
Shika : My mother didn't take part in the training. Next!
(07/12) Djamin schurr : Hard to explain because im dutch but..
the battles A and B are already decided when he devides them in groups of 4 so the only logical options are 0,1,2 and 3
but if there are two rounds and everyone can choose their opponents in the first round but eventually have to fight ALL the others in the second round then no matter how many acadamy students there
are.. it must be divided by 2
now if you for example pick a number like 15.. the students will be fighting 7 in the first round and in the second round another 7 wich is evenly devided so that iruka could just swap the names to
get a normal outcome and if you take 14 as an example the students will be fighting 6 in the first round and 7 in the second. so no swapping is possible.
up until now it seems that the option is 1 or 3 since it cannot be an even
number and somebody is always left out, so wich one is it..
for that i picked the numbers 5 (1 guy left) and 7 (3 guys left)
you can only have a combat if it comes from two sides so if nr. 1 fights nr. 3 then nr. 3 must also fight nr. 1
with 5: round A =
1 vs. 2,3
2 vs. 1,4
3 vs. 1,5
4 vs. 2,5
5 vs. 4,3
with 5: round B =
1 vs. 4,5
2 vs. 3,5
3 vs. 2,4
4 vs. 1,3
5 vs. 1,2
with 7: round A
1 vs. 2,3,6
2 vs. 1,4,7
3 vs. 1,5,4
4 vs. 2,6,3
5 vs. 3,7,6
6 vs. 4,1,5
7 vs. 5,2
with 7: round B
1 vs. 4,5,7
2 vs. 3,5,6
3 vs. 2,6,7
4 vs. 1,5,7
5 vs. 1,4,2
6 vs. 2,3,7
7 vs. 1,3,4,6
As you can see with 5 you can swap numbers easy and with 7 its impossible..
so my conclusion is that there is only 1 person that will be left out.
a bit long but i hope i explained it well :)
so am i right?
Shika : Nope. The answer is a range of values. And it's incredibly simple.
(07/12) Nico : Just a random guess:
Take a class of x nins.
One ninja may fight n nins in the first round. In the second round, that nin must then fight x - (n + 1) nins to complete the condition of fighting every nin.
If the table for round 2 can be rearranged to be exactly the same as round 1, that indicates that x - (n + 1) == n , which means x = 2n + 1. This indicates that x must be odd ( x - 1 cannot be equal
to 2 times any number, if x is even ) , hence the remainder when divided by 4 can also therefore only be odd, i.e. 1 or 3.
I don't think it could be narrowed down further than that, but hey. Much
like Shika, I'm far too lazy to go any further.
Shika : I see some sort of reasoning here, which leads to a wrong answer.
(06/12) PCay : Ok. Since 1 person goes against 5, 4 go against 8.
*shrugs* I say 8 persons can be left out.
Shika : I guess you mean 0 then(since the remainder of any number when divided by 4 is 0,1,2,3). Please refer to previous wrong answer.
(03/12) Puizui Puizui: Dear shika, i wanted to clarify a few things
1) In the table, is the students place in groups? (eg if there are 5 students,a and b fought c . so does the table show a and b together in a box ,with arrows leading to C; or 2 seperate boxes with
2) Does"If all the students are now divided into squads of 4, what is the
possible numbers of students that will be left over. " mean that the total
number of students in iruga's class is divided by 4 (after the training)
(and thus the qn asks for the remainder ,after dividing it by 4)
Shika : With regard to 1.) it is a & b in a box labelled c, to indicate c's opponent. 2.)yes
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Revisiting a Childhood Addition Code with Sketchpad
Posted on June 20, 2013 by Daniel Scher
As a fourth-grader in 1977, I had a love-hate relationship with my Addison-Wesley textbook. Its contents overflowed with arithmetic problems, but every so often an entertaining brainteaser appeared
to break the monotony of drill practice. These puzzles were clearly marked: Each appeared in a box set aside from the main text and featured a bespectacled fish to introduce the challenge.
The back pages of the textbook featured still more drill, but below each set of problems came a mysterious code. Here is an example:
There was no fish or other indication to suggest that these codes were there for my consumption, but I didn’t care. I needed to satisfy my curiosity and uncover their meaning. I soon realized that
the codes were a way to represent the answer to each addition problem with letters in place of the digits. If the answer to 420 + 189 was given as HDA, for example, then H = 6, D = 0, and A = 9.
I now had powerful technique for breezing through each set of drill problems without solving each and every one. By focusing on just enough problems to determine the values of the letters A through
J, I could complete the rest by applying the cracked code.
Unfortunately, my teacher didn’t share my enthusiasm for thwarting the computations and chided me to get back on course. To this day, I still wonder about the intent of the curriculum authors. Were
the codes an Easter egg intended for students to discover and use to their advantage? I’d like to think so.
Several months ago as part of the Dynamic Number project, I decided to create my own Sketchpad game that would focus on codes. The result is similar to my Addison-Wesley textbook, but with a key
difference: Now, both the sums and addends are encoded with letters so that no digits appear at all. Take a look at the example below.
A grid displays the letters from A to J, each of which represents a digit from 0 to 9. Red counters sit on B and I, and their sum, AD (where A is the tens digit and D is the ones digit) appears below
the grid. By dragging the counters onto different letter pairs, students’ goal is to determine the numerical values of all ten letters. Download the sketch and spend some time with the code, paying
attention to the strategies you use to decrypt it.
Unlike the codes from my elementary textbook where the mathematical connection between the codes and the addition problems was minimal, these Sketchpad codes require students to think about the
properties of addition in order to decipher them. Knowing, for example, that zero is the identity element for addition can help students to pinpoint which letter represents zero.
When you’ve solved the addition code, move on to the next page of the Sketchpad document. You’ll find a new code to decipher that replaces addition with multiplication.
Share with us in the comments section the strategies you devise for solving the codes. Unlike my fourth-grade teacher, I promise that I won’t disapprove of the time you spend with these puzzles!
Leave a Reply Cancel reply
About Daniel Scher
Daniel Scher, Ph.D., is a senior scientist at KCP Technologies, where he co-directs the NSF-funded Dynamic Number project. He has developed Sketchpad activities across the entire mathematics
curriculum, from elementary school through college. He received his Ph.D. in Mathematics Education at New York University.
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Volume of Sphere
May 2nd 2009, 04:32 PM #1
Nov 2007
Volume of Sphere
A spherically shaped balloon is being inflated so that the radius is changing at a constant rate of 2 in./sec. Find an algebraic representation for V(t), the volume in terms of time, t.
Would this be the right algebraic equation, $V(t) = \frac {4}{3} \pi (2t + r)^3$ ?
Adding r inside the parentheses is redundant because r=2t. You see what I mean? So everything is right, but I would write the eqaution like this:
Does that help you at all?
Last edited by VonNemo19; May 2nd 2009 at 04:43 PM. Reason: 1
Well if "r" is the original radius of the balloon, wouldn't you need the variable when it is at 0 seconds.
I doubt that the problem requires previous Knowlege of the original radius. I mean, It's a balloon. It could be laying down flat before it was inflated, or possibly wrinkled. Yeah, maybe, but if
I was your teacher, the answer I gave you is the one I would be looking for.
Ok thanks alot. I'm gonna ask my teacher to clarify it just to make sure, but thanks.
May 2nd 2009, 04:41 PM #2
May 2nd 2009, 05:36 PM #3
Nov 2007
May 2nd 2009, 07:52 PM #4
May 2nd 2009, 08:01 PM #5
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[Haskell-cafe] beginner's problem about lists
falseep at gmail.com falseep at gmail.com
Tue Oct 10 09:42:04 EDT 2006
Thanks for your reply. I tried a few ways but none worked.
One is like:
shorter as bs = f id id as bs where
f ca cb [] _ = ca []
f ca cb _ [] = cb []
f ca cb (a:as) (b:bs) = f (ca.(a:)) (cb.(b:)) as bs
However this will result in a non-terminating loop for shorter [1..] [2..],
since the first two patterns of f shall never match.
Another way, I could guarantee that the evaluation of
shorter [1..5] (shorter [1..] [2..])
terminate but I lose the information to figure out which list was the
shortest one.
Using zips:
shorter = zipWith (\a b -> undefined)
-- this returns the length, but not the content of the shorter list
(\a b -> undefined) could be replaced with something that encode the
contents of
the two lists, but it makes no difference since I won't know which one is
the answer.
The difficulty is that I cannot have these both:
A. if one list is finite, figure out the shorter one
B. if both are infinite, returning an infinite list could work
BTW, there IS an way to implement this functionality for a finite list of
(possibly infinite) lists:
shortest = measureWith [] where
measureWith ruler as = f matches
where ruler' = undefined : ruler
matches = filter p as
p a = length (zip ruler' a) == length (zip ruler a)
f [] = measureWith ruler' as
f matches = matches
which somehow makes it unnecessary to find the function "shorter",
but the original simple problem is interesting itself.
On 10/10/06, Neil Mitchell <ndmitchell at gmail.com> wrote:
> Hi,
> The trick is not call "length", since length demands the whole of a
> list, and won't terminate on an infinite list. You will want to
> recurse down the lists.
> Is this a homework problem? It's best to declare if it is, and show
> what you've managed to do so far.
> Thanks
> Neil
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Half Range Fourier Series
January 29th 2013, 10:52 AM #1
Nov 2012
Half Range Fourier Series
If $f(x)$ is defined on $[0,L]$ then it can be extended to be either odd or even giving formulas
$f(x)=\sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L}$ and
$f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}$ respectively
If $f(x)$ was not even or odd, is there another formula that can be used to express it?
I'm trying to get something of the form
$f(x)=\sum_{n=0}^{\infty}c_n\sin\frac{(2n+1)\pi x}{2L}$, where $c_n$ is some constant expressed as an integral of $f$, but I've no idea how to show this.
Last edited by Rachel123; January 29th 2013 at 10:59 AM.
Re: Half Range Fourier Series
The Fourier basis functions $\sin$ and $\cos$ are eigenvectors of the operator $abla^2$. Recall that eigenvectors satisfy $abla^2 u = -\lambda^2 u$. Then $u = a\cos(\lambda x)+b\sin(\lambda x)$.
If you have (zero) Dirichlet boundary conditions, you get eigenvectors $u = \sin(\lambda x)$ where $\lambda = {\pi n\over L}$. If you have (zero) Neumann boundary condition $u = \cos({\pi n\over
L} x)$ are eigenvectors. You can also have periodic boundary conditions, which yields the traditional Fourier series.
Now you ask about $u = \sin\left({\pi (2n+1)\over 2L} x\right)$ eigenvectors. I think you can make it work with $u(0)=0$ and ${du\over dx}(L)=0$.
Last edited by vincisonfire; January 29th 2013 at 05:57 PM.
Re: Half Range Fourier Series
I don't understand this, we have not yet covered eigenvectors relating to fourier series and Dirichlet boundary condition yet. Is there a simpler way to show this?
Re: Half Range Fourier Series
I think you can just break it up into odd and even parts - so $f(x) = f_{odd}(x)+f_{even}(x)=$
$\sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L} +\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}=$
$\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}+b_n\sin\frac{\pi nx}{L}$
And by the way, $f_{odd}(x)=\frac{f(x)-f(-x)}{2}$ and $f_{even}(x)=\frac{f(x)+f(-x)}{2}$.
- Hollywood
January 29th 2013, 11:37 AM #2
January 29th 2013, 03:48 PM #3
Nov 2012
January 30th 2013, 06:49 AM #4
Super Member
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(Sh,Sh-map) represents the category of sheaves on a stack.
up vote 5 down vote favorite
I'm trying to understand the following theorem, but I don't think I'm reading it correctly.
Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid of objects over $X\in \mathcal{C}$. Let $\mbox{Sh}:\mathcal{C}^{op} \rightarrow \mbox{Gpds}$ be
the functor taking $X$ to the category of sheaves on $\mathcal{C}/X$ and isomorphisms of sheaves, and let $\mbox{Sh-map}:\mathcal{C}^{op} \rightarrow \mbox{Gpds}$ be the functor taking $X$ to the
category whose objects are sheaf morphisms $\mathscr{F} \rightarrow \mathscr{G}$ and whose morphisms are commuting squares of sheaves determined by isomorphisms $\mathscr{F}_1 \stackrel{\sim}{\
rightarrow} \mathscr{F}_2$ and $\mathscr{G}_1 \stackrel{\sim}{\rightarrow} \mathscr{G}_2$. These are in fact both stacks on $\mathcal{C}$, and moreover they determine a category-object $(\mbox{Sh},\
mbox{Sh-map})$ in the category of stacks.
Theorem: The category of sheaves on a stack $\mathscr{M}$ is equivalent to the category of morphisms of stacks $\mathscr{M} \rightarrow (\mbox{Sh,Sh-map})$. That is, the objects are the
1-morphisms and the morphisms are the 2-morphisms.
I'd like to interpret this to mean that the objects of $Shv(\mathscr{M})$ are associated to 1-morphisms $\mathscr{M} \rightarrow \mbox{Sh}$, and that the morphisms of $Shv(\mathscr{M})$ are
associated to 2-morphisms in $Hom_{Stacks}(\mathscr{M},\mbox{Sh})$, which in turn should be the same as 1-morphisms $\mathscr{M} \rightarrow \mbox{Sh-map}$. But there a number of problems with this.
First, given a sheaf $\mathcal{F} \in Shv(\mathscr{M})$ I'm having trouble constructing a natural transformation $\mathscr{M} \rightarrow \mbox{Sh}$. Perhaps I shouldn't, but to check this I'm using
a test object $X\in \mathcal{C}$. By Yoneda, an object of $\mathscr{M}(X)$ is the same as a 1-morphism of stacks $f:X\rightarrow \mathscr{M}$, and so I obtain an object of $Sh(X)$ (i.e. a sheaf on $\
mathcal{C}/X$) via $(\alpha:Y\rightarrow X) \mapsto \mathcal{F}(f\alpha:Y \rightarrow X \rightarrow \mathscr{M})$. That's natural enough. Again by Yoneda, a morphism in $\mathscr{M}(X)$ is a
2-morphism between maps $f,g:X\rightarrow \mathscr{M}$ of stacks, i.e. a section $s:X\rightarrow X\times_\mathscr{M} X$ of the projection from the 2-category fiber product. Out of this, I'm supposed
to construct a natural transformation from the sheaf $(\alpha:Y\rightarrow X) \mapsto \mathcal{F}(f\alpha:Y \rightarrow X \rightarrow \mathscr{M})$ to the sheaf $(\alpha:Y\rightarrow X) \mapsto \
mathcal{F}(g\alpha:Y \rightarrow X \rightarrow \mathscr{M})$. But the only structure in place to give me such a thing is a morphism in $Stacks/\mathscr{M}$ between $f\alpha$ and $g\alpha$, and I
don't see how to construct this.
Second, a 2-morphism between 1-morphisms $f,g\in Hom_{Stacks}(\mathscr{M},\mbox{Sh})$ is a section $s:\mathscr{M} \rightarrow \mathscr{M} \times_{\mbox{Sh}} \mathscr{M}$. Thus for any $(\alpha:X\
rightarrow \mathscr{M})\in \mathscr{M}(X)$, we get an object $(\alpha,\beta:X \rightarrow \mathscr{M},\varphi:f\alpha \stackrel{\sim}{\rightarrow} g\alpha)\in (\mathscr{M}\times_{\mbox{Sh}}\mathscr
{M})(X)$. On the other hand, a 1-morphism $\mathscr{M} \rightarrow \mbox{Sh-map}$ is for each $\alpha:X \rightarrow \mathscr{M}$ an arbitrary morphism on sheaves on $\mathcal{C}/X$. These can't be
the same.
By the way, I've tried to do (what I think is) the right thing and work out the sheaf in $Shv(\mbox{Sh})$ associated to the 1-morphism $\mbox{Id}:\mbox{Sh} \rightarrow \mbox{Sh}$, following Yoneda
and all. From the above, it's easy to see what this sheaf should do to morphisms $X\rightarrow \mbox{Sh}$ from a representable stack. But it appears that I need to make choices if I want to say what
it does to arbitrary morphisms of stacks $\mathscr{N} \rightarrow \mbox{Sh}$. Perhaps instead I should take a limit or colimit over its application to the full subcategory of representable stacks
over $\mathscr{N}$?
ct.category-theory stacks
Where did you see this theorem? – S. Carnahan♦ Jul 12 '11 at 5:56
This is Theorem 11.2 on page 36 here: math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf – Aaron Mazel-Gee Jul 12 '11 at 16:06
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2 Answers
active oldest votes
The notes you are reading seem to disagree with more commonly accepted language (cf. SGA1 Exp 13, Vistoli's notes, or the Stacks project). Some of this seems to be an attempt at
expository ease, e.g., the parenthetical remark in example 8.2 ("We will mention the following technical difficulties but will ignore them for now:") where "for now" really means forever.
Oddly enough, one of the mentioned technical difficulties is more or less what prevents $\text{Sh}$ and $\text{Sh-map}$ from having natural stack structures in the sense of the notes -
pullback is not strictly functorial. This un-naturality is why the common definition of stack is different - the notion of stack in the notes corresponds to the usual notion of stack in
groupoids equipped with a splitting (or cleavage).
The use of the category object $(\text{Sh}, \text{Sh-map})$ is a kludge to replace the usual stack $Sh/\mathcal{C}$ (in categories rather than groupoids) whose objects are sheaves over
comma categories, and whose morphisms over any $f: U \to V$ in $\mathcal{C}$ are $f$-maps of sheaves - see Examples 3.20 and 4.11 in Vistoli. The author of the notes employs $\text
{Sh-map}$ in order to add non-invertible sheaf maps, because the 2-morphisms in $Hom_{Stacks}(\mathcal{M}, \text{Sh})$ are all invertible. In other words, you have to throw away the
up vote 3 2-morphisms that are given to you by $\text{Sh}$, and use the larger collection of possibly non-invertible two-morphisms afforded by $\text{Sh-map}$.
down vote
accepted Once you have done that, I think your main problems are resolved. You've already worked out the object part of getting from a sheaf on $Stacks/\mathcal{M}$ to a natural transformation
from $\mathcal{M}$ to $\text{Sh}$. If you have a morphism $\beta: X \to Z$ in $\mathcal{C}$, and $f: Z \to \mathcal{M}$, then $\beta$ induces a morphism of stacks over $\mathcal{M}$. If
I'm not mistaken, the sheaf $\mathcal{F}$ takes this to the map in $\text{Sh}$ given by base change: $$\left( (\alpha: Y \mapsto Z) \mapsto \mathcal{F}(f \circ \alpha) \right) \mapsto \
left( \beta^* \alpha: Y \times_Z X \to X) \mapsto \mathcal{F}(f \circ \beta \circ \beta^*\alpha) \right)$$
Similarly, you can get from a sheaf map on $Stacks/\mathcal{M}$ to a natural transformation from $\mathcal{M}$ to $\text{Sh-map}$. There seems to be a lot of additional checking necessary
for proving the equivalence, which I don't feel like doing for you (sorry).
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Let me try to strip off all the stack language, which confuses me, and recast what I think is your question just in terms of category theory. (If I have misinterpreted which part is your
question, I apologize. The only question mark in your post is in the very last line, but I don't think that's the main question.)
You are, I believe, in the following situation. You have some ambient Cartesian category ($Stacks(M)$). You have a test object $M$ in your category, which happens to be the terminal object,
if I'm not mistaken, but I don't think this matters. You have a category object $C_1 \rightrightarrows C_0$ internal to your category. Then in particular for any test object $M$, there is a
category (in SET) whose morphisms are the arrows $M \to C_1$, and whose objects are the arrows $M \to C_0$ --- indeed, being a category object is equivalent to being a representable presheaf
valued in categories. Then you also have a theorem that recognizing this category as some other more interesting category ($Sheaves(M)$).
This almost sums up your paragraph after the theorem. But complicating the matter is that you don't have some ambient Cartesian category. Rather, $Stacks(M)$ is a (Cartesian) 2-category. So
now we have two separate notions. Indeed, as a 2-category, it is among other things enriched in categories, so that you have in fact a category of morphisms $M \to C_0$, for example. In the
previous paragraph, I was only considering this as a set.
up vote
1 down So then perhaps your question is the following. You have an ambient (Cartesian) 2-category, and let's assume it to be strict, a test object $M$, and an object $C_0$. Then $\hom(M,C_0)$ is a
vote 1-category. The objects of this 1-category is the unenriched hom, which I will denote as $|\hom(M,C_0)|$. You'd like to recognize the morphisms of $\hom(M,C_0)$ as the set $|\hom(M,C_1)|$
for some particular $C_1$. Suppose that you have good Cartesian-closedness conditions, and an "inner hom" $\underline{\hom}$. Then what you'd like is a "walking arrow object" $Arr$ (which
you do have with even stronger conditions, with buzzwords like "tensored over CAT"), and then $C_1 = \underline{\hom}(Arr,C_0)$.
My guess is that the category of stacks on $M$ does have all of these strong closedness and tensored conditions. Moreover, my guess is that the correctly-implemented inner-hom construction
in the previous paragraph, applied when $C_0 = Sh(M)$, yields $C_1 = ShMap(M)$. Maybe these are originally the versions with target CAT, not GPD, but then the trick is to post-compose with
the 2-functor CAT$\to$GPD that keeps only the invertible morphisms, and make sure that this doesn't change too much.
I hope this recasting helps, and that I didn't utterly misinterpret your question. This is about the limit of my category theory, and well past what I know about stacks properly.
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An introduction to inductive definitions
Results 1 - 10 of 128
- Handbook of Logic in Computer Science , 1994
"... Least fixpoints as meanings of recursive definitions. ..."
- Annals of Pure and Applied Logic , 1991
"... The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of
computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and system ..."
Cited by 231 (10 self)
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The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation
introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of
programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying
logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of
information about its execution. These ideas lead to the following programme:
- In 10th IEEE Computer Security Foundations Workshop , 1997
"... Informal justifications of security protocols involve arguing backwards that various events are impossible. Inductive definitions can make such arguments rigorous. The resulting proofs are
complicated, but can be generated reasonably quickly using the proof tool Isabelle/HOL. There is no restriction ..."
Cited by 150 (7 self)
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Informal justifications of security protocols involve arguing backwards that various events are impossible. Inductive definitions can make such arguments rigorous. The resulting proofs are
complicated, but can be generated reasonably quickly using the proof tool Isabelle/HOL. There is no restriction to finite-state systems and the approach is not based on belief logics. Protocols are
inductively defined as sets of traces, which may involve many interleaved protocol runs. Protocol descriptions model accidental key losses as well as attacks. The model spy can send spoof messages
made up of components decrypted from previous traffic. Several key distribution protocols have been studied, including NeedhamSchroeder, Yahalom and Otway-Rees. The method applies to both
symmetrickey and public-key protocols. A new attack has been discovered in a variant of Otway-Rees (already broken by Mao and Boyd). Assertions concerning secrecy and authenticity have been proved.
CONTENTS i Contents 1 Intro...
, 1988
"... Three languages with polymorphic type disciplines are discussed, namely the *-calculus with Milner's polymorphic type discipline; a language with imperative features (polymorphic references);
and a skeletal module language with structures, signatures and functors. In each of the two first cases we ..."
Cited by 92 (2 self)
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Three languages with polymorphic type disciplines are discussed, namely the *-calculus with Milner's polymorphic type discipline; a language with imperative features (polymorphic references); and a
skeletal module language with structures, signatures and functors. In each of the two first cases we show that the type inference system is consistent with an operational dynamic semantics. On the
module level, polymorphic types correspond to signatures. There is a notion of principal signature. So-called signature checking is the module level equivalent of type checking. In particular, there
exists an algorithm which either fails or produces a principal signature.
- Abstract in Proc. PODS 90 , 1995
"... We study the expressive powers of two semantics for deductive databases and logic programming: the well-founded semantics and the stable semantics. We compare them especially to two older
semantics, the two-valued and three-valued program completion semantics. We identify the expressive power of the ..."
Cited by 86 (5 self)
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We study the expressive powers of two semantics for deductive databases and logic programming: the well-founded semantics and the stable semantics. We compare them especially to two older semantics,
the two-valued and three-valued program completion semantics. We identify the expressive power of the stable semantics, and in fairly general circumstances that of the well-founded semantics. In
particular, over infinite Herbrand universes, the four semantics all have the same expressive power. We discuss a feature of certain logic programming semantics, which we call the Principle of
Stratification, a feature allowing a program to be built easily in modules. The three-valued program completion and well-founded semantics satisfy this principle. Over infinite Herbrand models, we
consider a notion of translatability between the three-valued program completion and well-founded semantics which is in a sense uniform in the strata. In this sense of uniform translatability we show
the well-founded semantics to be more expressive than the three-valued program completion. The proof is a corollary of our result that over non-Herbrand infinite models, the well-founded semantics is
more expressive than the three-valued program completion semantics. 1
"... . We show that infinite objects can be constructively understood without the consideration of partial elements, or greatest fixedpoints, through the explicit consideration of proof objects. We
present then a proof system based on these explanations. According to this analysis, the proof expressions ..."
Cited by 83 (2 self)
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. We show that infinite objects can be constructively understood without the consideration of partial elements, or greatest fixedpoints, through the explicit consideration of proof objects. We
present then a proof system based on these explanations. According to this analysis, the proof expressions should have the same structure as the program expressions of a pure functional lazy
language: variable, constructor, application, abstraction, case expressions, and local let expressions. 1 Introduction The usual explanation of infinite objects relies on the use of greatest
fixed-points of monotone operators, whose existence is justified by the impredicative proof of Tarski's fixed point theorem. The proof theory of such infinite objects, based on the so called
co-induction principle, originally due to David Park [21] and explained with this name for instance in the paper [18], reflects this explanation. Constructively, to rely on such impredicative methods
is somewhat unsatisfactory (see fo...
- Logical Frameworks , 1991
"... Martin-Lof's type theory is presented in several steps. The kernel is a dependently typed -calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive
functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. ..."
Cited by 76 (13 self)
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Martin-Lof's type theory is presented in several steps. The kernel is a dependently typed -calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive
functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. At each step syntax, inference rules, and set-theoretic semantics are given. 1 Introduction
Usually Martin-Lof's type theory is presented as a closed system with rules for a finite collection of set formers. But it is also often pointed out that the system is in principle open to extension:
we may introduce new sets when there is a need for them. The principle is that a set is by definition inductively generated - it is defined by its introduction rules, which are rules for generating
its elements. The elimination rule is determined by the introduction rules and expresses definition by primitive recursion on the way the elements of the set are generated. (In this paper I shall use
the term ...
- Journal of Symbolic Logic , 1998
"... The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe à la Tarski. A set U0 of codes for small sets is generated inductively
at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursi ..."
Cited by 65 (10 self)
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The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the
same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying
general notion of simultaneous inductiverecursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions
in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe
constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a
realisability model ...
, 1988
"... This paper develops a transformational paradigm by which nonnumerical algorithms are treated as fixed point computations derived from very high level problem specifications. We begin by
presenting an abstract functional + problem specification language SQ , which is shown to express any partial re ..."
Cited by 59 (10 self)
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This paper develops a transformational paradigm by which nonnumerical algorithms are treated as fixed point computations derived from very high level problem specifications. We begin by presenting an
abstract functional + problem specification language SQ , which is shown to express any partial recursive function in a fixed point normal form. Next, we give a nondeterministic iterative schema that
in the case of finite iteration generalizes the 'chaotic iteration' of Cousot and Cousot for computing fixed points of monotone functions efficiently. New techniques are discussed for recomputing
fixed points of distributive functions efficiently. Numerous examples illustrate how these techniques for computing and recomputing fixed points can be incorporated within a transformational
programming methodology to facilitate the design and verification of nonnumerical algorithms. 1. Introduction In a recent survey article [25] Martin Feather has said that the current state of the art
of program...
, 2000
"... The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its
relationships to other logics in the area of non-monotonic reasoning, logic programming and deductiv ..."
Cited by 58 (38 self)
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The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its
relationships to other logics in the area of non-monotonic reasoning, logic programming and deductive databases, and to show its application for knowledge representation by giving a typology of
definitional knowledge. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=737224","timestamp":"2014-04-16T05:37:22Z","content_type":null,"content_length":"37752","record_id":"<urn:uuid:fb5c4069-01db-4c17-b699-5f06f557aa9a>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00240-ip-10-147-4-33.ec2.internal.warc.gz"} |
Course Descriptions
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Mathematics, Statistics, and Computer Science Olin-Rice Science Center, Room 222 651-696-6287 sburr@macalester.edu
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COMP 110 - Data and Computing Fundamentals
This course provides an introduction to the handling, analysis, and interpretation of "big data," the massive datasets now routinely being collected in science, commerce, and government. Students
achieve facility with a sophisticated, technical computing environment. The course aligns with techniques being used in several courses in the sciences, statistics, and mathematics. The course is
intended to be accessible to all students, regardless of background.
COMP 120 - Computing and Society
Topics course that introduces students to the field of computing by way of a central theme. Topics vary; offerings include Digital Humanities, Green Computing, and Social Media. Full description
given in advance of registration. This course is suitable for students with little or no experience with computing, but it can serve as a starting point for the Computer Science major.
Frequency: Typically offered in the fall as a first-year course.
COMP 121 - Introduction to Scientific Programming
This course is intended to give students from diverse areas of science-e.g., economics, biology, physics, chemistry, geography, geology, mathematics, engineering, statistics-an ability to write
software for solving problems and carrying out research in those disciplines. The course provides an introduction to programming and computation as well as to a number of important and widely used
techniques: scientific graphics, equation solving, function fitting, optimization, storing and searching data, and simulation. There is an emphasis on ways to represent and transform information on
the computer in addition to numbers and text: images, sound, graphs and databases.
Frequency: Offered spring semester.
COMP 123 - Core Concepts in Computer Science
This course introduces the field of computer science, including central concepts such as the design and implementation of algorithms and programs, testing and analyzing programs, the representation
of information within the computer, and the role of abstraction and metaphor in computer science. The exploration of these central ideas will draw examples from a range of application areas including
multimedia processing, robotics, turtle graphics, and text processing. Course work will use the Python programming language.
Frequency: Every semester.
COMP 124 - Object-Oriented Programming and Data Structures
This course introduces the principles of software design and development using the object-oriented paradigm (OOP) and the Java programming language. Students will learn to use data structures such as
lists, trees and hash tables and they will compare the efficiency of these data structures for a particular application. Students will learn to decompose a project using OOP principles. They will
work with integrated development environments (IDEs) and version control systems. Students will practice their skills by creating applications in areas such as graphics, games, simulations, and
natural language processing. There is a required 1.5 hour laboratory section associated with this course.
Frequency: Every semester.
COMP 154 - Ethics and the Internet
This course looks at ethical questions connected with the internet as we know it today: an online environment where content is generated and shared through user activities such as blogging, media
sharing, social neetworking, tagging, tweeting, virtual world gaming, wiki developing, and the like. We will start by considering debates over freedom of speech, privacy, surveillance, and
intellectual property: issues that pre-exist the development of the Internet, but which because of it have taken on new dimensions. From here we will go on to take up some ethical questions arising
from four different domains of activity on the social web: gaming, social networking, blog/wiki developing, and "hacktivism." In the third part of the course, we will consider broad questions
connected to the integration of the Internet with devices other than the personal computer and mobile phone and which open the prospect of a world of integrated networked systems. What are some of
the impacts of such integration on our everyday ethical relations with others and on the overall quality of our lives? How does being networked affect the meaning of being human?
Frequency: Offered alternate years.
COMP 194 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
COMP 221 - Algorithm Design and Analysis
An in-depth introduction to the design and analysis of algorithms. Topics may include algorithmic paradigms and structures, including recursion, divide and conquer, dynamic programming, greedy
methods, branch and bound, randomized, probabilistic, and parallel algorithms, non-determinism and NP completeness. Applications to searching and sorting, graphs and optimization, geometric
algorithms, and transforms.
Frequency: Every fall.
COMP 225 - Software Design and Development
This course builds upon the software design foundation started in COMP 124. Students will design and implement medium-sized software projects using modern software design principles such as design
patterns, refactoring, fault tolerance, stream-based programming, and exception handling. The concept of a distributed computing system will be introduced, and students will develop multithreaded and
networked applications using currently available software libraries. Advanced graphical user interface methods will be studied with an emphasis on appropriate human-computer interaction techniques.
Students will use operating systems services and be introduced to methods of evaluating the performance of their software.
Frequency: Every fall.
COMP 124 or permission of instructor.
COMP 240 - Computer Systems Organization
This course familiarizes the student with the internal design and organization of computers. Topics include number systems, internal data representations, logic design, microarchitectures, the
functional units of a computer system, memory, processor, and input/output structures, instruction sets and assembly language, addressing techniques, system software, and non-traditional computer
Frequency: Every spring.
COMP 124 , or permission of instructor.
COMP 261 - Theory of Computation
A discussion of the basic theoretical foundations of computation as embodied in formal models and descriptions. The course will cover finite state automata, regular expressions, formal languages,
Turing machines, computability and unsolvability, and the theory of computational complexity. Introduction to alternate models of computation and recursive function theory.
Frequency: Every spring.
COMP 294 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
COMP 302 - Introduction to Database Management Systems
This course will introduce students to the design, implementation, and analysis of databases stored in database management systems (DBMS). Topics include implementation-neutral data modeling,
database design, database implementation, and data analysis using relational algebra and SQL. Students will generate data models based on real-world problems, and implement a database in a
state-of-the-art DBMS. Students will master complex data analysis by learning to first design database queries and then implement them in a database query language such as SQL. Advanced topics
include objects in databases, indexing for improved performance, distributed databases, and data warehouses.
Frequency: Offered even-numbered spring semesters.
COMP 320 - Computational Biology
This course will examine selected topics in computational biology, including basic bioinformatics, algorithms used in genomics an genome analysis, computational techniques forsystems biology, and
synthetic biology. This is an interdisciplinary course that will often be cross-listed with a course in Biology.
Frequency: Offered odd-numbered spring semesters.
Students with either Biology or Computer Science or Math coursework may register for this interdisciplinary course.
COMP 325 - Compilers, Interpreters, and Programming Languages
This course will examine the techniques that underlie compiler and interpreter creation, including lexical analysis, parsing, and compiler generators. These tools will serve as a framework for
examining programming language design issues across a range of language types (procedural, object-oriented, modern programming languages with an eye to understanding the underlying philosophy of each
language, and how it influences and is influenced by the needs of a compiler or interpreter for the language. "Back-end" issues, including intermediate representations, code generation, and
optimization will be included as they relate to specific programming languages.
Frequency: Offered odd-numbered fall semesters.
COMP 340 - Digital Electronics
A survey of fundamental ideas and methods used in the design and construction of digital electronic circuits such as computers. Emphasis will be on applying the theoretical aspects of digital design
to the actual construction of circuits in the laboratory. Topics to be covered include basic circuit theory, transistor physics, logic families (TTL, CMOS), Boolean logic principles, combinatorial
design techniques, sequential logic techniques, memory circuits and timing, and applications to microprocessor and computer design. Three lectures and one three-hour laboratory per week.
Frequency: Offered alternate spring semesters.
MATH 137 or permission of instructor.
COMP 342 - Operating Systems and Computer Architecture
The basic principles related to the design and architecture of operating systems. Concepts to be discussed include sequential and concurrent processes, synchronization and mutual exclusion, processor
scheduling, time-sharing, multiprogramming, multitasking, and parallel processing. Memory management techniques. File system design. Security and protection systems. Performance evaluation.
Frequency: Offered odd-numbered spring semesters.
COMP 240 or permission of instructor.
COMP 346 - Internet Computing
This course will investigate the latest technology available for building web applications with dynamic content. It will look at all stages in the web application design process, including: 1) client
applications, 2) web applications that service client requests, 3) application servers that manage requests for information, update data, and serve client applets, and 4) the database management
system that holds the data. The course will be programming-intensive using aspects of the Java language available for designing and implementing Internet applications. The format of the course will
be mainly laboratory-based sessions where you learn to build these four components of a web application, supported by lectures and discussions. Students will research particular topics and present
their findings during these discussion sessions. The course will also investigate the usability of designs from a human factors standpoint and discuss privacy and other social consequences of this
Frequency: Offered even-numbered fall semesters.
COMP 225 or permission of instructor.
COMP 365 - Computational Linear Algebra
This course covers a central point of contact between mathematics and computer science. Many of the computational techniques important in science, commerce, and statistics are based on concepts from
linear algebra: subspaces, projection, matrix decompositions, etc. The course reviews these concepts, adopts them to large scales, and applies them in the core techniques of scientific computing;
solving systems of linear and nonlinear equations, approximation and statistical function estimation, optimization, interpolation, Monte Carlo techniques. Applications throughout the sciences and
Frequency: Every spring.
COMP 380 - Bodies/Minds: AI Robotics
This course examines two distinct aspects of work in robotics: the physical construction of the robot's "body" and the creation of robot control programs that form the robot's "mind." It will study
the strengths and weaknesses of a variety of robot sensors, including sonar, infrared, touch, GPS, and computer vision. It will also examine a variety of techniques for robot control programs,
including both reactive and deliberative approaches. The course will include hands-on work with multiple robots, and a semester-long course project in robotics. The course format will be a seminar,
with students reading and discussing the research literature.
Frequency: Offered even-numbered spring semesters.
COMP 221 or permission of instructor.
COMP 394 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
COMP 440 - Collective Intelligence
This course will explore how computers can analyze people's collective behavior to help them. Students will read and discuss recent academic research papers about collective behavior on sites such as
Wikipedia, Facebook, and Twitter. Students will use coputational simulation and data-mining techniques to analyze online datasets.
Frequency: Offered odd-numbered fall semesters.
COMP 445 - Parallel and Distributed Processing
Many current computational challenges, such as Internet search, protein folding, and data mining require the use of multiple processes running in parallel, whether on a single multiprocessor machine
(parallel processing) or on multiple machines connected together on a network (distributed processing). The type of processing required to solve such problems in adequate amounts of time involves
dividing the program and/or problem space into parts that can run simultaneously on many processors. In this course we will explore the various computer architectures used for this purpose and the
issues involved with programming parallel solutions in such environments. Students will examine several types of problems that can benefit from parallel or distributed solutions and develop their own
solutions for them.
Frequency: Offered odd-numbered fall semesters.
COMP 469 - Discrete Applied Mathematics
Topics in applied mathematics chosen from: cryptography; complexity theory and algorithms; integer programming; combinatorial optimization; computational number theory; applications of geometry to
tilings, packings, and crystallography; applied algebra. This course counts towards the capstone requirement.
Frequency: Offered even-numbered fall semesters.
COMP 484 - Introduction to Artificial Intelligence
An introduction to the basic principles and techniques of artificial intelligence. Topics will include specific AI techniques, a range of application areas, and connections between AI and other areas
of study (i.e., philosophy, psychology). Techniques may include heuristic search, automated reasoning, machine learning, deliberative planning and behavior-based agent control. Application areas
include robotics, games, knowledge representation, logic, perception, and natural language processing.
Frequency: Offered even-numbered fall semesters.
COMP 221, or permission of instructor.
COMP 490 - Senior Capstone Seminar
Working with their capstone supervisor, seminar coordinators, and other faculty, students will discuss their capstone project, make presentations of their progress, critique the work of other
students, and participate in the activities of the seminar. These activities will include instruction and discussion of strategies for research, writing, and presentation. The scheduled times will
include both group meetings with other seminar participants as well as individually arranged meetings with the student's capstone supervisor.
Frequency: Every semester.
COMP 494 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
COMP 601 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of computer science not available
through the regular offerings.
Frequency: Every semester.
Permission of instructor.
COMP 602 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of computer science not available
through the regular offerings.
Frequency: Every semester.
Permission of instructor.
COMP 603 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of computer science not available
through the regular offerings.
Frequency: Every semester.
Permission of instructor.
COMP 604 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of computer science not available
through the regular offerings.
Frequency: Every semester.
Permission of instructor.
COMP 611 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in computer science. Subject matter may complement but not duplicate material
covered in regular courses.
Frequency: Every semester.
Arrangements must be made with a department member prior to registration and permission of instructor.
COMP 612 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in computer science. Subject matter may complement but not duplicate material
covered in regular courses.
Frequency: Every semester.
Arrangements must be made with a department member prior to registration and permission of instructor.
COMP 613 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in computer science. Subject matter may complement but not duplicate material
covered in regular courses.
Frequency: Every semester.
Arrangements must be made with a department member prior to registration and permission of instructor.
COMP 614 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in computer science. Subject matter may complement but not duplicate material
covered in regular courses.
Frequency: Every semester.
Arrangements must be made with a department member prior to registration and permission of instructor.
COMP 621 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 622 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 623 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 624 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 631 - Preceptorship
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 632 - Preceptorship
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 633 - Preceptorship
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 634 - Preceptorship
Frequency: Every semester.
Available to junior and senior students with declared majors in computer science. Arrangements must be made prior to registration. Permission of instructor. Work with Internship Office.
COMP 641 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
COMP 642 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
COMP 643 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
COMP 644 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
MATH 116 - Math and Society
Topics course offered for non-majors aiming to fulfill distribution requirement. Topics changes, and offerings may include Math of Elections and Voting, Climate Modeling, Game Theory, and Sports
Statistics. Full descriptions given in advance of registration.
Frequency: Offered even-numbered fall semesters.
MATH 125 - Epidemiology
Epidemiology is the study of the distribution and determinants of disease and health in human populations and the application of this understanding to the solution of public health problems. Topics
include measurement of disease and health, the outbreak and spread of disease, reasoning about cause and effect, analysis of risk, detection and classification, and the evaluation of trade-offs. The
course is designed to fulfill and extend the professional community's consensus definition of undergraduate epidemiology. In addition to the techniques of modern epidemiology, the course emphasizes
the historical evolution of ideas of causation, treatment, and prevention of disease. The course is a required component of the concentration in Community and Global Health.
Frequency: Every semester.
MATH 135 - Applied Multivariable Calculus I
This course focuses on calculus useful for applied work in the natural and social sciences. There is a strong emphasis on developing scientific computing and mathematical modeling skills. The topics
include functions as models of data, differential calculus of functions of one and several variables, integration, differential equations, and estimation techniques. Case studies are drawn from
varied areas, including biology, chemistry, economics, and physics.
Frequency: Every semester.
MATH 136 - Discrete Mathematics
An introduction to the basic techniques and methods used in combinatorial problem-solving. Includes basic counting principles, induction, logic, recurrence relations, and graph theory.
Frequency: Every semester.
MATH 137 - Applied Multivariable Calculus II
This course focuses on calculus useful for both theoretical and applied work in the mathematical, natural, and social sciences at a more rigorous level than Math 135. Topics include: partial
derivatives, gradients, contour plots, constrained and unconstrained optimization, Taylor polynomials, and differential equations, interpretations of integrals via finite sums, the fundamental
theorem of calculus, double integrals over a rectangle. Attention is given to both symbolic and numerical computing.
Frequency: Every semester.
MATH 135 or a year of high school calculus at the level of AP calculus with an AB score of 4 or higher.
MATH 155 - Introduction to Statistical Modeling
An introductory statistics course with an emphasis on multivariate modeling. Topics include descriptive statistics, experiment and study design, probability, hypothesis testing, multivariate
regression, single and multi-way analysis of variance, logistic regression.
Frequency: Every semester.
MATH 194 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
MATH 212 - Philosophy of Mathematics
Why does 2 + 2 equal four? Can a diagram prove a mathematical truth? Is mathematics a social construction or do mathematical facts exist independently of our knowing them? Philosophy of mathematics
considers these sorts of questions in an effort to understand the logical and philosophical foundations of mathematics. Topics include mathematical truth, mathematical reality, and mathematical
justifications (knowledge). Typically we focus on the history of mathematics of the past 200 years, highlighting the way philosophical debates arise in mathematics itself and shape its future.
Frequency: Alternate years.
MATH 236 - Linear Algebra
This course blends mathematical computation, theory, abstraction, and application. It starts with systems of linear equations and grows into the study of matrices, vector spaces, linear independence,
dimension, matrix decompositions, linear transformations, eigenvectors, and their applications.
Frequency: Offered every semester.
MATH 237 - Multivariable Calculus
For Fall 2014 this course will be offered as Multivariable Calculus, with the following description:
Differentiation and integration of functions of two and three variables. Applications of these, including optimization techniques. Also includes introduction to vector calculus, with treatment of
vector fields, line and surface integrals, and Green's Theorem.
For Spring 2015 this course will be offered as Applied Multivariable Calculus III, with the following description:
This course focuses on calculus useful for the mathematical and physical sciences. Topics include: scalar and vector-valued functions and derivatives; parameterization and integration over regions,
curves, and surfaces; the divergence theorem; and Taylor series. Attention is given to both symbolic and numerical computing. Applications drawn from the natural sciences, probability, and other
areas of mathematics.
Frequency: Every semester.
MATH 137 or a strong high school calculus at the level of AP calculus with a BC score of 4 or higher.
Curricular Change
Permanent title/desc change effective Spring 2015 semester
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Statistics as applied to "big data," including large numbers of variables. The linear and logistic modeling techniques from Math 155 will be augmented with computer-based methods of data
exploration, visualization, data mining, supervised and unsupervised clustering, discriminant analysis, contemporary approaches to the problem of overfitting, and dimension reduction. The course
also deals with methods of combining and organizing data from diverse sources and the high-level statistical computer programming needed to carry out sophisticated data analysis.
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An introduction to basic probability concepts with an emphasis on applications in mathematical statistics: sample spaces, combinatorics, conditional probability, random variables, probability
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MATH 312 - Differential Equations
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MATH 313 - Advanced Symbolic Logic
A second course in symbolic logic which extends the methods of logic. A main purpose of this course is to study logic itself-to prove things about the system of logic learned in the introductory
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arithmetic; Turing computability; modal logic; and intuitionistic logic.
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MATH 353 - Survival Analysis
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practical reality: the precise values of data are often unknown. We will introduce the concepts of censoring and truncation, and discuss the Kaplan-Meier curve, parametric regression models, Cox's
proportional hazards model, and time-varying covariates. The course will have an applied focus. Examples may be drawn from a variety of fields including, but not restricted to, medicine, economics,
sociology, and political science. The course will count toward completion of the concentration in Community and Global Health.
Frequency: Offered every fall semester.
MATH 355 - Bayesian Statistics
Bayesian statistics, an alternative to the traditional frequentist approach taken in our other statistics courses, is playing an increasingly integral role in modern statistics. Highlighted by Nate
Silver of fivethirtyeight.com and Baseball Prospectus fame, Bayesian statistics has even reached a popular audience. This course explores the Bayesian philosophy, the Bayesian approach to statistical
analysis, Bayesian computing, as well as both sides of the frequentist versus Bayesian debate. Topics include Bayes' Theorem, prior and posterior probability distributions, Bayesian regression,
Bayesian hierarchical models, and an introduction to Markov chain Monte Carlo techniques.
MATH 361 - Theory of Computation
A discussion of the basic theoretical foundations of computation as embodied in formal models and descriptions. The course will cover finite state automata, regular expressions, formal languages,
Turing machines, computability and unsolvability, and the theory of computational complexity. Introduction to alternate models of computation and recursive function theory.
Frequency: Every spring.
MATH 365 - Computational Linear Algebra
This course covers a central point of contact between mathematics and computer science. Many of the computational techniques important in science, commerce, and statistics are based on concepts from
linear algebra: subspaces, projection, matrix decompositions, etc. The course reviews these concepts, adopts them to large scales, and applies them in the core techniques of scientific computing;
solving systems of linear and nonlinear equations, approximation and statistical function estimation, optimization, interpolation, Monte Carlo techniques. Applications throughout the sciences and
Frequency: Every spring.
MATH 371 - Modern Geometry
Topics in geometry selected by the instructor. Possible courses include classical Euclidean and non-Euclidean geometry (Hilbert's axioms; parallel postulate; hyperbolic, elliptic, spherical,
projective geometries; Poincare models), differential geometry (calculus on surfaces; curvature; minimal surfaces; geodesics; the Gauss-Bonet theorem), computational geometry (triangulation; point
location; Voronoi diagrams; linear programming).
Frequency: Offered even-numbered spring semesters.
MATH 373 - Number Theory
An introduction to the properties of and unsolved problems about the integers (whole numbers). This course is built around the problem of proving that a large integer is prime or finding its
factorization into primes. Topics include: divisibility and prime numbers, the Euclidean algorithm, modular arithmetic, quadratic residues, continued fractions, and public-key cryptosystems. This
course counts towards the capstone requirement.
Frequency: Even numbered fall semesters.
MATH 376 - Algebraic Structures
Introduction to abstract algebraic theory with emphasis on finite groups, rings, fields, constructibility, introduction to Galois theory.
Frequency: Every spring.
MATH 377 - Real Analysis
Basic theory for the real numbers and the notions of limit, continuity, differentiation, integration, convergence, uniform convergence, and infinite series. Additional topics may include metric and
normed linear spaces, point set topology, analytic number theory, Fourier series.
Frequency: Every fall.
MATH 379 - Combinatorics
Advanced counting techniques. Topics in graph theory, combinatorics, graph algorithms, and generating functions. Applications to other areas of mathematics as well as modeling, operations research,
computer science and the social sciences.
Frequency: Offered odd-numbered fall semesters.
MATH 394 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
MATH 432 - Mathematical Modeling
Draws on the student's general background in mathematics to construct models for problems arising from such diverse areas as the physical sciences, life sciences, political science, economics, and
computing. Emphasis will be on the design, analysis, accuracy, and appropriateness of a model for a given problem. Case studies will be used extensively. Specific mathematical techniques will vary
with the instructor and student interest. This course counts towards the capstone requirement.
Frequency: Odd numbered fall semesters.
MATH 437 - Continuous Applied Mathematics
Topics in applied mathematics chosen from: Fourier analysis; partial differential equations; wavelets; signal processing; time-frequency analysis; and more. Topics are examined in theoretical and
applied contexts, and from analytical and computational viewpoints. This course counts toward the capstone requirement.
Frequency: Odd numbered spring semesters.
MATH 455 - Mathematical Statistics
An introduction to the mathematical theory of statistics: sampling distributions, estimation, hypothesis testing, regression. Additional topics may include: analysis of variance and goodness of fit.
Emphasis on the theory underlying statistic, not on applications.
Frequency: Even numbered spring semesters.
MATH 469 - Discrete Applied Mathematics
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tilings, packings, and crystallography; applied algebra. This course counts towards the capstone requirement.
Frequency: Even numbered fall semesters.
MATH 471 - Topics in Topology/Geometry
An introduction to the topology of Euclidean, metric, and abstract spaces. Covers the fundamental ideas from point set topology - continuity, convergence, and connectedness - as well as selected
topics from knot theory, three-dimensional manifolds, fixed-point theory, the fundamental group, and elementary homotopy theory. This course counts towards the capstone requirement.
Frequency: Even numbered fall semesters.
MATH 476 - Topics in Algebra
Topics in algebra to be chosen from: group representations; algebraic coding theory and finite fields; Galois theory; algebraic and transcendental numbers; ring theory; applied algebra. This course
counts toward the capstone requirement.
Frequency: Odd numbered fall semesters.
MATH 477 - Topics in Analysis
A continuation of Real Analysis including discussion of basic concepts of analysis. Topics determined by professor and may include the development of the Riemann and Lebesgue integrals, functional
analysis, Fourier analysis. This course counts towards the capstone requirement.
Frequency: Odd numbered spring semesters.
MATH 478 - Complex Analysis
Algebra of complex numbers, analytic functions, the Cauchy-Riemann equations, Cauchy's theorem, the Cauchy integral formula, Taylor and Laurent series, the residue theorem, and conformal mapping.
This course counts towards the capstone requirement.
Frequency: Even numbered spring semesters.
MATH 494 - Topics Course
Varies by semester. Consult the department or class schedule for current listing.
MATH 601 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of mathematics not available through
the regular offerings.
Frequency: Every semester.
Permission of instructor.
MATH 602 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of mathematics not available through
the regular offerings.
Frequency: Every semester.
Permission of instructor.
MATH 603 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of mathematics not available through
the regular offerings.
Frequency: Every semester.
Permission of instructor.
MATH 604 - Tutorial
Closely supervised individual (or very small group) study with a faculty member in which a student may explore, by way of readings, short writings, etc., an area of mathematics not available through
the regular offerings.
Frequency: Every semester.
Permission of instructor.
MATH 611 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in mathematics. Subject matter may complement but not duplicate material
covered in regular courses.
Arrangement with faculty prior to registration, departmental approval, and permission of instructor.
MATH 612 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in mathematics. Subject matter may complement but not duplicate material
covered in regular courses.
Arrangement with faculty prior to registration, departmental approval, and permission of instructor.
MATH 613 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in mathematics. Subject matter may complement but not duplicate material
covered in regular courses.
Arrangement with faculty prior to registration, departmental approval, and permission of instructor.
MATH 614 - Independent Project
Individual project including library research, conferences with instructor, oral and written reports on independent work in mathematics. Subject matter may complement but not duplicate material
covered in regular courses.
Arrangement with faculty prior to registration, departmental approval, and permission of instructor.
MATH 621 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Junior and Senior standing. Arrangements must be made prior to registration. Departmental approval and permission of instructor required. Work with Internship Office.
MATH 622 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Junior and Senior standing. Arrangements must be made prior to registration. Departmental approval and permission of instructor required. Work with Internship Office.
MATH 623 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
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MATH 624 - Internship
Internships are offered only as S/D/NC grading option.
Frequency: Every semester.
Junior and Senior standing. Arrangements must be made prior to registration. Departmental approval and permission of instructor required. Work with Internship Office.
MATH 631 - Preceptorship
Frequency: Every semester.
Work with Internship Office.
MATH 632 - Preceptorship
Frequency: Every semester.
Work with Internship Office.
MATH 633 - Preceptorship
Frequency: Every semester.
Work with Internship Office.
MATH 634 - Preceptorship
Frequency: Every semester.
Work with Internship Office.
MATH 641 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
MATH 642 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
MATH 643 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor.
MATH 644 - Honors Independent
Independent research, writing, or other preparation leading to the culmination of the senior honors project.
Frequency: Every semester.
Permission of instructor. | {"url":"http://www.macalester.edu/academics/mscs/courses/","timestamp":"2014-04-18T00:58:02Z","content_type":null,"content_length":"121826","record_id":"<urn:uuid:180c364f-8855-4a6b-b640-2f83119ec8c6>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00359-ip-10-147-4-33.ec2.internal.warc.gz"} |
radians question.
March 9th 2008, 09:31 PM #1
Super Member
Oct 2007
radians question.
hey guys.. got a problem with this question.
there is a clock.. the minute hand measures 2 metres.
Question: through what angle in radians does the minute hand pass between 8:10am and 8:46am???
You know that a complete circle is 2pi radians
When the minute hand goes through a complete circle, it's 60 minutes.
You want to know what is the angle corresponding to 8:10->8:46, that is to say 36min.
So if x is the angle you want to find, we got : x/2pi = 36/60
yeah the way i did it was..
360/60 = 6
then 36 * 6 = 216 degrees. Now we convert degrees to radians by multiplying 216 by (pi/180). This should give us 6pi/5 radians
March 9th 2008, 11:45 PM #2
March 10th 2008, 04:56 AM #3
Super Member
Oct 2007 | {"url":"http://mathhelpforum.com/trigonometry/30529-radians-question.html","timestamp":"2014-04-21T14:54:08Z","content_type":null,"content_length":"35604","record_id":"<urn:uuid:6c458325-3457-4d8d-831d-1410547931e6>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00263-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rensselaer Magazine - December 1999 - Practical Applications
Diverse Applications
One of Holmes’ occasional colleagues at Rensselaer is Margaret Cheney, professor of mathematics, whose research has addressed such diverse phenomena as land mines, oil deposits, birch forests, and
pulmonary edema. What unifies these seemingly unrelated situations is Cheney’s mathematical specialty—inverse boundary-value problems. That’s what scientists are doing when they gather information
about an inaccessible region by probing it from the outside with fields, waves, or particles.
“They are ‘inverse,’” Cheney says, “because the order of cause and effect is reversed. The scientist observes an effect and tries to deduce the causes.”
“Boundary-value” refers to measurements made at the edge or surface of an object that reveal something about its interior. For example, by applying a current to the human skin, a doctor can measure
the resulting electrical field, deduce the amount of fluid in the patient’s lungs, and potentially diagnose a case of pneumonia.
Inverse boundary-value problems usually involve differential equations. For the mathematically challenged among us, those are equations in which the unknown quantity is not a number (the electric
potential at one point), but a function (the electric potential at every point). Such equations are usually too difficult to solve with paper, pencil, and unaided intellectual exertion, but solutions
can be approximated by computers. That doesn’t render mathematicians extinct. Rather, their job is to come up with the appropriate algorithms for computers to use.
Cheney described applied mathematics as “math that’s very closely connected to some physical or engineering problem. Most of these problems are already encapsulated in some mathematical equation.
That’s how we discover new truths about the physical world. The math is just everywhere in the world.”
Cheney has received funding from the Department of the Navy to research more effective means of detecting underwater mines (as in explosives, not ore deposits). “Present methods don’t work very well.
A dog’s nose or a dolphin’s sonar is more reliable. If they can do it, we should be able to figure it out,” Cheney says.
Numerical simulation of the stability of a converging circular shock wave subject to a small perturbation in its speed. The black curves show the shock positions at successive intervals of time
beginning with the outer circular shock. The colored contours indicate the speed, or Mach number, of the shock as it converges to form a pentagonal-shaped shock that ultimately focuses to a point.
(Contributed by D. Schwendeman.)
Joyce McLaughlin, Ford Foundation Professor of Mathematical Sciences, is a member, like Cheney, of the board of trustees of the 9,000-member Society for Industrial and Applied Mathematics. In fact,
she’s associate editor of the Journal of Mathematical Analysis and Applications.
Her work, funded by the Office of Naval Research and the National Science Foundation, has a number of ongoing projects, including a joint research project with a Hong Kong scientist on the
propagation of radio signals in the ocean.
“You can’t really compare our math department to most traditional math departments. We’re entirely applied math. We probably have the largest concentration of applied mathematicians in the same
department in the country,” McLaughlin says.
Isom Herron, professor of mathematics, concentrates on the theory of fluid flows and its application to such fields as oceanography, meteorology, and the motion of ships and aircraft. He taught at
Howard University for 18 years before joining the math department at Rensselaer in 1992.
Continued on next page | {"url":"http://www.rpi.edu/dept/NewsComm/Magazine/dec99/math3.html","timestamp":"2014-04-20T23:41:31Z","content_type":null,"content_length":"7016","record_id":"<urn:uuid:92666bcf-3cd6-46fb-9f5f-f18980487337>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00516-ip-10-147-4-33.ec2.internal.warc.gz"} |
Zoran Skoda
Širšov-Bergman diamond lemma in Bergman form
• George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218, MR81b:16001, doi
Suppose we are given the following data
• $k$ – associative ring
• $X$ – any set,
• $\left\{X\right\}$ – free semigroup with $1$ on $X$
• $k\left\{X\right\}$ – semigroup algebra of $\left\{X\right\}$
Let $S$ be a set of pairs of the form $\sigma =\left({W}_{\sigma },{f}_{\sigma }\right)$ where ${W}_{\sigma }\in \left\{X\right\}$ , ${f}_{\sigma }\in k\left\{X\right\}$.
For any $\sigma \in S$ and $A,B\in \left\{X\right\}$, let ${r}_{A\sigma B}$ denote the $k$-module endomorphism of $k\left\{X\right\}$ that fixes all elements of $\left\{X\right\}$ other than ${\
mathrm{AW}}_{\sigma }B$, and that sends this basis element to ${\mathrm{Af}}_{\sigma }B$. We shall call the given set $S$ a reduction system and the maps ${r}_{A\sigma B}:k\left\{X\right\}\to k\left\
We shall say that a reduction ${r}_{A\sigma B}$acts trivially on an element $a\in k\left\{X\right\}$ if the coefficient of ${\mathrm{AW}}_{\sigma }B$ in $a$ is zero, and we shall call $a$irreducible
(under $S$) if every reduction is trivial on $a$, i.e. if $a$ involves none of the monomials ${\mathrm{AW}}_{\sigma }B$. The $k$-submodule of all irreducible elements of $k\left\{X\right\}$ will be
denoted $k\left\{X{\right\}}_{\mathrm{irr}}$. A finite sequence of reductions ${r}_{1},\dots ,{r}_{n}$$\left({r}_{i}={r}_{{A}_{i}{\sigma }_{i}{B}_{i}}\right)$ will be said to be final on $a\in k\left
\{X\right\}$ if ${r}_{n}\cdots {r}_{1}\left(a\right)\in \left\{X{\right\}}_{\mathrm{irr}}$.
An element $a\in k\left\{X\right\}$ will be called reduction-finite if for every infinite sequence ${r}_{1},{r}_{2},\dots$ of reductions, ${r}_{i}$ acts trivially on ${r}_{i-1}\cdots {r}_{1}\left(a\
right)$ for all sufficiently large $i$. If $a$ is reduction-finite, then any minimal sequence of reductions ${r}_{i}$, such that each ${r}_{i}$ acts nontrivially on ${r}_{i-1}\cdots {r}_{1}\left(a\
right)$ will be finite, and hence a final sequence. It follows from their definition that the reduction-finite elements form a $k$-submodule of $k\left\{X\right\}$.
We shall call an element $a\in k\left\{X\right\}$reduction-unique if it is reduction-finite, and if its image under all final sequences of reductions are the same. This common value will be denoted $
Lemma. (i) The set of reduction-unique elements of $k\left\{X\right\}$ forms a $k$-submodule, and ${r}_{S}$ is a $k$-linear map of this submodule into $k\left\{X{\right\}}_{\mathrm{irr}}$.
(ii) Suppose $a,b,c\in k\left\{X\right\}$ are such that for all monomials $A,B,C$ occurring with nonzero coefficient in $a,b,c$ respectively, the product $\mathrm{ABC}$ is reduction-unique. (In
particular this implies that $\mathrm{abc}$ is reduction-unique.) Let $r$ be a finite composition of reductions. Then $\mathrm{ar}\left(b\right)c$ is reduction-unique, and ${r}_{S}\left(\mathrm{ar}\
Proof. (i) Say $a,b\in k\left\{X\right\}$ are reduction-unique, and $\alpha \in k$. We know $\alpha a+b$ is reduction-finite. Let $r$ be any composition of reductions final on this element. Since $a$
is reduction-unique, we can find a composition of reductions $r\prime$ such that $r\prime r\left(a\right)={r}_{S}\left(a\right)$, and similarly there is a composition of reductions $r″$ such that
$r″r\prime r\left(b\right)={r}_{S}\left(b\right)$. As $r\left(\alpha a+b\right)$ is irreducible, we have
(1)$r\left(\alpha a+b\right)=r″r\prime r\left(\alpha a+b\right)=\alpha r″r\prime r\left(a\right)+r″r\prime r\left(b\right)=\alpha {r}_{S}\left(a\right)+{r}_{S}\left(b\right),$r(\alpha a + b) = r'' r'
r(\alpha a + b) = \alpha r'' r' r(a) + r'' r' r(b) = \alpha r_S(a) + r_S(b),
from which our assertions follow.
(ii) By (i) and the way (ii) is formulated, it clearly suffices to prove (ii) in the case where $a,b,c$ are monomials $A,B,C,$ and $r$ is a single reduction ${r}_{D\sigma E}$. But in that case,
(2)${\mathrm{Ar}}_{D\sigma E}\left(B\right)C={r}_{\mathrm{AD}\sigma EC}\left(\mathrm{ABC}\right),$Ar_{D \sigma E}(B)C = r_{AD\sigma E C}(ABC),
which is the image of $\mathrm{ABC}$ under a reduction, hence is reduction-unique if $\mathrm{ABC}$ is, with the same reduced form. $\square$
Let us call a $5$-tuple $\left(\sigma ,\tau ,A,B,C\right)$ with $\sigma ,\tau \in S$ and $A,B,C\in \left\{X\right\}-\left\{1\right\},$ such that ${W}_{\sigma }=\mathrm{AB},$${W}_{\tau }=\mathrm{BC},$
an overlap ambiguity of $S$. We shall say the overlap ambiguity $\left(\sigma ,\tau ,A,B,C\right)$ is resolvable if there exist compositions of reductions, $r$ and $r\prime$, such that $r\left({f}_{\
sigma }C\right)=r\prime \left({\mathrm{Af}}_{\tau }\right)$ (the confluence condition on the results of the two indicated ways of reducing $\mathrm{ABC}$).
Similarly, a 5-tuple $\left(\sigma ,\tau ,A,B,C\right)$ with $\sigma e \tau \in S$ and $A,B,C\in k\left\{X\right\}$ will be called an inclusion ambiguity if ${W}_{\sigma }=B,$${W}_{\tau }=\mathrm
{ABC};$ and such an ambiguity will be called resolvable if ${\mathrm{Af}}_{\sigma }C$ and ${f}_{\tau }$ can be reduced to a common expression.
By a semigroup partial ordering on $k\left\{X\right\}$ we shall mean a partial order $″\le ″$ such that $B<B\prime ⇒\mathrm{ABC}<\mathrm{AB}\prime C$ ($A,B,B\prime ,C\in \left\{X\right\}$), and it
will be called compatible with $S$ if for all $\sigma \in S$, ${f}_{\sigma }$ is a linear combination of monomials $<{W}_{\sigma }$.
Let $I$ denote the two-sided ideal of $k\left\{X\right\}$ generated by the elements ${W}_{\sigma }-{f}_{\sigma }$ ($\sigma \mathrm{in}S$). As a $k$-module, $I$ is spanned by the products $A\left({W}_
{\sigma }-{f}_{\sigma }\right)B$.
If $\le$ is a partial order on $k\left\{X\right\}$ compatible with the reduction system $S$, and $A$ is any element of $\left\{X\right\}$, let ${I}_{A}$ denote the submodule of $k\left\{X\right\}$
spanned by all elements $B\left({W}_{\sigma }-{f}_{\sigma }\right)C$ such that ${\mathrm{BW}}_{\sigma }C<A$. We shall say that an ambiguity $\left(\sigma ,\tau ,A,B,C\right)$ is resolvable relative
to $\le$ if ${f}_{\sigma }C-{\mathrm{Af}}_{\tau }\in {I}_{\mathrm{ABC}}$ (or for inclusion ambiguities, if ${\mathrm{Af}}_{\sigma }B-{f}_{\tau }\in {I}_{\mathrm{ABC}}$). Any resolvable ambiguity is
resolvable relative to $\le$.
Theorem. Let $S$ be a reduction system for a free associative algebra $k\left\{X\right\}$ (a subset of $\left\{X\right\}×k\left\{X\right\}$), and $\le$ a semigroup ordering on $\left\{X\right\}$,
compatible with $S$, and having descending chain condition. Then the following conditions are equivalent:
(a) All ambiguities of $S$ are resolvable.
(b) All ambiguities of $S$ are resolvable relative to $\le$.
(c) All elements of $k\left\{X\right\}$ are reduction-unique under $S$.
Corollary. Let $k\left\{X\right\}$ be a free associative algebra, and $″\le ″$ a semigroup partial ordering of $\left\{X\right\}$ with descending chain condition.
If $S$ is a reduction system on $k\left\{X\right\}$ compatible with $\le$ and having no ambiguities, then the set of $k$-algebra relations ${W}_{\sigma }={f}_{\sigma }$ ($\sigma \in S$) is
More generally, if ${S}_{1}\subset {S}_{2}$ are reduction systems, such that ${S}_{1}$ is compatible with $\le$ and all its ambiguities are resolvable, and if ${S}_{2}$ contains some $\sigma$ such
that ${W}_{\sigma }$ is irreducible with respect to ${S}_{1}$, then the inclusion of ideals associated with these systems, ${I}_{1}\subset {I}_{2}$, is strict. | {"url":"http://ncatlab.org/zoranskoda/show/diamond+lemma","timestamp":"2014-04-17T00:49:30Z","content_type":null,"content_length":"39150","record_id":"<urn:uuid:8bab1c09-f60a-46da-a187-eb2e6098079d>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00615-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pathway analysis for family data using nested random-effects models
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BMC Proc. 2011; 5(Suppl 9): S22.
Pathway analysis for family data using nested random-effects models
Recently we proposed a novel two-step approach to test for pathway effects in disease progression. The goal of this approach is to study the joint effect of multiple single-nucleotide polymorphisms
that belong to certain genes. By using random effects, our approach acknowledges the correlations within and between genes when testing for pathway effects. Gene-gene and gene-environment
interactions can be included in the model. The method can be implemented with standard software, and the distribution of the test statistics under the null hypothesis can be approximated by using
standard chi-square distributions. Hence no extensive permutations are needed for computations of the p-value. In this paper we adapt and apply the method to family data, and we study its performance
for sequence data from Genetic Analysis Workshop 17. For the set of unrelated subjects, the performance of the new test was disappointing. We found a power of 6% for the binary outcome and of 18% for
the quantitative trait Q1. For family data the new approach appears to perform well, especially for the quantitative outcome. We found a power of 39% for the binary outcome and a power of 89% for the
quantitative trait Q1.
Testing for the joint effect of single-nucleotide polymorphisms (SNPs) located in a set of genes is a popular alternative to single-marker tests [1]. Typically these SNPs have small effect sizes, and
thus separate SNP analysis methods will be underpowered. On the other hand, approaches that consider sets of genes and test for the combined effect of multiple SNPs will be more powerful. Gene sets
can be defined on the basis of the biological function of the genes (pathways) and can contribute thereby to biologically interpretable results. Gene-set methods were originally proposed for gene
expression data and have recently been adapted to test for pathway effects using genetic data [2,3]. A fundamental difference between gene expression data and genetic data is that in genetic data
multiple SNPs within genes that are correlated are available. Current pathway-based methods for genetic data do not properly capture this correlation structure of the genetic data and therefore may
lose efficiency. Recently, two pathway approaches were proposed that take the correlation between SNPs into account [4,5]. Both approaches have two steps: (1) reducing the dimensionality of the
genetic data and producing gene-specific summaries and (2) introducing these summaries as covariates into the model for the phenotype.
The two-stage approach of Tsonaka et al. [5] models the correlation between SNPs in a pathway using a generalized linear mixed model for the SNPs with nested random effects. This approach uses a
pathway-level and a gene-level random effect to capture the correlation between genes and within each gene, respectively. The empirical Bayes estimates of the random effects per subject and gene are
used as summary measures of the SNP data and are included in the phenotype model to test for pathway association. Tsonaka and colleagues proposed this approach to test for pathway effects for disease
progression in a longitudinal study. They used a likelihood ratio test and a Wald statistic and showed by simulations that the test statistics follow a chi-square distribution under the null
hypothesis. The aim of this paper is to study the performance of this approach for the sequence data on the families of Genetic Analysis Workshop 17 (GAW17).
We knew the answers and took the simulation setup for these data into account. We considered that the genes that were simulated were associated with the phenotype Q1 and studied their association
with the quantitative traits Q1 and Q4 and the binary trait. Because an interaction between smoking status and the KDR gene was included in the simulation model, we also considered this
gene-environment interaction in the phenotype models.
Study sample
We considered data from 697 unrelated individuals and 311 subjects from 140 sibships forming 8 families. The sibship sizes vary from one to six siblings. The degree of relationships between members
of different sibships was larger than 0.25; that is, parents were removed when their offspring were included. For the pathway analysis we considered SNPs that belong to the following eight genes:
ARNT, ELAVL4, FLT1, HIF3A, KDR, FLT4, and VEGFA. Gene HIF1A was not considered because it does not contain SNPs that vary in the family data. In total, the analysis was restricted to 48 out of 125
SNPs in the Q1 (vascular endothelial growth factor [VEGF]) pathway because only these SNPs show variation in the families. The minor allele frequencies (MAFs) of the associated variants in these
genes vary from 0.000717 to 0.164933. As covariates we considered Age and Smoking status. For gene KDR, an interaction with Smoking was included in the phenotype models. We applied the two-stage
method to the 200 simulated GAW17 data sets [6] to study its power. In addition, we present the results of the analysis of data set 1. A description of data set 1 is given in Table Table11.
Description of data set 1
Model specification
Let y[ij] be the outcome variable for individual j from sibship i. Assume that a pathway is analyzed with G genes and that each gene g (g = 1, …, G) contains S[g] SNPs. Let w[ijgs] be the genotype at
SNP s (s = 1, …, S[g]) located in gene g (g = 1, …, G) for individual j of sibship i. The genotype w[ijgs] is coded 0, 1, or 2. For individual j of sibship i, let
Gene model
We assume that Hardy-Weinberg equilibrium holds. We consider three random levels: (1) a sibship’s random effect, (2) a subject’s random effect, and (3) a gene effect within a subject. Let b[i] be the
random effect for the sibship, b[ij] the random effect for subject j within sibship i, and b[ijg] the random effect of gene g of person j within sibship i, and let b[ij] represents for each
individual j the shared effect of the genes of the pathway. Given these random effects b[i], b[ij], and b[ijg] and the covariates w[ijgs] is assumed to follow a binomial distribution with n = 2
trails and probability π[ijgs]. The probability π[igs] is modeled as follows:
where b[i], b[ij], and b[ijg] follow normal distributions with zero mean and variances b[i].
For individuals and for each gene the empirical Bayes estimate is given by:
Intuitively the value of the empirical Bayes estimate will increase with the number of rare variants that a subject carries.
Phenotype model
The empirical Bayes estimates obtained from the first stage can be plugged into the models for the phenotypes to test for pathway effects and gene-specific effects. For the quantitative traits (i.e.,
Q1 and Q4) we use a linear mixed model:
where u[i] is a normally distributed random sibship effect and e[ij] is a normally distributed residual. For the binary outcome variable a generalized estimating equation (GEE) approach was used with
an exchangeable correlation structure for subjects within sibships:
where h is the logit function. The lme4 package in R was used to fit mixed models. The gee package in R [7] was used for the GEE approach. Based on models (3) and (4) we can test the null hypothesis
of no pathway effect, which is equivalent to testing the null hypothesis H[0]: γ[1] = … = γ[G] = 0. We used a Wald statistic with G degrees of freedom. In addition, gene-level effects can be tested.
Type I error
Using extensive simulations, Tsonaka et al. [5] showed that the test statistics preserve the type I error at a nominal level for pathway analysis for longitudinal data. We tested for association
between the Q1 pathway and the Q4 trait. Because Q4 should not be influenced by the genes of this pathway, the power should be equivalent to the type I error.
Type I error and power
We fitted model (1) to the 48 SNPs of the Q1 pathway, which showed variation in the families. We included the covariate Smoking in the model because an interaction between KDR and smoking status was
included in the simulations. Then we plugged the empirical Bayes estimates per gene and subject into models (3) and (4) for the quantitative and binary variables, respectively. Age and Smoking were
included as covariates. In addition, we included an interaction between smoking status and the empirical Bayes estimate for KDR.
In Table Table22 for Q1, Q4, and the binary outcome, we show the percentages of the data sets for which the null hypothesis of no Q1 pathway is rejected at the 5% level. The results are based on all
200 data sets. The genes of the Q1 pathway have a direct effect on Q1 and, through Q1, also have an effect on the binary outcome. These genes should not be associated with Q4; hence the percentages
for Q4 are estimates of the type I error. For unrelated subjects the type I error is near 5%, but for the family data the type I error is too high (10%). The novel test performs well for quantitative
traits observed in families: The power to detect the Q1 pathway for Q1 is 89%. The power to detect the Q1 pathway for disease in families is smaller, namely, 39%. The power to detect the Q1 pathway
in the set of unrelated subjects is small.
Percentage of data sets for which H[0]: “no Q1 pathway effect” is rejected at the 5% level
Analysis of data set 1
The results of the pathway analyses for data set 1 are given in Table Table3.3. In the families, we obtained a highly significant result for the pathway of eight genes for the quantitative trait Q1
(p = 4.9 × 10^−10). In the set of unrelated individuals the pathway was not significantly associated with Q1 (p = 0.06). Also the p-values per gene are presented in Table Table3.3. These p-values
correspond to tests for a gene effect conditional on the empirical Bayes estimate of the remaining genes in the pathway. For the family data and Q1 trait, the FLT4 and VEGFC genes were significant.
p-Values for testing pathway and gene effects for data sets
For the binary outcome in the families we found a significant association, although it was less strong than that for the quantitative trait (p = 0.002). None of the genes were significant, which
suggests that multiple SNPs in multiple genes have a joint effect on the outcome. The Q1 pathway was not significantly associated with the binary trait in the unrelated individuals.
The interaction between smoking status and the KDR gene was not significant for both outcomes either in the families or in the set of unrelated subjects (see Table Table33).
The pathway analysis applied to the family data resulted in more significant results than using the set of unrelated individuals, especially for the analysis of trait Q1. One reason for the better
performance in the family data compared to the set of unrelated subjects is probably the oversampling of rare variants in the families. For example, the SNP C4S4935 of the VEGFC gene has a MAF of
0.0290 in the families in contrast to a MAF of 0.0007 in the set of unrelated individuals. Also, two of the three SNPs of the FLT4 gene have a larger MAF in the families than in the unrelated set.
Another reason for the larger power in the families is the fact that we are testing against a smaller residual when a sibship effect is included. Finally, the power may be too high because the size
of the test is not correct. Indeed, based on the pathway analysis for trait Q4, we obtained a high type I error.
We further investigated whether the high type I error could be attributed to the fact that correlation between cousin pairs is not taken into account in our models (3) and (4). Thus we added an extra
random effect to these models, but the obtained type I error did not change. Uh et al. [8] also obtained too large type I errors when testing associations between Q1 genes and the Q4 trait in the
unrelated subjects. They showed that, by using simulations under the null hypothesis, the type I error of the test statistics are at a nominal level. One reason for the large percentage of rejections
using Q4 may be that Q1 and Q4 are not independent in the data sets. Indeed in all 200 data sets Pearson’s correlation coefficients between the two traits are smaller than 0 (mean, −0.31; minimum,
−0.39; and maximum, −0.24). Concerning the binary trait, the power to detect the Q1 pathway was smaller than for Q1 trait. The reasons are that the Q1 genes have an indirect effect on the binary
trait and that we used a less efficient approach, namely, the GEE approach instead of a likelihood-based approach.
The power to detect the pathway in unrelated subjects was disappointing. The rare variants have smaller frequencies in these samples and are not tagged by more common variants [8]. The power of a set
of unrelated individuals may be improved by combining estimates from different studies, for example, the GAW17 unrelated individuals and family data [9]. For GAW17 this approach could not be applied
because the rare variants are oversampled in the families. Another approach may be to focus on rare variants only. One of the reviewers pointed out that an alternative test for association between
the outcome and rare variants can be obtained by fitting model (1) to all rare variants of all genes of the pathway. By doing so, one can obtain an empirical Bayes estimate that represents all rare
variants of the pathway. The advantage of this approach is that by including this empirical Bayes estimate in the phenotype model, a one degree of freedom test for association between the rare
variants and phenotypes is obtained that may be more powerful than the G degrees of freedom test that was studied in this paper. A disadvantage of this approach is that the structure of SNPs within
genes is ignored. Therefore gene-environment interaction between specific genes and environmental factors cannot be modeled. For the GAW17 data, interaction between KDR and smoking status was
included in the simulation model.
Our novel method captures the correlation between SNPs within and between genes by using random effects. Another approach was proposed by Chen et al. [4]. They summarize the SNPs per gene by using
principal components (eigen-SNPs). They show that their approach is more powerful than methods that ignore the dependency structure between the SNPs. This approach cannot be directly applied to
family data because one of the principal components may capture the dependence between relatives. Therefore the selection of eigen-SNPs to represent a gene is not straightforward. Moreover, SNPs are
categorical variables, and therefore applying principal components analysis may not be optimal. Finally, principal components analysis cannot deal with missing genotypes. Hence missing genotypes
should be first imputed before this approach can be applied.
Application of the new method to sequence data in unrelated individuals shows that when rare variants are not tagged by common variants, the new method is not able to detect these rare variants.
Currently we are working on a method that jointly models the rare variants by using collapsing methods and the common variants.
Our novel pathway test is a powerful tool to detect pathways in family data. The advantages of this method are that it captures the correlation between SNPs, can deal with missing data, can adjust
for gene-gene or gene-environment interaction, can be applied to any phenotype model, and can be implemented in standard statistical software.
Competing interests
The authors declare that there are no competing interests.
Authors’ contributions
JJHD: method development and writing of manuscript; HWU: data management and contributed to discussion on methodology and results of data analyses; and RT: statistical analyses, method development
and helped to draft the manuscript
This work is supported by a grant from the Netherlands Organization for Scientific Research. The Genetic Analysis Workshop is supported by National Institutes of Health grant GM031575.
This article has been published as part of BMC Proceedings Volume 5 Supplement 9, 2011: Genetic Analysis Workshop 17. The full contents of the supplement are available online at http://
Articles from BMC Proceedings are provided here courtesy of BioMed Central
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physics HELPP!!!!!!!
Number of results: 108,848
urgent alg 1
add or subtract 2 2 (2t -8t)+ (8t + 9t) second question 3 2 3 (t +8t ) + (-3t ) please helpp its urgent i have the answer butt im not suree pleasee helpp
Wednesday, February 9, 2011 at 11:46pm by kimberly
what is g?
Monday, September 17, 2007 at 7:50pm by confusedd!HELPP!!!
HELPP !!
Copper(I) ions in aqueous solution react with NH3 according to Cu^+ + 2NH3 --> Cu(NH3)2^+ Kf=6.3x10^10 Calculate the solubility (in g/L) of CuBr (Ksp=6.3x10^-9) in 0.81 M NH3.
Thursday, May 10, 2012 at 7:58pm by HELPP !
Physics helpp please!
Sunday, April 3, 2011 at 9:21pm by Anonymous
It says it was incorrect even after I converted to rev/min
Monday, October 17, 2011 at 3:07pm by Helpp
It says it was incorrect even after I converted to rev/min
Monday, October 17, 2011 at 3:07pm by Helpp
helpp in physics! so tough!
How did you figure out the higest point I am still struggling eith that.
Tuesday, September 28, 2010 at 11:10pm by Am
physics need urgent helpp
i have tried everything, please helpp A snowmobile, with a mass of 530 kg, applied a force of 410 N backwards on the snow. What force is responsible for the snowmobile’s resulting forward motion?
(Hint: Think action–reaction force pairs.) If the force of friction on the ...
Monday, October 28, 2013 at 9:27pm by iyzi
Physics helpp!!
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Friday, July 30, 2010 at 6:35pm by Anonymous
A motorcycle with a mass of 350 kg is accelerated from rest to a velocity of 30 m/s by a force of 4000 N. calculate the distance over which this acceleration occurred.
Tuesday, April 19, 2011 at 5:17pm by helpp
Physics helpp
Newtons 2nd law The force of earths gravity is about 10 N downward. Whats the acceleration of a 15kg backpack if lifted with a 15N force??
Tuesday, May 14, 2013 at 10:49am by Kate
The three vectors in the figure below have magnitudes a = 2.45 m, b = 3.15 m, and c = 15.0 m and angle θ = 36.5° find the x and y component of vector a
Sunday, September 30, 2012 at 8:19pm by HELPP assignment Due at 10
physics PLEASE helpp
Find the net change in entropy when 213 g of water at 0.0°C is added to 213 g of water at 96.9°C.
Friday, January 6, 2012 at 4:57pm by melisa
Physics helpp
F-F(gr) = ma a= {F-F(gr)}/m = =(15-10)/15=0.33 m/s² (upward)
Tuesday, May 14, 2013 at 10:49am by Elena
Physics helpp please!
if the springs are in series then the combined spring constant is 1/k=1/1700 + 1/3700 k= (1700*3700)/5400=1168N/m 1/2 k x^2=1/2 m v^2 you have k, x (.45m), and mass m. Solve for v
Sunday, April 3, 2011 at 9:21pm by bobpursley
Wednesday, October 26, 2011 at 3:03pm by HELPP ME
ok thank you
Wednesday, October 26, 2011 at 5:23pm by HELPP
ok thanks
Wednesday, October 26, 2011 at 5:20pm by HELPP
Vector C has a magnitude 25.8 m and is in the direction of the negative y axis. Vectors A and B are at angles α = 41.9° and β = 27.2° up from the x axis respectively. If the vector sum A B C = 0,
what are the magnitudes of A and B?
Saturday, June 1, 2013 at 2:17pm by helpp
Vector C has a magnitude 25.8 m and is in the direction of the negative y axis. Vectors A and B are at angles α = 41.9° and β = 27.2° up from the x axis respectively. If the vector sum A B C = 0,
what are the magnitudes of A and B?
Saturday, June 1, 2013 at 2:18pm by helpp
math helpp! :|
You're welcome! =)
Saturday, November 1, 2008 at 7:08pm by Writeacher
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Friday, February 17, 2012 at 3:13pm by tori
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HELPP math
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Thursday, January 10, 2013 at 11:11pm by HELPP!!!
HELPP !!
Thursday, May 10, 2012 at 7:58pm by poop
a child twirls a yo-yo about in a horizontal circle. The yo-yo has a mass of 0.20 kg and is attached to a string 0.8 m long. what is the velocity of the yo-yo? thanks!
Tuesday, December 7, 2010 at 7:35pm by i neeeeed helpp!!
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Saturday, November 1, 2008 at 7:08pm by jazz
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Monday, April 12, 2010 at 7:02pm by Dave
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Thursday, February 2, 2012 at 8:45pm by s
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Wednesday, February 8, 2012 at 9:58pm by A Girl.
Tuesday, July 27, 2010 at 7:07pm by olivia
Geometry HELPP
find H(-5,5) and I (7,3)
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Thursday, October 25, 2012 at 3:06pm by Aya
math helpp
3/10 and 7/8
Wednesday, January 16, 2013 at 8:14pm by tesj
physics please helpp
Find the minimum diameter of a 51.5-m-long nylon string that will stretch no more than 1.01 cm when a load of 68.9 kg is suspended from its lower end. Assume that Ynylon = 3.51·108 N/m2
Tuesday, October 25, 2011 at 8:32pm by melisa
A car starts from rest and reaches a velocity of 8 m/s in 4 seconds. a) What is its acceleration? b)If its acceleration remains the same, what will be its velocity in 15 seconds? plsss helpp
Wednesday, December 7, 2011 at 12:36pm by bella
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You're welcome, Beeca.
Sunday, January 25, 2009 at 9:32pm by Ms. Sue
Your subject is not Helpp!
Tuesday, October 20, 2009 at 7:06pm by Ms. Sue
0.16g MnCl2
Monday, April 12, 2010 at 7:02pm by Kenny
calculus--please helpp
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Friday, October 14, 2011 at 1:54pm by krishna
ok iget 0.328 not 0.0984
Wednesday, October 26, 2011 at 2:33pm by HELPP ME
so i do the -log(3.4x10^-3)=2.46
Wednesday, October 26, 2011 at 5:20pm by HELPP
Algebra 1
I think that you're right too. Thank so much for your help.
Tuesday, October 9, 2012 at 7:37pm by HELPP!!!
Thank you so much that helps a lot!
Thursday, January 10, 2013 at 10:07pm by HELPP!!!
is 20 correct?
Tuesday, June 4, 2013 at 1:30am by pleas helpp
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Wednesday, June 5, 2013 at 2:11am by pleas helpp
why pi?
Saturday, July 20, 2013 at 3:13pm by Azal - pleaseee helpp
Physics helpp!!
A baseball is thrown horizontally off a cliff with a speed of 24 m/s. What is the horizontal distance, to the nearest tenth of a meter, from the face of the cliff after 2.8 seconds? To the nearest
tenth of a meter, how far has it fallen in that time? So is 24 m/s Vx or Vox? Do...
Friday, July 30, 2010 at 6:35pm by Anonymous
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more info?
Friday, September 21, 2007 at 7:58pm by Tyler
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Monday, October 29, 2007 at 9:12pm by helpp me!
Geometry HELPP
I will be happy to critique your thinking.
Saturday, November 7, 2009 at 2:59pm by bobpursley
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Wednesday, October 26, 2011 at 2:33pm by HELPP ME
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Tuesday, February 28, 2012 at 12:28am by Cassie
Find the Magnitude of the force exerted by each cable to support the 625 N punching bag. One cable is connected to the ceiling with an angle of 37 degrees and the other cable is connected
perpendicular to the side wall (Horizontal). HELPP
Thursday, August 9, 2012 at 10:33am by kacey
Find the Magnitude of the force exerted by each cable to support the 625 N punching bag. One cable is connected to the ceiling with an angle of 37 degrees and the other cable is connected
perpendicular to the side wall (Horizontal). HELPP
Thursday, August 9, 2012 at 10:42am by kacey
A student ties a 600.0 g rock to a 1.50 m-long string and swings it around her head in a horizontal circle. At what angular speed (in rev/min) does the string tilt down at a 13.1° angle, assuming
that the local acceleration due to gravity is -9.80 m/s2?
Monday, October 17, 2011 at 3:07pm by Helpp
They give answer 0.0984 mol and i get 3 something
Wednesday, October 26, 2011 at 2:33pm by HELPP ME
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Thursday, January 19, 2012 at 3:02pm by yolanda
MATH Is there no one who can helpp???????
its a fraction of -9 over 2 x to the power of -3/2... (-9/2)x^-3/2
Wednesday, February 15, 2012 at 2:51pm by Anonymous
english 3
I wanted to check my answers but I guess not. Thanks anyways.
Friday, February 24, 2012 at 6:32pm by helpp
i think autombile is a good one
Sunday, January 25, 2009 at 9:32pm by Anonymous
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the clam take to fall?
Monday, September 17, 2007 at 7:50pm by confusedd!HELPP!!!
physics someone please helpp
calculate the pressures at those depths. Then, use the Boyle's law (constant temp). Ptank*volumetank=pressuredepth*n*.4Liters volume of the tank in liters is 10liters. solve for n. On the pressure at
depth, be certain to add atmospheric pressure which is on top of the water.
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Chemistryy....PLEASE HELPP HAVE MIDTERMMM
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Sunday, October 12, 2008 at 5:55pm by Sara
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7th grade essay writing
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Thursday, January 20, 2011 at 1:00pm by Sabrina
Algebra 1
What is the slope, m, of the line that contains the points (-12, 16), (-3, 4) and (6,-8)?
Tuesday, October 9, 2012 at 7:37pm by HELPP!!!
Chemistry- helpp!
Tuesday, November 2, 2010 at 5:24pm by Anonymous
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How do peoples' lives change by living near a mountain?
Wednesday, September 10, 2008 at 7:49pm by LARRY!! HELPP ME!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
9th grade
find the unit rate 3 tablespoons for 1.5 serving how do u do it??? helpp thanks
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Plant Physiology
Okay in understand everything except D....SOMEONE PLEASE HELpp
Friday, February 13, 2009 at 7:14am by Saira
geo HELP!!
what are examples of formal economic activities? helpp plz and thanku :)
Thursday, February 26, 2009 at 9:23pm by beecca
evaluate the expression n + (-5) for each value of n. n=312 i dont know how to do this..please helpp..
Tuesday, August 31, 2010 at 7:54pm by paloma
SPANISH PLEAASSEE HELPP (im stuck)!!!!!!!!!!!!!!!!
the last answer is ellos, not este.
Thursday, September 15, 2011 at 9:20pm by cliff
im sorry , thats the question that i have problems with . HELPP , Please !
Tuesday, May 15, 2012 at 8:54am by Remy W
Solve 2/3(6x+30) = x+5(x+4)-2x and please show me the steps?
Thursday, January 10, 2013 at 11:11pm by HELPP!!!
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Monday, May 12, 2008 at 6:14am by Nolan
Yes. Spending money on a sweater and boots are economic activities.
Sunday, January 25, 2009 at 9:32pm by Ms. Sue
the big idea of energy
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Tuesday, December 5, 2006 at 2:18pm by Strawbeeryyy!!
CHEM HELPP!!!
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Tuesday, March 10, 2009 at 3:38am by Anonymous
evaluate 3 + x for 1 over 2 our teacher never taught us this, and i dont know how to do it
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simple easy math question PLEASEEE HELPP!!!
is this true? log^2 x = (logx)^2
Thursday, January 6, 2011 at 8:31am by Sabrina
Geometry HELPP
(-3,-5) and (5,-1) what ordered pair describes the location of the midpoint if the line segment
Saturday, November 7, 2009 at 2:59pm by Anonymous
Helpp? Graphs of functions !! I need to determine if these are odd or even? 1. f(x)=x^5-x 2. f(x)=5 3.f(x)=x^4+2x^3
Wednesday, August 29, 2012 at 2:31pm by Unknown
If (x, 3) is the solution to the equation 4x+2y=22, then what is the value of x? Can you solve it for me so i can see how you did it? a. 8 b. 6 c. 5 d. 4
Thursday, January 10, 2013 at 10:28pm by HELPP!!!
physics please helpp
A sinusoidal wave on a string is described by the equation y = (0.169 m) sin (0.713 x - 41.9 t), where x and y are in meters and t is in seconds. If the linear mass density of the string is 10.1 g/m
... a) ... the phase of the wave at x = 2.27 cm and t = 0.175 s. ? b)... the ...
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┃ ┃
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┃ & Tornado Alley ┃
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┃ ┃
┃ The Solenoid Terms of the ┃
┃ Vorticity Equation in Meteorology ┃
The purpose of this material is to show the relationships among the solenoids terms which arise in meteorological analysis. A solenoid term is of the form ∇f×∇g, where f and g are scalar field
variables. The remarkable fact is that when f and g are field variables such as temperature, pressure, density, specific volume and entropy the solenoid terms are proportional.
The magnitude of such a term is equal to |∇f||∇g|sin(φ), where φ is the angle between the two gradients. The solenoid vector lies in the line intersection of the two surfaces of constant levels of f
and g.
The blue vector perpendicular to the blue-outlined plane is the gradient vector for the f field. The red vector perpendicular to the red-outlined plane is the gradient vector for the g field. The
green vector in the line of intersection of the two planes is their vector cross product. If the two constant surfaces coincide then the angle between the gradients is zero and the solenoid term
vanishes. This condition prevails when the atmosphere is barotropic; i.e., when the atmospheric density is a function only of the pressure.
Derivation of the Equation for the Time Rate of Change of Vorticity
For an inertial frame of reference the equations of motion for a parcel of air are, in vector form,:
dv/dt = -(1/ρ)∇p - gk + f
where v is the velocity vector, ρ the density, p pressure, g the acceleration due to gravity, k the unit vertical vector and f the vector of friction forces.
The pressure gradient term
is especially important.
This term can be put into an interesting form by noting that from the definition of potential temperature θ:
ln(θ) = ln(T) + κln(p[0]) - κln(p)
and when the gradient operator ∇ is applied to this equation the result is
∇θ/θ = ∇T/T - κ∇p/p
From the defintion of entropy s it follows that
∇s = c[p]∇θ/θ
and hence from the previous equation
∇s = c[p]∇T/T − R∇p/p
since κ = R/c[p].
Multiplying through by T and noting that c[p]∇T is the same as ∇h, where h stands for enthalpy, results in
T∇s − ∇h = −(RT/p)∇p = −(1/ρ)∇p
Thus if the pressure gradient term in the equations of motion is replaced with T∇s - ∇h the result is
dv/dt = T∇s − ∇h −gk + f
Since k is the same as ∇z the above equation is equivalent to
dv/dt = T∇s − ∇h −g∇z + f
or, equivalently
dv/dt = T∇s − ∇(h + gz) + f
The motion-following derivative dv/dt is composed of an instaneous rate of change at a point and an advection term; i.e.,
dv/dt = ∂v/∂t + v·∇v
The advection term v·∇v can be expressed^1 as
∇(v^2/2) − v×(∇×v)
but ∇×v is just the vorticity vector q
v·∇v = ∇(v^2/2) - v×q
Thus the equations of motion for the atmosphere can be expressed in vector form as
(1) ∂v/∂t = v×q + T∇s − ∇(v^2/2 + h + gz) + f
The curl operator ∇× can be applied to this equation. The curl of any gradient of a scalar field vanishes; i.e., ∇×∇γ=0 for any scalar field γ because of the equality of cross derivatives. Therefore
under the curl operation ∇(v^2/2 + h + gz) vanishes.
Also, because the curl of a curl vanishes,
∇×(T∇×s) = ∇T×∇s.
The result of applying the curl operator to the left-hand side of the above equation of motion (1) and taking into account the interchangeability of the time and space derivatives is
∇×(∂v/∂t) = ∂(∇×v)/∂t = ∂q/∂t
Equating this to the result of applying the curl operation to the right-hand side of the equation (1) gives
∂q/∂t = −∇×(v×q) + ∇T×∇s + ∇×f
This form of the vorticity equation points out the role of the intersection or non-intersection of the isothermal surface and the isoentropic surface through the term ∇T×∇s.
Note that since ∇s = c[p]∇T/T - R∇p/p
∇T×∇s = ∇T×(c[p]∇T/T - R∇p/p)
= −∇T×(R∇p/p)
= −(R/p)(∇T×∇p)
Since by the ideal gas law
ln(T) = ln(p) − ln(ρ) − ln(R)
∇T/T = ∇p/p − ∇ρ/ρ
and hence
∇T×∇p = −(T/ρ)(∇p×∇ρ)
Thus the ∇T×∇s term in the vorticity equation can be replaced by a term involving ∇p×∇ρ. Generally all of these cross product terms, called solenoid terms, are proportional and they all vanish when
the atmosphere is barotropic; i.e.; when ∇ρ always has the same direction as ∇p.
Combining the last two derivations gives
∇T×∇s = (RT/pρ)(∇p×∇ρ)
= (1/ρ²)∇p×∇ρ
The −∇×(v×q) term in the vorticity equation reduces^2 to
q(∇·v) + [(v·∇)q − (q·∇)v]
These two terms are known as the divergence term and the twisting term, respectively.
Not much can be done analytically with the friction term except note that it is directed opposite to the velocity vector. It is of a low order of magnitude than the other terms and is usually
neglected in the analysis.
The term vorticity in meterology usually refers to the vertical component of the vorticity vector. The vertical component of a vector may be obtained analytically by take the dot product of the
vector with the unit vector in the vertical direction k; i.e., ζ = k·q. Thus the time rate of change of vorticity ζ is given by
dζ/dt = ζ(∇·v) + k·[(v·∇)q −(q·∇)v] + (1/ρ²)k·(∇p×∇ρ)
The terms on the right-hand side of the equation are known, respectively, as the divergence term, the twisting or tilting term and the solenoid term.
^1This follows from the vector identity
∇(A·B) = (A·∇)B + (B·∇)A + A×(∇×B) + B×(∇×A)
with A=B=v; i.e.,
∇(v·v) = 2(v·∇)v + 2v×(∇×v)
and hence
(v·∇)v = (1/2)∇(v·v) − v×(∇×v).
^2 The vector identity
∇×(A×B) = A(∇·B) − B(∇·A) − (A·∇)B + (B·∇)A
when A=v and B=q=∇×v reduces to
∇×(v×q) = −q(∇·v) − (v·∇)q + (q·∇)v
because the divergence of the curl of a vector vanishes. | {"url":"http://www.sjsu.edu/faculty/watkins/solenoids.htm","timestamp":"2014-04-17T09:52:38Z","content_type":null,"content_length":"14703","record_id":"<urn:uuid:1ac28b05-45af-42d1-841e-fd4bf8bf1c2d>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00162-ip-10-147-4-33.ec2.internal.warc.gz"} |
How much uranium-235 does a nuclear power generator consume to generate 1.5 GW?
1. The problem statement, all variables and given/known data
The total thermal power generated in a nuclear power reactor is 1.5 GW.
How much uranium-235 does it consume in a year?
2. Relevant equations
3. The attempt at a solution
Solving for m leaves me with .53 kg which come up incorrect in Mastering Physics.
The approach is correct, but some steps are missing.
One must determine the energy E used in a year. E = Power (average) * time, so J = W * s.
Then one must realize the energy per fission, fission consumes 1 atom and the mass of 1 atom. Fission of U-235 produces ~200-205 MeV/fission. (This is fine if one does not consider the contribution
of Pu-239/Pu-240/Pu-241 which builds up slowly during operation in commercial reactor.) | {"url":"http://www.physicsforums.com/showthread.php?t=360052","timestamp":"2014-04-17T10:01:18Z","content_type":null,"content_length":"36354","record_id":"<urn:uuid:69b20364-5b52-4708-b4bf-69ec76625d38>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
This lesson is on Grouping and Place Value using various objects
view a plan
This lesson is on Grouping and Place Value using various objects
1, 2
Title – Place Value
By – Scott Dan
Subject – Math
Grade Level – 1st – 2nd
N343 sec 02
Format: Guided Discovery
Concept: Grouping and Place Value
1. Place Value Mat
2. Beans
3. Cards with numbers on them (enough for everyone to have at least 10)
4. Overhead transparencies of tens units (made out of peppers)
5. Overhead transparency of the place value mat
6. 12 small Bowls (enough for every group of two to have one)
7. Base Ten Units made out of tongue depressors and ten beans
8. Overhead Projector
NCTM Curriculum Standards for School Mathematics:
1. Mathematics as Problem Solving
2. Mathematics as Communication
3. Mathematics as Reasoning
4. Mathematics as Connections
5. Estimation
6. Number Sense and Numeration
7. Concepts of Whole-Number Operations
1. The students will expand their awareness of numbers and knowledge of number sense by expressing a symbolic number in various ways.
2. Through the process of counting and manipulation of beans, children will discover various ways to group numbers so that they may become easier to recognize and count.
3. The students will begin to explore the concept of place value through the grouping of beans into sets of tens and their appropriate placement on a place value mat.
1. Make place value mats for the tens units and ones units.
2. On a separate day (before this lesson), have children make base ten bean sticks (thumb depressors with ten beans glued on). Each child should make ten of these sticks. Have them write their names
on the back of each depressor
1. Have the students clear off their desks so that they may have a good working surface.
2. Place one pepper on the overhead and turn it on for about 4 seconds. Turn off the machine and ask anyone if they can tell how many peppers were placed on the overhead. Take a child’s answer and
then turn on the machine to see if they were right. Do the same for 3 peppers and 5 peppers.
3. Then, randomly place 8 peppers on the overhead and ask the same question. If the children have trouble answering this question, turn on the machine and count how many peppers there are. Repeat the
number 8 one more time, but this time place the peppers into one group of five and one group of three.
If the children have an easier time answering this time, ask what helped them.
4. Place ten peppers on the overhead in no particular order. Have the children guess
the number and then repeat with one stick of ten peppers.
5. Reinforce this concept of grouping with the following numbers: 11, 15, 20, 25, 30 and 39. Turn on the machine with the numbers already organized into groups. For every ten, use a base ten pepper
rod. Ask the children to guess the number of peppers every time, and then turn on the machine to check their answers.
6. Ask the children, “How am I grouping the numbers?” Ask them, “Do you always have to group by tens?” Ask them, “Can someone come up and show me another way to group the number 20.” Take several
volunteers until no one else has a new way. Do this with a couple of numbers, but be aware of the children’s attention span and interest level in the activity. Also discuss which way is easier when
counting large numbers. Is it counting by two’s? By fives? Or by tens?
7. Give every child a pile of beans (peppers will burn their eyes) and their 10 bean sticks that they made the previous day. (*Note: To save time, the teacher should put every child’s bean sticks and
a pile of beans into a zip lock bag with their name on it. The teacher can then pass out the bags when it is time for the activity.)
8. Guide the children through a few more examples by writing a number on the overhead and allowing the children time to work on grouping that number so that it is easiest for them to read. After
every example, allow a couple of children to come up to the overhead to explain how they decided to group their numbers. Discuss any use of bean sticks if it is brought up by one of the children.
9. If no one brings up the use of bean sticks for grouping in sets of ten, then ask the children to do a fairly large number, such as fifty. Allow time for them to work, ask for volunteers to share
how they solved the problem and then share how you had divided them up; five sticks of ten (if no one brings up the idea first).
10. Pass out the number cards to the children and ask them to continue grouping the numbers so that they are able to read and recognize that number quickly. Allow this part of the lesson to continue
for about 10-15 minutes. Stop and visit each child in the room to see how they are doing.
This would be the end of this today’s lesson.
The following is what would be done on the next day.
The same materials and objectives apply, as it is a continuation of the previous day’s lesson.
1. Have the children separate into groups of two.
2. Pass out the place value mats and discuss what they see. Ask if anyone can guess what we are going to use the mats for today.
3. Pass out the bowls, each bowl containing no more than 99 beans. Also pass out the children’s zip lock bags containing their base ten rods that they made.
4. Explain to the children (as you demonstrate on the overhead with a transparency of the place value mat) that they may take turns in grabbing a handful of beans and laying them on them on their
place value mat. Tell the children that sometimes they may grad big handfuls or small handfuls of beans, so long as they don’t keep getting the same number over and over. Explain that they are to
count the number of beans that they grabbed and then are to group them using the place value mat that is in front of them. Remind them of the activity that they did yesterday, and ask them what they
did every time that they had ten of something (they used a stick). Ask them to place their sticks under the tens column on their place value mats. Ask the children to explain why they think we are
putting the groups of ten under this column.
5. Demonstrate a couple of numbers for the children. For each completed example ask the following three questions: “What was my number?” “How many tens did I have?” “How many ones did I have?” “If I
count by tens and add the ones do I get the number of beans I grabbed?” Demonstrate that last part out loud so the children can hear how you think it through. Then ask for a few volunteers to come up
to the overhead to complete one. Ask them the same questions as you asked before. Ask if they have any questions and then let them work in their groups for about 20-25 minutes.
6. The teacher should use this time to walk around the room and talk to the children about how they are doing. He/She should also encourage them to ask those three questions that were stated earlier.
7. At the end of the lesson, ask the children if they want to share anything that they found out while they were using the beans. Some children might point out that there were only 99 beans in the
tray. Other children might point other various interesting points that they enjoyed about the lesson.
Connections to the Students’ Lives:
1. Everyday continue to use the overhead to show both scrambled numbers and organized numbers to increase the students’ abilities in number recognition.
2. Continue this process both inside and outside the classroom by:
a. Using everyday materials (pencils, keys, misc. non-messy food) and spilling them on the floor. Challenge the students to count the materials by grouping them in their heads.
b. Taking trips outside and counting objects in nature (trees, bugs, shoes).
c. Ask children to count as many things as they can at home by grouping them into easy to recognize sets.
Work Cited
1. Reys, Robert E. et. Al. (1998). Helping Children Learn Mathematics. Chapter 6, “Development of Number Sense and Counting.” (pp 88-89). Needham Heights, MA: Allyn and Bacon, A Viacom Company.
2. Reys, Robert E. et. Al. (1998). Helping Children Learn Mathematics. Chapter 7, “Developing Number Sense with Numeration and Place Value.” (pp 115-123). Needham Heights, MA: Allyn and Bacon, A
Viacom Company.
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SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N)
up vote 7 down vote favorite
Since $\Gamma(N)$ is normal in $\mathrm{SL}(2,\mathbb{Z})$, the quotient group $\mathrm{SL}(2,\mathbb{Z}/N)$ acts on the spaces of cusp forms $S_k(\Gamma(N))$. How do these spaces decompose into
irreducible representations?
I can do the case $N=2$. I'm mostly interested in the case of $N$ a prime.
modular-forms rt.representation-theory
I wonder if this question of mine is related: mathoverflow.net/questions/2875/… – Ilya Nikokoshev Nov 9 '09 at 20:08
add comment
4 Answers
active oldest votes
See Theorem 1.0.3 of Jared Weinstein's phd thesis (it uses equivariant Riemann Roch).
up vote 4 down vote accepted
add comment
As usual, once I spot a question on here I have anything useful to say about, somebody has already answered it.
I can sum up that part of my thesis this way: let M be the induced representation of the character (-I) --> (-1)^k of the center up to all of SL(2,Z/NZ). Then S_k(Gamma(N)) is roughly k
up vote 7 /12 copies of M, plus some error term which can be given precisely, with some effort.
down vote
When you ask instead about Hilbert modular forms over a totally real field K, the "1/12" becomes the absolute value of zeta_K(-1).
Thank you moonface and Jared, that's perfect. I wish I could give more than one accepted answer. :) – Dan Petersen Nov 10 '09 at 8:35
add comment
If you think about this question in terms of automorphic representations then it sort of becomes trivial. The space $Sk(\Gamma(N))$ can be re-interpreted as the direct sum of $\pi^{U(N)}$,
where $\pi$ is running through the automorphic representations of $GL_2$ which are holomorphic of weight $k$. Each factor is $SL(2,Z/NZ)$-invariant and often irreducible but sometimes has
up vote 3 small finite length. The representation of $SL(2,Z/NZ)$ that shows up on $\pi{^U(N)}$ is the "type" of $\pi$. For explicit $\pi$s one will be able to explicitly determine this
down vote representation.
I'm not sure that they are irreducible (at least if I'm thinking about GL_2(F_p)) - e.g. can't I get things like the induction of the trivial character on the Borel showing up? – user1125
Nov 9 '09 at 23:48
add comment
You can almost do this with nothing more than Riemann-Hurwitz; in particular, by R-H you can compute the action of SL_2(Z/NZ) on H_1(X(N),C), which is just the sum of the representation
up vote 2 you want with its dual.
down vote
The great thing about this approach is that it's completely different to the automorphic forms approach, and if the two are used together then they give information about the number of
automorphic forms of a given "type". Sort of a generalisation of counting how many level 1 forms there are by computing the dimension of the space of level 1 forms by some global method.
– Kevin Buzzard Nov 10 '09 at 7:38
add comment
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[FOM] What produces certainty in mathematical proofs?
Martin Davis martin at eipye.com
Fri Sep 14 01:02:09 EDT 2007
Arnon Avron in recent posts has explained that
only proofs that meet criteria of predicativity
rise in his view to the level of full
acceptability. He acknowledges the interest of
proofs that go beyond this bound, but insists
that they can not provide him with full
confidence in the truth of the theorem proved.
Now, as long as this is expressed as a personal
feeling, no one can reasonably dispute the
matter. But it should be realized that capable
mathematicians who think about foundational
matters are all over the map with respect to
where they draw the line of full acceptability of
a proof. I had a colleague who refuses to accept
any non-constructive proof I don't have an
example at hand, but I suspect that there are
non-constructive proofs that are fully
predicative as that notion is usually explicated.
Going in the other direction, here in Berkeley
there are logicians who accept very large
cardinals without a qualm, whereas another
equally fine logician has devoted a great effort
in attempting to prove the inconsistency of the
existence of measurable cardinals with ZFC.
Harvey Friedman has emphasized the progression of
means of proof from the most basic to the highest
regions of the transfinite with a decision to
draw the line at at particular level a matter of
personal idiosyncrasy rather than the result of philosophical acumen.
There are two further points I'd like to make.
One is that, in the last analysis mathematics is
a social activity, and as time goes on, methods
that achieve desired results without leading to
catastrophe will just gain acceptance.
The other is that we know by Gödel that however
far we go out in this progression there will be
true propositions, even Pi-0-1 sentences, that
require that one proceed further up the
transfinite staircase in order to prove them.
Harvey's work gives tantalizing examples, but so
far there is no example of a true Pi-0-1 sentence
that number theorists have longed to prove but
which requires set theoretic, or even highly
transfinite, methods for its proof. Gödel himself
once suggested that the Riemann Hypothesis could be such a proposition.
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
(Add 1 and get 0)
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A note on the busy period of the M/G/1 queue with finite retrial group
Artalejo, Jesús R. and Gómez-Corral, Antonio (2007) A note on the busy period of the M/G/1 queue with finite retrial group. Probability in the Engineering and Informational Sciences, 21 (1). pp.
77-82. ISSN 0269-9648
Restricted to Repository staff only until 31 December 2020.
Official URL: http://journals.cambridge.org/action/displayFulltext?type=1&fid=600996&jid=PES&volumeId=21&issueId=01&aid=600992
We consider an M/G/1 retrial queue with finite capacity of the retrial group. We derive the Laplace transform of the busy period using the catastrophe method. This is the key point for the numerical
inversion of the density function and the computation of moments. Our results can be used to approach the corresponding descriptors of the M/G/1 queue with infinite retrial group, for which direct
analysis seems intractable.
Item Type: Article
Additional This work was supported by MEC grant no. MTM2005-01248
Subjects: Sciences > Mathematics > Operations research
ID Code: 15483
References: Artalejo, J.R., Economou, A., & Lopez-Herrero, M.J. (2006). Algorithmic approximations for the busy period of the M/M/c retrial queue. European Journal of Operational
Research, to appear.
Artalejo, J.R. & Falin,G.I.(1996). On the orbit characteristics of the M/G/1 retrial queue. Naval Research Logistics 43: 1147-1161.
Artalejo, J.R. & Lopez-Herrero, M.J. (2000). On the busy period of the M/G/1 retrial queue. Naval Research Logistics 47: 115-127.
Falin, G.I. & Templeton, J.G.C. (1997). Retrial queues. London: Chapman & Hall.
Leonard Kleinrock, Theory, Volume 1, Queueing Systems, Wiley-Interscience, 1975
Lopez-Herrero, M.J. (2006). A maximum entropy approach for the busy period of the M/G/1 retrial queue. Annals of Operations Research 141: 271-281.
Tijms, H.C. (2003). A first course in stochastic models. Chichester: Wiley.
Deposited On: 05 Jun 2012 09:26
Last Modified: 06 Feb 2014 10:25
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Buffer overflow asymptotics for a buffer handling many traffic sources
Results 1 - 10 of 40
- IN PROC. ACM SIGCOMM '96 , 1996
"... There has been a growing concern about the potential impact of long-term correlations (second-order statistic) in variable-bit-rate (VBR) video traffic on ATM buffer dimensioning. Previous
studies have shown that video traffic exhibits long-range dependence (LRD) (Hurst parameter large than 0.5). We ..."
Cited by 116 (9 self)
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There has been a growing concern about the potential impact of long-term correlations (second-order statistic) in variable-bit-rate (VBR) video traffic on ATM buffer dimensioning. Previous studies
have shown that video traffic exhibits long-range dependence (LRD) (Hurst parameter large than 0.5). We investigate the practical implications of LRD in the context of realistic ATM traffic
engineering by studying ATM multiplexers of VBR video sources over a range of desirable cell loss rates and buffer sizes (maximum delays). Using results based on large deviations theory, we introduce
the notion of Critical Time Scale (CTS). For a given buffer size, link capacity, and the marginal distribution of frame size, the CTS of a VBR video source is defined as the number of frame
correlations that contribute to the cell loss rate. In other words, second-order behavior at the time scale beyond the CTS does not significantly affect the network performance. We show that whether
the video source model i...
, 1999
"... We describe how a packet network with a simple pricing mechanism and no connection acceptance control may be used to carry a telephony-like service with low packet loss and some call blocking.
The packet network uses packet marking to indicate congestion and endsystems are charged a fixed small amou ..."
Cited by 104 (1 self)
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We describe how a packet network with a simple pricing mechanism and no connection acceptance control may be used to carry a telephony-like service with low packet loss and some call blocking. The
packet network uses packet marking to indicate congestion and endsystems are charged a fixed small amount per mark received. The end-systems are thus provided with the information and the incentive
to use the packet network efficiently. We demonstrate that algorithms embedded in the end-systems are able to synthesize a telephony-like service by blocking calls at times when the load in the
packet network is high. 1.
, 1997
"... In previous work [24], Fractal Point Processes (FPPs) have been proposed as novel tools for understanding, modeling and analyzing diverse types of self-similar traffic behavior. We apply the FPP
models in the context of network traffic modeling and performance analysis. Two qualitatively different f ..."
Cited by 50 (8 self)
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In previous work [24], Fractal Point Processes (FPPs) have been proposed as novel tools for understanding, modeling and analyzing diverse types of self-similar traffic behavior. We apply the FPP
models in the context of network traffic modeling and performance analysis. Two qualitatively different fractal data sets (Bellcore Ethernet traces) are characterized by FPP models. Comparison of
model-driven and trace-driven queueing simulation results show that the matched models yield close agreement with the traces over a wide range of system parameters. We also show that under suitable
conditions, the FPP models yield Gaussian processes. Queueing simulation shows that the FPP models can be computationally efficient alternatives for generating fractional Gaussian noise processes.
Finally, we divide fractal traffic into two types, applicationlevel fractal traffic and network-level fractal traffic, and argue that each type has radically different implications for the design and
control of fut...
- in Proceedings of IEEE INFOCOM , 2002
"... Abstract — A key criterion in the design of high-speed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled
as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumpti ..."
Cited by 39 (1 self)
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Abstract — A key criterion in the design of high-speed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as
point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumptions on the input processes we show that the steady state probability of the buffer content
exceeding a threshold � � tends to the corresponding probability assuming Poisson input processes. We verify the assumptions for a large class of long-range dependent sources commonly used to model
data traffic. Our results show that with superposition, significant multiplexing gains can be achieved for even smaller buffers than suggested by previous results, which consider Ç Ò buffer size.
Moreover, simulations show that for realistic values of the exceedance probability and moderate utilisations, convergence to the Poisson limit takes place at reasonable values of the number of
sources superposed. This is particularly relevant for high-speed networks in which the cost of high-speed memory is significant. Keywords—Long-range dependence, overflow probability, Poisson limit,
heavy tails, point processes, multiplexing.
- Queueing Systems , 1999
"... Consider a switch which queues traffic from many independent input flows. We show that in the large deviations limiting regime in which the number of inputs increases and the service rate and
buffer size are increased in proportion, the statistical characteristics of a flow are essentially uncha ..."
Cited by 35 (2 self)
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Consider a switch which queues traffic from many independent input flows. We show that in the large deviations limiting regime in which the number of inputs increases and the service rate and buffer
size are increased in proportion, the statistical characteristics of a flow are essentially unchanged by passage through the switch. This significantly simplifies the analysis of networks of
switches. It means that each traffic flow in a network can be assigned an effective bandwidth, independent of the other flows, and the behaviour of any switch in the network depends only on the
effective bandwidths of the flows using it. Keywords. Effective bandwidths, feedforward networks, large deviations, decoupling bandwidths, output of a switch, many sources. 1 Introduction A switch is
a device that routes traffic. A switch has several input flows of traffic, each of which is routed to a specified destination; and inside the switch, work from all of the inputs is queued together.
- In Proc. IEEE INFOCOM , 1996
"... A key problem for modern network designers is to characterize /model the "bursty" traffic arising in broadband networks with a view on predicting and guaranteeing performance. In this paper we
attempt to unify several approaches ranging from histogram/interval based methods to "frequency domain" app ..."
Cited by 28 (2 self)
Add to MetaCart
A key problem for modern network designers is to characterize /model the "bursty" traffic arising in broadband networks with a view on predicting and guaranteeing performance. In this paper we
attempt to unify several approaches ranging from histogram/interval based methods to "frequency domain" approaches by further investigating the asymptotic behavior of a multiplexer carrying a large
number of streams. This analysis reveals the salient traffic /performance relationships which should guide us in selecting successful methods for traffic management and network dimensioning. 1
Introduction Efficient methods for congestion control in high-speed communication networks will be based on reasonable characterizations for traffic flows and time scale decompositions of the network
dynamics. With a view on resolving the question of admission control, including bandwidth allocation and routing, as well as other traffic management activities, researchers have developed several
approaches to mod...
- Queueing Systems , 1998
"... We analyse the deviant behavior of a queue fed by a large number of traffic streams. In particular, we explicitly give the most likely trajectory (or `optimal path') to buffer overflow, by
applying large deviations techniques. This is done for a broad class of sources, consisting of Markov fluid sou ..."
Cited by 16 (8 self)
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We analyse the deviant behavior of a queue fed by a large number of traffic streams. In particular, we explicitly give the most likely trajectory (or `optimal path') to buffer overflow, by applying
large deviations techniques. This is done for a broad class of sources, consisting of Markov fluid sources and periodic sources. Apart from a number of ramifications of this result, we present
guidelines for the numerical evaluation of the optimal path.
, 2001
"... The main network solutions for supporting QoS rely on traffic policing (conditioning, shaping). In particular, for IP networks the IETF has developed Intserv (individual flows regulated) and
Diffserv (only aggregates regulated). The regulator proposed could be based on the (dual) leaky-bucket mechan ..."
Cited by 13 (0 self)
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The main network solutions for supporting QoS rely on traffic policing (conditioning, shaping). In particular, for IP networks the IETF has developed Intserv (individual flows regulated) and Diffserv
(only aggregates regulated). The regulator proposed could be based on the (dual) leaky-bucket mechanism. This explains the interest in network element performance (loss, delay) for leaky-bucket
regulated traffic. This paper describes a novel approach to the above problem. Explicitly using the correlation structure of the sources' traffic, we derive approximations for both small and large
buffers. Importantly, for small (large) buffers the short-term (long-term) correlations are dominant. The large buffer result decomposes the traffic stream in a stream of constant rate and a periodic
impulse stream, allowing direct application of the Brownian bridge approximation. Combining the small and large buffer results by a concave majorization, we propose a simple, fast and accurate
technique to statistically multiplex homogeneous regulated sources. To address heterogeneous inputs, we present similarly efficient techniques to evaluate the performance of multiple classes of
traffic, each with distinct characteristics and QoS requirements. These techniques, applicable under more general conditions, are based on optimal resource (bandwidth and buffer) partitioning. They
can also be directly applied to set GPS (Generalized Processor Sharing) weights and buffer thresholds in a shared resource system.
, 2003
"... In this paper, we show that significant simplicities can arise in the analysis of a network when link capacities are large enough to carry many flows. In particular, we prove that, when an
upstream queue serves a large number of regulated traffic sources, the queue-length of the downstream queue con ..."
Cited by 11 (0 self)
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In this paper, we show that significant simplicities can arise in the analysis of a network when link capacities are large enough to carry many flows. In particular, we prove that, when an upstream
queue serves a large number of regulated traffic sources, the queue-length of the downstream queue converges almost surely to the queue-length of a simplified queueing system (single queue) obtained
by removing the upstream queue. We provide similar results (convergence of the queue-length in distribution) for general (including non-regulated) traffic arrivals. In both cases, the convergence of
the overflow probability is uniform and at least exponentially fast. Through an extensive numerical investigation, we demonstrate several aspects and implications of our results in simplifying
network analysis.
- J. Appl. Probab , 2002
"... In this paper we consider a queue fed by a large number n of independent continuoustime Gaussian processes with stationary increments. After scaling the buffer exceedance threshold B and the
(constant) service capacity C bythenumber of sources (i.e., B nb and C nc), we present asymptotically e ..."
Cited by 9 (4 self)
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In this paper we consider a queue fed by a large number n of independent continuoustime Gaussian processes with stationary increments. After scaling the buffer exceedance threshold B and the
(constant) service capacity C bythenumber of sources (i.e., B nb and C nc), we present asymptotically exact results for the probability that the buffer threshold is exceeded. We both consider the
stationary overflow probability, and the (transient) probabilityofoverflow at a finite time horizon T . We give detailed results on the practically important cases in which the inputs are fractional
Brownian motion processes or integrated Gaussian processes. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=625297","timestamp":"2014-04-20T12:43:19Z","content_type":null,"content_length":"39979","record_id":"<urn:uuid:8b971038-849e-489a-9d09-60ba559d885a>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00614-ip-10-147-4-33.ec2.internal.warc.gz"} |
Interview with Persi Diaconis
A most unusual book by Persi Diaconis and Ron Graham has recently appeared, titled
Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks.
It relates some series of card manipulations and tricks with deep mathematics, of different kinds, but with a minimal degree of technicity, and beautifully shows how the two domains really intersect,
at least for both authors. Magical Mathematics is a wonderful book, potentially enjoyable by a large range of individuals and not only mathematicians. The following interview took place on the phone
on February 22, 2012.
Christian Robert: Persi, I was very glad to read this book because I had always wondered about this side of yours, mysterious as it should be because it’s magic, so I set to read the book and I
really loved it. My first question is, how did you find the time to write a general audience book in addition to doing research in mathematics?
Persi Diaconis: Well, why did I write the book, I don’t really know. My interaction between magic and the academic side has always been to keep the magic side out of the academic side. Magic was
something I used to do and now I’m an academic. On the other hand, magic has done a lot of good for me and every once in a while you have to give something back and I somehow was persuaded that it
would be a good idea. I have hopes that it’ll reach young people, like I was, who have some interest in magic, but could be swayed toward mathematics or stochastic things, and best thing for me would
be if over the years some 16 year old would find it in a library or a bookstore and says, hey, that’s cool!
The writing took forever. We signed a contract with Princeton University more than 25 years ago and they paid a big healthy advance and they were very, very nice about it. Every once in a while they
would say, how’s the book going? Once a year I’d get a call and I’d say, great, we’re working on it. And then my coauthor Ron Graham said, you know, maybe it’s time we should finish it. We worked
pretty hard for a couple of years, intensely flying up and down and getting photos and stuff, and when you decide to finish things then you’re in finishing mode and you can finish it.
Of course that entails taking all kinds of projects that we hope to work on and, deciding we’re not going to do that in this book or in this life and here’s what we have. How it came to be written is
just deciding, okay, I’m going to do it. And that is a decision, because it does mean a lot of work focused in the direction of writing for the public instead of doing our technical work and/or
analyzing data or doing teaching. So, I’m happy it’s done and onto the next!
Christian Robert: What is next?
Persi Diaconis: Well, I have a long-time project, even longer overdue than this one, about coincidences. I’ve written a little bit about it with Fred Mosteller. I promised Fred that I would do such a
book and I’m headed in that direction. I have a public talk, which is about how to think about some of the weird things that happen and then I stopped giving it because I just got tired. Now when I’m
asked to give public talks, which is plenty, I just sign up and say, well I’ll give this coincidence talk, I hope to trick myself into getting down to work now I see that it can be done. There’s a
real tension between contributing to our field and writing for the public and I have shied away from writing for the public. I did promise Fred I’d do it and if I wait too long, I won’t do it.
So, I think that that will be the next project but I wouldn’t hold my breath. It’s been more than 30 years. Fred Mosteller, who was a marvelous statistician, was fascinated by coincidences and
collected in a kind of focused way, articles and clippings and weird stories. When he’d read something, like Jesuits die in three, there would be a story in the paper that somebody noticed the
triples—he would make a little back of the envelope calculation or model and then say, it’s not so unlikely or that is unlikely and can I get the data and see if it’s really true? Over the years, he
had many giant loose leaf binders filled with these kinds of things, and we wrote an article, it was his presidential address for the American Statistical Association, on methods for studying
The New York Times did a front-page article about it, and one thing is, if you get on the front page of The New York Times as professors study coincidences, you get a lot of weird letters. So, we got
hundreds of stories. “This happened to me, how can you guys explain it? I’m haunted by that …” We saved every single one of them and made little calculations. All of that material is around, sitting
in my office and taunting me. I hope to make some sense out of it. It’s not easy, in the sense that, if you want to figure out what’s really going on, a lot of it is human psychology; why people are
surprised by things that aren’t actually so surprising and that’s not something that we’re so trained to do or understand.
Sam Behseta: Persi, one gets the impression that there is a decent number of mathematicians involved with magic.
Persi Diaconis: I wouldn’t say so many. I mean it doesn’t seem to be more than in the population of academics, for example. The most well-known one is Raymond Smullyan who writes about philosophical
paradoxes and chess problems and books called, “What is the name of this book.” He was a very serious amateurish magician and a good musician. I don’t know many mathematicians who are interested in
magic. Nor statisticians, it’s a handful. One of the things that happen with the book having published is that people come out of the woodwork and send you their tricks. I just returned from a trip
so I got a bunch of letters.
I received a letter saying, “here’s a trick with correcting codes and I wrote an article but nobody will publish it. Maybe we could write it up together since you seem to know how to get things
published.” I haven’t looked at it yet. Probably it’s an awful trick. One of the problems is that mathematical tricks are usually deadly. We did have a trick that I had hoped to put in the book which
was based on error correcting codes like the Hamming code where there’s an array of cards, face up on the table, say 48 cards in 6 rows of 8. Somebody thinks of a card and then points to the row the
card is in and calls out the colors and lies when he comes to his card. So you hear something like red, red, red, black, red. Because he has lied, there will be an error in that, and it’ll be an
error on the card he is thinking of and because it is an error correcting code you know when he lied and therefore you know what the card is. It’s not a great trick, the way I described it, but it is
a start. That’s a trick that didn’t make into the book but it could. Maybe volume two.
Christian Robert: Back to what you were saying about this tension between academic publishing and writing for the general public: Aren’t you afraid of getting too much pressure from writing this book
and for giving talks to the general public?
Persi Diaconis: I’m acutely aware of it. If one of our colleagues always has his name in the newspapers … people know that in order to communicate with the public, there is some extent to which you
have to lie. If you say the absolute truth with all the weasel words that make your conclusions carefully true, nobody will listen to you. So you do need to sort of simplify things and shortcut the
truth and if somebody has always have their name in the paper, I find, at least for myself, why is this person always has his name in front of the public? What is wrong? You trust them less.
I think you can get publicity about every two or three years, and still be a respected academic. You have to keep your head down in some way and, that is another thing. I am pretty good at saying no
to giving talks even though. If you ask my wife, Susan Holmes, she says, “you’re always saying yes.” But I say no, about three times for every time I say yes. The end result is still I wind up giving
about 50 outside talks a year and I would say about a quarter of them are talks for the general public like the talk on coincidences or what’s the nature of randomness or mathematics and magic
tricks. That has increased a little bit because of the publicity.
But I also say no. I had a call from some national organization for something like the Boy Scouts and some earnest person says “our kids don’t get to see any science and you can communicate and would
you come to our national meeting?” It is an opportunity of a sort but not for me. On the other hand, I am currently answering emails from having been away for two weeks and two of them are to give
big public lectures, I think I said yes to those two. It has been like this for 35 years and it cannot get much worse.
Sam Behseta: I really liked that chapter that you titled “Is this stuff actually good for anything?”
Persi Diaconis: Thank you. I was trying to explain to people what we do a little bit, once I had their attention with those tricks, to say it is a pretty good life, what we do. You know, we get to
talk to each other and find a few other people who care about the silly things we do and they get interested and then the progress gets made so I did try to communicate that doing research in
mathematics or statistics is an exciting activity.
Sam Behseta: Especially there’s a part that you talk about genetic applications, you mention de Finetti’s exchangeability, Bayes, and reversible Markov chains and all of that, and I found it
fascinating, but you kept it at the minimum technical explanation.
Persi Diaconis: Well it wasn’t for you, it was for the public. But for this interview I could say it more carefully. That development keeps going. That was David Freedman’s thesis actually. It is a
funny story: David Freedman was a kind of cantankerous guy, a little more honest than most of us and a difficult guy. He was a graduate student at Princeton. He had come to work with John Tukey but
he couldn’t figure out what he was talking about. John Tukey spoke his own language and he wouldn’t translate, so one day in 1960, David walked into Feller’s office and said “Professor Feller, I’ve
decided I would like to try to do a thesis in probability.” Feller, without looking up, said, all right young man, prove de Finetti’s theorem for Markov chains and then looked back down at his papers
and David walked out and went and found out, what is de Finetti’s theorem, what is a Markov chain. He figured out a way of detecting when a stochastic process is a mixture of Markov chains and
developed a notion of partial exchangeability.
In order to make his theorem go, he needed to assume that the process was stationary so that the Markov chain started in its stationary distribution. When I came out to the West Coast, I knew that
you did not need that assumption that you could let the Markov chain start at any place and still have the theorem. Also, I knew how to make finite versions of it. I met David and Lester Dubins at
one of these Berkeley-Stanford meetings—four times a year we have a joint meeting—we started talking and I said I knew about his thesis work. He then invited me to give a talk at Berkeley and I
remember Dubins being very hostile, “Who is this kid? He’s not going to tell us anything.” This is 1975. I was a starting assistant professor, but Friedman realized there was something there and we
started to work together and the condition of partial exchangeability all worked out. But when we went to making a finite version of the theorem without assuming that the process could be extended to
infinity, we had to develop some new theory.
Suppose I give you a sequence, say 0–1 sequence and you can record the transition counts in a 2 by 2 matrix: how many times did it go from 0 to 0, 0 to 1, etc. If I give you such a matrix, you can
ask conversely, how many sequences give rise to such a matrix? We know the answer to that by combinatorics. Then, how do I generate a uniform sequence of fixed length with a given transition count
matrix? That involves some funny work about trees and Eulerian paths and tricky combinatorics but we figured it out. That allowed us to have a finite version of the de Finetti’s theorem for Markov
About eight years later, a very similar problem came up in thinking about quantifying DNA string matching algorithms. As you know, in DNA, there are strings such as A-C-G-T, or often just 0s and 1s.
People might have two strings and then ask: are these two strings close together? There are various moves you are allowed to make: leading, reversing, blocks and rather complicated sets of moves that
would allow you to define a metric between strings. Any of the algorithms such as BLAST do that. Suppose you give it two strings. One is a string from your data and another is a string from some
library of strings, and then the algorithm finds the distance between your string and the closest matching string in the catalog to be say, 135. Well is that big or small? How do you calibrate that?
What the biologists wanted to do is to generate those strings at random, fixing the transition counts or the higher order transition counts, not just 0 to 1 but maybe the previous five symbols to 1.
As we know, that just can be seen as a Markov chain on a larger state-space, and the way they were doing it was by naïve simulation: just generate strings at random until you find one that had the
right transition counts. But because we had done our math, we could give a closed form algorithm that was exactly generating strings with a given transition count matrix.
That became part of the program BLAST which is used millions of times every day when somebody has a string and wants to go to the big database and find another string that is close. The program also
gives you the detail on how sure we are about it. That is a thing I did for a philosophy problem that is used millions of times a day in practice. It doesn’t happen all the time. I have certainly
done a lot of things that are not used a million times a day. This work we did on de Finetti’s theorem for Markov chains now has a very healthy development. Nice story, Christian will like: some
chemists are studying protein folding and this is the part of the IBM Blue Gene project which is IBM’s big new effort. They had Deep Blue, which played chess and now they are trying to do protein
They are actually doing the molecular dynamics very carefully. They have a molecule with 40 atoms and then 10,000 water molecules around it. Then they all interact with quantum mechanical forces. The
problem is, it’s a very, very high dimensional state space. To describe an atom is, to describe anything with 12 numbers: position, velocity, angular position, rotation, and three angular variables.
It’s a 100,000 dimensional dynamical system. They wanted to coarse-grain space and time and break it into something like 5000 to 50,000 boxes. Then there are images that the dynamical system will
kind of get random within the little boxes and it’s the transitions from box to box that matters.
They are modeling that by a Markov chain. It’s a standard thing to do in chemistry, but then they are Bayesians, hooray! it came as a surprise to me. They said—sort of sheepishly—we are Bayesians and
I said, great. And they wanted to do a Bayesian analysis but they were very insistent that they did not just want to put a prior on the Markov transition matrix, but because the laws of mechanics and
physics are time-reversible, they were sure that the Markov chain would be reversible so they wanted to put a prior on reversible Markov chains. Well, that is not an off-the-shelf distribution
because it is a curved subfamily of the big family.
But because I had done this work along de Finetti’s theorem for Markov chains I knew how to do it. I had written a paper in the Annals of Statistics with Silke Rolles a couple years before on how to
do that. There is something like a conjugate prior which updates reasonably neatly and does exactly what they wanted. So they had one of their graduate students working on it as his thesis. Anytime
theory meets practice, practice has all kinds of demands that the theory didn’t think about. In our theory, you have a long run of this Markov chain. What they had is many short runs of the Markov
chain and the theory needed to be changed and then they wanted to adapt it for Markov chains which were in some states Markovian but in other states second-order Markovian. A student, a marvelous
young man named Sergio Bacalado, became interested, did his thesis on the subject, and has two papers in the Annals that extended the work that Freedman and I did and now it is being used: that is
the nice thing. It is not just somebody writing papers in the Annals, they are actually doing the inference and they say it makes a big difference. So there is a little leaf out there growing which
comes from that so it’s alive today. And of course those sorts of stories are what keep us going. I wish it would happen more often but it is nice that it happens every once in a while that somebody
actually read the paper or cares about it.
Christian Robert: Is that related with the paper you wrote on importance sampling for contingency tables with fixed marginals?
Persi Diaconis: No, this is different because in that paper we did not know how to sample. We needed to use this fancy algebra in order to derive Markov chain Monte Carlo to sample from contingency
tables with given margins. Here, we have a string, say an ACGT string and the simulation task is the following: fix the transition matrix, how many times did it go A to A, A to G, A to C, etc. Then I
want to generate uniformly, at random, a string with that given transition matrix. We do not need to run an MCMC for that. The combinatorics is an analog of perfect sampling, but without iterating or
going backwards. It is like sampling from the hypergeometric or something you can just do it. You don’t have to run an MCMC.
So we had a closed form algorithm which we did in order to be able to write down something that we could do calculus on to do our asymptotics. Let me take the two state case: I have a two by two
transition matrix, I fixed those numbers and there should be at least one string that is compatible with them and then I want to generate a random string. Well here is what you do. You take two urns
and on the outside of one of the urns you write a giant 0 and on the outside of the other urn you write a giant 1. Then if you want to do A many 0 to 0 transitions, you put A balls labeled 0 in the 0
urn and if you want to do B many 0 to 1 transitions you put B balls labeled 1 in the 0 urn and so on. Same thing for the 1 urn.
Then you just do sampling without replacement in the following way: say you start at 0, you reach into the 0 urn, you pull out a ball, you write it down and you throw it away and then if that ball is
a 0 you reach into the 0 urn again. If the ball is a 1, the next time you reach into the 1 urn. So you are doing some analog of sampling without replacement but you will balance out and get the
transition counts that you want. Now one problem that occurs is if you just do it naïvely, you could get stuck. You could reach into a 0 urn and it is empty, and the math came in figuring out what to
do, how to adjust the totals a little bit, so you never get stuck, in that it works. But the adjustment didn’t foul things up, and we had our sharp bounds on sampling without replacement then that
enabled us to do the asymptotics, but it needed—going back to the magic book because that’s where it started—the math of the growing sequences and deBruijn, Ehrenfest, Smith, Tutte theorem and if I
had not done it earlier because of the magic, I wouldn’t have known that math. So it’s a nice circle.
Persi Diaconis: One other thing that might be an interesting story on the connections between statistics and magic and me is the way that I got to go to graduate school. I guess it is told in the
book but I’m not positive. Fred Mosteller who was a famous old statistician, great man, was this very serious amateur magician and, Martin Gardner wrote to him and that was my letter of reference. I
had poor grades, Martin Gardner wrote to him, saying “of the ten best card tricks invented in the last 10 years, this kid invented two of them and maybe you should give him a break.”
Fred was on the committee and said, “yeah, let’s take this kid.” And so that was a way in which magic made a big impact on my career. I got to be a graduate student at Harvard because of it. One of
the things that you might think, is that magic might enliven, teaching. Fred tried it. He used to do something where he would be lecturing for a class, he’d be writing a lecture and he’d put the
chalk down on the ledge and then he’d talk to the kids for a second and then he’s start writing and he had chalk in his hands again. Or he’d be writing on the blackboard and all of a sudden he’d
underline something but it would appear in red or something like that. What he said he found was, it was a disaster, the kids thought he wasn’t a professor but some actor or magician and they stopped
listening. You know, because they do take us with a certain kind of respect. He found that mixing show-business to liven things up, killed the academic beliefs of the kids. If we hear a lecture from
some senior person we listen. Even we listen with a sort of rapt attention. As if the truth is coming from on high or something. Some of that interaction between students and faculty and
entertainment part killed it so he stopped doing it. I don’t do it either; I mean I don’t mix my worlds.
I am going to have a problem this next quarter starting in April. I’m going to teach a course out of the book and I put a cap on it at 40 students and it is already completely filled and the students
are saying, “I came to Stanford because I wanted to take this course.” I don’t know what I am going to do about it, but, I am going to do a course that mixes mathematics and magic. There are
probability tricks that did not get in the book. There are tricks such that people don’t do calculations so well and there are tricks that, you know, work 90% of the time. When they work, folks say,
“that’s amazing” and when they don’t work you have to say, “you have to concentrate and believe.”
Christian Robert: You don’t associate directly these card tricks with statistics because there is this level of uncertainty that you wouldn’t want with card tricks?
Persi Diaconis: It’s true. The magicians did a survey: they put out a call in their journals saying, “Would everybody ask 10 people to name a card?” and then they recorded what cards people named and
people named very few cards. They are not good at randomizing and they just name the ace of spades or the king of hearts, and so if you know those ten cards, well you could be prepared for that, and
appear to do a miracle, whereas people think they just told me any card. Anyway, it is limited, but that was one of the places where statistics touched magic.
Sam Behseta: So are there tricks with coins? The reason I am asking is because you have done this wonderful work in which you studied if there is such a thing as a fair toss and ended up showing that
51% of times it lands on what you started with.
Persi Diaconis: Of course there are coin tricks, lots and lots of them but they’re mostly sleight of hand. There are some mathematical coin tricks which use parity, that are not very good. I’ll
answer with a recommendation for a good novel which is called American Gods and it’s a slightly strange book but very readable and, uh –
Christian Robert: From Neil Gaiman?
Persi Diaconis: Yes, that’s right. It’s filled with coin tricks; the guy who is the protagonist does coin magic on the side. I just asked a friend, a magician friend who is in Hollywood if he knows
the author and he said, yeah, he is a magic buff.
I don’t know a single magic trick that works using calculus, and there must be some tricks that use calculus, so I am hoping somebody gets annoyed at me and says, what about this?
The book has had some very nice reviews which is nice. It wasn’t just something that we knocked off. We worked very hard at it over a very long time. It’s a lifetime of mine being in magic that is
woven into it. I recommend the chapter on juggling, too. Ron Graham has taught thousands of people to juggle. He is a very good mathematician and often in his talks, after the talk, he will say, if
anybody brought three balls—and hundreds of people will have brought three balls—he will give a public juggling lesson. Well he has really learned how to teach people how to juggle three balls and
all of that is woven into that chapter. I don’t juggle, and the reason is because all my friends who juggle are really good at it.
Sam Behseta: Thank you Persi for the wonderful conversation.
Persi Diaconis: Sure, what a pleasure. It’s nice to hear your voice, both of you. Nice to meet you, Sam, and I think doing what you’re doing and keeping CHANCE lively and visible is great for the
field so I’m all in favor.
Christian Robert: Thank you, Persi.
Persi Diaconis: Okay, a pleasure and hope I see you both soon.
Visit Christian Robert’s blog to read a review of Magical Mathematics.
2 Comments
1. Nice interview! Readers interested in connecting magic to their statistics teaching may also enjoy this paper available online at: http://iospress.metapress.com/content/j730w4777h667125/.
Lesser, L. M. & Glickman, Mark E. (2009). Using Magic in the Teaching of Probability and Statistics. Model Assisted Statistics and Applications, 4(4), 265-274.
2. Doubtless Persi’s interest in magic explains his keen ability to scrutinize experiments in PSI.
I was disappointed, though, that Persi didn’t show me any magic tricks, either at that conference or others, as I’m a big magic fan. True, Freedman warned me not to ask Persi to do a trick, yet I
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Harald August Bohr
Born: 22 April 1887 in Copenhagen, Denmark
Died: 22 January 1951 in Copenhagen, Denmark
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to see four larger pictures
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Harald Bohr was a younger brother of Niels Bohr. Their father, Christian Bohr, was professor of physiology at the University of Copenhagen. Christian Bohr was famous for his work on the physical and
chemical aspects of respiration. Harald and Niels Bohr's mother, Ellen Adler Bohr, came from a wealthy Jewish family with family members who were important in banking and in politics in Denmark.
Harald studied mathematics at the University of Copenhagen. He entered the University in 1904 and quickly became a well known Danish personality, not for his mathematics but rather for his soccer
skills. He was in the Danish soccer team which was placed second in the 1908 Olympic games in London. When his doctoral dissertation was examined at the University of Copenhagen, there were more
soccer fans wishing to attend this public examination than there were mathematicians!
Mathematics soon became more important to Bohr than soccer and he became professor of mathematics in the Polytechnic Institute in Copenhagen in 1915. Then, in 1930, he was appointed professor of
mathematics at the University of Copenhagen. Although he never quite attained the fame of his brother Niels (except as a soccer player!), he did produce some mathematics of the very highest
importance. It is perhaps surprising that Harald and Niels did not collaborate more frequently. They only published one joint paper.
Harald Bohr worked on Dirichlet series, and applied analysis to the theory of numbers. He collaborated with Edmund Landau, who was at this time at Göttingen, in studying the Riemann zeta function. In
1914 they proved the Bohr-Landau theorem on the distribution of zeros of the zeta function.
Some of this important work on the zeta function was due to Bohr alone, some came from the collaboration with Landau. Some of the most impressive from the many striking results which they proved were
major steps towards a proof of the Riemann hypothesis (which, however, is still unproved). Bohr and Landau proved that all but an infinitesimal proportion of the zeros of the zeta function lie in a
small neighbourhood of the line s = ^1/[2] .
Bohr's interest in which functions could be represented by a Dirichlet series led him to devise the theory of almost periodic functions. He founded this theory between the years 1923 and 1926 and it
is with this work that his name is now most closely associated. Roughly speaking an almost periodic function is one which, after a period, takes values within e of its values in the previous period.
Bohr published three major works on this topic in Acta Mathematica between 1924 and 1926.
The fundamental theorem for almost periodic functions is a generalisation of the Parseval identity for Fourier series. This result lead Bohr to a result on the uniform approximation to almost
periodic functions by exponential functions.
Titchmarsh, writing in [10], sums up his work on almost periodic functions:-
The general theory was developed for the case of a real variable, and then, in the light of it, was developed the most beautiful theory of almost periodic functions of a complex variable. ... The
creation of the theory of almost periodic functions of a real variable was a performance of extraordinary power, but was not based on the most up-to-date methods, and the main results were soon
simplified and improved. However, the theory of almost periodic functions of a complex variable remains up to now in the same perfect form in which it was given by Bohr.
After setting up the theory of almost periodic functions, Bohr's mathematical work became devoted exclusively to furthering the subject. He continued his work until shortly before his death, in fact
he attended the International Congress of Mathematicians in Cambridge, Massachusetts four months before his death.
Besicovitch writes in [3]:-
For most of his life Bohr was a sick man. He used to suffer from bad headaches and had to avoid all mental effort. Bohr the man was not less remarkable than Bohr the mathematician. He was a man
of refined intellect, harmoniously developed in many directions. He was also a most humane person. His help to his pupils, to his colleagues and friends, and to refugees belonging to the academic
world was generous indeed. Once he had decided to help he stopped at nothing and he seldom failed. He was very sensitive to literature. His favourite author was Dickens; he had a deep admiration
of Dickens' love of the human being and deep appreciation of his humour.
Article by: J J O'Connor and E F Robertson
Click on this link to see a list of the Glossary entries for this page
List of References (10 books/articles) A Quotation
A Poster of Harald Bohr Mathematicians born in the same country
Additional Material in MacTutor
Honours awarded to Harald Bohr
(Click below for those honoured in this way)
Speaker at International Congress 1932
Cross-references in MacTutor
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JOC/EFR © October 2003 School of Mathematics and Statistics
Copyright information University of St Andrews, Scotland
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Logically Smooth Density Estimation
Results 1 - 10 of 51
- IEEE Trans. Inform. Theory , 1998
"... Abstract — We review the principles of Minimum Description Length and Stochastic Complexity as used in data compression and statistical modeling. Stochastic complexity is formulated as the
solution to optimum universal coding problems extending Shannon’s basic source coding theorem. The normalized m ..."
Cited by 305 (12 self)
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Abstract — We review the principles of Minimum Description Length and Stochastic Complexity as used in data compression and statistical modeling. Stochastic complexity is formulated as the solution
to optimum universal coding problems extending Shannon’s basic source coding theorem. The normalized maximized likelihood, mixture, and predictive codings are each shown to achieve the stochastic
complexity to within asymptotically vanishing terms. We assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of
statistical inference. Context tree modeling, density estimation, and model selection in Gaussian linear regression serve as examples. Index Terms—Complexity, compression, estimation, inference,
universal modeling.
- IEEE Transactions on Information Theory , 1990
"... Abstract-In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect
to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian densit ..."
Cited by 107 (10 self)
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Abstract-In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a
prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter
vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher
information matrix evaluated at the true parameter value. Consequences are given for density estima-tion, universal data compression, composite hypothesis testing, and stock-market portfolio
selection. 1.
- in Advances in Minimum Description Length: Theory and Applications. 2005
"... ..."
"... . The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet A) from observation of a sample path x 1 ; x 2 ; : : : ; x n , as that value k = k that
minimizes the sum of the negative logarithm of the k-th order maximum likelihood and the penalty term jAj ..."
Cited by 55 (3 self)
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. The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet A) from observation of a sample path x 1 ; x 2 ; : : : ; x n , as that value k = k that
minimizes the sum of the negative logarithm of the k-th order maximum likelihood and the penalty term jAj k (jAj\Gamma1) 2 log n: We show that k equals the correct order of the chain, eventually
almost surely as n ! 1, thereby strengthening earlier consistency results that assumed an apriori bound on the order. A key tool is a strong ratio-typicality result for Markov sample paths. We also
show that the Bayesian estimator or minimum description length estimator, of which the BIC estimator is an approximation, fails to be consistent for the uniformly distributed i.i.d. process. AMS 1991
subject classification: Primary 62F12, 62M05; Secondary 62F13, 60J10 Key words and phrases: Bayesian Information Criterion, order estimation, ratiotypicality, Markov chains. 1 Supported in part by a
joint N...
, 1997
"... The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, I arrive at a
systematic series for the Bayesian posterior probability of a model family that significantly extends known ..."
Cited by 54 (3 self)
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The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, I arrive at a
systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, I arrive at a precise understanding of how
Occam’s razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory. These results require a measure on
the space of model parameters and I derive and discuss an interpretation of Jeffreys ’ prior distribution as a uniform prior over the distributions indexed by a family. Finally, I derive a
theoretical index of the complexity of a parametric family relative to some true distribution that I call the razor of the model. The form of the razor immediately suggests several interesting
questions in the theory of learning that can be studied using the techniques of statistical mechanics.
- Annals of Statistics , 1999
"... Key words and phrases. Complexity regularization, classi cation, pattern recognition, regression estimation, curve tting, minimum description length. 1 Given n independent replicates of a
jointly distributed pair (X; Y) 2R d R, we wish to select from a xed sequence of model classes F1; F2;:::a deter ..."
Cited by 36 (8 self)
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Key words and phrases. Complexity regularization, classi cation, pattern recognition, regression estimation, curve tting, minimum description length. 1 Given n independent replicates of a jointly
distributed pair (X; Y) 2R d R, we wish to select from a xed sequence of model classes F1; F2;:::a determin-istic prediction rule f: R d! R whose risk is small. We investigate the possibility of
empirically assessing the complexity of each model class, that is, the actual di culty of the estimation problem within each class. The estimated complexities are in turn used to de ne an adaptive
model selection procedure, which is based on complexity penalized empirical risk. The available data are divided into two parts. The rst is used to form an empirical cover of each model class, and
the second is used to select a candidate rule from each cover based on empirical risk. The covering radii are determined empirically to optimize a tight upper bound on the estimation error.
- J. Theoret. Probab , 1998
"... this paper we investigate the asymptotic behavior of recurrence and waiting times for finite-valued stationary processes, under various mixing conditions. Let X = fX n ; n 2 Zg be a stationary
ergodic process on the space of infinite sequences (S ..."
Cited by 22 (6 self)
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this paper we investigate the asymptotic behavior of recurrence and waiting times for finite-valued stationary processes, under various mixing conditions. Let X = fX n ; n 2 Zg be a stationary
ergodic process on the space of infinite sequences (S
- PROC. ISIT 2004 , 2004
"... We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ)
type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978 ..."
Cited by 22 (6 self)
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We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ) type
if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978). We show that the empirical probability distributions of any finite order of two sequences of the
same universal type converge, in the variational sense, as the sequence length increases. Consequently, the normalized logarithms of the probabilities assigned by any kth order probability assignment
to two sequences of the same universal type, as well as the kth order empirical entropies of the sequences, converge for all k. We study the size of a universal type class, and show that its
asymptotic behavior parallels that of the conventional counterpart, with the LZ78 code length playing the role of the empirical entropy. We also estimate the number of universal types for sequences
of length n, and show that it is of the form exp((1+o(1))γ n/log n) for a well characterized constant γ. We describe algorithms for enumerating the sequences in a universal type class, and for
drawing a sequence from the class with uniform probability. As an application, we consider the problem of universal simulation of individual sequences. A sequence drawn with uniform probability from
the universal type class of x n is an optimal simulation of x n in a well defined mathematical sense.
- IEEE Trans. Inform. Theory , 1999
"... We characterize the achievable pointwise redundancy rates for lossy data compression at a fixed distortion level. "Pointwise redundancy" refers to the difference between the description length
achieved by an nth-order block code and the optimal nR(D) bits. For memoryless sources, we show that the be ..."
Cited by 21 (13 self)
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We characterize the achievable pointwise redundancy rates for lossy data compression at a fixed distortion level. "Pointwise redundancy" refers to the difference between the description length
achieved by an nth-order block code and the optimal nR(D) bits. For memoryless sources, we show that the best achievable redundancy rate is of order O( p n) in probability. This follows from a
second-order refinement to the classical source coding theorem, in the form of a "one-sided central limit theorem." Moreover, we show that, along (almost) any source realization, the description
lengths of any sequence of block codes operating at distortion level D exceed nR(D) by at least as much as C p n log log n, infinitely often. Corresponding direct coding theorems are also given,
showing that these rates are essentially achievable. The above rates are in sharp contrast with the expected redundancy rates of order O(log n) recently reported by various authors. Our approach is
based on showing that... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=81437","timestamp":"2014-04-19T03:00:34Z","content_type":null,"content_length":"36582","record_id":"<urn:uuid:dbf14f4c-243e-4a76-a00a-f9a35e850790>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00448-ip-10-147-4-33.ec2.internal.warc.gz"} |
$S^1$-action in three dimensions
up vote 6 down vote favorite
Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action.
What does this imply for $M$? What are examples except for (products of) spheres?
3-manifolds dg.differential-geometry group-actions
2 The standard examples are Seifert fibre spaces, which admit a free $S^1$ action. The wikipedia page gives a reasonable description and some references. – HJRW Oct 10 '11 at 18:26
2 @HW: they are standard examples, but not the only examples. In particular, if you allow fixed points, you get direct sums of lens spaces (proved in the Raymond paper, where a complete
classification is given). – Igor Rivin Oct 10 '11 at 19:32
add comment
2 Answers
active oldest votes
You will find the complete answer in the papers
Frank Raymond, Classification of the actions of the circle on 3-manifolds. Trans. Amer. Math. Soc. 131 1968 51–78.
up vote 10 down vote and
Peter Orlik and Frank Raymond, Actions of SO(2) on 3-manifolds. 1968 Proc. Conf. on Transformation Groups (New Orleans, La., 1967) pp. 297–318 Springer, New York
2 He will find the complete answer only if he finds the second paper, which is nontrivial. – Igor Rivin Oct 10 '11 at 18:30
Only nontrivial if you don't know where to look :) – José Figueroa-O'Farrill Oct 10 '11 at 19:19
Is interlibrary loan really nontrivial these days? – Richard Kent Oct 10 '11 at 19:39
You don't know the OP's situation. He might be thinking these things in the jungles of amazon (though his web page indicates that he is in Bonn, and the book is, with high
probability, in the MPI library, I admit). – Igor Rivin Oct 10 '11 at 19:42
You've got me there. – Richard Kent Oct 10 '11 at 20:12
add comment
If the action has no fixed points, that class of manifolds has a name: Seifert-fibred 3-manifold with an orientation on the fiber. Seifert classified a slightly larger class of manifolds
(Seifert-fibered ones -- no orientability constraint on the fibers). This is available in either the Jaco or Hatcher 3-manifolds lecture notes, also Orlik's book on Seifert-Fibered
up vote 8 Such examples include things like lens spaces, and relatively complicated 3-manifolds like the double of a torus knot complement.
down vote
When you allow fixed points, the orbit decomposition theorem gives you a stratification of the manifold into the fixed-point set (a link) together with its Seifert-fibred complement. So
you could view these manifolds as certain Dehn Fillings on Seifert-fibred manifolds with boundary. As the references above point out, they're abundant.
add comment
Not the answer you're looking for? Browse other questions tagged 3-manifolds dg.differential-geometry group-actions or ask your own question. | {"url":"http://mathoverflow.net/questions/77715/s1-action-in-three-dimensions","timestamp":"2014-04-21T15:59:08Z","content_type":null,"content_length":"60817","record_id":"<urn:uuid:c2d4a350-3613-48d8-9143-a8c0f680a7fa>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00217-ip-10-147-4-33.ec2.internal.warc.gz"} |
Statistics Formulas
Important Statistics Formulas
This web page presents statistics formulas described in the Stat Trek tutorials. Each formula links to a web page that explains how to use the formula.
Unless otherwise noted, these formulas assume simple random sampling.
Simple Linear Regression
Random Variables
In the following formulas, X and Y are random variables, and a and b are constants.
Sampling Distributions
Standard Error
Discrete Probability Distributions
Linear Transformations
For the following formulas, assume that Y is a linear transformation of the random variable X, defined by the equation: Y = aX + b.
Hypothesis Testing
Degrees of Freedom
The correct formula for degrees of freedom (DF) depends on the situation (the nature of the test statistic, the number of samples, underlying assumptions, etc.).
Sample Size
Below, the first two formulas find the smallest sample sizes required to achieve a fixed margin of error, using simple random sampling. The third formula assigns sample to strata, based on a
proportionate design. The fourth formula, Neyman allocation, uses stratified sampling to minimize variance, given a fixed sample size. And the last formula, optimum allocation, uses stratified
sampling to minimize variance, given a fixed budget. | {"url":"http://www.stattrek.com/statistics/formulas.aspx","timestamp":"2014-04-18T10:34:50Z","content_type":null,"content_length":"104668","record_id":"<urn:uuid:54424ce1-73de-42f8-b24c-1e648aa309f1>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00331-ip-10-147-4-33.ec2.internal.warc.gz"} |
CH Robotics
The CH Robotics content library provides an overview of some of the fundamental topics needed to use AHRS and Orientation Sensors like the UM6 and the UM6-LT.
We begin with a brief description of Euler Angles, Quaternions, and types of sensors to give the user a practical understanding of how CH Robotics sensors work and how the data that they provide can
be used.
More advanced tutorials about using CH Robotics Orientation Sensors for specific applications are also provided for users who want to dig in. We will be adding content to the library, so visit again
for updates.
The Basics
• Understanding Euler Angles -
An introduction to Euler Angles and an overview of how to interpret the angle estimates produced by the UM6 and the UM6-LT.
• Understanding Quaternions -
Learn how to take advantage of the quaternion outputs of CH Robotics sensors.
• Sensors for Orientation Estimation -
Learn about accelerometers, rate gyros, and magnetometers.
Advanced Topics
• Fundamentals of Attitude Estimation
An introduction to the basic idea behind attitude estimation. This chapter describes exactly how the different types of sensors are used to produce reliable orientation estimates.
• Using Accelerometers to Estimate Velocity and Position
Learn about the mathematics involved in estimating velocity and position using accelerometers. This chapter discusses the “gotchas” that make velocity and position estimation difficult. | {"url":"http://www.chrobotics.com/library","timestamp":"2014-04-20T08:19:12Z","content_type":null,"content_length":"13402","record_id":"<urn:uuid:74898c7b-c2d0-461a-acc2-ce1577cddd78>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00411-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Verticle Jump
1. The problem statement, all variables and given/known data
In order for a volleyball player to jump vertically upward a distance of 0.8 meters, his initial velocity must be?
2. Relevant equations
s=v[0]t + .5at^2
vf= v[0] + at
3. The attempt at a solution
I missed the question the other day on a practice and it has been forever since I messed with this material. Those equations were given as well as the answer, the only problem is I do not know what
the letters represent or even how to solve for anything with only information of 0.8 meters.
Could someone help me learn this and what the expressions mean so I can solve it? | {"url":"http://www.physicsforums.com/showpost.php?p=1999677&postcount=1","timestamp":"2014-04-18T08:30:59Z","content_type":null,"content_length":"9137","record_id":"<urn:uuid:a82154e7-b9d9-4d27-8b90-1ceb97966eea>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00529-ip-10-147-4-33.ec2.internal.warc.gz"} |
My particular emphasis is on the use of time series analysis tools to answer questions such as:
Movement ecology- How can we use wildlife location time series data, combined with habitat and climate information, to detect and disentangle multiscale relationships between behavior, sociality,
environment, ecology, and spatiotemporal patterns of population distribution?
Botany- How can we quantify and compare phenology patterns such as mast seeding within and across plant species and communities? What are the consequences of different phenology patterns to the
broader community of plant consumers?
Population ecology- What regulates and determines population abundance through time?
Statistical ecology- How can we best use time-frequency domain (e.g. wavelets) and time-domain (e.g. state-space models) in a complimentary manner to probe data variance structure and then model
relationships parametrically?
Mathematical and statistical formulations of models that appropriately capture ecological relationships and stochasticity often involve complex likelihood functions and computations. Additionally,
ecological data can be somewhat limited by sample size. As a result, my work often involves both simulation studies to evaluate how methods will perform in typically complex and noisy ecological data
with limited sample sizes, combined with analysis of empirical data. In this way I believe we can have the most confidence in moving from statistical inference of models to conclusions about biology. | {"url":"http://www.eva.mpg.de/primat/staff/leo_polansky/research.html","timestamp":"2014-04-17T09:51:59Z","content_type":null,"content_length":"4475","record_id":"<urn:uuid:407a6029-78be-4e11-a467-005f3c201936>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00414-ip-10-147-4-33.ec2.internal.warc.gz"} |
Electronics Measurement: Ohm's Law
The term Ohm's law refers to one of the fundamental relationships found in electronic circuits: that, for a given resistance, current is directly proportional to voltage. In other words, if you
increase the voltage through a circuit whose resistance is fixed, the current goes up. If you decrease the voltage, the current goes down.
Ohm's law expresses this relationship as a simple mathematical formula:
In this formula, V stands for voltage (in volts), I stands for current (in amperes), and R stands for resistance (in ohms).
Here's an example of how to calculate voltage in a circuit with a lamp powered by the two AA cells. Suppose you already know that the resistance of the lamp is 12 Ω, and the current flowing through
the lamp is 250 mA, which is the same as 0.25 A. Then, you can calculate the voltage as follows:
Ohm's law is incredibly useful because it lets you calculate an unknown voltage, current, or resistance. In short, if you know two of these three quantities you can calculate the third.
Go back (if you dare) to your high-school algebra class and remember that you can rearrange the terms in a simple formula such as Ohm's law to create other equivalent formulas. In particular:
• If you don't know the voltage, you can calculate it by multiplying the current by the resistance.
• If you don't know the current, you can calculate it by dividing the voltage by the resistance.
• If you don't know the resistance, you can calculate it by dividing the voltage by the current.
To convince yourself that these formulas work, look again at the circuit with a lamp that has 12 Ω of resistance connected to two AA batteries for a total voltage of 3 V. Then you can calculate the
current flowing through the lamp as follows:
If you know the battery voltage (3 V) and the current (250 mA, which is 0.25 A), you can calculate the resistance of the lamp like this:
Wasn't going back to high school algebra fun? Next thing you know, you're going to start looking for a prom date.
The most important thing to remember about Ohm's law is that you must always do the calculations in terms of volts, amperes, and ohms. For example, if you measure the current in milliamps (which you
usually will in electronic circuits), you must convert the milliamps to amperes by dividing by 1,000. For example, 250 mA is 0.25 A.
Here are a few other things you should keep in mind concerning Ohm's law:
• Remember that the definition of one ohm is the amount of resistance that allows one ampere of current to flow when one volt of potential is applied to it? This definition is based on Ohm's law.
If V is 1 and I is 1, then R must also be 1.
• If you wonder why the symbols for voltage and resistance are V and R, which make perfect sense, but the symbol for current is I, which makes no sense, it has to do with history.
The unit of measure for current — the ampere — is named after André-Marie Ampère, a French physicist who was one of the pioneers of early electrical science.
The French word he used to describe the strength of an electric current was intensité – in English, intensity. Thus, amperage is a measure of the intensity of the current. Hence the letter I.
• In the interest of international cooperation, the term volt is named for the Italian scientist Alessandro Volta, who invented the first electric battery in 1800. (Actually, his full name was
Count Alessandro Giuseppe Antonio Anastasio Volta.) | {"url":"http://www.dummies.com/how-to/content/electronics-measurement-ohms-law.navId-810962.html","timestamp":"2014-04-19T21:04:48Z","content_type":null,"content_length":"58095","record_id":"<urn:uuid:f9b373cd-65e2-4757-a439-7d0ef24903c7>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00616-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cracking the language code: NAPLAN numeracy tests in
years 7 and 9.
Background of NAPLAN tests
NAPLAN is an acronym for National Assessment Program: Literacy and Numeracy which came into being nationally in Australia in 2008. Year 3, 5, 7 and 9 students have taken NAPLAN tests every year since
2008. Test results give information about how students are performing in Literacy and Numeracy and are used to inform parents/carers of student's progress and to provide information to schools to
help rectify literacy and numeracy problems in different stages of schooling.
Australian Curriculum Assessment and Reporting Authority (ACARA) manage the national testing and Queensland Studies Authority (QSA) administers, marks and reports on the testing in Queensland.
Previous test papers, sample questions, a summary of previous results and other relevant information can be found at the following website: http://www. naplan.edu.au/tests/tests_landing_page.html
In this article the language used in Year 7 and 9 NAPLAN numeracy tests will be investigated. In both Years 7 and 9 students complete two numeracy papers of 32 questions with a time limit of 40
minutes. In one paper they may use a calculator and in the other calculator use is not permitted.
Referring to previous NAPLAN numeracy tests it is evident that every question demands an understanding of everyday language and mathematical language which includes specific mathematics terminology
and the concise use of vocabulary as well as symbols, graphs and other representations of mathematical operations and concepts.
Focusing on the use of language is a crucial strategy in good mathematics teaching. Issues of language in the Year 7 and 9 NAPLAN numeracy tests will be discussed with reference to previous research
in the teaching and learning of mathematical language.
Language use in NAPLAN Numeracy Tests
'Language is a medium through which students learn mathematics' (Leach & Bowling, 2000, p. 24) and a large amount of the literature has focused on language leading to problems with the learning of
Many authors have discussed the difficulties of words that have multiple meanings (Pierce & Fontaine, 2009; Saxe, 1988). For instance, words used in English often have a different meaning in
mathematics. Teachers need to 'recognise and make explicit the difference between 'mathematical' English and 'natural' English' (Dawe & Mulligan, 1997 cited in Frigo, 1999, p. 15). Examples of such
words from the Year 7 and 9 tests are mean, grid, rate, volume, balance, scale, key, face, head, tail, capacity, mode, range, die, positive, product, expression, prime, regular, right and rule. Even
within mathematics words such as scale, cube and square have more than one meaning. In some cases, the same word functions as a different part of speech, for instance square can be a noun, verb or
Often many different words can be used to describe the same concept in mathematics (Spanos, Rhodes, Dale & Crandall, 1988). For instance subtraction, minus, take-away and difference are used for the
same concept but the word difference is most often used in the NAPLAN testing. It should also be noted that the word chance in lower years becomes increasingly replaced by the word probability and
the word average becomes mean in later years.
Knowledge of the meanings of prefixes like bi- and di- both meaning two, tri- meaning three, quad-meaning four are also important. This can be confusing because often two prefixes indicate the same
number for instance both sex- and hex- represent six (sex- originating from Latin and hex- from Greek). Examples of words using hex- and sex- are hexagon, hexagonal and sextant, sextet, sextuplet.
Bi- and di- are found in binary, bisect, binomial, bias, biannual, bilateral, bilingual, bicycle, bipolar and diagonal, diameter, divide, dialogue. Other important prefixes are circum- meaning around
tetra- meaning four, pent(a)- meaning five, kilomeaning thousand, mega- meaning million and micro- meaning a millionth. More prefixes are shown in the table below, many of which are utilised in the
NAPLAN numeracy tests.
The word altogether is used in a number of questions. Altogether is often taken to imply addition. However, this is not always the case in these questions. In the same way, the word total does not
necessarily imply addition. This indicates the pitfalls of teaching that a particular word always implies a specific mathematics operation.
Mathematical text is lexically dense which means that it contains a minimum of redundant words, that is, contextual clues (NSW Department of School Education, 1997). Students' attention needs to be
drawn to dense phrases which contain multiple concepts which can pose difficulties to students for example reflex angle, closest to, possible outcomes, exactly halfway, number sentence, per person,
satisfies equations, best estimate, number line, equal length, regular hexagon, percentage decrease, square based pyramid, positive whole number, satisfy the inequality, average daily saving, stem
and leaf plot, standard six sided die, product of prime factors, right angled isosceles triangle, four consecutive whole numbers, three quarter turn clockwise, sum of dots on opposite faces.
Symbols in mathematics can also pose problems for students. In the NAPLAN numeracy tests for Year 7 and 9 many symbols or words for symbols or combinations of words and symbols are used. Examples of
the included symbols in one or both Year 7 and 9 are: the four operations (+, -, +, x), am and pm, symbols for units such as mL, kg, mg, m, km, cm, mm, [m.sup.2], [cm.sup.2], [degrees]C, symbols such
as $, <, [check], [10.sup.3] and %,--and [degrees] for percentage, negative and degrees, 3D for three dimensional, N, S, E, W for the four directions, x, h for variables, fractions written as numbers
and grid references such as A3. A combination of words and symbols is seen in 6-sided, 6 metres, 3 times, litres per 100 km and words are used when writing cubic metres, minutes, metres per minute,
cents. Square centimetres, litres, grams, kilograms, dollars and grams and numbers such as seventy-five are written as either words or symbols. Large numbers are often written in prose form.
In addition to words and symbols causing misunderstandings, mathematics also introduces graphs, diagrams and other representations which add to the complexity (Lowrie & Diezmann, 2009). Lowrie and
Diezmann (2009) pointed out that students may have problems interpreting graphics in word questions and they stated, 'Students' performance may thus be a measure of their ability to comprehend the
graphical (or linguistic) components of a task rather than their knowledge of the mathematics within the task' (Lowrie & Diezmann, 2009, p. 146). Students need practice interpreting information in
the many different visuals in NAPLAN tests and they need to examine these representations carefully for help with meaning. However, in many cases the words and visual image duplicate each other and
it is possible to answer the question without understanding every aspect of the question,
Common themes in the NAPLAN tests are shopping involving prices (costs), sale prices, percentage increases, travelling questions involving maps, distances, speeds, petrol consumptions and directions,
areas and perimeters of land (paddocks), recipes involving masses, volumes, capacities and ratios, questions based on sport/game scores giving rise to scores, means, medians and so on and probability
questions based on marbles or lollies such as jelly beans, questions and movies, timetables, ticket sales, temperatures, surface areas, population figures. Teachers need to expose students to word
questions, including reading the questions, especially on the above topics.
The NAPLAN tests allow just over a minute per question which does not seem long for students who are struggling to understand and complete the questions in the allocated time. However, special
provisions such as readers, scribes, use of dictionaries and extra time are available for some students for instance ESL and special needs students if applications are forwarded to the relevant
bodies. Current information on this can be found in the Test preparation handbook on the NAPLAN website.
Students should be introduced to resources such as dice, coins (emphasising the head and the tail), spinners, money, maps, recipes, games, calendars and clocks as they are referred to in the test
As expected, more context-based questions appear in the Year 9 than the Year 7 tests. Examples in the Year 7 and 9 NAPLAN questions refer to recipes, floodlights, smudges, conveyor belts, satellite
dishes, air-conditioners, paddocks, planks, rockets, aeroplane seating, exchange rates, films, sticks, watches, alarm clocks, DVD players, leaflets and the summit of a mountain all of which are
probably unfamiliar to some students especially rural and LBOTE students. Some of the vocabulary and several contexts may be unfamiliar to certain groups of students, raising equity questions. This
issue has been highlighted by many researchers such as Zevenbergen, Dole and Wright (2004). Although the NAPLAN tests have been attractively presented, they have not been designed to be inclusive of
all Australians. Questions which contain familiar content and are attractive to indigenous peoples should be included in each paper.
Numeracy is more than straightforward mathematical computation and includes the interpretation of information presented in printed and other forms. It is reasonable that NAPLAN numeracy tests present
students with age-appropriate challenges in interpreting information in print form. However, Abedi argued that
There is a difference between language that is an essential part of the content of the question and language that makes the question incomprehensible to many students ... While it is important to
understand and value the richness of language in an assessment system; it is also important to make sure that ... students ... not be penalised for their lack of English proficiency in areas where
the target of assessment is not language. Though we understand the views of some language modification critics in not 'dumbing down' assessment questions by simplifying the language, we also
recognise the distinction between necessary and unnecessary linguistic complexity. (Abedi, 2009, p. 173)
NAPLAN tests from 2008, 2009 and 2010 have similar layouts, questions and use of language. The NAPLAN test papers and some additional trial questions are freely available on the web making it
possible for students to practice on previous tests.
Teachers need to ensure that students become familiar with the necessary mathematical language. This article should not be seen as implying that teachers should aim to teach primarily for test
purposes, but rather that vocabulary is an important part of mathematics teaching and learning. In terms of the NAPLAN testing, it is important that students have a sound knowledge of mathematics
language in order to feel well prepared for the testing. This must occur in mathematics lessons--English teachers cannot be expected to teach the nuances of mathematical language. This means that
mathematics teachers must also become teachers of language and literacy.
Teachers' guidance can assist students to master the language of mathematics. And as Murray, 2009 stated, 'The language, and progressively the more specialist language, is necessary for learning and
even more, understanding mathematics' (Murray, 2009, p. 6).
Abedi, J. (2009). Validity of assessments for English language learning students in a national/ international context. Estudiossobre Education, 16, 167-183.
Frigo, T. (1999). Resources and teaching strategies to support Aboriginal children's numeracy learning. A review of the literature. Retrieved October 18, 2010, from http://research.acer.edu.au/
Leach, S. & Bowling, J. (2000). A classroom research project : ESL students and the language of mathematics. Australian Primary Mathematics Classroom, 5(1), 24-27.
Lowrie, T. & Diezmann, C.M. (2009). National numeracy tests: A graphic tells a thousand words. Australian Journal of Education, 53 (2), 141-158.
Ministerial Council for Education Early Childhood Development and Youth Affairs (MCEECDYA). (2010). National assessment program literacy and numeracy. Viewed 10 March, 2010, at http://
Murray, J. (2009). Two heads are better than one. Retrieved October 18, 2010, from http://nrich. maths.org/public/viewer.php?obj_id=6383
NSW Department of School Education. (1997). Teaching literacy in mathematics in Tear 7. Sydney: Author.
Pierce, M.E. & Fontaine, L.M. (2009). Designing vocabulary instruction in mathematics. Reading Teacher, 63(3), 239-243.
Saxe, G.B. (1988). Linking language with mathematics achievement. In R.R. Cocking & J.P. Mestre (Eds.), Linguistic and cultural influences on learning mathematics. Hillsdale, NJ: Lawrence Erlbaum
Spanos, G., Rhodes, N.C., Dale, T.C. & Crandall, J. (1988). Linguistic features of mathematical problem solving: Insights and applications. In R.R. Cocking & J.P. Mestre (Eds.), Linguistic and
cultural influences on learning mathematics (pp. 221-240). Hillsdale NJ: Lawrence Erlbaum Associates.
Zevenbergen, R., Dole, S. & Wright, R. (2004). Teaching mathematics in primary schools. Crows Nest, NSW: Allen and Unwin.
Table 1. More examples of prefixes
Prefix and meaning Examples of words which include the
mon(o)- and uni- meaning one or monolith , monorail , monotony ,
single monocle, monocycle, monologue,
monotreme, unit, uniform,
unification, unicorn, unique
tri- meaning three triangle, trillion, triomino,
trinomial, third, tripod, triplets,
tricycle, tricolour, triple, triad
quadr- or quart- meaning four quadrilateral, quadrillion,
quarter; quartile, quadrant,
quadratic, quartet, quart,
quadbike, quadruplets, quadrangle
oct- meaning eight octagon, octopus, octane, octopod
dec(a) or dek(a)- meaning ten decade, decagon, decillion
deci- meaning one tenth decimal system, decimetre,
cent- or centi- meaning hundred cents, centimetre, century
or hundredth centenary, centennial, centipede
midcentury, percentage, percent,
milli- meaning thousand or millimetre, millennium
de- meaning taking something decrease, deduce, descend, decline
away the opposite decentralisation , deforestation
poly- meaning many polygon(2D), polyhedron (3D) ,
polymer polyomino, polynomial,
Teachers should encourage students to take note of common roots in
words, for example, rectangle and rectangular, cylinder and
cylindrical, parallel and parallelogram, clock and clockwise or
anticlockwise, reflect and reflection, half and halfway and division
and divisible or divisor. | {"url":"http://www.freepatentsonline.com/article/Literacy-Learning-Middle-Years/249223413.html","timestamp":"2014-04-20T00:40:43Z","content_type":null,"content_length":"35743","record_id":"<urn:uuid:bea7f263-a37c-4ac2-a0ed-8ca6b8e65906>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00328-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ribet, Kenneth A. - Department of Mathematics, University of California at Berkeley
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• ``Pokerhontas drops the King of ---, the King of ~ and the Ace of into a large bag. She shakes the bag vigorously and then removes two of the cards without looking at them.
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• (x1.6, #15) Suppose that g is a function from A to B and f is a function from B to C. a) Show that if both f and g are onetoone functions, then f ffi g is also onetoone.
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Dragan Stevanovic and Vladimir Brankov
Prirodno-matematicki fakultet, 18000 Nis, Serbia and Matematicki institut SANU, Kneza Mihaila 35, 11000 Beograd, Serbia
Abstract: A vertex of a simple graph is called large if its degree is at least 3. It was shown recently that in the class of starlike trees, which have one large vertex, there are no pairs of
cospectral trees. However, already in the classes of trees with two or three large vertices there exist pairs of cospectral trees. Thus, one needs to employ additional graph invariant in order to
characterize such trees. Here we show that trees with two or three large vertices are characterized by their eigenvalues and angles.
Classification (MSC2000): 05C50
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Electronic fulltext finalized on: 10 Oct 2006. This page was last modified: 27 Oct 2006.
© 2006 Mathematical Institute of the Serbian Academy of Science and Arts
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Help with differential systems dealing with linearity and linear systems
February 13th 2011, 11:40 AM #1
Junior Member
Feb 2011
Help with differential systems dealing with linearity and linear systems
1) Verify the linearity principle for linear, nonautonomous systems of differential equations?
2) Consider all linear systems with two zero eigenvalues. which of these systems are conjugate? prove this.
3) Prove that a linear map T : R2 -> R2 is a homeomorphism if and only if it is nondegenerate?
I'm trying to figure these questions out for practice for an upcoming midterm. I don't know where to start or how to start any of them
I know that nondegenerate just means that the det =\= 0 and that the linearity principle is "if X' = AX is a planar linear system for which Y_1(t) and Y_2(t) are both solutions, then the function
aY_1(t)+bY_2(t) is also a solution to this system."
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Results 1 - 10 of 62
- Appl. Comp. Harm. Anal
"... Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical
analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery probl ..."
Cited by 142 (14 self)
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Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical
analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery problem. We show that the algorithm has the following properties (made more precise in the main
text of the paper) • It gives near-optimal error guarantees. • It is robust to observation noise. • It succeeds with a minimum number of observations. • It can be used with any sampling operator for
which the operator and its adjoint can be computed. • The memory requirement is linear in the problem size. Preprint submitted to Elsevier 28 January 2009 • Its computational complexity per iteration
is of the same order as the application of the measurement operator or its adjoint. • It requires a fixed number of iterations depending only on the logarithm of a form of signal to noise ratio of
the signal. • Its performance guarantees are uniform in that they only depend on properties of the sampling operator and signal sparsity.
- SIAM J. Imaging Sci , 2008
"... Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman
iterative regularization, and they give a very accurate solution after solving only a very small number o ..."
Cited by 59 (13 self)
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Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman
iterative regularization, and they give a very accurate solution after solving only a very small number of 1 instances of the unconstrained problem minu∈Rn μ‖u‖1 + 2 ‖Au−fk ‖ 2 2 for given matrix A
and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six
iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving A and A ⊤ can be computed by fast
transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of
compressed sensing problems on a standard PC.
"... We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and of the
Compressive Sampling Matching Pursuit or Subspace Pursuit algorithms, is called Hard Thresholding Pursuit. ..."
Cited by 22 (0 self)
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We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and of the
Compressive Sampling Matching Pursuit or Subspace Pursuit algorithms, is called Hard Thresholding Pursuit. We study its general convergence, and notice in particular that only a finite number of
iterations are required. We then show that, under a certain condition on the restricted isometry constant of the matrix of the system, the Hard Thresholding Pursuit algorithm indeed finds all
s-sparse solutions. This condition, which reads δ3s < 1 / √ 3, is heuristically better than the sufficient conditions currently available for other Compressive Sensing algorithms. It applies to fast
versions of the algorithm, too, including the Iterative Hard Thresholding algorithm. Stability with respect to sparsity defect and robustness with respect to measurement error are also guaranteed
under the condition δ3s < 1 / √ 3. We conclude with some numerical experiments to demonstrate the good empirical performance and the low complexity of the Hard Thresholding Pursuit algorithm.
- ArXiv
"... Abstract—This paper revisits the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This
problem is an extension of single channel sparse recovery, which lies at the heart of compressed sensing. Ins ..."
Cited by 15 (4 self)
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Abstract—This paper revisits the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem
is an extension of single channel sparse recovery, which lies at the heart of compressed sensing. Inspired by the links to array signal processing, a new family of MMV algorithms is considered that
highlight the role of rank in determining the difficulty of the MMV recovery problem. The simplest such method is a discrete version of MUSIC which is guaranteed to recover the sparse vectors in the
full rank MMV setting, under mild conditions. This idea is extended to a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case
while also providing guaranteed recovery in the full rank setting. In contrast, popular MMV methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques are
shown to be rank blind in terms of worst case analysis. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple
measurements. Index Terms—Compressed sensing, multiple measurement vectors, rank, sparse representations. I.
"... © 2011 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for
resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other w ..."
Cited by 9 (4 self)
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© 2011 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale
or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Abstract—Formulated as a least square problem under an 0
constraint, sparse signal restoration is a discrete optimization problem, known to be NP complete. Classical algorithms include, by increasing cost and efficiency, matching pursuit (MP), orthogonal
matching pursuit (OMP), orthogonal least squares (OLS), stepwise regression algorithms and the exhaustive search. We revisit the single most likely replacement (SMLR) algorithm, developed in the
mid-1980s for Bernoulli–Gaussian signal restoration. We show that the formulation of sparse signal restoration as a limit case of Bernoulli–Gaussian signal restoration leads to an 0-penalized least
square minimization problem, to which SMLR can be straightforwardly adapted. The resulting algorithm, called single best replacement (SBR), can be interpreted as a forward–backward extension of OLS
sharing similarities with stepwise regression algorithms. Some structural properties of SBR are put forward. A fast and stable implementation is proposed. The approach is illustrated on two inverse
problems involving highly correlated dictionaries. We show that SBR is very competitive with popular sparse algorithms in terms of tradeoff between accuracy and computation time. Index
Terms—Bernoulli-Gaussian (BG) signal restoration, inverse problems, mixed 2- 0 criterion minimization, orthogonal least squares, SMLR algorithm, sparse signal estimation, stepwise regression
algorithms. I.
"... Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing
allows to recover this signal from much fewer samples than the Shannon-Nyquist theory requires. Many images ..."
Cited by 8 (3 self)
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Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows
to recover this signal from much fewer samples than the Shannon-Nyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms
and others. Generally, wavelets represent point-like features while curvelets represent line-like features well. For a suitable recovery of images, we propose models that contain weighted sparsity
constraints in two different frames. Given the incomplete measurements f = Φu + ɛ with the measurement matrix Φ ∈ R K×N, K<<N, we consider a jointly sparsity-constrained optimization problem of the
form argmin{‖ΛcΨcu‖1 + ‖ΛwΨwu‖1 + u 1 2‖f − Φu‖22}. Here Ψcand Ψw are the transform matrices corresponding to the two frames, and the diagonal matrices Λc, Λw contain the weights for the frame
coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved
by showing that they can be understood as special cases of the Douglas-Rachford Split algorithm. Numerical experiments for compressed sensing based Fourier-domain random imaging show good
performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
- In Proc. IEEE International Symposium on Biomedical Imaging: from Nano to Macro. IEEE , 2008
"... Visualization and analysis of the micro-architecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment
of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, ..."
Cited by 7 (0 self)
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Visualization and analysis of the micro-architecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of
various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, the conventional diffusion tensor imaging (DTI) used for estimating the local orientations of
neural fibers, is incapable of performing reliably in the situations when a voxel of interest accommodates multiple fiber tracts. In this case, a much more accurate analysis is possible using the
high angular resolution diffusion imaging (HARDI) that represents local diffusion by its apparent coefficients measured as a discrete function of spatial orientations. In this note, a novel approach
to enhancing and modeling the HARDI signals using multiresolution bases of spherical ridgelets is presented. In addition to its desirable properties of being adaptive, sparsifying, and efficiently
computable, the proposed modeling leads to analytical computation of the orientation distribution functions associated with the measured diffusion, thereby providing a fast and robust analytical
solution for q-ball imaging.
- IEEE Trans Signal Processing , 2011
"... Abstract—We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although
these constituents are drawn from a continuous family, most current signal decomposition methods rely on a �ni ..."
Cited by 6 (0 self)
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Abstract—We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these
constituents are drawn from a continuous family, most current signal decomposition methods rely on a �nite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of
basic waveforms), and apply sparse optimization methods to select and solve for the relevant coef�cients. Here, we generate a dictionary that includes auxiliary interpolation functions that
approximate translates of features via adjustment of their coef�cients. We formulate a constrained convex optimization problem, in which the full set of dictionary coef�cients represents a linear
approximation of the signal, the auxiliary coef�cients are constrained so as to only represent translated features, and sparsity is imposed on the primary coef�cients using an L1 penalty. The basis
pursuit denoising | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=4754176","timestamp":"2014-04-23T12:27:12Z","content_type":null,"content_length":"39736","record_id":"<urn:uuid:089f5e5d-cd66-4bc0-97be-5174c5ec7f2e>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00079-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fundamentals of Hydraulic Engineering Systems 4th Edition Chapter 8.9 Solutions | Chegg.com
7162-8.9-1P AID: 1825 | 10/05/2013
Find the possible head loss
Denote height above the culvert invert as H, bottom slope of the culvert as L, and diameter of culvert as D.
Substitute the values of 3.2 m for H, 0.003 for L, and 1.25 m for D in equation (1) to get
Find the design head loss using another formula
Denote entrance coefficient as n, hydraulic radius as g, and discharge as Q.
Substitute the values of 0.2 for n, 40 m for L, 1.25 m for D, Q, and g to get
Design head loss required is higher than possible head loss.
The smaller diameter culvert will not satisfy the requirement.
The slope
Substitute the values of 3.2 m for H, 0.01 for L, and 1.25 m for D in equation (1) to get
Design head loss required is higher than possible head loss.
The smaller diameter culvert with increased slope will not satisfy the requirement. | {"url":"http://www.chegg.com/homework-help/fundamentals-of-hydraulic-engineering-systems-4th-edition-chapter-8.9-solutions-9780136016380","timestamp":"2014-04-16T08:48:50Z","content_type":null,"content_length":"28265","record_id":"<urn:uuid:2ff99f4d-a606-435f-a22d-7432706aa894>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00514-ip-10-147-4-33.ec2.internal.warc.gz"} |
What to plot as the errors for a moving average plot of temperature?
Well, all I want is the value of the error, i.e. [itex] E(t_i) = E_{moth}(t_i) + E_{meas}(t_i) [/itex] . In that case, can I simply add the standard deviation (which I am taking to be the error
caused by mother nature, i.e. [itex] E_{moth}(t_i) [/itex]) to the measurement error, [itex] E_{meas}(t_i) [/itex] ?
OK, but the standard deviation of the error [itex] E(t_i) [/itex] isn't a direct indicator of how the moving average curve itself would vary if you could run the experiment over and over again under
the same conditions.
You have a situation where it's almost irresistible to do certain arithmetic and this urge would drive most people to make the following assumptions.
1. Assume the [itex] E_{moth}(t_i) [/itex] each have mean zero and are independent and identically distributed random variables for all [itex] t_i [/itex].
2. Assume [itex] E_{meas}(t_i) [/itex] each have mean zero and are independent and identically distributed random variables for all [itex] t_i [/itex].
and we may have to make more assumptions , but the general idea is to use the deviations we can measure [itex] D_m(t_i) = T_m(t_i) - A_m(t_i) [/itex] and argue that the variance of these deviations
is approximately the variance of [itex] D(t_i) = T_m(t_i) - A(t_i) [/itex].
Compute [itex] D_m [/itex] for all [itex] t_i [/itex] and consider this one big set of data for the same random variable [itex] D_m [/itex]. Find the variance of [itex] D_m [/itex].
It is the variances of sums of independent random variables that can be computed as sums (not their standard deviations). So [itex] Var(D_m) = Var(D) = Var(E_{moth}) + Var(E_{meas}) [/itex].
Then you have the problem of translating the measurement device specifications into a numerical value for [itex] Var(E_{meas}) [/itex].
If you can do that, you can solve for [itex] Var(E_{moth}) [/itex].
A person with very serious intentions would check things like whether [itex] D_m [/itex] really appears to have mean zero and whether [itex] D_m(t_i) [/itex] really is identically distributed for all
times [itex] t_i [/itex]. For example, do the devations tend to be positive when temperatures are falling?
My aim is to see which mirror surfaces are more likely to experience condensation on cold, clear nights.
That's a clear goal. It focuses attention on a certain range of temperatures. I assume your data is for temperatures in that range. But is the question of interest a yes-or-no question? Or is there a
more-or-less aspect to the question of condensation?
trying to tweak my model until it simulates the observations. So far I have a model which allows me to vary emissivity, thermal conductivity, mass and area of the mirrors
According to physical laws, should the temperature-vs-environment behavior of the mirror change once some condensation does form on it? Is the formation of "some" condensation something that promotes
even more condensation at the same temperature? | {"url":"http://www.physicsforums.com/showthread.php?p=3763733","timestamp":"2014-04-18T03:13:38Z","content_type":null,"content_length":"51159","record_id":"<urn:uuid:14e09d0e-28e2-43e0-8718-c4d595dd0fbb>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hunting Down the Old References
Hunting Down the Old References
While writing the research papers one quite often needs to get back to the full texts of old (pre-Internet or at least pre-arXiv) references. Of course, having access to a good library and/or the
interlibrary loan usually solves the problem but can be somewhat time- and cost-consuming.
It is not that well known, however, that there is a fair chance to find the old paper or preprint you need online for free. Of course, the first thing to try is Google or perhaps another search
engine of your choosing. However, if this does not work, you still have a fighting chance, at least as far physics and mathematics are concerned. The places to try are:
All items but KISS are purely mathematical databases (to be precise, MathNet.Ru includes several physics, mechanics and mathematical physics journals as well).
If you know of other similar databases (be it in physics, mathematics, life sciences,…), please feel free to drop a comment with the relevant link(s).
4 Responses to Hunting Down the Old References
1. I can’t think of any other databases than what you pasted but there is one more thing that has helped me in the past. Which is to go to the homepage of the author(s) of the paper and see whether
they have posted a copy of the paper there (often scanned if the paper is old). In my experience, the chances of success are higher if the author(s) happens to be a well-known/important figure in
the field :)
2. Your suggestion about going for the homepages is certainly correct but Google (and Google Scholar) is quite good in finding the papers available from such pages, so this is often unnecessary.
3. I have some other relevant links on my math journal resource page.
4. Sorry, I forgot the link: http://homotopical.wordpress.com/links/journals/ | {"url":"http://aclinks.wordpress.com/2009/06/05/hunting-down-the-old-references/","timestamp":"2014-04-19T09:24:35Z","content_type":null,"content_length":"67474","record_id":"<urn:uuid:44fa5cfb-baee-4e90-8499-f9f1b3aef84a>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00142-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] Question about bounds checking
Rich E rjel@ceh.ac...
Mon Aug 10 10:08:27 CDT 2009
Dear all,
I am having a few issues with indexing in numpy and wondered if you could help
me out.
If I define an array
a = zeros(( 4))
array([ 0., 0., 0., 0.])
Then I try and reference a point beyond the bounds of the array
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
IndexError: index out of bounds
but if I use the slicing format to reference the point I get
array([ 0., 0., 0., 0.])
array([ 0., 0., 0., 0.])
it returns a[ 0 : 3 ], with no error raised. If I then ask for the shape of the
array, I get
but if I use
which is one less than I would have expected.
This is numpy 1.2.1 on python 2.5
Thanks in advance for you help,
More information about the NumPy-Discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2009-August/044551.html","timestamp":"2014-04-16T04:26:15Z","content_type":null,"content_length":"3272","record_id":"<urn:uuid:0c0cdb1c-1cc3-4940-acc7-83b0ca25b1b4>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00402-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to check if a number is a palindrome or not in Java - Example
How to check if a number is a palindrome or not is a variant of popular String interview question how to check if a String is a palindrome or not. A number is said to be a palindrome if number itself
is equal to reverse of number e.g. 313 is a palindrome because reverse of this number is also 313. On the other hand 123 is not a palindrome because reverse of 123 is 321 which is not equal to 123,
i.e. original number. In order to check if a number is a palindrome or not we can reuse the logic of How to reverse number in Java. Since in most of interview, you are supposed to solve this question
without taking help from API i.e. only using basic programming construct e.g. loop, conditional statement, variables, operators and logic. I have also seen programmer solving this question by first
converting integer to String and than reversing String using reverse() method of StringBuffer and than converting String back to Integer, which is not a correct way because you are using Java API.
Some programmer may think that this is just a trivial programming exercise but it’s not. Questions like this or Fibonacci series using recursion can easily separate programmers who can code and who
can’t. So it’s always in best interest to keep doing programing exercise and developing logic.
Java program to check if number is palindrome or not
Here is a simple Java program which finds if a number is a palindrome or not. This program does not use any API method instead it uses division and remainder operator of Java programming language to
determine if number is palindrome or not. Programming logic to reverse a number is encapsulate in reverse() method and isPalindrome(int number) reuse that logic to test if a number is palindrome or
import java.util.Scanner;
* This Java program takes an input number from command line and integer array
* and check if number is palindrome or not. A number is called palindrome
* if number is equal to reverse of number itself.
* @author Javin Paul
public class PalindromeTest {
public static void main(String args[]){
Scanner scanner = new Scanner(System.in);
//int number = scanner.nextInt();
int[] numbers = {1, 20, 22, 102, 101, 1221, 13321, 13331, 0, 11};
for(int number: numbers){
System.out.println("Does number : "
+ number +" is a palindrome? " + isPalindrome(number));
private static boolean isPalindrome(int number) {
if(number == reverse(number)){
return true;
return false;
private static int reverse(int number){
int reverse = 0;
while(number != 0){
reverse = reverse*10 + number%10;
number = number/10;
return reverse;
Does number : 1 is a palindrome? true
Does number : 20 is a palindrome? false
Does number : 22 is a palindrome? true
Does number : 102 is a palindrome? false
Does number : 101 is a palindrome? true
Does number : 1221 is a palindrome? true
Does number : 13321 is a palindrome? false
Does number : 13331 is a palindrome? true
Does number : 0 is a palindrome? true
Does number : 11 is a palindrome? true
That's all on how to check if a number is a palindrome or not. As I said this is a good programming exercise particularly for beginners who have just started learning Java programming language, as it
teaches how to use division and remainder operator in Java. Once again Fibonacci series, Palindrome are classical coding question and should not be missed during preparation.
Other Java programming exercise for practice
No comments : | {"url":"http://javarevisited.blogspot.sg/2012/12/how-to-check-if-number-is-palindrome-or-not-example.html","timestamp":"2014-04-19T11:57:42Z","content_type":null,"content_length":"117395","record_id":"<urn:uuid:7f7176b3-683d-4759-ab3d-714fa88fe52d>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00333-ip-10-147-4-33.ec2.internal.warc.gz"} |
Determining distances through the moving cluster method
There is another method that can help confirm the results of the parallax method. (This method is not included in our book, but it is quite ingenious.) It works for stars that are further away if
they are in a cluster. It is necessary to suppose that the stars are all moving together. This is is based on the idea that the stars in the cluster all came from the same gas cloud.
This method has been successfully applied to (just) one cluster, the Hyades (the head of Taurus the Bull), which is found to be 40 pc away.
To understand the method, we need to proceed step by step. We consider first one star in the cluster. Let d be the distance to the star in pc. We don't know d. We want to know it.
From photographs taken, say, 10 years apart, we can see that the star has moved. (It has proper motion.) Let us suppose that the proper motion is 3 arc sec in ten years.
We can conclude that the star has moved 3 times d astronomical units across our line of sight. (But we don't know d).
From measurments of the doppler shift of light from our star, we can determine how fast it is moving toward us or away from us. Let us suppose that the speed is 2 AU per year away from us, so that in
the ten years it has moved 20 AU further away:
Now if we knew the direction in which the star is moving, we could determine d.
How can we determine the direction of motion? Here is where the cluster comes in. Our star is in a cluster and they are all moving in the same direction. (That is, we have to assume that they are all
moving in the same direction.).
If there were some way that we could determine the direction in which the cluster is moving, we would be done!
The art of finding the direction of the cluster
Here is a famous surrealistic painting by Magritte. It may be surrealistic, but part of it is realistic. That has to do with how parallel lines in three dimensional space look when represented on a
two dimensional canvas. Here is what Magritte did.
All artists know this, and astronomers know it too. From our two photographs of the star cluster taken 10 years apart, we can plot the directions that the stars are moving as seen on the ``canvas''
of the sky.
Since the paths of the stars are parallel lines, they seem to converge on a vanishing point. The direction from Earth to the vanishing point is the direction of motion of the cluster.
We use the direction of motion that we now know to determine the distance to the stars in the cluster.
Astronomers use trigonometry, but you could just make a scale drawing to determine the length of the purple line in AU. Dividing this length by 3 in our example gives the distance to the cluster in
Updated 22 Octobber 2007
Davison E. Soper, Institute of Theoretical Science, University of Oregon, Eugene OR 97403 USA | {"url":"http://zebu.uoregon.edu/~soper/Stars/movingcluster.html","timestamp":"2014-04-16T10:46:08Z","content_type":null,"content_length":"3512","record_id":"<urn:uuid:7f00a1a8-a251-48f7-9f6f-c9bda6985518>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00202-ip-10-147-4-33.ec2.internal.warc.gz"} |
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st: Interpreting 3 way dummy interaction with margins
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st: Interpreting 3 way dummy interaction with margins
From Lauren Beresford <lberesfo@hotmail.com>
To Statalist <statalist@hsphsun2.harvard.edu>
Subject st: Interpreting 3 way dummy interaction with margins
Date Mon, 6 Feb 2012 17:16:37 +0000
Dear statlisters,
I am estimating logistic regression models (with svy command) with three way interactions between binary ivs.
My final model without controls (for simplicity...) is:svy:logit mgr i.female i.baplus i.firm1024 i.firm2599 i.firm100499 i.firm500999 i.firm1000 female##firm1024##baplus female##firm2599##baplus female##firm100499##baplus female##firm500999 ##baplus female##firm1000##baplusTo interpret the 3-way interaction I want to calculate the predicted probability of mgr for men and women, treating men as men and women as women (female=0 and1), with and without degrees higher than a BA (baplus=1 and 0) for each category of firm size (e.g., firm1024 - as indicated by the series of binary predictors for firm size). I then want to calculate the difference that going from baplus=0 to baplus=1 makes for men and then for women respectively for each category of firm size. I also want to calculate the difference that going from female=0 to female=1 makes across educational categories (baplus=0 and baplus=1). Finally I want to test that these differences are statistically significant.
I use the margins command as follows:
margins baplus, over(female) at(firm1024==1) vce(unconditional) post
I then calculate the difference that education makes for men as a group and then women as a group using lincom commands, and test the differences in these effects using a lincom command. I do this for each firm size category.
My problem is that I have sifted through extensive documentation and I am confused about which margins command to use.
If I follow the logic put forth in Maarten Buis' Stata tip 87 "Interpretation of interactions in non-linear models" my margins command would look like this:
margins, over(female firm1024 baplus) expression(exp(xb())) vce(unconditional) postI assume if I wanted predicted probabilities I would just omit expression(exp(xb())).
If I try to emulate the suggestions put forth by ATS UCLA in "How can I understand categorical by categorical interaction in logistic regression? (Stata 11) this leads me to the following commands:margins baplus, at(female=(0 1) firm1024==1) vce(unconditional) postmargins female, dydx(baplus) at(firm1024==1) vce(unconditional) postmargins baplus, dydx(female) at(firm1024==1) vce(unconditional) post
Then again, looking at ATS UCLA "How can I use margins to understand categorical by categorical interactions" I am lead to believe I want something like this:
margins female#firm1024, dydx(baplus) vce(unconditional) postFinally, from the statslist archive I found the following command (msg01503):
margins, over(female firm1024) dydx(baplus) at female=(0 1) firm1024=(0 1)) vce(unconditional) post
Therefore, I am looking for direction as to which margins command is the most accurate to decompose my three way interaction terms. I appreciate any help.
Kind regards,Lauren Beresford
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American Mathematical Society
Bulletin Notices
AMS Sectional Meeting Program by Day
Current as of Tuesday, April 12, 2005 15:09:59
Program | Deadlines | Registration/Housing/Etc. | Inquiries: meet@ams.org
1999 Spring Eastern Sectional Meeting
Buffalo, NY, April 24-25, 1999
Meeting #943
Associate secretaries:
Lesley M Sibner
, AMS
Sunday April 25, 1999
• Sunday April 25, 1999, 8:00 a.m.-12:00 p.m.
Exhibit and Book Sale
Rotunda, Diefendorf Hall
• Sunday April 25, 1999, 8:00 a.m.-12:00 p.m.
Meeting Registration
Room 114, Diefendorf Hall
• Sunday April 25, 1999, 8:00 a.m.-10:50 a.m.
Special Session on Knot and 3-Manifolds, IV
Room 2, Diefendorf Hall
Thang T.Q. Le, State University of New York at Buffalo letu@math.buffalo.edu
William W. Menasco, SUNY at Buffalo menasco@tait.math.buffalo.edu
Morwen B. Thistlethwaite, University of Tennessee morwen@math.utk.edu
• Sunday April 25, 1999, 8:00 a.m.-10:50 a.m.
Special Session on Representations of Lie Algebras, III
Room 8, Diefendorf Hall
Duncan J. Melville, Saint Lawrence University dmel@music.stlawu.edu
• Sunday April 25, 1999, 8:00 a.m.-10:40 a.m.
Special Session on Complex Geometry, III
Room 103, Diefendorf Hall
Terrence Napier, Lehigh University tjn2@lehigh.edu
Mohan Ramachandran, State University of New York at Buffalo ramac-m@newton.math.buffalo.edu
□ 8:00 a.m.
On the fundamental group of an elliptic fibration.
Stephen S.Y. Lu*, Pure Math. Dept., University of Waterloo.
□ 9:00 a.m.
Hermitian-Einstein Metrics for Vector Bundles on Complete Kahler Manifolds.
Lei Ni*, Purdue University
□ 10:00 a.m.
Extended bisectors in complex projective space.
William M. Goldman*, University of Maryland
• Sunday April 25, 1999, 8:30 a.m.-10:50 a.m.
Special Session on Combinatorics and Graph Theory, III
Room 6, Diefendorf Hall
Harris Kwong, SUNY College at Fredonia kwong@cs.fredonia.edu
• Sunday April 25, 1999, 8:40 a.m.-10:50 a.m.
Special Session on Integrable Systems, III
Room 104, Diefendorf Hall
Mich\'ele Audin, Universit\'e Louis Pasteur et NCRS maudin@math.u-strasbg.fr
Lisa Claire Jeffrey, University of Toronto jeffrey@math.toronto.edu
• Sunday April 25, 1999, 9:00 a.m.-10:50 a.m.
Special Session on Smooth Categories in Geometry and Mechanics, III
Room 5, Diefendorf Hall
F. William Lawvere, SUNY at Buffalo wlawvere@acsu.buffalo.edu
• Sunday April 25, 1999, 9:00 a.m.-10:45 a.m.
Special Session on Mathematical Physics, III
Room 7, Diefendorf Hall
Jonathan Dimock, SUNY at Buffalo dimock@acsu.buffalo.edu
□ 9:00 a.m.
Heat kernel analysis on infinite dimensional complex groups.
Maria Gordina*, McMaster University
□ 9:30 a.m.
Boundary conformal fiel theory on the upper half plane.
J\"urg Fr\"ohlich, ETHZ Z\"urich
Olivier Grandjean*, Harvard University
Andreas Recknagel, MPI Leipzig
Volker Schomerus, Hamburg University
□ 10:00 a.m.
Supersymmetry with a Twist.
Arthur M. Jaffe*, Harvard University
• Sunday April 25, 1999, 9:00 a.m.-10:50 a.m.
Special Session on Operads, Algebras, and Their Applications, III
Room 4, Diefendorf Hall
Alexander A. Voronov, Michigan State University voronov@math.mit.edu
□ 9:00 a.m.
Homology of $A_{\infty}$ and $L_{\infty}-$algebras.
Masoud Khalkhali*, University of Western Ontario
□ 9:40 a.m.
Formality conjectures for chains and characteristic classes.
Boris L Tsygan*, Penn State University
□ 10:20 a.m.
A master identity for homotopy Gerstenhaber algebras.
Fusun Akman*, Coastal Carolina University
• Sunday April 25, 1999, 11:00 a.m.-11:50 a.m.
Invited Address
Operad Theory and Some Applications.
Room 148, Diefendorf Hall
Alexander A. Voronov*, Michigan State University
• Sunday April 25, 1999, 1:30 p.m.-2:20 p.m.
Invited Address
Harmonic Algebra.
Room 148, Diefendorf Hall
Gregg J. Zuckerman*, Yale University
• Sunday April 25, 1999, 2:30 p.m.-4:00 p.m.
Special Session on Smooth Categories in Geometry and Mechanics, IV
Room 5, Diefendorf Hall
F. William Lawvere, SUNY at Buffalo wlawvere@acsu.buffalo.edu
□ 2:30 p.m.
Round Table Discussion.
• Sunday April 25, 1999, 2:30 p.m.-4:50 p.m.
Special Session on Mathematical Physics, IV
Room 7, Diefendorf Hall
Jonathan Dimock, SUNY at Buffalo dimock@acsu.buffalo.edu
• Sunday April 25, 1999, 2:30 p.m.-5:20 p.m.
Special Session on Knot and 3-Manifolds, V
Room 2, Diefendorf Hall
Thang T.Q. Le, State University of New York at Buffalo letu@math.buffalo.edu
William W. Menasco, SUNY at Buffalo menasco@tait.math.buffalo.edu
Morwen B. Thistlethwaite, University of Tennessee morwen@math.utk.edu
• Sunday April 25, 1999, 2:30 p.m.-5:50 p.m.
Special Session on Representations of Lie Algebras, IV
Room 8, Diefendorf Hall
Duncan J. Melville, Saint Lawrence University dmel@music.stlawu.edu
• Sunday April 25, 1999, 2:30 p.m.-4:20 p.m.
Special Session on Operads, Algebras, and Their Applications, IV
Room 4, Diefendorf Hall
Alexander A. Voronov, Michigan State University voronov@math.mit.edu
□ 2:30 p.m.
A Tale of Two Quantizations.
Jim Stasheff*, UNC-CH
□ 3:10 p.m.
Semi-infinite forms and topological vertex operator algebras.
Yi-Zhi Huang*, Rutgers University
Wenhua Zhao, University of Chicago
□ 3:50 p.m.
The Operads of locally trivialized holomorphic $G$-bundles over Riemann sphere.
Wenhua Zhao*, University of Chicago
• Sunday April 25, 1999, 2:30 p.m.-4:40 p.m.
Special Session on Integrable Systems, IV
Room 104, Diefendorf Hall
Mich\'ele Audin, Universit\'e Louis Pasteur et NCRS maudin@math.u-strasbg.fr
Lisa Claire Jeffrey, University of Toronto jeffrey@math.toronto.edu
• Sunday April 25, 1999, 3:00 p.m.-4:50 p.m.
Special Session on Combinatorics and Graph Theory, IV
Room 6, Diefendorf Hall
Harris Kwong, SUNY College at Fredonia kwong@cs.fredonia.edu
• Sunday April 25, 1999, 3:00 p.m.-3:40 p.m.
Special Session on Complex Geometry, IV
Room 103, Diefendorf Hall
Terrence Napier, Lehigh University tjn2@lehigh.edu
Mohan Ramachandran, State University of New York at Buffalo ramac-m@newton.math.buffalo.edu
□ 3:00 p.m.
A Generalized Boundary Schwarz's Lemma.
Dov N. Chelst*, Drew University
Inquiries: meet@ams.org | {"url":"http://ams.org/meetings/sectional/2039_program_sunday.html","timestamp":"2014-04-18T03:34:29Z","content_type":null,"content_length":"60248","record_id":"<urn:uuid:235ff878-1439-49dc-a90d-c7ceda366b62>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00654-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: binary search with arbitrary or random split
Replies: 2 Last Post: Jul 25, 2012 5:39 AM
Messages: [ Previous | Next ]
Re: binary search with arbitrary or random split
Posted: Jul 24, 2012 9:26 PM
pam <pamelafluente@libero.it> writes:
> I am doing this to make the split point explicitly dependent on n
> or else we have experts like Ben coming out and saying that it's
> O(n) for reason due to the split point being independent of n ;-)
What does the wink signify to you? To me it means, "I know this not
what Ben said but I hope I get a rise from him by saying it". If that's
want you were going for, you nailed it; if not, you may want to re-think
you use of smilies.
> Ben seemed to agree at one point that vanishing events should not be
> considered ("vanishing" meaning
> with a limiting probability 0), but then said he is in agreement with
> Patricia.
> So at the moment i have still not understood what is the answer ;-)
That's a flat-out lie. I said I did not know what a "vanishing event"
was, so how could you read that as agreeing to anything about them at
all? I don't think you are confused about what I said, I think you want
to misrepresent it for your own purposes.
> Well in any case i might say that is O(logn) "almost certain". Can we
> agree on that ? Or no ?
Why do you care? No one would use a random slit when a binary split is
simple and efficient. What's the actual algorithm you have in mind
whose worst-case cost you are so keen to be logarithmic? There's more
to this that you are letting on, and it's starting to have a hint of
obsession about it. Spill the beans.
Date Subject Author
7/24/12 Re: binary search with arbitrary or random split Ben Bacarisse
7/25/12 Re: binary search with arbitrary or random split pamela fluente | {"url":"http://mathforum.org/kb/message.jspa?messageID=7853095","timestamp":"2014-04-21T15:34:13Z","content_type":null,"content_length":"18681","record_id":"<urn:uuid:d2205bcb-a5e6-49ef-978a-1a16f3226de7>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00123-ip-10-147-4-33.ec2.internal.warc.gz"} |
115 projects tagged "BSD Revised"
UFOClock draws an astronomical clock. From it you can read the time of day, phase of the lunar month, ratio of day to night, time until a solstice or equinox, and time until the end or beginning of
twilight. The time of day depends on your location on the surface of the Earth, which you can enter on the command line or in a dotfile in your home directory. Location can be given as latitude/
longitude or ZIP code. It is called UFO clock because it sort of looks like crop circles. It is based on Sundial by George Williams. It requires GLUT installed to run and the NOVAS library to build. | {"url":"http://freecode.com/tags/bsd-license-revised?page=1&sort=vitality&with=2776&without=","timestamp":"2014-04-25T04:29:58Z","content_type":null,"content_length":"98296","record_id":"<urn:uuid:52f59c91-4340-45f7-882f-0de68c2e014d>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00035-ip-10-147-4-33.ec2.internal.warc.gz"} |
Building a Pinhole Camera
06-24-2009 03:46 PM #11
Join Date
Mar 2009
Central Coast, CA
At some of the art supply stores, Ive seen some of the balsa in thicker sheets, so that may be an option. I think I might look for some to make one, or just use foamcore since it's easy to glue
together and is really light.
I too have been bitten by the pin-hole camera bug. This is a project for my 11 year old daughter and self. we are using a left over Christmas popcorn tin- 9.5in.diameter by 10.5in. tall. By my
calculations, we can get a 5x7 negative using the bottom for the pin hole and the lid for the negative . this is dictated by the lid size. However, we can get an 8x10 panorama using the side
forour set up.
if you want to do it the easy way ...
go to pinhole resource . com
or look for pinholebilly on eBoo they make custom laser cut holes ...
the easy rule of thumb for pinhole coverage is 3 or 3 1/2" x ( relative) focal length of lens will tell you coverage / image circle.
have fun!
Yes, I found the Pinholebilly method much easier. For 25 or so bucks you get a stack of them in different sizes.
You do miss out on the cutting up pop cans, or going to the hardware store and asking for "shim stock" and getting blank stares part.
And the figuring out how to just dimple the metal and sanding it just right part.
But there is plenty fun to be had in building the rest of the camera or fitting one of the holes onto a suitable camera body.
Thanks for all the replies. The more I think about building a pinhole camera, the more I want to do it. Hopefully I'll have a rough sketch of my camera and the dimensions later tonight
Any idea what size pinhole I would need for 6 x 9 120?
I used the pinhole calculator here http://www.mrpinhole.com/holesize.php to find what the image circle and the f/stop would be. Using the focal length of 746.5 that I came up with earlier, the
calculator told me the image circle diameter would be 1433mm and the f/stop would be 648. What caught my eye was the pinhole size. According to my formula (which I'll post at the bottom), the
pinhole diameter should be 1mm. According to the calculator, the pinhole diameter should be 1.15mm.
The calculator must be using a different formula than me. What formula is the calculator using, and is that more accurate than mine?
the formula I use is "pinhole diameter=0.0366(square root of the focal length)"
I know I posted my formula on my first post. If someone knew the answer to my question, I didnt want that person to have to go to page one of the thread to find it
I played around with the focal length on the calculator to find what focal length I sould use if I have a pinhole of 1mm. The calculator told me 565mm. I know that would be a freaking huge
pinhole camera, and I really have no intention of building one this big. I'll just use a smaller pinhole and make a smaller camera. I'm just using the measurments I came up with earlier as an
Yes, I found the Pinholebilly method much easier. For 25 or so bucks you get a stack of them in different sizes.
I might pay 25 dollars for an assortment of pinholes, but pinholebilly isn't selling any on the auction site right now.
f/22 and be there.
I might pay 25 dollars for an assortment of pinholes, but pinholebilly isn't selling any on the auction site right now.
the other nice thing about a laser-cut hole
is the pinhole is exact and "relative to f16" so you can
take a light meter reading and figure out the exposure ...
i am sure if you go to a completed auction
and ask a question of the seller
you can find out when his next auction will be ..
or you can ask him what his email address is
and buy from him directly ...
making the hole is kind of fun, but billy makes it painless ...
i bought mine from him pre-ebay .. he has been making
these a very long time, and he is a very nice guy.
The calculator must be using a different formula than me. What formula is the calculator using, and is that more accurate than mine?
the formula I use is "pinhole diameter=0.0366(square root of the focal length)"
Methinks accuracy is a somewhat subjective term here.
One of the formulas often referenced is:
d = c * sqrt( f * l)
Where d is the pinhole diameter
c is a constant (that's where the trouble starts)
f is the focal length
and l is the wavelength of the light (more trouble)
Folks seem to use 0.00055 for the light wavelength (which might be a green) but the spectral sensitivity of the film could come into play. Ortho film would likely produce sharper results if one
optimized for the appropriate wavelength.
The constant --- oooh --- I think Lord Rayleigh, an early tinkerer with these matters, came up with 1.9. But I've seen people using numbers as low as 1.5. Obviously this cauld bend the results
quite a bit. In my quick perusal of the MrPinhole calculator, I didn't see an indication of what he used, although it may be lurking there somewhere. Pinhole Designer defaults to .00055 for the
light wavelength and 1.9 for the constant, but gives a user the option to edit those values.
Edit: The 0.0366 is simply bundling numbers, the product of the constant, c, and sqrt( wavelength) for metric values; 0.0073 for inches.
When one considers the issues of fabrication, pinhole cutting, reciprocity failure and possible film flatness problems, it's probably not worth doing three decimal place calculations here.
Last edited by DWThomas; 06-24-2009 at 09:24 PM. Click to view previous post history.
06-24-2009 03:56 PM #12
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Examples of Average and Instantaneous Rate of Change
(a) Find the average rate of change of
(b) Find the instantaneous rate of change of
(a) For Average Rate of Change:
We have
Again Put
The average rate of change over the interval
(b) For Instantaneous Rate of Change:
We have
Now, putting
The instantaneous rate of change at the point
A particle moves on a line away from its initial position so that after
(a) Find the average velocity of the particle over the interval
(b) Find the instantaneous velocity at
(a) For Average Velocity:
We have
Again Put
The average velocity over the interval
(b) For Instantaneous Velocity:
We have
Now putting
The instantaneous velocity at
Use the formula
(a) the average rate of which the area of a circle changes with
(b) the instantaneous rate at which the area of a circle changes with
(a) For Average Velocity:
We have
Again Put
The average rate of change from
(b) For Instantaneous Rate of Change:
We have
Now putting
The instantaneous rate of change at the point | {"url":"http://www.emathzone.com/tutorials/calculus/examples-of-average-and-instantaneous-rate-of-change.html","timestamp":"2014-04-20T00:39:12Z","content_type":null,"content_length":"27199","record_id":"<urn:uuid:83008e21-0195-486f-b1d1-af7dac31570b>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00225-ip-10-147-4-33.ec2.internal.warc.gz"} |
In econometrics course we always say to our students that "if you fit a linear model with no constant, then you might have trouble. For instance, you might have a negative R-squared". So I tried to
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Ventana Research analyst David Menninger was on the judging panel for the Applications of R in Business contest. In a post on the Ventana research blog, he offers his perspectives on the contest,
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Confirming SSR, SSE, and SST using matrix in R
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In mixture experiments there is a constraint that the variables are the proportions of components that are mixed together with the consequence that these proportions sum to one. When fitting
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would go even further and argue that The Art of R programming is a
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This Article
Bibliographic References
Add to:
ASCII Text x
Y.N. Srikant, "Parallel Parsing of Arithmetic Expressions," IEEE Transactions on Computers, vol. 39, no. 1, pp. 130-132, January, 1990.
BibTex x
@article{ 10.1109/12.46288,
author = {Y.N. Srikant},
title = {Parallel Parsing of Arithmetic Expressions},
journal ={IEEE Transactions on Computers},
volume = {39},
number = {1},
issn = {0018-9340},
year = {1990},
pages = {130-132},
doi = {http://doi.ieeecomputersociety.org/10.1109/12.46288},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Computers
TI - Parallel Parsing of Arithmetic Expressions
IS - 1
SN - 0018-9340
EPD - 130-132
A1 - Y.N. Srikant,
PY - 1990
KW - parallel algorithms; arithmetic expressions; parsing; mesh; shuffle; cube; cube-connected cycle parallel computers; mesh-connected model; wrap-around row-major ordering; shuffle computer;
grammars; parallel algorithms.
VL - 39
JA - IEEE Transactions on Computers
ER -
Parallel algorithms for parsing expressions on mesh, shuffle, cube, and cube-connected cycle parallel computers are presented. With n processors, it requires O( square root n) time on the
mesh-connected model and O(log/sup 2/ n) time on others. For the mesh-connected computer, the author uses a wrap-around row-major ordering. For the shuffle computer, he uses an extra connection
between adjacent processors, and thus four connections per processor are required.
[1] S. G. Akl,Parallel Sorting Algorithms. Orlando, FL: Academic, 1985.
[2] I. Bar-On and U. Vishkin, "Optimal generation of a computation tree form,"ACM Trans. Programming Lang. Syst., vol. 7, no. 2, pp. 659-663, 1985.
[3] E. Dekel and S. Sahni, "Parallel generation of postfix and tree forms,"ACM Trans. Programming Languages Syst., pp. 300-317, July 1983.
[4] K. Hwang and F. A. Briggs,Computer Architecture and Parallel Processing. New York: McGraw-Hill, 1984.
[5] D. Nassimi and S. Sahni, "Data broadcasting in SIMD computers,"IEEE Trans. Comput., vol. C-30, pp. 101-107, Feb. 1981.
[6] M. C. Pease, "The indirect binary n-cube microprocessor array,"IEEE Trans. Comput., vol. C-26, pp. 458-473, May 1977.
[7] F. P. Preparata and J. Vuillemin, "The cube-connected cycle: A versatile network for parallel computation,"Commun. ACM, vol. 24, pp. 300-309, May 1981.
[8] J. T. Schwartz, "Ultracomputers,"ACM Trans. Programming Languages Syst., vol. 2, no. 4, pp. 484-521, Oct. 1980.
[9] Y. N. Srikant and P. Shankar, "A new parallel algorithm for parsing arithmetic infix expressions,"Parallel Comput., vol. 4, pp. 291-304, 1987.
Index Terms:
parallel algorithms; arithmetic expressions; parsing; mesh; shuffle; cube; cube-connected cycle parallel computers; mesh-connected model; wrap-around row-major ordering; shuffle computer; grammars;
parallel algorithms.
Y.N. Srikant, "Parallel Parsing of Arithmetic Expressions," IEEE Transactions on Computers, vol. 39, no. 1, pp. 130-132, Jan. 1990, doi:10.1109/12.46288
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Construct Knowledge and Mathematical Concepts
Since the mid-1980s, constructivism has played a major role in mathematics education, and constructivist approaches to learning - which are supported in the two NCTM Standards documents - are
beginning to influence the teaching of mathematics. Two hallmarks of the constructivist position (Van de Walle, 1995) help guide effective mathematics teaching and learning. First, constructing
knowledge is a highly active endeavor on the part of the learner (Baroody, 1987). Constructing and understanding a new idea involves making connections between old ideas and new ideas. Teachers might
help make this connection by asking reflective questions such as the following:
• How does this idea fit with what you already know?
• In what ways is this problem like other problems/situations you've experienced?
• What is it about this problem that reminds you of yesterday's problem? (Cook & Rasmussen, 1991)
Constructing knowledge requires reflective thought.
Second, networks or "cognitive schemas" that exist in the learner's mind are the principal determining factors for how an idea will be constructed. These networks are the product of both constructing
knowledge and developing mathematical concepts.
Copyright © North Central Regional Educational Laboratory. All rights reserved.
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the first resource for mathematics
Sample path properties of the local time of multifractional Brownian motion.
(English) Zbl 1138.60032
The paper is devoted to various properties of the multifractal Brownian motion (mBm) ${B}^{H}=\left({B}^{H\left(t\right)}\left(t\right)$, $t>0\right)$, which is the Gaussian process with the Hurst
exponent depending on time. The representation of the mBm was introduced by J. Lévy-Vehel and R. F. Peltier [“Multifractional Brownian motion: Definitions and preliminary results”, Technical Report
PR-2645, INRIA (1995)] by
$\begin{array}{c}{B}^{H\left(t\right)}\left(t\right)=\frac{1}{{\Gamma }\left(H\left(t\right)+1/2\right)}\left({\int }_{-\infty }^{0}\left[{\left(t-u\right)}^{H\left(t\right)-1/2}-{\left(-u\right)}^{H
\left(t\right)-1/2}\right]W\left(du\right)\hfill \\ \hfill +{\int }_{0}^{t}{\left(t-u\right)}^{H\left(t\right)-1/2}W\left(du\right)\right),\phantom{\rule{1.em}{0ex}}t\ge 0,\end{array}$
where $H$ is some Hölder continuous function and $W$ is the standard Brownian motion on $\left(-\infty ,\infty \right)$. Independently, mBm was defined by A. Benassi, S. Jaffard and D. Roux [Rev.
Mat. Iberoam. 13, No. 1, 19–90 (1997; Zbl 0880.60053)] by
$\stackrel{^}{B}{\phantom{\rule{0.166667em}{0ex}}}^{H\left(t\right)}\left(t\right)={\int }_{ℝ}\frac{{e}^{it\xi }-1}{{|\xi |}^{H\left(t\right)+1/2}}d\stackrel{^}{W}\left(d\xi \right)·$
First, the authors establish the local and uniform moduli of continuity for the local time of the mBm. In contrast to the classical result, in the case of mBm the moduli of continuity depends on the
point at which the regularity is studied [see also M. Csörgö, Z.-Y. Lin and Q.-M. Shao, Stochastic Processes Appl. 58, No. 1, 1–21 (1995; Zbl 0834.60088) and N. Kono, Proc. Japan Acad., Ser. A 53,
84–87 (1977; Zbl 0437.60057)]. Secondly, the authors establish the law of iterated logarithm in the Chung’s form.
Further, the authors establish the asymptotic self-similarity for the local times of the mBm. Then this results is used to prove some local limit theorems for mBm.
60G17 Sample path properties
60G15 Gaussian processes
60J55 Local time, additive functionals | {"url":"http://zbmath.org/?q=an:1138.60032","timestamp":"2014-04-16T22:29:47Z","content_type":null,"content_length":"25289","record_id":"<urn:uuid:d8a9ab02-316d-40fe-bc59-adaa55d5c6c9>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00582-ip-10-147-4-33.ec2.internal.warc.gz"} |
Search results 1-20 of 52 for
analytical geometry calculating the shortest distance between a point and a line 1 part 1
URL for Part 2
3-Dimensional Geometry - Shortest Distance between Two Lines
This video lecture from Geometry (F.Sc. second year Mathematics) covers: Question number 28, 29 and 30 of exercise 4.3. Questions 28 and 29 are about…
This video shows how to find the shortest distance from a point to a line using analytic geometry. I have also uploaded another video that…
Let there be a line (d) and a point (A). To find the distance between (d) and (A), 1. create a plane perpendicular to (d) going by (A); 2. find the…
This is for my SDSU geometer's sketchpad class. Fall2013.
This video demonstrates a method for finding the shortest distance from a point to a line using some analytic geometry and also right angled…
This is no simple task. We need to find the shortest distance from a point to a line. There are a few steps, but if we reason it out, things should…
Determine the shortest distance from a point to a line.
Finding the distance between a point and a line using a formula based on the "Normal Form" of a line. This is for a class I am missing.
This tutorial will demonstrate how to construct a perpendicular from a point on a given line. With audio.
Made with Doodlecast Pro from the itunes App Store.
Part B of a two-part recording discussing some quesions from Analytical (coordinate) geometry.
Video to download:
analytical geometry calculating the shortest distance between a point and a line 1 part 1 | {"url":"http://www.videorolls.com/search/analytical-geometry-calculating-the-shortest-distance-between-a-point-and-a-line-1-part-1","timestamp":"2014-04-18T10:44:42Z","content_type":null,"content_length":"34423","record_id":"<urn:uuid:99894cef-f052-4348-a2e1-8a6824b6f71b>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00365-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find a[1] and d for the arithmetic sequence with the given terms.
a[20] = 13, and a[25] = 33
We have
a[25] – a[20] = 20.
There are 5 steps from a[20] to a[25], so we divide by 5 to get
d = 4.
Then we proceed as before:
a[1] + 19d = a[20]
a[1] + 19(4) = 13
a[1] = -63.
Once we know we have an arithmetic sequence, if we start at a[1] and take (n – 1) steps of size d, we end up on term a[n].
We can write this idea in symbols as
a[1] + (n – 1)d = a[n].
This formula has three unknowns: a[1], n and d. Like a buy two, get one free deal on cups of key lime pie yogurt, any two pieces of information about an arithmetic sequence tells us the third and
final piece of information. So, if we know a[1] and d we can find any term a[n] we like. We can also work backwards from d and some term a[n] (which gives us n) to find a[1]. | {"url":"http://www.shmoop.com/sequences/arithmetic-sequence-exercises-26.html","timestamp":"2014-04-17T10:24:30Z","content_type":null,"content_length":"29581","record_id":"<urn:uuid:48c419b3-2c45-42a1-9345-39a69ce2fef9>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00089-ip-10-147-4-33.ec2.internal.warc.gz"} |
problem with fractions and proportions
May 21st 2009, 08:27 PM #1
May 2009
i have a mixture which has 87% of x and 13 % of y , which cannot be seperated.
if i need to make a mixture with 99.5 % of x and 0.5 % of y, how much of x and y should i mix.
pls answer quickly.
Hello arch
Welcome to Math Help Forum!
You don't give us any quantity of the original mixture, but let me assume that there's 1 litre, and show you how to do that. Then you can work out how to do it for any other volume.
In 1 litre of the mixture, 13%, or 0.13 litre, is y. We want to add a quantity of x so that this 0.13 litre is 0.5% of the new total volume. If we add $v$ litres of x, the new total volume is $
(1+v)$ litres, and 0.5% of this is $\frac{0.5}{100}(1+v)$. So we have:
$\frac{0.5}{100}(1+v) = 0.13$
$\Rightarrow 0.5(1+v) = 13$
$\Rightarrow 1 + v = 26$
$\Rightarrow v = 25$
So for every litre of the original mixture, we must add 25 litres of x, in order to make the new mixture 99.5% x, 0.5% y.
May 21st 2009, 10:01 PM #2 | {"url":"http://mathhelpforum.com/algebra/90024-problem-fractions-proportions.html","timestamp":"2014-04-21T14:13:34Z","content_type":null,"content_length":"35394","record_id":"<urn:uuid:1c9f091c-6ce3-4898-bfcc-5b7f6da1b26f>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00032-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability Densities for Fluorescent Photons Emitted by a Two-State Atom Driven by a Laser
ISRN Optics
Volume 2012 (2012), Article ID 745871, 6 pages
Research Article
Probability Densities for Fluorescent Photons Emitted by a Two-State Atom Driven by a Laser
Department of Physics and Astronomy, Mississippi State University, P.O. Drawer 5167, Mississippi State, MS 39762-5167, USA
Received 20 September 2012; Accepted 14 November 2012
Academic Editors: J. M. Girkin, D. Poitras, and M. Potasek
Copyright © 2012 Robertsen A. Riehle and Henk F. Arnoldus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Fluorescent photons emitted by a two-state atom in a laser beam are correlated. We have obtained the probability density for the emission of the th photon after a random initial time . It is shown
that the correlations between the photons lead to a deviation from the poissonian value for this function (the probability density for independent events), although the deviation is not as
significant as one may expect.
1. Introduction
When a two-state atom, with energy level separation , is immersed in a laser beam with an angular frequency , then photons are exchanged between the atom and the field in stimulated absorption and
emission, provided that is near . We shall consider the case of resonance, so . In addition, photons are emitted in spontaneous transitions from the excited state to the ground state, and these
photons are emitted in all direction as electric dipole radiation. The emission of these fluorescent photons can be considered as a random event process on the time axis, and we assume that the
duration of each event is negligible (dots on the time axis). This interpretation can be justified with the theory of photon detection from an electromagnetic field [1, 2]. Let be the Einstein
coefficient for spontaneous transitions from the excited state to the ground state [3] and let be the population of the excited state. Then, the number of emitted photons per second is equal to and
this is the intensity of the random process. We shall assume that the atom is in the steady state, so that is independent of time. The temporal statistics of photon emissions can be represented by
the probability densities , with , 2,…. Let be the time at which the th photon is emitted, after an initial time . Since photons are emitted randomly, is a random variable, and its probability
density is : We shall evaluate for the emission of fluorescent photons.
If photons were emitted independently of each other, then the emission process would be a Poisson process, and the probability for the emission of photons in would be The probability for the emission
of the th photon in is . If this emission would be independent of previous emissions, then so that
2. Photon Correlations in Resonance Fluorescence
Correlations between random events are expressed through the intensity correlation functions [4, 5]: For resonance fluorescence, these correlation functions take the form [6, 7] involving the
function . This function equals the population of the excited state at time , under the condition that the atom is in the ground state at . Therefore, From (7), we then see that an intensity
correlation function vanishes when two consecutive time arguments are the same. Consequently, immediately after the emission of a photon, there can be no emission of the next photon. This phenomenon
is called antibunching and has been observed experimentally [8–10]. Physically, this can be understood from the fact that after the emission of a photon the atom is in the ground state, whereas a
photon can only be emitted when the atom is in the excited state. It takes a finite time for the atom to make the transition from the ground state to the excited state, and, therefore, the
probability for an emission immediately after a previous one is zero. We also have since is the population of the excited state in the steady state.
The function can be evaluated by solving the equation of motion for the system. The free parameter is the the Rabi frequency , which is assumed to be positive. This parameter depends on the atomic
transition dipole moment, and is proportional to the power of the laser. We find that [11] where we have set for the time in units of the life time of the excited state. The parameter is defined as
with being the Rabi frequency in units of . For , the function reaches its steady state exponentially, but for , the hyperbolic functions become trigonometric functions, and oscillations appear.
Figure 1 illustrates the behavior of . The steady state is found to be
3. Laplace Transform of the Probability Densities
The intensity correlation functions contain all information about the random process, and the probability densities can be found from these functions [12]. It appears convenient to adopt a Laplace
transform of : We then obtain that [13] in terms of the Laplace transform of .
From (10), we find that and this yields where we have set
4. Probability Densities
In order to evaluate the Laplace inverse of (16), we first shift the parameter with the attenuation theorem: The inverse on the right-hand side can be computed with the Bromwich integral [14]. We
find that The functions are defined as and the coefficients and are for , and . A variety of properties of these coefficients have been derived elsewhere [15]. For small and , these coefficients can
be computed from (21), and the results are tabulated in Tables 1 and 2. For , only the term with survives, and with various sumrules for the coefficients and , we derive the initial values On general
grounds, this relation holds for any probability density of a random event process.
5. Deviation from Poisson Statistics
The expression on the right-hand side of (19) for is the main result of this paper. From this rather formidable formula, it is not readily clear what the significance of this result is. In order to
discover the main features of the probability densities, we work out the cases for and . We find that Photons are emitted randomly, but the photons are correlated, as can be seen from Figure 1. For
uncorrelated photons, the function would be a constant: , and would be the Poisson distribution given by (5) (with ). The solid curves in Figures 2, 3, and 4 show for , 2, and 8, respectively, and
the dashed curves are for a Poisson distribution with the same intensity. These parameters are the same as for the various functions in Figure 1. We see from Figure 2 that for small laser power the
probability density for the emitted photons is almost indistinguishable from the Poisson distribution. For larger , oscillations set in, and deviates more noticeably from the Poisson curve, as
illustrated in Figure 3. For even larger , as in Figure 4, the oscillations get faster, but the amplitude of the oscillations diminishes, and the probability density is again nearly poissonian. The
probability density in Figure 4 is determined by curve for in Figure 1, and this function deviates very strongly from its Poisson value . We notice that the strong oscillations in result only in
small ripples on the function in Figure 4, which appears quite remarkable. Figures 5–7 show for the same values of as in Figures 2–4. It appears that deviates more from Poisson statistics, in
particular for moderate laser power, as in Figure 6.
6. Conclusions
A two-state atom in a laser beam emits fluorescent photons. We have obtained the probability density for the emission of the th photon after a random initial time . It is assumed that the laser
frequency is on resonance with the atomic transition. The result is given by (19), and it only has the dimensionless Rabi frequency as free parameter. Of particular interest is the deviation from the
probability density for independent events (Poisson process), since this reflects the correlations between emitted photons. The results are illustrated in Figures 2–7, and we observe that the
functions only deviate moderately from the corresponding functions for independent photon emissions.
1. R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds., pp. 65–185, Gordon and Breach, New York, NY, USA,
2. P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Physical Review, vol. 136, no. 2A, pp. A316–A334, 1964. View at Publisher · View at
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3. R. Loudon, The Quantum Theory of Light, Clarendon Press, Oxford, UK, 2nd edition, 1983.
4. R. L. Stratonovich, Topics in the Theory of Random Noise, vol. 1, chapter 6, Gordon and Breach, New York, NY, USA, 1963.
5. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, chapter 2, Elsevier, Amsterdam, The Netherlands, 3rd edition, 2007.
6. G. S. Agarwal, “Time factorization of the higher-order intensity correlation functions in the theory of resonance fluorescence,” Physical Review A, vol. 15, no. 2, pp. 814–816, 1977. View at
Publisher · View at Google Scholar · View at Scopus
7. D. Lenstra, “Photon-number statistics in resonance fluorescence,” Physical Review A, vol. 26, no. 6, pp. 3369–3377, 1982. View at Publisher · View at Google Scholar · View at Scopus
8. H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Physical Review Letters, vol. 39, no. 11, pp. 691–695, 1977. View at Publisher · View at Google Scholar
· View at Scopus
9. M. Dagenais and L. Mandel, “Investigation of two-time correlations in photon emissions from a single atom,” Physical Review A, vol. 18, no. 5, pp. 2217–2228, 1978. View at Publisher · View at
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10. F. Diedrich and H. Walther, “Nonclassical radiation of a single stored ion,” Physical Review Letters, vol. 58, no. 3, pp. 203–206, 1987. View at Publisher · View at Google Scholar · View at
11. D. F. Walls and G. J. Milburn, Quantum Optics, Springer, Berlin, Germany, 1994, p. 221.
12. H. F. Arnoldus and R. A. Riehle, “Waiting times, probabilities and the Q factor of fluorescent photons,” Journal of Modern Optics, vol. 59, no. 11, pp. 1002–1015, 2012. View at Publisher · View
at Google Scholar
13. H. F. Arnoldus and G. Nienhuis, “Photon statistics of fluorescence radiation,” Optica Acta, vol. 33, no. 6, pp. 691–702, 1986. View at Scopus
14. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, Calif., USA, 4th edition, 1995.
15. H. F. Arnoldus and R. A. Riehle, “Conditional probability densities for photon emission in resonance fluorescence,” Physics Letters A, vol. 376, pp. 2584–2587, 2012. View at Publisher · View at
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4-H Military Partnerships - Curriculum Resources>4-H 101
4-H 101
4-H 201
Adult Babysitting
Character Education
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Youth Babysitting
4-H 101>Lesson 1 Introducing Youth Development 4-H 101>Lesson 2 The Essential Elements of 4-H Youth Development 4-H 101>Lesson 3 The Organizational Structure and History of 4-H 4-H 101>Lesson 4
Understanding the Culture of 4-H 4-H 101>Lesson 5 Understanding 4-H Youth Development Delivery 4-H 101>Lesson 6 Life Skills and the Experiential Learning Model 4-H 101>Lesson 7 Knowing and Using 4-H
Curricula 4-H 101>Lesson 8 Putting the Experiential learning Model to Practice 4-H 101>Lesson 9 4-H Club Basics 4-H 101>Lesson 10 Starting 4-H Clubs 4-H 101>Lesson 11 Conducting Club Meetings 4-H
101>Lesson 12 Planning the 4-H Club Year 4-H 101>Lesson 13 Marketing 4-H Clubs 4-H 101>Lesson 14 Recruiting, training and Recognizing Volunteers 4-H 101>Lesson 15 Recognizing 4-H'ers Accomplishments
4-H 101>Lesson 16 Making Action Plans
4-H 101>Lesson 2 The Essential Elements of 4-H Youth Development
4-H 101>Lesson 3 The Organizational Structure and History of 4-H
4-H 101>Lesson 6 Life Skills and the Experiential Learning Model
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[New-bugs-announce] [issue13535] Improved two's complement arithmetic support: to_signed() and to_unsigned()
[New-bugs-announce] [issue13535] Improved two's complement arithmetic support: to_signed() and to_unsigned()
Nick Coghlan report at bugs.python.org
Tue Dec 6 01:49:34 CET 2011
New submission from Nick Coghlan <ncoghlan at gmail.com>:
This RFE proposes two new methods "to_signed" and "to_unsigned" on 'int' objects and on the numbers.Integral ABC.
Semantics (and number.Integral implementation):
def to_unsigned(self, bits):
"Convert this integer to its unsigned two's complement equivalent for the given bit length"
if self.bit_length() >= bits:
raise ValueError("{} is too large for {}-bit two's complement
precision".format(self, bits))
if self >= 0:
return self
return 2**bits + self # self is known to be negative at this point
def to_signed(self, bits):
"Convert an integer in two's complement format to its signed equivalent for the given bit length"
if self < 0:
raise ValueError("{} is already signed".format(self))
if self.bit_length() > bits:
raise ValueError("{} is too large for {}-bit two's complement
precision".format(self, bits))
upper_bound = 2**bits
if self < (upper_bound / 2):
return self
return upper_bound - self
To add these methods to numbers.Integral, a concrete numbers.Integral.bit_length() operation will also be needed. This can be implemented simply as:
def bit_length(self):
return int(self).bit_length()
(Initial concept from this python-ideas thread: http://mail.python.org/pipermail/python-ideas/2011-December/012989.html)
components: Interpreter Core, Library (Lib)
messages: 148896
nosy: ncoghlan
priority: normal
severity: normal
stage: needs patch
status: open
title: Improved two's complement arithmetic support: to_signed() and to_unsigned()
type: feature request
versions: Python 3.3
Python tracker <report at bugs.python.org>
More information about the New-bugs-announce mailing list | {"url":"https://mail.python.org/pipermail/new-bugs-announce/2011-December/012362.html","timestamp":"2014-04-19T03:25:13Z","content_type":null,"content_length":"5426","record_id":"<urn:uuid:7e9c7279-eed6-472f-8524-5bc4ee9c271d>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00649-ip-10-147-4-33.ec2.internal.warc.gz"} |
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AA (2001), pp. 79-94
Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001
Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.)
DMTCS Conference Volume AA (2001), pp. 79-94
author: André Barbé and Fritz von Haeseler
title: Periodic Patterns in Orbits of Certain Linear Cellular Automata
keywords: cellular automata, p-fold bifurcation of periods, finite fields
abstract: We discuss certain linear cellular automata whose cells take values in a finite field. We investigate the periodic behavior of the verticals of an orbit of the cellular automaton and
establish that there exists, depending on the characteristic of the field, a universal behavior for the evolution of periodic verticals.
If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files.
reference: André Barbé and Fritz von Haeseler (2001), Periodic Patterns in Orbits of Certain Linear Cellular Automata, in Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001,
Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AA, pp. 79-94
bibtex: For a corresponding BibTeX entry, please consider our BibTeX-file.
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pdf-source: dmAA0105.pdf (649 K)
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arallelism in
, 1995
"... A degree of parallelism is an equivalence class of Scott-continuous functions which are relatively definable each other with respect to the language PCF (a paradigmatic sequential language). We
introduce an infinite ("bi-dimensional") hierarchy of degrees. This hierarchy is inspired by a representat ..."
Cited by 8 (1 self)
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A degree of parallelism is an equivalence class of Scott-continuous functions which are relatively definable each other with respect to the language PCF (a paradigmatic sequential language). We
introduce an infinite ("bi-dimensional") hierarchy of degrees. This hierarchy is inspired by a representation of first order continuous functions by means of a class of hypergraphs. We assume some
familiarity with the language PCF and with its continuous model. Keywords: sequentiality, stability, strong stability, logical relations, sequentiality relations. 1 Introduction A natural notion of
relative definability in the continuous type hierarchy is given by the following definition: Definition 1 Given two continuous functions f and g, we say that f is less parallel than g (f par g) if
there exists a PCF-term M such that [jM j]g = f . A degree of parallelism is a class of the equivalence relation associated to the preorder par . In this paper we deal with degrees of parallelism of
first ord...
- Annals of Pure and Applied Logic
"... In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided
by a form of generalized recursion theory which uses schemata for explicit definition, conditional definiti ..."
Cited by 7 (2 self)
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In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a
form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate
structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type. 1
"... The aim of this work is to show how hypergraphs can be used as a sistematic tool in the classication of continous boolean functions according to their degree of parallelism. Intuitively f is \
less parallel" than g if it can be dened by a sequential program using g as its only free variable. It turn ..."
Cited by 3 (1 self)
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The aim of this work is to show how hypergraphs can be used as a sistematic tool in the classication of continous boolean functions according to their degree of parallelism. Intuitively f is \less
parallel" than g if it can be dened by a sequential program using g as its only free variable. It turns out that the poset induced by this preorder is (as for the degrees of recursion) a
sup-semilattice. Although hypergraphs had already been used in [6] as a tool for studying degrees of parallelism, no general results relating the former to the latter have been proved in that work.
We show that the sup-semilattice of degrees has a categorical counterpart: we dene a category of hypergraphs such that every object \represents" a monotone boolean function; nite coproducts in this
category correspond to lubs of degrees. Unlike degrees of recursion, where every set has a recursive upper bound, monotone boolean functions may have no sequential upper bound. However the ones which
do have a sequential upper bound can be nicely characterised in terms of hypergraphs. These subsequential functions play a major role in the proof of our main result, namely that f is less parallel
than g if there exists a morphism between their associated hypergraps. 1
- Proceedings of the 7th International Conference of Foundations of Software Science and Computation Structures – FOSSACS 2004 , 2004
"... Abstract. In order to study relative PCF-definability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [3, 5]). We introduce the notion of timed hypergraph
morphism and show that it is: – Sound: if there exists a timed morphism from Hf to Hg then f is PCF-defin ..."
Cited by 1 (0 self)
Add to MetaCart
Abstract. In order to study relative PCF-definability of boolean functions, we associate a hypergraph Hf to any boolean function f (following [3, 5]). We introduce the notion of timed hypergraph
morphism and show that it is: – Sound: if there exists a timed morphism from Hf to Hg then f is PCF-definable relatively to g. – Complete for subsequential functions: if f is PCF-definable relatively
to g, and g is subsequential, then there exists a timed morphism from Hf to Hg. We show that the problem of deciding the existence of a timed morphism between two given hypergraphs is NP-complete. 1
, 2005
"... The focus of this thesis is the study of relative definability of first-order boolean functions with respect to the language PCF, a paradigmatic typed, higher-order language based on the
simply-typed λ-calculus. The basic core language is sequential. We study the effect of adding construct that embo ..."
Add to MetaCart
The focus of this thesis is the study of relative definability of first-order boolean functions with respect to the language PCF, a paradigmatic typed, higher-order language based on the simply-typed
λ-calculus. The basic core language is sequential. We study the effect of adding construct that embody various notions of parallel execution. The resulting set of equivalence classes with respect to
relative definability forms a supsemilattice analoguous to the lattice of degrees in recursion theory. Recent results of Bucciarelli show that the lattice of degrees of parallelism has both infinite
chains and infinite antichains. By considering a very simple subset of Sieber’s sequentiality relations, we identify levels in the lattice and derive inexpressiblity results concerning functions on
different levels. This allows us to explore further the structure of the lattice of degrees of parallelism and show the existence of new infinite hierarchies. We also identify four subsemilattices of
this structure, all characterized by a simple property. ii Résumé Dans ce mémoire nous nous concentrons sur l’étude de la definition relative de fonctions | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1586691","timestamp":"2014-04-25T02:11:48Z","content_type":null,"content_length":"25866","record_id":"<urn:uuid:857ae24a-78f9-4a87-90cd-b5565185ed7d>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00437-ip-10-147-4-33.ec2.internal.warc.gz"} |
Type Specifier SATISFIES
Compound Type Specifier Kind:
Compound Type Specifier Syntax:
satisfies predicate-name
Compound Type Specifier Arguments:
predicate-name---a symbol.
Compound Type Specifier Description:
This denotes the set of all objects that satisfy the predicate predicate-name, which must be a symbol whose global function definition is a one-argument predicate. A name is required for
predicate-name; lambda expressions are not allowed. For example, the type specifier (and integer (satisfies evenp)) denotes the set of all even integers. The form (typep x '(satisfies p)) is
equivalent to (if (p x) t nil).
The argument is required. The symbol * can be the argument, but it denotes itself (the symbol *), and does not represent an unspecified value.
The symbol satisfies is not valid as a type specifier.
The following X3J13 cleanup issue, not part of the specification, applies to this section:
Copyright 1996-2005, LispWorks Ltd. All rights reserved. | {"url":"http://clhs.lisp.se/Body/t_satisf.htm","timestamp":"2014-04-20T08:16:29Z","content_type":null,"content_length":"4736","record_id":"<urn:uuid:9f62788b-8f04-48c8-a3fe-f820ce62793c>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00647-ip-10-147-4-33.ec2.internal.warc.gz"} |
CFD Online Discussion Forums - Re: About wall shear stress
Mike November 5, 2003 17:04
Re: About wall shear stress
Hi--Markus. Glad to see your response. I wrote the following code.I am confused when I saw the udm-0 is are all 0 on the zone I applied with XYplot, but it is not zero in the data file
"Current4.dat". And they are not the same as the wall shear stress in Fluent. Anyway, I am not good at hacking. Any comment will be appreciated.
Best Regards, Mike
Thread *th;
FILE *fp;
cell_t c_Cell;
face_t f;
Thread *t_Cell;
int ID =3;
Domain *domain;
real Tauxy, Tauyx;
real Wall_shear;
real NV_VEC(A),NV_VEC(es),NV_VEC(dr0);
real ds,A_by_es;
domain = Get_Domain(1);
th = Lookup_Thread(domain,ID);
fp=fopen ("Current4.dat","w");
if(fp == NULL)
{ Message("\n Warning: unable to open %s for reading\n", "Current4.dat"); }
begin_f_loop(f, th) {
c_Cell = F_C0(f,th);
t_Cell = th->t0 ;
Tauxy = (F_U(f,t_Cell)-C_U(c_Cell,t_Cell))/dr0[1];
Tauyx = (F_V(f,t_Cell)-C_V(c_Cell,t_Cell))/dr0[0];
Wall_shear = C_MU_EFF(c_Cell,t_Cell)*(Tauxy+Tauyx);
F_UDMI(f,th,0) = Wall_shear;
fprintf(fp,"%f %f %f\n", Tauxy,Tauyx,Wall_shear);
}end_f_loop(f, th)
fclose (fp); } | {"url":"http://www.cfd-online.com/Forums/fluent/32433-re-about-wall-shear-stress-print.html","timestamp":"2014-04-24T09:01:18Z","content_type":null,"content_length":"12191","record_id":"<urn:uuid:4c2197ae-7c2d-494a-8c03-8f2a81cfc69d>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00253-ip-10-147-4-33.ec2.internal.warc.gz"} |