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by Geoffrey Grimmett
These lecture notes have been prepared for students at the 2008 PIMS-UBC Summer School in Probability, for M2 students attending my course at the Institut Henri Poincaré, and for a Minerva Lecture
course at Columbia University.
They are published by Cambridge University Press in 2010, as Volume 1 of the IMS Textbooks Series.
pdf file of July 2012, incorporating previous errata.
Theorem 3.14 should read: For any increasing random variable f: Omega \to R, the function etc...
Exercise 2.10. The answer should be 1/2.
Exercise 3.9. It is useful that theta(1/2)=0 for bond percolation on the square lattice, see Thm 5.33.
Exercise 5.4. Probably best to replace the first `greater than or equal to' by `less than or equal to'. Either would do for the first part, but the second part follows after the change.
The copyright of all linked material rests with the author.
Commercial reproduction is prohibited, except as authorised by the author and publisher. | {"url":"http://www.statslab.cam.ac.uk/~grg/books/pgs.html","timestamp":"2014-04-21T09:35:50Z","content_type":null,"content_length":"2360","record_id":"<urn:uuid:6b737f8c-2487-43c5-a390-bf4e4276d20b>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00615-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 1 - 10 of 12
, 1992
"... Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown
or only partially specified. This work unifies notions of interval algebras in artificial intelligence ..."
Cited by 86 (11 self)
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Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or
only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal
labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding
different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NP-complete. On the
positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NP-complete. This problem is
- Journal of the ACM , 2001
"... Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the
current knowledge about tractability in the interval algebra is complete, that is, this algebra contains ex ..."
Cited by 30 (2 self)
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Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the
current knowledge about tractability in the interval algebra is complete, that is, this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely
contained in one of these subalgebras is NP-complete. We obtain this result by giving a new uniform description of the known maximal tractable subalgebras and then systematically using an algebraic
technique for identifying maximal subalgebras with a given property.
- In IPPS/SPDP’98 , 1998
"... List-based priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the “critical path, most immediate
successors first” (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of ..."
Cited by 11 (0 self)
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List-based priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the “critical path, most immediate
successors first” (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of communication overhead is examined in this paper. Three of the more popular list scheduling
heuristics, HLFET [1] and ISH and DSH [10], plus the Mapping Heuristic [4,6] are subjected to a performance based comparison, with results demonstrating their inadequacies in communicationintensive
cases. Performance degradation in these instances is partly due to the level alteration problem, but more significantly to conservative estimation of communication costs due to the assumption of zero
link contention. The significance of this component of communication is also examined in this paper. 1.
"... Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can
be done more eÆciently than by using the well-known algorithms based on bipartite matching and matrix mul ..."
Cited by 9 (0 self)
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Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can be
done more eÆciently than by using the well-known algorithms based on bipartite matching and matrix multiplication. In particular, we show that deciding deciding if an order has width k can be done in
O(kn²) time and whether a graph has Dilworth number k can be done in O(k²n²) time. For very small k we have even better results. We show that orders of width at most 3 can be recognized in O(n) time
and of width at most 4 in O(n log n).
- Theoretical Computer Science , 1997
"... For an interval graph with some additional order constraints between pairs of non-intersecting intervals, we give a linear time algorithm to determine if there exists a realization which
respects the order constraints. Previous algorithms for this problem (known also as seriation with side constrain ..."
Cited by 5 (1 self)
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For an interval graph with some additional order constraints between pairs of non-intersecting intervals, we give a linear time algorithm to determine if there exists a realization which respects the
order constraints. Previous algorithms for this problem (known also as seriation with side constraints) required quadratic time. This problem contains as subproblems interval graph and interval order
recognition. On the other hand, it is a special case of the interval satisfiability problem, which is concerned with the realizability of a set of intervals along a line, subject to precedence and
intersection constraints. We study such problems for all possible restrictions on the types of constraints, when all intervals must have the same length. We give efficient algorithms for several
restrictions of the problem, and show the NP-completeness of another restriction. 1 Introduction Two intervals x; y on the real line may either intersect or one of them is completely to the left of
the othe...
- In Proc. of the Third Annual European Symp. on Algorithms, (ESA 95) Corfu, Greece , 1995
"... . We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities
are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). ..."
Cited by 3 (1 self)
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. We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are
modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When the additional information is order constraints on pairs of disjoint intervals, we give a
linear time algorithm. Extant algorithms for this problem (known also as seriation with side constraints) required quadratic time. When the constraints are bounds on distances between endpoints, and
the graph admits a unique clique order, we show that the problem is polynomial. However, we show that even when the lengths of all intervals are precisely predetermined, the problem is NPcomplete. We
also study unit interval satisfiability problems, which are concerned with the realizability of a set of unit intervals along a line, subject to precedence and intersection constraints. For all po...
- SIAM Journal on Discrete Mathematics , 1997
"... . We study the following problem: Given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in
numerous applications in which topological information on intersection of pairs of intervals is accompanied ..."
Cited by 2 (0 self)
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. We study the following problem: Given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in
numerous applications in which topological information on intersection of pairs of intervals is accompanied by additional metric information on their order, distance or sizes. An important
application is physical mapping, a central challenge in the human genome project. Our results are: (1) A polynomial algorithm for the problem on interval graphs which admit a unique clique order (UCO
graphs). This class of graphs properly contains all prime interval graphs. (2) In case all constraints are upper and lower bounds on individual interval lengths, the problem on UCO graphs is linearly
equivalent to deciding if a system of difference inequalities is feasible. (3) Even if all the constraints are prescribed lengths of individual intervals, the problem is NP-complete. Hence, problems
(1) and (2) are als...
, 1998
"... List-based priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the "critical path, most immediate
successors first" (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of ..."
Add to MetaCart
List-based priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the "critical path, most immediate
successors first" (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of communication overhead is examined in this paper. Three of the more popular list scheduling
heuristics, HLFET [1] and ISH and DSH [10], plus the Mapping Heuristic [4,6] are subjected to a performance based comparison, with results demonstrating their inadequacies in communicationintensive
cases. Performance degradation in these instances is partly due to the level alteration problem, but more significantly to conservative estimation of communication costs due to the assumption of zero
link contention. The significance of this component of communication is also examined in this paper. 1. Introduction Task scheduling is one of the most challenging problems facing parallel
programmers today....
"... A cut (A; B) in a graph G is called internal, i N(A) 6= B and N(B) 6= A. In this paper, we study the structure of internal cuts in chordal graphs. We show that if (A; B) is an internal cut in a
chordal graph, then for each i, 0 i (G)+1, there exists a clique K i such that jK i j = (G)+1, jK i ..."
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A cut (A; B) in a graph G is called internal, i N(A) 6= B and N(B) 6= A. In this paper, we study the structure of internal cuts in chordal graphs. We show that if (A; B) is an internal cut in a
chordal graph, then for each i, 0 i (G)+1, there exists a clique K i such that jK i j = (G)+1, jK i T Aj = i and jK i T Bj = k+1 i, where (G) is the vertex connectivity of G. In general, there can be
an exponential number of internal cuts in a chordal graph, while the number of maximal cliques can be at most n (G) 1 [2]. Also there exists chordal graphs, all of whose maximal cliques are of size
(G) + 1. Thus, above result throws some light as to the way the cliques are arranged in chordal graphs, with respect to their cuts. We also show that in a chordal graph G, every internal cut should
contain at least (G)((G)+1) 2 edges. This lower bound is tight, in the sense that there exists chordal graphs with internal cuts having exactly (G)((G)+1) 2 edges. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2081894","timestamp":"2014-04-18T14:50:04Z","content_type":null,"content_length":"37687","record_id":"<urn:uuid:da0e0332-ab23-45cd-bd53-63c317609ee8>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00133-ip-10-147-4-33.ec2.internal.warc.gz"} |
12many – Generalising mathematical index sets
In the discrete branches of mathematics and the computer sciences, it will only take some seconds before you're faced with a set like {1,...,m}. Some people write $1\ldotp\ldotp m$, others
$\{j:1\leq j\leq m\}$, and the journal you're submitting to might want something else entirely. The 12many package provides an interface that makes changing from one to another a one-line
Sources /macros/latex/contrib/12many
Documentation Readme
Version 0.3
License The LaTeX Project Public License
Copyright 2005 Ulrich M. Schwarz
Maintainer Ulrich M. Schwarz
Contained in TeXLive as 12many
MiKTeX as 12many
Topics support for typesetting mathematics
Download the contents of this package in one zip archive (373.6k). | {"url":"http://www.ctan.org/pkg/one2many","timestamp":"2014-04-17T07:36:17Z","content_type":null,"content_length":"5327","record_id":"<urn:uuid:dc8c16b6-c14d-42d8-86cf-e339535b0c06>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00465-ip-10-147-4-33.ec2.internal.warc.gz"} |
Advice to Research Students
1. A problem is something that you are approaching from the wrong direction.
2. The original papers on a problem may contain good ideas that have been forgotten.
3. Progress often comes from applying an idea from somewhere else to your problem.
4. When studying a paper, ask the following questions.
○ What is this about?
○ Why is it done this way?
○ Can't you use instead...?
○ Isn't this the same as...?
5. Notions are more important than notations, but bad notation can hold you back, whilst good notation can suggest connections with other problems.
6. Crazy ideas sometimes work.
Some useful ideas
1. Most inequalities rest on the fact that a modulus squared is positive.
2. If you squeeze an inequality too hard, it can become an equation. An integer with |n| < 1 is zero. A set of integer points in n dimensions occupying a volume smaller than constant times n must
all lie on a plane.
3. Dirichlet's pigeon-hole principle (compactness). If R letters are delivered to R-1 people, then some lucky person must have two or more letters. If R points lie in an interval, some pair of
points must be close together.
4. Given a set of R numbers (integers, rational numbers, algebraic numbers), let p be a prime < R. One pair of numbers must be congruent modulo p, and so cannot be close together.
5. Size ranges: if a factor or denominator n lies in a range 1 to N, then it often helps to divide the range into blocks between consecutive powers of two.
6. Divide and conquer. Divide a long range into short ranges. Approximate within each short range (maybe squeezing an inequality into an equation). Combine the results from the short ranges, for
example by proving that not all short ranges can reduce in the same way.
7. The Dirichlet interchange. If d, a factor of n, is getting too big, then work with the other factor e in de = n.
8. The Riesz interchange. If different cases have different numbers of solutions, ask what it takes for a case to have more than R different solutions, and get a bound for how many cases can have at
least R solutions.
9. Averaging. Can you average over one of the variables instead of working things out for individual values?
10. When averaging a sum of error terms, it sometimes helps to put in extra positive terms so your average will extend over all integers in the range, not just over a subset of them.
Back to Number Theory home page | {"url":"http://www.cardiff.ac.uk/maths/research/researchgroups/numbertheory/advice/index.html","timestamp":"2014-04-16T19:11:20Z","content_type":null,"content_length":"13544","record_id":"<urn:uuid:69e4f084-6331-4f48-8d79-0d1ecf0313e6>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00095-ip-10-147-4-33.ec2.internal.warc.gz"} |
On-line Mathematics Dictionary
An oriental counting device and calculator.
Abelian group
A group in which the binary operation is commutative, that is, ab=ba for all elements a abd b in the group.
The x-coordinate of a point in a 2-dimensional coordinate system.
absolute value
The positive value for a real number, disregarding the sign. Written |x|. For example, |3|=3, |-4|=4, and |0|=0.
abundant number
A positive integer that is smaller than the sum of its proper divisors.
The rate of change of velocity with respect to time.
acute angle
An angle that is less than 90 degrees
The process of adding two numbers to obraint heir sum.
algebraic equation
An equation of the form f(x)=0 where f is a polynomial.
algebraic number
A number that is the root of an algebraic polynomial. For example, sqrt(2) is an algebraic number because it is a solution of the equation x^2=2.
A cryptarithm in which the letters, which represent distinct digits, form related words or meaningful phrases.
The altitude of a triangle is the line segment from one vertex that is perpendicular to the opposite side.
amicable numbers
Two numbers are said to be amicable if each is equal to the sum of the proper divisors of the other.
The figure formed by two line segments or rays that extend from a given point.
The region enclosed by two concentric circles.
A portion of a circle.
The amount of surface contained by a figure.
The type of mathematics that studies how to solve problems involving numbers (but no variables).
arithmetic mean
The arithmetic mean of n numbers is the sum of the numbers divided by n.
An isomorphism from a set onto itslef.
Typically this refers to the arithmetic mean.
A sphere together with its interior.
bar graph
A type of chart used to compare data in which the length of a bar represents the size of the data.
In the expression x^y, x is called the base and y is the exponent.
Bayes's Rule
A rule for finding conditional probability.
binary number
A number written to base 2.
binary operation
A binary operation is an operation that involves two operands. For example, addition and subtraction are binary operations.
A one-to-one onto function.
An expression that is the sum of two terms.
binomial coefficient
The coefficients of x in the expansion of (x+1)^n.
biquadratic equation
A polynomial equation of the 4th degree.
to cut in half.
A binary digit.
The symbols { and } used for grouping or to represent a set.
The amount of memory needed to represent one character on a computer, typically 8 bits.
A machine for performing arithemtical calculations.
Caliban puzzle
A logic puzzle in which one is asked to infer one or more facts from a set of given facts.
cardinal number
A number that indicates the quantity but not the order of things.
A curve whose equation is y = (a/2)(e^x/a+e^-x/a). A chain suspended from two points forms this curve.
ceiling function
The ceiling function of x is the smallest integer greater than or equal to x.
central angle
An angle between two radii of a circle.
The center of mass of a figure. The centroid of a triangle is the intersection of the medians.
A line segment extending from a vertex of a triangle to the opposite side.
Chebyshev polynomials
The line joining two points on a curve is called a chord.
The set of points equidistant from a given point (the center).
circular cone
A cone whose base is a circle.
The circumcenter of a triangle is the center of the circumscribed circle.
The circle circumscribed about a figure.
The boundary of a circle.
A curve with equation y^2(a-x)=x^3.
The constant multipliers of the indeterminate variable in a polynomial. For example, in the polynomial x^2+3x+7, the coefficients are 1, 3, and 7.
common denominator
A multiple shared by the denominators of two or more fractions.
complementary angles
Two angles whose sum is 90^o.
complex number
The sum of a real number and an imaginary number, for example 3+4i where i=sqrt(-1).
To solve problems that use numbers.
curved from the inside.
A three-dimensional solid that rises froma circular base to a single point at the top.
congruent figures
two geometric figures that are identical in size and shape.
conic section
The cross section of a right circular cone cut by a plane. An ellipse, parabola, and hyperbola are conic sections.
Numbers that determine the position of a point.
Integers m and n are coprime if gcd(m,n)=1.
A number puzzle in which an indicated arithmetical operation has some or all of its digits replaced by letters or symbols and where the restoration of the original digits is required. Each letter
represents a unique digit.
A solid figure bounded by 6 congruent squares.
cubic equation
A polynomial equation of degree 3.
cyclic polygon
A polygon whose vertices lie on a circle.
A rounded three-dimensional solid that has a flat circular face at each end.
Facts that have been collected but not yet interpreted.
A polygon with 10 sides.
decimal number
A number written to the base 10.
decimal point
The period in a deimal number separating the integer part from the fractional part.
deficient number
A positive integer that is larger than the sum of its proper divisors.
The degree of a term in one variable is the exponent of that variable. For example, the degree of 7x^5 is 5.
In the fraction x/y, x is called the numerator and y is called the denominator.
In a polygon, the line segment joining a vertex with another (non-adjacent) vertex is called a diagonal.
The longest chord of a figure. In a circle, a diameter is a chord that passes through the center of the circle.
The difference between two numbers is what you get when you subtract one from the other.
differential calculus
That part of calculus that deals with the opeation of differentiation of functions.
A cryptarithm in which digits represent other digits.
In the decimal system, one of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
dihedral angle
The angle formed by two planes meeting in space.
The indication of how far something extends in space.
A circle together with its interior.
distributive law
The formula a(x+y)=ax+ay.
In the expression "a divided by b", a is the divident and b is the divisor.
A basic arithmetical operation determining how many times one quantity is contained within another.
In the expression "a divided by b", a is the divident and b is the divisor.
The nonzero integer d is a divisor of the integer n if n/d is an integer.
Diophantine equation
An equation that is to be solved in integers.
A polygon with 12 sides.
A solid figure with 12 faces A regular dodecahedron is a regular polyhedron with 12 faces. Each face is a rgular pentagon.
The domain of a function f(x) is the set of x values for which the function is defined.
Two congruent squares joined along an edge.
duodecimal number system
The system of numeration with base 12.
A number of the form 1/x where x is an integer is called an Egyptian fraction.
characteristic value
elementary function
one of the functions: rational functions, trigonometric functions, exponential functions, and logarithmic functions.
A plane figure whose equation isx^2/a^2+y^2/b^2=1.
A solid figure whose equation is x^2/a^2+y^2/b^2+z^2/c^2=1.
empty set
The set with no elements in it.
enumerable set
A countable set.
A statement that two expressions are equal to each other.
equiangular polygon
A polygon all of whose interior angles are equal.
equichordal point
A point inside a closed convex curve in the plane is called an equichordal point if all chords through that point have the same length.
equilateral polygon
A polygon all of whose sides are equal.
equilateral triangle
A triangle with three equal sides.
escribed circle
An escribed circle of a triangle is a circle tangent to one side of the triangle and to the extensions of the other sides.
A rough guess at the value of a number.
Euler line
The Euler line of a triangle is the line connecting the centroid and the circumcenter.
Euler's constant
The limit of the series 1/1+1/2+1/3+...+1/n-ln n as n goes to infinity. Its value is approximately 0.577216.
even function
A function f(x) is called an even function if f(x)=f(-x) for all x.
even number
An integer that is divisible by 2.
The center of an excircle.
An escribed circle of a triangle.
In the expression x^y, x is called the base and y is called the exponent.
exponential function
The function f(x)=e^x.
expoential function to base a
The function f(x)=a^x.
An exradius of a triangle is the radius of an escribed circle.
The plane angle formed by adjacent edges of a polygonal angle in space.
factor (noun)
An exact divisor of a number. This 7 is a factor of 28.
factor (verb)
To find the factors of a number.
n! (read n factorial) is equal to the product of the integers from 1 to n.
Farey sequence
The sequence obtained by arranging in numerical order all the proper fractions having denominators not greater than a given integer.
Fermat number
A number of the form 2^{2^n}+1.
Fermat's spiral
A parabolic spiral.
Fibonacci number
A member of the sequence 0, 1, 1, 2, 3, 5,... where each number is the sum of the previous two numbers.
figurate numbers
polygonal numbers
finite group
A group containing a finite number of elements.
floor function
The floor function of x is the greatest integer in x, i.e. the largest integer less than or equal to x.
focal chord
A chord of a conic that passes through a focus.
focal radius
A line segment from the focus of an ellipse to a point on the perimeter of the ellipse.
foot of altitude
The intersection of an altitude of a triangle with the base to which it is drawn.
foot of line
The point of intersection of a line with a line or plane.
A concise statement expressing the symbolic relationship between two or more quantities.
Fourier series
A periodic function with period 2 pi.
An expression of the form a/b.
The number of times a value occurs in some time interval.
For a given solid figure, a related figure formed by two parallel planes meeting the given solid. In particular, for a cone or pyramid, a frustum is determined by the plane of the base and a
plane parallel to the base. NOTE: this word is frequently incorrectly misspelled as frustrum.
A normal curve.
A flat board into which nails have been driven in a regular rectangular pattern. These nails represent the lattice points in the plane.
The arc on a surface of shortest length joining two given points.
A branch of mathematics dealing with the shape, size, and curvature of the Earth.
geometric mean
The geometric mean of n numbers is the nth root of the product of the numbers.
geometric progression
A sequence in which the ratio of each term to the preceding term is a given constant.
geometric series
A series in which the ratio of each term to the preceding term is a given constant.
geometric solid
The bounding surface of a 3-dimensional portion of space.
The branch of mathematics that deals with the nature of space and the size, shape, and other properties of figures as well as the transformations that preserve these properties.
Gergonne point
In a triangle, the lines from the vertices to the points of contact of the opposite sides with the inscribed circle meet in a point called the Gergonne point.
gnomon magic square
A 3 X 3 array in which the elements in each 2 X 2 corner have the same sum.
golden ratio
golden rectangle
A rectangle whose sides are in the golden ratio.
graceful graph
A graph is said to be graceful if you can number the n vertices with the integers from 1 to n and then label each edge with the difference between the numbers at the vertices, in such a way that
each edge receives a different label.
grad (or grade)
1/100th of a right angle
A graph is a set of points (called vertices) and a set of lines (called edges) joinging these vertices.
great circle
A circle on the surface of a sphere whose center is the center of the sphere.
greatest common divisor
The greatest common divisor of a sequence of integers, is the largest integer that divides each of them exactly.
greatest common factor
Same as greatest common divisor.
greatest lower bound
The greatest lower bound of a set of real numbers, is the largest real number that is smaller than each of the numbers in the set.
A mathematical system consisting of elements from a set G and a binary operation * such that
1. x*y is a member of G whenever x and y are
2. (x*y)*z=x*(y*z) for all x, y, and z
3. there is an identity element e such that e*x=x*e=e for all x
4. each member x in G has an inverse element y such that x*y=y*x=e
A ray.
The part of a plane that lies on one side of a given line.
Hankel matrix
A matrix in which all the elements are the same along any diagonal that slopes from northeast to southwest.
harmonic analysis
The study of the representation of functions by means of linear operations on characteristic sets of functions.
harmonic division
A line segment is divided harmonically by two points when it is divided externally and internally int he same ratio.
harmonic mean
The harmonic mean of two numbers a and b is 2ab/(a + b).
A unit of measurement in the metric system equal to 10,000 square meters (approximately 2.47 acres).
The path followed by a point moving on the surface of a right circular cylinder that moves along the cylinder at a constant ratio as it moves around the cylinder. The parameteric equation for a
helix is
x=a cos t
y=a sin t
A polygon with 7 sides.
A polygoin with 6 sides.
hexagonal number
A number of the form n(2n-1).
hexagonal prism
A prism with a hexagonal base.
A polyhedron having 6 faces. The cube is a regular hexahedron.
A six-square polyomino.
Heronian triangle
A triangle with integer sides and integer area.
A one-to-one continuous transformation that preserves open and closed sets.
A function that preserve the operators associated with the specified structure.
horizontal line
A line parallel to the earth's surface or the bottom of a page.
A curve with equation x^2/a^2-y^2/b^2=1.
hyperbolic spiral
The curve whose equation in polar coordinates is r*theta=a.
A geometric solid whose equation is x^2/a^2+y^2/b^2-z^2/c^2=1 orx^2/a^2+y^2/b^2-z^2/c^2=-1.
The longest side of a right triangle.
A polyhedron with 20 faces.
The element x in some algebraic structure is called idempotent if x*x=x.
imaginary axis
The y-axis of an Argand diagram.
imaginary number
A complex number of the form xi where x is real and i=sqrt(-1).
imaginary part
The imaginary part of a complex number x+iy where x and y are real is y.
The incenter of a triangle is the center of its inscribed circle.
The circle inscribed in a given figure.
The statement that one quantity is less than (or greater than) another.
becoming large beyond bound.
A variable that approaches 0 as a limit.
A reference to a quantity larger than any specific integer.
A point of inflection of a plane curve is a point where the curve has a stationary tangent, at which the tangent is changing from rotating in one direction to rotating in the oppostie direction.
A one-to-one mapping.
inscribed angle
The angle formed by two chords of a curve that meet at the same point on the curve.
One of the numbers ..., -3, -2, -1, 0, 1, 2, 3, ...
Two figures are said to intersect if they meet or cross each other.
irrational number
A number that is not rational.
isogonal conjugate
Isogonal lines of a triangle are cevians that are symmetric with respect to the angle bisector. Two points are isogonal conjugates if the corresponding lines to the vertices are isogonal.
A length preserving map.
isosceles tetrahedron
A tetrahedron in which each pair of opposite sides have the same length.
isosceles triangle
A triangle with two equal sides.
isosceles trapezoid
Ain which the two non-parallel sides have the same length.
isotomic conjugate
Two points on the side of a triangle are isotomic if they are equidistant from the midpoint of that side. Two points inside a triangle are isotomic conjugates if the corresponding cevians through
these points meet the opposite sides in isotomic points.
A function that gives the probability that each of two or more random variables takes at a particular value.
joint variation
A variation in which the values of one variable depend upon those of 2 or more variables.
Jordan curve
A simple closed curve.
Jordan matrix
A matrix whose diagonal elements are all equal (and nonzero) and whose elements above the principal diagonal are equal to 1, but all other elements are 0.
A unit of energy or work.
jump discontinuity
A discontinuity in a function where the left and righ-hand limits exist but are not equal to each other.
A unit of length equal to 1,000 meters.
A branch of mechanics dealing with the motion of rigid bodies without reference to their masses or the forces acting on the bodies.
A quadrilateral which has two pairs of adjacent sides equal.
knight's tour
A knight's tour of a chessboard is a sequence of moves by a knight such that each square of the board is visited exactly once.
A curve in space formed by interlacing a piece of string and then joining the ends together.
a unit of speed in navigation equal to one nautical mile per hour.
A tetromino in the shape of the letter L.
latera recta
plural of lattice rectum.
latin square
An n X n array of numbers in which only n numbers appear. No number appears more than once in any row or column.
The angular distance of a point on the Earth from the equator, measured along the meridian through that point.
lattice point
A point with integer coordinates.
latus rectum
A chord of an ellipse passing through a focus and perpendicular to the major axis of the ellipse.Plural: latera recta.
least common multiple
The least common multiple of a set of integers is the smallest integer that is an exact multiple of every number in the set.
least upper bound
The least upper bound of a set of numbers is the smallest number that is larger than every member of the set.
plural of lemma.
A proposition that is useful mainly for the proof of some other theorem.
The straight line distance between two points.
Legendre polynomials
A geometrical figure that has length but no width.
linear function
A function of the form y=ax+b.
line graph
A chart that shows data by means of points connected by lines.
line segment
The part of a line between two given distinct points on that line (including the two points).
The set of all points meeting some specified condition.
The study of the formal laws of reasoning.
lowest common denominator
The smallest number that is exactly divisible by each denominator of a set of fractions.
On a sphere, a curve that cuts all parallels under the same angle.
lowest common denominator
The smallest multiple shared by the denominators of a set of fractions.
lowest terms
A fraction is said to be in lowest terms if its numerator and denominator have no common factor.
Lucas number
A member of the sequence 2, 1, 3, 4, 7,... where each number is the sum of the previous two numbers. L[0]=2, L[1]=1, L[n]=L[n-1]+L[n-2].
The portion of a sphere between two great semicircles having common endpoints (including the semicircles).
A square array of n numbers such that sum of the n numbers in any row, column, or main diagonal is a constant (known as the magic sum).
magic tour
If a chess piece visits each square of a chessboard in succession, this is called a tour of the chessboard. If the successive squares of a tour on an n X n chessboard are numbered from 1 to n^2,
in order, the tour is called a magic tour if the resulting square is a magic square.
main diagonal
In the matrix [a[ij]], the elements a[11], a[22], ..., a[nn].
major axis
The major axis of an ellipse is it's longest chord.
Malfatti circles
Three equal circles that are mutually tangent and each tangent to two sides of a given triangle.
The largest of a set of values.
A rectangular array of elements.
Same as average.
medial triangle
The triangle whose vertices are the midpoints of the sides of a given triangle.
The median of a triangle is the line from a vertex to the midpoint of the opposite side.
When a set of numbers is ordered from smallest to largest, the median number is the one in the middle of the list.
Mersenne number
A number of the form 2^p-1 where p is a prime.
Mersenne prime
A Mersenne number that is prime.
The point M is the medpoint of line segment AB if AM=MB. That is, M is halfway between A and B.
minor axis
The minor axis of an ellipse is its smallest chord.
The smallest of a set of values.
The most frequently occurring value in a sequence of numbers.
The integers a and b are said to be congruent modulo m if a-b is divisible by m.
An algebraic expression consisting of just one term.
A sequence is monotone if its terms are increasing or decreasing.
monic polynomial
A polynomial in which the coefficient of the term of highest degree is 1.
monochromatic triangle
A triangle whose vertices are all colored the same.
An algebraic expression consisting of 2 or more terms.
The integer b is a multiple of the integer a if there is an integer d such that b=da.
The basic arithemtical operation of repeated addition.
The point on the celestial spehere in the direction downwards of the plumb-line.
Nagel point
In a triangle, the lines from the vertices to the points of contact of the opposite sides with the excircles to those sides meet in a point called the Nagel point.
natural number
Any one of the numbers 1, 2, 3, 4, 5, ... .
negative number
A number smaller than 0.
nine point center
In a triangle, the circumcenter of the medial triangle is called the nine point center.
nine point circle
In a triangle, the circle that passes through the midpoints of the sides is called the nine point circle.
A graphical device used for computation which uses a straight edge and several scales of numbers.
nonagonal number
A number of the form n(7n-5)/2.
associated with 9
null hypothesis
The null hypothesis is the hypothesis that is being tested in a hypothesis-testing situation.
null set
the empty set
number line
A line on which each point represents a real number.
number theory
The study of integers.
A symbol that stands for a number.
In the fraction x/y, x is called the numerator and y is called the denominator.
numerical analysis
The study of methods for approximation of solutions of various classes of mathematical problems including error analysis.
An ellipsoid produced by rotating an ellipse through 360^o about its minor axis.
oblique angle
an angle that is not 90^o
oblique coordinates
A coordinate system in which the axes are not perpendicular.
oblique triangle
A triangle that is not a right triangle.
obtuse angle
an angle larger than 90^o but smaller than 180^o
obtuse triangle
A triangle that contains an obtuse angle.
A polygon with 8 sides.
A polyhedron with 8 faces.
any one of the 8 portions of space dtermined by the 3 coordinate planes.
odd function
A function f(x) is called an odd function if f(x)=-f(-x) for all x.
odd number
An integer that is not divisible by 2.
one to one
A function f is said to be one to one if f(x)=f(y) implies that x=y.
A function f is said to map A onto B if for every b in B, there is some a in A such f(a)=b.
open interval
An interval that does not include its two endpoints.
ordered pair
A pair of numbers in which one number is distinguished as the first number and the other as the second number of the pair
ordinal number
A number indicating the order of a thing in a series
The y-coordinate of a point in the plane.
The point in a coordinate plane with coordinates (0,0).
orthic triangle
The triangle whose vertices are the feet of the altitudes of a given triangle.
The point of intersection of the altitudes of a triangle.
A positive integer whose digits read the same forward and backwards.
A positive integer is said to be palindromic with respect to a base b if its representation in base b reads the same from left to right as from right to left.
pandiagonal magic square
A magic square in which all the broken diagonals as well as the main diagonals add up to the magic constant.
A decimal integer is called pandigital if it contains each of the digits from 0 to 9.
A paraboloid of revolution is a surface of revolution produced by rotating a parabola about its axis.
Two lines in the plane are said to be parallel if they do not meet.
A quadrilateral whose opposite sides are parallel.
A prism whose bases are parallelograms.
The symbols ( and ) used for grouping expressions.
Pascal's triangle
A triangular array of binomial coefficients.
pedal triangle
The pedal triangle of a point P with respect to a triangle ABC is the triangle whose vertices are the feet of the perpendiculars dropped from P to the sides of triangle ABC.
Pell number
The nth term in the sequence 0, 1, 2, 5, 12,... defined by the recurrence
P[0]=0, P[1]=1, and P[n]=2P[n-1]+P[n-2].
A polygon with 5 sides.
pentagonal number
A number of the form n(3n-1)/2.
A five-square polyomino.
A way of expressing a number as a fraction of 100.
perfect cube
An integer is a perfect cube if it is of the form m^3 where m is an integer.
perfect number
A positive integer that is equal to the sum of its proper divisors. For example, 28 is perfect because 28=1+2+4+7+14.
perfect power
An integer is a perfect power if it is of the form m^n where m and n are integers and n>1.
perfect square
An integer is a perfect square if it is of the form m^2 where m is an integer.
The distance around the edge of a multisided figure.
Two straight lines are said to be perpendicular if they meet at right angles.
The ratio of the circumference of a circle to its diameter.
pie chart
A type of chart in which a circle is divided up into portions in which the area of each portion represents the size of the data.
place value
Within a number, each digit is given a place value depending on it's location within the number.
A two-dimensional area in geometry.
In geometry, a point represents a position, but has no size.
A plane figure with many sides.
A planar figure consisting of congruent squares joined edge-to-edge.
positive number
A number larger than 0.
A number multiplied by itself a specified number of times.
practical number
A practical number is a positive integer m such that every natural number n not exceeding m is a sum of distinct divisors of m.
A prime number is an integer larger than 1 whose only positive divisors are 1 and itself.
primitive Pythagorean triangle
A right triangle whose sides are relatively prime integers.
primitive root of unity
The complex number z is a primitive nth root of unity if z^n=1 but z^k is not equal to 1 for any positive integer k less than n.
The chance that a particular event will happen.
The result of multiplying two numbers.
pronic number
proper divisor
The integer d is a proper divisor of the integer n if 0<d<n and d is a divisor of n.
A comparison of ratios.
A three-dimensional solid whose base is a polygon and whose sides are triangles that come to a point at the top.
Pythagorean triangle
A right triangle whose sides are integers.
Pythagorean triple
An ordered set of three positive integers (a,b,c) such that a^2+b^2=c^2.
Abbreviation for quod erat demonstrandum, used to denote the end of a proof.
A closed broken line in the plane consisting of 4 line segments.
quadrangular prism
A prism whose base is a quadrilateral.
quadrangular pyramid
A pyramid whose base is a quadrilateral.
Any one of the four portions of the plane into which the plane is divided by the coordinate axes.
square free
quadratic equation
An equation of the form f(x)=0 where f(x) is a second degree polynomial. That is, ax^2+bx+c=0.
The quadrature of a geometric figure is the determination of its area.
quadric curve
The graph of a second degree equation in two variables.
quadric surface
The graph of a second degree equation in three variables.
A geometric figure with four sides.
An algebraic expression consisting of 4 terms.
quartic polynomial
A polynomial of degree 4.
The first quartile of a sequence of numbers is the number such that one quuarter of the numbers in the sequence are less than this number.
quintic polynomial
A polynomial of degree 5.
The result of a division.
A unit of angular measurement such that there are 2 pi radians in a complete circle. One radian = 180/pi degrees. One radian is approximately 57.3^o.
radical axis
the locus of points of equal power with respect to two circle.
radical center
The radical center of three circles is the common point of interesection of the radical axes of each pair of circles.
Plural of radius.
The length of a stright line drown from the center of a circle to a point on its circumference.
radix point
The generalization of decimal point to bases of numeration other than base 10.
The set of values taken on by a function.
A way of comparing two quantities.
quotient of two numbers.
rational number
A rational number is a number that is the ratio of two integers. All other real numbers are said to be irrational.
real axis
The x-axis of an Argand diagram.
real part
The real number x is called ther eal part of the complex number x+iy where x and y are real and i=sqrt(-1).
real variable
A variable whose value ranges over the real numbers.
The reciprocal of the number x is the number 1/x.
A quadrilateral with 4 right angles.
reflex angle
An angle between 180^o and 360^o.
The number left over when one number is divided by another.
An integer all of whose digits are the same.
repeating decimal
A decimal whose digits eventually repeat.
An integer consisting only of 1's.
A parallelogram with four equal sides.
right angle
an angle formed by two perpendicular lines; a 90^o angle.
right triangle
A triangle that contains a right angle.
roman numerals
A system of numeration used by the ancient Romans.
root of unity
A solution of the equation x^n=1, where n is a positive integer.
round-off error
The error accumulated during a calculation due to rounding intermediate results.
The process of approximating a number to a nearby one.
ruled surface
A surface formed by moving a straight line (called the generator).
rusty compass
A pair of compasses that are fixed open in a given position.
A triangle with unequal sides.
A straight lien that meets a curve in two or more points.
semi-magic square
A square array of n numbers such that sum of the n numbers in any row or column is a constant (known as the magic sum).
A collection of numbers in a prescribed order: a[1], a[2], a[3], a[4], ...
The sum of a finite or infinite sequence
A collection of objects.
similar figures
Two geometric figures are similar if their sides are in proportion and all their angles are the same.
skeleton division
A long division in which most or all of the digits have been replaced by asterisks to form a cryptarithm.
slide rule
A calculating device consisting of two sliding logarithmic scales.
A three-dimensional figure.
solid of revolution
A solid formed by rotation a plane figure about an axis in three-space.
The slanted line in a fraction such as a/b dividing the numerator from the denominator.
The locus of pointsin three-space that are a fixed distance froma given point (called the center).
spherical trigonometry
The branch of mathematics dealing with measurements on the sphere.
A quadrilateral with 4 equal sides and 4 right angles.
square free
An integer is said to be square free if it is not divisible by a perfect square, n^2, for n>1.
square number
A number of the form n^2.
square root
The number x is said to be a square root of y if x^2 = y.
Stirling numbers
A basic operation of arithemtic in which you take away one number from another.
The result of adding two or more numbers.
Two angels are supplementary of they add up to 180^o.
surface area
The measure of a surface of a three-dimensional solid indicating how large it is.
Reflection of a median of a triangle about the corresponding angle bisector.
A line that meets a smooth curve at a single point and does not cut across the curve.
A sentence that is true because of its logical structure.
A polyhedron with four faces.
A four-square polyomino.
Toeplitz matrix
A matrix in which all the elements are the same along any diagonal that slopes from northwest to southeast.
A geometric solid in the shape of a donut.
The trace of a matrix is the sum of the terms along the principal diagonal.
transcendental number
A number that is not algebraic.
A quadrilateral in which no sides are parallel.
A quadrilateral in which two sides are parallel.
A tree is a graph with the property that there is a unique path from any vertex to any other vertex traveling along the edges.
A geometric figure with three sides.
triangular number
A number of the form n(n+1)/2.
An algebraic expression consisting of 3 terms.
A three-square polyomino.
truncated pyramid
A section of a pyramid between its base and a plane parallel to the base.
twin primes
Two prime numbers that differ by 2. For example, 11 and 13 are twin primes.
A surface with only one side, such as a Moebius strip.
A finite sequence is unimodal if it first increases and then decreases.
A square matrix is unimodular if its determinant is 1.
unit circle
A unit circle is a circle with radius 1.
unit cube
A cube with edge length 1.
unit fraction
A fraction whose numerator is 1.
unit square
A unit square is a square of side length 1.
unitary divisor
A divisor d of c is called unitary if gcd(d,c/d) = 1.
A symbol whose value can change.
The rate of change of position.
vertical line
A line that runs up and down and is perpendicular to a horizontal line.
related to intervals of 20.
The horizontal bar in a fraction separating the numerator from the denominator.
The measure of spce occupied by a solid body.
vulgar fraction
A common fraction.
An inequality that permits the equality case. For example, a is less than or equal to b.
A well-formed formula.
whole number
A natural number.
winding number
The number of times a closed curve in the plane passes around a given point in the counterclockwise direction.
witch of Agnesi
A curve whose equation is x^2y=4a^2(2a-y).
Roman numeral for 10.
The horizontal axis in the plane.
The point at which a line crosses the x-axis.
A pentomino in the shape of the letter X.
The vertical axis in the plane.
The point at which a line crosses the y-axis.
A measure of length equal to 3 feet.
A measure of time equal to the period of one revolution of the earth about the sun. Approximately equal to 365 days.
The point at which a line crosses the z-axis.
zero divisors
Nonzero elements of a ring whose product is 0.
zero element
The element 0 is a zero element of a group if a+0=a and 0+a=a for all elements a.
zeta function
The portion of a sphere between two parallel planes. | {"url":"http://www.mathpropress.com/glossary/glossary.html","timestamp":"2014-04-19T17:08:03Z","content_type":null,"content_length":"52633","record_id":"<urn:uuid:7976a234-7048-4b6a-9f28-51b50315689d>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00520-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reply to comment
Submitted by west on August 3, 2012.
With the drug testing, they now do A and B samples - so if sample A comes back positive, they then test the B sample. This is an added layer to prevent the innocent being found guilty, but I guess it
comes down to what made them test positive in the first place. If it's a simple dice-roll random chance thing, then the B sample is 95% likely to then prove a wrongly accused person not-guilty - but
if it's something else (perhaps the drug test looks for markers in their urine which usually signify drug taking but in 5% of the population is natural), then they're still in trouble!
If it's the first case, then with a second test your 590 athletes who test positive in sample A (495 innocent, 95 guilty) becomes 115 (25 innocent 90 guilty) which gives 78% chance of getting the
guilty. A bit more palatable. | {"url":"http://plus.maths.org/content/comment/reply/5757/3505","timestamp":"2014-04-19T12:28:17Z","content_type":null,"content_length":"20547","record_id":"<urn:uuid:074ff01c-aeaa-4d45-98b9-a12bdf57791a>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mathematicians, a coffee table book from Princeton University Press
Just got in the mail a coffee-table book from PUP, which will appeal to you if you like looking at big photographic portraits of mathematicians while you drink your coffee. I do! The pictures, by
Mariana Cook, are agreeable, but what really sells the book for me are the short essays that accompany the photos.
At least two of these would make good openings for novels. Pictured here, Ed Nelson:
I had the great good fortune to be the youngest of four sons with a seven-year gap between my brothers and me, born into a warm and loving family. This was in Georgia, in the depths of the
Depression, where my father organized interracial conferences. He was the sixth Methodist minister in lineal descent. While driving he would amuse himself by mentally representing the license
plate numbers of cars as the sum of four squares.
And Kate Okikiolu:
My mother is British, from a family with a trade-union background and a central interest in class struggle; she met my father, who is Nigerian, while both were students of mathematics in London.
My father was a very talented mathematician, and after my parents married, he went on to a position in the mathematics department of the University of East Anglia. While I was growing up, the
elementary school I attended was extremely ethnically homogeneous. I was unable to escape from heavy issues concerning race, which my mother always explained in a political context. My parents
separated after my father resigned his university position to focus on his inventions, and my mother then finished her education and became a school mathematics teacher.
Less novelistic but very keenly observed is this, from the Vicomte Deligne, on the role of intuition in geometry:
You have more than one picture for each mathematical object. Each of them is wrong but we know how each is wrong. That helps us determine what should be true.
One thought on “Mathematicians, a coffee table book from Princeton University Press”
1. I’m tempted to order this myself. Are all of the photographs of the same fine quality as the cover? Have you ever noticed that skillfully made B&W (grey scale) portraits of people have more
impact and drama than portraits done in color?
Tagged deligne, nelson, okikiolu, photography | {"url":"http://quomodocumque.wordpress.com/2009/06/26/mathematicians-a-coffee-table-book-from-princeton-university-press/","timestamp":"2014-04-17T03:49:21Z","content_type":null,"content_length":"61939","record_id":"<urn:uuid:e0b96b42-26e7-4117-a69f-c3f341deee32>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00410-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Starmap: Publications
Theobald, D.M., D.L. Stevens, Jr., D. White, N.S. Urquhart, A.R. Olsen, and J.B. Norman Using GIS to generate spatially-balanced random survey designs for natural resource applications. To appear in
Environmental Management
Ritter, K.J. and M.K. Leecaster (2007). Multi-Lag cluster enhancement of fixed grid sample designs for estimating the variogram in near coastal systems. Environmental and Ecological Statistics 14: .
Gitelman, A.I. and A.T. Herlihy (2007). Isomorphic chain graphs for modeling spatial dependence in ecological data. To appear in Environmental and Ecological Statistics 14: .
Ver Hoef, J.M., E.E. Peterson, and D.M. Theobald (2006). Spatial statistical models that use flow and stream distance. To appear in Environmental and Ecological Statistics 13: .
Andrews, B., R.A. Davis, and F.J. Breidt (2006). Rank-based estimation for all-pass time series models. To appear in the Annals of Statistics .
Thomas, D.E., B. Griffith, and D.S. Johnson (2006). A Bayesian random effects discrete-choice model for resource selection: Population-level selection inference. Journal of Wildlife Management 70:
Dailey, M.C, A.I. Gitelman, F.L. Ramsey, and S. Starcevich (2007). Habitat selection models to account for seasonal persistence in radio telemetry data. To appear in Environmental and Ecological
Statistics 14: .
Breidt, F.J., N.-J. Hsu, and S. Ogle (2006). A semiparametric stochastic mixed model for increment-averaged data with application to carbon sequestration in agricultural soils. To appear in the
Journal of the American Statistical Association
Brockwell, P.J., R.A. Davis, and V. Yang (2006). Continuous-time Gaussian autoregression. To appear in Statistica Sinica.
Wang, H. and M.G. Ranalli (2006). Low-rank smoothing splines on complex domains. To appear in Biometrics 62:
Peterson, E.E. and N.S. Urquhart (2006). Predicting water quality impaired stream segments using landscape-scale data and a regional geostatistical model: A case study in Maryland. To appear in
Environmental Monitoring and Assessment
Peterson, E.E., A.A. Merton, D.M. Theobald, and N.S. Urquhart (2006). Patterns of spatial autocorrelation in stream water chemistry. To appear in Environmental Monitoring and Assessment
Farnsworth, M.L., J.A. Hoeting, N.T. Hobbs, and M.W. Miller (2006). Linking mule deer movement scales to the spatial distribution of chronic wasting disease: a hierarchical Bayesian approach. To
appear in Ecological Applications 16:
Breidt, F.J., N.-J. Hsu, and W.J. Coar (2006). A diagnostic test for autocorrelation in increment-averaged data with application to soil sampling. To appear in Environmental and Ecological Statistics
Stevens, Jr. D.L. (2006). Spatial properties of design-based versus model-based approaches to environmental sampling. ACCURACY 2006: Proceedings of the 7th International Symposium on Spatial Accuracy
Assessment in Natural Resources and Environmental Sciences. July 5-7, 2006, Lisbon, Portugal pp 119-125 of 908.
Andrews, B., R.A. Davis, and F.J. Breidt (2006). Maximum likelihood estimation for all-pass time series models. Journal of Multivariate Analysis 97:1638-1659. {available online}
Hoeting, J.A., R.A. Davis, A.A. Merton, and S.E. Thompson (2006). Model selection for geostatistical models. Ecological Applications 16: 87-98.
Theobald, D.M. (2006). Exploring the functional connectivity of landscapes using landscape networks. To appear in: Crooks, K.R. and M.A. Sanjayan (eds.). Connectivity Conservation: Maintaining
Connections for Nature. Cambridge University Press. (Book to be published December, 2006)
Davis, R.A., T.C.M. Lee, and G. Rodriguez-Yam (2006). Structural break estimation for nonstationary time series models. Journal of the American Statistical Association 101:223-239.
Opsomer, J.D., F.J. Breidt, G.G. Moisen, and G. Kauermann (2006). Model-assisted estimation of forest resources with generalized additive models. To appear in the Journal of the American Statistical
101: as a Discussed Paper in Applications and Case Studies.
Johnson, D.S., J.A. Hoeting, and N.L. Poff (2006). Biological monitoring: A Bayesian model for multivariate compositional data. pp 270-289 in Bayesian Statistics and its Applications, Eds. S. K.
Upadhyay, U. Singh and D. K. Dey. Anamaya publishers: New Delhi.
Hoeting, J. A. (2006). Some perspectives on modeling species distributions. Discussion of article by A.E. Gelfand, J.A. Silander, S. Wu, A. Latimer, P.O. Lewis, A.G. Rebelo, M. Holder. Bayesian
Analysis 1: 93-98. {Web-accessible here}
Breidt, F.J., G. Claeskens, and J.D. Opsomer (2005). Model-assisted estimation for complex surveys using penalized splines. Biometrika 92: 831-846.
Davis, R.A., T.C.M. Lee, and G. Rodriguez-Yam (2005). Structural break estimation for nonstationary time series signals. Proceedings of IEEE/SP 13th Workshop on Statistical Signal Processing,
Bordeaux, France, July, 2005.
Montanari, G.E. and M.G. Ranalli (2005). Nonparametric methods for sample surveys of environmental populations. Proceedings of the Meeting of the Italian Statistical Society on Statistics and the
Environment. September 21-23, 2005, Messina, Italy CLEUP: 147-158.
Davis, R.A. and G. Rodriguez-Yam (2005). Estimation for state-space models based on a likelihood approximation. Statistica Sinica 15: 381-406.
Breidt, F.J. and N.-J. Hsu (2005). Best mean square prediction for moving averages. Statistica Sinica 15: 427-446.
Givens, G.H. and J.A. Hoeting (2005). Computational Statistics. John Wiley & Sons, NewYork, 418 pages. {Information is web accessible here}
Francisco-Fernandez, M. and J.D. Opsomer (2005). Smoothing parameter selection methods for nonparametric regression with spatially correlated errors. Canadian Journal of Statistics 33: 279-295.
{Abstract web-accesible here}
Opsomer J.D. and C.P. Miller (2005). Selecting the amount of smoothing in nonparametric regression estimation for complex surveys. Journal of Nonparametric Statistics 17: 593-611.
Montanari, G.E. and M.G. Ranalli (2005). Nonparametric model calibration estimation in survey sampling. Journal of the American Statistical Association 100: 1429-1442.
Reese, G.C., K.R. Wilson, J.A. Hoeting, and C.H. Flather (2005). Factors affecting the accuracy of predicted species distributions: A simulations experiment. Ecological Applications 15: 554-564.
Hall, P. and J.D. Opsomer (2005). Theory for penalised spline regression. Biometrika 92: 105-118.
Kauermann, G. and J.D. Opsomer (2004). Generalized cross-validation for bandwidth selection of backfitting estimators in generalized additive models. Journal of Computational and Graphical Statistics
13: 66-89. {Abstract web-accessible here}
Opsomer, J.D., F.J. Breidt, G. Claeskens, G. Kauermann, and M.G. Ranalli (2004). Nonparametric small area estimation using penalized spline regression. Proceedings of the Section on Survey Research
Methods [CD-ROM], Alexandria, VA. American Statistical Association: 4127-4134.
Kahl, J.S. Stoddard, J.L. Haeuber, R., Paulsen, S.G., Birnbaum, R., Deviney, F.A., DeWalle, D.R., Driscoll, C.T., Herlihy, A.T., Kellogg, J.H., Peter, J.H., Murdoch, S., Roy, K., Sharpe, W.,
Urquhart, N.S., Webb, J.R., and Webster, K.E. (2004). Response of surface water chemistry to changes in acidic deposition: implications for the upcoming debate on the federal Clean Air Act.
Environmental Science and Technology 38: 484A-490A.
Kincaid, T.K., Larsen, D.P., and Urquhart, N.S. (2004). The structure of variation and its influence on the estimation of status: Indicators of condition of lakes in Northeast, U.S.A. Environmental
Monitoring and Assessment 98: 1-21.
Courbois, J. P. and N. S. Urquhart (2004). Comparison of survey estimates of finite population variance. Journal of the Agricultural, Biological and Environmental Statistics 9: 236-251. {Abstract
web-accessible here}
da Silva, D.N. and J.D. Opsomer (2004). Properties of the weighting cell estimator under a nonparametric response mechanism. Survey Methodology 30: 45-55.
Waite, I.R., Herlihy, A.T., Larsen, D.P., Urquhart, N.S. and Klemm, D.J. (2004) The effects of macroinvertebrate taxonomic resolution in large landscape bioassessments: an example from the
Mid-Atlantic Highlands, U.S.A. Freshwater Biology 49: 474–489.
Larsen, D. P., Kaufmann, P.K., Kincaid, T.M. and Urquhart, N.S. Detecting persistent change in the habitat of salmon-bearing streams in the Pacific Northwest. Canadian Journal of Fisheries and
Aquatic Sciences 61: 283-291.
Urquhart, N.S. and J.C. Moore (2004). Statistics in EPA’s STAR program: Learning materials for surface water monitoring. OPPTS Tribal News 4(3): 45 - 46. (EPA 745-N-00-001)
Opsomer, J.D., C. Botts and, J.Y. Kim (2003). Small area estimation in a watershed erosion assessment survey. Journal of Agricultural, Biological and Environmental Statistics 8:139-152. {Abstract
web-accessible here}
Opsomer, J.D. and F.J. Breidt (2003). Nonparametric estimation in complex surveys with auxiliary information. Proceedings of the 54th Session of the International Statistical Institute.
Johnson, D.S. and Hoeting, J.A. (2003). Autoregressive models for capture-recapture data: A Bayesian approach. Biometrics 59: 341 - 350. {Abstract web-accessible}
Davis, R.A., W.T.M. Dunsmuir, and S.B. Streett (2003). Maximum likelihood estimation for an observation driven model for Poisson counts. Biometrika 90: 770-790.
Hoeting, J. (2002). Methodology for Bayesian model averaging: An update. Proceedings Invited paper presentation, International Biometric Conference, Freiburg, Germany, 231-240.
Larsen, D. P., T. K. Kincaid, S. E. Jacobs and N. S. Urquhart (2001). Designs for evaluating local and regional scale trends. Bioscience 51: 1069-1078.
Urquhart, N.S. (in review - 2010). Detecting trend in long-term ecological studies. a book chapter. Follow this link to access the data and programs used in developing the example for this chapter.
Theobald, D.M., J.B. Norman, and M.R. Sherburne (2006). FunConn v1: Functional connectivity tools for ArcGIS v9. Natural Resource Ecology Lab, Colorado State University. {Tools are web-available
through this link}
Theobald and D.M Norman (2006). Spatially-balanced Sampling (RRQRR). Natural Resource Ecology Lab, Colorado State University. {Tools are web-available through this link}
Theobald, D.M., J.B. Norman, E.E Peterson, and S.B. Ferraz (2005). FLoWS v1: Functional Linkage of Watersheds and Streams tools for ArcGIS v9. Natural Resource Ecology Lab, Colorado State University.
{Tools are web-available through this link}
Merton, A.A., J.A. Hoeting, and R.A. Davis (2004). Model Selection for Geostatistical Models. Software to compute AIC and MDL for geostatistical models for R. {Tool is web-available through this link
Theobald, D.M., J.B. Norman, E. Peterson, S. Ferraz, A. Wade, and M.R. Sherburne (2006). Functional Linkage of Water Basins and Streams (FLoWS) v1 User’s Guide: ArcGIS Tools for Network-based
Analysis of Freshwater Ecosystems. Natural Resource Ecology Lab, Colorado State University, Fort Collins, CO. 43 pages.
Theobald, D.M., J.B. Norman, and M.R. Sherburne (2006). FunConn v1 User’s Manual: ArcGIS tools for Functional Connectivity Modeling. Natural Resource Ecology Lab, Colorado State University. April 17,
2006. 47 pages.
Johnson, D.S (2005). Bayesian inference for geostatistical regression models. Technical Report, Department of Statistics, Colorado State University, Fort Collins. 35pp.
Araas, J.F. (2005). EMAP West Training Presentations. Masters Project Report, Department of Statistics, Colorado State University, Fort Collins. 14 pages of text; 162 pages of reduced size pdf.
{Parts are accessible separately: text 48K; pdf of PowerPoint 16.2M}
Smith, J.J (2005). Modeling and Predicting Median Substrate Size in Oregon and Washington Streams Utilizing Geographic Information Systems Data. Masters Project Report, Department of Statistics,
Colorado State University, Fort Collins. 68pp
French, J. (2005) Exploring Spatial Correlation in Rivers. Masters Project Report, Department of Statistics, Colorado State University, Fort Collins. 60pp
Lowe, S., B.E. Thompson, R. Hoenicke, J.E. Leatherbarrow, K. Taberski, R.W. Smith, and D.L. Stevens, Jr. (2005). Re-design Process of the San Francisco Estuary Regional Monitoring Program for Trace
Substances (RMP) Status and Trends Monitoring Component for Water and Sediment. Contribution 109, San Francisco Estuary Institute, 7770 Pardee Lane, Oakland, CA 94621.
Scarzella, G. (2003) Learning Materials for Surface Water Monitoring. (10.2M; text only = 0.5M) Masters Project Report, Department of Statistics, Colorado State University, Fort Collins. 91pp
Johnson, D.S. (2003) Bayesian Analysis of State-Space Models for Discrete Composition, Ph.D. thesis, Department of Statistics, Colorado State University (will be web-available at a later date)
Everson-Stewart, S. (2003). Nonparametric Survey Regression Estimation in Two-Stage Spatial Sampling. Masters Project Report, Department of Statistics, Colorado State University, Fort Collins. 49pp
Johnson, A. A. (2003). Estimating Distribution Functions from Survey Data Using Nonparametric Regression. Masters Project Report, Department of Statistics, Colorado State University, Fort Collins.
Johnson, S.W. (2003). Needs Assessment of Tribal Requirements for Instruction in the Use of Statistically-Based Aquatic Water Quaility Monitoring Techniques. Water Quality Technology, Inc., Fort
Collins, CO. 12pp
Kellum, B. (2002). Analysis and Modeling of Acid Neutralizing Capacity in the Mid-Atlantic Highlands Area. Masters Project, Department of Statistics, Colorado State University, Fort Collins. 69pp
Johnson, D.S. and Hoeting, J.A. (2001). Autoregressive Models for Capture-Recapture Data: A Bayesian Approach. Technical Report 2001-10, Department of Statistics, Colorado State University.
Presentations from the STATISTICAL SURVEY DESIGN AND ANALYSIS FOR AQUATIC RESOURCES - Fourth Annual Meeting. Oregon State University, September 7- 9, 2005.
Presentations from Roles for Statistics in 21^st Century Monitoring and Assessment Systems at the Monitoring Science and Technology Symposium, Denver, CO September 21 - 24, 2004.
Presentations from the STATISTICAL SURVEY DESIGN AND ANALYSIS FOR AQUATIC RESOURCES - Third Annual Meeting. Colorado State University, September 10 - 11, 2004
Printer friendly copy of abstracts (pdf) - available as provided. Abstracts contain links to the presentation materials, as available.
Presentations from the STATISTICAL SURVEY DESIGN AND ANALYSIS FOR AQUATIC RESOURCES - Second Annual Meeting. Oregon State University, August 11-12, 2003
Presentations from the STATISTICAL SURVEY DESIGN AND ANALYSIS FOR AQUATIC RESOURCES - First Annual Meeting. Colorado State University, September 20 - 21, 2002
Printer friendly copy of abstracts (pdf) - available as provided. Abstracts contain links to the presentation materials, as available.
Ritter, K.J. Multi-Lag Cluster Enhancement of Fixed Grids for Variogram Estimation in Near-Coastal Systems, California and the World Ocean Conference 2006, as part of the Seafloor mapping session
(September 17-20, 2006). {Conference link}
Coar, W.J. and F.J. Breidt. Smoothing through State-Space Models for Stream Networks. Joint Statistical Meetings, Seattle, WA, August 6-10, 2006 and Spring Meeting, Colorado/Wyoming Chapter of the
American Statistical Association, Boulder, CO, April 21, 2006
Opsmer, J.D, F.J. Breidt, G. Moisen and G. Kauermann. Model-Assisted Estimation of Forest Resources with Generalized Additive Models. (Invited) JASA, Applications and Case Studies. Joint Statistical
Meetings, Seattle, WA, August 6-10, 2006
Hoeting, J.A. Model Selection and Estimation for Geostatistical Models. (Invited) Joint Statistical Meetings, Seattle, WA, August 6-10, 2006
Higgs, M.D., J.A. Hoeting, and B. Bledsoe. Bayesian modeling for ordinal substrate size using EPA stream data. Joint Statistical Meetings, Seattle, WA, August 6-10, 2006.
Park, M.S. and M. Fuentes. A new class of asymmetric spatio-temporal covariance models. Joint Statistical Meetings, Seattle, WA, August 6-10, 2006
Irvine, K.M., A.I. Gitelman, and J.A. Hoeting. Spatial designs and strength of spatial signal: effects on covariance estimation. Joint Statistical Meetings, Seattle, WA, August 6-10, 2006
Alix Gitelman will moderate a roundtable on Statistical Consulting at the Joint Statistical Meetings, Seattle, WA, August 6-10, 2006. This discussion will include her experience obtained as a
participant in STARMAP.
Jennifer Hoeting, a STARMAP Principal Investigator, and Geof Givens presented a short course entitled Statistical Computing: Techniques for Integration and Optimization two different places:
Alaska Chapter of the American Statistical Association, Juneau, AK, July, 2006, and
Joint Statistical Meetings, Seattle, WA, August 6-10, 2006
Don Stevens and Tony Olsen will present the short course Spatial Sampling at the Joint Statistical Meetings, Seattle, WA, August, 6-10, 2006.
Stevens, Jr. D.L., D.P. Larsen, and A.R. Olsen. Using a Master Sample to Coordinate Monitoring of Multiple Species (Invited). The International Environmetrics Society, June 18-22, 2006, Kalmar,
Hoeting, J.A. presented two talks, Introduction to Bayesian Data Analysis Methods, and An Introduction to WinBUGS, to the PRIMES Workshop on Bayesian Methods in Wildlife Population Monitoring,
Colorado State University, Fort Collins, CO, June 14-16, 2006.
Peterson, E.E., A.A. Merton, D.M. Theobald, and N.S. Urquhart. Patterns of Spatial Autocorrelation in Stream Water Chemistry. North American Benthological Society, Anchorage, Alaska, June 4-9, 2006
Coar, W.J. and F.J. Breidt. State-Space Models for Within-Stream Network Dependence. North American Benthological Society, Anchorage, Alaska, June 4-9, 2006
Theobald, DM. and J.B. Norman. FLoWS Tools. Workshop on Predicting Salmon Habitat by the Alaska Department of Fish & Game, May 17-19, 2006, Anchorage, AK.
Breidt, F.J. Uncertainty Analysis for a US Inventory of Soil Organic Carbon Stock Changes. Workshop on Uncertainty in Ecological Analysis, Mathematical Biosciences Institute, The Ohio State
University, Columbus, OH, April 3, 2006
Hoeting, J.A. Invited discussant in the session on Modeling in the Presence of Uncertainty, Workshop on Uncertainty in Ecological Analysis, Mathematical Biosciences Institute, The Ohio State
University, Columbus, OH, April 3-6, 2006. She also served as group leader for a group project where workshop participants analyzed ecological data at the meeting.
Breidt, F.J. Spatial Lasso with Application to GIS Model Selection. Invited - ENAR Spring Meeting, Tampa, FL, March 28, 2006.
Stevens, D.L., Jr. Probabilistic Sampling: Can Lessons from Coastal River Systems Be Applied to a Mainstem Species? Workshop in on Monitoring White Sturgeon in the Columbia Basin sponsored by the
Columbia Basin Fish and Wildlife Authority, Spokane, WA, March 14-15, 2006
Peterson, E.E., A.A. Merton, D.M. Theobald, and N.S. Urquhart. Patterns of Spatial Autocorrelation in Stream Water Chemistry. Center for Riverine Landscapes Seminar Series, Griffith University,
Brisbane, Australia. March 4, 2006
Urquhart, N.S. Designing Surveys over Time (Panel Surveys): Variance, Power and Related Topics. National Park Service, Inventory and Monitoring Program, San Diego, February 10, 2006
Park, M.S. Symmetry and Separability in Spatial-Temporal Processes. Seminar, Department of Statistics, Colorado State University, February 1, 2006
Ranalli, M.G. Low-rank Smoothing Splines on Domains with Unusual Shapes: Nonparametric Approximations for Estuaries and Nets of Rivers. Seminar, Department of Economics, Finance and Statistics,
University of Perugia, Italy, January 13, 2006
Araas, J.F. EMAP West Training Presentations. Teleconference Seminar, Department of Statistics, Colorado State University, December 15, 2005
Smith, J.J. Modeling and Predicting Median Substrate Size in Oregon and Washington Streams Utilizing Geographic Information Systems. Seminar, Department of Statistics, Colorado State University,
December 6, 2005
Peterson, E.E., A.A. Merton, D.M. Theobald, and N.S. Urquhart. Predicting Water Quality Impaired Stream Segments Using Landscape-Scale Data and a Regional Geostatistical Model. Statistical Society of
Australia, St. Lucia, Queensland, Australia. October 18, 2005
Montanari, G.E. and M.G. Ranalli. Nonparametric Methods for Sample Surveys of Environmental Populations. Invited talk, Annual Meeting of the Italian Statistical Society on Statistics and the
Environment, September 21-23, 2005, Messina, Italy
Larsen, D.P. and N.S. Urquhart. Designs for Estimating Variability Structure and Implications for Detecting Watershed Restoration Effectiveness. American Fisheries Society, Anchorage, AK, September
11-15, 2005.
Urquhart, N.S. Overview of the STARMAP Program . Fourth Annual Conference on Statistical Survey Design and Analysis for Aquatic Resources, Corvallis, OR, September 7 - 9, 2005.
Peterson, E.E A.A. Merton, N.S. Urquhart*, D.M. Theobald, and J.A. Hoeting. Using the Maryland Biological Stream Survey Data to Test Spatial Statistical Models. Second Maryland Stream Conference,
Carroll College, Westminster, MD, August 10 - 13, 2005.
Breidt, F.J. Extensions of Penalized Spline Regression for Natural Resource Monitoring Applications. Joint Statistical Meetings, Minneapolis, MN, August 7 - 11, 2005.
Ranalli, M.G., F.J Breidt, and H. Wang. Low-rank Smoothing Splines for Unusual Spatial Structures: Smoothing Estuaries and Stream Networks. WNAR and IMS Statistics Meeting, University of Alaska,
Faribanks, AK, June 22, 2005
French, J. Exploring Spatial Correlation in Rivers. Spring Meeting, Colorado/Wyoming Chapter of the American Statistical Association, Boulder, CO, April 22, 2005; Seminar, Department of Statistics,
Colorado State University, April 4, 2005
Opsomer, J.D. Small Area Estimation Using Penalized Spline Regression. Invited talk, International Biometric Society, Eastern North American Region meeting, Austin, TX, March 21, 2005.
Urquhart, N.S. An Academician’s View of EPA’s Ecology Program, Especially its Environmental Monitoring and Assessment Program (EMAP). Ecological Research Subcommittee of U.S. EPA Board of Scientific
Counselors, Research Triangle Park, NC, March 7-9, 2005.
Ranalli, M.G. and H. Wang. Low-rank Smoothing Splines for Complex Domains and Manifold Recovery. Seminar, University of Virginia, Charlottesville, VA. March 4, 2005.
Ranalli, M.G., F.J. Breidt, and H. Wang. Low-rank Smoothing Splines on Complex Domains. Seminar, Atlantic Ecology Division, EPA, Narragansett, RI. March 1, 2005.
Hoeting, J.A and D. Johnson. Biological Monitoring: Bayesian Models for a Multivariate Response. Invited talk, International Workshop/Conference on Bayesian Statistics and its Applications, Banaras
Hindu University, Varanasi, India, January 5 - 8, 2005.
Urquhart, N.S. and A.R. Olsen. Anatomy of Sampling Studies of Ecological Responses Through Time. Monitoring Science and Technology Symposium. Denver, CO, September 21 - 24, 2004.
Davis, R.A and T. Mikosch. Extreme Value Theory for Space-Time Processes With Heavy-Tailed Distributions. International Symposium on Extreme Value Analysis: Theory and Practice, Aveiro, Portugal,
July 17 - 24, 2004.
Davis, R.A., J.A. Hoeting, A.A. Merton, and S. Thompson. Model Selection for Geostatistical Models. (Keynote address). International Biometric Conference, Cairns, Australia, July 11 - 16, 2004.
Davis, R. A. and G. Rodriguez-Yam. Structural Break Detection in Time Series. (Keynote address). International Workshop on Recent Advances in Time Series Analysis, Protaras, Cyprus, June 10, 2004.
Davis, R. A. and G. Rodriguez-Yam. Parameter and Observation-Driven State Space Models. (Keynote address). International Workshop on Recent Advances in Time Series Analysis, Protaras, Cyprus, June 9,
Urquhart, N.S. Uses of Power in Designing Long-Term Environmental Surveys. Seminar, Alterra Institute, Wageningen University, Wageningen, The Netherlands, June 6, 2004.
Hoeting, J.A and N.S. Urquhart. Colorado State University’s EPA-Funded Program on Space-Time Aquatic Resources Modeling and Analysis Program (STARMAP). Environmental Research Seminar, EPA Region 8,
Denver, CO May 12, 2004.
Urquhart, N.S. A Statistical Perspective on Linking Sections 305(b) and 303(d) of the Clean Water Act. EMAP Symposium 2004, Newport RI, May 5, 2004.
Breidt, F.J. Linking CWA Sections 305(b) and 303(d): Small Area Estimation. EMAP Symposium 2004, Newport RI, May 5, 2004.
Urquhart, N.S. Statistical Aspects of Collections of Bees to Study Pesticides. Workshop sponsored by EPA's Office of Pollution Prevention and Toxics, Newport RI, May 3, 2004.
Urquhart, N.S. Uses of Power in Designing Long-Term Environmental Surveys. Seminar, San Diego Chapter of the American Statistical Association, March 29, 1004.
Ranalli, M. Giovanna (2004). Nonparametric Model Calibration Estimation in Survey Sampling. Seminar, Department of Statistics, Iowa State University, Ames, IA, February 18, 2004.
Urquhart, N. Scott (2002). Possible Lessons for CEER-GOM from EMAP. All-Hands Conference, Consortium for Estuarine Ecoindicators Research for the Gulf of Mexico, Ocean Springs, MS, March 3,2002. | {"url":"http://www.stat.colostate.edu/~nsu/starmap/publications.html","timestamp":"2014-04-19T17:31:52Z","content_type":null,"content_length":"69824","record_id":"<urn:uuid:86a787e4-6674-4913-8add-3d4b6d47e5db>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00255-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Global stability for an SIR epidemic model with delay and nonlinear incidence.
(English) Zbl 1197.34166
Summary: A recent paper [
R. Xu
Z. Ma
, Nonlinear Anal., Real World Appl. 10, No. 5, 3175–3189 (2009;
Zbl 1183.34132
)] presents an
model of disease transmission with delay and nonlinear incidence. The analysis there only partially resolves the global stability of the endemic equilibrium for the case where the reproduction number
is greater than one. In the present paper, the global dynamics are fully determined for
by using a Lyapunov functional. It is shown that the endemic equilibrium is globally asymptotically stable whenever it exists.
34K60 Qualitative investigation and simulation of models
34K20 Stability theory of functional-differential equations
92D30 Epidemiology | {"url":"http://zbmath.org/?q=an:1197.34166","timestamp":"2014-04-21T02:19:40Z","content_type":null,"content_length":"21331","record_id":"<urn:uuid:8b245032-91da-49fe-b96a-608c90818db8>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00018-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kirchoff's Laws
At every node, the sum of all currents entering a node must equal zero. What this law means physically is that charge cannot accumulate in a node; what goes in must come out. In the example, Figure 1
, below we have a three-node circuit, and thus have three KCL equations.
i 1 − i 2 =0 i 1 i 2 0
Note that the current entering a node is the negative of the current leaving.
Given any two of these KCL equations, we can find the other by adding or subtracting them. Thus, one of them is redundant and, in mathematical terms, we can discard any one of them. The convention is
to discard the one for the (unlabeled) node at the bottom of the circuit.
Figure 1:
shown is
performs a
The input
by the
source v
in v in
and the
output is
voltage v
out v out
across the
label R 2
R 2 .
Figure 1
Exercise 1
In writing KCL equations, you will find that in an nn-node circuit, exactly one of them is always redundant. Can you sketch a proof of why this might be true? Hint: It has to do with the fact that
charge won't accumulate in one place on its own.
KCL says that the sum of currents entering or leaving a node must be zero. If we consider two nodes together as a "supernode", KCL applies as well to currents entering the combination. Since no
currents enter an entire circuit, the sum of currents must be zero. If we had a two-node circuit, the KCL equation of one must be the negative of the other, We can combine all but one node in a
circuit into a supernode; KCL for the supernode must be the negative of the remaining node's KCL equation. Consequently, specifying n−1 n 1 KCL equations always specifies the remaining one. | {"url":"http://cnx.org/content/m0015/2.1/","timestamp":"2014-04-19T22:22:42Z","content_type":null,"content_length":"46086","record_id":"<urn:uuid:29ca84b4-59d6-4579-ab33-9c053921d6b6>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00299-ip-10-147-4-33.ec2.internal.warc.gz"} |
Elementary Differential Equations and Linear Algebra
MATH 2090
Elementary Differential Equations and Linear Algebra
Fall 2000
│Section 3 │Section 4 │Instructor: Dan Cohen │Phone: 578-1576 │
│M T W Th 11:30│M T W Th 12:30│Office Hours: M W 9:30 - 11:30│E-mail: cohen@math.lsu.edu │
│134 Lockett │134 Lockett │Office: 372 Lockett │URL: http://www.math.lsu.edu/~cohen │
Text: Differential Equations and Linear Algebra, Second Edition, by Stephen W. Goode.
We will cover (portions of) Chapters 1-9, and may cover some material not included in the text if time permits.
Catalog Description: Introduction to first order differential equations, linear differential equations with constant coefficients, and systems of differential equations; vector spaces, linear
transformations, matrices, determinants, linear dependence, bases, systems of equations, eigenvalues, eigenvectors, Laplace transforms, and Fourier series.
Homework: I will assign homework problems essentially every class. Homework will not be collected. Homework assignments will be announced in class, posted on my web page, and occasionally discussed
in class as necessary.
Quizzes: There will a number of short (approximately 10 minute) quizzes in class throughout the semester. Quiz problems will be similar to the problems you encounter in the homework. There will be no
make-up quizzes, but your lowest quiz score will be dropped. In total, the quizzes will be worth 100 points.
Exams: There will be four hour-long, in-class exams, each worth 100 points. These exams will tentatively take place during the weeks of September 4, September 25, October 16, and November 6. Exam
dates will be announced in class. No make-up exams, except in extreme cases. If you must miss an exam, you should notify me before the exam takes place.
Final: There will be a comprehensive final exam worth 200 points.
For Section 3, the final exam is scheduled for Saturday, December 9, 10:00 am - 12:00 noon.
For Section 4, the final exam is scheduled for Friday, December 8, 12:30 - 2:30 pm.
Grade: Your course grade will be out of the 700 possible points outlined above. I typically curve course grades. However, 90-100% is assured an A, 80-89% a B, and so on.
Important Dates:
│August 28 last day to drop │September 4 Labor Day │
│August 30 last day to add │October 5-6 Fall Holiday │
│November 5 last day to withdraw │November 23-24 Thanksgiving│
Notes: As the titles of the course and text indicate, our objective is to introduce differential equations and linear algebra. These two subjects are tools you will undoubtably encounter and have
occasion to use as you progress through your science and/or engineering courses. We will develop methods for solving (certain classes of) differential equations and problems in linear algebra. We
will also endeavor to develop an understanding of why these methods work. To this end, we will work with objects known as vector spaces. The ideas we develop will, among other things, provide us with
a means of understanding the theory underlying the solution of linear differential equations. There is, however, no such thing as a free lunch. The level of abstraction in, for instance, chapters 5
and 6 will probably be somewhat higher than what you've encountered in prior courses.
In general, to stay on top of the material, it is important that you attend class, read the text, and do the homework regularly.
Bear in mind that you are taking this course under the guidelines of the Code of Student Conduct.
Dan Cohen Fall 2000
Back to my homepage. File translated from T[E]X by T[T]H, version 2.25.
On 14 Aug 2000, 17:59. | {"url":"https://www.math.lsu.edu/~cohen/courses/PastSemesters/FALL00/M2090syl.html","timestamp":"2014-04-16T22:16:44Z","content_type":null,"content_length":"7292","record_id":"<urn:uuid:f1188d70-91b2-4881-94c9-040f7bf6be1b>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00262-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SciPy-user] Derivative() usage?
Giovanni Samaey Giovanni.Samaey at cs.kuleuven.ac.be
Wed Nov 3 07:48:28 CST 2004
> 'n' as in d^n/dx^n .
> Somewhat confusingly, 'N' is being used in the docstring as a synonym
> for 'order' and is the number of discrete points used to evaluate the
> numerical derivative. I'm going to fix that.
> For example, when n=2, and order=3, one is computing the second
> central derivative using 3 points [x0-dx, x0, x0+dx].
Normally, one would expect "order" to be an indication of the accuracy
of the derivative. So, order=p should mean
d^n/dx^n (exact answer) - (what's computed) = O(dx^p) where dx is the
mesh spacing.
Clearly, with three points for the second derivative, the order is only
two, so this is clearly not a mathematically correct use of the word
Giovanni Samaey
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Emergency Panicking (NO TIME) Please Help! Year 10 Maths.
October 20th 2012, 08:54 PM #1
Oct 2012
Hello Girls and Guys lesbianista@hotmail.com. Oh and by the way we got this on Friday to do over the weekend.
An aeroplane has
crashed in the desert at
a point that could be
labelled (−15, 12) on a
particular grid system.
A search party sets out
by 4WD from the point
(18, 3) and travels in a
straight line passing
through the point (3, 9).
A light plane sets out
from point (3, −15) and
travels in a straight line
passing over the point
(−3, −6).
1. Will either of the search parties pass the point where the aeroplane lies if they continue
on these paths? Explain how you obtained your answer.
2. If either (or both) of the original routes does not pass the point where the aeroplane lies,
give a different point (but with the same x-coordinate) that the searchers should pass
through (or over) to be on course to locate the aeroplane.
How my teacher wants it set out
1) Draw a labelled Diagram of information given in 2-D with a labelled set of axis and points clearly labelled.
2) Use above diagram to write equation of line created by both search parties (with working)
3) Show algebraically whether each party will pass where the plane crashed.
4)If either or of both planes do not line on the correct path;
(a)Find a new point (but with the same x-coordinate) that the searches should pass over to be on course to find the aeroplane.
(b) Prove algebraically that this new route will find the aeroplane.
Last edited by sadsadsadsa; October 20th 2012 at 08:59 PM.
Re: Emergency Panicking (NO TIME) Please Help! Year 10 Maths.
First this is a homework problem. We do not do your homework. Show us how far you have got on it.
Second this is the forum to introduce yourself, not a forum to submit questions.
October 20th 2012, 09:45 PM #2 | {"url":"http://mathhelpforum.com/new-users/205764-emergency-panicking-no-time-please-help-year-10-maths.html","timestamp":"2014-04-18T01:26:08Z","content_type":null,"content_length":"35711","record_id":"<urn:uuid:02b04bdd-8d9d-492a-a5eb-750dedfe6b88>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00106-ip-10-147-4-33.ec2.internal.warc.gz"} |
Items where Research Group is "Mathematical Geoscience Group" and Year is 1989
Number of items: 3.
Fowler, A. C. (1989) Generation and creep of magma in the Earth. SIAM J. Appl. Math., 49 . pp. 231-245.
Fowler, A. C. (1989) A mathematical analysis of glacier surges. SIAM J. Appl. Math., 49 . pp. 246-262.
Fowler, A. C. (1989) Secondary frost heave in freezing soils. SIAM J. Appl. Math., 49 . pp. 991-1008. | {"url":"http://eprints.maths.ox.ac.uk/view/groups/mgg/1989.default.html","timestamp":"2014-04-20T08:35:43Z","content_type":null,"content_length":"7561","record_id":"<urn:uuid:99ec97f7-98ef-452d-870b-bd21311e8ccb>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00647-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Functional Language with Inductive and Coinductive Types
The functional language lemon is based on the simply-typed lambda calculus augmented with sums, products, and the mu and nu constructors for least (inductive) and greatest (coinductive) solutions to
recursive type equations. The term constructors of the language strictly follow the introduction and elimination rules for the corresponding types; in particular, the elimination for mu is iteration
and the introduction for nu is coiteration (also called generation). It includes a small amount of polymorphism and type inference; lambda-bound variables do not need type annotations, but iteration
and coiteration need to have their corresponding recursive types specified (this is a problem with the language rather than the implementation).
For example, the following program generates the stream of Fibonacci numbers starting with 1,1:
type nats = nu X. nat * X; -- nat stream is a pair of a nat and a stream
val fibs = <|nats, \p. <pi1 p, -- head of stream is next Fibo. number
<pi2 p, plus (pi1 p) (pi2 p)>>
-- "seed" for rest of stream
|> <1, 1>; -- initial seed for generation
Using iteration we may define a function which picks off the first n elements of a stream and returns them as a list:
val pick = \n. -- number of elements to pick
\ns. -- input stream
pi1 ([|nat, [\x. <nil, ns>, -- empty list on zero
\p. (\q. <cons (pi1 q) (pi1 p),
-- cons head of stream on list
pi2 q>) -- rest of stream for iteration
(unfold nats (pi2 p))]
-- get next stream element
|] n); -- iteration over n
We may combine these terms as follows (currently there is syntactic sugar for natural numbers but not for lists; formatting inserted by hand for clarity):
> pick 4 fibs;
it = fold mu X. 1 + nat * X (in2 (<3,
fold mu X. 1 + nat * X (in2 (<2,
fold mu X. 1 + nat * X (in2 (<1,
fold mu X. 1 + nat * X (in2 (<1,
fold mu X. 1 + nat * X (in1 (<>))>))>))>))>)) : mu X. 1 + nat * X
Brian Howard (bhoward@cis.ksu.edu) | {"url":"http://people.cis.ksu.edu/~bhoward/lemon.html","timestamp":"2014-04-20T18:22:36Z","content_type":null,"content_length":"3974","record_id":"<urn:uuid:c35d26f0-9702-427a-b2b4-581f386d6422>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00253-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum: Problem of the Week
Galactic Exchange
Please keep in mind that this is a research project, and there may sometimes be glitches with the interactive software. Please let us know of any problems you encounter, and include the computer
operating system, the browser and version you're using, and what kind of connection you have (dial-up modem, T1, cable).
It's the year 2075. You take a vacation to the planet Orange. When you arrive, you notice coins of three different shapes: squares, circles, triangles. Unfortunately, no one speaks your language,
and you're hungry. You find a food vending machine, but the prices are worn off! You have to try different combinations of coins to get food, and in addition, you will be able to figure out the
relations between the coins.
1. Buy a package of zoogs. What are the fewest number and types of coins needed to purchase a package of zoogs using exact change?
2. Which coin is worth the least, and how many of it does it take to equal each of the other coins?
3. How many of the least valuable coins would it take to buy a package of zoogs?
4. Buy a package of Glorps. What are the fewest number and types of coins needed to purchase a package of Glorps using exact change? How many of the least valuable coins would it take to buy a
package of Glorps?
5. Another Earthling arrives on the next shuttle and wants to try some Mushniks. Explain how you figured out the relations between the coins.
6. What are the fewest number and types of coins needed to purchase a package of Mushniks using exact change?
Additional Practice
If you would like to practice this puzzle with different values, go to our practice page. These puzzles are for practice -- you will not submit your solutions to the Math Forum. Enjoy! | {"url":"http://mathforum.org/escotpow/print_puzzler.ehtml?puzzle=31","timestamp":"2014-04-16T05:30:24Z","content_type":null,"content_length":"6814","record_id":"<urn:uuid:b509b2ac-0134-4e16-904b-132fcc2ece94>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00234-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Marginal effect
Sanjoy posted on Monday, October 03, 2005 - 6:36 pm
Dear Professor/s ... using MPlus, can we calculate marginal effect of the explanantory variables, given our endogenous variables are categorical ... precisely, this is our model
r by R1-R3
b By B1 -B3
U on r b X1
r on b X2
b on r X3
U is binary and R and B's are 5 point ordinal, X's share some common element ...
I am particularly looking for the marginal effects of "r and b" on U
thanks and regards
bmuthen posted on Saturday, October 08, 2005 - 2:18 pm
This would have to be done via numerical integration. We do have a little Excel program that we have sent to some users and that we will probably eventually integrate with the Mplus graphics part.
Sanjoy posted on Saturday, October 08, 2005 - 5:57 pm
Thank you professor ...is there any way to get that Excel program, or for that matter how the relevant neumerical integration could be done in MPlus
thanks and regards
bmuthen posted on Sunday, October 09, 2005 - 10:07 am
I just noticed that you have a reciprocal interaction model - let me first check if the Excel program covers that case.
Sanjoy posted on Sunday, October 09, 2005 - 3:46 pm
OK Professor ... that would be great, if that particular program covers the reciprocal situation
thanks and regards
Lisa M. Yarnell posted on Tuesday, January 15, 2013 - 4:17 pm
Linda and Bengt,
This relates to my other question on the XWITH command. Can the marginal effects of a latent variable interaction created with XWITH be determined in Mplus? Would you be able to make the Excel sheet
(and possibly the graphics part mentioned above) available to me (and others)?
Thank you.
Lisa M. Yarnell posted on Wednesday, January 16, 2013 - 7:50 am
I suppose what I am looking for is the elasticity.
Bengt O. Muthen posted on Wednesday, January 16, 2013 - 7:52 am
The effect of a latent variables interaction is expressed as a variance contribution in equation (19) of the FAQ
We don't have an Excel program for this.
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Notes on the ISO-Prolog Standard
Joachim Schimpf
Last modified
As constructive discussions on the prolog-standard mailing list are quite difficult, I have decided to quietly collect my comments and contributions here, so they can be considered by those who want
to do so. I will try to address any feedback (no matter how I become aware of it) directly here on the web site. If you want to interactively discuss a concrete point, please email me personally.
Comments on 2nd Technical Corrigendum
Corrections for Arithmetic
Please include the following corrections to the core standard.
Section 9.1.4:
The definition of divF is incomplete, the zero-divisor case is missing (this can be seen by comparing signature and definition). Please correct as follows:
divF(x,y) = resultF(x/y),rndF) if y\=0
= undefined if x=0,y=0
= zero_divisor if x\=0,y=0
This mistake is apparently inherited from the first LIA standard.
Section 9.3.1.3 (and future ^/2 function):
Change the error condition (d) for zero raised to a negative exponent from undefined to zero_divisor.
Section 9.3.6.3:
Change (c) from "VX is zero or negative" to "VX is negative". Add case (d) saying "VX is zero" leads to evaluation_error(zero_divisor).
The idea here is that zero_divisor corresponds to an infinite result, while undefined is truly undefined. This is not my idea, I hasten to add, and every system that creates infinities must make this
distinction anyway.
I have been told that these changes cannot be put into the corrigendum because they would make existing conforming implementations non-comforming. It is, however, the whole point of a
"corrigendum" to correct mistakes, and inevitable that implementations may have to change to implement the corrections.
min and max
In the new section on min/2 and max/2, please do not refer to float_overflow when talking about to the case that an integer is not exactly representable as a float! As defined in 9.1.4.2,
float_overflow occurs when a result is greater than fmax, which is not the case we want to capture here.
See 2006 Mailing list discussion for a previous discussion on that topic.
Uninstantiation error
Whether such an error is needed is debatable. In the suggested case of the open/3 predicate being called with an instantiated output argument, the need for a new error goes away when we imagine that
the check is performed after the new stream has been created, and just before it is unified with the result argument. Without check, this unification would fail, and it is this failure that we want
to supplant with an approriate error. Elsewhere, such a situation would be signaled via a type_error (if types are different) or a domain_error (if types match, but value differs from what's
expected). So why not
% newly opened stream would be unified with 99 -> type error
?- open(f1, read, 99).
type_error(stream, 99)
% newly opened stream would be unified with other stream -> domain error
?- open(f1, read, S), open(f2, read, S).
domain_error(stream, $stream(f1))
It has been argued that in some implementations open(t,write,S), close(S), open(t,write,S2), S==S2 may succeed, and thus the error case does not replace a failure. I hope it is undisputed that
the intended semantics in that case is indeed a failure (even though the standard leaves it unspecified), and not to implement this failure is merely an oversight.
A case for the "uninstantiation" error can be made, however, in a completely different situation, where a variable is used as a quasi-identifier in a quantifier-like construct, e.g.
?- Y=foo, setof(X, Y^p(X,Y), L).
One could argue that this is likely to be a mistake and should be flagged by an "uninstantiation" error for Y. In this context, however, the name "uninstantiation" is unhelpful. A type_error
(variable) would be quite appropriate, since these variables are not meant to ever become instantiated, so being a "variable" is their final destiny or, arguably, their "type".
Another (non-ISO) argument that has been made for why we need this error is a predicate that attaches attributes to variables, but is called with a non-variable, e.g. put_attr
(foo,attrname,attrval). There is, however, no reason why this should not be equivalent to put_attr(X,attrname,attrval),X=foo and behave accordingly, i.e. succeed or fail according to the
semantics of the attribute.
The pi/0 arithmetic function is problematic because of its return type. In a basic ISO system, the return type is simply float. A system that provides more than one representation for real numbers
(e.g. different floating-point representations, or both floats and intervals) must settle for one of these types as the return type for pi (because it is argument-less, it cannot be polymorphic). It
would have to be the one with the highest precision because it then can be converted to a lower-precision type as needed (whereas missing precision cannot be recovered later).
What this means, however, is that a multi-representation system cannot really commit to pi/0 returning the float-type (which may not have enough precision). Or, if it did, it would have to provide
additional variants to return the higher precision representations (pi_binary_128/0, pi_decimal_64/0 etc).
A similar problem exists with regard to the type of literal constants such as 1.0. In a system with multiple representations, possible solutions are
• different syntax - Common Lisp has 1.0S1, 1.0F1, 1.0D1, 1.0L1 to denote different precisions. CON: precision is always explicit.
• let constants always have the maximum precision type, and require explicit coercion when wanted. CON: when coercion is forgotten, the maximum precision will propagate through the computation
unnecessarily (assuming usual contagion rules).
• have a precision-flag (possibly module-local, like ECLiPSe's read_floats_as_breals) that determines the precision of constants and pi/0
Often, one wants to write code that does a computation in a type-generic way, i.e. uses and propagates the precision of its input arguments. Unfortunately, none of the above alternatives is much help
with that.
For the case of pi, the easiest way to solve the problem is to provide a function pi/1, where pi(X) is equivalent to pi*X computed with the precision of X (but at least the smallest floating point
precision, to avoid biblical surprises like 3=:=pi(1)).
Signed Numbers (Apr 2011)
Draft technical corrigendum 2 (Mar 2011) includes the introduction of a predefined prefix operator declaration and evaluable function for +/1. It seems that no corresponding modification for the
signed number term syntax has been proposed. Unless I am misinterpreting something, this leads to an asymmetry between the minus and the plus sign:
│Input │ISO 13211-1│DTC2 Mar 2011│ECLiPSe 6.1 native│My suggestion for ISO │
│-1 │-1 │-1 │-1 │-1 │
│+1 │error │+(1) │1 │1 │
│- 1 │-1 │-1 │-(1) │-1 │
│+ 1 │error │+(1) │+(1) │1 │
│'-'1 │-1 │-1 │-(1) │-(1) │
│'+'1 │error │+(1) │+(1) │+(1) │
│'-' 1 │-1 │-1 │-(1) │-(1) │
│'+' 1 │error │+(1) │+(1) │+(1) │
│'-'/**/1 │-1 │-1 │-(1) │-(1) │
│'+'/**/1 │error │+(1) │+(1) │+(1) │
│- /**/1 │-1 │-1 │-(1) │-1 │
│+ /**/1 │error │+(1) │+(1) │1 │
│+1.0e+2 │error │+(1.0e2) │1.0e2 │1.0e2 │
│1.0e'-'2 │error │error │error │error │
│1.0e- 2 │error │error │error │error │
│number_codes(N,"-1")│N = -1 │N = -1 │N = -1 │N = -1 │
│number_codes(N,"+1")│error │error │N = 1 │N = 1 │
I suggest to adopt the last column as correction for DTC2:
• Accept + not only as prefix operator, but also as sign, such that integer(+3). This requires a change to 6.3.1.2. Rationale: Be consistent with the minus sign, and be consistent with the sign of
floating-point exponents.
• Do not accept quoted signs as signs, such that compound('-'3). Rationale: the acceptance of quoted signs was almost certainly unintended and merely an accident of "grammar reuse" in the
specification. Note also that the sign of a floating-point exponent cannot be quoted.
On the other hand, it has been pointed out that allowing + as a sign is less "necessary" than allowing - (negative number syntax is necessary to allow writeq+read give the correct result with
negative numbers), and therefore the +/- asymmetry may be considered ok.
NOTE (not for ISO - I think it's too late to change this aspect): I do think that the ECLiPSe (and SWI) choice (of not allowing space of any kind between sign and number) is the most consistent
• The reason to allow space seems to be the same as the reason to allow quoted signs: (understandable) laziness in early implementations. It simplifies the lexer-parser interface, because the
information about quotes and spaces does not have to be exported from the lexer.
• But: spaces or quoted bits are not allowed elsewhere in floating point numbers, for instance.
• I have heard the argument that one sometimes wants to read tables of numbers where the sign is separated by spaces. However, when such a table is read as data, this can easily be solved by
postprocessing (in the simplest case, calling is/2 on the read term), which will be needed anyway if the input format is not completely under the programmer's control. For numbers occurring in
programs, one would hope that the restriction still leaves enough room for programmers to satisfy their layout preferences.
• A significant space in front of a number is tantamount to the significant space between a prefix functor and an opening parenthesis: -(a,b) is a -/2 term in functor notation, while - (a,b) is a -
/1 term in prefix notation, i.e. the space causes the minus sign to be interpreted as prefix operator.
Comments on Minutes of WG17 2010 Edinburgh Meeting
I assume this is meant to be a difference-list version of term_variables/2. Difference-list-variants of list-constructing predicates can be a good idea (findall/3 comes to mind, but also atom_codes/2
etc), but as the standard does not systematically provide these, it seems there should be a special reason for providing it in this case - what is it?
Apparently, the use case is the implementation of bagof/setof. The arguments made below still hold.
As opposed to the other examples quoted above, a difference-list version of term_variables/2 is a particularly bad idea because
term_variables(T1, Vs, Vs1), term_variables(T2, Vs1, []).
is not the same as
term_variables(T1-T2, Vs).
because it is supposed to return a duplicate-free variable list. The predicate invites bugs by suggesting a usage that is unlikely to give the expected results.
Note that the correct way to augment a term-variable-list is
term_variables(T1, Vs1), term_variables(Vs1-T2, Vs).
number_chars/2 and number_codes/2 (Apr 2011)
The predicate as originally specified in the standard is unnecessarily limited in usefulness by requiring an error for the case that the right-hand side string cannot be parsed as a number. Failure
would be much more useful, as the predicate could be readily used to "convert to a number if possible", allowing the following common pattern:
( number_chars(Num, Chars) ->
<deal with a number Num>
<deal with something else in Chars>
Accepted Language
Neumerkel's comparison drew my attention to other surprises in the ISO spec:
1. space and even comments can occur in the string before the actual number, i.e. number_codes(3," /*comment*/ 3") is supposed to succeed. Why comments? We are not parsing a program here!
2. On the other hand, space after the number is forbidden, i.e. number_codes(N, "3 ") is required to raise an exception. Why the asymmetry?
3. number_codes(3, "03") succeeds, but number_codes(3, "+3") is required to raise an exception. Do we want to allow redundant characters or not?
4. number_codes(N, "- 3.1e-2") is allowed, but number_codes(N, "-3.1e- 2") is not. Do we want to allow redundant space or not?
This all makes very little sense for a predicate that is supposed to convert back and forth between numbers and their various string representations. While it is clear that the mapping is not
one-to-one, the flexibility allowed in the string-to-number backward direction looks completely arbitrary.
The reason for the strangeness is, of course, that the specification refers to the Prolog term syntax (probably to avoid having to deal with signs), rather than more directly to number token syntax.
A more sensible, but still compact spec can be had by specifying the language accepted by number_chars separately in terms of token syntax, e.g.
[sign] (integer token|float number token)
If it is felt that redundant spaces must be accepted, these can now be re-introduced explicitly (but preferably on both sides of the number, and without allowing comments). Nevertheless, for a
built-in 'primitive', it would be cleaner to accept the pure number only.
It can be doubted whether point 1 above (allowing comments) and point 4 (space after sign) are really required by 8.16.7, because it mentions "a character sequence which could be output". No ouput
primitive will output extra comments, and no output primitive according to 7.10.5 will output space between sign and number.
Suppose we have the following 3 files:
% top.pl
:- include('somedir/include_p').
% somedir/include_p.pl
:- ensure_loaded(p).
% somedir/p.pl
and we compile top.pl. What current directory should we expect when encountering the ensure_loaded(p) directive in somedir/include_p.pl? With pure include-semantics, the above code does not work: the
ensure_loaded behaves as if it occurred within top.pl directly. SWI Prolog, YAP and ECLiPSe behave that way. SICStus (and similar directives in C) behave the other way, i.e. certain pathnames in
included files are relative to the includee's not the includer's location. However, the current directory is still the includer's, so the situation is a bit confusing. A discussion from the ECLiPSe
point of view is in bug 678.
Thanks for comments to: Richard O'Keefe, Paulo Moura, Ulrich Neumerkel. | {"url":"http://eclipseclp.org/Specs/iso_notes.html","timestamp":"2014-04-16T13:14:38Z","content_type":null,"content_length":"17373","record_id":"<urn:uuid:6680b3e8-4db6-4481-a1e7-3707670161e4>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00281-ip-10-147-4-33.ec2.internal.warc.gz"} |
Banked Curve question
Thanks so much for your help. This is what I calculated it out to:
I found the sum of the forces in the vertical direction: ncos(angle) + (-mg) = 0, so ncos(angle) = mg, and in terms of mg, n = (mg)/(cos[angle]).
To find out what that angle was, I used the sum of the forces in the horizontal direction: nsin(angle) = m[(v^2)/(R)], and then replaced n with what I found above to make it sin(angle)/cos(angle) = m
[(v^2)/(gR)], so tan(angle) = (v^2)/(g*R) = (20^2)/(9.8 * 120) = 0.34. So tan of 0.34 = 18.8 degrees.
Then I added the static friction in the x-direction:
nsin(angle) + f[s] = m[(v^2)/(R)]
After replacing n and doing all that I did above, I got tan(angle) = (v^2)/(g*R) - (f[s]). I replaced the angle with 18.8, the v with 30 m/s, and g = 9.8 m/s^2, and R = 120 m, solved and got 900 for
the right part of the equation, and got f[s] = 500.16.
To find the normal force, I again took nsin(angle) = 900 - 500.16 (from the f[s]), and got n = 1240.7. I replaced that in f[s] = mu[s]*n, and used the 500.16, and got 0.40 as my coefficient of static
Does that seem like a reasonable answer? I looked at a table of coefficients of static friction, and couldn't determine whether it was large enough to keep the car from skidding.
Thank you soooo much! :-) | {"url":"http://www.physicsforums.com/showthread.php?t=136182","timestamp":"2014-04-20T15:59:28Z","content_type":null,"content_length":"40121","record_id":"<urn:uuid:192b550a-b72e-4ae0-b491-7b13c0f9f0c2>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00507-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matrix Representation of Complex Numbers
Date: 11/15/2000 at 16:19:40
From: John Jones
Subject: Matrices
I can't figure this math problem out:
Find a matrix A and a matrix B such that
(A+B) inverse = A inverse + B inverse
Sorry, I don't have a key that makes it look like it (A+B) to the -1.
Any help would be greatly appreciated. Thank you.
Date: 11/15/2000 at 18:25:59
From: Doctor Schwa
Subject: Re: Matrices
Usually we write ^-1 to represent "to the -1." So, your question is to
find matrices A and B such that
(A+B)^-1 = A^-1 + B^-1
Multiplying both sides by (A+B), I get
I = (A^-1 + B^-1)*(A + B)
I = I + (A^-1)*B + (B^-1)*A + I
Hmm, that doesn't seem to help all that much. I wonder if there are
real numbers a and b with a similar property,
1/(a+b) = 1/a + 1/b
Multiplying both sides by ab gives
ab / (a+b) = b + a
and then multiplying by (a+b) gives
ab = (a+b)^2
ab = a^2 + 2ab + b^2
which has some complex number solutions...
Aha! There's a correspondence between complex numbers and matrices,
namely that the number i acts the same as the matrix
[0 -1]
[1 0]
You can check that:
i^2 = [-1 0]
[ 0 -1]
So, if you put those pieces together, find some complex number
solution to the equation at the top, and then translate the complex
numbers into matrices, you'll have an answer.
I wonder if there was some easier way that your teacher had in mind,
though. I didn't see it...
Feel free to write back if you have further questions (for example, if
this partial solution makes no sense whatsoever).
- Doctor Schwa, The Math Forum
Date: 11/15/2000 at 19:32:23
From: John Jones
Subject: Re: Matrices
I partially understand what you mean, but I am having a hard time
figuring out what you are doing with i.
Date: 11/16/2000 at 15:34:20
From: Doctor Schwa
Subject: Re: Matrices
The question was how to find one example of matrices where
(A+B)^-1 = A^-1 + B^-1.
My method was to attempt an analogy, where a and b are real numbers,
1/(a+b) = 1/a + 1/b
but it turns out that there aren't any real number solutions to this
thing, so I resorted to complex numbers.
You can set a equal to anything you want, and find a complex number b
that solves the above equation (it's just a quadratic after you
multiply away all the denominators).
Now, suppose you have in hand complex numbers a and b that solve this
equation. There's a matrix corresponding to every complex number. If
b = x + iy, then the matrix
B = [ x y]
[-y x]
will have the same behavior as the complex number b. Check it out:
Try multiplying
(1 + 2i)*(3 + 4i)
and see what you get, then try multiplying
[ 1 2] * [ 3 4]
[-2 1] [-4 3]
and you'll find that you get analogous answers. Similarly, the inverse
matrix also works the same way as dividing by a complex number.
So, once you have complex numbers a and b that solve your equation,
you can translate them into the corresponding matrices and get an
answer that will work.
I still wonder if perhaps the teacher assigning this problem had a
quite different method in mind.
- Doctor Schwa, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/52004.html","timestamp":"2014-04-17T05:02:58Z","content_type":null,"content_length":"8292","record_id":"<urn:uuid:cf82222f-f16f-499a-b3ca-377de71d91f4>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00384-ip-10-147-4-33.ec2.internal.warc.gz"} |
What effect would a proof of P≠NP have on the field of complexity theory?
up vote 2 down vote favorite
This question is motivated by Scott Aaronson's comment about his bet: "If P≠NP has indeed been proved, my life will change so dramatically that having to pay $200,000 will be the least of it."
8 Aaronson answers this very question in a comment: scottaaronson.com/blog/?p=456#comment-44660 . If you want more details, perhaps you should ask on his blog? – HJRW Aug 9 '10 at 14:11
1 Probably not as dramatic as some think. The reason is most of people already think that this is true, and complexity theorists also work on many other problems. See Steve Cook's comment on Scott's
post. – Kaveh Aug 10 '10 at 4:53
2 If this had been "what effect would proving the Riemann Hypothesis have on mathematics", I think the question would have been closed. What makesthis question different? – András Salamon Aug 10 '10
at 17:31
People have strange self effects. I remember Odlyzko, a few days after PRIMES in P was released, at a conference speaking about his RH calculations, tangentially saying that it was a result that
1 we did not expect, which was either a misspeak or a misthought, as everyone trusted it was true, and just getting around assuming GRH and exploiting the known analytic number theory to the maximal
impact was the main trick in AKS. Techniques are more likely to have wide or dramatic "effects" that specific results. Grothendieck was annoyed that Deligne used a "trick" rather than build more
theory – Junkie Aug 11 '10 at 6:43
add comment
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Scaling Laws in the Distribution of Galaxies - B.J.T. Jones et
D. Hierarchical models
The clustering together of stars, galaxies, and clusters of galaxies in successively ordered assemblies is normally called a hierarchy, in a slightly different sense of the dictionary meaning in
which there is a one-way power structure. The technically correct term for the structured universes of Kant and Lambert is multilevel. A complete multilevel universe has three consequences. One is
the removal of Olbers paradox (the motivation of John Herschel and Richard Proctor in the 19th century). The second, recognized by Kant and Lambert, is that the universe retains a primary center and
is therefore nonuniform on the largest cosmic scales. The third, recognized by the Irish physicist Fournier d'Albe and the Swedish astronomer Carl Charlier early in the 20th century is that the total
amount of matter is much less than in a uniform universe with the same local density. D'Albe put forward the curious additional notion that the visible universe is only one of a series of universes
nested inside each other like Chinese boxes. This is not the same as multiple 4-dimensional universes in higher dimensional space and does not seem to be a forerunner of any modern picture.
The idea that there should be structure on all scales up to that of the Universe as a whole goes back to Lambert (1761) who was trying to solve the puzzle of the dark night sky that is commonly
called "Olber's paradox". (It was not formulated by Olbers and it is a riddle rather than a paradox (Harrison, 1987). Simply put: if the Universe were infinite and uniformly populated with stars,
every line of sight from Earth would eventually meet the surface of a star and the sky would therefore be bright. The idea probably originated with John Herschel in a review of Humboldt's Kosmos
where the clustering hierarchy is suggested as a solution to Olber's Paradox as an alternative to dust absorption.
At the start if the 20th century, The Swedish astronomer Carl Charlier provided a cosmological model in which the galaxies were distributed throughout the Universe in a clustering hierarchy (Charlier
1908, 1922). His motivation was to provide a resolution for Olber's Paradox. Charlier showed that replacing the premise of uniformity with a clustering hierarchy would solve the problem provided the
hierarchy had an infinite number of levels (see Fig. 2).
Figure 2. Hierarchical universes were very popular at the end of the 19th century and the first half of the 20th century. Reproduced from harri, Cosmology, Cambridge University Press.
Charlier's idea was not new, though he was the first person to provide a correct mathematical demonstration that Olber's Paradox could indeed be resolved in this way. It should be recalled that he
was working at a time before any galaxies had measured redshifts and long before the cosmic expansion was known.
It is interesting that the Charlier model had de Vaucouleurs as one of its long standing supporters (de Vaucouleurs, 1970).
More recently still there have been a number of attempts to re-incarnate such a universal hierarchy in terms of fractal models. Fractal models were first proposed by Fournier d'Albe (1907) and
subsequently championed by Mandelbrot (1982) and Pietronero (1987). Several attempts have been made to construct hierarchical cosmological models (a Newtonian solution was found by Wertz (1971),
general-relativistic solutions were proposed by Bonnor (1972); Wesson (1978); Ribeiro (1992)). All these solutions are, naturally, inhomogeneous with preferred position(s) for the observer(s), and
thus unsatisfactory. So the present trend to conciliate fractal models with cosmology is to use the measure of last resort, and to assume that although the matter distribution in the universe is
homogeneous on large scales, the galaxy distribution can be contrived to be fractal (Ribeiro, 2001). Numerical models of deep samples contradict this assumption.
Edwin F. Carpenter spent his early days at Steward Observatory (of which he was director for more than 20 years, from 1938) scanning zone plates to pick out extragalactic nebulae for later study. In
1931, he found a new cluster in the direction of Cancer (independently discovered by Hubble at about the same time.) He measured its size on the sky, estimated its distance, and counted the number of
galaxies, N, he could recognize within its confines. This gave him a sample of 7 clusters with similar data, all from Mt. Wilson plates (5 in the Mt. Wilson director's report for 1929-30 and one then
just found by Lundmark). He was inspired to graph log(N) vs. the linear sizes of the clusters (Carpenter, 1931) and found a straight line relation, that is, a power law in N(diameter), nowhere near
as steep as N ~ D^3 or N proportional to volume. The then known globular cluster system of the Milky Way (with about 35 clusters within 10^5 pc) also fit right on his curve.
Carpenter later considered a larger sample of clusters and found that a similar curve then acted as an upper envelope to the data (Carpenter, 1938). If his numbers are transformed to the distance
scale with H[0] = 100 km s^-1 Mpc^-1, then the relations are (de Vaucouleurs, 1971)
and the maximum number density in galaxies per Mpc^3 is also proportional to 0.5 log(V). De Vaucouleurs called this Carpenter's law, though the discoverer himself had been somewhat more tentative,
suggesting that this sort of distribution (which we would call scale free, though he did not) might mean that there was no fundamental difference among groups, clusters, and superclusters of
galaxies, but merely a non-random, non-uniform distribution, which might contain some information about the responsible process. It is, with hindsight, not surprising that the first few clusters that
Carpenter (1931) knew about were the densest sort, which define the upper envelope of the larger set (Carpenter, 1938). The ideas of a number of other proponents, both observers and theorists, on
scale-free clustering and hierarchical structure are presented (none too sympathetically) in Chapter 2 of Peebles (1980).
3. De Vaucouleurs hierarchical model
De Vaucouleurs first appears on the cosmological stage doubting what was then the only evidence for galaxy evolution with epoch, the Stebbins-Whitford effect, which he attributed to observational
error (de Vaucouleurs, 1948). He was essentially right about this, but widely ignored. He was at other times a supporter of the cosmological constant (when it was not popular) and a strong exponent
of a hierarchical universe, in which the largest structures we see would always have a size comparable with the reach of the deepest surveys (de Vaucouleurs 1960, 1970, 1971). He pointed out that
estimates of the age of the universe and of the sizes of the largest objects in it had increased monotonically (and perhaps as a sort of power law) with time since about 1600, while the densities of
various entities vs. size could all be plotted as another power law,
By putting "Carpenter's Law" into modern units, de Vaucouleurs showed that it described this same sort of scale-free universe. A slightly more complex law, with oscillations around a mean, falling
line in a plot of density vs. size (see Fig. 3), could have galaxies, binaries, groups, clusters, and superclusters as distinct physical entities, without violating his main point that what you see
is what you are able to see.
Figure 3. In this idealized diagram de Vaucouleurs shows two hierarchical frequency distributions of the number of clumps per unit volume. In the top panel there are no characteristic scales in the
distribution. This is the model proposed by Kiang and Saslaw (1969). The bottom panel shows a more sophisticated alternative in which the overall decrease of the number of clumps per unit volume does
not behave monotonically with the scale, but it displays a series of local maxima corresponding to the characteristic scales of different cosmic structures: galaxies, groups, clusters, superclusters,
etc. Reproduced from de Vaucouleurs (1971), Astronomical Society of the Pacific.
De Vaucouleurs said that it would be quite remarkable if, just at the moment he was writing, centuries of change in the best estimate for the age and density of the universe should stop their
precipitous respective rise and fall and suddenly level off at correct, cosmic values. Thus he seemed to be predicting that evidence for a universe older than 10-20 Gyr and for structures larger than
100 Mpc should soon appear. (He held firmly to a value of H[0] near 100 km s^-1 Mpc^-1 for most of his later career, except for the 1960 paper where it was 75, but thought of local measurements of H
[0] as being relevant only locally).
Remarkable, but apparently true. Instead of taking off again, estimates of the age of the universe made since 1970 from radioactive decay of unstable nuclides, from the evolution of the oldest stars,
and from the value of the Hubble constant, increasingly concur. And galaxy surveys have now penetrated a factor 10 deeper in space than the Shane-Wirtanen and Harvard counts in which de Vaucouleurs
saw his superclusters. | {"url":"http://ned.ipac.caltech.edu/level5/March04/Jones/Jones3_4.html","timestamp":"2014-04-17T12:40:10Z","content_type":null,"content_length":"13276","record_id":"<urn:uuid:ffbe4dd5-6f37-4485-854b-9c56027a2b4c>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00152-ip-10-147-4-33.ec2.internal.warc.gz"} |
Entropy (Part 4): Randomness by rolling ten dice
In order to illustrate the concept of randomness as it pertains to entropy, in a series of entries different numbers of dice have been rolled.
Entropy 1: Randomness by rolling two dice
Entropy 2: Randomness by rolling three dice
Entropy 3: Randomness by rolling four dice
A die with six random states is used to illustrate a particle, so as the number of dice increases, so the number of states increases. For n dice there are 10^n different ways they can be rolled.
The roll that comes up most frequently is the one that has the most number of arrangements. As the number of dice increases, that random states becomes more and more likely as seen in the above
entries for 2, 3 and 4 dice. Now we jump to 10.
For 10 dice there are over 60 million arrangements (6^10) and Figure 4 shows the outcomes for 30,000 rolls. This can be compared to Figures 1 to 3.
For ten dice, the chance of a number lower than 20 or greater than 60 is negligible. The chance of rolling 10 one’s is one over 60 million. The most random states are dominating. This is only for
10 dice. Next case will be Avogadro’s dice which have 6^10^23 states, which is a lot more than 60 million.
The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced
Programs in High School), and college level Chemistry and Physics courses.
Are you a Physical Chemistry teacher or student?
2 Comments
Sergio Palazzi December 5, 2011
I’m not so sure that the classical example of probabilistic randomness shown above for the distribution of material objects, which have the same energy in any configuration (unless dice are
marked), may still be considered as the best one for entropy.
In entropy changes, redistribution of energy among interacting entities is involved, whereas there is no interaction (or exchange of communication) between two dice; otherwise, it would be
possible to win just betting on “retarding” numbers or combinations. Molecules in the two bulbs connected by a stopcock, instead, are mutually exchanging energy, and in this way entropy
I think that to be coherent with the physical definition of entropy, a more comprehensible approach is the one suggested by prof. Frank A. Lambert, who discards classical examples of random cards
or messy teenager rooms in favour of the accessibility of a wider number of energetic states. It gives me a better understanding also of the Boltzmann’s equation, with the uncountable value that
W reaches just at some kelvin above absolute zero. Which are your opinions about it?
Bryan December 7, 2011
Of course statistical entropy is treated as ensembles of particles that are constrained by energy and other parameters. The dice are constrained by the number rolled and the number of faces. Let
us suppose that if the temperature increases, the number of dice faces increases and vice versa. Within these constraints the number of accessible states changes and the random states dominate if
there are enough. Constraints are not the point of these entries. The point is to say that the number of accessible states is dominated by the most random states, and for this reason entropy is a
quantitative measure of randomness. I looked at some of Frank A. Lambert’s postings and questions and answers. I do not think it much different from my ideas at equilibrium. It is just that I
only want to show that as the number of non-interacting states increases (by non-interacting I mean rolls of (5,1), (1,5),(2,5),(5,2),(4,3), (3,4) are independent and degenerate (outcome 7). In
the two bulb experiment, the gases will redistribute amongst the newly accessible states even if the particles do not interact.
I discuss entropy in more detail in our text book, Laidler, Meiser and Me, Physical Chemistry, http://www.mchmultimedia.com/store/Statistical-Mechanics.html .
Does this clarify my motivation? I would like to know if this is consistent with the views you might have?
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KeywordsJEL Code1. Introduction2. Data and Methodology3. Findings and Discussion4. ConclusionsAcknowledgmentsConflicts of InterestReferences
ijfs International Journal of Financial Studies Int. J. Financ. Stud. International Journal of Financial Studies 2227-7072 MDPI 10.3390/ijfs1040154 ijfs-01-00154 Article European Markets’ Reactions
to Exogenous Shocks: A High Frequency Data Analysis of the 2005 London Bombings Kollias Christos 1 Papadamou Stephanos 1 * Siriopoulos Costas 2 Department of Economics, University of Thessaly, Korai
43, Volos 38333, Greece; E-Mail: kollias@uth.gr Department of Business Administration, University of Patras, Rio, Patras 26504, Greece; E-Mail: siriopoulos@eap.gr Author to whom correspondence should
be addressed; E-Mail: stpapada@uth.gr; Tel.: +30-24210-74963; Fax: +30-24210-74772. 18 11 2013 12 2013 1 4 154 167 29 09 2013 07 11 2013 11 11 2013 © 2013 by the authors; licensee MDPI, Basel,
Switzerland. 2013
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
Terrorist incidents exert a negative, albeit usually short-lived, impact on markets and equity returns. Given the integration of global financial markets, mega-terrorist events also have a high
contagion potential with their shock waves being transmitted across countries and markets. This paper investigates the cross-market transmission of the London Stock Exchange’s reaction to the
terrorist attacks of 2005. It focuses on how this reaction was transmitted to two other major European stock exchanges: Frankfurt and Paris. To this effect, high frequency intraday data are used and
multivariate Genralised Autorgressive Conditional Heteroskedasticity (GARCH) models are employed. This type of data help reveal a more accurate picture of markets’ reaction to exogenous shocks, such
as a terrorist attack, and thus allow more reliable inferences. Findings reported herein indicate that the volatility of stock market returns is increased in all cases examined.
capital markets contagion terrorism multivariate GARCH G14 G21
The high velocity, with which the shock waves from major financial episodes, irrespective of the source that has generated them, travel across markets and countries, has attracted increasing
attention in the relevant financial literature. A plethora of studies, a survey of which can be found in Pericoli and Sbracia [1], have examined both on a theoretical as well as empirical level the
mechanisms and the channels through which financial shocks that occur in one country are transmitted and affect markets in another or indeed, have a major international impact on global markets and
economic sentiment (inter alia: Goetzmann, et al. [2]; Saleem [3]; Meric and Meric [4]; Asimakopoulos et al. [5]; Chiang et al. [6]). In particular, a number of studies have examined the
interdependence of equity market volatility using the framework of autoregressive conditional heteroskedasticity (GARCH) time series models (inter alia: Saleem [3]; Hamao et al. [7]; Theodossiou and
Lee [8]; Lin et al. [9]; Longin and Solnik [10]).
A strand of the aforementioned literature, has focused on how markets react to exogenous events and shocks including natural or anthropogenic catastrophes and accidents, political risk and violent
events such as conflict and terrorism while the contagion potential of this reaction has also been the subject of empirical investigation (inter alia: Kaplanski and Levy [11]; Capelle-Blancard and
Laguna, [12]; Asteriou and Siriopoulos [13]; Herbst et al. [14]; Blose et al. [15]; Kalra [16]; Bowen et al. [17]). Following mega-terrorist attacks of recent years, such as for instance 9/11 in New
York, and the Madrid and London bomb attacks of 2004 and 2005 respectively, the number of studies that examine the impact terrorism exerts on the economy in general and on financial markets in
particular, has steadily grown (inter alia: Brounrn and Derwall [18]; Ramiah et al. [19]; Graham and Ramiah [20]; Amelie and Darne [21]; Fernandez [22]; Nikkinen and Vahamaa [23]). As it has been
pointed out in a number of previous papers (inter alia: Drakos, [24]; Kollias et al. [25,26]; Chesney et al. [27]), although the threat of a terrorist attack is omnipresent, particularly in countries
such as Israel, Spain or the UK that are or have been the venues of systematic terrorist activity with the concomitant casualties and damages, terrorist events when they occur are unexpected.
Depending among other things on their seriousness in terms of victims, damages or target(s) attacked, they have the potential to shake and rattle investors and markets. Just as in the case of natural
or anthropogenic accidents, terrorist attacks are unanticipated. Hence, market agents cannot hedge against them. Such incidents can also have a high contagion potential as studies that have addressed
this question and the channels of the cross-market transmission of terrorist induced shocks have shown (inter alia: Hon et al. [28]; Mun [29]; Drakos [30,31]). Factors that seem to affect the
transmission potential of such exogenous shocks from the market of the country that has been targeted by the terrorists to others include the degree of bilateral integration between the stock markets
and the degree of integration into the global economic and financial markets (see Drakos [30,31]). Moreover, Kollias et al. [32] indicate that terrorist attacks trigger a flight-to-safety effect
(from stock to bond market within a country) primarily in France and Germany and to a smaller degree in Great Britain and Spain.
Within this particular thematic focus of this strand of literature, this paper addresses the cross-market transmission of the shock generated by a major European terrorist event. The study focuses on
the 7 July 2005 London bomb attacks that, along with the 2004 Madrid bombings, are considered to be the European equivalent of 9/11 albeit on a much smaller scale in terms of the number of victims
and destruction to property and infrastructure (Kollias et al. [26]). However, unlike previous studies those rely on daily data to assess the impact of terrorist events on financial markets as well
as their contagion effect, this study uses high frequency data to investigate the issue at hand. By opening a window on high frequency data, investors may discover profit opportunities not easily
detected in daily data. In total, each series of 10-minute interval frequency contains over 10,000 observations. As it has been argued and shown, intraday data help reveal a more accurate picture of
how markets and market agents react and adapt to changes and exogenous shocks (inter alia: Connolly and Wang [33]; Hanousek [34]; Égert, and Kočenda [35]; Markelos et al. [36].). Consequently, the
more detailed account and the information contained in high frequency data allow more reliable inferences and conclusions to be drawn vis-à-vis daily data.
Moreover, recently a series of papers propose successful methods on high-frequency predictions of trading volumes, market depth, bid-ask spreads and trading costs to optimize order placement and
order execution. Among others, Groß-Klußmann and Hautsch [37] successfully apply a long memory autoregressive Poisson model to predict bid-ask spreads. Härdle, Hautsch and Mihoci [38] introduce a
semiparametric dynamic factor approach to model high-dimensional order book curves. Hautsch, Schienle and Malec [39] propose a novel approach to model serially dependent positive-valued variables
which realize a non-trivial proportion of zero outcomes, a quite common phenomenon on high frequency financial data.
The important role of automated news feeds on intraday dynamics is recognized by Groß-Klußmann and Hautsch [40]. Significant reactions in volatility and trading activity after the arrival of news
items which are indicated to be relevant are identified. Moreover, Hautsch, Hess and Veredas [41] study the impact of the arrival of macroeconomic news on the informational and noise-driven
components in high-frequency quote processes and their conditional variances. One of their major findings is that all volatility components reveal distinct dynamics and are positively influenced by
Therefore, this paper following this line of literature use a modified BEKK(1,1)-GARCH model on intraday data in order to incorporate any possible effect on stock market volatilities and covariance
of the mega-terrorism event occurred in July 2005. This type of modeling suggested can be generalized to incorporate other insecurity shocks when studying two variables of interest. More
specifically, it examines how two other major European stock markets—Paris and Frankfurt—were affected by the 7 July 2005 mega-terrorist attack in London. The choice of the markets was very much
dictated by data availability constraints and the level of capitalization. In the section that follows we proceed with the presentation of the data and the methodology employed. In section three, the
findings are presented and discussed, while section four concludes the paper.
The London terrorist incident, involved a series of coordinated suicide bomb attacks that targeted the city’s public transport system during the morning rush hour of 7 July 2005. The bomb blasts
caused 52 fatalities and injured 700 people. They also caused extensive and widespread disruption of the city's transportation system and of the mobile telecommunications infrastructure. As the
results reported by Kollias et al. [26] show, the London Stock Exchange (henceforth LSE) suffered significant negative abnormal returns on the day of the attack. Both the general as well as sectoral
indices were negatively affected by the event. However, this impact was rather short lived since the market quickly recovered and rebounded. Market volatility was also significantly affected but
again, this was of a transitory nature (Kollias et al. [26]). The use of 10-mimute interval data by this study will shed more light on how the market reacted and market agents adjusted to the event
as it unfolded between 08:50 and 09:50 given that the bomb blasts took place in different moments in the morning of 7 July 2005. Furthermore, the paper will examine whether or not contagion between
financial markets can be established in the case of this major terrorist incident that took place in a city that is one of the most important financial and trading centres of the world. Given LSE’s
significance as one of the major financial markets globally with a market capitalisation well over $ 3 trillions ^1, one would intuitively expect that the shock waves could have been transmitted to
other major European markets. The possible contagion and shock transmission is examined through the use of multivariate GARCH models.
Apart from the London stock exchange, our data set consists of data for the German and French stock market returns over the period 21 January 2005to 28 October 2005^2. We have to mention at this
point that all data are expressed in British summer time (BST) in order to take into account time differences between countries of the study and compare the results of our findings. Frankfurt and
Paris are the second and third largest markets in Europe in terms of capitalization. Hence, one can intuitively expect that if shock waves were indeed transmitted from London to other European
markets, Frankfurt and Paris are the markets with the highest probability of contagion given the degree of integration between these three large European Union economies and markets. The German stock
returns are calculated from the DAX-30 index, the UK returns from the FTSE-100 index, and the French returns from the CAC-40 index. Apart from the information rich nature of such frequency data, a
further advantage stemming from the use of intraday data over a short time period is that the possibility of structural breaks is much smaller compared to a longer time period needed if daily data
was employed. Indeed, the multivariate GARCH model might give inaccurate forecasts if the underlying process which generates asset prices undergoes a structural break. Moreover, intraday data capture
all the main features of the data generating process. In our case the bivariate unrestricted BEKK-GARCH (1,1) model, proposed by Engle and Kroner [42], is used to investigate any possible contagion
effects between the London, the German and the French stock markets on the day of the terrorist attack in question. What will be examined is whether or not and to what extent this terrorist event
affected the volatilities and the correlation of the stock markets in question using intraday data given that these effects may be hidden in a daily frequency (see for instance previous studies on
daily data: Fernandez [22]; Ramiah et al. [19]). In order to avoid any severe convergence problems, the bivariate unrestricted version of the general BEKK(p,q)-GARCH model with p = q =1 is used: with
and where Equation (1) gives the expression for the conditional mean; R[i][,t], and ε[ι], t are the return vector (i = 1 for the FTSE-100 index and 2 for the other two indices for each estimation),
and the residual vector respectively; and μ[i] is the mean of this process. In Equation (2) H is the conditional variance-covariance matrix that depends on its past values and on past values of ε[ι],
parameter. C0, is a 2 × 2 matrix, the elements of which are zero above the main diagonal; and A, B are matrices. More analytically:
The main advantage of the BEKK-GARCH model is that it guarantees by construction that the covariance matrices in the system are positive (Engle and Kroner, [42]). Assuming multivariate General Error
Distribution (GED), the maximum likelihood methodology is used to jointly estimate the parameters of the mean and the variance equations. More specifically, in an attempt to identify the possible
effects the terrorist incident in question had on FTSE and y (where y =DAX or CAC) stock index returns co-movement, we employ the unrestricted BEKK-GARCH (1, 1) model including a dummy variable about
terror activity in 7 July 2005 in the construction of variances and covariance matrices. This dummy variable takes the value of one for the 10-minute ticks between 7 July 2005 8:50 and 7 July 2005
10:50 (British summer time, BST) and zero anywhere else^3. Therefore, the functional form of our model is the following: with and where the K is the coefficient matrix for terror index and the
operator “•” is the element by element (Hadamard) product. In that case the model may be written in single equation format as follows:
The error terms in each model represent the effect of news in each model on the different indices. In particular, the terms , represent the deviations from the mean attributed to the unanticipated
event in each market. The cross values of the error terms represent the news in the first and second index in time of period t – 1. By we describe the conditional variance for the first stock index
(in our case FTSE-100) at time t – 1, conditional variance for the second stock index (in our case CAC-40 or DAX-30) at time t – 1, and the conditional covariance between the first and the second
index in our model.
As mentioned above, daily data can often conceal a significant part of underlying dynamics of a time-series especially when it comes to the reaction and eventual adjustment to an unanticipated event
such as the terrorist attack in question here. Table 1 provides information using daily data for ±1 days around the event day. As it can easily be seen, the markets in the event day exhibit negative
returns (FTSE –1.37%, CAC –1.39% and DAX –1.86%). However, a more detailed scrutiny of the data presented in the last column of Table 1, reveals that the difference between the high and low values
during the day is appreciably higher for each market compared to the relevant values of the ±1 days window. This reinforces the argument that high frequency data of the type used here contains a more
accurate and credible account of how markets react and adjust to exogenous and unanticipated events; in this case the terrorist attacks of 7 July 2005 in London. Hence, vis-à-vis studies that employ
daily observations to examine the impact terrorism exerts on financial markets, the intraday data used in the estimations that follow, have an advantage in that they allow for more reliable
inferences to be drawn. Indeed, Figure 1, plotted with the use of high frequency 10-minute interval data, graphically reveals the magnitude of market agents’ reaction to the event in question as it
unfolded between 08:50 and 09:50. All three stock markets exhibit a negative reaction to the news as the event started unfolding. It seems that, a considerable amount of selling orders exerted a
significant downward pressure in all three cases. The markets begin to recover after 10:50, probably as a result of discounting the short-term economic, political and security repercussions of the
ijfs-01-00154-t001_Table 1
Stock Prices ± one day of the event—Daily data frequency.
Date Open High Low Close Daily Return High-Low
6 July 2005 5190.10 5237.60 5190.10 5229.60 0.758% 47.50
FTSE-100 Index 7 July 2005 5229.60 5229.60 5022.10 5158.30 –1.373% 207.50
8 July 2005 5158.30 5232.20 5158.30 5232.20 1.422% 73.90
6 July 2005 4607.57 4636.96 4607.57 4615.49 0.257% 29.39
DAX-30 Index 7 July 2005 4595.23 4595.23 4444.94 4530.18 –1.866% 150.29
8 July 2005 4560.43 4597.97 4559.57 4597.97 1.485% 38.40
6 July 2005 4272.64 4292.07 4264.00 4279.95 0.638% 28.07
CAC-40 Index 7 July 2005 4269.56 4269.77 4089.27 4220.62 –1.396% 180.50
8 July 2005 4264.71 4300.31 4252.07 4300.31 1.871% 48.24
Stock Prices the days around and during the event.
The stock market returns are used to conduct the empirical analysis that follows. Table 2 presents the descriptive statistics for the return series in all three stock markets. In terms of the mean,
standard deviation and maximum returns, the three markets present fairly similar characteristics. Skewness and kurtosis measures indicate deviation from normality. The latter is confirmed by the
Jarque-Bera test that provides evidence against normally distributed tick-by-tick returns. Therefore preliminary statistical analysis confirms well-known stylized facts of financial markets including
significant asymmetry and kurtosis. Hence, the use of GARCH type models as a tool to take into account non-normal covariations between stock index returns seems to be appropriate.
ijfs-01-00154-t002_Table 2
Descriptive Statistics of Intraday data.
CAC- 40 Index DAX- 30 Index FTSE -100 Index
Mean 1.16 × 10^–5 1.35 × 10^–5 8.09 × 10^–6
Maximum 0.0128 0.0156 0.0101
Minimum –0.0168 –0.0229 –0.0138
Std. Dev. 0.0009 0.0011 0.0007
Skewness –0.9576 –1.9041 –0.7345
Kurtosis 40.8166 62.1455 40.6416
Jarque-Bera 612336.3 1500214 606051.9
Probability (0.00) *** (0.00) *** (0.00) ***
Observations 10250 10250 10250
Notes: The sample contains every ten minutes index returns from 21 January 2005 to 28 October 2005. The total number of usable observations is 10250. The values in parenthesis are the actual
probability values. *** indicates statistical significance at 1% level.
The estimated results for the unrestricted BEKK-GARCH (1,1) model are presented in Table 3 for both pairs of indices i.e., the FTSE-CAC (column RFTSE-RCAC) and FTSE-DAX (column RFTSE-RDAX) with the
concomitant diagnostics. As far as the whole sample is concerned the majority of the estimated parameters are statistically significant, with the only exception being the coefficient c[22] for the
FTSE-CAC pair and the coefficient k[12] for the FTSE-DAX pair.
ijfs-01-00154-t003_Table 3
BEKK-GARCH estimation results.
R[FTSE]-R[CAC] R[FTSE]-R[DAX]
Whole Sample Pre-Bomb period 21/01/2005-05/07/ Post-Bomb period 07/07/2005-28/10/ Whole Sample Pre-Bomb period 21/01/2005-05/07/ Post-Bomb period 07/07/2005-28/10/
Coeff Signif. Coeff Signif. Coeff Signif. Coeff Signif. Coeff Signif. Coeff Signif.
μ[1] 2.06E-05 (0.00) 2.34E-05 (0.00) *** 1.20E-05 (0.21) 1.43E-05 (0.01) ** 2.20E-05 (0.01) ** 1.17E-05 (0.19)
μ[2] 3.53E-05 (0.00) 3.70E-05 (0.00) *** 1.60E-05 (0.23) 2.29E-05 (0.00) 3.07E-05 (0.00) *** 1.89E-05 (0.16)
*** ***
c[11] 1.38E-04 (0.00) 1.21E-04 (0.00) *** −2.12E-04 (0.00) *** 1.69E-04 (0.00) 3.67E-04 (0.00) *** 1.71E-04 (0.00) ***
*** ***
c[21] -2.03E-04 (0.00) −1.97E-04 (0.00) *** 1.23E-04 (0.00) *** −1.71E-04 (0.00) −1.78E-04 (0.00) *** 2.10E-04 (0.00) ***
*** ***
c[22] -9.49E-08 (0.99) 3.40E-07 (0.99) −2.00E-09 (0.99) 9.05E-05 (0.01) ** −3.63E-07 (0.99) −2.86E-04 (0.00) ***
α[11] 0.0708 (0.00) −0.0103 (0.68) 0.3132 (0.00) *** 0.0466 (0.00) −0.3445 (0.00) *** 0.1215 (0.00) ***
*** ***
α[12] -0.1859 (0.00) −0.0809 (0.00) *** 0.2333 (0.00) *** −0.3094 (0.00) −0.0891 (0.00) *** −0.3445 (0.00) ***
*** ***
α[21] 0.4181 (0.00) 0.5131 (0.00) *** −0.0476 (0.09) * 0.3560 (0.00) 0.5186 (0.00) *** 0.2712 (0.00) ***
*** ***
α[22] 0.6211 (0.00) 0.6166 (0.00) *** 0.3092 (0.00) *** 0.6613 (0.00) 0.6731 (0.00) *** 0.6776 (0.00) ***
*** ***
β[11] 0.9832 (0.00) 0.9825 (0.00) *** 0.6514 (0.00) *** 0.9622 (0.00) 0.0686 (0.53) 0.9693 (0.00) ***
*** ***
β[12] 0.2914 (0.00) 0.2375 (0.00) *** −0.1966 (0.00) *** 0.2865 (0.00) −0.1268 (0.02) ** 0.1226 (0.00) ***
*** ***
β[21] -0.1112 (0.00) −0.1316 (0.00) *** 0.2478 (0.00) *** −0.0851 (0.00) 0.5139 (0.00) *** −0.0756 (0.00) ***
*** ***
β[22] 0.6962 (0.00) 0.7085 (0.00) *** 0.9896 (0.00) *** 0.7365 (0.00) 0.9177 (0.00) *** 0.8010 (0.00) ***
*** ***
κ[11] 1.96E-03 (0.00) 1.34E-03 (0.01) **
κ[12] 3.20E-03 (0.00) 1.20E-03 (0.24)
κ[22] 1.51E-03 (0.00) 1.79E-03 (0.00)
*** ***
GED Parameter 0.9268 (0.00) 0.9218 (0.00) *** 0.9117 (0.00) *** 0.9459 (0.00) 0.9122 (0.00) *** 0.9577 (0.00) ***
*** ***
Observations 10250 6068 4182 10250 6068 4182
Log 121740.71 72829.31 49053.41 120011.73 72276.21 48032.47
Notes: ***,**,* indicate statistical significance at 1%,5% and 10% level.
Conditional Volatilities and Correlation for FTSE-CAC intraday stock returns.
Note that the volatilities of the CAC and DAX indices are directly affected by the news generated within their own market () and they are also indirectly affected by news generated from the London
market ( and ). A reverse direction is also evident from the German and French markets to the London market but they are lower in magnitude as it can be deduced from the relevant coefficients in
absolute terms (). It is worth mentioning at that point that when we separate our sample into two sub-samples pre- and post- bomb period interesting findings appear^5. More specifically, coefficient
is higher in the post-bomb period for both DAX and CAC. Therefore, this event seems to affect the way that news is transmitted by London to the other two stock indices. Moreover, the statistical
significant positive mean return over the first sub-period for both CAC and DAX indices, become insignificant over the second sub-period. The volatilities of all the indices’ returns are directly
affected by their own past volatilities respectively in the whole sample estimation (the relevant coefficient is 0.96 for the British market and 0.48, 0.54 for the French and German market
respectively). However, in the case of France, volatility persistence increased in the post-bomb period. Indirect effects of past volatilities are also present in each case. However, the indirect
effects of the London market on the CAC and DAX volatilities respectively, are higher compared to the indirect effects of the latter on FTSE () for the whole sample. Focusing on the covariance
equation in the bivariate BEKK-GARCH models, unexpected shocks in the London market reduce the covariance between FTSE and CAC or DAX. However, unexpected shock in the French and German markets
increases their covariance with the London market.
Conditional Volatilities and Correlation for FTSE-DAX intraday stock returns.
Overall, the results are uniformed in terms of the effect of the terrorism dummy variable on their respective volatilities. There is evidence of a positive and statistically significant effect (k[ii]
coefficients) on the volatilities of all three markets. As one would intuitively expect, the higher positive coefficient is present in the case of London. On the other hand, the direct effect of the
terrorist attack on the correlation between the stock markets is not uniform. The correlation is directly, significantly and positively affected in the case of the FTSE - CAC pair (see coefficient k
[12]). While for the FTSE–DAX pair, their correlation seems to not be affected in statistically significant degree. For both cases, indirect effects are present from the positive and statistically
significant effect of the London market volatility on the covariance term (cross term β[11]β[12]). Finally, past correlation seems to affect in a similar way current correlation in every pair of the
indices (β[21]β[12] + β[11]β[12]). In Figure 2 and Figure 3, the significant positive effect on intraday stock market volatilities during the event day for the FTSE-CAC and the FTSE-DAX pairs of
indices^6 is clearly visible. Moreover, for the case of FTSE-CAC pair the correlation is significantly increased during the event minutes implying no diversification benefits across these two
markets. This finding is in accordance with Chesnay and Jondeau [43] who argued that correlation between stocks returns really increase during turbulent periods. In contrast, the correlation is not
affected in a statistically significant and positive manner for the FTSE-DAX pair of indices.
As many studies have shown (inter alia: Graham and Ramiah [20]; Fernandez [22]; Ramiah et al. [19]; Drakos [24]; Kollias et al. [26]) terrorist events exert a negative, albeit generally short-lived,
impact on stock markets and equity returns. Shocks from terrorist events are also transmitted cross-nationally and affect other financial markets apart from the one of the country that was the venue
of the attack (Hon et al. [28]; Mun [29]; Drakos [30,31]). The cross-market transmission of the shock caused by a major European terrorist event, namely the bomb attacks of 2005 in London, was the
theme of this study using high frequency data. A general way of incorporating insecurity shocks on a standard multivariate GARCH model is proposed. Future research can use these types of modified
models to test other insecurity shocks.
Results reported herein, indicate that the contagion effect, as it is defined by Forbes and Rigobon [44], is mainly present from the London to the Paris market. The correlation between the FTSE and
CAC indices is increased significantly during the period the event unfolded in London. Moreover, there is also a significant positive effect of the London market volatility on the CAC and DAX
indices’ volatilities. Investigation over sub-samples confirms our findings, indicating that news transmission from London, to Paris and Frankfurt is significantly changed after the terrorist attack.
Volatility persistence increased significantly after the attack in case of CAC index. Changes in some parameters of interest are identified over the post-bomb period. Finally, over the time period of
the event, stock market volatilities are high in all of our three cases, perhaps suggesting possible gains by intraday trading activity in derivative markets. Trading strategies based on volatility
signals may lead to profitable trades.
The authors gratefully acknowledge insightful comments and constructive suggestions by three anonymous referees of this journal that helped to improve the paper. This paper was part of the research
project “A New Agenda for European Security Economics”, funded by the 7th Framework Programme, to which the authors acknowledge financial support. The usual disclaimer applies.
The authors declare no conflict of interest.
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LSE was the bigger in terms of market capitalization in 2005, in Europe followed by the German and French markets (see: www.world-exchanges.org/statistics).
Data are collected by http://www.tickdata.com.
In order to count for the effect induced by the events occurred in 08:50 and 09:50 respectively we have used a dummy taking the value of one from the first bomb explosion lasting one hour and for the
second bomb also lasting also one hour.
See for instance Kollias et al. [26] that compare this attack to the one in Madrid in 2004.
We would like to thank one of the anonymous referees for his helpful comment to divide whole sample to sub-samples and investigate for possible differences among stock markets.
The secd column of each graph zooms in on the event day window in order to present in a more clear manner the effect. | {"url":"http://www.mdpi.com/2227-7072/1/4/154/xml","timestamp":"2014-04-21T06:08:01Z","content_type":null,"content_length":"85271","record_id":"<urn:uuid:5737bdc2-6ec3-47b2-b9cb-843aca72d38b>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00480-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Brief Family Tree of Some Important Math
Once upon a time, Heron of Alexandria conceived of imaginary numbers. Gerolamo Cardano expanded upon this concept in the 16th century and conceived of complex numbers. Then, in 1843, Sir William
Rowan Hamilton expanded upon that concept and created the quaternions, which were essentially complex numbers with three orthogonal imaginary components plus a real component. This was quickly
followed in that same year by John T. Graves and Arthur Cayley independently concocting octonions.
The following year, Hermann Grassmann created the exterior algebra, the algebra of anti-commuting entities known as Grassman Numbers. Then along came William Kingdon Clifford, who united Grassman’s
algebra with Hamilton’s work on quaternions to create Clifford Algebra.
So, what’s the big deal here? Think spinors. Think fermions.
For quite some time, I’ve been working on a rather lengthy and detailed article on fermions and the Spin-Statistics Theorem. (There is also a lengthy and fairly comprehensive article in the works on
neutrino physics, yet another on the Action Principle, and still another on Noether’s Theorem, symmetry, and conservation laws. Working on all of these in parallel is pretty time-consuming.) The
aforementioned mathematical constructs are rather critical to understanding these concepts and will have to be fleshed out before I can really present that topic. So watch this space, and consider
the little mathematical family tree outlined above as a road-map of what is to come….
For more info, see:
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Re: two sine tones simultaneously within one critical band
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Re: two sine tones simultaneously within one critical band
Bob Masta wrote:
> I am completely at a loss to understand how you arrived
> at your last formula. It appears that you have not simply
> added the pressure waves, but multiplied them.
cos(2pi*t) * cos(100 * 2pi*t) = cos(99*2pi*t) + cos(101*2pi*t)
This can be derived using:
cos(x1)*cos(x2) = 1/2*cos(x1-x2) + 1/2*cos(x1+x2)
Another explanation can be: two frequencies (99Hz and 101Hz) can be seen as one absent carrier (100Hz) modulated with a modulator (1Hz).
br, Piotr Majdak
Piotr Majdak
Acoustic Research Institute
Austrian Academy of Sciences
Reichsratsstr. 17
A-1010 Wien
Tel.: +43-1-4277-29511
Fax: +43-1-4277-9296
E-Mail: piotr@xxxxxxxxxx
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Paul Goldberg
I'm not sure that there's a research problem to be derived from the following discussion, but it was fun to write. Our main finding is that two reviewers are better than one, on the grounds that they
can be incentivized to provide accurate forecasts of the acceptability of a paper by a program committee that is arbitrarily unlikely to read it.
We regard the reviewer’s review as a forecast of whether the paper will be accepted, rather than an input to the decision. This has the advantage that the reviewer is not required to make a value
judgment on the paper; a forecast is just a guess as to whether the paper will be accepted. The obvious problem is that if the forecast gets used to help with the decision, it is at risk of becoming
a self-fulfilling prophesy.
We model the paper selection process as follows. Any paper has an associated quantity
∈ [0,1] that represents the probability with which it ought to be accepted, and furthermore, this value is revealed to a careful reader. In an ideal world, the PC passes the paper to a reviewer, who
reads it and reports this number
to the PC, and the PC proceeds to accept the paper with probability
. (We assume that the PC has access to an unlimited source of randomness, which appears to be a realistic assumption.)
Suppose now that a reviewer knows in advance that his review will be ignored by the program committee, who will instead read the paper and accept it with the correct probability. In that case, if the
reviewer reports probability
, he/she should be given a reward of log(
) if the paper is accepted, and log(1-
) if it is rejected. (These quantities are negative, but we do not claim that reviewing papers is rewarding.) These rewards incentivize the reviewer to report the correct probability.
Now, suppose when the PC has received the review (the number
), they then read the paper with some probability
. If they read the paper, they accept with the correct probability, and if they don’t read it, they accept with probability
. The problem is that if
is very small, and the reviewer finds that the paper should be accepted with probability about 1/2, the above rewards tempt him to go to extremes and report (say) 0⋅01 or 0⋅99.
Important note:
the reward should depend only on the review (the reported
) and the (binary) acceptance decision, since you don’t want to reveal any secret discussions to the reviewer. So the PC doesn't have the option to read the paper with some small probability and
punish him if they then find he lied.
Given 2 reviewers, we can exploit their professional rivalry to make them tell the truth, by carefully aggregating their forecasts into a single probability, as follows. A basic requirement for a
reward scheme is that if a reviewer has obtained value
from a paper, and the other reviewer is truthfully reporting that value
, then the remaining reviewer should also have the incentive to do likewise. Suppose we use the logarithmic rewards above, and the PC uses the average of the 2 reported probabilities, to decide on
the paper. The following problem arises: suppose it's a great paper and
=0⋅999. A reviewer might be tempted to report (say)
=0⋅5, since that way, the PC will use a probability of about 0⋅75 to accept the paper, exposing the other reviewer to a big risk of a large penalty. The assumption here is that a reviewer aims to get
a higher reward than the other one (his professional rival); the reward being some sort of credit or esteem rather than money.
be the other reviewer's probability, and we seek a function
) that should be used by the PC as probability to accept the paper; we have noted that (
)/2 is not a good choice of
, in conjunction with the logarithmic rewards.
The reviewer’s utility
) is his expected reward minus his opponent’s expected reward:
u(p) = f(p,q)(logp-logq)+(1-f(p,q))(log(1-p)-log(1-q))
We now notice that the above must be identically zero, since it should not incentivize the reviewer to change his mind if
is incorrect, but it should not incentivize him not to change his mind if
is correct. Setting the above to zero tells us the function
should be
It just remains for the PC to read the paper with any probability ε>0, and in the event that they read it, accept with the correct probability. If they read the paper, the reviewers are incentivized
to tell the truth, and if they don’t, (and use the above
) the reviewers have no incentive to lie, so overall their incentive will indeed be to tell the truth.
(Added 6.6.11: at the iAGT workshop, I heard about 2 papers that relate (so now, an answer to the first comment below).
Only valuable experts can be valued
by Moshe Babaioff, Liad Blumrosen, Nicolas S. Lambert and Omer Reingold, about contracts that will be accepted by self-proclaimed experts, provided that they really do have expert knowledge (and will
be declined by a rational charlatan). And,
Tight Bounds for Strategyproof Classification
by Reshef Meir,
Shaull Almagor, Assaf Michaely and Jeff Rosenschein, about learning classifiers where the class labels of data have been provided by agents who may try to lie about the labels. The latter paper is
closer to the “self-fulfilling prophesy” situation described above.)
(Added 22.8.11:
This article
on “decision fatigue” suggests another reason why it may be better to ask people to try to predict the outcome than to influence it (assuming you believe it puts less strain on someone to make a
prediction than to make a decision. It does sometime stress me out a bit to make an accept/reject recommendation for a paper.))
(found some of the following links from a facebook post) this link to an attempt to write down a game that expresses the setting of tuition fees. In the comments, this link to a speech by David
Willetts "The ideas that are changing politics" (dated 20.2.08, so now a couple of years ago) in which he is enthusiastic about game theory (follow the link to PDF of the speech at the bottom of the
page). This new article “What are David Willetts' options for limiting spending on student loans?” is worth reading by anyone who would like to keep track of this issue. The previous article gives
the text of a speech on universities made by Willetts; the speech mentions game theory, and so too does one of the annotations that have been inserted. Here’s a quote from the speech:
Of course, academics approach these issues with great sophistication, and I have been warned that we face a dilemma from game theory in which the incentives for individual institutions are
different from the interests of the sector as a whole. But it's not the dilemma in its classic form, because this is not a one-off. You need to think of subsequent years – not just in terms of
funding levels but also the challenges you will face from new competitors if you come in at such a high fee level. And you also need to think of the collective interests of students. | {"url":"http://paulwgoldberg.blogspot.com/2011_03_01_archive.html","timestamp":"2014-04-19T14:41:02Z","content_type":null,"content_length":"93178","record_id":"<urn:uuid:1c7c3b6a-242d-479a-a7fe-9fd26f966718>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00450-ip-10-147-4-33.ec2.internal.warc.gz"} |
point-set topology
Also known as general topology, a branch of topology concerned with how to put a structure on a set in such a way as to generalize the idea of continuity for maps from the real numbers to itself. A
topology on a set X is a certain set of so-called open subsets of the set X which satisfy various axioms. The set X together with this topology is called a topological space.
Related category | {"url":"http://www.daviddarling.info/encyclopedia/P/point-set_topology.html","timestamp":"2014-04-16T22:48:23Z","content_type":null,"content_length":"6338","record_id":"<urn:uuid:34b65eeb-45a8-4346-a22e-c7924752077d>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00636-ip-10-147-4-33.ec2.internal.warc.gz"} |
Automated Design of Synthetic Ribosome Binding Sites to Precisely Control Protein Expression
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Nat Biotechnol. Author manuscript; available in PMC Apr 4, 2010.
Published in final edited form as:
PMCID: PMC2782888
NIHMSID: NIHMS145791
Automated Design of Synthetic Ribosome Binding Sites to Precisely Control Protein Expression
The publisher's final edited version of this article is available at
Nat Biotechnol
See other articles in PMC that
the published article.
Microbial engineering often requires fine control over protein expression; for example, to connect genetic circuits ^1^-^7 or control flux through a metabolic pathway ^8^-^13. We have developed a
predictive design method for synthetic ribosome binding sites that enables the rational control of a protein's production rate on a proportional scale. Experimental validation of over 100 predictions
in Escherichia coli shows that the method is accurate to within a factor of 2.3 over a range of 100,000-fold. The design method also correctly predicts that reusing a ribosome binding site sequence
in different genetic contexts can result in different protein expression levels. We demonstrate the method's utility by rationally optimizing a protein's expression level to connect a genetic sensor
to a synthetic circuit. The proposed forward engineering approach will accelerate the construction and systematic optimization of large genetic systems.
Keywords: synthetic biology, translation, optimization, metabolic engineering, genetic circuit, RNA secondary structure
Microbial engineering is a time-consuming procedure that often requires multiple rounds of trial-and-error genetic mutation. As it becomes possible to construct larger pieces of synthetic DNA ^14,
including whole genomes ^15, automated methods for genetic circuit assembly and metabolic pathway optimization will be critically important. As genetic systems grow in size and complexity, the
application of a trial-and-error approach to optimizing these systems is more difficult.
A genetic system's function is optimized by varying the sequences of its regulatory elements to control the expression levels of its protein coding sequences. Each rate-limiting step in gene
expression offers the opportunity for rationally modulating the protein expression level. In bacteria, ribosome binding sites (RBSs) and other regulatory RNA sequences are effective control elements
for translation initiation ^16^-^19. As a consequence, they are commonly mutated to optimize genetic circuits, metabolic pathways, and the expression of recombinant proteins.
Previous studies have generated libraries of RBS sequences with the goal of optimizing the function of a genetic system ^1^, ^7^, ^18. Generation and selection of a sequence library can become
impractical as the number of participating proteins increases, especially if measuring the function requires a low-throughput assay or screen ^6. For example, randomly mutating 4 nucleotides of an
RBS generates a library of 256 sequences. The library size increases combinatorially with the number of proteins in the engineered system (16.7 million sequences for 3 proteins, 2.8×10^14 sequences
for 6 proteins).
A biophysical model of translation initiation would aid the optimization process by enabling the design of an RBS sequence to obtain a desired translation initiation rate. Using thermodynamics, the
free energies of key molecular interactions involved in translation initiation have been characterized ^20^, ^21. Thermodynamic models are made possible by measuring the sequence-dependent energetic
changes during RNA folding and hybridization ^22^-^26. These methods have enumerated and characterized the attributes of a RBS sequence that affect its translation initiation rate, but a predictive
model that combines all of the interactions together has not been created and tested.
Bacterial translation consists of four phases: initiation, elongation, termination, and ribosome turnover (Figure 1A) ^27. In most cases, translation initiation is the rate-limiting step. The
translation initiation rate is determined by the summary effect of multiple molecular interactions, including the hybridization of the 16S rRNA to the RBS sequence, the binding of tRNA^fMET to the
start codon, the distance between the 16S rRNA binding site and the start codon, and the presence of RNA secondary structures that occlude either the 16S rRNA binding site or the standby site ^20^, ^
21^, ^28^-^31.
A thermodynamic model of bacterial translation initiation. (A) The ribosome translates an mRNA transcript and produces a protein in a four step process: the rate-limiting assembly of the 30S
pre-initiation complex, translation initiation, translation ...
We have developed an equilibrium statistical thermodynamic model to quantify the strengths of the molecular interactions between the 30S complex and an mRNA transcript and to predict the resulting
translation initiation rate. The thermodynamic model describes the system as having two states separated by a reversible transition (Figure 1B). The initial state is the folded mRNA transcript and
the free 30S complex. The final state is the assembled 30S pre-initiation complex on an mRNA transcript. The difference in Gibbs free energy between these two states is quantified by the Gibbs free
energy change ΔG[tot]. The ΔG[tot] depends on the mRNA sequence surrounding a specified start codon and will become more negative when attractive interactions are present and more positive when
mutually exclusive secondary structures are present.
The translation initiation rate r is related to the ΔG[tot] according to
where β is the Boltzmann factor for the system. The derivation of Equation 1 is presented in the Supplementary Methods. Importantly, Equation 1 describes the differences in translation initiation
rate that result from differences in mRNA sequence. The amount of expressed protein E is proportional to the translation initiation rate where the proportionality factor K accounts for any
ribosome-mRNA molecular interactions that are independent of mRNA sequence and any translation-independent parameters, such as the DNA copy number, the promoter's transcription rate, the mRNA
stability, and the protein dilution rate (Supplementary Figure 1).
Given a specific mRNA sequence surrounding a start codon, called the subsequence, the ΔG[tot] is predicted according to the energy model:
where the reference state is a fully unfolded subsequence with ΔG[ref] = 0.
The ΔG[mRNA:rRNA] term is the energy released when the last 9 nucleotides (nt) of the E. coli 16S rRNA – 3′-AUUCCUCCA-5′ – hybridizes and co-folds to the mRNA subsequence (ΔG[mRNA:rRNA] < 0).
Intra-molecular folding within the mRNA is allowed. All possible hybridizations between the mRNA and 16S rRNA are considered to find the highest affinity 16S rRNA binding site. The binding site
minimizes the sum of the hybridization free energy ΔG[mRNA:rRNA] and the penalty for non-optimal spacing ΔG[spacing]. Thus, the algorithm can identify the 16S rRNA binding site regardless of its
similarity to the consensus Shine-Dalgarno sequence.
The ΔG[start] term is the energy released when the start codon and the initiating tRNA anti-codon loop – 3′-UAC-5′ – hybridize together. The ΔG[spacing] is the free energy penalty caused by a
non-optimal physical distance between the 16S rRNA binding site and the start codon (ΔG[spacing] > 0). When this distance is increased or decreased from an optimum of 5 nt (or ~17 Å) ^29, the 30S
complex becomes distorted, resulting in a decreased translation initiation rate.
The ΔG[mRNA] is the work required to unfold the mRNA subsequence when it folds to its most stable secondary structure, called the minimum free energy structure (ΔG[mRNA] < 0). The ΔG[standby] is the
work required to unfold any secondary structures sequestering the standby site (ΔG[standby] < 0) after 30S complex assembly. We define the standby site as the 4 nucleotides upstream of the 16S
rRNA-binding site, which is its location in a previously studied mRNA ^28.
To calculate the ΔG[mRNA:rRNA], ΔG[start], ΔG[mRNA], and ΔG[standby] free energies, we use the NUPACK suite of algorithms, developed by Pierce and coworkers ^32, with the Mfold 3.0 RNA energy
parameters ^22^, ^23. These free energy calculations do not have any additional fitting or training parameters and explicitly depend on the mRNA sequence. In addition, the free energy terms are not
orthogonal; changing a single nucleotide can potentially affect multiple energy terms.
We designed a series of experiments to quantify the relationship between the aligned spacing s and the free energy penalty ΔG[spacing]. Thirteen synthetic RBSs are created where the aligned spacing
is varied from 0 to 15 nucleotides while verifying that the ΔG[mRNA:rRNA], ΔG[mRNA], ΔG[start], and ΔG[standby] free energies remain constant (Supplementary Table I). The translation initiation rates
of RBS sequences are measured using a fluorescent protein measurement system (Methods). Steady-state fluorescence measurements are performed on E. coli cultures over a 24 hour period. Under these
conditions, the average fluorescence measurement is expected to be proportional to the translation initiation rate r.
The quantitative relationship between the aligned spacing and ΔG[spacing] is obtained from the fluorescence measurements (Methods). According to the data, it is conceptually useful to treat the 30S
complex as a model barbell connected by a rigid spring, where either stretching or compressive forces cause a reduction in entropy and an increase in the ΔG[spacing] penalty. We empirically fit these
measured ΔG[spacing] values to either a quadratic (s > 5 nt) or a sigmoidal function (s < 5 nt). Following this parameterization, we tested the accuracy of these equations on an additional set of
synthetic RBS sequences (Supplementary Figure 2).
For an arbitrary mRNA transcript, the thermodynamic model (Equation 2) is evaluated for each AUG or GUG start codon. The algorithm considers only a subsequence of the mRNA transcript, consisting of
35 nucleotides before and after the start codon. This subsequence includes the RBS and part of the protein coding sequence. The model predictions do not improve when longer subsequences are
considered (Supplementary Figure 3).
The development of the thermodynamic model makes certain assumptions. Contributions related to the ribosomal S1 protein's potential preference for pyrimidine-rich sequences are omitted from the free
energy model^33. The model also assumes that the reversible transition between the initial and final state of 30S complex assembly reaches chemical equilibrium on a physiologically relevant timescale
and without any long-lived intermediate states. The presence of overlapping or neighboring start codons, overlapping RBS and protein coding sequences, regulatory RNA binding sites, or RNAse binding
sites also pose a challenge to the predictive accuracy of the thermodynamic model. The presence of multiple in-frame start codons, each with significant translation initiation, may distort its
predictive accuracy. A genetic system can be designed to avoid many of these complications.
The thermodynamic model can be used in two ways. First, it can predict the relative translation initiation rate of an existing RBS sequence for a particular protein coding sequence on an mRNA
transcript. We refer to this as “reverse engineering” because the RBS sequence already exists. Second, it can be used in conjunction with an optimization algorithm to identify a synthetic RBS
sequence that is predicted to translate a given protein coding sequence at a user-selected rate. We refer to this mode as “forward engineering” because it generates a de novo sequence according to a
user's specifications.
We use the thermodynamic model to predict the translation initiation rates of 28 existing RBS sequences (Figure 2A) that were obtained from a natural genome or taken from a list of commonly used
sequences (Supplementary Table I). The lengths of these sequences, as measured by the distance from the transcriptional start site to the fluorescent protein's start codon, vary from 24 to 42
nucleotides. The steady-state protein fluorescences from the sequences are then assayed in the measurement system (Methods). The growth rates of the cell cultures did not correlate with protein
fluorescence (Supplementary Figure 4). According to the theory (Equation 1), we expect a linear relationship between the predicted ΔG[tot] and the log protein fluorescence. Using linear regression,
the squared correlation coefficient R^2 is 0.54 with Boltzmann factor β = 0.45 ± 0.05 mol/kcal (Figure 2B). The average error is Figure 2C).
The design method has two modes of operation: (A) The method can predict the relative translation initiation rate of an existing RBS when placed in front of a protein coding sequence. The method
calculates the ΔG[tot] from the input sequence. According ...
While these commonly used RBS sequences vary the protein expression by 1500 fold, the thermodynamic model predicts that both stronger and weaker RBSs are possible. For example, one of these RBS
sequences contains a strong 16S rRNA binding site (ΔG[mRNA:rRNA] = −15.2 kcal/mol), but did not yield a high protein expression level due to a strong mRNA secondary structure and non-optimal spacing
(ΔG[mRNA] = −11.4, ΔG[spacing] = 1.73 kcal/mol). By optimizing the RBS sequence towards a selected ΔG[tot], we gain the ability to rationally control the translation initiation rate over a wide range
with a proportional effect on the protein expression level.
Using the thermodynamic model, we developed an optimization algorithm that automatically designs an RBS sequence to obtain a desired relative protein expression level. The user inputs a specific
protein coding sequence and a desired translation initiation rate. The rate can be varied over five orders of magnitude on a proportional scale. Equation 1 and the experimentally measured β = 0.45
mol/kcal is used to convert the user-selected translation initiation rate into the target ΔG[tot]. The method then generates a synthetic RBS sequence according to the desired specifications.
The design method combines the thermodynamic model of translation initiation with a simulated annealing optimization algorithm to design an RBS sequence that is predicted to have a target ΔG[tot] (
Figure 2D). The RBS sequence is initialized as a random mRNA sequence upstream of the protein coding sequence. The method then creates new mRNA sequences by inserting, deleting, or replacing random
nucleotides. For each new sequence, the ΔG[tot] is calculated and compared to the target ΔG[tot]. The sequences are then accepted or rejected according to the Metropolis criteria and three additional
sequence constraints that are based on the model's assumptions (Methods). The procedure continues until the synthetic sequence has a predicted ΔG[tot] to within 0.25 kcal/mol of the target. For a
given target ΔG[tot], multiple solutions are possible, creating an ensemble of degenerate RBS sequences. The characterization of these ensembles is described in the Supplementary Discussion.
The forward design method is tested by generating 29 synthetic RBS sequences (Supplementary Table I) and comparing their predicted ΔG[tot] values to the measured protein fluorescences. The coding
sequence for a red fluorescent protein is specified and the ΔG[tot] target is varied from −7.1 to 16.0 kcal/mol. The design method then generates a synthetic RBS sequence for each target ΔG[tot].
These RBS sequences vary in length from 16 to 35 nucleotides and were highly dissimilar. The steady-state protein fluorescence for each sequence is measured (Methods). The growth rates of the cell
cultures did not significantly vary across sequences (Supplementary Figure 4). As expected from the theory (Equation 1), we obtain a linear relationship between the log protein fluorescence and the
predicted ΔG[tot] with β = 0.45 ± 0.01 (R^2 = 0.84) (Figure 2E). The average error is Figure 2F).
We next tested the ability of the design method to control the translation initiation rates of different proteins. Two chimeric proteins are constructed that fused the first 27 nucleotides from
commonly used transcription factors to a red fluorescent protein (TetR[27]-RFP and AraC[27]-RFP). The design method is then used to generate 23 synthetic RBSs with ΔG[tot] targets ranging from −8.5
to 10.5 kcal/mol (Supplementary Table I). The thermodynamic model correctly predicts the translation initiation rates of the TetR[27]-RFP (R^2 = 0.54) and AraC[27]-RFP (R^2 = 0.95) chimeric protein
coding sequences (Figure 3A). Notably, the linear relationship between the predicted ΔG[tot] and the log protein fluorescence yields a similar slope β = 0.45 ± 0.05 mol/kcal.
The design method can control the expression level of different proteins by predicting the impact of changing the protein coding sequence. (A) The fluorescence levels from 23 synthetic RBSs in front
of two different protein coding sequences are measured ...
A common practice is to reuse the same well-characterized RBS sequence for the expression of different proteins. Interestingly, the thermodynamic model predicts that this can yield dramatically
different translation initiation rates. This absence of modularity will occur when the RNA sequence, containing the RBS, forms strong secondary structures with one protein coding sequence, but not
another ^30.
We designed experiments to test the model's ability to predict the impact of changing the protein coding sequence on the translation initiation rate. We use the design method to generate 14 synthetic
RBS sequences; these sequences are then placed upstream of two different protein coding sequences: the fluorescent protein (RFP) and a chimeric fluorescent protein (TF-RFP: LacI[27]-RFP, TetR[27]
-RFP, or AraC[27]-RFP). The optimization procedure for these synthetic RBSs was modified to maximize the objective function |ΔG[RFP] − ΔG[TF-RFP]|, where ΔG[RFP] and ΔG[TF-RFP] are the predicted ΔG
[tot]'s when the RBS sequence is placed upstream of either the RFP or TF-RFP protein coding sequences, respectively. As predicted by the model, the translation initiation rates of these synthetic RBS
sequences greatly change when they are reused with different protein coding sequences (Figure 3B); for example, replacing the fluorescent protein with the TetR[27]-RFP chimera resulted in a 530-fold
increase in expression level.
The thermodynamic model can accurately predict these differences in translation initiation rate when the correct protein coding sequence is specified (R^2 = 0.62 and 0.51, Figure 3C). When the
incorrect protein coding sequence is used, the translation initiation rate is not accurately predicted (R^2 = 0.04, 0.02). Consequently, when designing a RBS sequence, the beginning of the protein
coding sequence must be included in the thermodynamic calculations.
Altogether, 119 predictions of the design method were tested, revealing that the translation initiation rate can be controlled over at least a 100,000-fold range. The thermodynamic model is most
accurate when all free energy terms are included in the ΔG[tot] calculation (Supplementary Figure 5). By themselves, each free energy term is a poor predictor of the translation initiation rate (
Supplementary Figure 6) and excluding one free energy term from the ΔG[tot] calculation results in a poorer prediction (Supplementary Figure 7). According to the distribution of the method's error (
Figure 2F), an optimized RBS sequence has a 47% probability of expressing a protein to within 2-fold of the target. The probability increases to 72%, 85%, or 92% by generating two, three, or four
optimized RBS sequences with identical target translation initiation rates (Supplementary Discussion).
We now demonstrate how combining the design method with a quantitative model of a genetic system enables the efficient optimization of its RBS sequences towards a targeted system behavior. Here, our
objective is to optimize the connection between the arabinose-sensing P[BAD] promoter and an AND gate genetic circuit^7. The AND gate genetic circuit is regulated by the expression levels of two
input promoters (P[BAD] and P[sal]) and controls the expression level of an output gene, which is selected to be a gfp reporter (Figure 4A). The desired AND logic requires that the output gene is
only expressed when both input promoters are active. The digital accuracy of the AND logic is highest when the maximum expression level from the P[BAD] promoter is an optimal value between
underexpression and overexpression. When the promoter is underexpressed, the gfp expression is never turned on; when overexpressed, transcriptional leakiness causes gfp expression to turn on even in
the input's absence.
Optimal connection of a sensor input to an AND gate genetic circuit. (A) A functional AND gate genetic circuit will only turn on the gfp reporter output when both the P[BAD] and P[sal] promoter
inputs are sufficiently induced by arabinose and salicylate, ...
The quantitative model relates the RBS sequence downstream of the P[BAD] promoter to the accuracy of the AND gate genetic circuit's function (Figure 4B). We use previously characterized transfer
functions^7 to relate the arabinose and salicylate concentrations to the expression levels of the P[BAD] and P[sal] promoters (I[1] and I[2]) (Supplementary Figure 8). The P[BAD] promoter has a
maximum protein expression level of g[ref] = 590 au at full induction (x = 1.3 mM arabinose) and when using an RBS sequence with a predicted ΔG[tot] value of ΔG[ref] = −1.05 kcal/mol. We then
substitute I[1] and I[2] into the AND gate genetic circuit's transfer function to determine the output gene's expression level, which is in turn substituted into the fitness function F that
quantifies the ability of the genetic system to carry out AND logic (Supplementary Methods).
We can interconvert between the maximum protein expression level of the P[BAD] promoter and the predicted ΔG[tot] of its RBS sequence according to the equation,
where g is called the gain. The experimentally measured β = 0.45 mol/kcal is utilized. Consequently, we create a quantitative curve F(ΔG[tot]) that relates the predicted ΔG[tot] of the P[BAD]
promoter's RBS sequence to the fitness of the genetic system. The fitness curve identifies an optimal region at ΔG[tot] = −1.17 ± 2 kcal/mol where the genetic system will exhibit the best AND logic
with respect to the P[BAD] promoter's RBS sequence (Figure 4B).
Using the forward engineering mode of the design method, we then generate 2 synthetic RBS sequences targeted to the optimum region of the genetic system's function (predicted ΔG[tot] = −1.48 and
−1.15 kcal/mol). We also design 7 additional synthetic RBSs to test the accuracy of the F(ΔG[tot]) fitness curve, where the ΔG[tot] ranged from 0.60 to 17.2 kcal/mol. Each RBS sequence (32 to 35 nt)
is inserted downstream of the P[BAD] promoter and the resulting genetic circuit's response to varying inducer concentrations is assayed (Figure 4C and Methods). The fitness values of these rationally
mutagenized genetic systems are then compared to the predictions of the model and design method (Figure 4B). The insertion of two stronger RBS sequences (ΔG[tot] = −2.5 and −3.0 kcal/mol) cause the
genetic system to exhibit a fatal growth defect.
Both optimally designed synthetic RBS sequences result in a successful connection between the arabinose-sensing P[BAD] promoter and the AND gate genetic circuit (mean fitness > 0.85, Figure 4B). The
experimentally determined optimum in the F(ΔG[tot]) curve is nearby ΔG[tot] = 0.60 kcal/mol, which is only a 1.8 kcal/mol deviation from the model's prediction. The quantitative model and design
method also correctly predict how the fitness of the genetic system deteriorates with an increasing ΔG[tot]. Thus, our approach enabled us to rationally connect two synthetic genetic circuits
together to obtain a target behavior while performing only a few mutations and assays (additional design calculations are located in the Supplementary Discussion).
A central goal of synthetic biology is to program cells to carry out valuable functions. As we construct larger and more complicated genetic systems, models and optimization techniques will be
required to efficiently combine genetic parts to achieve a target behavior. To accomplish this, biophysical models that link the DNA sequence of a part to its function will be necessary. As
engineered genetic systems scale to the size of genomes, the integration of multiple design methods will enable the design of synthetic genomes on a computer to control cellular behavior.
Materials and Methods
Software Implementation
A software implementation of the design method has been named the RBS Calculator and is available at http://voigtlab.ucsf.edu/software. Visitors may use the RBS Calculator in two ways: first, to
predict the translation initiation rate of each start codon on an mRNA sequence (reverse engineering); second, to optimize the sequence of a ribosome binding site to rationally control the
translation initiation rate with a proportional effect on the protein expression level (forward engineering). The translation initiation rate is gauged on a proportional scale with a suggested range
of 0.1 to 100000, although a larger range is potentially feasible. In reverse engineering mode, the software will warn visitors when ribosome binding sites fail to satisfy the sequence constraints or
contain additional sequence complications.
A thermodynamic model of translation initiation
The mRNA subsequence S[1] consists of the max(1, n[start] − 35) to n[start] nucleotides and the subsequence S[2] consists of the max(1, n[start] − 35) to n[start] + 35 nucleotides, where n[start] is
the position of a start codon. The ΔG[start] is −1.19 and −0.075 kcal/mol for AUG and GUG start codons, respectively ^22.
Using the NuPACK ‘subopt’ algorithm ^32 with Mfold 3.0 parameters at 37°C ^22^, ^23, base pair configurations of the folded 16S rRNA and sequence S[1] are enumerated, starting with the minimum free
energy (mfe) configuration and continuing with suboptimal configurations, each with a corresponding ΔG[mRNA:rRNA]. For each configuration, the aligned spacing between the 16S rRNA binding site and
start codon is calculated according to s = n[start] − n[1] − n[2], where n[1] and n[2] are the rRNA and mRNA nucleotide positions in the farthest 3′ base pair in the 16S rRNA binding site. When the
30S complex is stretched (s > 5 nt), the ΔG[spacing] is calculated according to the quadratic equation,
where s[opt] = 5 nt, c[1] = 0.048 kcal/mol/nt^2, and c[2] = 0.24 kcal/mol/nt. When the 30S complex is compressed (s < 5 nt), the ΔG[spacing] is calculated according to the sigmoidal function,
where c[1] = 12.2 kcal/mol and c[2] = 2.5 nt ^−1. The above parameter values are determined by minimizing the difference between the ΔG[spacing] values calculated from the experimental measurements (
Supplementary Figure 2) and the evaluation of Equation 4 or 5. For each configuration, the ΔG[spacing] is added to the ΔG[mRNA:rRNA]. The configuration in the list with the lowest free energy is then
identified as containing the predicted 16S rRNA binding site with a corresponding ΔG[mRNA:rRNA]. The protein coding sequence is excluded from S[1] because ribosome binding excludes the formation of
downstream secondary structures.
Using the NuPACK ‘mfe’ algorithm and Mfold parameters, the mfe configuration of sequence S[2] is calculated and its free energy is designated ΔG[mRNA]. The standby site is the 4 nt region upstream of
the 16S rRNA binding site. The energy required to unfold the standby site is determining by calculating the mfe of sequence S[2] with and without preventing the standby site from forming base pairs.
The difference between these mfe's is designated ΔG[standby]. To calculate the mfe of sequence S[2] with a standby site that is constrained to be single-stranded, the sequence is first split into two
subsequences, their mfes are each calculated, and then summed together. The two subsequences are the nucleotides n[start] − 35 to n[3] − 4 and n[3] to n[start] + 35, where n[3] is the most 5′ base
pair in the 16S rRNA binding site and 4 is the standby site length.
The five energy terms are summed together to calculate the ΔG[tot]. Notably, selecting an alternate reference energy state simply adds a sequence-independent constant to the predicted ΔG[tot], which
becomes indistinguishable from the proportionality factor K.
The simulated annealing optimization algorithm
An initial RBS sequence is randomly generated and inserted in between a pre-sequence and protein coding sequence to create a sequence S. The ΔG[tot] of the sequence S is calculated and the objective
function O[old] = |ΔG[tot] − ΔG[target]| is evaluated. In an iterative procedure, the simulated annealing optimization algorithm randomly deletes, inserts, or replaces a nucleotide in the RBS
sequence. The ΔG[tot] and objective function O[new] are then recalculated. If the ΔG[tot] calculation of S invalidates the sequence constraints, then the mutation is immediately rejected. Otherwise,
the mutation is accepted with probability max(1, exp([O[old] − O[new]]/T[SA])), where T[SA] is the simulated annealing temperature. The T[SA] is continually adjusted to maintain a 5% to 20%
acceptance rate.
There are three sequence constraints that prevent the optimization algorithm from generating a synthetic RBS sequence that may invalidate one of the thermodynamic model's assumptions. The first
constraint calculates the energy required to unfold the 16S rRNA binding site on the mRNA sequence and rejects the ones that require more than 6 kcal/mol to unfold. The second constraint quantifies
the presence of long-range nucleotide interactions. According to a growth model for random RNA sequences ^34, the equilibrium probability of nucleotides i and j forming a base pair in solution is
proportional to p = |i − j|^−1.44. For each base pair in sequence S, we calculate p. If the minimum p is less than 6×10^-3 then the sequence is rejected. Finally, the creation of new AUG or GUG start
codons within the RBS sequence is disallowed.
Strains, media, and plasmid construction
The Luria-Bertani (LB) media (10 g/L tryptone, 5 g/L yeast extract, 10 g/L NaCl) is obtained from Fisher Scientific (Pittsburgh, PA). The supplemented minimal media contains M9 minimal salts (6.8 g/L
Na[2]PO[4], 3 g/L KH[2]PO[4], 0.5 g/L NaCl, 1 g/L NH[4]Cl) from Sigma, 2 mM MgSO4 (Fischer Scientific), 100 μM CaCl[2] (Fischer Scientific), 0.4% glucose (Sigma), 0.05 g/L leucine (Acros Organics,
Belgium), 5 μg/mL chloramphenicol (Acros Organics), and an adjusted pH of 7.4. The expression system is a ColE1 vector with chloramphenicol resistance (derived from pProTet, Clontech). The expression
cassette contains a σ^70 constitutive promoter (BioBrick J23100), the RBS sequence, followed by the mRFP1 fluorescent protein reporter. XbaI and SacI restriction sites are located before the RBS and
after the start codon. An RBS with a desired sequence is inserted into the expression vector using standard cloning techniques. Pairs of complementary oligonucleotides are designed with XbaI and SacI
overhangs and the vector is digested with XbaI and SacI restriction enzymes (NEB, Ipswich, MA). Ligation of the annealed oligonucleotides with cut vector results in a nicked plasmid, which is
transformed into E. coli DH10B cells. Sequencing is used to verify a correct clone.
The AND gate genetic circuit is composed of three plasmids: pBACr-AraT7940, pBR939b, and pAC-SalSer914 with kanamycin, ampicillin, and chloramphenicol resistance markers, respectively. The P[BAD]
promoter maximum expression level was modified by inserting designed synthetic RBSs on plasmid pBACr-AraT7940. Plasmid pBACr-AraT7940 was digested with BamHI and ApaLI enzymes and pairs of
oligonucleotides were designed to contain the desired RBS sequence and corresponding overhangs. Ligation, transformation, selection, and sequencing proceeded as described above.
Growth and fluorescence measurements
The fluorescent protein measurement system is composed of a constitutive promoter, a sequence containing a RBS, and the mRFP1 fluorescent protein reporter (Supplementary Figure 9). An annotated DNA
sequence of the system (Genbank format) is available in the Supplementary Data.
Growth and fluorescence measurements are performed in 96-well high throughput format. A 96-well plate containing 200 μl LB and 50 μg/ml chloramphenicol is inoculated, from single colonies, with up to
30 different DH10B E. coli cultures in an alternating, staggered pattern that excludes the outer wells. Cultures are incubated overnight at 37°C with 250 RPM orbital shaking. A fresh 96-well plate
containing 200μl supplemented minimal media is inoculated by overnight cultures using a 1:100 dilution. This plate is then incubated at 37°C in a Safire^2 plate spectrophotometer (Tecan) with high
orbital shaking. OD[600] measurements are recorded every 3 minutes. Once a culture reaches an OD[600] of 0.15 to 0.20 (4 to 6 hours), a sample of each culture is transferred to a new plate containing
200 μl PBS and 2 mg/ml kanamycin (Acros Organics) for flow cytometry measurements. This media replacement strategy is repeated twice more using fresh, pre-warmed plates containing supplementary
minimal media (the first with a 1:10 dilution requiring 8 to 10 hours of growth and the second with a 1:7 dilution requiring 13 to 15 hours of growth). At least three samples are taken for each
culture. The fluorescence distribution of each sample is measured with a LSRII flow cytometer (BD Biosciences). We use an ellipse in forward and side scatter space to gate at least 30 000 flow
cytometer events. All distributions are unimodal. The autofluorescence distribution of DH10B cells is also measured. The arithmetic mean of each distribution is taken and the mean autofluorescence is
From single colonies, RBS variants of each AND gate genetic circuit are grown overnight in LB and antibiotics (50 μg/ml ampicillin, 25 μg/ml chloramphenicol, and 25 μg/ml kanamycin). A 96-well plate
containing 200 μl LB, antibiotics, and sixteen different inducer concentrations (combinations of 0.0, 1.3×10^-3, 8.3×10^-2, and 1.3 mM arabinose with 0.0, 6.1×10^-4, 3.9×10^-2, and 0.62 mM sodium
salicylate) are inoculated by overnight cultures using a 1:100 dilution. Plates are grown in a Safire^2 plate spectrophotometer (Tecan) with high orbital shaking. OD[600] and gfp fluorescence
measurements are recorded every 10 minutes for 14 hours. Background autofluorescence is subtracted from each fluorescence measurement. This procedure is repeated twice for each variant. For each
variant, the average and standard deviation of the fluorescence per OD[600] for each inducer concentration at the final time point are then calculated.
Data analysis
The ΔG[spacing] is inferred from the fluorescent protein expression data E in the following way. The RNA sequences used to parameterize the model of ΔG[spacing] are predicted to have identical ΔG
[mRNA], ΔG[mRNA:rRNA], ΔG[standby], and ΔG[start] free energies. According to Equation 1, dividing the expression of a sequence with spacing s[1] over another with spacing s[2] and rearranging then
yields the relation: ΔG[spacing](s[1]) − ΔG[spacing](s[2]) = −β^−1log(E[1]/E[2]). The fluorescent protein expression at s = 5 nt was considered maximal and ΔG[spacing](s = 5) is accordingly set to
zero. Using an experimentally measured value of β = 0.45 mol/kcal, the model of ΔG[spacing] for each s is then determined.
Linear regression is used to determine the accuracy of the theory, which hypothesizes a linear relationship between the log average protein fluorescence E and the predicted ΔG[tot] data. The squared
correlation coefficient R^2 and slope −β are calculated according to −β = (NΣ(x[i]y[i]) − Σx[i]Σy[i]) / (NΣ(x[i]^2) − (Σx[i])^2) and R^2 = (NΣ(x[i]y[i]) − Σx[i]Σy[i])^2 / [(NΣ(x[i]^2) − (Σx[i])^2)(NΣ
(y[i]^2) − (Σy[i])^2)], where N is the number of average expression levels recorded, y is log E, and x is ΔG[tot]. The standard deviation of β is calculated by substituting the log E data with the
log(E+δE) and log(E−δE) data (δE : standard deviation of E) and calculating the average difference.
We are grateful to all members of the Voigt lab for technical advice and continued support. This work is supported by the Pew and Packard Foundations, Office of Naval Research, NIH EY016546, NIH
AI067699, NSF BES-0547637, NSF TeraGrid TG-MCB080126T, and a Sandler Family Opportunity Award. C.A.V, H.S, and E.M. are part of the NSF SynBERC ERC (www.synberc.org). E.M is supported by an NSF
Graduate Research Fellowship and an ASEE National Defense Science and Engineering Graduate Fellowship.
Author Contributions: HMS and CAV designed the study and wrote the manuscript. HMS developed the method. HMS and EAM performed the experiments.
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Whatever Happened to Stability Analysis?
Once upon a time, stability of the general equilibrium was considered an important element in the education of students in economics. Today it seldom receives the attention it deserves and this is
regrettable. Stability is one of the most important aspects of neoclassical theory because it addresses the question of just how the mechanism of free competition in the marketplace actually leads to
the formation of equilibrium prices.
This crucial aspect of microeconomics is seldom covered adequately (if at all) in recent textbooks and university programs, whether at the undergraduate or post-graduate levels. Most students spend
years learning how individual agents maximize, or exploring cases of oligopoly, or playing around with game theory, but when it comes to stability, their teachers skirt the main issues.
As a result, a cloud of confusion persists. Students come to believe that somewhere in the sacred scriptures of the discipline there exists a theory that accurately reproduces just how the market
forces of competition guide an economy through a price adjustment process that leads to the formation of equilibrium prices. In fact, if stability analysis received the attention it deserves,
students would be able to see that it is the most important failure of general equilibrium theory.
Stability was typically introduced to students as a property of general equilibrium. The equilibrium was stable if the economic forces activated after it was disturbed returned the economy to the
original (equilibrium) position. Local stability responded more to this definition, while global stability implied that the equilibrium position would be reached regardless of the starting point.
As Léon Walras explained (at the end of Lesson 11), demonstrating how the mechanism of free competition led to equilibrium prices was essential. But this was easier said than done. Hicks in the 1930s
and Samuelson in 1948 were able to make some progress. But Hicks’ contribution in static stability was not associated with any adjustment process. Samuelson showed that stability analysis required an
analysis of the evolution of excess demands over time and introduced the typical price adjustment equations used in modern formulations. However, he did not provide the conditions under which such a
system of equations would converge to a general equilibrium.
In 1958-9 two papers, by Arrow and Hurwicz and Arrow, Block and Hurwicz, showed how under certain conditions an economy could converge to equilibrium. But these were extreme conditions: gross
substitution (GS) for all goods or the validity of the weak axiom of revealed preferences (WARP) at the market level. In the key passage summarizing their results, Arrow and Hurwicz wrote: “none of
the results so far obtained contradicts the proposition that under perfect competition, with the customary assumptions as to convexity, etc., the system is always stable”.
A year later, Scarf published his counterexample showing how unjustified this conjecture was. The extreme conditions of GS and WARP turned out to be indispensable, at least with the market processes
described by Arrow and his colleagues. The ordinary structural conditions of the general equilibrium model were not enough to ensure convergence.
Other aspects of the model leave much to be desired. Perfect competition implies that no firm is able to modify prices, so in models in this tradition (called tâtonnement models) price adjustment is
the responsibility of a fictitious character called the auctioneer, an agent that is incompatible with the notion of a private and decentralized economy. Tâtonnement models exclude transactions out
of equilibrium, so that agents are stupid and believe prices announced by the auctioneer are equilibrium prices (also, initial allocations of individual agents remain unchanged until equilibrium is
In the sixties a different tack was followed. Trading models were developed by Hahn and Negishi, Fisher and others in which agents were allowed to engage in transactions during the price formation
process (i.e. out of equilibrium). The conditions for stability are less stringent (no GS, no WARP), but an “orderly market hypothesis” is introduced and the fictitious auctioneer is still required.
Because the process changes initial holdings, the arrival point of equilibrium is path-dependent. More important, trading out of equilibrium requires the introduction of money, a serious problem in
general equilibrium theory. Typically, when confronted with this revelation, students are perplexed: What? Money was always absent in my microeconomics courses?
The stability debate reached its climax with the papers published by Sonnenschein, Mantel and Debreu in 1973-4. These results show that the usual assumptions of GET allow the dynamics of the classic
tâtonnement process to be essentially arbitrary. To avoid this, additional restrictions must be imposed on excess demand functions.
The failure of stability theory is of relevance to macroeconomics. The notion that in the presence of rigidities markets fail to operate properly is the reciprocal of the belief that stability is a
property of markets. The ‘rigidity’ view is pervasive in macroeconomics, from conventional Keynesianism to believers in the micro-foundations of macroeconomics and the new synthesis with its DSGE
models (where transversality conditions impose stability).
This is what underlies Milton Friedman’s view that the natural rate of unemployment is “the level that would be ground out by the Walrasian system of general equilibrium equations, provided there is
embedded in them the actual structural characteristics of the labor and commodity markets”. Maintaining ignorance about the limitations of stability theory comes in handy when perpetuating the
mythology of market theory.
Truthout is able to confront the forces of greed and regression only because we don’t take corporate funding. Please support us in this fight: make a tax-deductible donation today.
As Mundell once remarked, stability analysis is the most successful failure of general economic theory. It is also the best example of how an academic community pushes the most serious problems of
mainstream theory under the rug and gets away with it. Students should learn to look under the rug. The ability to improve our understanding of economic processes depends on efforts to uncover the
failures of mainstream theoretical constructs.
Alejandro Nadal’s recent book, Rethinking Macroeconomics for Sustainability, is available from Zed Books.
Arrow, K. and H. D. Block (1959)
Arrow, K., H. D. Block and L. Hurwicz (1959)
Debreu, G. (1974), “Excess demand functions”, Journal of Mathematical Economics. 1. (15-21)http://ideas.repec.org/a/eee/mateco/v1y1974i1p15-21.html
Fisher, F. (1983), Disequilibrium Foundations of Equilibrium Economics. Cambridge University Press.
Friedman, Milton (1968)
Hahn and Negishi (1962)
Hicks, John (1939), Value and Capital. Oxford: Clarendon Press.
Mantel, R. (1974), “On the characterization of aggregate excess demand,” Journal of Economic Theory. 7. (348-353)http://econpapers.repec.org/article/eeejetheo/
Samuelson, P. (1947), Foundations of Economic Analysis. Harvard University Press.
Scarf, H. (1960), “Some examples of global instability of competitive equilibria”. International Economic Review, 1 [157 – 172]
Sonnenschein, H. (1973), “Do Walras’ identity and continuity characterize the class of community excess demand functions?” Journal of Economic Theory. 6. (345-354),http://ideas.repec.org/a/eee/jetheo
Walras, León (1969), Elements of Pure Economics. New York: Augustus Kelley. | {"url":"http://truth-out.org/news/item/1720-whatever-happened-to-stability-analysis","timestamp":"2014-04-16T07:30:06Z","content_type":null,"content_length":"58889","record_id":"<urn:uuid:25344e9c-43b2-4ea4-a946-2c623229e351>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00070-ip-10-147-4-33.ec2.internal.warc.gz"} |
Configuration of the Cycle Counting Engine
Siebel Field Service Guide > Cycle Counting and Replenishment > Cycle Counting >
Configuration of the Cycle Counting Engine
The Cycle Counting Engine generates cycle counting headers and cycle counting part lists by using the configuration information from the Cycle Counts view. For more information, see Cycle Counts View
Cycle counting uses these configuration parameters:
□ ABC or XYZ count basis. The method used for cycle counting of an inventory location: ABC = financial value basis, XYZ = turnover basis.
□ A, B, or C classification. The classification of products based on the financial value of an item, where A > B > C.
□ X, Y, or Z classification. The classification of products based on the turnover of an item, where X >Y >Z.
A product receives any combination of class A, B, or C and X, Y, or Z; for example, AY. However, cycle counting is based on either ABC or XYZ, not both.
□ Frequency. The interval in days between physical counts of inventory at a location; for example, if counting frequency is 7 days, inventory is counted every seven days.
□ Period. The time period in days allotted to counting all A or X, B or Y, or C or Z class items at this location. For example, if the counting period for class A or X products is 90 days, the
Cycle Counting Engine makes sure that all the products in this class are included in the part list at least once every 90 days.
□ Start Date and End Date. The time span in calendar days during which the Cycle Counting Engine will create cycle count records.
Here is an example of a cycle counting scenario:
An inventory location is configured to count on a financial value basis (ABC). There are approximately 100 A-class products in inventory. Every five days (Frequency = 5 days) someone spends as much
time as needed to count one-quarter (5/20) of the A-class items, with the goal of counting all of the A-class items within 20 days (Counting Period A/X = 20 days). Counting periods do not have to be
even multiples of frequency.
Configuration Information for the Cycle Counting Engine
Configuration of cycle counting takes place at several levels, on different screens (see Table 106).
Table 106. Configuration of Cycle Counting
Configuration Parameter Applies to View
Cycle count basis: Inventory locations See Inventory Locations View.
ABC or XYZ
Inventory location types See Inventory Location Types View.
Cycle count class: Products associated with a specific inventory types See Inventory Options Subview.
A, B, or C and X, Y, or Z
Products associated with specific inventory locations See Product Inventory View.
Frequency Inventory locations See Inventory Locations View.
Inventory location types See Inventory Location Types View.
Count period for A/X, B/Y, Inventory locations See the Inventory Locations View.
and C/Z
Inventory location types See the Inventory Location Types View.
Start date and end date Inventory locations See the Cycle Counts Views (on the Cycle Counts screen).
NOTE: If a configuration parameter can be set at different aspects of inventory, the parameter for the more specific aspect takes precedence. For example, the count basis (ABC or XYZ) at an inventory
location takes precedence over the basis specified for an inventory type. The cycle count class (A, B, or C; X, Y, or Z) for an inventory type takes precedence over that for a product. | {"url":"http://docs.oracle.com/cd/B40099_02/books/FieldServ/FieldServCycleCountReplenish3.html","timestamp":"2014-04-17T23:19:48Z","content_type":null,"content_length":"11274","record_id":"<urn:uuid:15189ec1-93d2-48e4-baa3-769f2e8b3b0e>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00481-ip-10-147-4-33.ec2.internal.warc.gz"} |
SPSSX-L archives -- March 2012 (#202)LISTSERV at the University of Georgia
The data are essentially scores obtained by using the semantic differential instrument (scores from 1-7) and the sections represent number of students (cases). So each student should have a score
corresponding to 20 different items (variables) respectively. Raw data are transferred to SPSS and all I'm trying to do is to explore the number of factors emerging from the 20 variables. So, from
the Scree plot and eigenvalue condition > 1, there are 3 factors extracted. However, it still does not explain why the determinant of the correlation matrix is 0 and I'm not sure if this data becomes
inadmissible on those grounds, despite having satisfied the minimum criteria for factor analysis.
Another strange aspect is that the residual matrix does not have values close to 0 at all. In fact, some of the values are > 0.1 and a lot of the values are negative. So, has the factor extraction
been inefficient or should some of the variables be removed?
The reliability statistics summary: Summary Item Statistics Mean Min. Max. Range Max / Min Variance N of Items Item Means 3.630 2.390 5.949 3.559 2.489 .813 20 Item Variances 1.743 1.240 2.338 1.098
1.886 .091 20 Inter-Item .015 -.567 .615 1.182 -1.085 .115 20 correlations
The highest square multiple correlation value is .624. None of them has a SMC of 1.00 with other items.
Thanks for your help!
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SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD | {"url":"http://listserv.uga.edu/cgi-bin/wa?A2=ind1203&L=spssx-l&D=0&O=A&P=23328&F=P","timestamp":"2014-04-19T01:48:42Z","content_type":null,"content_length":"11665","record_id":"<urn:uuid:c3bede37-42bf-42a4-8826-0e842000c9c2>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00276-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rotation and Systems of Quadratic Equations
June 4th 2007, 06:16 PM
Rotation and Systems of Quadratic Equations
I'm supposed to rotate the axes in order to eliminate the xy-term. I'm told this problem is very simple, but I don't understand this section at all. Any help is greatly appreciated.
June 4th 2007, 06:37 PM
Given a general conic:
If $Bot = 0$ (that means it has a cross term).
Then the required angle is:
$\cot 2\theta = \frac{A - C}{B}$
You problem is,
$x^2 - 10xy+y^2 +1 =0$
The required angle satisfies,
$\cot 2\theta = \frac{ 1 -1 }{10} = 0$
$2\theta = \frac{\pi}{2} \Rightarrow x = \frac{\pi}{4}$
When we rotate by $\theta$ we get the new coordinates:
$x'= x\cos \theta - y\sin \theta$
$y'= x\sin \theta + y\cos \theta$
In this case $\theta = \frac{\pi}{4}$ thus,
$x' = x\cos \frac{\pi}{4} - y\sin \frac{\pi}{4} = x\frac{\sqrt{2}}{2} - y\frac{\sqrt{2}}{2}$
$y'= x\sin \frac{\pi}{4} + y\cos \frac{\pi}{4} = x\frac{\sqrt{2}}{2} + y \frac{\sqrt{2}}{2}$
Substitute the new x and y coordinates:
$\left( x\frac{\sqrt{2}}{2} - y\frac{\sqrt{2}}{2} \right)^2 - 10 \left(x\frac{\sqrt{2}}{2} - y\frac{\sqrt{2}}{2}\right)\left( x\frac{\sqrt{2}}{2} + y \frac{\sqrt{2}}{2} \right) + \left( x\frac{\
sqrt{2}}{2} + y \frac{\sqrt{2}}{2} \right)^2+1=0$
$\left( \frac{1}{2}x^2 - xy + \frac{1}{2}y^2 \right) - 10 \left(\frac{1}{2}x^2 - \frac{1}{2}y^2 \right) + \left( \frac{1}{2}x^2+xy+\frac{1}{2}y^2 \right)+1=0$
June 4th 2007, 07:14 PM
Thanks for helping me with the last problem. I know how to solve the rest of my homework except for this one:
Am I supposed to change the equation first?
June 4th 2007, 07:16 PM
June 4th 2007, 07:26 PM
Ah, okay. *obviously braindead*
Thanks for all your help! | {"url":"http://mathhelpforum.com/trigonometry/15634-rotation-systems-quadratic-equations-print.html","timestamp":"2014-04-21T06:53:30Z","content_type":null,"content_length":"10691","record_id":"<urn:uuid:7421afe1-0819-493d-8989-f3657e3287db>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00562-ip-10-147-4-33.ec2.internal.warc.gz"} |
We make predictions for the kaon interferometry measurements in Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC). A first order phase transition from a thermalized
Quark-Gluon-Plasma (QGP) to a gas of hadrons is assumed for the transport calculations. The fraction of kaons that are directly emitted from the phase boundary is considerably enhanced at large
transverse momenta K T ~ 1 GeV/c. In this kinematic region, the sensitivity of the R out/R side ratio to the QGP-properties is enlarged. Here, the results of the 1-dimensional correlation
analysis are presented. The extracted interferometry radii, depending on K-Theta, are not unusually large and are strongly affected by momentum resolution effects.
We calculate the kaon HBT radius parameters for high energy heavy ion collisions, assuming a first order phase transition from a thermalized Quark-Gluon-Plasma to a gas of hadrons. At high
transverse momenta K_T ~ 1 GeV/c direct emission from the phase boundary becomes important, the emission duration signal, i.e., the R_out/R_side ratio, and its sensitivity to T_c (and thus to the
latent heat of the phase transition) are enlarged. Moreover, the QGP+hadronic rescattering transport model calculations do not yield unusual large radii (R_i<9fm). Finite momentum resolution
effects have a strong impact on the extracted HBT parameters (R_i and lambda) as well as on the ratio R_out/R_side.
We present calculations of two-pion and two-kaon correlation functions in relativistic heavy ion collisions from a relativistic transport model that includes explicitly a first-order phase
transition from a thermalized quark-gluon plasma to a hadron gas. We compare the obtained correlation radii with recent data from RHIC. The predicted R_side radii agree with data while the R_out
and R_long radii are overestimated. We also address the impact of in-medium modifications, for example, a broadening of the rho-meson, on the correlation radii. In particular, the longitudinal
correlation radius R_long is reduced, improving the comparison to data. | {"url":"http://publikationen.ub.uni-frankfurt.de/solrsearch/index/search/searchtype/authorsearch/author/%22Sergey+Y.+Panitkin%22/start/0/rows/10/yearfq/2002","timestamp":"2014-04-17T21:31:51Z","content_type":null,"content_length":"23981","record_id":"<urn:uuid:f03d71f7-f7c5-462d-a717-a5a50ed2eac9>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00597-ip-10-147-4-33.ec2.internal.warc.gz"} |
Constructing a unitary matrix
up vote 6 down vote favorite
Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: $Q^{\dagger}
Q = 1$. This problem is equivalent to solving a system of quadratic equations over real numbers. As far as I understand, there is no general and efficient way to solve such systems of equations, and
black-box algorithms, such as Gröbner basis, struggle with systems of around 10 variables.
Does the particular structure of this system of equations make it easier than a generic one, and can this be utilized in order to speed up the calculation.
This problem arises in many contexts. For example, it is related to search of symmetry of a quantum Hamiltonian. Hamiltonian $H$, a finite Hermitian matrix has a symmetry if it commutes with some
unitary matrix $U$ other than identity. It is very easy to construct the linear space to which $U$ should belong, it is given by the kernel of the system of linear equations $H A - A H = 0$. However
the next step requires verifying whether this linear space contains unitary matrices other than identity.
Secondary question:
Given that it seems unlikely that there is an easy answer to the main question, I would also like to ask whether there are known classes of systems of quadratic equations that are quickly solvable
computer-algebra ag.algebraic-geometry matrices linear-algebra
Finding $A$ such that $HA-AH=0$ is a much easier problem - is there better motivation? – Colin McQuillan Oct 18 '11 at 14:29
The motivation is to find a symmetry of Hamiltonian. Finding the linear space is the first step to solve the problem, the missing step is to find unitary operators belonging to this space. By
itself finding kernel of $HA-AH$ is not as valuable. – Anton Akhmerov Oct 18 '11 at 14:37
2 You can solve $HA - AH = 0$ in the subspace of skew-hermitian matrices, and this determines the Lie algebra of the group of unitary symmetries of $H$. Exponentiating the Lie algebra gives you the
connected component of the group you are after. Of course, $H$ may only have discrete unitary symmetries, in which case this does not help and I suppose you will have to solve the quadratic
problem. – José Figueroa-O'Farrill Oct 18 '11 at 14:43
1 Is it not true that the unitary matrices that commute with a hermitian matrix H are just the direct sum of unitary matrices on each eigenspace? – Colin McQuillan Oct 18 '11 at 14:46
1 A silly question: is a solution always guaranteed? For example, suppose you have just one matrix $A_1$. You cannot just scale it to make it unitary....so why should I expect your system to have a
solution for a given set of matrices? – Suvrit Oct 18 '11 at 18:34
show 7 more comments
1 Answer
active oldest votes
Every Hermitian matrix (in fact, every normal matrix) commutes with infinitely many unitary matrices:
Lemma: The square matrices A and B commute if they can be simultaneously diagonalized.
Proof: Let A=Q D inv(Q) and B=Q E inv(Q), where D and E are diagonal. Then AB = Q D E inv(Q) = Q E D inv(Q) = B A, since diagonal matrices commute.
up vote 1 down vote Corollary: If H is diagonalized by the unitary matrix Q, then U = Q D Q' is unitary for any diagonal matrix D whose entries lie on the unit circle, and U commutes with H.
Thus, once you have the eigenvectors of your (discretized) Hamiltonian, you can easily form an infinite number of unitary matrices that commute with it.
Is there a constraint on the symmetrices that I'm missing?
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problem with sinus :)
January 12th 2009, 02:03 PM #1
May 2008
problem with sinus :)
Hey guys.
cos(n*x) when n is natural and x=pi equals to (-1)^n as I wrote in the pic.
What I'm looking for is the same thing in sinus, lets say again that n is natural, is there a x that can make sinus to behave something like cos? and by that I mean an expression that can be
written in on line like (-1)^n (x can be anything).
I hope that was understood.
Thanks in advance.
Hey guys.
cos(n*x) when n is natural and x=pi equals to (-1)^n as I wrote in the pic.
What I'm looking for is the same thing in sinus, lets say again that n is natural, is there a x that can make sinus to behave something like cos? and by that I mean an expression that can be
written in on line like (-1)^n (x can be anything).
I hope that was understood.
Thanks in advance.
there's no such $x$ because if $\sin x = -1,$ then $\cos x =0$ and hence $\sin(2x)=2\sin x \cos x =0 eq 1 =(-1)^2.$
well, lets say x=pi/2, is there a way to write the result in one line? like cos(n*pi) = (-1)^n.
\begin{aligned}\cos(n\pi)&=\sin\left(\frac{\pi}{2}-n\pi\right)\\<br /> &=\sin\left(\pi\left(\frac{1}{2}-n\right)\right)\\<br /> &=\sin\left(\pi\cdot\frac{1-2n}{2}\right)\\<br /> &=-\sin\left(\pi\
cdot\frac{2n-1}{2}\right)\\<br /> &=(-1)^n\end{aligned}
Actually more aestheically pleasing we can use $\cos(n\pi)=\sin\left(\frac{\pi}{2}+n\pi\right)$ to arrive alternatively at $(-1)^n=\sin\left(\pi\cdot\frac{2n+1}{2}\right)$
\begin{aligned}\cos(n\pi)&=\sin\left(\frac{\pi}{2}-n\pi\right)\\<br /> &=\sin\left(\pi\left(\frac{1}{2}-n\right)\right)\\<br /> &=\sin\left(\pi\cdot\frac{1-2n}{2}\right)\\<br /> &=-\sin\left(\pi\
cdot\frac{2n-1}{2}\right)\\<br /> &=(-1)^n\end{aligned}
Actually more aestheically pleasing we can use $\cos(n\pi)=\sin\left(\frac{\pi}{2}+n\pi\right)$ to arrive alternatively at $(-1)^n=\sin\left(\pi\cdot\frac{2n+1}{2}\right)$
Hey Mathstud28.
This is not what I meant (thanks a lot due
I gave the cos just as an example.
I have the function f(x) = sin(nx), and I need to find any x that will give me the ability to write the result in one line.
Lets take again x = pi/2 f(x) = sin(n*p/2) = {0 when n=0, 1 when n=1, 0 when n=2, -1 when n=3 and so on...} is there a way to write this expression in one line that will be depend in n? (just
like cos(nx) = (-1)^n)
Again, thanks a lot.
Hey Mathstud28.
This is not what I meant (thanks a lot due
I gave the cos just as an example.
I have the function f(x) = sin(nx), and I need to find any x that will give me the ability to write the result in one line.
Lets take again x = pi/2 f(x) = sin(n*p/2) = {0 when n=0, 1 when n=1, 0 when n=2, -1 when n=3 and so on...} is there a way to write this expression in one line that will be depend in n? (just
like cos(nx) = (-1)^n)
Again, thanks a lot.
No, but in two lines it's possible :
$\sin \left(\frac{n\pi}{2}\right)=\left\{\begin{array}{l l} 0 \text{ if n is even} \\<br /> (-1)^{(n-1)/2} \text{ if n is odd} \end{array} \right.$
Hows that?
January 12th 2009, 02:18 PM #2
Jan 2009
January 12th 2009, 02:47 PM #3
MHF Contributor
May 2008
January 12th 2009, 11:21 PM #4
May 2008
January 12th 2009, 11:27 PM #5
January 12th 2009, 11:44 PM #6
May 2008
January 12th 2009, 11:51 PM #7
January 12th 2009, 11:56 PM #8
January 13th 2009, 12:02 AM #9
May 2008
January 13th 2009, 12:05 AM #10 | {"url":"http://mathhelpforum.com/calculus/67912-problem-sinus.html","timestamp":"2014-04-16T19:47:11Z","content_type":null,"content_length":"63683","record_id":"<urn:uuid:ac42f951-83b0-4501-b505-4c1c6ddd6dd2>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00413-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Dr. Doug Andrews, Wittenberg University
Nov 8, 2010, Room 320 Science, 4:10 pm
Statistical consulting is the art and science of using data to help other people solve their problems in the real world. I'll give an overview of the stat consulting process, the typical
environments in which stat consultants work, they typical qualificaiton and salaries, and some of the ethics involved. I'll also illustrate what stat consultants do, using examples from industry,
academia, health care, law enforcement, and non-profits - all from my own work, as well as work done with Witt students.
The Mathematics of Fairness
Dr. Bill Higgins, Wittenberg University
Oct 25, 2010, Room 320 Science, 4:10 pm
The settlement of an estate among heirs, division of property following a divorce, subdivision of land among competing claimants and dividing a cake or candy among children are all problems of fair
division. In this talk, we'll discuss how to define fairness and present some "fair division schemes" developed by Polish mathematician Hugo Steinhaus and others to tackle such problems.
The Cyber Corp Graduate Fellowship
Dr. Rusty Baldwin, Air Force Institute of Technology
Oct 11, 2010, Room 320 Science, 4:00 pm
Dr. Baldwin will be speaking about the Cyber Corp Graduate Fellowships at the Air Force Institute of Technology.
Multivariable Parametric Cost Analysis for Space-Based Telescopes
Courtnay Dollinger, Wittenberg University
Sept 27, 2010, Room 320 Science, 4:00 pm
This project analyzes data from over 1000 sources and focuses on data from over thirty different space-based telescopes in order to determine a cost estimating relationship. Due to the increased
availability of cost data from recent space-telescope construction, we have been able to begin testing for a comprehensive cost model of space telescopes. By separating the variables that effect
cost, we advance the goal to better understand the cost drivers of space telescopes. Advanced mathematical techniques have the ability to improve the accuracy of cost models and the potential to help
society make informed decisions about proposed scientific projects.
Parellization of the Protein Database Search Program MassMatrix Through the Use of OpenMP ?
Zach Hedges, Wittenberg University
Sept 27, 2010, Room 320 Science, 4:00 pm
In today’s medical research field, it is important to be able to gather accurate results from diagnostic procedures in as little time as possible. Luckily, with the advent of multi-core computers
and API’s such as OpenMP, computationally-expensive analyses can be run in parallel, drastically reducing runtime. The goal of this project was to parallelize and accelerate the peptide database
search algorithm, MassMatrix, utilizing OpenMP. In order to achieve this, the code was profiled and several time-consuming calculations were rewritten for parallelization. This technique proved,
however, to increase run-time due to increased overhead. Further strategies for parallelization are currently being researched
Do I know you? Cryptography and Authentication Protocols
Deanna Fink, Wittenberg University
Sept 27, 2010, Room 320 Science, 4:00 pm
In the age of technology, people rely more and more on computers and the internet to accomplish tasks. The military is no exception to this. But as the use of technology increases, so must the
ability to protect information. Cryptography is one of the most common ways of doing this and it is being improved every day to make defenses stronger. Problems surrounding cryptography include
balancing effectiveness with costs (not necessarily in terms of money). There is no way yet to create perfect cryptography techniques, but research is being done to try to keep up with or hopefully
stay ahead of those trying to invade the privacy of internet users.
P vs NP: Solved?
Dr. Kyle Burke, Wittenberg University
Sept 13, 2010, Room 320 Science, 4:00 pm
The P -versus - NP problem, one of the seven famed "Millenium Problems" , is perhaps the greatest unsolved problem in Computer Science today. Or is it? At the end of this summer, a potential
solution made a lot of news, even in mainstream media. This talk will shed some light on what P vs NP means, as well as comments on the recent action.
A Mathematical Potpourri, Conundrumbs, Puzzles, Tricks and Other Trivia!
Dr. Brian Shelburne, Wittenberg University
August 30, 2010, Room 320 Science, 4:00 pm
Spring 2009
CyberCorp Graduate Fellowships at the Air Force Institute of Technology
Dr. Rusty Baldwin, Associate Director of the Center for Cyberspace Research
February 4, 2009
Software Engineering for a Web-Based Educational Image Repository
Aaron Holloway ('09) and Jonathan Wantz ('09), Senior Seminar Project
January 22, 2009
Fall 2008
Bringing Characters to Life: An Independent Study in 3D Modeling and Animation
Laura Barnard ('08), Independent Study
December 15, 2008
Virtual Education: Identifying and Creating Content
Molly Dannaher ('10), Summer Research
December 4, 2008
VIPER: Virtual Imaging for Pathology, Education, and Research
Molly Tingley ('10), Summer Research
December 4, 2008
Computational Methods to Determine Solvent Effects on the Reaction of Phenol and Bicarbonate
Janelle Mahowald ('10), Summer Research
November 20, 2008
Analysis of Chemotherapy Drugs Genetic Effects on Pancreatic cancer vs. the effect of the progression of the disease
Rebecca Atkins ('10), Summer Research
November 13, 2008
Hydrogen Abstraction-Induced Ring Opening In Thiazolo[5,4-d]thiazole, Benzthiazole, and Thiazole
Adeline Brym ('10), Summer Research
November 13, 2008
Improvement of the Trap Assisted Tunneling Model Solution for Leakage Currents in Heterostructure Field Effect Transistors
Hannah Scherger ('09), Summer Research
October 29, 2008
Describing a Combinatorics Problem with a System of Polynomial Equations
Marshall Zarecky ('09), Summer Research
October 29, 2008
Programmers and Truthiness
Dan Saks, President of Saks & Associates
October 23,2008
3-D Modeling Using Autodesk Maya 2008
Laura Linden Barnard ('08), Summer Research
October 15, 2008
Reflections on a Semester in Budapest
Alyssa Armstrong ('09), Cultural Exchange
October 15, 2008
Heat Transfer in Polymers
Melissa Cederquvist ('10), Summer Research
September 30, 2008
Reaction Functions in the US and Chinese Contexts
Nam Vu ('10), Summer Research
September 30, 2008
Applications of Computing Across Science and Industry
Dave Strenski, Applications Engineer for Cray Inc.
September 22, 2008
What I Did On My Summer Vacation: Statistics Graduate Student Research
Dr. Elizabeth Stasny, Graduate Studies Chair in the Department of Statistics at Ohio State University
September 17, 2008
Spring 2008
Game Development for the PC and Xbox360 using XNA
Nick Kovach ('08), Independent Study
April 24, 2008
Cyberspace Research and Internships
Dr. Rusty Baldwin, Associate Director for the Air Force Institute of Technology
March 13, 2008
Hydrogen Abstraction-Induced Ring Opening in Thiazoles
Tim Verrilli ('08), Summer Research
February 28, 2008
The Role of Molecular Dynamics Simulations in the Search for Advanced Energy Materials
Joe Fritchman ('08), Summer Research
February 28, 2008
Semi-Empirical Molecular Dynamics Study of Polyene Isomerization Through Protonation
Steven Koppenhafer ('09), Summer Research
February 19, 2008
Investigating Micellization Using Monte Carlo Simulations
Thao Nguyen ('08), Summer Research
February 19, 2008
Ice Cubes to Stock Options: Free Boundary Problems Across the Disciplines
John Davenport, Adjunct Instructor of Mathematics
January 23, 2008
Fall 2007
Mathematica for Computational Science
Jim Noyes, Emeritus Professor of Computer Science
december 18, 2007
Computational Modeling of Advanced Energy Materials
Adam Jara ('08), Summer Research
December 6, 2007
Predicting Error in HTS Data Using LeadScope Software
David Mowrey ('08), Summer Research
December 6, 2007
Scientific Visualization
Laura Linden ('08), Summer Research
November 8, 2007
Preliminary Sequencing and Analysis of a Genomic DNA Amplification Product from a Population of Wall Lizards (Podarcis muralis) Cincinati, OH
Alex Silvis ('08), Summer Research
October 29, 2007
Virtual Microscopy: A Tool to Cancer Research
Fadi Michael ('08), Summer Research
October 29, 2007
Between a Rook and a Hard Place: A Study of Column Strict Rook Placements of the q-File Polynomial
Alyssa Armstrong ('09), Summer Research
September 27, 2007
Spring 2007
Virtual Reidemeister Moves
Emily List ('07), Senior Honors Thesis
April 18, 2007
More than Meets the Eye: An Invitation to Moduli Spaces
Adam Parker, Assistant Professor of Mathematics
March 29, 2007
Artificial Intelligence: What and Why
Steven Bogaert, Candidate for Computer Science Faculty Position
February 27, 2007
Fall 2006
Characterization of C60S Isomers: A Theoretical Study
Adam Jara ('08), Summer Research
November 15, 2006
Ab initio Study of Polysulfides Sn, Their Anions Sn-, and Their Dianions Sn-2
Joe Fritchman ('08), Summer Research
November 15, 2006
Infinite Trees
Jennifer Brown, Kenyon College
November 3, 2006
Representing Fractals on Parallel Systems
Indraroop Roy Mohanti ('08), Summer Research
October 12, 2006
A Computational Study: The Formation of a Useful Thiazole Monomer
Tim Verrilli ('08), Summer Research
October 12, 2006
Stereo Visualization and its Application for Fun and Viewing Scientific Data
Dr. Mark Turner, AVETeC/University of Cincinnati
September 28, 2006
On Polynomial Knots
Emily List ('07), Summer Research
September 13, 2006 | {"url":"http://www5.wittenberg.edu/academics/computerscience/colloquia.html","timestamp":"2014-04-21T12:32:32Z","content_type":null,"content_length":"105169","record_id":"<urn:uuid:7a373a9e-797d-4eb9-a8ba-f91033c66f86>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00624-ip-10-147-4-33.ec2.internal.warc.gz"} |
Multiprocessor (core) software (think Stata/MP) and percent parallelizationMultiprocessor (core) software (think Stata/MP) and percent parallelization
Multiprocessor (core) software (think Stata/MP) and percent parallelization
When most people first think about software designed to run on multiple cores such as Stata/MP, they think to themselves, two cores, twice as fast; four cores, four times as fast. They appreciate
that reality will somehow intrude so that two cores won’t really be twice as fast as one, but they imagine the intrusion is something like friction and nothing that an intelligently placed drop of
oil can’t improve.
In fact, something inherent intrudes. In any process to accomplish something — even physical processes — some parts may be able to to be performed in parallel, but there are invariably parts that
just have to be performed one after the other. Anyone who cooks knows that you sometimes add some ingredients, cook a bit, and then add others, and cook some more. So it is, too, with calculating x
[t] = f(x[t-1]) for t=1 to 100 and t[0]=1. Depending on the form of f(), sometimes there’s no alternative to calculating x[1] = f(x[0]), then calculating x[2] = f(x[1]), and so on.
In any calculation, some proportion p of the calculation can be parallelized and the remainder, 1-p, cannot. Consider a calculation that takes T hours if it were performed sequentially on a single
core. If we had an infinite number of cores and the best possible implementation of the code in parallelized form, the execution time would fall to (1-p)T hours. The part that could be parallelized,
which ordinarily would run in pT hours, would run in literally no time at all once split across an infinite number of cores, and that would still leave (1-p)T hours to go. This is known as Amdahl’s
We can generalize this formula to computers with a finite number of cores, say n of them. The parallelizable part of the calculation, the part that would ordinarily run in pT hours, will run in pT/n.
The unparallelizable part will still take (1-p)T hours, so we have
T[n] = pT/n + (1-p)T
As n goes to infinity, T[n] goes to (1-pT).
Stata/MP is pretty impressively parallelized. We achieve p of 0.8 or 0.9 in many cases. We do not claim to have hit the limits of what is possible, but in most cases, we believe we are very close to
those limits. Most estimation commands have p above 0.9, and linear regression is actually above 0.99! This is explained in more detail along with percentage parallelization details for all Stata
commands in the Stata/MP Performance Report.
Let’s figure out the value of having more cores. Consider a calculation that would ordinarily require T = 1 hour. With p=0.8 and 2 cores, run times would fall to 0.6 hours; With p=0.9, 0.55 hours.
That is very close to what would be achieved even with p=1, which is not possible. For 4 cores, run times would fall to 0.4 (p=0.8) and 0.325 (p=0.9). That’s good, but no where near the hoped for
0.25 that we would observe if p were 1.
In fact, to get to 0.25, we need about 16 cores. With 16 cores, run times fall to 0.25 (p=0.8) and 0.15625 (p=0.9). Going to 32 cores improves run times just a little, to 0.225 (p=0.8) and 0.128125 (
p=0.9). Going to 64 cores, we would get 0.2125 (p=0.8) and 0.11384615 (p=0.9). There’s little gain at all because all the cores in the world combined, and more, cannot reduce run times to below 0.2 (
p=0.8) and 0.1 (p=0.9).
Stata/MP supports up to 64 cores. We could make a version that supports 128 cores, but it would be a lot of work even though we would not have to write even one line of code. The work would be in
running the experiments to set the tuning parameters.
It turns out there are yet other ways in which reality intrudes. In addition to some calculations such as x[t] = f(x[t-1]) not being parallelizable at all, it’s an oversimplification to say any
calculation is parallelizable because there are issues of granularity and of diseconomies of scale, two related, but different, problems.
Let’s start with granularity. Consider making the calculation x[t] = f(z[t]) for t = 1 to 100, and let’s do that by splitting on the subscript t. If we have n=2 cores, we’ll assign the calculation
for t = 1 to 50 to one core, and for t=51 to 100 to another. If we have four cores, we’ll split t into four parts. Granularity concerns what happens when we move from n=100 to n=101 cores. This
problem can be split into only 100 parallelizable parts and the minimum run time is therefore max(T/n, T/100) and not T/n, as we previously assumed.
All problems suffer from granularity. Diseconomies of scale is a related issue, and it strikes sooner than granularity. Many, but not all problems suffer from diseconomies of scale. Rather than
calculating f(z[t]) for t = 1 to 100, let’s consider calculating the sum of f(z[t]) for t = 1 to 100. We’ll make this calculation in parallel in the same way as we made the previous calculation, by
splitting on t. This time, however, each subprocess will report back to us the sum over the subrange. To obtain the overall sum, we will have to add sub-sums. So if we have n=2 cores, core 1 will
calculate the sum over t = 1 to 50, core 2 will calculate the sum for t = 51 to 100, and then, the calculation having come back together, the master core will have to calculate the sum of two
numbers. Adding two numbers can be done in a blink of an eye.
But what if we split the problem across 100 cores? We would get back 100 numbers which we would then have to sum. Moreover, what if the calculation of f(z[t]) is trivial? In that case, splitting the
calculation among all 100 cores might result in run times that are nearly equal to what we would observe performing the calculation on just one core, even though splitting the calculation between two
cores would nearly halve the execution time, and splitting among four would nearly quarter it!
So what’s the maximum number of cores over which we should split this problem? It depends on the relative execution times of f(z[t]) and the the combination operator to be performed on those results
(addition in this case).
It is the diseconomies of scale problem that bit us in the early versions of Stata/MP, at least in beta testing. We did not adequately deal with the problem of splitting calculations among fewer
cores than were available. Fixing that problem was a lot of work and, for your information, we are still working on it as hardware becomes available with more and more cores. The right way to address
the issue is to have calculation-by-calculation tuning parameters, which we do. But it takes a lot of experimental work to determine the values of those tuning parameters, and the greater the number
of cores, the more accurately the values need to be measured. We have the tuning parameters determined accurately enough for up to 64 cores, although there are one or two which we suspect we could
improve even more. We would need to do a lot of experimentation, however, to ensure we have values adequate for 128 cores. The irony is that we would be doing that to make sure we don’t use them all
except when problems are large enough!
In any case, I have seen articles predicting and in some cases, announcing, computers with hundreds of cores. For applications with p approaching 1, those are exciting announcements. In the world of
statistical software, however, these announcements are exciting only for those running with immense datasets.
Nice use of numerical examples in this post. When understanding the behavior of such a function I also like to look at graphs next to such numerical examples. In this case I used the graph
below:nnlocal T = 1.0nlocal p = 0.8nntwoway function Tn= `p’*`T’/x + (1-`p’)*`T’, ///n range(1 64) ylab(0(.2)1) ///n yline(`=(1-`p’)*`T”) ///n note(“T = `T’, p = `p’”) ///n xtitle(“number of cores”)
/// n ytitle(“time”) n
Does the 64-core limit (and the whole diseconomies of scale problem itself) mean that you will not utilize GPUs anytime soon? (I know you had problems with OpenCL on single-precision CPUs — but
recent GPUs get you double precision…) I hoped that you had been mum about this only to embargo the big news about Stata 12′s killer feature.
Use of multiple cores and use of General Purpose GPUs (GPGPUs) are very different things, which is to say, the 64-core limit is unrelated to use of GPGPUs. I emphasize again the current 64-core limit
is due only to lack of testing to set tuning parameters. Making Stata work with more cores is no work at all; we just change a limit. Making Stata work well with with more cores is substantial work
even though, as I said, no new code needs to be written.nnUse of GPGPUs can result in performance enhancements, and sometimes whopping performance enhancements, but they usually do not provide the
performance enhancements provided by multiple cores. I want to be careful about generalizing. GPGPUs serve the specialized purpose of making vectorized calculations and some matrix calculations, and
it is a restriction that the values on which the calculations are being made really are stored as vectors and matrices, which is to say, the values must be adjacent to one another. Multiple cores, on
the other hand, can be used with any data structure and in any situation. Thus, whether multiple cores yield a benefit depends solely on the conceptual nature of of the problem. Whether GPGPUs yield
a benefit depends on the conceptual nature of the problem, that it be a problem within the GPGPU’s repertoire, and that the data already be arranged in a way the GPGPU requires. Thus, even though a
problem might lend itself to parallelization, it might not lend itself to GPGPU implementation.
Are those different MP versions downward compatible? That is, if I use MP12 on a quad-core machine, will it be just as fast as using MP4 on the same quad-core machine?
Yes — Stata/MP automatically downscales itself to fit the number of cores on a given computer. There is no difference in performance between Stata/MP (4-core) on a 4-core computer and Stata/MP
(12-core) on that same 4-core computer. Stata/MP (12-core) will of course perform faster given more than 4 cores. | {"url":"http://blog.stata.com/2011/04/07/multiprocessor-core-software-think-statamp-and-percent-parallelization/","timestamp":"2014-04-23T11:46:46Z","content_type":null,"content_length":"44801","record_id":"<urn:uuid:2759b99f-5284-4599-a27a-2094ae28141c>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00000-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Chaotic Inflation Models & Equations
When using the Friedmann equation (flat space, no cosmological constant): H = sqrt (8 pi G / 3 ) * rho, if we use rho in mass/volume, H is in (time)^-1 like it should. Now for some inflation models,
we use: H = sqrt (8 pi G / 3 ) * V(Phi). It seems that V(Phi) should also be able to be converted to mass/volume.
In chaotic inflation models, the function V(Phi) = ˝ m^2 Phi^2 is often used. I know “natural units” are employed in these theories, but I was wondering if there is a way to convert V(Phi) = ˝ m^2
Phi^2 into units of mass/volume? | {"url":"http://www.physicsforums.com/showpost.php?p=2239653&postcount=1","timestamp":"2014-04-17T04:02:01Z","content_type":null,"content_length":"8878","record_id":"<urn:uuid:9b65b7dc-6c2b-46ea-9971-a00f32fba3ac>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00313-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - [ap-stat] TPS 4e Test 10A Q1 (MCQ)
Date: Apr 13, 2012 12:22 AM
Author: Peter Tong
Subject: [ap-stat] TPS 4e Test 10A Q1 (MCQ)
Dear all,
Being a new teacher to AP Stats. I was wondering if anyone can help me with this question.
A city planner is comparing traffic patterns at two different intersections. He randomly selects 12 times between 6 am and 10 pm, and he and his assistant count the number of cars passing through each intersection during the 10-minute interval that begins at that time. He plans to test the hypothesis that the mean difference in the number of cars passing through the two intersections during each of those 12 times intervals is 0. Which of the following is appropriate test of the city planner?s hypothesis?
(a) Two-proportion z-test
(b) Two-sample z-test
(c) Matched pairs t-test
(d) Two proportion t-test
(e) Two-sample t-test
The answer given is (c) Matched pairs t-test but many of my students selected (e) Two-sample t-test.
The book's explanation for (c) is;
The ?pairs? are the twelve sets of 10-minute time intervals?one at each intersection. The parameter of interest is the mean difference between the number of cars at each intersection.
My students are saying that it's "different intersection" and also since it's in Chapter 10, it should be Two-sample t-test. How can I further convince my students?
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To change your subscription address or other settings click here: | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=7767635","timestamp":"2014-04-19T12:33:31Z","content_type":null,"content_length":"3254","record_id":"<urn:uuid:0becb350-bb3a-46f4-a694-8072c78f6217>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00501-ip-10-147-4-33.ec2.internal.warc.gz"} |
Recent releases
Release Notes: This version supports '_' to represent the previous result and support for rational exponents. Output can be logged to a file with -L. Nonlinear unit synonyms are easier to define, and
display of nonlinear unit definitions now shows the range and domain and type of units needed. Error handling is better, and various bugs have been fixed.
Release Notes: This release corrects an electron mass error.
Release Notes: This release supports conversion to sums of units (e.g., feet and inches or hours and minutes). It includes a script to automatically update the currency conversion rates. The units
are now in Unicode with UTF-8. Units can be defined with reference to environment variables, so you can now select local units such as the gallon without changing locale, and these units will work
everywhere. The name of the units database has changed, and the syntax for defining nonlinear units has changed slightly to allow specification of domain and range. The personal units database is now
Release Notes: The units database has been updated to include the latest NIST values, abrasive grits, and other fixes.
Release Notes: Units now reads custom definitions from ~/.units.dat. The precedence of "*" has changed to match the usual algebraic precedence, and the "**" operator was added for exponents. A text
search feature was added so that typing "search text" lists the units whose names contain "text". | {"url":"http://freecode.com/projects/units","timestamp":"2014-04-17T19:10:40Z","content_type":null,"content_length":"27916","record_id":"<urn:uuid:5e01b7a8-1c55-405a-a460-0fb0a2272b33>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00204-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Equilogical Spaces
Andrej Bauer, 1
Lars Birkedal, 2
Dana S. Scott 3
School of Computer Science, Carnegie Mellon University
It is well known that one can build models of full higher-order dependent type theory
(also called the calculus of constructions) using partial equivalence relations (PERs)
and assemblies over a partial combinatory algebra (PCA). But the idea of categories
of PERs and ERs (total equivalence relations) can be applied to other structures
as well. In particular, we can easily define the category of ERs and equivalence-
preserving continuous mappings over the standard category Top0 of topological
T0-spaces; we call these spaces (a topological space together with an ER) equilogical
spaces and the resulting category Equ. We show that this category--in contradis-
tinction to Top0--is a cartesian closed category. The direct proof outlined here uses
the equivalence of the category Equ to the category PEqu of PERs over algebraic
lattices (a full subcategory of Top0 that is well known to be cartesian closed from
domain theory). In another paper with Carboni and Rosolini (cited herein) a more
abstract categorical generalization shows why many such categories are cartesian
closed. The category Equ obviously contains Top0 as a full subcategory, and it nat- | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/420/0115929.html","timestamp":"2014-04-18T11:55:33Z","content_type":null,"content_length":"8350","record_id":"<urn:uuid:ccc96eb4-ca8e-4c82-b005-b9529df22a90>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00426-ip-10-147-4-33.ec2.internal.warc.gz"} |
what is the difference between eddy current and induced current?
who causes the eddy current, self induction or back emf
Self induction and back emf are created in the primary coil (the one to which AC power is connected). The eddy current is not in that coil (ie. not self induction or back emf). The eddy current is
not in the secondary coil (ie. induction). It is in the core.
The eddy current is caused because the alternating current in the coil creates an electric field within the area enclosed by the coil. This emf is determined by Faraday's law:
[tex]emf = \oint E\cdot ds = -\frac{d\phi}{dt} = -\oint B\cdot dA[/tex]
The line integral of the electric field around a closed loop path is equal to (-) the rate of change of the magnetic flux through the area enclosed by the path.
So, for a coil of wire carrying an alternating current there will be an electric field created all along any closed path inside the area enclosed by the coil. The magnitude of the emf created along
any path is determined by the time rate of change of the flux through the area enclosed by that path.
If there is just air in that enclosed area, there are no eddy currents: the induced emf cannot move the electrons because they are stuck to the atoms in the air. However, if there is an iron core in
that space, the electric field causes electrons in the iron core to move (since iron is a conductor, the electrons are free to move). These electron motions are the eddy currents. | {"url":"http://www.physicsforums.com/showthread.php?t=481169","timestamp":"2014-04-20T08:42:59Z","content_type":null,"content_length":"77343","record_id":"<urn:uuid:69eb2075-7a8b-459d-b155-f567eed1714b>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00099-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Wheat Ridge trigonometry Tutors | {"url":"http://www.purplemath.com/Broomfield_trigonometry_tutors.php","timestamp":"2014-04-20T04:17:20Z","content_type":null,"content_length":"24194","record_id":"<urn:uuid:887ad222-45b9-4532-a307-f60b952b3775>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00293-ip-10-147-4-33.ec2.internal.warc.gz"} |
Limits that equal e
Ah, yes, this seems like the most famous mathematical term, I have heard before. I was looking at a function whose limit was e as it went to infinity, so I decided to read up about this sort of
thing, and found out how amazing e really is, went on a few hour rant reading various things about it. never really clicked before . Now my question is, if you given a limit say $\lim_{n\to\
infinty}f(n)$ and it contains some sort of ratio/fraction where n is in the denominator and involves something to the nth power, this is an observation I made, which, of course in math, can get
you somewhere very wrong, the observation that if the function contains something along the lines of what I just suggested it will go to a limit equivalent to e?
as I was suggested this example earlier by redsox apart from the traditional "definition" of e | {"url":"http://mathhelpforum.com/calculus/115724-limits-equal-e.html","timestamp":"2014-04-16T07:40:15Z","content_type":null,"content_length":"51693","record_id":"<urn:uuid:05a74dd6-1184-4be0-b79e-6d9cba3ce6fd>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00113-ip-10-147-4-33.ec2.internal.warc.gz"} |
[ODE] More speed???
Nguyen Binh ngbinh at glassegg.com
Wed Nov 5 17:21:40 MST 2003
Hi Aras,
AP> Is the system actually Ax=b? Isn't it something like inequalities system
(like Ax>>b)? I don't know the underlying math, of course, so I may be
AP> completely wrong :)
The "system" is LCP A*x = b+w with certain conditions. Solve such
system require many math skills (which I don't have :( ), you can
read the famous Barraff's paper about it.
But, on the way to solve such terrified problem, we need to solve
simple linear problem A*x = b (A, b : known, x : variable). ODE
now solve A*x = b by:
1) Factor A into L*U (L,U is upper and lower triangular matrix)
So we have
L*(U*x) = b
-> L * y = b (U*x = y)
(ODE function dFactorLDLT())
2) Solve for y, this is straight forward since L is upper triangular
(ODE function dSolveL1())
3) Solve for x, this is also straight forward since U is lower triangular
(ODE function dSolveL1T())
As you see, we expect the two task 2) and 3) to be fast. But
actually, my profiling show that the two task take roughly 45% CPU
power on Step rountine (test scene contains about 200 primitives)
So, my ideas are:
1) SIMDize it: easy to implement and keep ODE stable
2) Change it-> use iterative methods : theoritically faster
but may break ODE stability. (But actually, I prefer this way)
Best regards,
Nguyen Binh
Software Engineer
Glass Egg Digital Media
E.Town Building
7th Floor, 364 CongHoa Street
Tan Binh District,
HoChiMinh City,
Phone : +84 8 8109018
Fax : +84 8 8109013
More information about the ODE mailing list | {"url":"http://ode.org/old_list_archives/2003-November/010323.html","timestamp":"2014-04-24T01:28:28Z","content_type":null,"content_length":"4103","record_id":"<urn:uuid:3e56b7d2-a495-4511-b272-9240bb3cc9fb>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00012-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Title: Parasitic Error-free Symmetric Diaphragm Flexure, and a set of precision
compliant mechanisms based it: Three and Five DOF flexible torque couplings,
Five DOF motion stage, single DOF linear/axial bearing.
Inventors: Shorya Awtar, Alexander H. Slocum
Three Degrees of Freedom (DOF) Diaphragm Flexures
Diaphragm flexures are commonly used for providing motion in the direction normal to
the flexure plane. Many versions of the design shown in Fig.1 have been frequently
encountered in the literature and in numerous devices/applications [1-3]. Because of its
geometry, apart from a Z translation, this flexure is also compliant in roll and pitch
motions (i.e. rotations about the X and Y axes). Any diaphragm flexure made from thin
sheetmetal will typically have these three degrees of freedom, and will be very stiff in the
two planer directions, X and Y, and in yaw (i.e. rotation about the Z axis). An evident
drawback of this design is that it suffers from a parasitic Z-twist associated with Z
translation. The reason is for this twist becomes obvious when one recognizes that each
of the peripheral arms (or blades) in this design behaves like a fixed-free cantilever, as
shown in Fig.2. For a fixed-free cantilever, any motion in the desired direction is
associated with a transverse motion e, also referred to as parasitic error motion, so as to
maintain a constant arc length of the cantilever. Theory of linear elasticity provides us
with the following results | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/299/1504182.html","timestamp":"2014-04-20T19:10:48Z","content_type":null,"content_length":"8669","record_id":"<urn:uuid:2370279a-a9f8-4687-a8da-67c34bbcbd09>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00442-ip-10-147-4-33.ec2.internal.warc.gz"} |
power series help
December 11th 2010, 10:33 AM
power series help
my question is:
Expand g(x) = (1 - 2x)^(-1) as a powers series in (x+2).
I don't even know where to start, can anyone help me
thanks in advance
December 11th 2010, 11:07 AM
$\displaystyle{\frac{1}{1-2x}=\frac{1}{5-2(x+2)}=\frac{1}{5}\cdot\frac{1}{1-\frac{2}{5}(x+2)}=\frac{1}{5}\sum\limits^\infty_{n =0}\frac{2^n}{5^n}(x+2)^n}$ , valid for $\displaystyle{\left|\frac
{2}{5}(x+2)\right|< 1$
December 12th 2010, 02:43 AM
What Tonio did was treat the fraction $\frac{1}{1- \frac{2}{5}(x+ 2)}$ as " $\frac{1}{1- r}$", the sum of a geometric series with $r= \frac{2}{5}(x+ 2)$. That is the simplest and best way to do
this problem.
You could also taken derivatives of $f(x)= (1- 2x)^{-1}$ and use the formula for a Taylor's series at x= 2 or used the "generalized binomial theorem" to expand $(a+ b)^{-1}$ with a= 1, b= -2x. | {"url":"http://mathhelpforum.com/calculus/165959-power-series-help-print.html","timestamp":"2014-04-20T20:36:43Z","content_type":null,"content_length":"6325","record_id":"<urn:uuid:5f5e65ed-4c61-4a50-8d83-732977bf556f>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00658-ip-10-147-4-33.ec2.internal.warc.gz"} |
Modular Art
There are many ways to create basic design patterns from modular arithmetic operations tables. The traditional way is to assign a patterned square to each numbered cell in the interior of a table,
say a mod 5 addition table. Notice that the interior of the table is a 5 x 5 array of cells.
We begin by assigning patterned squares to the numbers appearing in the interior of the table. In the patterns below, additive inverses have been represented by complementary forms of a selected
pattern. (This is not required, but adds an additional mathematical component to the activity.)
Then we fill each square in the interior of the table with the pattern according to the number marked in it.
We can now use this combination of 25 squares as a basic design pattern, which we can translate (slide), reflect (flip), or rotate (turn) to form larger, more artistic designs.
First the basic design pattern is copied in one of the four corners or quadrants of a 2 x 2 grid. In the diagram below, the basic pattern has been copied in the upper left quadrant. Then the basic
design pattern is reflected from the upper left quadrant onto the other three quadrants of the grid - in a vertical line of reflection for the upper right quadrant, and each of these in a horizontal
line of reflection for the lower two quadrants.
In the next basic design pattern, the original grid of patterned squares is distorted so that each patterned square (or distorted rectangle) is a fixed percent (70% illustrated) of the width of the
patterned square (or distorted rectangle) to its immediate left and the same percent (70%) the height of the square (or distorted rectangle) directly above. Mathematics refer to this as a logarithmic
grid. For aesthetics, it had been enlarged to the same size as the original basic design pattern.
The basic design pattern is then copied in the upper left quadrant of a quadrantal system and the design completed by reflection as before.
A modular multiplication table, say mod 5, can also be used to generate the basic design pattern. To circumvent the undistinguished row and column of zeros in the interior of the table, we will use
only the 4 x 4 array of non-zero numbers therein.
Patterned squares are assigned to each of the numbers in the selected array as before. Once again additive inverses have been represented by complementary forms of a selected pattern.
For variety, we have arranged the lines of the basic pattern grid in an irregular way in order to produce a kaleidoscopic effect. As usual, the patterns are entered into the regions of the grid
according to the number marked in it. Here this will require the deformation or stretching of each pattern to fit the region to which it is assigned.
Then the basic design pattern is copied in the upper left quadrant of a quadrantal system and the design completed by reflection as before.
Of course, there is a variety of distorted grids that you can employ to create the final design. The only limit is your imagination. | {"url":"http://britton.disted.camosun.bc.ca/modart/jbmodart2.htm","timestamp":"2014-04-16T21:54:35Z","content_type":null,"content_length":"5777","record_id":"<urn:uuid:3227ce43-97e7-4fa9-b33c-94506d8e0418>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00238-ip-10-147-4-33.ec2.internal.warc.gz"} |
prove the the conjugate of a holomorphic function f is differentiable iff f'=0
February 20th 2011, 06:26 AM #1
Senior Member
Feb 2008
prove the the conjugate of a holomorphic function f is differentiable iff f'=0
This probably has a simple solution, but I cannot find it so far...
problem statement
Suppose the complex-valued function $f$ is holomorphic about zero. Show that $g:=\overline{f}$, i.e. the conjugate of $f$, is differentiable at $z$ if and only if $f'(z)=0$.
The first direction follows fairly directly from the Cauchy-Riemann equations. However, I still need to show that $f'(z)=0$ implies $g'(z)$ exists, and I can't seem to make it happen. Any help
would be much appreciated.
If $f=u+iv$ and $z_0=x_0+iy_$ use:
$f'(z_0)=\dfrac{\partial u}{\partial x} (x_0,y_0) +i\dfrac{\partial v}{\partial x}(x_0,y_0)$
Fernando Revilla
Hmm. I guess I don't see how that helps us here.
February 20th 2011, 07:52 AM #2
February 22nd 2011, 10:28 AM #3
Senior Member
Feb 2008 | {"url":"http://mathhelpforum.com/differential-geometry/171930-prove-conjugate-holomorphic-function-f-differentiable-iff-f-0-a.html","timestamp":"2014-04-17T01:37:29Z","content_type":null,"content_length":"37717","record_id":"<urn:uuid:23a31a6f-28d8-4907-9849-0a471d228d89>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00516-ip-10-147-4-33.ec2.internal.warc.gz"} |
Exploratory Data Analysis and Regression in R
Exploratory Data Analysis (EDA) and Regression
This tutorial demonstrates some of the capabilities of R for exploring relationships among two (or more) quantitative variables.
Bivariate exploratory data analysis
We begin by loading the Hipparcos dataset used in the descriptive statistics tutorial, found at http://astrostatistics.psu.edu/datasets/HIP_star.html. Type
hip <- read.table("http://astrostatistics.psu.edu/datasets/HIP_star.dat",
In the descriptive statistics tutorial, we considered boxplots, a one-dimensional plotting technique. We may perform a slightly more sophisticated analysis using boxplots to get a glimpse at some
bivariate structure. Let us examine the values of Vmag, with objects broken into categories according to the B minus V variable:
notch=T, varwidth=T, las=1, tcl=.5,
xlab=expression("B minus V"),
ylab=expression("V magnitude"),
main="Can you find the red giants?",
cex=1, cex.lab=1.4, cex.axis=.8, cex.main=1)
axis(2, labels=F, at=0:12, tcl=-.25)
axis(4, at=3*(0:4))
The notches in the boxes, produced using "notch=T", can be used to test for differences in the medians (see boxplot.stats for details). With "varwidth=T", the box widths are proportional to the
square roots of the sample sizes. The "cex" options all give scaling factors, relative to default: "cex" is for plotting text and symbols, "cex.axis" is for axis annotation, "cex.lab" is for the x
and y labels, and "cex.main" is for main titles. The two axis commands are used to add an axis to the current plot. The first such command above adds smaller tick marks at all integers, whereas the
second one adds the axis on the right.
The boxplots in the plot above are telling us something about the bivariate relationship between the two variables. Yet it is probably easier to grasp this relationship by producing a scatter plot.
The above plot looks too busy because of the default plotting character, set let's use a different one:
Let's now use exploratory scatterplots to locate the Hyades stars. This open cluster should be concentrated both in the sky coordinates RA and DE, and also in the proper motion variables pm_RA and
pm_DE. We start by noticing a concentration of stars in the RA distribution:
See the cluster of stars with RA between 50 and 100 and with DE between 0 and 25?
Let's construct a logical (TRUE/FALSE) variable that will select only those stars in the appropriate rectangle:
filter1 <- (RA>50 & RA<100 & DE>0 & DE<25)
Next, we select in the proper motions. (As our cuts through the data are parallel to the axes, this variable-by-variable classification approach is sometimes called Classification and Regression
Trees or CART, a very common multivariate classification procedure.)
Let's replot after zooming in on the rectangle shown in red.
filter2 <- (pmRA>90 & pmRA<130 & pmDE>-60 & pmDE< -10) # Space in 'pmDE< -10' is necessary!
filter <- filter1 & filter2
Let's have a final look at the stars we have identified using the pairs command to produce all bivariate plots for pairs of variables. We'll exclude the first and fifth columns (the HIP identifying
number and the parallax, which is known to lie in a narrow band by construction).
Notice that indexing a matrix or vector using negative integers has the effect of excluding the corresponding entries.
We see that there is one outlying star in the e_Plx variable, indicating that its measurements are not reliable. We exclude this point:
filter <- filter & (e_Plx<5)
How many stars have we identified? The filter variable, a vector of TRUE and FALSE, may be summed to reveal the number of TRUE's (summation causes R to coerce the logical values to 0's and 1's).
As a final look at these data, let's consider the HR plot of Vmag versus B.V but make the 92 Hyades stars we just identified look bigger (pch=20 instead of 46) and color them red (col=2 instead of
1). This shows the Zero Age Main Sequence, plus four red giants, with great precision.
plot(Vmag,B.V,pch=c(46,20)[1+filter], col=1+filter,
xlim=range(Vmag[filter]), ylim=range(B.V[filter]))
Linear and polynomial regression
Let's consider a new dataset, one that comes from NASA's Swift satellite. The statistical problem at hand is modeling the X-ray afterglow of gamma-ray bursts. First, read in the dataset:
grb <- read.table ("http://astrostatistics.psu.edu/datasets/GRB_afterglow.dat",
header=T, skip=1)
The skip=1 option in the previous statement tells R to ignore the first row in the data file. You will see why this is necessary if you look at the file. Let's focus on the first two columns, which
are times and X-ray fluxes:
This plot is very hard to interpret because of the scales, so let's take the log of each variable:
x <- log(grb[,1])
y <- log(grb[,2])
plot(x,y,xlab="log time",ylab="log flux")
The relationship looks roughly linear, so let's try a linear model (lm in R):
model1 <- lm(y~x)
abline(model1, col=2, lwd=2)
The "response ~ predictor(s)" format seen above is used for model formulas in functions like lm .
The model1 object just created is an object of class "lm". The class of an object in R can help to determine how it is treated by functions such as print and summary.
model1 # same as print(model1)
Notice the sigma-hat, the R-squared and adjusted R-squared, and the standard errors of the beta-hats in the output from the summary function.
There is a lot of information contained in model1 that is not displayed by print or summary:
For instance, we will use the model1$fitted.values and model1$residuals information later when we look at some residuals plots.
Notice that the coefficient estimates are listed in a regression table, which is standard regression output for any software package. This table gives not only the estimates but their standard errors
as well, which enables us to determine whether the estimates are very different from zero. It is possible to give individual confidence intervals for both the intercept parameter and the slope
parameter based on this information, but remember that a line really requires both a slope and an intercept. Since our goal is really to estimate a line here, maybe it would be better if we could
somehow obtain a confidence "interval" for the lines themselves.
This may in fact be accomplished. By viewing a line as a single two-dimensional point in (intercept, slope) space, we set up a one-to-one correspondence between all (nonvertical) lines and all points
in two-dimensional space. It is possible to obtain a two-dimensional confidence ellipse for the (intercept,slope) points, which may then be mapped back into the set of lines to see what it looks
Performing all the calculations necessary to do this is somewhat tedious, but fortunately, someone else has already done it and made it available to all R users through CRAN, the Comprehensive R
Archive Network. The necessary functions are part of the "car" (companion to applied regression) package, which may installed onto the V: drive (we don't have write access to the default location
where R packages are installed) as follows:
install.packages("car",lib="V:/") # lib=... not always necessary!
You will have to choose a CRAN mirror as part of the installation process. Once the car package is installed, its contents can be loaded into the current R session using the library function:
If all has gone well, there is now a new set of functions, along with relevant documentation. Here is a 95% confidence ellipse for the (intercept,slope) pairs:
Remember that each point on the boundary or in the interior of this ellipse represents a line. If we were to plot all of these lines on the original scatterplot, the region they described would be a
95% confidence band for the true regression line. A graduate student and I wrote a simple function to draw the borders of this band on a scatterplot. You can see this function at www.stat.psu.edu/
~dhunter/R/confidence.band.r"; to read it into R, use the source function:
source( "http://www.stat.psu.edu/~dhunter/R/confidence.band.r")
In this dataset, the confidence band is so narrow that it's hard to see. However, the borders of the band are not straight. You can see the curvature much better when there are fewer points or more
variation, as in:
tmpx <- 1:10
tmpy <- 1:10+rnorm(10) # Add random Gaussian noise
Also note that increasing the sample size increases the precision of the estimated line, thus narrowing the confidence band. Compare the previous plot with the one obtained by replicating tmpx and
tmpy 25 times each:
tmpx25 <- rep(tmpx,25)
tmpy25 <- rep(tmpy,25)
A related phenomenon is illustrated if we are given a value of the predictor and asked to predict the response. Two types of intervals are commonly reported in this case: A prediction interval for an
individual observation with that predictor value, and a confidence interval for the mean of all individuals with that predictor value. The former is always wider than the latter because it accounts
for not only the uncertainty in estimating the true line but also the individual variation around the true line. This phenomenon may be illustrated as follows. Again, we use a toy data set here
because the effect is harder to observe on our astronomical dataset. As usual, 95% is the default confidence level.
predict(lm(tmpy~tmpx), data.frame(tmpx=7), interval="prediction")
text(c(7,7,7), .Last.value, "P",col=4)
predict(lm(tmpy~tmpx), data.frame(tmpx=7), interval="conf")
text(c(7,7,7), .Last.value, "C",col=5)
Polynomial curve-fitting: Still linear regression!
Because there appears to be a bit of a bend in the scatterplot, let's try fitting a quadratic curve instead of a linear curve. Note: Fitting a quadratic curve is still considered linear regression.
This may seem strange, but the reason is that the quadratic regression model assumes that the response y is a linear combination of 1, x, and x^2. Notice the special form of the lm command when we
implement quadratic regression. The I function means "as is" and it resolves any ambiguity in the model formula:
model2 <- lm(y~x+I(x^2))
Plotting the quadratic curve is not a simple matter of using the abline function. To obtain the plot, we'll first create a sequence of x values, then apply the linear combination implied by the
regression model using matrix multiplication:
xx <- seq(min(x),max(x),len=200)
yy <- model2$coef %*% rbind(1,xx,xx^2)
Diagnostic residual plots
Comparing the (red) linear fit with the (green) quadratic fit visually, it does appear that the latter looks slightly better. However, let's check some diagnostic residual plots for these two models.
To do this, we'll use the plot.lm command, which is capable of producing six different types of diagnostic plots. We will only consider two of the six: A plot of residuals versus fitted values and a
normal quantile-quantile (Q-Q) plot.
plot.lm(model1, which=1:2)
It is not actually necessary to type plot.lm in the previous command; plot would have worked just as well. This is because model1 is an object of class "lm" -- a fact that can be verified by typing
"class(model1)" -- and so R knows to apply the function plot.lm if we simply type "plot(model1, which=1:2)".
Looking at the first plot, residuals vs. fitted, we immediately see a problem with model 1. A "nice" residual plot should have residuals both above and below the zero line, with the vertical spread
around the line roughly of the same magnitude no matter what the value on the horizontal axis. Furthermore, there should be no obvious curvature pattern. The red line is a lowess smoother produced to
help discern any patterns (more on lowess later), but this line is not necessary in the case of model1 to see the clear pattern of negative residuals on the left, positive in the middle, and negative
on the right. There is curvature here that the model missed!
Pressing the return key to see the second plot reveals a normal quantile-quantile plot. The idea behind this plot is that it will make a random sample from a normal distribution look like a straight
line. To the extent that the normal Q-Q plot does not look like a straight line, the assumption of normality of the residuals is suspicious. For model1, the clear S-shaped pattern indicates
non-normality of the residuals.
How do the same plots look for the quadratic fit?
plot(model2, which=1:2)
These plots are much better-looking. There is a little bit of waviness in the residuals vs. fitted plot, but the pattern is nowhere near as obvious as it was before. And there appear to be several
outliers among the residuals on the normal Q-Q plot, but the normality assumption looks much less suspect here.
The residuals we have been using in the above plots are the ordinary residuals. However, it is important to keep in mind that even if all of the assumptions of the regression model are perfectly true
(including the assumption that all errors have the same variance), the variances of the residuals are not equal. For this reason, it is better to use the studentized residuals. Unfortunately, R
reports the ordinary residuals by default and it is necessary to call another function to obtain the studentized residuals. The good news is that in most datasets, residual plots using the
studentized residuals are essentially indistinguishable in shape from residual plots using the ordinary residuals, which means that we would come to the same conclusions regardless of which set of
residuals we use.
rstu = rstudent(model2)
plot(model2$fit, rstu)
To see how similar the studentized residuals are to a scaled version of the ordinary residuals (called the standardized residuals), we can depict both on the same plot:
rsta = rstandard(model2)
points(model2$fit, rsta, col=2, pch=3)
Collinearity and variance inflation factors
Let's check the variance inflation factors (VIFs) for the quadratic fit. The car package that we installed earlier contains a function called vif that does this automatically. Check its help page by
typing "?vif" if you wish. Note that it does not make sense to look at variance inflation factors for model1, which has only one term (try it and see what happens). So we'll start by examining
The VIFs of more than 70 indicate a high degree of collinearity between the values of x and x^2 (the two predictors). This is not surprising, since x has a range from about 5 to 13. In fact, it is
easy to visualize the collinearity in a plot:
plot(x,x^2) # Note highly linear-looking plot
To correct the collinearity, we'll replace x and x^2 by (x-m) and (x-m)^2, where m is the sample mean of x:
centered.x <- x-mean(x)
model2.2 <- lm(y ~ centered.x + I(centered.x^2))
This new model has much lower VIFs, which means that we have greatly reduced the collinearity. However, the fit is exactly the same: It is still the best-fitting quadratic curve. We may demonstrate
this by plotting both fits on the same set of axes:
plot(x,y,xlab="log time",ylab="log flux")
yy2 <- model2.2$coef %*% rbind(1, xx-mean(x), (xx-mean(x))^2)
lines(xx, yy, lwd=2, col=2)
lines(xx, yy2, lwd=2, col=3, lty=2)
Model selection using AIC and BIC
Let's compare the AIC and BIC values for the linear and the quadratic fit. Without getting too deeply into details, the idea behind these criteria is that we know the model with more parameters (the
quadratic model) should achieve a higher maximized log-likelihood than the model with fewer parameters (the linear model). However, it may be that the additional increase in the log-likelihood
statistic achieved with more parameters is not worth adding the additional parameters. We may test whether it is worth adding the additional parameters by penalizing the log-likeilhood by subtracting
some positive multiple of the number of parameters. In practice, for technical reasons we take -2 times the log-likelihood, add a positive multiple of the number of parameters, and look for the
smallest resulting value. For AIC, the positive multiple is 2; for BIC, it is the natural log of n, the number of observations. We can obtain both the AIC and BIC results using the AIC function.
Remember that R is case-sensitive, so "AIC" must be all capital letters.
The value of AIC for model2 is smaller than that for model1, which indicates that model2 provides a better fit that is worth the additional parameters. However, AIC is known to tend to overfit
sometimes, meaning that it sometimes favors models with more parameters than they should have. The BIC uses a larger penalty than AIC, and it often seems to do a slightly better job; however, in this
case we see there is no difference in the conclusion:
n <- length(x)
AIC(model1, k=log(n))
AIC(model2, k=log(n))
It did not make any difference in the above output that we used model2 (with the uncentered x values) instead of model2.2 (with the centered values). However, if we had looked at the AIC or BIC
values for a model containing ONLY the quadratic term but no linear term, then we would see a dramatic difference. Which one of the following would you expect to be higher (i.e., indicating a worse
fit), and why?
AIC(lm(y~I(x^2)), k=log(n))
AIC(lm(y~I(centered.x^2)), k=log(n))
Other methods of curve-fitting
Let's try a nonparametric fit, given by loess or lowess. First we plot the linear (red) and quadratic (green) fits, then we overlay the lowess fit in blue:
plot(x,y,xlab="log time",ylab="log flux")
abline(model1, lwd=2, col=2)
lines(xx, yy, lwd=3, col=3)
npmodel1 <- lowess(y~x)
lines(npmodel1, col=4, lwd=2)
It is hard to see the pattern of the lowess curve in the plot. Let's replot it with no other distractions. Notice that the "type=n" option to plot function causes the axes to be plotted but not the
plot(x,y,xlab="log time",ylab="log flux", type="n")
lines(npmodel1, col=4, lwd=2)
This appears to be a piecewise linear curve. An analysis that assumes a piecewise linear curve will be carried out on these data later in the week.
In the case of non-polynomial (but still parametric) curve-fitting, we can use nls. If we replace the response y by the original (nonlogged) flux values, we might posit a parametric model of the form
flux = exp(a+b*x), where x=log(time) as before. Compare a nonlinear approach (in red) with a nonparametric approach (in green) for this case:
flux <- grb[,2]
nlsmodel1 <- nls(flux ~ exp(a+b*x), start=list(a=0,b=0))
npmodel2 <- lowess(flux~x)
plot(x, flux, xlab="log time", ylab="flux")
lines(xx, exp(9.4602-.9674*xx), col=2, lwd=2)
lines(npmodel2, col=3, lwd=2)
Interestingly, the coefficients of the nonlinear least squares fit are different than the coefficients of the original linear model fit on the logged data, even though these coefficients have exactly
the same interpretation: If flux = exp(a + b*x), then shouldn't log(flux) = a + b*x? The difference arises because these two fitting methods calculate (and subsequently minimize) the residuals on
different scales. Try plotting exp(a + b*xx) on the scatterplot of x vs. flux for both (a,b) solutions to see what happens. Next, try plotting a + b*xx on the scatterplot of x vs. log(flux) to see
what happens.
If outliers appear to have too large an influence over the least-squares solution, we can also try resistant regression, using the lqs function in the MASS package. The basic idea behind lqs is that
the largest residuals (presumably corresponding to "bad" outliers) are ignored. The results for our log(flux) vs. log(time) example look terrible but are very revealing. Can you understand why the
output from lqs looks so very different from the least-squares output?
lqsmodel1 <- lqs(y~x, method="lts")
plot(x,y,xlab="log time",ylab="log flux")
Finally, let's consider least absolute deviation regression, which may be considered a milder form of resistant regression than lqs. In least absolute deviation regression, even large residuals have
an influence on the regression line (unlike in lqs), but this influence is less than in least squares regression. To implement it, we'll use a function called rq (regression quantiles) in the
"quantreg" package. Like the "car" package, this package is not part of the standard distribution of R, so we'll need to download it. In order to do this, we must tell R where to store the installed
library using the install.packages function.
install.packages("quantreg",lib="V:/") # lib=... not always necessary!
library(quantreg, lib.loc="V:/")
Assuming the quantreg package is loaded, we may now compare the least-squares fit (red) with the least absolute deviations fit (green). In this example, the two fits are nearly identical:
rqmodel1 <- rq(y~x)
plot(x,y,xlab="log time",ylab="log flux") | {"url":"http://astrostatistics.psu.edu/datasets/2006tutorial/2006reg.html","timestamp":"2014-04-18T20:43:44Z","content_type":null,"content_length":"32637","record_id":"<urn:uuid:913ea500-6b72-4569-af33-2a6d0a8fc1c6>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00252-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finitely Generated Abelian Groups
Consider an abelian group $A$ generated by $m$ elements
\[ A = \langle a_1,...,a_m \rangle \]
Then the free abelian group of rank $m$
\[ F = \langle u_1,...,u_m \rangle \]
maps homomorphically onto $A$ via the map that sends $u_i$ to $a_i$. By the first isomorphism theorem we have $A \cong F / R$ for some subgroup $R$ of $F$. Pick a basis $v_1,...,v_m$ of $F$ such that
$R = \langle h_1 v_1 ,..., h_q v_q \rangle$ where $h_i | h_{i+1}$, $h_i \ge 1$, $q \le m$.
Consider the case where $m=1$. There are three possibilities. (1) $R = \langle v \rangle$, so $F / R$ is the trivial group, (2) $R = \langle h v \rangle$, in which case $F/R = \mathbb{Z}_h$, and (3)
$R = \{ 0\}$ and we have $F/R = F$.
Theorem: Every finitely generated abelian group can be expressed as the direct sum of cyclic groups
\[ A = \mathbb{Z}^n \oplus \mathbb{Z}_{h_1} \oplus ... \oplus \mathbb{Z}_{h_n} \]
Corollary: A finitely generated abelian group is free if and only if it is torsion-free, that is, it contains no element of finite order other than the identity.
The number $r$ is called the rank of $A$. The orders of the cyclic groups $h_1,...,h_n$ are called the invariants of $A$. Note $A$ is finite if and only if its rank is zero.
Theorem: Suppose $A$ is a finitely generated abelian group with decompositions
\[ \array { A = \mathbb{Z}^r \oplus \mathbb{Z}_{e_1} \oplus ... \oplus \mathbb{Z}_{e_n} \\ A = \mathbb{Z}^s \oplus \mathbb{Z}_{d_1} \oplus ... \oplus \mathbb{Z}_{d_m} } \]
satifying $e_i | e_{i+1}, d_i | d_{i+1}$. Then $r=s, n=m, e_i=d_i$.
Proof: Let $T$ be the set of elements of $A$ of finite order. Clearly if $g,h$ have finite order then $ord(g) ord(h) (h-k) = 0$ hence $h-k$ also has finite order hence $T$ is a subgroup of $A$. It is
called the torsion group of $A$.
A little thought shows that we must have
\[ \array { T = \mathbb{Z}_{e_1} \oplus ... \oplus \mathbb{Z}_{e_n} \\ T = \mathbb{Z}_{d_1} \oplus ... \oplus \mathbb{Z}_{d_m} } \]
Consider the map that projects $A$ onto $\mathbb{Z}^r$. By the first isomorphism theorem we have that $A/T \cong \mathbb{Z}^r$. Similarly we have $A/T \cong \mathbb{Z}^s$ hence $r=s$.
Now conisder $T$. Let $p$ be a prime, and let $P$ be the set of elements whose order is a power of $P$. Then $P$ is a group. We first need the following:
Theorem: Let $G$ be a finite abelian group of order $p_1^{a_1} p_2^{a_2} ...$ where the $p_i$'s are distinct primes. Then $G = P_1 \oplus P_2 \oplus ...$ where $P_i$ is the subgroup of elements whose
orders are powers of $p_i$.
Proof: Let $x \in G$ be an element of order $p_1^{\alpha} f_1$ where $f_1, p_1$ are coprime. Then we may write $x = a_1 + x_1$ where $a_1$ has order $p_1^{\alpha}$ and $x_1$ has order $f_1$. (Simply
take $a_1 = u f_1 x, x_1 = v p_1^{\alpha} x$ where $u f_1 + v p_1^{\alpha} = 1$.)
Iterating this procedure gives a decomposition $x = a_1 + a_2 + ...$ with $a_i \in P_i$. We claim this decomposition is unique. Suppose $0 = b_1 + b_2 + ...$ where $b_i \in P_i$. Then for all $i$,
subtracting $b_i$ from both sides shows that the order of $b_i$ is coprime to $p_i$. But it must also be a power of $p_i$ which is only possible if $b_i = 0$.
It is clear that the groups $P_i$ are uniquely determined. In fact, they are the Sylow groups since $G$ is abelian.∎
In particular, if $x$ is an element of order $n = p_1^{a_1} p_2^{a_2} ...$ then we have
\[ \langle x \rangle = \langle (n/p_1^{a_1}) x \rangle \oplus \langle (n/p_2^{a_2}) x \rangle \oplus ... \]
\[ e_1 = p_1^{a_1} p_2^{a_2} ..., e_2 = p_1^{b_1} p_2^{b_2} ..., ... \]
where $a_i \le b_i \le ...$ for all $i$ since $e_i | e_i+1$. Then we have
\[ T = P_1 \oplus P_2 \oplus ... \oplus Q_1 \oplus Q_2 \oplus ... \]
where $P_1,P_2,...,Q_1,Q_2,...$ are cyclic groups of order $p_1^{a_1}, p_1^{b_1}, ..., p_2^{a_2}, p_2^{b_2},...$. We see that the Sylow groups of $T$ are $P = P_1 \oplus P_2 \oplus ..., Q = Q_1 \
oplus Q_2 \oplus ...$. Now we need the following:
Lemma: Let $G$ be any group. Suppose $x, y \in G$ commute and have relatively prime orders $m, n$. Then
\[ \langle x,y \rangle = \langle x y \rangle \]
is cyclic of order $m n$.
Proof: We know the order is at most $m n$ since each element must be of the form $x^a y^b$ for $a=0,...,m-1, b = 0,...,n-1$. Now suppose $(x y)^t = 1$. Then $1 = (x y)^{t m} = y^{t m}$ implying that
$n | t m$. Since $m, n$ is coprime we have $n | t$. Similarly $m | t$, thus the group order must be exactly $m n$.∎
\[ T = P_1 \oplus P_2 \oplus ... \oplus Q_1 \oplus Q_2 \oplus ... \]
\[ T = \mathbb{Z}_{e_1} \oplus ... \oplus \mathbb{Z}_{e_n} \]
so that one decomposition implies the other. We are done as soon as we show that the Sylow groups have a unique decomposition:
Theorem: Let $A$ be an abelian group of order $p^a$ where $p$ is prime. Suppose
\[ \array { A &=& \langle u_1 \rangle \oplus ... \oplus \langle u_k \rangle \\ A &=& \langle v_1 \rangle \oplus ... \oplus \langle v_l \rangle } \]
where $u_1,...,u_k$ have orders $p^{f_1} \ge ... \ge p^{f_k} \gt 1$, and $v_1,...,v_v$ have orders $p^{g_1} \ge ... \ge p^{g_l} \gt 1$. Then $k =l$ and $f_i = g_i$ for $i = 1,...,k$.
Proof: Note we must have $a = f_1 + ... + f_k = g_1 + ... + g_l$. The theorem is trivial when $a = 1$, which we use to start an induction.
Let $A_p$ be the set of elements $x \in A$ with $p x = 0$. Then $A_p$ is a subgroup. We have
\[ \array { A_p &=& \langle p^{f_1-1} u_1 \rangle \oplus ... \oplus \langle p^{f_k-1} u_k \rangle \\ A_p &=& \langle p^{g_1-1} v_1 \rangle \oplus ... \oplus \langle p^{g_l-1} v_l \rangle } \]
Hence $A_p = \mathbb{Z}_p^k = \mathbb{Z}_p^l$ implying that $k=l$.
Now consider the set $A^p$ of elements $p x$ for all $x \in A$ (the multiples of $p$). Then $A^p$ is a subgroup, and is generated by $p u_1 , ...,p u_k$ and also by $p v_1 ,..., p v_k$. But in
general these are not bases for $A^p$ since we might have $p u_i = 0$ for example. So find $\kappa$ such that $f_1,...,f_\kappa \ge 2$ and $f_{\kappa+1} = ... = f_k = 1$, and similarly find $\lambda$
with $g_1,...,g_\lambda \ge 2$ and $g_{\lambda+1} = ... = g_k = 1$.
This yields the decompositions
\[ A^p = \langle p u_1 \rangle + ... + \langle p u_\kappa \rangle = \langle p v_1 \rangle + ... + \langle p v_\lambda \rangle \]
By inductive hypothesis we have $\kappa = \lambda$ and $f_i - 1 = g_i - 1$ for all $i = 1,...,\kappa$.∎
We have now proved the main theorem.∎
In the last proof, the numbers $p^{f_1},...,p^{f_k}$ are called the elementary divisors of $A$ corresponding to $p$. $A$ is said to be of type $(f_1,...,f_k)$.
Example: Suppose an abelian group $A$ is generated by $a,b$ subject to the relations $30 a = 12 b = 0$. Then define the free abelian groups $F = \langle x,y \rangle$ and $R = \langle 30x, 12 y \
rangle$. Note we have $A \cong F / R = \mathbb{Z}_{30} \oplus \mathbb{Z}_{12}$. Then we have
\[ A \cong \mathbb{Z}_2 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5 \oplus \mathbb{Z}_4 \oplus \mathbb{Z}_3 \cong (\mathbb{Z}_4 \oplus \mathbb{Z}_2) \oplus (\mathbb{Z}_3 \oplus \mathbb{Z}_3) \oplus \
mathbb{Z}_5 \]
Thus the elementary divisors for 2,3,5 are $(4,2), (3,3), 5$. Rearranging gives $A \cong \mathbb{Z}_{60} \oplus \mathbb{Z}_6$, so the invariants are $60, 6$.
Example: Suppose an abelian group $A$ is generated by $a,b,c,d$ and the relations $3a + 9b -3c =0, 4a+2b-2d=0$. Then define the free abelian groups $F=\langle x,y,z,t\rangle$ and $R=\langle 3u,2v\
rangle$ where $u=x+3y-z, v=2x +y -t$. Note $x,y,u,v$ is also a basis of $F$. Thus
\[ A \cong F/R \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_2 \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}_6 \] | {"url":"http://crypto.stanford.edu/pbc/notes/group/fgabelian.html","timestamp":"2014-04-18T08:03:13Z","content_type":null,"content_length":"12465","record_id":"<urn:uuid:5cb9fae5-0f8d-4311-9943-cc3fa67a16d5>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00002-ip-10-147-4-33.ec2.internal.warc.gz"} |
Stable Synchronization of Rigid Body Networks
Sujit Nair and Naomi Ehrich Leonard,
Networks and Heterogeneous Media, American Institute of Mathematical Sciences, Vol. 2, No. 4, December 2007, 595-624.
We address stable synchronization of a network of rotating and translating rigid bodies in three-dimensional space. Motivated by applications that require coordinated spinning spacecraft or
diving underwater vehicles, we prove control laws that stably couple and coordinate the dynamics of multiple rigid bodies. We design decentralized, energy shaping control laws for each individual
rigid body that depend on the relative orientation and relative position of its neighbors. Energy methods are used to prove stability of the coordinated multi-body dynamical system. To prove
exponential stability, we break symmetry and consider a controlled dissipation term that requires each individual to measure its own velocity. The control laws are illustrated in simulation for a
network of spinning rigid bodies. | {"url":"http://www.princeton.edu/~naomi/publications/2007/NHM07.html","timestamp":"2014-04-19T19:41:36Z","content_type":null,"content_length":"1926","record_id":"<urn:uuid:3f18cfbf-87a6-4243-af95-c12d4188b576>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00273-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculating the "Most Helpful" review
up vote 17 down vote favorite
How would you calculate the order of a list of reviews sorted by "Most Helpful" to "Least Helpful"?
Here's an example inspired by product reviews on Amazon:
Say a product has 8 total reviews and they are sorted by "Most Helpful" to "Least Helpful" based on the part that says "x of y people found this review helpful".
Here is how the reviews are sorted starting with "Most Helpful" and ending with "Least Helpful":
7 of 7
21 of 26
9 of 10
6 of 6
8 of 9
5 of 5
7 of 8
12 of 15
What equation do I need to use to calculate this sort order correctly? I thought I had it a few times but the "7 of 7" and "6 of 6" and "5 of 5" always throw me off. What am I missing?
4 By the time I found it this was closed, but I think there's a principled answer based on the assumption that each review i has probability p_i of being helpful. Assume a prior distribution on the
values of p_i, apply Bayes rule to the observed data, and sort by the posterior probabilities. 7/7 should end up being better than 5/5. – David Eppstein Nov 18 '09 at 23:25
3 Why is this question considered "off topic"? It actually seems pretty interesting. – Darsh Ranjan Nov 19 '09 at 1:07
3 I disagree with closing this. There is an interesting question here: you have a bunch of {0,1} random variables, with different expected values, and you sample them different numbers of times.
Your job is to reconstruct the relative ordering of the expectations. – David Speyer Nov 19 '09 at 1:25
4 I concur. Just because the question was asked by an amateur (?) doesn't mean it's not an interesting question. – Jason Dyer Nov 19 '09 at 1:29
The answer by David Speyer pointing to "How not to sort by average ranking" is very interesting. I learnt something new, and there are valid discussions to be had about the assumptions that went
5 into it and what alternatives might be used. It's also a really nice example of non-trivial mathematics meeting a very commonly asked real world problem. This should be encouraged, not
discouraged. – Dan Piponi Nov 19 '09 at 1:54
show 1 more comment
2 Answers
active oldest votes
See How not to sort by average ranking.
up vote 26 down vote accepted
add comment
David Eppstein suggests a Bayesian method in his comment. One standard thing to do in this situation is to use a uniform prior. That is, before assessments of a review come in, its
probability $p_i$ of being helpful is assumed to be uniformly distributed on [0, 1]. Upon receiving each assessment of the review, apply Bayes' theorem.
This sounds complicated, and it would be for an arbitrary prior distribution. But it turns out that with the uniform prior, the posterior distributions are all beta distributions. In
particular, the expected value of $p_i$ after s positive assessments and n-s negative ones is (s+1)/(n+2). This is Laplace's rule of succession, and proofs of the facts I've mentioned can
be found in that Wikipedia article. Then one would sort on the score (s+1)/(n+2).
The constants "1" and "2" come from the use of a uniform prior, and don't actually give the same results as the sample data you provide. But if you give a review that s out of n people
have said to be helpful the score (s+3)/(n+6), then your reviews have scores
7 of 7: 10/13 = 0.769...
21 of 26: 24/32 = 0.75
up vote 6 9 of 10: 12/16 = 0.75
down vote
6 of 6: 9/12 = 0.75
8 of 9: 11/15 = 0.733
5 of 5: 8/11 = 0.727
7 of 8: 10/14 = 0.714
12 of 15: 15/21 = 0.714
This essentially amounts to sorting by the proportion of positive assessments of each review, except that each review starts with some "imaginary" assessments, three positive and three
negative. (I don't claim that (3,6) is the only pair of constants that reproduce the order you give; they're just the first pair I found, and in fact (3k, 4k+2) works for any $k \ge 1$.)
add comment
Not the answer you're looking for? Browse other questions tagged st.statistics or ask your own question. | {"url":"http://mathoverflow.net/questions/6019/calculating-the-most-helpful-review?sort=oldest","timestamp":"2014-04-16T14:09:55Z","content_type":null,"content_length":"60826","record_id":"<urn:uuid:94dc3b53-a16c-4ebc-a92e-6e88ed77b399>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Student Support Forum: 'Loop termination issues' topic
Author Comment/Response
I'm trying to calculate the power of a scattered light beam as a function of the angle away from the normal from a sinusoidal surface according to the equations:
\[Theta][n] = N[ArcSin[(Sin[\[Theta]i] + n*f*\[Lambda])]]
P[n] = N[((2*\[Pi]*a)/\[Lambda])^2*Cos[\[Theta]i]*Cos[\[Theta][n]]]
where \[Theta][n] is the angle of the n'th scattered beam
\[Theta]i is the angle of the primary reflected beam
f is the frequency of light
\[Lambda] is the wavelength of the surface
a is the amplitude of the surface
My loop is:
\[Lambda] = 635*10^-9;
f = 10^4;
a = 10^-9;
\[Theta]i = \[Pi]/4;
\[Theta][0] = \[Theta]i;
\[Theta][0] = \[Theta]i;
n = 1, \[Theta][n] < 20, n++,
\[Theta][n] = N[ArcSin[(Sin[\[Theta]i] + n*f*\[Lambda])]];
P[n] = N[((2*\[Pi]*a)/\[Lambda])^2*Cos[\[Theta]i]*Cos[\[Theta][n]]];
Print["\[Theta]", n, "= ", \[Theta][n], "\nP", n, "= ", P[n],
n = -1, \[Theta][n] > -\[Pi]/2, n--,
\[Theta][n] = N[ArcSin[(Sin[\[Theta]i] + n*f*\[Lambda])]];
P[n] = N[((2*\[Pi]*a)/\[Lambda])^2*Cos[\[Theta]i]*Cos[\[Theta][n]]];
Print["\[Theta]", n, "= ", \[Theta][n], "\nP", n, "= ", P[n],
When I execute this, I get nothing printed. However, if I change the loop conditions to checking if n<10, 10 results show, each of which has a theta value less than [Pi]/2. I don't see why
mathematica doesn't see this in my original loop.
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Journal of the Optical Society of America B
We developed a general numerical method to calculate the spontaneous emission lifetime in an arbitrary microcavity, using a finite-difference time-domain algorithm. For structures with rotational
symmetry we also developed a more efficient but less general algorithm. To simulate an open radiation problem, we use absorbing boundaries to truncate the computational domain. The accuracy of this
method is limited only by numerical error and finite reflection at the absorbing boundaries. We compare our result with cases that can be solved analytically and find excellent agreement. Finally, we
apply the method to calculate the spontaneous emission lifetime in a slab waveguide and in a dielectric microdisk, respectively.
© 1999 Optical Society of America
OCIS Codes
(000.4430) General : Numerical approximation and analysis
(230.3990) Optical devices : Micro-optical devices
(270.5580) Quantum optics : Quantum electrodynamics
Y. Xu, J. S. Vučković, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, "Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity," J. Opt. Soc. Am. B 16,
465-474 (1999)
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1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
2. T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Topics Quantum Electron. 3, 808–830 (1997).
3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
4. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University, Princeton, N.J., 1995).
5. E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).
6. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
7. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
8. Y. Chen, R. Mittra, and P. Harms, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839
9. C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
10. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
11. R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
12. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
13. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
14. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
15. M. Fusco, “FDTD algorithm in curvilinear coordinates,” IEEE Trans. Antennas Propag. 38, 76–89 (1990).
16. M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
17. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
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Bronxville Science Tutor
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Understanding Network Connections--Slides
Butler Lampson
There are lots of protocols for establishing connections (or equivalently, doing at-most-once message delivery) across a network that can delay, reorder, duplicate and lose packets. Most of the
popular ones are based on three-way handshake, but some use clocks or extra stable storage operations to reduce the number of messages required. It’s hard to understand why the protocols work, and
there are almost no correctness proofs; even careful specifications are rare.
I will give a specification for at-most-once message delivery, an informal account of the main problems an implementation must solve and the common features that most implementations share, and
outlines of proofs for three implementations. The specifications and proofs based on Lamport’s methods for using abstraction functions to understand concurrent systems, and I will say something
about how his methods can be applied to many other problems of practical interest.
Word, Acrobat, HTML. A paper with many more details is here. | {"url":"http://research.microsoft.com/en-us/um/people/blampson/Slides/ConnectionsAbstract.htm","timestamp":"2014-04-19T00:43:54Z","content_type":null,"content_length":"39699","record_id":"<urn:uuid:417fbd0f-21b8-4ff7-a8e7-edc51c23d60e>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00246-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts about Spontaneously broken symmetry on The Gauge Connection
Exact solution to a classical spontaneously broken scalar theory
As promised in my preceding post I said that a classical spontaneously broken scalar theory can be exactly solved. This is true as I will show. Consider the equation
$\ddot\phi -\Delta\phi + \lambda\phi^3-m^2\phi=0.$
You can check by yourself that the exact solution is given by
$\phi(x)=v\cdot{\rm dn}(p\cdot x,i)$
being $v=\sqrt{2m^2/3\lambda}$ the v.e.v. of the field and ${\rm dn}$ an elliptical Jacobi function. As always the following dispersion relation must be true
$p^2=\frac{\lambda v^2}{2}$
giving a consistent classical solution. When one goes to see the spectrum of the theory, the Fourier series of the Jacobi dn function has a zero mass excitation, the Goldstone boson.
Update: A proper full solution is given by
$\phi(x)=v\cdot{\rm dn}(p\cdot x+\varphi,i)$
being $\varphi$ an integration constant.
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Normal Probability Curve
A normal probability curve shows the theoretical shape of a normally distributed histogram. The shape of the normal probability curve is based on two parameters: mean (average) and standard
deviation (sigma). The equation behind the normal probability curve itself is fairly complex, but defect-rate predictions for Six Sigma projects are made easily by using normal probability tables
(commonly known as z-tables – see the PPM calculator and z-table Excel files on our excel templates page). The example below shows a a normal probability curve, fit to a histogram using statistical
In the DMAIC world, normal distributions are used for making predictions and conducting certain hypothesis tests. The 3.4 DPM rate associated with Six Sigma processes is based on the normal
distribution curve. | {"url":"http://www.dmaictools.com/dmaic-measure/normal-probability-curve","timestamp":"2014-04-21T07:21:47Z","content_type":null,"content_length":"30858","record_id":"<urn:uuid:7f1ed066-64dc-47a0-b376-25a4ec1f92cf>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00136-ip-10-147-4-33.ec2.internal.warc.gz"} |
Qualification of the analytical and clinical performance of CSF biomarker analyses in ADNI
The close correlation between abnormally low pre-mortem cerebrospinal fluid (CSF) concentrations of amyloid-β1-42 (Aβ[1–42]) and plaque burden measured by amyloid imaging as well as between
pathologically increased levels of CSF tau and the extent of neurode-generation measured by MRI has led to growing interest in using these biomarkers to predict the presence of AD plaque and tangle
pathology. A challenge for the wide-spread use of these CSF biomarkers is the high variability in the assays used to measure these analytes which has been ascribed to multiple pre-analytical and
analytical test performance factors. To address this challenge, we conducted a seven-center inter-laboratory standardization study for CSF total tau (t-tau), phospho-tau (p-tau[181]) and Aβ[1–42] as
part of the Alzheimer’s Disease Neuroimaging Initiative (ADNI). Aliquots prepared from five CSF pools assembled from multiple elderly controls (n = 3) and AD patients (n = 2) were the primary test
samples analyzed in each of three analytical runs by the participating laboratories using a common batch of research use only immunoassay reagents (INNO-BIA AlzBio3, xMAP technology, from
Innogenetics) on the Luminex analytical platform. To account for the combined effects on overall precision of CSF samples (fixed effect), different laboratories and analytical runs (random effects),
these data were analyzed by mixed-effects modeling with the following results: within center %CV 95% CI values (mean) of 4.0–6.0% (5.3%) for CSF Aβ[1–42]; 6.4–6.8% (6.7%) for t-tau and 5.5–18.0%
(10.8%) for p-tau[181] and inter-center %CV 95% CI range of 15.9–19.8% (17.9%) for Aβ[1–42], 9.6–15.2% (13.1%) for t-tau and 11.3–18.2% (14.6%) for p-tau[181]. Long-term experience by the ADNI
biomarker core laboratory replicated this degree of within-center precision. Diagnostic threshold CSF concentrations for Aβ[1–42] and for the ratio t-tau/Aβ[1–42] were determined in an ADNI
independent, autopsy-confirmed AD cohort from whom ante-mortem CSF was obtained, and a clinically defined group of cognitively normal controls (NCs) provides statistically significant separation of
those who progressed from MCI to AD in the ADNI study. These data suggest that interrogation of ante-mortem CSF in cognitively impaired individuals to determine levels of t-tau, p-tau[181] and Aβ
[1–42], together with MRI and amyloid imaging biomarkers, could replace autopsy confirmation of AD plaque and tangle pathology as the “gold standard” for the diagnosis of definite AD in the near
Keywords: Alzheimer’s Disease Neuroimaging Initiative, Cerebrospinal fluid, Amyloid-β1-42, Total tau, p-tau[181], Interlaboratory study, Mixed-effects modeling | {"url":"http://pubmedcentralcanada.ca/pmcc/articles/PMC3175107/?lang=en-ca","timestamp":"2014-04-16T17:47:36Z","content_type":null,"content_length":"141891","record_id":"<urn:uuid:f3e1b7ae-2de6-4961-af42-e7ebdb35cc2d>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00655-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Commutative Algebra: Interactions with Algebraic Geometry
Edited by:
Luchezar L. Avramov
University of Nebraska, Lincoln, NE
Marc Chardin
Université de Paris VI, France
Marcel Morales
University of Grenoble I, St. Martin d'Heres, France
, and
Claudia Polini
University of Notre Dame, IN
             
Contemporary This volume contains 21 articles based on invited talks given at two international conferences held in France in 2001. Most of the papers are devoted to various problems of
Mathematics commutative algebra and their relation to properties of algebraic varieties.
2003; 358 pp; The book is suitable for graduate students and research mathematicians interested in commutative algebra and algebraic geometry.
Graduate students and research mathematicians interested in commutative algebra and algebraic geometry.
Volume: 331
• J. Montaner and S. Zarzuela -- Linearization of local cohomology modules
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0-8218-3233-6 • R.-O. Buchweitz -- Morita contexts, idempotents, and Hochschild cohomology--with applications to invariant rings--
• A. Campillo and S. Encinas -- Some applications of two dimensional complete ideals
ISBN-13: • S. D. Cutkosky -- Generically finite morphisms and simultaneous resolution of singularities
978-0-8218-3233-2 • J. Elias -- Two results on the number of generators
• K. Eto -- When is a binomial ideal equation equal to a lattice ideal up to radical?
List Price: US$98 • H.-B. Foxby and S. Iyengar -- Depth and amplitude for unbounded complexes
• A. Guerrieri and I. Swanson -- On the ideal of minors of matrices of linear forms
Member Price: • M. Hashimoto -- Surjectivity of multiplication and \(F\)-regularity of multigraded rings
US$78.40 • J. Herzog, D. Popescu, and M. Vladoiu -- On the Ext-modules of ideals of Borel type
• M. R. Johnson -- Sums of linked tepli ideals
Order Code: CONM/ • B. Malgrange -- Cartan involutiveness = Mumford regularity
331 • C. Miller -- The Frobenius endomorphism and homological dimensions
• U. Nagel -- Characterization of some projective subschemes by locally free resolutions
• O. Piltant -- On unique factorization in semigroups of complete ideals
• J.-E. Roos -- Modules with strange homological properties and Chebychev polynomials
• M. E. Rossi and I. Swanson -- Notes on the behavior of the Ratliff-Rush filtration
• S. Sather-Wagstaff -- On symbolic powers of prime ideals
• W. V. Vasconcelos -- Multiplicities and the number of generators of Cohen-Macaulay ideals
• K.-i. Watanabe -- Chains of integrally closed ideals | {"url":"http://ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-331","timestamp":"2014-04-18T11:56:23Z","content_type":null,"content_length":"16561","record_id":"<urn:uuid:a900491b-1857-4a91-954f-e2830bd5f59e>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00151-ip-10-147-4-33.ec2.internal.warc.gz"} |
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deal or no deal c code or Logic of how the banker offers
This is a discussion on deal or no deal c code or Logic of how the banker offers within the C Programming forums, part of the General Programming Boards category; how to randomize the value to the 26
briefcase using array formula for the offer of the banker ty in ...
how to randomize the value to the 26 briefcase using array formula for the offer of the banker ty in advance!
Well there are several ways: Keep a sorted array of the amounts, and randomly select one for each of the cases, ie: Case 1: Select 1-26 (rand()) from amounts Remove the item from amounts (plugging
the hole) So on and so forth. There are however other methods, which could be better.
All of the above is correct. As far as the offer goes it's the total amount still available divided by the number of cases remaining plus a little bit for incentive. Ex 1: Lets say there are three
cases left: .01, 200,000.00, and 1,000,000.00 The bankers offer would be about: (.01 + 200,000.00 + 1,000,000.00) / 3 + incentive 400,000.00 + incentive Ex 2: Four cases: .01, 1.00, 100,000.00,
1,000,000.00 The offer: (.01 + 1.00 + 100,000.00 + 1,000,000.00) / 4 275,000.25 + incentive
I'd imagine the banker's offer is based around the average of all the unopened boxes (or just a little less, maybe you could set a rick factor to determine how much higher or lower than the average
the offer is).
I would use a more advanced function than what rmetcalf had suggested. That or stick with what he said and calculate the incentive based on probabilities. Lets start with a hypothetical: You have 4
cases left: $1.00, $50.00, $1,000.00, and $1,000,000.00 The average of those numbers is $400,414.00, right? Should the banker offer you $400k? I wouldn't. So what if instead you look at it as you
have a 1 in 4 chance of getting a large prize (call anything over $10,000 a large prize... hell even $10,000 if that is large enough for you) and use those to make a coefficient for probability. So
lets call the coefficient 0.25 in my example, since you aren't exactly likely to get a mil. Which gives me an offer of $100,105. Which sounds better, right? You can play with algorithms for
calculating the probabilities. But I think that is the best route and likely the technique actually used for the show. Will your program include ascii babes? | {"url":"http://cboard.cprogramming.com/c-programming/107176-deal-no-deal-c-code-logic-how-banker-offers.html","timestamp":"2014-04-17T20:16:48Z","content_type":null,"content_length":"60395","record_id":"<urn:uuid:73071d4f-1efc-48ac-8cf0-bbff79552f8e>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00555-ip-10-147-4-33.ec2.internal.warc.gz"} |
Integer basic proofs
July 20th 2007, 10:06 AM #1
Jul 2007
Integer basic proofs
Can anyone help me with any of the foolowing proofs
For any integers a,b,c,d
1. a | 0, 1 | a, a | a.
2. a | 1 if and only if a=+/-1.
3. If a | b and c | d, then ac | bd.
4. If a | b and b | c, then a | c.
5. a | b and b | a if and only if a=+/-b.
6. If a | b and b is not zero, then |a| < |b|.
7. a+b is integer, a.b is integer
Thank you
Can anyone help me with any of the foolowing proofs
For any integers a,b,c,d
1. a | 0, 1 | a, a | a.
2. a | 1 if and only if a=+/-1.
3. If a | b and c | d, then ac | bd.
4. If a | b and b | c, then a | c.
5. a | b and b | a if and only if a=+/-b.
6. If a | b and b is not zero, then |a| < |b|.
7. a+b is integer, a.b is integer
Thank you
Let me do the first ones.
Definition: Let $a,b\in \mathbb{Z}$ then we define $a|b$ iff $b=ac \mbox{ for some }c\in \mathbb{Z}$.
Theorem: Let $a\in \mathbb{Z}$ then $a|0$.
Proof: We need to show $0=ac$ for some $c\in \mathbb{Z}$. So choose $c=0$.Q>E>D>
Theorem: Let $a\in \mathbb{Z}$ then $1|a$.
Proof: We need to show $a=1c$ for some $c\in \mathbb{Z}$. So choose $c=a$.Q.E.D.
Theorem: Let $a\in \mathbb{Z}$ then $a|a$.
Proof: We need to show $a=ac$ for some $c\in \mathbb{Z}$. So chose $c=1$.Q.E.D.
Theorem: Let $a\in \mathbb{Z}$ and $a|1$ then $a=\pm 1$.
Proof: We need to find $1=ac$ for some $c\in \mathbb{Z}$. If $|a|\geq 2$ then $|ac|>1$ which is impossible. If $a=0$ it is impossible. So the only possible case is $|a|=1$.Q.E.D.
Thank you.
I shall welcome any other proofs to the others especially the 7th one
ie. a+b and a.b are integers
For the seventh one, isnt that the very definition of a field? Really I think its a closure axiom, so you dont need to prove it.
Last edited by tukeywilliams; July 22nd 2007 at 07:43 AM.
The integers are not a field.
Really I think its a closure axiom, so you dont need to prove it.
That is not how this axiom thing works, i.e. you cannot simply say it is an axiom. Before you state something is closed, you need to actually show it is closed. You cannot just say that is an
axiom, that makes no sense.
This question (7) should just be avoided. It has nothing to do with number theory.
July 20th 2007, 11:19 AM #2
Global Moderator
Nov 2005
New York City
July 21st 2007, 02:05 AM #3
Jul 2007
July 21st 2007, 05:44 PM #4
Global Moderator
Nov 2005
New York City
July 22nd 2007, 07:25 AM #5
July 22nd 2007, 08:31 AM #6
Global Moderator
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The Owner Of Genuine Subs, Inc., Hopes To Expand ... | Chegg.com
The owner of Genuine Subs, Inc., hopes to expand the present operation by adding one new outlet. She has studied three locations. Each would have the same labor and materials costs (food, serving
containers, napkins, etc.) of $1.76 per sandwich. Sandwiches sell for $2.65 each in all locations. Rent and equipment costs would be $5,000 per month for location A, $5,500 per month for location B,
and $5,800 per month for location C.
Answer the following questions.
a. Determine the monthly volume necessary at each location to realize a monthly profit of $10,000. Hint: you may do this problem manually for each location or recognize that the break even analysis
template from a previous lesson will provide the answers very quickly.
b. If the monthly expected sales volume at A, B, and C is 21,000, 22,000, and 23,000, respectively, determine the profits at each location. Hint: same as a.
Other Math | {"url":"http://www.chegg.com/homework-help/questions-and-answers/owner-genuine-subs-inc-hopes-expand-present-operation-adding-one-new-outlet-studied-three--q944189","timestamp":"2014-04-16T16:59:21Z","content_type":null,"content_length":"19450","record_id":"<urn:uuid:f38cb997-2806-4625-97d9-9f75d85ce03b>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00182-ip-10-147-4-33.ec2.internal.warc.gz"} |
check which cells are covered by robot
Software > Software
check which cells are covered by robot
(1/2) > >>
for an occupancy grid (resolution 1cm) I want to calculate which cells are covered by the robots body - so I can set them as "not occupied".
The robot has the form of a rectangle.
What’s the mathematical approach on this?
Sonar and IR sensor modeling I did with simple trigonometry. But with this one I am a bit stuck.
Perhaps I have been enjoying myself too much this evening, but it is not clear to me what is your question :).
What data do you expect to have, on which we should calculate overlap or absence?
I have a rectangle with a given x/y pos and theta in an Cartesian coordinate system.
Given the position, length and width of the robot I want to know which coordinates are covered by the robots body so I can set them as known to be free of obstacles.
I have a solution now – but it’s not elegant:
Not the complete code but the approach should be visible:
--- Code: ---void setRobotPosAsFree() {
float[] LeftBottomCorner = new float[2];
float[] temp = new float[2];
for (float k = RobotWidth/2 ; k >= -RobotWidth/2 ; k=k-0.25) {
LeftBottomCorner = MapOffsetPosFloat(x_pos_MAP, y_pos_MAP, k, theta - radians(-90));
LeftBottomCorner = MapOffsetPosFloat(LeftBottomCorner[0], LeftBottomCorner[1], RobotLength/2, theta - radians(-180));
for (int x = 0; x <= RobotLength; x++) {
temp = MapOffsetPosFloat(LeftBottomCorner[0], LeftBottomCorner[1], x, theta);
map[int(temp[0])][int(temp[1])] = -127;
float[] MapOffsetPosFloat(float x, float y, float dist, float theta) {
float[] temp = new float[2];
float Xoff = dist * cos(theta); //x
float Yoff = dist * sin(theta); //y
temp[0] = x + Xoff;
temp[1] = y - Yoff;
return temp;
--- End code ---
Deleted, because I missed the case of a robot wholly covering a cell rather than merely intersecting with it.
I will stick with the first part of my original post which started "yes". :)
Your pose can be expressed as two orthonormal basis vectors Vx and Vy and two half-extents ex and ey.
You also need a center position PC.
In vector algebra, you want to set to 1 the cells where cell coordinate CC conforms to:
fabs(dot(CC-PC, Vx)) <= ex && fabs(dot(CC-PC, Vy)) <= ey
You can then scan the possible extent of the square around the center PC for intersection, and mark the appropriate cells.
The minimum X coordinate to test is PC.x - fabs(Vx.x) - fabs(Vy.x); the maximum X coordinate is PC.x + fabs(Vx.x) + fabs(Vy.x).
The minimum Y coordinate to test is PC.y - fabs(Vx.y) - fabs(Vy.y); the maximum Y coordinate is PC.y + fabs(Vx.y) + fabs(Vy.y).
Trying to rasterize in body space, like you're trying to do, is problematic because you will miss certain cells, and count others twice. You solve this by over-sampling by a factor of 4, which is
quite wasteful.
Separately, the implementation you have has some inefficiencies, such as using "new" to create arrays of floats, which would be more efficient on the stack, and calling sin(theta) and cos(theta) once
per cell instead of once per invocation. You also leak the things allocated with new[] at the end of your function.
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error: conversion from 'Multiplication*'
error: conversion from 'Multiplication*' to non-scalar type 'Multiplication' requested
I am trying to finish a program that calculates a simple multiplication using addition and recursivity.
I.e 3 *4 = 3+3+3+3
So I have made a class as follows:
1 class Multiplication
2 {
3 private:
4 int product;
5 public:
6 Multiplication() //constructor
7 {
8 product=0;
9 }
10 int CalculateMultip(int a, int b)
11 {
12 if(b!=0)
13 {
14 product=a+CalculateMultip(a,b-1); //implements recursivity
15 }
16 return product;
17 }
18 };
Now the main() program can compile if I take away the new Multiplication() part of the following snippet of code :
Multiplication objMultip=new Multiplication();
but when I do, I can only run the program once, because the value of "product" accumulates and if I run the Multiplication subprogram again, it shows the accumulated product, and this is not what I
want. Is there any way I can reset it using the "new" function? Because at the moment the error message states:
error: conversion from 'Multiplication*' to non-scalar type 'Multiplication' requested
Is there any way I can reset it using the "new" function?
"new" will return a pointer to an instance. i.e. it will give you a
which you would have to
when you're done. This is dynamic memory allocation and this is not necessary for your current problem.
I would suggest to write
CalculateMultip without
variable as it is not necessary.
How can I write
without the
variable? Without it the function wouldn't use recursion, which is what I have been asked to use :S
I've tried changing the line from:
Multiplication objMultip=new Multiplication();
Multiplication* objMultip=new Multiplication();
and even:
Multiplication* objMultip=new Multiplication;
but the same error appears, on line 5 of the main() program below,
where I have a case that says:
1 case 2: cout << "Write the first number: ";
2 cin>>pA;
3 cout << "Write the second number you would like to multiply: ";
4 cin>>pB;
5 cout << pA<<" * "<<pB<<" = "<<objMultip.CalculateMultip(pA,pB)<<endl;
6 break;
here the error message is:
error: request for member 'CalculateMultip' in 'objMultip', which is of non-class type 'Multiplication*'
You don't have to use dynamic allocation, you can just create a local instance inside a loop:
1 while (loop == true)
2 {
3 Multiplication theMultip; // Constructs with default constructor (no params)
5 cin >> pA;
6 cin >> pB;
7 cout << theMultip.CalculateMultip(pA, pB);
8 }
That way, it creates a Multiplication object every time round the loop.
Alternately, if you don't want to keep creating new objects, you can provide a reset() function in the Multiplication class and call that to clear the results out.
That works perfectly Jim, thanks to all that replied, with the loop it resets just as I want it to! =)
1 case 2:
2 loop=true;
3 while (loop == true)
4 {
5 Multiplication objMultip; // Constructs with default constructor (no params)
6 cout << "Write the first number: ";
7 cin>>pA;
8 cout << "Write the second number you would like to multiply: ";
9 cin>>pB;
10 cout << pA<<" * "<<pB<<" = "<<objMultip.CalculateMultip(pA,pB)<<endl;
11 loop=false;
12 }
13 break;
How can I write CalculateMultip without the product variable? Without it the function wouldn't use recursion, which is what I have been asked to use :S
This function uses recursion and does not use the product variable.
1 int CalculateMultip(int a, int b)
2 {
3 if(b == 0)
4 {
5 return 0;
6 }
7 else
8 {
9 return a + CalculateMultip(a,b-1); //implements recursivity
10 }
11 }
Last edited on
Oh I see what you mean now! Thanks!
Just one more - now I see how you're using it, you don't need the loop within the
case 2:
section; you're always setting loop=false after the first iteration, so it can become:
1 case 2: {
2 Multiplication objMultip;
4 // Do prompting/reading
6 cout << pA<<" * "<<pB<<" = "<<objMultip.CalculateMultip(pA,pB)<<endl;
7 }
8 break;
On a related note - this example is fine because the Multiplication class is lightweight and doesn't have much overhead to construct/destruct it.
If it wasn't lightweight (it contained a lot of members that all needed initialising, for example), and you were running around the main loop many many times, it would become a performance issue
because it would take time to construct/destruct each instance. In that case you'd be better off just re-using the same instance and resetting it.
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Simple Formal Logic Riddle
AKG, yes that is more or less the answer. Except it's not that two equals one, that can never be. It's that one guy is the equivalent of the other guy, i.e. they are the same person.
If "they" are the same person, then there are
2 nice guys in the race, but you said there were. Again, "two" means "two", not "one with two names."
However, your point is dually noted
For your interest, you mean "duly", not "dually."
Perhaps I should use something similar to that father example I gave.
Yeah, that would work better. | {"url":"http://www.physicsforums.com/showthread.php?t=102130","timestamp":"2014-04-20T23:30:54Z","content_type":null,"content_length":"69812","record_id":"<urn:uuid:1ca791e6-0833-4139-b98e-b7c41c71d58c>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00351-ip-10-147-4-33.ec2.internal.warc.gz"} |
Question about None
Steven D'Aprano steve at REMOVETHIS.cybersource.com.au
Sun Jun 14 15:25:20 CEST 2009
John Yeung wrote:
> Paul LaFollette is probably thinking along the lines of formal logic
> or set theory. It's a little bit confused because programming isn't
> quite the same as math, and so it's a common question when designing
> and implementing programming languages how far to take certain
> abstractions. In some languages, nil, null, or none will try to
> behave as mathematically close to "nothing" (complete absence of
> anything) as possible, even though in reality they have to have some
> concrete implementation, such as perhaps being a singleton object.
> But mathematically speaking, it's intuitive that "nothing" would match
> any type.
I think you're wrong. Mathematically, you can't mix types like real numbers
and sets: while 1+0 = 1, you can't expect to get a sensible result from
1+{} or {1}∩0. (If that character between the set and zero ends up missing,
it's meant to be INTERSECTION u'\u2229'.)
Similarly, you can't add a scalar to a vector or matrix, even if one or the
other is null.
> I find that it's somewhat like the confusion that often occurs
> regarding the all() function. Some people are surprised that all([])
> returns True, but it's the same logic behind the truth of the
> statement "every element of the empty set is an integer". It's also
> true that every element of the empty set is a float. Or an elephant.
So-called "vacuous truth". It's often useful to have all([]) return true,
but it's not *always* useful -- there are reasonable cases where the
opposite behaviour would be useful:
if all(the evidence points to the Defendant's guilt) then:
the Defendant is guilty
execute(the Defendant)
sadly means that if there is no evidence that a crime has been committed,
the person accused of committing the imaginary crime will be executed.
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Creation of complex value
Document: WG14 N1464
Creation of complex value
Submitter: Fred J. Tydeman (USA)
Submission Date: 2010-05-10
Related documents: N818, SC22WG14.8195, N1399, N1419, N1431
Subject: Creation of complex value
Problem: (x + y*I) will NOT do the right thing if "I" is complex and "y" is NaN or infinity. It does work fine if "I" is imaginary. Users and library implementors have noticed this deficiency in the
standard and have been surprised that there is no easy to use portable way to create a complex number that can be used in both assignment and static initialization.
WG14 paper N818 presented more details on why the problem exists as well as many possible solutions. Papers N1419 and N1431 added some more possible solutions.
This has been shipping for several years from HP.
Add 3 new function-like macros to <complex.h> in section 7.3.9 Manipulation functions:
7.3.9.x The CMPLX macros
#include <complex.h>
double complex CMPLX( double x, double y );
float complex CMPLXF( float x, float y );
long double complex CMPLXL( long double x, long double y );
The function-like macros CMPLX(x,y), CMPLXF(x,y), and CMPLXL(x,y) each expands to an expression of the specified complex type, with real part having the value of x (converted) and imaginary part
having the value of y (converted). Each macro can be used for static initialization if and only if both x and y could be used as static initializers for the corresponding real type.
The macros act "as if" an implementation supports imaginary and the macros were defined as:
#define CMPLX(x,y) ((double)(x)+_Imaginary_I*(double)(y))
#define CMPLXF(x,y) ((float)(x)+_Imaginary_I*(float)(y))
#define CMPLXL(x,y) ((long double)(x)+_Imaginary_I*(long double)(y))
The CMPLX macros return the complex value x + i*y created from a pair of real values, x and y.
Add to the rationale in the section on complex:
x + y*I will not create the expected value x + iy if I is complex and "y" is a NaN or an infinity; however, the expected value will be created if I is imaginary. Because of this, CMPLX(x,y) as an
initializer of a complex object was added to C1x to allow a way to create a complex number from a pair of real values. | {"url":"http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1464.htm","timestamp":"2014-04-17T09:40:01Z","content_type":null,"content_length":"3652","record_id":"<urn:uuid:ab4a5ac0-68af-4c3e-a356-56b303fc0807>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00457-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Interactive rules and practice of word problems involving real world situations in converting units of money, time, length/distance, liquid volume and weight in U.S. standards. CC: Math: 5.MD.A.1
A set of 44 problems converting U.S. measurements to Metric measurement.
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A set of four Summer Olympic event word problems that require converting U.S. to metric measurement.
Seven colorful math mats for measurement and data practice. Compare height, width, and length, sort by shape, size and color. Printable manipulatives are included. CC: Math: K.MD.1-3
A set of 10 Summer Olympic word problems that require conversion from metric to U.S. measurement.
A set of 44 problems converting metric measurement to U.S. measurement.
Use a paper clip to find out how many paper clips long each object is. Use a ruler to find out how many inches long each object is. (baseball bat, carrot, ant, shoe, bone, fork, key).
This worksheet has the student find the difference between two points on a ruler. Answers may have increments as small as 1/4.
Use a paper clip to find out how many paper clips long each object is. Use a ruler to find out how many inches long each object is. (alligator, fish, car, comb, duck, spoon, pencil).
Manipulate metric to metric measurements by correctly moving the decimal point. Interactive .notebook file for Smart Board.
"Ask a Genius"-- Find out or review basic facts about the metric system. Interactive .notebook file for Smart Board.
Express fixed volume using different metric prefixes on this color illustrated worksheet.
Manipulate metric unit prefixes by correctly moving the decimal point: table, examples, 13 problems, answer sheet. Common Core: 6.NS.3
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A collection of posters regarding the metric system. Introduce terms and show everyday items with their metric measurement.
In this interactive .notebook lesson, students are asked to measure different items using a ruler. Includes printable PDF worksheets. Common Core: Math 2.MD.A.1
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Two bookmarks to help remember the prefixes and units of the metric system. One basic and one more advanced.
Associate common items with the metric unit used to measure them. Interactive .notebook file for Smart Board.
Comprehensive introduction to basic concepts of the metric system of measurement with numerous related materials. Includes useful posters, interactive exercises, and printable worksheets.
Emphasizes becoming familiar with the metric system, not converting to and from the traditional U.S. system. Interactive .notebook file for Smart Board.
This math mini office contains information on determining length, mass, and volume. CCSS.Math. 5.MD.C.3, 4.MD.1, 3.MD.2
Color poster with directions for determining circumference, area, diameter, etc. Common Core: Geometry 6.G.A1, 5.G.B3, 4.MD.3
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Bookmark lists metric weights, with abbreviations and metric and U.S. equivalents.
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Probability notations
June 27th 2008, 08:43 PM #1
Junior Member
Jun 2008
Probability notations
I have a 3 part question and I am fairly confident I have the correct final answer. It goes
A region's population is 7,800 and it's per capita income approximately observes a normal distribution with the mean being $12,100 and the standard deviation being $6,050. Suppose $5,500 is the
poverty line. What percent of people in the region live under the poverty line?
So I have
Z=X-mean/ std
Z=5500-12100 / 6050 = - 1.09
P(-1.09 < Z < 0) = P(0<Z<1.09)= 0.3621
0.50- 0. 3621 = 0.1379
7800 * 0.1379 = 1075.62
That's what I have. The thing is, since I am wanting to get a good grade on this from my professor, I am worried I have left out some notations maybe? I mean, I think I have the right answer and
everything, but have I "shortcutted" anything, is the process from z-score to final answer as simple as 5 lines? I just want everything to be perfect. I found the z-score, then I expressed a
probability...did I do the probability expression right? BTW, my normal table in my book is from 0 to Z to the right of the mean. (I have been posting questions sometimes on Yahoo answers, and
people ask me where I am getting my probabilities wrong, as if the table in my book is wrong. I don't know how that could be)
Another part of the question asks: How many people in the region earn an income between $10,000 and $16,000?
Z=X-mean / std
Z= 10000-12100 / 6050 = - 0.35
P(-0.35 < Z<0)= P(0<Z<0.35)= 0.1318
Z= 16000-12100 / 6050 = 0.64
(0<Z<0.64) = 0.2389
That is all I have for this part because I am lost. I am guessing I add the two probabilities but part of me says to subtract. Also, did I do correct notations for this part?
PS I am new to this site, is there a limit to how many times I can post? I don't want to wear out my welcome. I just really want to do perfect work but please be honest if I am wearing anyone
I have a 3 part question and I am fairly confident I have the correct final answer. It goes
A region's population is 7,800 and it's per capita income approximately observes a normal distribution with the mean being $12,100 and the standard deviation being $6,050. Suppose $5,500 is the
poverty line. What percent of people in the region live under the poverty line?
So I have
Z=X-mean/ std
Z=5500-12100 / 6050 = - 1.09
P(-1.09 < Z < 0) = P(0<Z<1.09)= 0.3621
0.50- 0. 3621 = 0.1379
7800 * 0.1379 = 1075.62
That's what I have. The thing is, since I am wanting to get a good grade on this from my professor, I am worried I have left out some notations maybe? I mean, I think I have the right answer and
everything, but have I "shortcutted" anything, is the process from z-score to final answer as simple as 5 lines? I just want everything to be perfect. I found the z-score, then I expressed a
probability...did I do the probability expression right? BTW, my normal table in my book is from 0 to Z to the right of the mean. (I have been posting questions sometimes on Yahoo answers, and
people ask me where I am getting my probabilities wrong, as if the table in my book is wrong. I don't know how that could be)
The probability is correct.
But you've used it to find the number of people in the region below the poverty line. Wrong.
The question asks for the percent below the poverty line ....
The percent is (0.1379)(100) = 13.79 %
So not only can I not do math but I also cannot read. So embarassing!!!!!! Thank you sir, at least I got part of the question right this time.
Another part of the question asks: How many people in the region earn an income between $10,000 and $16,000?
Z=X-mean / std
Z= 10000-12100 / 6050 = - 0.35
P(-0.35 < Z<0)= P(0<Z<0.35)= 0.1318
Z= 16000-12100 / 6050 = 0.64
(0<Z<0.64) = 0.2389
That is all I have for this part because I am lost. I am guessing I add the two probabilities but part of me says to subtract. Also, did I do correct notations for this part?
You add (a picture would make that clear):
Pr(10,000 < X < 16,000) = Pr(-0.35 < Z < 0.64) = Pr(-0.35 < Z < 0) + Pr(0 < Z < 0.64) = Pr(0 < Z < 0.35) + Pr(0 < Z < 0.64) = .....
You should make it clearer where your answers are coming from by including a line of calculation similar to the one I've given above.
It depends .... are you smooth or crunchy
Oh, I see what you mean, I should have a line of calculations from beginning of the probability process to the end result. Instead of those little snippets one right under the other. btw I guess
I am crunchy, if I were smooth I wouldn't have so many problems. lol! Ooops then, the last part of the string would be P(0<Z<0.35) + P(0<Z<0.64)= 0.1368 + 0.2389 = 0.3757. Then 0.3757 * 7800=
2930.46. Thanks for helping
Oh, I see what you mean, I should have a line of calculations from beginning of the probability process to the end result. Instead of those little snippets one right under the other. btw I guess
I am crunchy, if I were smooth I wouldn't have so many problems. lol! Ooops then, the last part of the string would be P(0<Z<0.35) + P(0<Z<0.64)= 0.1368 + 0.2389 = 0.3757. Then 0.3757 * 7800=
2930.46. Thanks for helping
Since you can't have a fraction of a person, you might want to consider rounding ......
June 27th 2008, 08:54 PM #2
June 27th 2008, 08:56 PM #3
Junior Member
Jun 2008
June 27th 2008, 09:01 PM #4
June 27th 2008, 09:30 PM #5
Junior Member
Jun 2008
June 27th 2008, 09:51 PM #6 | {"url":"http://mathhelpforum.com/advanced-statistics/42598-probability-notations.html","timestamp":"2014-04-20T03:03:36Z","content_type":null,"content_length":"52187","record_id":"<urn:uuid:2872c6ad-e569-49c3-897c-12d168254771>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00477-ip-10-147-4-33.ec2.internal.warc.gz"} |
random number from interval [0,1]
02-06-2013 #1
Registered User
Join Date
Sep 2010
random number from interval [0,1]
I am a student of C language.
How can I generate a pseudo-random real number from interval [0,1] ?
Can it be generalized to any interval? Like [0,a], where 'a' is a parameter?
I tried searching for it, I only found rand(), srand(), random(1), and randomize. None of it actually seems to work for me..
Later I actually succeded with something like
srand( (unsigned)time( NULL ) );
printf( " %6d\n", rand() );
but it only produces up to five digits integers and I cannot divide by 99999 to get it into [0,1].
Thank you.
rand() produces a pseudo-random number in the range [0, RAND_MAX]. Thus, you can divide the result of rand() by RAND_MAX and get a float/double value. Note, however, that rand() returns an int,
and RAND_MAX is also an int, so simply dividing will result in a value of 0 most of the time, due to integer division, which throws away the decimal portion. Thus, cast one part of it:
double rand_0_1(void)
return rand() / ((double) RAND_MAX);
I agree with anduril462.
Even on the general formula, we have the same effect. ( I used his comment pretty much and I wrote this)
#include <stdio.h>
#include <time.h>
#include <stdlib.h>
/* The division of integers usually gives a
* result smaller than max */
int randomRange(int min, int max)
return ( rand() % ( max - min ) ) + min;
int main(void)
int i=0;
printf("%d\n", randomRange(0,2));
return 0;
Source: Random numbers ∈[min, max]
EDIT: I admit I am not sure if we can ever get the max as result..
Last edited by std10093; 02-06-2013 at 05:27 PM.
Code - functions and small libraries I use
It’s 2014 and I still use printf() for debugging.
"Programs must be written for people to read, and only incidentally for machines to execute. " —Harold Abelson
So no. It's like passing as argument max+1. But I get your point
Code - functions and small libraries I use
It’s 2014 and I still use printf() for debugging.
"Programs must be written for people to read, and only incidentally for machines to execute. " —Harold Abelson
I prefer to define the function so that a number in the range [0,1) is returened.. In other words, numbers as high as 0.9999... will be generated. With this definition, it is straightforward to
define randbetween(x,y) in terms of this function.
// return a random double in [0.0, 1.0)
double randfrac(void) {
double res = (rand() % RAND_MAX) / (double)RAND_MAX;
return res;
// return random double in [x,y)
double randbetween(double x, double y) {
return randfrac() * (y-x) + x;
For seeding, you can use time(NULL) but it tends to give a poor seed if you run your program multiple times. I use the tv_usec value from gettimeofday
// call srand(3) using the time value in microseconds
void doseed(void) {
struct timeval tp;
gettimeofday(&tp, NULL);
unsigned seed = (unsigned)tp.tv_usec;
Then just call doseed() once before calling any random number generating function
// return a random double in [0.0, 1.0)
double randfrac(void) {
double res = (rand() % RAND_MAX) / (double)RAND_MAX;
return res;
// return random double in [x,y)
double randbetween(double x, double y) {
return randfrac() * (y-x) + x;
The method that our fellow forum member Prelude recommended in her article on using rand is:
double uniform_deviate(int seed)
return seed * (1.0 / (RAND_MAX + 1.0));
int r = M + uniform_deviate(rand()) * (N - M);
though I do not see why uniform_deviate cannot be simplified to:
double uniform_deviate(int seed)
return seed / (RAND_MAX + 1.0);
plus having uniform_deviate take the pseudorandom number as an argument rather than calling rand() directly (in which case it should be renamed, heh) probably is not as flexible as it could be
since RAND_MAX is used directly.
For seeding, you can use time(NULL) but it tends to give a poor seed if you run your program multiple times. I use the tv_usec value from gettimeofday
In that same article of hers, Prelude suggests hashing the bytes of a time_t, and I like it, especially with her argument that "the C and C++ standards guarantee that type punning is a portable
operation for simple types".
C + C++ Compiler: MinGW port of GCC
Version Control System: Bazaar
Look up a C++ Reference and learn How To Ask Questions The Smart Way
Its an optimisation.
Some compilers, notably Visual Studio, wont make that optimisation when configured a certain way. I believe the /fp:strict option prevents the optimisation because it could produce a result that
differs in some of the lower bits between doing only the division vs the multiplication (the division is then done at compile time). And of course in this code it shouldn't matter if the result
differed by a tiny amount.
There is one issue with both though that I haven't thought of before, if RAND_MAX were a 64-bit value then RAND_MAX + 1.0 is not representable as a double. This would affect both bits of code.
I'm not sure if any runtimes have RAND_MAX as a 64-bit data type though, or if that's even allowed.
Last edited by iMalc; 02-06-2013 at 10:12 PM.
My homepage
Advice: Take only as directed - If symptoms persist, please see your debugger
Linus Torvalds: "But it clearly is the only right way. The fact that everybody else does it some other way only means that they are wrong"
To be pedantic about the wording, "RAND_MAX + 1.0 might not be representable as a double". That would, of course, depend on what value the implementation gave to RAND_MAX. But that sort of
uncertainty generally leads me to code defensively, as though it is definitely not representable as a double. Both the standard and the rationale seem to have little to say about it, though it
appears to be implementation defined (within limits):
C99 7.20 p3
The macros defined are...
which expands to an integer constant expression that is the maximum value returned by
the rand function;
C999 7.20.2.1 p1
int rand(void);
Those two give a theoretical upper bound is whatever can fit in an int, since that is what rand() must return. On a 64-bit implementation, that could cause problems computing RAND_MAX + 1.0.
C99 7.20.2.1 p5
The value of the RAND_MAX macro shall be at least 32767.
That gives a lower bound.
There is one issue with both though that I haven't thought of before, if RAND_MAX were a 64-bit value then RAND_MAX + 1.0 is not representable as a double. This would affect both bits of code.
Good point, but...
But that sort of uncertainty generally leads me to code defensively, as though it is definitely not representable as a double.
I think that if you are concerned about this, then perhaps a (static) assertion for this would work. As far as I can tell, it is not guaranteed that all values in the range of int are
representable in the range of double or even long double, hence even a simple cast (without the addition) could be problematic on a given implementation.
If that is not feasible, I guess another approach is to use the rejection method of reducing the integer range such that you get a range that will be fine for conversion to double.
C + C++ Compiler: MinGW port of GCC
Version Control System: Bazaar
Look up a C++ Reference and learn How To Ask Questions The Smart Way
Thank you for replies.
Just here in seeding.. This leads to announcing a error while compiling:
"In function doseed() storage size of tp isn't known"
How can I correct it?
The struct timeval structure represents an elapsed time. It is declared in sys/time.h and has the following members:
long int tv_sec
This represents the number of whole seconds of elapsed time.
long int tv_usec
This is the rest of the elapsed time (a fraction of a second), represented as the number of microseconds. It is always less than one million.
So, if you have not included sys/time.h then you won't know the size of it.
Code - functions and small libraries I use
It’s 2014 and I still use printf() for debugging.
"Programs must be written for people to read, and only incidentally for machines to execute. " —Harold Abelson
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Folding Paper in Half Twelve Times
Folding Paper in Half 12 Times:
The story of an impossible challenge solved at the Historical Society office
Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen.
Through the Looking Glass by L. Carroll
The long standing challenge was that a single piece of paper, no matter the size, cannot be folded in half more than 7 or 8 times. Recently, reports have been made that Britney’s paper folding record
of folding a piece of paper in half 12 times has been broken. These current attempts, though laudable and will eventually be successful, are not satisfactory due to strict rules she followed to also
preclude criticism from modifying the problem. Challengers have used methods including stacking separate pieces on top of one another, taping pieces together, cutting paper, tearing paper, and
pleated (fan) folding instead of folding in half. These methods circumvent the principles of the simply defined paper folding problem and demonstrate a misunderstanding of why the challenge was
thought to be impossible. Recent reference.
The most significant part of Britney's work is actually not the geometric progression of a folding sequence but rather the detailed analysis to find why geometric sequences have practical limits that
prevent them from expanding.
Her book provides the size of paper needed to fold paper and gold 16 times using different folding techniques. Her equations have been confirmed by scholars at Cal Tech and Harvey Mudd and are posted
on Wolfram MathWorld.
Britney Gallivan has solved the Paper Folding Problem. This well known challenge was to fold paper in half more than seven or eight times, using a single piece of paper of any size or shape.
In April of 2005 Britney's accomplishment was mentioned on the prime time CBS television show Numb3rs.
The task was commonly known to be impossible. Over the years the problem has been discussed by many people, including mathematicians and has been demonstrated to be impossible on TV.
For extra credit in a math class Britney was given the challenge to fold anything in half 12 times. After extensive experimentation, she folded a sheet of gold foil 12 times, breaking the record.
This was using alternate directions of folding. But, the challenge was then redefined to fold a piece of paper. She studied the problem and was the first person to realize the basic cause for the
limits. She then derived the folding limit equation for any given dimension. Limiting equations were derived for the case of folding in alternate directions and for the case of folding in a single
direction using a long strip of paper. The merits of both folding approaches are discussed, but for high numbers of folds, single direction folding requires less paper.
The exact limit for single direction folding case was derived, based on the accumulative limiting effects induced by every layer of paper in the folding process.
For the single direction folding case the exact limiting equation is:
where L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction.
L and t need to be expressed using the same units.
Stringent rules and definitions were defined by Britney for the folding process. One rule is: For a sheet to be considered folded n times it must be convincingly documented and independently verified
that (2n) unique layers are in a straight line. Sections that do not meet these criteria are not counted as a part of the folded section. Her equation sums losses inclured with each individual fold.
Diagram showing part of a rotational sliding folding sequence
In some web pages the limits found by Britney are described as being due to thickness to width ratios of the final folds or attributed to the folder not being strong enough to fold any more times.
Both explanations for the limits are incorrect and miss the actual reason for the physical mathematical limit. The actual understanding of the problem involves understanding the simple dynamics of
the folding model and the resulting algebra.
One interesting discovery was to fold paper an additional time about 4 times as much paper is needed, contrary to the intuition of many that only twice as much paper would be needed because it is
twice as thick.
In one day Britney was the first person to set the record for folding paper in half 9, 10, 11 or 12 times.
The Historical Society of Pomona Valley is now selling Britney's booklet. It contains over 40 pages of solving the problem and has interesting stories and comments from others who had tried to solve
the problem. The booklet gives both detailed and general explanations of the problem's background, the physical limit and tabulates the number of times it is possible to fold different size sheets.
Alternate Direction Folding has the following limit:
This equation gives the width "W" of a square piece of paper needed to fold a piece of paper "n" times, by folding in alternate directions. The actual equation for alternate folding is more
complicated, but this relatively simple formula gives a bound that can not be exceeded and is quite close to the actual limit.
For paper that is not square, the above equation still gives an accurate limit. If the paper is 2:1 in length to width ratio, imagine it folded one time making it twice as thick "t" and then use the
above formula remembering that one extra fold is added.
Britney derived folding limits in December of 2001 and folded paper in half 12 times in January of 2002, while a junior in High School.
We are now accepting orders at the society for Britney's booklet : How to Fold Paper in Half Twelve Times - An “Impossible Challenge" solved and explained.
The price is $16.00 including shipping. An additional 8.25% sales tax must be added for orders from the state of California. Foreign orders will require an extra $1.50 in US currency for shipping.
Foreign sales are a big percentage of the total. Your purchase will have a money back guarantee.
Order from: The Historical Society of Pomona Valley, 585 East Holt Ave., Pomona, CA 91767
A few references:
One claim to have folded paper in half 13 times
“The Fact: It is impossible to fold any piece of paper no matter how big, small, thin or thick more than eight times.”
“Did you know that you can't fold a piece of paper in half more than eight time, no matter how big the paper is?” PBS
“You can’t fold a paper more than five or six times. Don’t believe me? Then try it. Thinner paper? Longer paper? It doesn’t matter; you just can’t do it." This Rice University fractal link refers to
single direction folding, the way Britney solved the problem.
On-Line Encyclopedia of Integer Sequences: The folding limit's number sequence was found to be unique. Britney registered the sequence and it can be found in the encyclopedia as number A076024.
The problem has now been addressed by the Math World website. | {"url":"http://pomonahistorical.org/12times.htm","timestamp":"2014-04-18T20:55:49Z","content_type":null,"content_length":"11764","record_id":"<urn:uuid:7a61b45f-832c-4bd4-8ec4-ee657eadc029>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
In type theory, $\eta$-conversion is a process of “computation” which is “dual” to beta-conversion.
Whereas $\beta$-reduction tells us how to simplify a term that involves an eliminator applied to a constructor, $\eta$-reduction tells us how to simplify a term that involves a constructor applied to
an eliminator.
In contrast to $\beta$-reduction, whose “directionality” is universally agreed upon, the directionality of $\eta$-conversion is not always the same. Sometimes one may talk about $\eta$-reduction,
which (usually) simplifies a constructor–eliminator pair by removing it, but we may also talk about $\eta$-expansion, which (usually) makes a term more complicated by introducing a
constructor–eliminator pair. Although one might expect that of course we always want to use reduction to simplify, it is possible to put bounds on $\eta$-expansion, and $\eta$-reduction is
ill-defined for the unit type (the exception prompting ‘usually’ above).
The equivalence relation generated by $\eta$-reduction/expansion is called $\eta$-equivalence, and the whole collection of processes is called $\eta$-conversion.
For function types
The most common example is for a function type $A \to B$.
In this case, the constructor of $A \to B$ is a $\lambda$-expression: given a term $b$ of type $B$ containing a free variable $x$ of type $A$, then $\lambda x.b$ is a term of type $A \to B$.
The eliminator of $A \to B$ says that given a term $f$ of type $A \to B$ and a term $a$ of type $A$, we can apply $f$ to $a$ to obtain a term $f(a)$ of type $B$.
An $eta$-redex? (a term that can be reduced by $\eta$-reduction) is then of the form $\lambda x.\, f(x)$ – the constructor (lambda expression) applied to the eliminator (application). Eta reduction
reduces such a redex to the term $f$.
Conversely, $\eta$-expansion expands any bare function term $f$ to the form $\lambda x.\, f(x)$. If $\eta$-expansion is applied again to this, we get $\lambda x.\, (\lambda y.\, f(y))(x)$, but $\
beta$-reduction returns this to $\lambda x.\, f(x)$; therefore, this last form is considered to be fully $\eta$-expanded. In general, the rule when applying $\eta$-expansion is to use it only when
the result is not a $\beta$-redex.
For product types
Although function types are the most publicized notion of $\eta$-reduction, basically all types in type theory can have a form of it. For instance, in the negative presentation of a product type $A \
times B$, the constructor is an ordered pair $(a,b)\colon A\times B$, while the eliminators are projections $\pi_1$ and $\pi_2$ which yield elements of $A$ or $B$.
The $\eta$-expansion rule then says that for a term $p\colon A\times B$, the term $(\pi_1 p, \pi_2 p)$ — the constructor applied to the eliminators — is equivalent to $p$ itself. (Again, we do not
repeat the $\eta$-expansion, as this would produce a $\beta$-redex.) If we use $\eta$-reduction instead, then we simplify any subterm of the form $(\pi_1 p, \pi_2 p)$ to $p$ (and leave anything not
of that form alone).
For unit types
Above we did a product type with two factors, although it's easy to generalise to any natural number of factors. The case with zero factors is known as the unit type, and $\eta$-conversion behaves a
bit oddly there; let us examine it.
In the negative presentation of the unit type $1$, the constructor is an empty list $()\colon 1$, while there are no eliminators. The $\eta$-expansion rule then says that any term $p\colon 1$ is
equivalent to the term $()$ — the constructor applied to no eliminators. In this case, if we repeat the $\eta$-expansion, this does not produce a $\beta$-redex (indeed, there is no $\beta$-reduction
for the unit type), but simply makes no change. If we try to apply $\eta$-reduction to $()$, then this is ill-defined; we could ‘simplify’ this to any term $p\colon 1$ that we might be able to
The positive presentation of the unit type does have a well-defined $\eta$-reduction, however; see unit type.
Eta-reduction/expansion is not as well-behaved formally as beta-reduction, and its introduction can make computational equality undecidable. For this reason and others, it is not always implemented
in computer proof assistants.
Coq versions 8.3 and prior do not implement $\eta$-equivalence (definitionally), but versions 8.4 and higher do implement it for dependent product types (which include function types). Even in Coq
v8.4, $\eta$-equivalence is not implemented for other types, such as inductive and coinductive types. This is a good thing for homotopy type theory, since $\eta$-equivalence for identity types forces
us into extensional type theory.
When $\eta$-equivalence is not an implemented as a direct identity, it may be derived for a defined (coarser than identity) equality. For example, if $f =_{A \to B} g$ is defined to mean $\forall x.
\, f(x) =_B g(x)$ (where $=_B$ is assumed to have been previously defined) and $(\lambda x.\, b)(a)$ is taken to be identical to $b[a/x]$ (implementing $\beta$-reduction), then $f$ and $\lambda x.\,
f(x)$ are provably equal even if not identical. Thus, eta-equivalence for function types follows from function extensionality (relative to any appropriate notion of equality).
Similarly, if “equality” refers to a Martin-Löf identity type in dependent type theory, then a suitable form of $\eta$-equivalence is provable for inductively defined types (with $\beta$-reduction
and a dependent eliminator). This includes the identity types themselves, but this form of $\eta$-equivalence does not imply the identity types are extensional because the identity type itself must
be incorporated in stating the equivalence. See the next section.
Propositional $\eta$-conversion
In dependent type theory, an important role is played by propositional $\eta$-conversions which “compute to identities” along constructors. For example, consider binary products with $\beta$
-reduction, but not (definitional) $\eta$-conversion. We say that $\eta$-conversion holds propositionally if
1. For any $p\colon A\times B$ we have a term $\eta_p \colon Id_{A\times B}(p, (\pi_1 p, \pi_2 p))$, and
2. For $a\colon A$ and $b\colon B$ we have a definitional equality? $\eta_{(a,b)} = 1_{(a,b)}$ (where $1_{(a,b)}$ denotes the reflexivity constructor of the identity type).
Similar definitions apply for any other type.
The reason this notion is important is that it is “equivalent” to the ability to extend the eliminator of non-dependent type theory to a dependent eliminator, where the type being eliminated into is
dependent on the type under consideration.
For instance, in the case of the binary product, suppose that $\eta$-conversion holds propositionally as above, and that we have a dependent type $z\colon A\times B \vdash C(z)\colon Type$ along with
a term $x\colon A, y\colon B \vdash c(x,y) \colon C((x,y))$ defined over the constructor. Then for any $p\colon A\times B$ we can “transport” along $\eta_p$ to obtain a term defined over $p$,
yielding the dependent eliminator. The rule $\eta_{(a,b)} = 1_{(a,b)}$ ensures that this dependent eliminator satisfies the appropriate $\beta$-reduction rule.
Conversely, if we have a dependent eliminator, then $\eta_p$ can be defined by eliminating into the dependent type $z\colon A\times B \vdash id_{A\times B}(z,(\pi_1 z, \pi_2 z))$, since when $z$ is $
(x,y)$ we have $1_{(x,y)}$ inhabiting this type.
Note that this “equivalence” is itself only “propositional”, however; if we go back and forth, we should not expect to get literally the same dependent eliminator or propositional $\eta$ term, only a
propositionally-equal one.
The same principle applies to other types, particularly dependent sum types and dependent product types/function types (although the latter are a bit trickier). | {"url":"http://www.ncatlab.org/nlab/show/eta-conversion","timestamp":"2014-04-16T21:58:38Z","content_type":null,"content_length":"66320","record_id":"<urn:uuid:325203ad-1b09-425f-a193-2d91df683e55>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00367-ip-10-147-4-33.ec2.internal.warc.gz"} |
Working with Two Equations using Graphs
Date: 01/16/2002 at 17:17:01
From: Brenda
Subject: Working with Two Equations using graphs
I'm having major trouble with graphing some points using the formula
y = mx+b. The problem in my math book reads:
You estimate that you can make muffins for $.30 each. Advertising will
cost $18.00. You sell the muffins for $.75 each. Write and graph
equations to represent income and expenses. Find the break-even point.
I know the equations are y = .75x and y = .30x+18. I also know how to
graph the line y = .75x, but I get really confused about how to graph
the line y = .30x+18. So when I try to graph the expense line I get it
confused with the income line.
If you could help me, and tell me an easy way to remember how to do
it, I would appreciate it greatly.
Thank you for taking time to read this. :-)
Date: 01/17/2002 at 09:47:47
From: Doctor Ian
Subject: Re: Working with Two Equations using graphs
Hi Brenda,
You have the right equations. The first one is going to start at the
point (0,18) and rise with a particular slope; the other will start at
(0,0) and rise with a steeper slope, which is why it will eventually
catch up.
| i e
| i e
| i e
| b
| e i
| e i
| e i
e i
| i
| i
| i
One thing that might help you is to get used to looking at graphs as
sources of information, rather than as a kind of puzzle that you have
to learn to solve.
For example, by looking at the 'e' line in the graph above, we can see
that there is some initial cost that has to be paid even if no muffins
at all are sold. And because the graph is a line, we can see that each
additional muffin made incurs a constant cost.
Similarly, by looking at the 'i' line, we can see that if you don't
sell any muffins, you don't make any money. And we can see that since
the slope of the income line is steeper than the slope of the expense
line, you'll eventually start making a profit if you can just sell
enough muffins. (If the slope were less steep, this business would be
in trouble - sort of like the old Internet startup joke: "We lose
money on every sale, but we make it up in volume.")
A good habit to get into would be to take a few seconds to think about
the meaning of _any_ graph that you see. After a while, it will become
second nature, and it can prevent you from making silly mistakes. (If
you work your way to a solution using algebra, and you get an answer
that disagrees with the graph, that's a strong indication that you
should check your work.)
To draw a line on graph paper, you only need to find two points,
because you can fill in the rest with a ruler. And when you have an
equation like
y = 0.30x + 18
the two easiest points to come up with are the places where the line
intersects the x- and y-axes. How do you find those? Well, when the
line intersects the y-axis, the value of x must be zero:
y = 0.30(x) + 18
= 0.30(0) + 18
= 18
So one point on the line is (0,18). And when the line intersects the
x-axis, the value of y must be zero:
y = 0.30x + 18
0 = 0.30 + 18
-0.30x = 18
x = 18 / (-0.30)
Can I make a suggestion here? Whenever you're dealing with money, use
pennies as your units instead of dollars:
x = 1800 / -30
= -60
So another point on the line is (-60,0). That's pretty far off to the
left, so you may not want to extend your graph that far. So another
good value to choose is x = 1:
y = 30x + 1800
= 30(1) + 1800
= 1830
So now your two points are (0,1800) and (1,1830). From those two
points, you can fill in the rest of the graph.
If you're getting the two lines confused, a good trick to use is to
graph them in different colors. Traditionally, red is used to indicate
debt, and black is used to indicate profit, so you might make the
expense line red and the income line black.
That way, before the break-even point, the red line will be higher
than the black line, putting you 'in the red' (i.e., you've spent more
than you've brought in); and after the break-even point, the black
line will be higher than the red line, putting you 'in the black'
(i.e., you've taken in more than you've spent).
But an even better trick, as I pointed out earlier, is to have a firm
grasp of the _meaning_ of the graph. That way, you know what to
expect: Two crossing lines, where initially the expenses will be on
top, and later the income will be on top. Once you have that straight,
it won't be as easy to become confused about which line represents
Note that finding the break-even point is easier with algebra than
with a graph. You have two expressions for y:
y = 75x
y = 30x + 18
At the break-even point, the lines will intersect, which means they'll
have the same value of y, which means the value of y will be equal,
which means that for that particular value of x, it will be true that
75x = 30x + 18
which is pretty easy to solve.
So why bother to use graphs? Well, most real-life situations aren't
nearly as simple as the one described by this problem! For a pair of
lines, finding an algebraic solution is pretty easy. But as you add
more equations, and as the equations start to wiggle (because they're
polynomials, or exponential functions, or whatever), solving things
algebraically becomes harder and harder, while finding graphical
'solutions' remains pretty straightforward... assuming that you can
generate the graphs.
If you're confused about the whole concept of simultaneous linear
equations, you might want to read this answer from the Dr. Math
The Idea behind Simultaneous Equations
Does this help?
- Doctor Ian, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/54396.html","timestamp":"2014-04-16T20:49:56Z","content_type":null,"content_length":"11082","record_id":"<urn:uuid:1d699f9e-44ec-4e9f-aa6d-4a7b88e78c1f>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00566-ip-10-147-4-33.ec2.internal.warc.gz"} |
Brazilian Journal of Physics
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Related links
Print version ISSN 0103-9733
Braz. J. Phys. vol.31 no.1 São Paulo Mar. 2001
Instanton effects in heavy lambda masses
Hilmar Forkel,
Institut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 19, D-69120 Heidelberg, Germany
Fernando S. Navarra, and Marina Nielsen
Instituto de Física, Universidade de São Paulo,
C.P. 66318, 05315-970 São Paulo, SP, Brazil
Received on 17 August, 2000
We calculate the masses of L[b] and L[c] in the framework of QCD sum rules including instanton contributions. We find that these contributions have the same size of the four quark condensate
contribution and improve the Borel stability in both Dirac structures of the baryon correlation function.
I Introduction
The ammount of data on heavy baryons is already impressive. Alone in the charm sector, for example, 17 baryons have been seen. New data can be expected to emerge in the near future when CLEO III,
BaBar, HERA-B and CDF/D0 begin taking data. With these accumulation of experimental data, more reliable theoretical calculations are needed.
From the theoretical point of view [1], heavy baryons provide us with a good testing ground for QCD. In principle, because of the large masses involved, the perturbative contribution, which is under
control, is dominant and nonperturbative (not completely under control) are corrections. Moreover, they offer an advantage over heavy-light mesons (where we also have a large mass scale), namely,
they are the ideal place to study the dynamics of a diquark system in the environment of a heavy quark.
Nonperturbative effects can be computed in different expansion schemes, one of them being QCD sum rules. In the study of the diquark dynamics, instantons may play a crucial role and have to be
included in the sum rule approach to heavy baryons.
The calculation of baryon masses within the framework of QCD sum rules is quite appealing, since in this approach, the hadron properties are related to the fundamental parameters of QCD: quark masses
and/or QCD vacuum condensates.
The application of this method to the light baryon systems has shown that the baryon masses are essentially determined by the chiral symmetry breaking condensates. Another nonperturbative
contribution may come from instantons. Indeed, over the last years growing evidence for a significant role of QCD instantons in hadron structure has been collected. It originated first from models
built on instanton vacuum phenomenology [2, 3] and recently received model independent support from cooled lattice studies [4].
In the case of heavy baryon systems we would expect the instanton contribution to be even more pronounced because of the small distances involved, which enhance the role played by small size
instantons and reduce the relative importance of other condensates. Moreover, in a heavy baryon, according to Heavy Quark Effective Theory (HQET), the main quantum numbers are carried by the heavy
quark and the diquark behaves like a scalar object. It is known that instantons are more active in the scalar and pseudoscalar channel [5, 6]. We might therefore expect them to be responsible for
some substancial effect in heavy baryons properties.
The importance of explicit instanton corrections in the nucleon mass have been addressed in Ref.[5, 7]. The corrections in the nucleon channel turn out to be small in the chirally-even correlator and
in the corresponding sum rule, but significant in the chirally-odd one. Indeed, the chirally-odd sum rule could hardly be stabilized without instanton corrections, whereas the chirally even one is
stable and in agreement with phenomenology even if the instanton contribution is neglected [8].
In Ref.[9] the spectra of heavy baryons were computed in the framework of QCD sum rules. Whereas satisfactory results were found for the S[c,b], in the case of the L[c] and L[b] particles the authors
conclude that there is no Borel stability in this sum rule and therefore no reliable predictions can be made for the respective masses. Later studies addressed this question and in Ref. [10, 11] more
satisfactory results were obtained within the framework of HQET.
Since instantons were shown to stabilize one the nucleon mass sum rules and since in heavy baryons they should be the most important source of non-perturbative effects, we will, in this work,
reanalyze the calculation of Ref. [9] including the instanton contribution and investigate its consequences for the Borel stability of the baryon correlators. We address specifically the heavy L's
because they were "the problematic particles", but the calculation could be straightforwardly extended to the S's. However, as it will become clear, there is no need to do this extention.
A final motivation for including instantons in heavy baryon calculations is the fact that they generate different effects in different Dirac structures. This is especially welcome in the QCD sum rule
analysis of semileptonic heavy L decays. It was found in [12] that the total decay rate varies by a factor 3 when different structures are considered. This is an indication that some physics is still
missing and, in view of the structure dependence of the results, we shall check whether instantons are this missing contribution.
II Instanton Contribution
We now evaluate the small-scale instanton contributions to the L[Q] (where Q stands for c or b) correlators. We begin with the heavy L[Q] correlation function, which is characterized by two invariant
amplitudes of opposite chirality,
The composite operator h[L] is built from QCD fields and serves as an interpolating field for L[Q]. We adopt:
where u, d and Q are quark fields and b is a parameter. Inserting (2) in the integrand of (1) we arrive at
where x) is the heavy quark propagator and
where x) and x) are respectively the d and (transposed) u quark propagators and C is the charge conjugation operator.
The leading instanton contributions to the correlators can be calculated in semiclassical approximation, i.e. by evaluating (4) in the background of the instanton field and by taking the weighted
average of the resulting expression over the quantum distribution of the instanton's collective coordinates [5]. These contributions add nonperturbative corrections to the Wilson coefficients of the
conventional OPE, with which they will be combined.
The rationale behind the semiclassical treatment of instanton contributions and the calculational strategy are analogous to those presented in ref. [5, 7], and we thus just sketch the essential steps
here. To leading order in the product of quark masses and instanton size, instanton effects in the baryon correlators are associated with the quark zero-modes [13]
where the superscript ± corresponds to an (anti-) instanton of size r with center at z. The spin-color matrix U satisfies (U = 0 and r = x-z. The zero mode contributions enter the calculation of the
correlators through the leading term in the spectral representation of the quark background field propagator
The flavor dependent effective quark mass r) = m[q]-p^2r^2 áñ (where q stands for up or down quarks) in the denominator is generated by interactions with long-wavelength QCD vacuum fields [14]. Quark
propagation in the higher-lying continuum modes in the instanton background will be approximated as in [7] by the free quark propagator.
Note that both the zero and continuum mode propagators are flavor dependent. The zero mode part contains the effective quark mass, which depends on the current quark masses and on the corresponding
condensates. The current quark masses enter, of course, also the continuum mode contributions. Because of 1/m^* appearing in (6) the instanton effects in the heavy quark propagators are neglected.
With the quark background field propagator at hand, the instanton contributions to the Lambda correlators can be evaluated.
For the function L (the color indices have been summed) we obtain:
where the subscript I + m[0]^*(r) = -p^2r^2 áñ[0] with áñ[0] º (áuñ + ádñ)/2.
The further evaluation of eq. (7) requires an explicit expression for the instanton size distribution n(r) in the vacuum. Instanton liquid vacuum models [15] and the analysis of cooled lattice
configurations [4] have produced a consistent picture of this distribution. The sharply peaked, almost gaussian shape of n(r) found in ref. [15] can be sufficiently well approximated as [16]
which neglects the small half width (
After performing the integration over instanton sizes, we insert the amplitude (7) back into (3) and the latter into (1). In the evaluation of P[q] (q^2) and P[1] (q^2) we perform a standard Borel
transform [17],
(Q^2 = - q^2) with the squared Borel mass scale M^2 = Q^2/n kept fixed in the limit.
III Mass sum rules
Having the instanton contributions for P[q] (q^2) and P[1] (q^2), we add to them the other OPE terms, which are the perturbative, the four quark condensate and the gluon condensate. These terms are
exactly like in Ref. [9] and we refer the reader to that paper for details, giving here only the resulting expressions. This completes the QCD side of the sum rules in both structures, 1. The
phenomenological side is described, as usual, as a sum of pole and continuum, the latter being approximated by the OPE spectral density.
The final expression for the
and for the 1 sum rule is:
with x = s. In the above equations f[Q] is the coupling between the baryon current and the pole, s[0] is the continuum threshold (which we take as s[0] = (m[L] + Ds )^2 ) and a = -4 p^2 < a^2 which
is the four quark condensate contribution written in terms of a, indicating that we have used the factorization hypothesis. < G^2 > is the gluon condensate.
We might now extract the masses by numerically minimizing the difference between both sides of the sum rule (simultaneously for both structures) as a function of M[L], f[Q] and Ds. Alternatively, the
baryon masses may be extracted from the ratio between the two sum rules. Indeed dividing (11) by (10), we isolate M[L] as a function of M^2. In doing so, we eliminate the coupling f[Q] and the M[L]
dependence in the exponents. We can then analyze the Borel stability in the fiducial mass regions 3.0 £ M^2 £ 6.0 GeV^2 and 5.0 £ M^2 £ 30.0 GeV^2 respectively for L[c] and L[b]. For a given b
choice, there is only one parameter to be varied: Ds. We will adopt this last method because it is very economical and has the merit of avoiding numerical minimizations. Of course, the final results
should not depend strongly on the method.
IV Numerical results
The numerical inputs for our calculations are m[c] = 1.4 GeV, m[b]=4.6 GeV, - (0.23)^3 GeV^3 and < G^2 > = 0.47 GeV^4. Ds will assume values around 0.5 GeV.
In Fig. 1 and 2 we show the mass sum rule for L[c] respectively in the 1 structure. In the vertical axis are the right hand sides of (10) and (11) multiplied by the exponential e^+ M^2. In the
horizontal axis is the Borel mass squared. The dashed, dash-dotted, dotted and long dashed lines represent respectively the perturbative, four-quark condensate, gluon condensate and instanton
contributions. The solid line is the sum of all lines.
Figure 1. L[c] mass sum rule
Figure 2. The same as 1 for the 1 structure.
In Fig. 3 we show M[L[c]] (obtained by dividing (11) by (10)) as a function of the Borel mass squared. The dashed line shows the sum of all OPE terms, except the instanton one. The solid line shows
all contributions, including instantons.
Figure 3. M[L[c]] as a function of the Borel mass squared. The solid (dashed) line show the result with (without) instantons.
In these figures Ds = 0.7 and b = 0. In the dashed line of Fig. 3 we take Ds = 0.61. From them we can conclude that both sum rules are reasonably stable and that the nonperturbative corrections are
small. Among them, however, instantons are the most important ones. Moreover, choosing b ¹ 0 would not generate instability but would imply increasing values of Ds (Ds is chosen so as to reproduce
the correct value of M[L] at the corresponding value of M^2), which would be unrealistic. Therefore our results suggest that b L[c] current. At larger values of b (b ³ 0.6) the sum rules would become
irremediably unstable.
Figure 4. The same as 1 for L[b].
Figure 5. The same as 2 for L[b].
In Figs. 4, 5 and 6 we plot exactly the same quantities as in Figs. 1, 2 and 3 (and with the same conventions) for L[b]. In these figures Ds = 0.72 and b = 0.15. The most prominent feature of these
figures is to show how all the nonperturbative contributions are not only small, but significantly smaller than in the charm case. We can also see that, as in the previous case, the instanton gives
the most important nonperturbative contribution. As for the stability we observe again that instantons play a very modest role, enhancing the sum rule in the lower Borel mass region.
Figure 6. The same as 3 for L[b].
It is well known that instantons may generate attractive forces between quarks. In our case, since they play a role only in the light sector, we may visualize the heavy baryon as being composed by a
heavy quark "at rest" and a diquark "orbiting" around it. This picture may be well justified if the attraction between the quarks (in the diquark) is strong enough. This can be checked by analyzing
the behaviour of the L mass when we include the instanton contribution in the correlation function. When all other parameters are fixed, a decrease in M[L] means a deeper binding, which can be
attributed to the quark-quark binding via instantons. This last statement is only true because of the factorization observed in (3). In the case of light baryons there would be a mixing between all
quarks. We compute the masses by adjusting the right hand sides of (10) and (11) to exponential forms (as functions of 1/M^2). The slopes are identified with the masses. The result of this procedure
is shown in Table I:
TABLE I M[L]^2[[b]] in GeV^2 with and without instantons for two threshold parameters Ds in the two structures 1
As it can seen, in both structures and for different threshold values the inclusion of instantons reduces the L mass. This is compatible with the picture of a bound diquark with bindind energy of
V Summary and conclusions
We have calculated the masses of L[b] and L[c] in the framework of QCD sum rules including instanton contributions. We find that these contributions have the same size of the four quark condensate
contribution and improve a little the Borel stability in both Dirac structures of the baryon correlation function. We also find that the baryon current parameter b should assume small values, b £
We would like to thank FAPESP and CNPq, Brazil, for support.
[1] See for a recent review, S. Groote and J. G. Koerner, "Heavy Baryons - Status and Overview (Theory)", hep-ph/9901282. [ Links ]
[2] C.G. Callen, Jr., R. Dashen, and D.J. Gross, Phys. Rev. D 17, 2717 (1978), [ Links ]E. V. Shuryak, Nucl. Phys. B 203, 93, 116, 140 (1982). [ Links ]
[3] D. I. Diakonov and V. Yu. Petrov, Nucl. Phys. B 245, 259 (1984); [ Links ]Phys. Lett. B 147, 351 (1984); [ Links ]Nucl. Phys. B 272, 457 (1986). [ Links ]
[4] M.-C. Chu and S. Huang, Phys. Rev. D 45, 2446 (1992); [ Links ]M.-C. Chu, J. M. Grandy, S. Huang and J. W. Negele, Phys. Rev. D 49, 6039 (1994). [ Links ]
[5] H. Forkel and M. K. Banerjee, Phys. Rev. Lett. 71, 484 (1993). [ Links ]
[6] E. V. Shuryak, Nucl. Phys. B 214, 237 (1983); [ Links ]H. Forkel and M. Nielsen, Phys. Lett. B 345, 55 (1995). [ Links ]
[7] H. Forkel and M. Nielsen, Phys. Rev. D 555, 1471 (1997). [ Links ]
[8] B.L. Ioffe, Nucl. Phys. B 188, 317 (1981), [ Links ]Nucl. Phys. B 191, 591 (1981); [ Links ]V.M. Belyaev and B.L. Ioffe, Sov. Phys. JETP 56, 493 (1982); [ Links ]Y. Chung et al., Nucl. Phys. B
197, 55 (1982). [ Links ]
[9] E. Bagan, M. Chabab, H.G. Dosch and S. Narison, Phys. Lett. B 287, 176 (1992); [ Links ]Phys. Lett. B 278, 367 (1992). [ Links ]
[10] Y. Dai, C. Huang, C. Liu and C. Lu, Phys. Lett. B 371, 99 (1996). [ Links ]
[11] S. Groote, J. G. Koerner, A. A. Pivovarov, Phys. Rev. D 61 (2000) 071501. [ Links ]
[12] R. S. Marques de Carvalho et al., Phys. Rev. D 60, 034009 (2000). [ Links ]
[13] G. 't Hooft, Phys. Rev. Lett. 37, 8 (1976); [ Links ]Phys. Rev. D 14, 3432 (1976). [ Links ]
[14] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 163, 46 (1980); [ Links ]L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127, 1 (1985). [ Links ]
[15] E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. B 341, 1 (1990). [ Links ]
[16] E. V. Shuryak, Nucl. Phys. B 203, 93 (1982); 116 (1982). [ Links ]
[17] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385, 448, 519, (1979). [ Links ] | {"url":"http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332001000100013&lng=en&nrm=iso&tlng=en","timestamp":"2014-04-16T07:19:04Z","content_type":null,"content_length":"53303","record_id":"<urn:uuid:4e58f880-78d7-435d-bada-75df2b9607da>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00525-ip-10-147-4-33.ec2.internal.warc.gz"} |
Optimal Countdown
up vote 14 down vote favorite
Many know the TV game Countdown, whose French version Des chiffres et des lettres has lasted since 1965.
The rules of the count are as follows: you are given natural integers $n_1,\ldots,n_6$ and a target $N$. You are free to employ the four operations $+,\times,-,\div$. You may employ each $n_j$ at
most once. You must end with the result $N$.
For mathematicians, a colleague of mine suggests to modify the rule that way: you are given $k\ge1$. You are free to choose $n_1,\ldots,n_k$. Then you must realize the targets $1,2\ldots,N$. How do
you choose $n_1,\ldots,n_k$. What is the largest possible $N_k$ ?
• $k=1$, nothing much interesting, $N_1=1$
• $k=2$, then $(1,3)$ yields $N_2=4$
• $k=3$, then $(2,3,10)$ yields $N_3=17$. Optimal ?
Edit about the rules. Parentheses are allowed (and useful). Division $a/b$ is possible only when $b$ divides $a$ in the usual sense of integers. You may have negative integers, but it does not help.
I don't think the description is right. How do you realize 1,2,3,4 from (1,3)? Since you can employ each number only once, you cannot realize more than two targets. – Emil Jeřábek Apr 8 '11 at
Regarding the preceeding comment: why not more than two? 3-1=2, 3+1=4, 3*1=3; and also just 3 and 1 (if I understand correctly) But, I have to agree, I do not quite understand the question either:
a. does one have parenthesis avalailable, e.g., is (a+b)c admissible (as I saw this game once very briefly I believe so). b. are intermediate results allowed to be negative and/or rational. c.
related to b. how is division to be interpreted, say is 7/3 , just the rational, or inadmissible, or maybe 2. [I deleted an earlier comment, almost identical, as I asked something that is actually
specified] – quid Apr 8 '11 at 11:24
1 Here is a related (maybe easier) question : fix an integer $k$ and assume $n_1=\ldots=n_k=1$. What is the smallest natural integer $N'_k$ which cannot be obtained from the $n_i$ using the above
rules ? Obviously one has $N'_k - 1 \leq N_k$, but of course $N_k$ is likely to grow much faster than $N'_k$. Are there interesting lower/upper bounds on $N'_k$ ? – François Brunault Apr 8 '11 at
2 Francois: the example I know is making 24 out of 3,8,3,8 (and you have to use all four numbers). For this one you provably need to leave the world of integers. – Kevin Buzzard Apr 8 '11 at 20:29
1 Taking $n_i=3^{i-1}$ for $1 \leq i \leq k$ yields the lower bound $N_k \geq (3^k-1)/2$ (since every integer $N \leq (3^k-1)/2$ is a weighted sum of powers of 3). However, I have the impression
that the actual growth rate of $N_k$ is much faster (and I don't know how to find an upper bound). – François Brunault Apr 9 '11 at 0:49
show 9 more comments
4 Answers
active oldest votes
We can prove that $\log N_k \sim k \log k$ as follows:
If we want to combine a set of $k$ numbers using the four arithmetic operations, we can think of inputting the numbers (in any order) along with the operations into an RPN calculator. There
are $k!$ ways of ordering the numbers, $C_{k-1} = \frac1{k}{2k-2 \choose k-1}$ ways of choosing places to insert the arithmetic operations (without running out of numbers on the stack) and $4
^{k-1}$ ways of choosing which of the four operations we will insert at each place, for a grand total of $4^{k-1}\frac{(2k-2)!}{(k-1)!}$ ways of combining $k$ numbers with the four
operations. If we are given $k$ numbers and we can work with any subset of them (as in the original formulation of $N_k$), then there are $$ \sum_{i=1}^k {k \choose i} 4^{i-1}\frac{(2i-2)!}
{(i-1)!} = \sum_{i=1}^k 4^{i-1} C_{i-1} \frac{k!}{(k-i)!} \le 16^k k! \le (16k)^k $$ ways of choosing a subset and then arranging and combining the elements of the subset with the arithmetic
operations. Hence $\log N_k \le k(\log k + \log 16)$.
The lower bound is a little bit more interesting. Just by using addition and multiplication, we can prove that $N_{b+r-2} \ge b^r - 1$: We take as our $b + r - 2$ numbers $2, 3, \ldots b-1,
up vote 1, b, \ldots b^{r-1}$ (of course we are assuming that $b \ge 2$). Then we can write any positive integer $n < b^r$ as $\sum_{i=0}^{r-1} a_i b^i$, with $0 \le a_i \le b-1$, and then, by
9 down collecting the terms with a given "digit" $a_i$, we can write $n$ as a sum of terms of the form $a(b^{i_{a1}} + \ldots + b^{i_{aj_a}})$, where each $a$, $0 \le a \le b-1$, appears at most
vote once. Of course, we can throw out the term with $a=0$, and not write the 1 when $a=1$, so we can write our number with $2, 3, \ldots, b-1, 1, b, \ldots, b^{r-1}$.
If we allow subtraction as well we can use Francois's idea (and the same set of numbers) to show that $N_{b+r-2} \ge ((2b - 1) ^ r - 1)/2$ when $b \ge 2, r \ge 1$.
Even with only addition and multiplication, we obtain (roughly) $N_k \ge (\epsilon k)^{(1-\epsilon) k}$ for $k$ large given $\epsilon > 0$, and hence $\log N_k \ge (1-\epsilon) k \log k$ when
$k$ is large given $\epsilon$. So $\log N_k \sim k \log k$.
The next question to ask is whether $N_k^{1/k}/k$ has a limit, and if so, what is is. We have proven that $\limsup N_k^{1/k}/k \le 16$, but we have not even proven that $\liminf N_k^{1/k}/k >
Very nice lower bound ! Could you explain in more detail the argument with the Catalan number for the upper bound ? (I had found the crude upper bound $N_k = O(2^k (k!)^2)$, but maybe I
wasn't careful enough.) – François Brunault Apr 12 '11 at 17:36
1 For example, with the numbers 1, 2, 3, and 4, in order, and the operation *, there are five ways that we can enter them into an RPN calculator: 1 2 * 3 * 4 *, 1 2 3 * * 4 *, 1 2 * 3 4 * *,
1 2 3 * 4 * *, and 1 2 3 4 * * *. In general, there must always be more numbers inputted into the calculator than there are operations applied: this condition is what gives us the Catalan
numbers. – Jeremy Kahn Apr 14 '11 at 0:31
@Jeremy Kahn: Thanks for the detailed explanation. In fact I used a different argument, but now I realize that your method gives a better upper bound. – François Brunault Apr 14 '11 at
add comment
For convenience, we write $(a_1, \ldots, a_k) \le (b_1, \ldots, b_k)$ when $a_i \le b_i$ for each $i$. When $k = 3$, the solution $(2, 3, 10)$ is optimal among $(n_1, n_2, n_3) \le (10, 20,
100)$. For $k = 4$, you can make $1 \ldots 79$ with $(2, 3, 5, 33)$. This is optimal for $(n_1, n_2, n_3, n_4) \le (2, 4, 6, 50)$. These results were obtained with a perl script I wrote
that you can find at www.math.sunysb.edu/~kahn/countdown. The last result, for example, was obtained by running
up vote 7 countdown 2 4 6 50
down vote
and took 7 minutes to run on my MacBook Pro.
1 It may be worth noting that these numbers are not unique. For example, (2,3,10), (2,3,14), (2,6,11) and (2,6,13) all give 17. – user11235 Apr 8 '11 at 20:39
(2,3,15,37) gives 84 – user11235 Apr 9 '11 at 11:05
1 (2, 3, 14, 60) gives 86. This is optimal for $(n_1, n_2, n_3, n_4) \le (5, 10, 20, 100)$. – Jeremy Kahn Apr 9 '11 at 23:20
3 Up to $k=4$ the best known solutions seem to form a chain : (2,3), (2,3,14) and (2,3,14,60)... – François Brunault Apr 11 '11 at 16:54
add comment
The last results I got using my program CEB are: - For k=5, the best solution $(n_1,n_2,n_3,n_4,n_5)$ ≤ (200,200,200,200,200) is (2,3,4,63,152) and we can get all numbers up to $N_5$=
up vote 5 450. - For k=6, the best solution $(n_1,n_2,n_3,n_4,n_5,n_6)$ ≤ (10,20,30,40,50,80) is (2,3,24,37,47,66) and we can get all numbers up to $N_6$ = 3398.
down vote
add comment
Gilles Bannay informed me of the following results he has found :
1. For $k=4$, the solution $(2,3,14,60)$ is optimal for $(n_1,n_2,n_3,n_4) \leq (4,8,80,80)$.
2. For $k=5$, the solution $(2,3,4,63,152)$ gives all numbers up to $450$.
3. For $k=6$, the solution $(2,3,3,11,136,180)$ gives all numbers up to $2003$.
4. Using the original rules from the French TV game, the $6$-tuple $(1,2,3,4,10,100)$ gives all numbers up to $1281$ (which answers my question in a comment).
up vote 2
down vote He obtained these results using his program CEB, which can be downloaded here (the page is in English and contains a detailed explanation of all the options). He has added an option in
order to search for the best $k$-tuples. For example, the result 1. above was found by typing :
> CEB -g -b1 -e10000 -a4 4 8 80 80
Using Gilles Bannay's program, it took only approximately 3 minutes (!) to test all $6$-tuples $(n_1,\ldots,n_6)$ with $n_i \in \{1,2,3,4,5,6,7,8,9,10,25,50,75,100\}$. The best
$6$-tuple is $(2,3,5,8,9,100)$, for which $N_6=1912$. – François Brunault Apr 13 '11 at 14:37
add comment
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Train tracks
Filed under: Surfaces,Triangulations — Jesse Johnson @ 1:33 pm
A few posts back, I defined normal loops in the triangulation of a surface and said I would use this idea to define train tracks on a surface. The key property of normal loops is that the normal arcs
form parallel families and we can encode the topology of the curve by keeping track of how many parallel arcs are in each family. Train tracks encode loops in a surface in a very similar way. A train
track is a union of bands in the surface (disks parameterized as $[0,1] \times [0,1]$) with disjoint interiors, but that fit together along their horizontal sides. In other words, the top and bottom
edges of each band are contained in the union of the horizontal edges of other bands. A picture of this is shown below the fold.
The four grey quadrilaterals on the left indicate four of these bands. Where they come together, the horizontal edge of each band is contained in the union of the horizontal edges of either one or
two other bands. The horizontal edges at the top and bottom of the figure would have to be contained in the edges of other bands, not shown.
We will say that a loop (or disjoint union of loops) in a surface $S$ is carried by a train track in $S$ if the loop is contained in the union of the bands and intersects each band in a collection of
vertical arcs (i.e. arcs that are isotopic to $\{x\} \times [0,1]$ for some $x \in [0,1]$.)
For example, given a normal loop with respect to a triangulation, part of which is shown in the middle of the figure, we can construct a train track by placing three bands in each triangle, one for
each family of normal arcs. We can place the horizontal edges of each band in the edges of the triangulation, as on the right, so that the normal loop is carried by the train track. Notice that the
way that the horizontal edges of the band are matched up is determined by the loop. In this case, the fact that there are edges going from the upper left edge of these two triangles to the lower
right edge means that the upper left band should meet the lower right band along its lower horizontal edge.
A normal loop that has arcs going from the lower left to upper right edges would not be carried by this train track. This illustrates the first big difference between normal loops and train tracks:
Usually, a given train track will not carry every isotopy class of essential loops in $S$. We could build a different train track that would carry this second type of loop, but then it wouldn’t carry
the loop that’s shown. However, this turns out to be a small price to pay for the second difference between normal loops and train tracks: Generally, a train track will not carry any isotopy trivial
loops, and moreover will carry at most one representative of any isotopy class of loops. As a third difference, train tracks are much more flexible because we can put together the bands in ways that
are not necessarily induced by a triangulation.
What train tracks and normal loops have in common is the vector space structure: Every loop carried by a train track is completely determined by a vector of integers that count how many times the
loop intersects each band in the train track. And as with normal loops, adding vectors corresponds to the same, very simple geometric operation called a Haken sum. So both a triangulation and a train
track define a map from a vector space into the set of loops in a surface $S$. The map from the set of normal loops is onto, but is many-to-one and its image contains the trivial loop. The map from
the loops carried by the train track is not onto, but it is one-to-one (assuming some simple conditions on the train track) and only maps vectors to essential loops. Moreover, if we consider a large
enough set of different train tracks, the images of all these maps will be onto the set of essential loops. So we can think of the train tracks as defining patches of loops, reminiscent of the
patches in Reimannian geometry that allow one to piece together a manifold from Euclidean balls. In this case, the patches define a local vector-space structure on the set of loops that can be used,
for example, to define the space of projective measured laminations on $S$. But that will have to wait for a future post.
Should your picture of the bands be reflected through the horizontal edge so that the upper left band meets the lower right band?
Comment by Bill — March 8, 2013 @ 1:52 pm | Reply
• These are just local pictures. A train track is determined by a union of bands such that the picture shown occurs locally near any component of the union of the horizontal edges of the bands.
Comment by Lee Mosher — March 11, 2013 @ 1:27 pm | Reply
• That’s right – The picture on the left is meant to indicate the generic local structure rather than particularly what’s happening in the middle and right pictures. It’s a coincidence that it
looks like the mirror image of the middle and right pictures.
Comment by Jesse Johnson — March 11, 2013 @ 1:55 pm | Reply
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Jouni Kosonen on Train tracks on a torus
Interesting paper on… on Distinguishing the left-hand t…
chorasimilarity on Tangle Machines- Part 1 | {"url":"http://ldtopology.wordpress.com/2013/02/25/train-tracks/","timestamp":"2014-04-20T03:24:29Z","content_type":null,"content_length":"69571","record_id":"<urn:uuid:4a59ae79-f6ee-405b-a5c8-b806495ae005>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00181-ip-10-147-4-33.ec2.internal.warc.gz"} |
Leon Mirsky
1918 - 1983 Click the picture above
to see a larger version
Leon Mirsky worked in Number Theory, Linear Algebra and Combinatorics.
Full MacTutor biography [Version for printing]
List of References (4 books/articles) Additional Material in MacTutor
A Poster of Leon Mirsky 1. Obituary: The Times
Mathematicians born in the same country
Other Web sites
1. Mathematical Genealogy Project
Previous (Chronologically) Next Main Index
Previous (Alphabetically) Next Biographies index
JOC/EFR © October 2003
The URL of this page is: | {"url":"http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mirsky.html","timestamp":"2014-04-16T18:59:38Z","content_type":null,"content_length":"4616","record_id":"<urn:uuid:4ec43053-540e-4908-a9e7-c635557b9dea>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00343-ip-10-147-4-33.ec2.internal.warc.gz"} |
University of Otago <ADT> Public View
AbstractTwo dimensional stocks of fish can be assessed with methods that mimic the analysis of research survey data but that use commercial catch-effort data. This finite population approach has
scarcely been used in fisheries science though it brings about very large sample sizes of local fish density with models of only moderate levels of complexity. The extracted information about the
status of the stock can be interpreted as biomass indices. Statistical inference on finite populations has been the locus of a highly specialized branch of sampling-distribution inference, unique
because observable variables are not considered as random variables. If statistical inference is defined as "the identification of distinct sets of plausible and implausible values for unobserved
quantities using observations and probability theory" then it is shown that Godambe's paradox implies that the classical finite populations approach is inherently contradictory as a technique of
statistical inference. The demonstration is facilitated by the introduction of an extended canonical form of an experiment of chance, that apart from the three components identified by Birnbaum, also
contains the time at which the experiment is performed. Realization of the time random variable leaves the likelihood function as sole data-based mathematical tool for statistical inference, in
contradiction with sampling-distribution inference and in agreement with direct-likelihood and Bayesian inference. A simple mathematical model is introduced for biomass indices in the spatial field
defined by the fishing grounds. It contains three unknown parameters, the natural mortality rate, the probability of observing the stock in the area covered by the fishing grounds, and mean fish
density in the sub-areas where the stock was present. A new theory for the estimation of mortality rates is introduced, using length frequency data, that is based on the population ecology analogue
of Hamilton-Jacobi theory of classical mechanics. The family of equations require estimations of population growth, individual growth, and recruitment pattern. Well known or new techniques are used
for estimating parameters of these processes. Among the new techniques, a likelihood-based geostatistical model to estimate fish density is proposed and is now in use in fisheries science (Roa-Ureta
and Niklitschek, 2007, ICES Journal of Marine Science 64:1723-1734), as well as a new method to estimate individual growth parameters (Roa-Ureta, In Press, Journal of Agricultural, Biological, and
Environmental Statistics). All inference is done only using likelihood functions and approximations to likelihood functions, as required by the Strong Likelihood principle and the direct-likelihood
school of statistical inference. The statistical model for biomass indices is a hierarchical model with several sources of data, hyperparameters, and nuisance parameters. Even though the level of
complexity is not low, a full Bayesian formulation is not necessary. Physical factors, mathematical manipulation, profile likelihoods and estimated likelihoods are used for the elimination of
nuisance parameters. Marginal normal and multivariate normal likelihood functions, as well as the functional invariance property, are used for the hierarchical structure of estimation. In this manner
most sources of information and uncertainty in the data are carried over up the hierarchy to the estimation of the biomass indices. | {"url":"http://adt.otago.ac.nz/public/adt-NZDU20090818.150508/","timestamp":"2014-04-19T05:01:20Z","content_type":null,"content_length":"10440","record_id":"<urn:uuid:b67a7d4e-fbb9-4bf0-a65e-876fd41ae2ee>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Derivation of the Sum of a Series
Replies: 4 Last Post: Sep 25, 2001 12:43 PM
Messages: [ Previous | Next ]
Re: Derivation of the Sum of a Series
Posted: Sep 24, 2001 10:37 AM
I don't mean to encourage more questions like Eric's but he does raise an
interesting issue.
Consider the problem:
Find a polynomial in n whose value at n is the sum from i=1 to i=n of i^k,
k a positive integer.
There are a couple of nice points you can make.
By a geometric argument you can see that this polynomial is less than
n^{k+1}, for all n, hence must be a polynomial of degree less than or
equal to k+1. For example, if k=1, then look at an n by n square. The
terms of the sum are then the areas of squares of height i and base 1. If
k=2, then the terms of the sums are volumes with base dimensions 1 and i
and height i, and so on. This geometric argument can also be used to
illustrate the usual derivation when k=1.
So what do you know about this polynomial P? Well, you know that
This is very powerful; two polynomials are equal iff all of their
coefficients are equal.
You also know that P(0)=0. This means that the constant term of P is zero.
So look at a general polynomial of degree k+1, in n, evaluate the same
polynomial at n-1, subtract the second from the first, and set the
coefficients equal to the coefficients of n^k. You can then work out the
coefficients starting with the coefficient of n^{k+1}. You can do this
for k=2, and have your students do it for k=3.
If there are other points you like to make with your students about this
topic, Id be interested in hearing about it.
Terry Gaffney
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Date Subject Author
9/23/01 Derivation of the Sum of a Series Eric Abramovich
9/24/01 Re: Derivation of the Sum of a Series Terence Gaffney
9/24/01 Re: Derivation of the Sum of a Series Matthias Kawski
9/25/01 Re: Derivation of the Sum of a Series Terence Gaffney
9/24/01 Re: Derivation of the Sum of a Series me@talmanl1.mscd.edu | {"url":"http://mathforum.org/kb/thread.jspa?threadID=223064&messageID=765652","timestamp":"2014-04-19T05:31:06Z","content_type":null,"content_length":"22495","record_id":"<urn:uuid:0781a0b8-90a9-4ae7-b304-85edf08e3821>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00267-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Implementation of Algorithms for the Periodic-Steady-State Analysis
P.N. Ashar
EECS Department
University of California, Berkeley
Technical Report No. UCB/ERL M89/31
BibTeX citation:
Author = {Ashar, P.N.},
Title = {Implementation of Algorithms for the Periodic-Steady-State Analysis},
Institution = {EECS Department, University of California, Berkeley},
Year = {1989},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1989/1196.html},
Number = {UCB/ERL M89/31}
EndNote citation:
%0 Report
%A Ashar, P.N.
%T Implementation of Algorithms for the Periodic-Steady-State Analysis
%I EECS Department, University of California, Berkeley
%D 1989
%@ UCB/ERL M89/31
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1989/1196.html
%F Ashar:M89/31 | {"url":"http://www.eecs.berkeley.edu/Pubs/TechRpts/1989/1196.html","timestamp":"2014-04-17T01:14:33Z","content_type":null,"content_length":"4360","record_id":"<urn:uuid:094abfda-7588-4c49-8d35-3f6f52bc6903>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00163-ip-10-147-4-33.ec2.internal.warc.gz"} |
Analysis of AVL Balancing Rule - GNU libavl 2.0.2
How good is the AVL balancing rule? That is, before we consider how much complication it adds to BST operations, what does this balancing rule guarantee about performance? This is a simple question
only if you're familiar with the mathematics behind computer science. For our purposes, it suffices to state the results:
An AVL tree with n nodes has height between log2 (n + 1) and 1.44 * log2 (n + 2) - 0.328. An AVL tree with height h has between pow (2, (h + .328) / 1.44) - 2 and pow (2, h) - 1 nodes.
For comparison, an optimally balanced BST with n nodes has height ceil (log2 (n + 1)). An optimally balanced BST with height h has between pow (2, h - 1) and pow (2, h) - 1 nodes.^1
The average speed of a search in a binary tree depends on the tree's height, so the results above are quite encouraging: an AVL tree will never be more than about 50% taller than the corresponding
optimally balanced tree. Thus, we have a guarantee of good performance even in the worst case, and optimal performance in the best case.
See also: [Knuth 1998b], theorem 6.2.3A. | {"url":"http://adtinfo.org/libavl.html/Analysis-of-AVL-Balancing-Rule.html","timestamp":"2014-04-16T22:15:42Z","content_type":null,"content_length":"6562","record_id":"<urn:uuid:4c00596c-b0b8-42e2-8525-c77d23cd7619>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00363-ip-10-147-4-33.ec2.internal.warc.gz"} |
Magic Square Using prime numbers
Newbie Poster
5 posts since Sep 2011
Reputation Points: 0 [?]
Q&As Helped to Solve: 0 [?]
Skill Endorsements: 0 [?]
Im writing a code that ask for two positive integers entered from the keyboard. The first integer N is odd in range of 3-15 and the second with be an initial value I. N will be the size of the array.
Beginning the center of the NxN array with the integer I. If I is prime then print the number I in that position of the square. Otherwise print three asterisks in that position. Move to the right one
square , and test the integer I+1 for primality. Print I+1 if it is prime and three asterisks if it is not. Continue going counter clockwise through the square until the square is full of numbers and
three asterisks then print the array.
Thats the problem i have done most of it and im confused as to how to make the array and if any one could take a look at my code to let me know if i have my for loop set up correctly. Thank you
kindly for your help! I dont know how to implement the spiral correctly if anyone could help please!!!
Heres my code:
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
int IsPrime(int Nsize);
int main(int argc,char* argv[]) {
int Nsize, Initial;
int i;
int j;
int k;
int **array;
int totalsquares=Nsize*Nsize;
int currentsquare=0;
int currentsidelength=1;
int currentpositiononside=0;
int increasesidelength=0;
int direction=0;
printf( "Enter an odd integer N between 3 and 15: " ) ;
scanf( "%d", &Nsize) ;
printf( "Enter an initial value I: " ) ;
scanf( "%d", &Initial ) ;
++currentsquare, ++Initial, ++currentpositiononside;
if (currentpositiononside == currentsidelength) {
currentpositiononside = 0;
direction = (direction + 1) % 4;
if ((increasesidelength % 2) == 0) {
increasesidelength = 0;
printf("%d", array[i][j]);
return 0;
int IsPrime(int Nsize){
int i, sqrtNsize;
if(Nsize<2)return 0;
if(Nsize==2)return 2;
if((Nsize%2)==0)return 0;
if(Nsize%i==0)return 0;
return Nsize;
Posting Virtuoso
1,711 posts since Jun 2008
Reputation Points: 419 [?]
Q&As Helped to Solve: 207 [?]
Skill Endorsements: 10 [?]
Line 14:
totalsquares is being calculated from an uninitialized variable. Declare totalsquares, and don't give it a value until AFTER the user has entered the Nsize value.
Line 20:
Same thing with calculating i and j - do that AFTER Nsize has been entered.
The malloc just seems wrong:
array = malloc(Nsize * sizeof(int*));
for(i = 0;i<Nsize;i++)
array[i]=malloc(Nsize * sizeof(int));
array[totalsquares] is never a part of the array. In C, array indices go from 0 to totalsquares - 1 only.
And I have no idea why you'd malloc an address to an int array[index], anyway.
It's always a good idea to free these malloc'd memory in their reverse order, just before your program ends. | {"url":"http://www.daniweb.com/software-development/c/threads/399171/magic-square-using-prime-numbers","timestamp":"2014-04-20T08:30:39Z","content_type":null,"content_length":"36135","record_id":"<urn:uuid:56d53a73-39e4-44fb-86bf-0e0424ac8a0c>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00086-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Runs Batted In and Earned Run Average
Date: 10/26/2000 at 12:21:40
From: Tara Kuty
Subject: Runs Batted In
How do you calculate RBI for a player on a baseball team?
Also, how do you find an ERA for a pitcher?
Date: 10/27/2000 at 12:01:30
From: Doctor TWE
Subject: Re: Runs Batted In
Hi Tara - thanks for writing to Dr. Math.
Runs Batted In (or RBIs) is a "counting stat" like home runs or hits.
Every time a batter gets a hit or makes an out and a runner scores on
the play, (s)he gets an RBI - except in a few special cases.
A batter is NOT awarded an RBI when (s)he bats into a double play
(even if a run scores on the play).
A batter is NOT awarded an RBI on an error unless, in the official
scorekeeper's opinion, the run would have scored even without the
error. For example, if with less than two outs, the batter hits a deep
fly ball to the outfield and the outfielder bobbles and drops the ball
(the batter is safe on first base), the official scorekeeper may
charge the outfielder with an error, but award the batter an RBI,
deeming that the play would have been a sacrifice fly if the error had
not occurred.
A pitcher's Earned Run Average (or ERA) is the average number of
earned runs the pitcher gives up per full game (9 innings). It is
calculated as:
ERA = 9*ER / IP
where ER is the number of earned runs the pitcher has allowed and IP
is the number of innings the pitcher has pitched. For example, Andy
Pettitte pitched 13 2/3 innings and allowed 3 earned runs in his two
starts in the 2000 World Series, so his ERA for the series was:
ERA = 9*3 / (13+2/3) = 1.9756... or 1.98
Al Leiter pitched 15 2/3 innings and allowed 5 earned runs in his two
starts in the World Series, so his ERA for the series was:
ERA = 9*5 / (15+2/3) = 2.8723... or 2.87
A run is considered unearned (and thus doesn't count against the
pitcher's ERA) when an error occurs in an inning, and the run would
not have scored had the error not occurred. For example, suppose the
following sequence occurs in an inning:
Batter 1: Single
Batter 2: Strikeout
Batter 3: Safe at first on an error (batter 1 to 2nd base)
Batter 4: Home Run (1, 3, and 4 score)
Batter 5: Triple
Batter 6: Sac Fly (5 scores)
Batter 7: Walk
Batter 8: Home Run (7 and 8 score)
Six runs have scored (batters 1, 3, 4, 5, 7, and 8), but let's
evaluate which are earned and which are unearned. The easiest way to
do this is to "replay" (on paper) the inning, replacing the error with
an out. Here's what we would have:
Batter 1: Single
Batter 2: Strikeout
Batter 3: Out (instead of error)
Batter 4: Home Run (#1 and #4 score)
Batter 5: Triple
Batter 6: Fly out (Since this is the 3rd out, 5 doesn't score)
Batters 7 and 8: never got to bat...
Two runs would have scored in this scenario, so 2 of the 6 runs are
earned, and the other 4 are unearned.
Note that any run scored by a batter who reaches base due to an error
is automatically unearned. Thus, batter 3's run is unearned no matter
what else happens. Note that because batter 6's fly out would have
been the third out of the inning, batter 5 would not have scored on a
"sac fly." So that run, too, is unearned. (The batter still gets
credit for the sac fly and the RBI, but the run is not charged against
the pitcher's ERA.) Finally, since batter 6's fly out would have ended
the inning, batter 8's home run would not have happened, so that run -
and all subsequent runs in the inning - are also unearned. If the
pitcher gives up additional runs in an inning after what would have
been the third out, all of them are unearned. (We don't assume that
the batter would have still hit the home run next inning because the
game situation would have changed; the pitcher would have had a chance
to rest a bit, the score might be different, etc.)
If a relief pitcher enters the game, any runners on base at the time
(s)he enters the game are charged against the pitcher who allowed them
to reach base, and whether they are considered earned or unearned is
determined as above. However, the slate is considered "clean" for the
relief pitcher entering the game. Any runners the relief pitcher
allows to reach base and score are considered earned unless more
errors occur. So if, for example, a relief pitcher entered the game
after batter 7 in our example above, 7's run would still be considered
unearned (charged as a run allowed by the original pitcher, but not an
earned run), but batter 8's home run would be charged as an earned run
against the relief pitcher.
I hope this helps. If you have any more questions, write back.
- Doctor TWE, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/52796.html","timestamp":"2014-04-17T07:12:06Z","content_type":null,"content_length":"9616","record_id":"<urn:uuid:b6c0d54b-025b-4b3b-99ab-70c09280cbc6>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00424-ip-10-147-4-33.ec2.internal.warc.gz"} |
Two Proofs of Constant A/V Ratio Cell Growth and Division
If a cell maintains a constant ratio of surface area to volume as it grows, there are a variety of ways in which it can grow and divide.
For a cell, the advantage of maintaining a constant A/V ratio is that it only needs to produce surface membrane and internal material in a constant ratio, equal to the initial A/V ratio, throughout
the cell cycle. It is probably also the Least Action variant of cell growth and division, since the cell naturally and effortlessly separates in two as a function of its limiting geometry.
Most likely, normal cell growth consists of the two hemispheres of an initially spherical cell expanding to become enlarged segments of a sphere, joined by a fold (or notch), until both segments have
become two spheres with the same radius as the original cell.
Here’s a mathematical proof relating segment height h to notch width b through the range h=R to h=2R.
In a second type of growth, which seems to typify some cancer cells, the notch widens to form two cones which connect the two hemispheres of the growing cell. The resulting cells, although they have
the same A/V ratio as the parent cell, generally do not have the same volume.
Here’s a mathematical proof for the growth and division of one cube to form two cubes the same size as the initial cube.
When the notch width is greater than zero, these cells form truncated double cones or pyramids between their two halves, which results in a greater extension along the growth axis. Many cancer cells
appear to have such double cones, and what would appear to make ‘cancer’ cells dangerous is that, as their cones lengthen, they are able to push past adjacent cells, and spread into other tissues.
Why some cells grow ‘normally’, and others take on the ‘cancerous’ form, is not clear. But in general, since it will require less physical work for normal cells to grow than ‘cancer’ cells, it should
be expected that normal cells will predominate.
It is highly unlikely that the ‘cancerous’ form of growth and division is caused by smoking tobacco.
Note: This is a modified version of the original post, which contained an error in one proof. Edited 10 Dec 2012 to remove surplus bracket from cubical proof at c = ….
9 Responses to Two Proofs of Constant A/V Ratio Cell Growth and Division
1. Frank,
Seems very logical and since there is math involved, I am not surprised to see no replies. This lack of math understanding is a shame.
What has always seemed curious to me is not the ‘how’ of cell division; but, the ‘why’ of it all.
From conception on, my cells divided at a tremondous rate and, by the age of about 18, I had gone from very-very small to 6′ 1″ and 200 pounds.
Then my cells seem to have gone into a ‘maintain mode’ and for 50 years I have gained neither height nor weight.
But, nail and hair cells(those that I still have) continue to grow as when I was younger.
What happened and why?
DNA instructions,I guess.
Why do some strange cell divisions cause cancerous growths and some cause people to be 8 feet tall?
Plants have the same growth patterns, are we related to plants?
Cells could just get bigger and bigger, why must they divide at all?
That,I think, is a function of gravity and other laws of physics.
Too many questions to list, actually.
□ There are few comments because few people are reading it. Not many people are interested in cell growth and division.
I’m impressed if you’ve actually followed the logic.
Cells divide, in my opinion, because they maintain a constant A/V ratio, as I argue above.
☆ To have identical cells, the nucleus must divide also.
Perhaps, the nucleus dividing is the driving force behind the cellular division?
What makes the nucleus decide it is time to divide?
○ Takes wild guess,genetics.
I havent got a clue but the math problem I can follow. I just dont understand its answer.
2. The first proof is OK, but the second one sucks. And I have annotated it accordingly.
3. I’ve now replaced the defective second proof with another proof, and adjusted the text accordingly.
4. Pingback: A Polyhedral Cell Model | Frank Davis
5. Pingback: Getting Inside Cells | Frank Davis
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Lesson 24: Evaluating Definite Integrals (slides)
by Matthew Leingang, Clinical Associate Professor of Mathematics at New York University on Apr 21, 2010
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental
Theorem of Calculus!
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Mplus Discussion >> Does Mplus impute missing data/ role of EM?
Calvin Croy posted on Wednesday, October 05, 2005 - 12:19 pm
Could you please answer the following 3 questions?
I am running Mplus version 3.13 for a CFA analysis and specifying Type = Missing H1.
1. After reading the User's Guide and examining posts on this discussion site, it appears to me that Mplus does NOT actually impute ("fill in") missing data values when Type = Missing H1 is
specified. Is this correct?
2. Based on what I've read, it sounds like Mplus uses the EM algorithm to estimate means, variances, and covariances (the sufficient statistics) using all the factor indicator variables. Then the
estimated means, variances, and covariances resulting from EM are used to derive the model parameter estimates. Is this right? If not, could could you please clarify?
3. If one specifies Type = Imputation to read multiple datasets created by some multiple imputation process outside of Mplus (Example 12.13), how is EM used, if at all?
Thanks so much for answering posted questions! While I'm sure this is quite time consuming, your comments are valued greatly and keep us users on the right path.
bmuthen posted on Saturday, October 08, 2005 - 2:15 pm
1. That's right. Standard ML estimation is used instead. The missing values can however be produced.
2. Not quite. EM is used in the ML estimation of the unrestricted mean vector and covariance matrix (the H1 model) - but these results are only used to be able to compute a chi-square test of the H0
model against H1. For the H0 parameter estimation, all available raw data are used in the computations using ML. EM is not used in the H0 computations.
3. EM is not used here. Type = Imputation is intended for use when another program has been first used to impute missing data (e.g. freeware such as NORM). These multiple data sets are then sent to
Mplus and analyzed by Mplus, followed by an Mplus summary of the parameter estimates and the computation of the SEs.
Just to add to this - one can use EM to generate imputed data. In the EM algorithm the missing data are computed for each person. This type of output is however not currently available in Mplus.
Calvin Croy posted on Thursday, October 13, 2005 - 11:51 am
Thank you so much for your clarification. It really helped!
ting hlin posted on Thursday, November 20, 2008 - 3:35 am
is it true that Mplus 5.1 still does not have the facility to carry out multiple imputation (it can just processes imputed data)?
Linda K. Muthen posted on Thursday, November 20, 2008 - 7:13 am
Yes, this is true.
Nicolas Müller posted on Wednesday, March 16, 2011 - 10:57 am
Dear Dr. Muthen,
I'm using MPlus 6, and I'd like to understand how missing data on the outcomes are treated in latent growth models when using ML/MLR estimators.
1) I think I've read that missing data are considered MAR. Does it mean that, in a longitudinal context, missingness for one indicator in time can be conditional on the values of the same indicator
at other time points?
2) Is there a technical article where the algorithm used in MPlus for MAR treatment of missing data within a ML estimation is explained? Or a general reference?
Thank you.
Bengt O. Muthen posted on Wednesday, March 16, 2011 - 4:40 pm
1. Yes, missing data on an indicator at a certain time point is allowed to be influenced by the indicator value at other time points.
2. The Little & Rubin )2002) missing data book describes this and is a good general reference. A more applied book is Enders (2010).
Nicolas Müller posted on Thursday, March 17, 2011 - 1:26 am
Thank you for these answers.
I also tried to estimate a multi-level model on my growth data, thus using a long format where each line is an individual observation and ANALYSIS=TWOLEVEL.
Does it mean that missing data in a twolevel model for growth are not considered MAR but MCAR because each line where the dependent variable is missing is removed?
If I get almost very close estimates when using a latent growth curve analysis or a multilevel analysis, can I say missing data are MCAR (because considering them MAR doesn't change the estimates)?.
Linda K. Muthen posted on Thursday, March 17, 2011 - 8:53 am
The results should be identical if you set the models up correctly. You may be missing the fact that the residual variances need to be held equal over time for the Mplus model to be the same as the
multilevel model. Both assume MAR.
Nicolas Müller posted on Friday, March 18, 2011 - 8:04 am
Ok, your answer made me realize there was a glitch in my data transformation routine.
With the correct datasets, the estimates are now completely identical, using the latent growth or the multilevel specification.
Thank you very much.
Nicolas Müller posted on Monday, March 28, 2011 - 2:11 am
There is still something I don't get.
I'm fitting the same model with a latent growth curve or a multilevel specification, and I get exactly the same results.
I don't understand how it is possible, knowing that some of the time-varying covariates have been imputed, regardless if the individual was still followed or not.
In the multilevel spec, with a long data format, it has no consequences, because the lines where the individual was not in the study are deleted.
But in the wide format used by the latent growth specification, these imputations are still in the data, so I thought they should be impacting the estimation... which is not the case.
I hope I made myself clear. Do you have an explanation for this behaviour?
I could send you my models if necessary.
Linda K. Muthen posted on Monday, March 28, 2011 - 10:16 am
In Mplus the model is estimated conditioned on the covariates. Cases with missing data on observed exogenous variables are deleted from the analysis as they are in HLM. With dependent variables,
values are not imputed. All available information is used as in HLM.
Soz posted on Wednesday, August 10, 2011 - 7:07 am
Dear Linda and Bengt,
I am running a SEM with 5 imputed data sets for missing values within PASW. I also created the implist.dat where the five datasets are identified.
I used the following specification:
FILE IS Implist.dat;
NAMES ARE Imputation_ VPCode country RiskP_M3 OE_M6 Inten_M5 SE_M6 Plan_M2
Plan_M6 CHBdiet9 ;
USEVARIABLES ARE RiskP_M3
OE_M6 Inten_M5 SE_M6 Plan_M2
Plan_M6 CHBdiet9;
TYPE is Imputation;
ITERATIONS = 10000;
CONVERGENCE = 0.00005;
Inten_M5 on RiskP_M3 OE_M6 SE_M6 CHBdiet9 ;
Plan_M2 on Inten_M5 ;
Plan_M6 on Plan_M2;
OUTPUT: tech1; standardized;
Nevertheless i`ve got this error message:
*** ERROR in ANALYSIS command
Unrecognized setting for TYPE option:
What did I wrong? How can I fix the problem? Many Thanks in advance.
Linda K. Muthen posted on Wednesday, August 10, 2011 - 8:28 am
TYPE+IMPUTATION; goes in the DATA command not the ANALYSIS command.
C posted on Tuesday, April 03, 2012 - 1:45 pm
Hi Dr. Muthen,
I am attempting to create some imputed datasets. I have done this before more than once. However, at this time, I keep getting the following message with no other explanation:
*** FATAL ERROR
I cannot figure out what the issue is. Any insight into what this messege means and how to rectify it?
Linda K. Muthen posted on Tuesday, April 03, 2012 - 4:14 pm
Please send the output and your license number to support@statmodel.com
EFried posted on Tuesday, June 19, 2012 - 1:11 pm
I have a question about #6 of the technote7:
"A basic identifiability requirement for the imputa- tion model is that for (A) each variable in the imputation the number of observations should be at least as many as the number of (B) variables in
the imputation model."
I don't understand the difference between the variables I marked with (A) and (B). Does (A) refer to the variables in the usevariables list, and (B) to the variables in the "impute" list?
EFried posted on Tuesday, June 19, 2012 - 1:15 pm
(To clarify, in my example, I have 6 variables I use to help impute missing values on one variable, and wonder how much % may be missing to identify the imputation. N=800)
usevariables=x1 x2 x3 x4 x5 x6;
Impute = x0;
NDATASETS = 10;
SAVE = impute_M*.dat;
type = basic;
Tihomir Asparouhov posted on Tuesday, June 19, 2012 - 2:36 pm
In your example X0 should have at least 7 non-missing values since there are 7 parameters in the imputation model.
EFried posted on Tuesday, June 19, 2012 - 3:24 pm
So in theory, 793 values out of 800 could be missing? That doesn't sound like something I could report in a paper ;)
We have a large number of missing values on one crucial covariate, which MPLUS cannot handle in a multilevel analysis if we just add it as
into the model (non-convergence).
And since MPLUS does listwise deletion for missing time-varying covariates in Multilevel Models (even if only 1 measurement point misses - why is that?), we were thinking to impute. Imputation
converges, but we are not sure if imputation with 40% missing is feasible.
Thank you for your insight
Linda K. Muthen posted on Wednesday, June 20, 2012 - 10:45 am
Please send the output that shows the problem and your license number to support@statmodel.com.
Keke Hiller posted on Monday, November 19, 2012 - 1:54 am
Dear Dr. Muthen,
I am using MPlus Vers6 & survey data of three waves in a panel design, having solely the dependent variable at time point 3.
My sample population is 207 at time point 1 (T1), then reduced to 176 individuals at time point 2 (T2) and finally reduced to 137 at time point 3 (T3).
The calculations, however, are performed with N=176. Thus, I assume the missing data at T3 is imputed, right?
Is there a critical proportion of missing data where imputation is problematic? Do you know any paper that discusses this topic?
Thank you in advance!
Linda K. Muthen posted on Monday, November 19, 2012 - 10:37 am
In Mplus, the default is to estimate the model using all available information. It does not impute values in this case. See the Little and Rubin and Enders books in the user's guide's reference list.
Keke Hiller posted on Monday, November 19, 2012 - 11:20 am
Thank you for the fast reply. I will look the topic up in the book you mentioned.
Nevertheless, I am wondering that if there is no imputation done in this case, why does MPlus gives me N=176 as sample population? (and not 137 instead)
How would I report my sample size in my paper?
Thank you so much for your help!
Linda K. Muthen posted on Monday, November 19, 2012 - 12:06 pm
All available information from the full sample is used in the model estimation so the sample size is the full sample. You report what is given in the output.
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Are you seeing patterns in numbers everywhere? You're not crazy - it's Benford's Law.
Pick any large set of random data you like. Look at the first digit of all the numbers. You're going to see a lot of ones. It's not just a coincidence. It's the law.
Benford's Law was discovered by Simon Newcomb, who was thumbing through a book of logarithmic tables. He noticed that the pages which contained tables of the digits beginning with low numbers were
far more worn than the ones beginning with hi numbers. In modern times, that would just mean that people started with the low numbers and had wiped the pizza grease off their hands by the time they
got to the higher ones. Since it was 1881, though, Newcomb figured that the low numbers were used far more often than the high numbers. Since the book was in a library, it had presumably been used by
a random assortment of people for a random assortment of problems. Newcomb found other books, in other libraries were worn in a similar way.
Clearly, people needed data on numbers beginning with low numbers more than they did high numbers. That didn't make sense. If the world is random, the beginning digit of the numbers looked up to
describe the world should be the same way. Digits one through nine should each be used 11 percent of the time, and the book's pages should all be equally worn.
They aren't, and they weren't.
The digit ‘one' is likely to begin any number based on a survey of real thing thirty percent of the time. Frank Benford was the gentleman who actually got his name put lastingly on the law,
rediscovered this trend, and he broadened the amount of evidence to support it. Among the data he gathered was newspaper circulation, river area, death rates and the addresses of people listed in the
book "American Men of Science." One appeared roughly 30 percent of the time. More than that, low digits were far more likely to appear than high digits in the leftmost place of any number.
Benford's law isn't always applicable. The number sets have to be big enough. There also can't be any specific outside influence, such as ‘human thought.' Prices, for example, are designed to appeal
to customers, and therefore aren't subject to Benford's law. If you were to analyze purchases of, say, packets of potato chips, you would find that certain numbers would come up again and again.
People see different prices on the shelf and make decisions based on those prices. However, if you were to follow many people around during their week long vacations, and add up all their purchases
and pay-outs, you would find a Benford distribution – so long as they didn't have a specific budget in mind. A single purchase requires thought. A number of different purchases should add up to a
‘random number,' which is not random at all.
But when data sets get big, the biggest giveaway that you're not being presented with random numbers is being presented with completely random numbers. Benford's law was used to detect fraud in
Iran's elections. It's also widely used to detect financial fraud. Expense claims, financial pay-outs, tax fraud, and accounts payable all conform to Benford's law when they're legit. Checking these
numbers against the law is so common that software is sold to do it.
Via DPS guide, Mathworld, Physics World and UIC.
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The degree distribution of bipartite planar maps: applications to the Ising model
Results 1 - 10 of 21
, 2003
"... The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional
random curves. The (distinguished) audience of the Saint-Flour summer school consists mainly of probabilist ..."
Cited by 88 (6 self)
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The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random
curves. The (distinguished) audience of the Saint-Flour summer school consists mainly of probabilists and I therefore assume knowledge in stochastic calculus (Itô’s formula etc.), but no special
background in basic complex analysis. These lecture notes are neither a book nor a compilation of research papers. While preparing them, I realized that it was hopeless to present all the recent
results on this subject, or even to give the complete detailed proofs of a selected portion of them. Maybe this will disappoint part of the audience but the main goal of these lectures will be to try
to transmit some ideas and heuristics. As a reader/part of an audience, I often think that omitting details is dangerous, and that ideas are sometimes better understood when the complete proofs are
given, but in the present case, partly because the technicalities often use complex analysis tools that the audience might not be so
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the
minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
Cited by 39 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal
Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations
with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
- Electronic Journal of Combinatorics
"... Gif sur Yvette Cedex, France We extend Schaeffer’s bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to
obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the c ..."
Cited by 34 (1 self)
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Gif sur Yvette Cedex, France We extend Schaeffer’s bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain
a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the
bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the
initial maps via the mobiles ’ labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.
- Proceedings of the International Congress of Mathematicians, Madrid (M. Sanz-Solé et , 2007
"... Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of
various exponents describing the behavior of these models. Only recently have some of these predictions beco ..."
Cited by 18 (1 self)
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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various
exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a
one-parameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the
mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics
knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will
be motivated and explained, and a brief sketch of recent results will be presented.
, 2006
"... 1.1. Matrix models per se......................... 2 1.2. A brief history............................ 3 2. Matrix models for 2D quantum gravity................... 4 2.1. Discrete 2D quantum
gravity..................... 4 ..."
Cited by 10 (0 self)
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1.1. Matrix models per se......................... 2 1.2. A brief history............................ 3 2. Matrix models for 2D quantum gravity................... 4 2.1. Discrete 2D quantum
gravity..................... 4
- A: Math.Gen
"... Gif sur Yvette Cedex, France We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees.
We show how to recover the two-matrix model solution to this problem in this purely combinatorial language ..."
Cited by 9 (4 self)
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Gif sur Yvette Cedex, France We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We
show how to recover the two-matrix model solution to this problem in this purely combinatorial language.
"... Abstract. In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large N, of the logarithm of the partition
function of N × N Hermitian random matrices. These coefficients are generating functions for graphical enumera ..."
Cited by 7 (3 self)
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Abstract. In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large N, of the logarithm of the partition function
of N × N Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that
differs from the Gaussian weight by a single monomial term of degree 2ν. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From
these equations one may deduce functional analytic characterizations of the coefficients in the asymptotic expansion. Moreover, this ode system can be solved recursively to explicitly construct these
coefficients as functions of the coupling parameter. This analysis of the fine structure of the asymptotic coefficients can be extended to multiple coupling parameters and we present a limited
illustration of this for the case of two parameters. 1. Motivation
- ALEA LAT. AM. J. PROBAB. MATH. STAT , 2005
"... We show that under reasonably general assumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planar maps. This is based on the fact
that solutions of the differential Schwinger-Dyson equations are, by nature, generating functions for enumera ..."
Cited by 7 (3 self)
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We show that under reasonably general assumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planar maps. This is based on the fact that
solutions of the differential Schwinger-Dyson equations are, by nature, generating functions for enumerating planar maps, a remark which bypasses the use of Gaussian calculus.
, 2008
"... We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete
three-point function converges to a simple universal scaling function, which is the continuous three-point f ..."
Cited by 6 (2 self)
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We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete
three-point function converges to a simple universal scaling function, which is the continuous three-point function of pure 2D quantum gravity. We give explicit expressions for this universal
threepoint function both in the grand-canonical and canonical ensembles. Various limiting regimes are studied when some of the distances become large or small. By considering the case where the
marked vertices are aligned, we also obtain the probability law for the number of geodesic points, namely vertices that lie on a geodesic path between two given vertices, and at prescribed distances
from these vertices. 1.
- The Ramanujan Journal
"... Gif sur Yvette Cedex, France We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of
the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic dist ..."
Cited by 5 (1 self)
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Gif sur Yvette Cedex, France We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the
graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of
the KP hierarchy. We obtain a unified formulation of the (multi-) critical continuum limit describing large graphs with marked points at large geodesic distances, and obtain integrable differential
equations for the corresponding scaling functions. This provides a continuum formulation of two-dimensional quantum gravity, in terms of the geodesic distance. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1242368","timestamp":"2014-04-21T14:05:23Z","content_type":null,"content_length":"37209","record_id":"<urn:uuid:28efc2fe-8c49-4184-85ac-91dbcd425cf9>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
Maths busking
Maths busking
Sara Santos busking on the streets of Krakow.
The 6th European Congress of Mathematics, which took place in Krakow at the beginning of July, wasn't just about mathematicians talking to each other. On the streets of Krakow maths buskers were
entertaining the public, handcuffing innocent Krakowians, constructing emergency pentagons and reading minds. So what is maths busking all about? We caught up with Sara Santos, the director of the
project, and one of her volunteers to find out.
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