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Newton's Laws
Mechanics: Newton's Laws of Motion
Newton's Laws of Motion: Problem Set Overview
There are 20 ready-to-use problem sets on the topic of Newton's Laws of Motions. These problem sets focus on situations in which the forces and accelerations are directed along the traditional
coordinate axes. The problems target your ability to distinguish between mass and weight, determine the net force from the values of the individual forces, relate the acceleration to the net force
and the mass, analyze physical situations to draw a free body diagram and solve for an unknown quantity (acceleration or individual force value), and to combine a Newton's second law analysis with
kinematics to solve for an unknown quantity (kinematic quantity or a force value). Problems range in difficulty from the very easy and straight-forward to the very difficult and complex.
Mass versus Weight
Many problems target your ability to distinguish between mass and weight. Mass is a quantity which is dependent upon the amount of matter present in an object; it is commonly expressed in units of
kilograms. Being the amount of matter possessed by an object, the mass is independent of its location in the universe. Weight, on the other hand, is the force of gravity with which the Earth attracts
an object towards itself. Since gravitational forces vary with location, the weight of an object on the Earth's surface is different than its weight on the moon. Being a force, weight is most
commonly expressed in the metric unit as Newtons. Every location in the universe is characterized by a gravitational field constant represented by the symbol g (sometimes referred to as the
acceleration of gravity). Weight (or F[grav]) and mass (m) are related by the equation:
F[grav] = m • g
You may find our video tutorial on the topic of Mass versus Weight to be useful.
Newton's Second Law of Motion
Newton's second law of motion states that the acceleration (a) experienced by an object is directly proportional to the net force (F[net]) experienced by the object and inversely proportional to the
mass of the object. In equation form, it could be said that a = F[net]/m. The net force is the vector sum of all the individual force values. If the magnitude and direction of the individual forces
are known, then these forces can be added as vectors to determine the net force. Attention must be given to the vector nature of force. Direction is important. An up force and a down force can be
added by assigning the down force a negative value and the up force a positive value. In a similar manner, a rightward force and a leftward force can be added by assigning the leftward force a
negative value and the rightward force a positive value.
The a = F[net]/m equation can be used as both a formula for problem solving and as a guide to thinking. When using the equation as a formula for problem solving, it is important that numerical values
for two of the three variables in the equation be known in order to solve for the unknown quantity. When using the equation as a guide to thinking, thought must be given to the direct and inverse
proportionalities between acceleration and the net force and mass. A two-fold or a three-fold increase in the net force will cause the same change in the acceleration, doubling or tripling its value.
A two-fold or three-fold increase in the mass will cause an inverse change in the acceleration, reducing its value by a factor of two or a factor of three. Watch this video to learn more about using
the equation as a guide to thinking.
Free Body Diagrams
Free body diagrams and force diagrams represent the forces that act upon an object at a given moment in time. The individual forces that act upon an object are represented by vector arrows. The
direction of the arrows indicate the direction of the force and the approximate length of the arrow represents the relative strength of the force. The forces are labeled according to their type. A
free body diagram can be a useful aid in the problem-solving process. It provides a visual representation of the forces exerted upon an object. If the magnitudes of all the individual forces are
known, the diagram can be used to determine the net force. And if the acceleration and the mass are known, then the net force can be calculated and the diagram can be used to determine the value of a
single unknown force.
Coefficient of Friction
An object that is moving (or even attempting to move) across a surface encounters a force of friction. Friction force results from the two surfaces being pressed together closely, causing
intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. The
maximum amount of friction force (F[frict]) can be calculated using the equation:
F[frict] = µ• F[norm]
The symbol µ (pronounced "mew") represents of the coefficient of friction and will be different for different surfaces. The F[norm] in the equation represents the normal force. On flat surfaces, it
is often equal to the weight of the object. This video explains more about the force of friction and how to use the formula to solve problems.
Blending Newton's Laws and Kinematic Equations
Kinematics pertains to a description of the motion of an object and focuses on questions of how far?, how fast?, how much time? and with what acceleration? To assist in answering such questions, four
kinematic equations were presented in the One-Dimensional Kinematics unit. The four equations are listed below.
• d = v[o] • t + 0.5 • a • t^2
• v[f] = v[o] + a • t
• v[f]^2 = v[o ]^2 + 2 • a • d
• d = (v[o ]+ v[f])/ 2 • t
• d = displacement
• t = time
• a = acceleration
• v[o] = original or initial velocity
• v[f] = final velocity
Newton's laws and kinematics share one of these questions in common: with what acceleration? The acceleration (a) of the F[net] = m•a equation is the same acceleration of the kinematic equations.
Common tasks thus involve:
1. using kinematics information to determine an acceleration and then using the acceleration in a Newton's laws analysis, or
2. using force and mass information to determine an acceleration value and then using the acceleration in a kinematic analysis.
When analyzing a physics word problem, it is wise to identify the known quantities and to organize them as either kinematic quantities or as F-m-a type quantities. Once identified and written down in
an organized fashion, the problem solution becomes more clear.
Habits of an Effective Problem-Solver
An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach,
they all have habits which they share in common. These habits are described briefly here. An effective problem-solver...
• ...reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it.
• ...identifies the known and unknown quantities in an organized manner, often times recording them on the diagram itself. They equate given values to the symbols used to represent the
corresponding quantity (e.g., v[o] = 0 m/s, a = 2.67 m/s/s, v[f] = ???).
• ...plots a strategy for solving for the unknown quantity; the strategy will typically centers around the use of physics equations and is heavily dependent upon an understanding of physics
• ...identifies the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
• ...performs substitutions and algebraic manipulations in order to solve for the unknown quantity.
Additional Readings/Study Aids:
The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.
Watch a Video
We have developed and continue to develop Video Tutorials on introductory physics topics. You can find these videos on our YouTube channel. We have an entire Playlist on the topic of Newton's Laws of
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What is the quotient t+3/t+4/(t^2+7t+12) - Factor Calculator
What is the quotient t+3/t+4/(t^2+7t+12)
In this article, we’ll explore how to simplify the complex expression
Step 1: Rewrite the Expression
We begin by recognizing that the given expression is a quotient of two terms. The expression can be rewritten as:
This shows that we are dividing the fraction
Step 2: Factor the Quadratic Expression
The next step is to factor the quadratic expression in the denominator,
The two numbers are (3) and (4) because:
Thus, we can factor
Step 3: Substitute the Factored Form
We now substitute this factored form of
Step 4: Simplify the Expression
Now that we have the expression
At this point, we can cancel the common factor of (t+3) from both the numerator and the denominator, leaving us with:
Final Answer
The simplified quotient is:
The expression
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Clebsch–Gordan coefficients
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Clebsch–Gordan coefficients
Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum
eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct
sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation) into irreducible representations, in cases where the numbers and types of irreducible
components are already known abstractly. The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory.
From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates.
The addition of spins in quantum-mechanical terms can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and
the integrals correspond to the Hilbert space inner product.^[2] From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist
complicated explicit formulas for their direct calculation.^[3]
Angular momentum operators
Angular momentum operators are self-adjoint operators jx, jy, and jz that satisfy the commutation relations
where ε**klm is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator,
It also known as a spherical vector, since it is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators.
By developing this concept further, one can define another operator j2 as the inner product of j with itself:
This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra so(3) ≅ su(2). This is physically
interpreted as the square of the total angular momentum of the states on which the representation acts.
One can also define raising (j+) and lowering (j−) operators, the so-called ladder operators,
Spherical basis for angular momentum eigenstates
It can be shown from the above definitions that j2 commutes with jx, jy, and jz:
When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, j2 and jz are chosen. From the commutation relations, the possible eigenvalues can be found. These
eigenstates are denoted |j m⟩ where j is the angular momentum quantum number and m is the angular momentum projection onto the z-axis.
They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations,
The raising and lowering operators can be used to alter the value of m,
where the ladder coefficient is given by:
In principle, one may also introduce a (possibly complex) phase factor in the definition of. The choice made in this article is in agreement with theCondon–Shortley phase convention. The angular
momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized,
Here the italicizedj and mdenote integer or half-integerangular momentumquantum numbers of a particle or of a system. On the other hand, the romanj[x],j[y],j[z],j[],j[−], andj^2denote operators. The
symbols areKronecker deltas.
We now consider systems with two physically different angular momentaj[1]andj[2]. Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or
the orbital angular momenta of two electrons. Mathematically, this means that the angular momentum operators act on a spaceof dimensionand also on a spaceof dimension. We are then going to define a
family of "total angular momentum" operators acting on thetensor productspace, which has dimension. The action of the total angular momentum operator on this space constitutes a representation of the
su(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory.
and V2 the (2 j2 + 1)-dimensional vector space spanned by the states
The tensor product of these spaces, V3 ≡ V1 ⊗ V2, has a (2 j1 + 1) (2 j2 + 1)-dimensional uncoupled basis
Angular momentum operators are defined to act on states in V3 in the following manner:
where 1 denotes the identity operator.
The total^[1] angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on V1⊗V2,
The total angular momentum operators can be shown to satisfy the very same commutation relations,
where k, l, m ∈ {x, y, z}. Indeed, the preceding construction is the standard method^[5] for constructing an action of a Lie algebra on a tensor product representation.
Hence, a set of coupled eigenstates exist for the total angular momentum operator as well,
for M ∈ {−J, −J + 1, …, J}. Note that it is common to omit the [j1 j2] part.
The total angular momentum quantum number J must satisfy the triangular condition that
such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle.^[6]
The total number of total angular momentum eigenstates is necessarily equal to the dimension of V3:
As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension, whereranges fromtoin increments of 1.
^[7] As an example, consider the tensor product of the three-dimensional representation corresponding towith the two-dimensional representation with. The possible values ofare thenand. Thus, the
six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional representation.
The goal is now to describe the preceding decomposition explicitly, that is, to explicitly describe basis elements in the tensor product space for each of the component representations that arise.
The total angular momentum states form an orthonormal basis of V3:
These rules may be iterated to, e.g., combine n doublets (s=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle),
whereis the integerfloor function; and the number preceding the boldface irreducible representation dimensionality (2j+1) label indicates multiplicity of that representation in the representation
reduction.^[8] For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, .
Formal definition of Clebsch–Gordan coefficients
The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis
The expansion coefficients
are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as ⟨j1 j2; m1 m2|J M⟩. Another common notation is ⟨j1 m1 j2 m2 | J M⟩ = CJMj1m1j2m2.
to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when
The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem in 1941.
Applying the total angular momentum raising and lowering operators
to the left hand side of the defining equation gives
Applying the same operators to the right hand side gives
where C± was defined in 1. Combining these results gives recursion relations for the Clebsch–Gordan coefficients:
Taking the upper sign with the condition that M = J gives initial recursion relation:
In the Condon–Shortley phase convention, one adds the constraint that
(and is therefore also real).
The Clebsch–Gordan coefficients ⟨j1 m1 j2 m2 | J M⟩ can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to
the requirement that the norm of the state |[j1 j2] J J⟩ must be one.
The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with M = J − 1. Repeated use of that equation gives all coefficients.
This procedure to find the Clebsch–Gordan coefficients shows that they are all real in the Condon–Shortley phase convention.
These are most clearly written down by introducing the alternative notation
The first orthogonality relation is
(derived from the fact that 1 ≡ ∑x |x⟩ ⟨x|) and the second one is
For J = 0 the Clebsch–Gordan coefficients are given by
For J = j1 + j2 and M = J we have
For j1 = j2 = J / 2 and m1 = −m2 we have
For j1 = j2 = m1 = −m2 we have
For j2 = 1, m2 = 0 we have
A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using 3. The symmetry properties of Wigner 3-j symbols are much simpler.
Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore (−1)2k is not necessarily 1 for a given quantum number k unless it can be
proven to be an integer. Instead, it is replaced by the following weaker rule:
for any angular-momentum-like quantum number k.
Nonetheless, a combination of ji and mi is always an integer, so the stronger rule applies for these combinations:
This identity also holds if the sign of either ji or mi or both is reversed.
It is useful to observe that any phase factor for a given (ji, mi) pair can be reduced to the canonical form:
where a ∈ {0, 1, 2, 3} and b ∈ {0, 1} (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this
form is only locally canonical: it fails to take into account the rules that govern combinations of (ji, mi) pairs such as the one described in the next paragraph.)
An additional rule holds for combinations of j1, j2, and j3 that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:
This identity also holds if the sign of any ji is reversed, or if any of them are substituted with an mi instead.
Relation to Wigner 3-j symbols
Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations.
The factor (−1)2 j2 is due to the Condon–Shortley constraint that ⟨j1 j1 j2 (J − j1)|J J⟩ > 0, while (–1)J − M is due to the time-reversed nature of |J M⟩.
Relation to Wigner D-matrices
Relation to spherical harmonics
In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics:
It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms a single spherical
SU(n) Clebsch–Gordan coefficients
For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the special unitary group are
known.^[9]^[10] In particular, SU(3) Clebsch-Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists that
relates the up, down, and strange quarks.^[11]^[12] A web interface for tabulating SU(N) Clebsch–Gordan coefficients ^[17] is readily available.
• 3-j symbol
• 6-j symbol
• 9-j symbol
• Racah W-coefficient
• Spherical harmonics
• Spherical basis
• Tensor products of representations
• Associated Legendre polynomials
• Angular momentum coupling
• Total angular momentum quantum number
• Azimuthal quantum number
• Table of Clebsch–Gordan coefficients
• Wigner D-matrix
• Wigner–Eckart theorem
• Angular momentum diagrams (quantum mechanics)
• Clebsch–Gordan coefficient for SU(3)
• Littlewood-Richardson coefficient
Citation Linkopenlibrary.orgThe word "total" is often overloaded to mean several different things. In this article, "total angular momentum" refers to a generic sum of two angular momentum operators
j1 and j2. It is not to be confused with the other common use of the term "total angular momentum" that refers specifically to the sum of orbital angular momentum and spin.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgGreiner, Walter; Müller, Berndt (1994). Quantum Mechanics: Symmetries (2nd ed.). Springer Verlag. ISBN 978-3540580805.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgEdmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-07912-7.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgCondon, Edward U.; Shortley, G. H. (1970). "Ch. 3". The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 978-0-521-09209-8.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgHall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer,
ISBN 978-3319134666 Section 4.3.2
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgMerzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). John Wiley. pp. 428–9. ISBN 978-0-471-88702-7.
Sep 22, 2019, 12:06 AM
Citation Link//doi.org/10.1142%2FS0217732392001270Zachos, C K (1992). "Altering the Symmetry of Wavefunctions in Quantum Algebras and Supersymmetry". Modern Physics Letters. A7 (18): 1595–1600.
arXiv:hep-th/9203027. Bibcode:1992MPLA....7.1595Z. doi:10.1142/S0217732392001270.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgAlex, A.; Kalus, M.; Huckleberry, A.; von Delft, J. (2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math.
Phys. 82 (2): 023507. arXiv:1009.0437. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgKaplan, L. M.; Resnikoff, M. (1967). "Matrix products and explicit 3, 6, 9, and 12j coefficients of the regular representation of SU(n)". J. Math. Phys. 8 (11): 2194.
Bibcode:1967JMP.....8.2194K. doi:10.1063/1.1705141.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgde Swart, J. J. (1963). "The Octet model and its Clebsch-Gordan coefficients". Rev. Mod. Phys. (Submitted manuscript). 35 (4): 916. Bibcode:1963RvMP...35..916D.
Sep 22, 2019, 12:06 AM
Citation Linkopenlibrary.orgKaeding, Thomas (1995). "Tables of SU(3) isoscalar factors". Atomic Data and Nuclear Data Tables. 61 (2): 233–288. arXiv:nucl-th/9502037. Bibcode:1995ADNDT..61..233K.
Sep 22, 2019, 12:06 AM
Citation Linkpdg.lbl.gov"Review of Particle Physics: Clebsch-Gordan coefficients, spherical harmonics, and d functions"
Sep 22, 2019, 12:06 AM
Citation Linkwww.volya.netClebsch–Gordan, 3-j and 6-j Coefficient Web Calculator
Sep 22, 2019, 12:06 AM
Citation Linkphys.csuchico.eduDownloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows
Sep 22, 2019, 12:06 AM
Citation Linkhomepages.physik.uni-muenchen.deWeb interface for tabulating SU(N) Clebsch–Gordan coefficients
Sep 22, 2019, 12:06 AM
Citation Linkhomepages.physik.uni-muenchen.deweb interface for tabulating SU(N) Clebsch–Gordan coefficients
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In a Binary Tree, find sum of all the nodes below certain level
Given a binary tree, find sum of all the nodes below certain level.
Root is at level 0.
All the children of root are at level 1.
All the grand children of root are at level 2.
and so on.
Given a level n, find some of all the nodes that are at level n or more.
int Sum(Node root, level n)
return 0;
return Sum(root->left, n-1) + Sum(root->right, n-1);
return root->data + Sum(root->left, n-1) + Sum(root->right, n-1);
No comments:
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Section VO Vector Operations
In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of
complex numbers, written vertically (Definition CV). The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition.
Subsection CV Column Vectors
Definition VSCV Vector Space of Column Vectors
The vector space $\complex{m}$ is the set of all column vectors (Definition CV) of size $m$ with entries from the set of complex numbers, $\complex{\null}$.
When a set similar to this is defined using only column vectors where all the entries are from the real numbers, it is written as ${\mathbb R}^m$ and is known as Euclidean $m$-space.
The term vector is used in a variety of different ways. We have defined it as an ordered list written vertically. It could simply be an ordered list of numbers, and perhaps written as $\left\langle
2,\,3,\,-1,\,6\right\rangle$. Or it could be interpreted as a point in $m$ dimensions, such as $\left(3,\,4,\,-2\right)$ representing a point in three dimensions relative to $x$, $y$ and $z$ axes.
With an interpretation as a point, we can construct an arrow from the origin to the point which is consistent with the notion that a vector has direction and magnitude.
All of these ideas can be shown to be related and equivalent, so keep that in mind as you connect the ideas of this course with ideas from other disciplines. For now, we will stick with the idea that
a vector is just a list of numbers, in some particular order.
Subsection VEASM Vector Equality, Addition, Scalar Multiplication
We start our study of this set by first defining what it means for two vectors to be the same.
Definition CVE Column Vector Equality
Suppose that $\vect{u},\,\vect{v}\in\complex{m}$. Then $\vect{u}$ and $\vect{v}$ are equal, written $\vect{u}=\vect{v}$ if \begin{align*} \vectorentry{\vect{u}}{i}&=\vectorentry{\vect{v}}{i} &&1\leq
i\leq m \end{align*}
Now this may seem like a silly (or even stupid) thing to say so carefully. Of course two vectors are equal if they are equal for each corresponding entry! Well, this is not as silly as it appears. We
will see a few occasions later where the obvious definition is not the right one. And besides, in doing mathematics we need to be very careful about making all the necessary definitions and making
them unambiguous. And we have done that here.
Notice now that the symbol “=” is now doing triple-duty. We know from our earlier education what it means for two numbers (real or complex) to be equal, and we take this for granted. In Definition SE
we defined what it meant for two sets to be equal. Now we have defined what it means for two vectors to be equal, and that definition builds on our definition for when two numbers are equal when we
use the condition $u_i=v_i$ for all $1\leq i\leq m$. So think carefully about your objects when you see an equal sign and think about just which notion of equality you have encountered. This will be
especially important when you are asked to construct proofs whose conclusion states that two objects are equal. If you have an electronic copy of the book, such as the PDF version, searching on
“Definition CVE” can be an instructive exercise. See how often, and where, the definition is employed.
OK, let us do an example of vector equality that begins to hint at the utility of this definition.
We will now define two operations on the set $\complex{m}$. By this we mean well-defined procedures that somehow convert vectors into other vectors. Here are two of the most basic definitions of the
entire course.
Definition CVA Column Vector Addition
Suppose that $\vect{u},\,\vect{v}\in\complex{m}$. The sum of $\vect{u}$ and $\vect{v}$ is the vector $\vect{u}+\vect{v}$ defined by \begin{align*} \vectorentry{\vect{u}+\vect{v}}{i} &=\vectorentry{\
vect{u}}{i}+\vectorentry{\vect{v}}{i} &&1\leq i\leq m \end{align*}
So vector addition takes two vectors of the same size and combines them (in a natural way!) to create a new vector of the same size. Notice that this definition is required, even if we agree that
this is the obvious, right, natural or correct way to do it. Notice too that the symbol `+' is being recycled. We all know how to add numbers, but now we have the same symbol extended to double-duty
and we use it to indicate how to add two new objects, vectors. And this definition of our new meaning is built on our previous meaning of addition via the expressions $u_i+v_i$. Think about your
objects, especially when doing proofs. Vector addition is easy, here is an example from $\complex{4}$.
Example VA Addition of two vectors in $\complex{4}$
Our second operation takes two objects of different types, specifically a number and a vector, and combines them to create another vector. In this context we call a number a scalar in order to
emphasize that it is not a vector.
Definition CVSM Column Vector Scalar Multiplication
Suppose $\vect{u}\in\complex{m}$ and $\alpha\in\complex{\null}$, then the scalar multiple of $\vect{u}$ by $\alpha$ is the vector $\alpha\vect{u}$ defined by \begin{align*} \vectorentry{\alpha\vect
{u}}{i} &=\alpha\vectorentry{\vect{u}}{i} &&1\leq i\leq m \end{align*}
Notice that we are doing a kind of multiplication here, but we are defining a new type, perhaps in what appears to be a natural way. We use juxtaposition (smashing two symbols together side-by-side)
to denote this operation rather than using a symbol like we did with vector addition. So this can be another source of confusion. When two symbols are next to each other, are we doing regular old
multiplication, the kind we have done for years, or are we doing scalar vector multiplication, the operation we just defined? Think about your objects — if the first object is a scalar, and the
second is a vector, then it must be that we are doing our new operation, and the result of this operation will be another vector.
Notice how consistency in notation can be an aid here. If we write scalars as lower case Greek letters from the start of the alphabet (such as $\alpha$, $\beta$, …) and write vectors in bold Latin
letters from the end of the alphabet ($\vect{u}$, $\vect{v}$, …), then we have some hints about what type of objects we are working with. This can be a blessing and a curse, since when we go read
another book about linear algebra, or read an application in another discipline (physics, economics, …) the types of notation employed may be very different and hence unfamiliar.
Again, computationally, vector scalar multiplication is very easy.
Subsection VSP Vector Space Properties
With definitions of vector addition and scalar multiplication we can state, and prove, several properties of each operation, and some properties that involve their interplay. We now collect ten of
them here for later reference.
Theorem VSPCV Vector Space Properties of Column Vectors
Suppose that $\complex{m}$ is the set of column vectors of size $m$ (Definition VSCV) with addition and scalar multiplication as defined in Definition CVA and Definition CVSM. Then
• ACC Additive Closure, Column Vectors
If $\vect{u},\,\vect{v}\in\complex{m}$, then $\vect{u}+\vect{v}\in\complex{m}$.
• SCC Scalar Closure, Column Vectors
If $\alpha\in\complex{\null}$ and $\vect{u}\in\complex{m}$, then $\alpha\vect{u}\in\complex{m}$.
• CC Commutativity, Column Vectors
If $\vect{u},\,\vect{v}\in\complex{m}$, then $\vect{u}+\vect{v}=\vect{v}+\vect{u}$.
• AAC Additive Associativity, Column Vectors
If $\vect{u},\,\vect{v},\,\vect{w}\in\complex{m}$, then $\vect{u}+\left(\vect{v}+\vect{w}\right)=\left(\vect{u}+\vect{v}\right)+\vect{w}$.
• ZC Zero Vector, Column Vectors
There is a vector, $\zerovector$, called the zero vector, such that $\vect{u}+\zerovector=\vect{u}$ for all $\vect{u}\in\complex{m}$.
• AIC Additive Inverses, Column Vectors
If $\vect{u}\in\complex{m}$, then there exists a vector $\vect{-u}\in\complex{m}$ so that $\vect{u}+ (\vect{-u})=\zerovector$.
• SMAC Scalar Multiplication Associativity, Column Vectors
If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in\complex{m}$, then $\alpha(\beta\vect{u})=(\alpha\beta)\vect{u}$.
• DVAC Distributivity across Vector Addition, Column Vectors
If $\alpha\in\complex{\null}$ and $\vect{u},\,\vect{v}\in\complex{m}$, then $\alpha(\vect{u}+\vect{v})=\alpha\vect{u}+\alpha\vect{v}$.
• DSAC Distributivity across Scalar Addition, Column Vectors
If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in\complex{m}$, then $(\alpha+\beta)\vect{u}=\alpha\vect{u}+\beta\vect{u}$.
• OC One, Column Vectors
If $\vect{u}\in\complex{m}$, then $1\vect{u}=\vect{u}$.
Many of the conclusions of our theorems can be characterized as “identities,” especially when we are establishing basic properties of operations such as those in this section. Most of the properties
listed in Theorem VSPCV are examples. So some advice about the style we use for proving identities is appropriate right now. Have a look at Proof Technique PI.
Be careful with the notion of the vector $\vect{-u}$. This is a vector that we add to $\vect{u}$ so that the result is the particular vector $\zerovector$. This is basically a property of vector
addition. It happens that we can compute $\vect{-u}$ using the other operation, scalar multiplication. We can prove this directly by writing that \begin{equation*} \vectorentry{\vect{-u}}{i} =-\
vectorentry{\vect{u}}{i} =(-1)\vectorentry{\vect{u}}{i} =\vectorentry{(-1)\vect{u}}{i} \end{equation*} We will see later how to derive this property as a consequence of several of the ten properties
listed in Theorem VSPCV.
Similarly, we will often write something you would immediately recognize as “vector subtraction.” This could be placed on a firm theoretical foundation — as you can do yourself with Exercise VO.T30.
A final note. Property AAC implies that we do not have to be careful about how we “parenthesize” the addition of vectors. In other words, there is nothing to be gained by writing $\left(\vect{u}+\
vect{v}\right)+\left(\vect{w}+\left(\vect{x}+\vect{y}\right)\right)$ rather than $\vect{u}+\vect{v}+\vect{w}+\vect{x}+\vect{y}$, since we get the same result no matter which order we choose to
perform the four additions. So we will not be careful about using parentheses this way.
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Insegnamento a.a. 2019-2020 - Universita' Bocconi
Insegnamento a.a. 2019-2020
20594 - ECONOMETRICS FOR BIG DATA
Department of Economics
Course taught in English
Mission & Content Summary
The main goal of this course is to give students a working knowledge of the most used econometric techniques. The key concepts of statistical theory underlying each method are covered, but special
emphasis is placed on implementation of each method in actual applications. Students also receive an introduction on how to conduct practical econometric analysis of actual data using software and
programming languages such as Stata and Matlab. The course is divided in two parts. The first, deals with regression and causal inference methods used in the analysis of cross-sectional and
longitudinal data, typically used in micro-econometrics (where the focus is on the individual behavior of individuals, households, firms and so on). The second, deals with time series data and
methods, typically used in macro-economic applications (where the focus is on the interaction of macroeconomic variables). As observational data, most commonly used in non-experimental sciences such
as economics, hardly tell the researcher what is the effect of a certain treatment variable on a given outcome variable of interest, economists have devised a variety of approaches to address
questions of cause-and-effect among economic variables both in microeconomics and macroeconomics. The unifying theme of the two parts of the course is a focus on understanding causality.
Part I (Microeconometrics):
• Introduction to Microeconometrics: The experimental ideal.
• Core Regression: Fundamentals of linear regression.
• Core Regression: Regression and causality.
• Core Regression: Heterogeneity and nonlinearity.
• Core Regression: Some Extensions to weighting and limited dependent variables.
• Instrumental Variables (IV): Classical IV and Two Stages Least Squares.
• Instrumental Variables (IV): Heterogeneous Potential Outcomes and Local Average Treatment Effect.
• Parallel Worlds: Individual fixed effects; Differences-in-differences; Panel data methods.
• Regression Discontinuity Designs (RDD): Sharp and Fuzzy RDD; relationship with IV.
• More on Regressions: Types and choice of loss functions; Quantile regression.
• More on Regressions: Parametric vs. non-parametric specification and estimation.
• Impact of Machine Learning and Big Data on Microeconometric Analysis.
Part II (Macroeconometrics):
• Why multivariate models.
• Simultaneous equation bias. The problem of identification.
• The Sims critique to old Keynesian macroeconometrics.
• VAR models.
• Granger causality (application: Sims, 1972).
• Structural VAR and identification (applications: a) Sims, 1980, b) Blanchard-Quah, 1989 and Gali, 1999. c) news shocks).
• External instruments.
• A first pass on Big Data: principal components and factor models.
• Dealing with causality. A microfounded model: the Real Business Cycle model.
• Estimating a microfounded model: using the Kalman filter to build the likelihood function.
Intended Learning Outcomes (ILO)
At the end of the course student will be able to...
Part I (Microeconometrics):
• Define key concepts in econometrics, for instance “econometric causality;” “fundamental problem of causal inference;” “average treatment effect;” “instrumental variable;” “loss function.”
• Explain key differences and links between distinct but related econometric concepts, for instance “identification and statistical inference;” “potential, realized, and counterfactual outcomes;”
“average treatment effect, average treatment effect on the treated, and average treatment effect on the untreated;” “classical instrumental variables estimator and local average treatment effect
estimator;” “sharp and fuzzy regression discontinuity designs;” “parametric, semiparametric, and nonparametric specification of an econometric model.”
• For each causal method covered during the course, describe the roles of the key assumptions underlying the method in yielding identification of the causal effect parameter of interest.
Distinguish untestable assumptions from testable ones, and describe how to test the latter. Discuss the main consequence/s of the failure of each assumption, illustrating with specific examples
or applications.
• For each causal method covered during the course, describe the basic data requirements for application of each method and discuss advantages and disadvantages of each method.
Part II (Macroeconometrics):
• Be familiar with the main concepts and tools of time series analysis and use them in other contexts.
• Understand a vast majority of the scientific literature on time-series and macroeconometrics.
• Identify what are the modelling assumptions underlying any structural macroeconometric model.
• Translate the main assumptions in economic theories into restrictions on the empirical statistical model.
At the end of the course student will be able to...
Part I (Microeconometrics):
• Design a simple randomized experiment to quantify the average causal effect of a treatment variable on an outcome variable of interest.
• Given a specified microeconomic application, a data set, and a causal parameter of interest, select the most appropriate microeconometric method among those covered in class.
• Implement the chosen method to quantify the causal parameter of interest and test hypotheses about the parameter’s true value, using actual data and a statistical software covered in class.
• Evaluate the causal effect(s) of a policy intervention or program, using given microdata and the most appropriate econometric method among those covered in class.
Part II (Macroeconometrics):
• Perform empirical analysis to uncover the effects of shocks in the economy.
• Design a well-functioning empirical macroeconometric model.
• Communicate effectively the empirical results of his/her analysis.
• Use programming software, Matlab, to perform different kind of time-series analyses.
• Do empirical analysis in a constructive way and think critically.
Teaching methods
• Face-to-face lectures
• Exercises (exercises, database, software etc.)
The learning experience of the course includes:
• Face-to-face lectures, introducing and illustrating the main topics of the course.
• Interactive in-class discussions around stylized microeconomic and macroeconomic applications, focusing on specific aspects of their implementation and interpretation.
• Students also be exposed to statistical/econometric software, such as Stata and Matlab. This allow them to get a hands-on introduction to the programming techniques that are a necessary part of
any econometric study.
Assessment methods
Continuous assessment Partial exams General exam
• Written individual exam (traditional/online) x x
To the end of measuring the acquisition of the above-mentioned learning outcomes, the students’ assessment is based on a written final exam accounting for 100% of the final grade.
• The exam consists of exercises, open-ended and/or closed-ended questions.
• The exam assesses students’ knowledge and understanding of the topics and students’ ability to carry out and interpret results from empirical econometric work.
Teaching materials
The main course material for both attending and non-attending students is:
Main textbook for Part I (Microeconometrics):
• J.D. ANGRIST, S. PISCHKE, Mostly Harmless Econometrics, Princeton University Press, 2009.
Main textbook for Part II (Macroeconometrics):
• L. SALA, Lecture Notes on Time Series Analysis, (available on Bboard).
• W. ENDERS, Applied Econometric Time Series, last edition, selected chapters.
• J.D. HAMILTON, Time Series Analysis, Princeton University Press, 1994, selected chapters.
The slides of the course, additional readings and support material are uploaded to the Bboard platform of the course.
Last change 05/06/2019 21:26
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Why is math important and How it is Applicable in Real Life
Math is not only crucial for passing a grade, but it is also beneficial practically. From performing complex tasks such as tracking your finances to shopping and cooking, math makes your life easy.
In short, it is a language that helps you describe the world around you and make sense of it.
10 Reasons That Tell Us ‘Why is Math Important?’
Of course, math is used to solve problems in science, technology, and engineering. It’s also used to design, build and test new technologies such as computers, cell phones, and cars. Here are other
uses that emphasize the importance of math.
Understanding Abstract Concepts
Math is a universal language that is used to understand many things. It helps in understanding the world around us and our place in it, which can provide a sense of comfort or peace. Math also allows
us to see how we fit into this more extensive system of nature and life around us. It helps you concretely understand abstract concepts like love or beauty.
Builds your brain’s capacity for critical thinking
Math is an excellent tool for building your brain’s capacity for problem-solving, critical thinking, and creativity. These skills are essential in any career but instrumental in STEM fields (science,
technology, engineering, and math).
Keeping your finances in check
It gets challenging to manage your money without math. Suppose you are applying for a loan. You will need math to understand its interest rate, monthly payments, and total cost. For example: If you
borrow $10,000 at 5% APR for 60 months (five years), it will cost you $1,000 per month ($120 per week). The total amount paid back by the end of this period will be $11,955 – about $600 more than
what was borrowed!
Math Helps in Solving Real-world Problems
You can use math to solve so many real-world problems. From simple ones like finding your way home after work to complex issues like designing an airplane wing or treating cancer with radiation
therapy. Some math skills focus on logic, while others focus on patterns or relationships between numbers. Students often move from learning basic arithmetic to solving complex problems using
calculus or linear algebra.
It’s a Great Way to Release Stress and Anxiety
Math is a great way to release stress and anxiety. It’s also an excellent way to clear your mind, as math can help keep you from dwelling on it. By building your brain’s capacity for problem-solving
and critical thinking skills, math will make you better at solving real-world problems. Therefore, it will keep the stress and anxiety away!
Helps in doing things efficiently
You can only get through the day by working with numbers. Whether you’re planning your schedule or keeping track of your finances, math is all around us. It’s so prevalent that we don’t even realize
it anymore. You might be surprised by just how much math plays a role in our daily lives, even if we don’t consider ourselves particularly mathematically inclined.
For example, you use statistics when deciding which route to take home after work. One road has fewer traffic lights than another but not as many stop signs; therefore, it will likely take less time
overall if you choose that route.
It Expands Our Understanding of How Things Work
Math is the language of science. It’s used to do science, explain science, predict science, and study all the different fields today. Without math, there would be no way for scientists to communicate
their findings with each other or even understand what they are observing in their experiments.
Math also helps us understand how things work in our everyday lives. For example, if you’re baking cookies and want them ready at exactly 8:30 p.m., you’ll set the temp of the oven at 350 degrees F
to cook it in less than 30 minutes. Wait, how do you know that? The math, of course!
Math Helps You Do Things Efficiently
It is used in the real world, like playing sports, traveling by car, and more. For example, you can use math to tell how far a ball will go when you throw it. The distance depends on how fast or slow
you throw it and where exactly you release the ball from your hand.
Furthermore, math helps us calculate distances between two points on Earth and how high an object has traveled above Earth.
Math Helps in Creating New Things
You can’t create anything without math. It is the language of science, and it’s used in every field, from engineering to architecture to music and art. Even if you’re not a scientist or engineer,
math will help you understand how things work and how they can improve.
Useful for carrying out various tasks efficiently
• Indeed, you can use math for so many things, including:
• For calculating the square footage
• To measure things like lumber and carpeting.
• For determining how much paint you need, fertilizer for your plants, or water your plants need.
• When building things like furniture or putting together
• You can use math to measure how much material your project needs and determine where it needs to go. You can also use it as a guide for putting things together and ensuring they’re level and
• Math is used to predict future events based on past data and to show relationships between different elements.
Tips and Tricks to get better at math
Not everyone is a math person. There are probably more people out there who aren’t math people than those who are! If you don’t feel like you have a knack for understanding numbers and equations,
don’t worry, you’re not alone. However, it doesn’t mean you can’t do anything about it, as we have some fantastic tips and tricks for you to get better at math.
Use Mnemonics or Other Ways of Remembering Math Facts
A system is one of the best ways to learn math facts. A good example is flashcards, which help you memorize the numbers and their corresponding operations. You can also use mnemonic devices like
“Every Good Boy Does Fine,” which helps you remember that 2 + 3 = 5 and more of such.
Moreover, if you’re having trouble remembering the difference between square roots and radicals, try square root = rt (root), Radical = rad. You can also use flashcards to practice problem-solving
techniques and standard formulas.
Improve Your Basic Understanding
To get better at math, you need to learn the basics first. You can do this by memorizing multiplication and division facts (e.g., 2 x 3 = 6) until they’re second nature. If you have trouble
remembering them all, try using mnemonics to help you remember them. It’s also important not to get frustrated if it takes time: learning new things takes practice!
Find a Study Buddy
Studying with a friend is a great way to learn. You can help each other understand the material, and you’ll have someone to challenge when it comes time for exams. If you’re struggling with a
particular concept or problem, your friend can point out where things went wrong so that you can avoid making the same mistake again. On top of that, if one person has already mastered a topic, but
their partner still needs to learn it, they can motivate their friend by offering encouragement and praise when they get answers right.
Have Fun with Math
You might think math is boring, but you can apply it to real-life scenarios. You might not realize this as a child, but math is used in sports and games like baseball or basketball. Math is also used
in puzzles and riddles we see daily on websites.
Many artists have used math, such as Leonardo Da Vinci (who made models of helicopters) or Pablo Picasso (who drew inspiration from geometry). Musicians rely heavily on numbers when composing music
because they know how specific notes will sound together based on their ratios with other notes within an octave scale system.
Invest More Time in Practicing your Skills
If you struggle with math, remember that practice and patience is the only way to improve at something. It may take some time, but you’ll get better at it if you keep pushing yourself more each day.
It’s also important to note that there are no stupid questions regarding math (or any subject). Feel free to ask your teacher or classmates for clarification if something doesn’t make sense at first
Don’t be Afraid to Ask for Help
There are many ways to improve math; some involve asking for help. Be bold and ask your teacher or a friend if they can walk you through a problem that troubles you. The more often you ask for help
when needed; the better equipped you’ll be in the future when facing similar problems on your own.
The best part is once someone shows us how something works; we tend not only to remember it but also understand why specific steps are necessary for solving some problems. It makes us more confident
about tackling new challenges because now we know where all those numbers come from!
Ask questions
Asking questions is a great way to ensure you understand the lesson but keep it from getting boring. Ask questions that can be answered by the teacher, and try to ask ones that will help you learn
more. Furthermore, when you’re done working on a math problem, always check the solution and ensure you did it right. It helps prevent careless mistakes and ensures errors are caught before they
become a big deal.
When checking your answers, ask yourself: Did I use the correct formula? Did I do everything correctly? Does this make sense? Is there anything else I should consider when looking at this problem? If
so, what could it be?
Hold on, as there’s more on the importance of math on your way as you have marched to the end of this exciting guide. You may need free math assignments if you are paying almost nothing and availing
expert mathematicians’ help in return.
Math is everywhere. It’s a universal language we all speak, and it can help us navigate our lives in many ways. Whether figuring out how much money we have left over at the end of the month or doing
some basic arithmetic while cooking dinner, math is always there when we need it most. This exciting blog post was about why math is important and how to improve your mathematical skills. We hope you
have enjoyed reading it to the end.
Furthermore, if you feel that the recent math assignment is too challenging to tackle, consider pressing the order now to get our professional writer to handle it.
Frequently Asked Questions
Why is Math Important?
Math is a way of expressing and understanding the world around us. It helps us see natural patterns, understand how things work, and predict what will happen next. Math also teaches you to think
critically and logically, essential for success in any field. It is crucial because it's how we communicate with each other, make sense of things and understand the world around us.
How to get better at math?
The most important thing you can do to improve your math skills is to keep doing it. But it's also essential that you practice correctly. Work on problems with similar structures repeatedly until
they feel natural. Learning new things helps our brains build connections between different concepts so that, eventually, one concept triggers another without us being aware of the connection
How to learn Math Skills Faster?
When you first start learning math, it can feel like a boring subject just there to make your life harder. But once you get past the basics and learn more about how math works in the real world,
you'll see it's helpful! Knowing how numbers and equations interact is one of the essential real-life skills anyone can have. Remember that it's not just about memorizing formulas and facts; it's
also about using those skills to solve real-world problems.
What is Algebra?
Algebra is a branch of mathematics that deals with symbols and their relationships. In other words, algebra is the language of science. We can understand and study anything with algebra because it's
used everywhere, from physics to biology to sociology!
Why should students learn math?
Students should learn math for:
- Developing their brain
- Improving their analytical skills
- Becoming more practical and beneficial in different situations
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Mayer f-function
The Mayer f-function, or f-bond is defined as (Ref. 1 Chapter 13 Eq. 13.2):
${\displaystyle f_{12}=f({\mathbf {r} }_{12}):=\exp \left(-{\frac {\Phi _{12}(r)}{k_{B}T}}\right)-1}$
In other words, the Mayer function is the Boltzmann factor of the interaction potential, minus one.
Diagrammatically the Mayer f-function is written as
Hard sphere model[edit]
For the hard sphere model the Mayer f-function becomes:
${\displaystyle f_{12}=\left\{{\begin{array}{lll}-1&;&r_{12}\leq \sigma ~~({\rm {overlap}})\\0&;&r_{12}>\sigma ~~({\rm {no~overlap}})\end{array}}\right.}$
where ${\displaystyle \sigma }$ is the hard sphere diameter.
1. Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940)
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Who Invented the Digital Computer?
I want to take you back to 1830, and tell you an amazing story from the time of the industrial revolution.
A well known British mathematician, named Charles Babbage, came up with the idea of a machine that could compute polynomials of a variable, i.e. functions like a[0 ]+ a[1]x + a[2]x^2+…+a[n]x^n. By
that time the importance of such functions had already been demonstrated in the work of the French physicist J.B. Fourier (1822). Fourier was studying the problem of heat propagation. He discovered
what is now referred to in the fields of physics and mathematics as the ‘Fourier series’ or ‘Fourier transform’. His discovery was that one can approximate a general function using sums of periodic
functions like sin(nx) and cos(nx), in which n is an integer. In a sense, these periodic functions serve as a basis for the approximation. It was well known then, that polynomial functions may be
used in a similar way. Therefore, a machine that computes polynomial functions could be used to compute an approximation for almost any function.
In the machine’s design, Babbage used a method of differences. For example, to compute f(x)= 2x^2 +3x +1 for any x, it is enough to compute the value of the function for some low x’s like f(1), f(2),
f(3) and f(4), the differences between those values: f(2)-f(1), f(3)-f(2), f(4)-f(3), and the differences of the differences: (f(3)-f(2))-(f(2)- f(1)) and (f(4)-f(3))- (f(3)-f(2)). In this way, it is
possible to compute the value of the function at any high x. To compute f(15), the machine works through 15 cycles, at each cycle only adding ‘differences’. The machine is activated by a crank, the
operator rotating a handle the necessary amount of times (in this example, 15 times). In the spirit of his time, Babbage thought of using a steam engine to work the crank and handle, so to make the
operation of the machine easier.
In order to help clarify this principle, we bring the following table for our function
f(x)= 2x^2 +3x +1 :
│x│f(x)│First difference │Second difference │
│1│6 │ │ │
│2│15 │9 │ │
│3│28 │13 │4 │
│4│45 │17 │4 │
│5│66 │21 │4 │
The machine has three cogwheels. The operator places the left wheel at state 6, the middle wheel at state 9 and the right wheel at state 4 (this corresponds to the upper diagonal in the table above).
At the first half cycle, a clutch (like the one in your car) joins the left and middle wheels. The machine turns the middle wheel to zero position, that is, 9 steps clockwise, moving the left wheel 9
steps counterclockwise – bringing the left wheel to state 15. At this point, the clutch is drawn back and a spring returns the middle wheel back to its original position – state 9. At the next half
cycle, the clutch joins the middle and right wheels. The machine turns the right wheel 4 steps clockwise, moving the middle wheel 4 steps counterclockwise to position 13. A spring now returns the
right wheel to its original position – state 4. This completes a full cycle and brings each of the cogwheels to its next position (the second diagonal from the top).
A portion of a reconstructed
Difference Engine, built from
Babbage’s working drawings in 1991
Babbage called the machine a “difference engine “. He made a small model of it and presented it at his evening cocktail parties, as the “thinking machine”. However, he could not build the complete
machine. At the time, even the most advanced technology available was far from satisfactory for the needs of such a machine. Babbage himself manufactured some of the parts necessary for the machine.
In addition to a generous fund granted to him by the British government for the purpose of building such a machine, Babbage also invested all of his savings in the project.
In 1832, Babbage was a highly influential scientist in England, a well known professor at Cambridge, a member of Parliament, a member of the Royal Society (of sciences), and a famous figure in the
world of economy. The next machine he was about to invent, the ‘analytical engine’, was an information- processing machine, which reflected his background in economy.
Babbage was thinking about the principle of ‘division of labor’, an economic term that had been in use since the time of Adam Smith (see the “Wealth of Nations”, 1776). According to this principle,
the most efficient way to manufacture a product is to divide the work between several workers, each specializing in a certain aspect of the process. The worst, least efficient way is to let each one
of the workers do the whole job.
A portion of the ‘mill’ of the
analytical engine built in 1871 by
Henry Babbage in honor of his father
Towards the end of the 18th century this idea was taken seriously by Gaspard de Prony, who was in charge of the new tax and measurement system in France. De Prony conceived the idea of building a
huge information ‘machine’. This ‘machine’ was based on distributing the mathematical work necessary for computing between three groups of people. At the top of the computing pyramid De Prony placed
several famous French mathematicians of that period, including Legendre and Carno, who gave the mathematical formulas and instructions for the computing process. At the middle of the pyramid he
placed less famous mathematicians, whose task was to interpret the formulas to simple computations, and to collect results from the computing section. At the bottom of the pyramid were the ‘human
computers’, whose main task was to compute, mainly adding numbers. At that time, the word ‘computer’ referred to a person whose job was to compute, not to a machine.
Babbage learned about the project during a visit to France. But De Prony’s information machine was not one of a kind. 19^th century banks used ‘Clearing Houses’. These were rooms to which agents from
every branch of each bank in London would arrive in at the end of each day in order to balance checks and transfer money. Babbage was visiting one of these clearing houses just before he came up with
the idea of the analytic engine. This gave him a strong sense that computation includes also manipulation of symbols.
In the analytic engine, the labor of computing is divided between several parts. There is the ‘mill’, where the machine actually calculates. Babbage used notions from the cotton industry. He was
inspired by the automatic looming machines of his time, especially the Jacquard loom.
Another part in the analytical engine is the ‘store’ where numbers are stored, waiting to be used by the machine. A third part of the machine includes a drum. On top of the drum there is a roll of
punched cards. The cards include codes of operations and codes for the places in the ‘store’ where the variables lie. Suppose now the first card includes a code of an operation, such as subtraction
or multiplication, and the second card includes a code for the locations of the variables in the ‘store’. Having read the first card, the machine tunes itself to the appropriate mode (subtraction or
multiplication). After reading the second card, the machine looks for the correct place in the ‘store’. Using the cogwheels, the machine reads numbers from the ‘store’ and multiplies (or subtracts)
them. The next card tells the machine where to store the result. The ‘mill’ uses the cogwheels to write the result in the ‘store’…
Looking at the machine from a distance, one could see the drum revolving and the ‘mill’ moving to and fro in front of the ‘store’. When the computation was completed the result could be read from a
certain point at the ‘store’.
Babbage’s analytical engine includes all the components of a modern computer. There is the memory- the ‘store’, the central processing unit- the ‘mill’, and the program or software- the punched
cards. Babbage understood that one can use numbers as codes or as data. He spent two years figuring out this scheme of computations.
It was a struggle to build the difference engine at that time, so you can imagine how building the analytical engine was virtually impossible. The British government stopped financing the project but
Babbage was obsessed by his invention and seeked recognition outside England. In 1840 he traveled to Italy in order to deliver a lecture about his engines. In Italy Babbage met a mathematician named
Luigi Federico Menabrea, later to become the Italian Prime Minister. Menabrea wrote a paper describing Babbage’s analytical machine, and Babbage’s old friend Ada Lovelace, a mathematician and
daughter of Lord Byron, translated Menabrea’s paper into English and added a mathematical appendix (longer than the paper itself) explaining the operation of the engine. In the appendix Lovelace
wrote that Babbage’s machine weaves numbers like a loom (you may have heard the name ADA, it’s a programming language named after Lady Lovelace).
I would like to conclude with Menabrea’s vision about the future role of the analytical engine, probably quoting Babbage:
“Again, who can foresee the consequences of such an invention? In truth, how many precious observations remain practically barren for the progress of the sciences, because there are not powers
sufficient for computing the results! And what discouragement does the perspective of a long and arid computation cast into the mind of a man of genius, who demands time exclusively for meditation,
and who beholds it snatched from him by the material routine of operations! Yet it is by the laborious rout of analysis that he must reach truth: but he cannot pursue this unless guided by numbers;
for without numbers it is not given us to raise the veil which envelopes the mysteries of nature. Thus the idea of constructing an apparatus capable of aiding human weakness in such researches, is a
conception which, being realized, would mark a glorious epoch in the history of the sciences.”
Babbage never lost his interest in the machine, but became angry, bitter, and frustrated with the British government for what he called a loss of sight. Well, I guess he was right!
Digital computers were forgotten for about a hundred years, from the failure of Babbage’s project to their resurrection in the 1940’s.
The difference engine, Charles Babbage and the quest to build the first computer, by Doron Swade, Penguin Group, 2000.
Jacquard’s web, How a hand loom led to the birth of the information age, by James Essinger, Oxford University Press, 2004.
About the author: Boaz Tamir holds a Ph.D. degree in Mathematics from the Weizmann Institute of Science. In 2008 he recived a second Ph.D. from the department of History and Philosophy of Science at
the Bar Ilan University.
Tagged with : ADA , analytical engine , Charles Babbage , computer , difference engine , digital computer , Gaspard de Prony , Lady Lovelace
About Dr. Boaz Tamir
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Computer Science Algorithms Exam Questions. - Get Neural Net
Computer Science Algorithms Exam Questions
Are you studying computer science and seeking practice questions to solidify your understanding of algorithms? This article provides a comprehensive collection of exam questions that cover various
topics in computer science algorithms.
Key Takeaways:
• Prepare for your computer science algorithms exams with these practice questions.
• Deepen your understanding of fundamental algorithms and their implementation.
• Gain confidence by solving different types of algorithmic problems.
Question 1:
In the Merge Sort algorithm, explain the following steps:
1. Divide the unsorted list into sublists.
2. Repeatedly merge the sublists together.
3. Continue merging until there is only one sorted list.
Question 2:
What are the time and space complexities of the QuickSort algorithm?
The time complexity is O(n log n) in average and best cases, while the worst case time complexity is O(n^2). The space complexity is O(log n).
Table 1: Sorting Algorithms Comparison
Algorithm Time Complexity Stability
Bubble Sort O(n^2) Stable
Selection Sort O(n^2) Unstable
Insertion Sort O(n^2) Stable
Question 3:
What is the purpose of Dijkstra’s algorithm and how does it work?
Dijkstra’s algorithm is used to find the shortest path between two vertices (or nodes) in a graph. It works by iteratively selecting the vertex with the smallest distance from the source vertex and
updating the distances of its neighbors.
Question 4:
Compare and contrast Breadth-First Search (BFS) and Depth-First Search (DFS).
• BFS explores nodes level by level, while DFS explores nodes branch by branch.
• BFS uses a queue data structure, while DFS uses a stack or recursion.
• BFS guarantees the shortest path, while DFS may not find the shortest path.
Table 2: Sorting Algorithms Time Complexity
Algorithm Best Case Average Case Worst Case
Merge Sort O(n log n) O(n log n) O(n log n)
QuickSort O(n log n) O(n log n) O(n^2)
Heap Sort O(n log n) O(n log n) O(n log n)
Question 5:
What is the purpose of the Knapsack problem and how does it relate to dynamic programming?
The Knapsack problem involves optimizing the selection of items to include in a knapsack given their weights and values. It is commonly solved using dynamic programming techniques, such as
memoization or tabulation.
Question 6:
How does the Floyd-Warshall algorithm work and what is its time complexity?
The Floyd-Warshall algorithm is used to find the shortest paths between all pairs of vertices in a weighted graph. It works by constructing a distance matrix and updating it iteratively. The time
complexity of the algorithm is O(V^3), where V is the number of vertices.
Table 3: Sorting Algorithms Stability
Algorithm Stable
Merge Sort Yes
QuickSort No
Heap Sort No
With these practice questions, you can strengthen your knowledge and enhance your problem-solving skills in computer science algorithms. Remember to thoroughly understand the concepts and analyze the
pros and cons of different algorithms. Happy studying!
Common Misconceptions
Misconception 1: Computer Science Algorithms Exam Questions are Only for Geniuses
Contrary to popular belief, computer science algorithms exam questions are not only for geniuses. While these exams may require logical thinking and problem-solving skills, anyone can excel with the
right amount of preparation and practice. It is important to understand that success in this field is not solely determined by innate intelligence.
• Preparation and practice are key to success in computer science algorithms exams.
• Logical thinking and problem-solving skills can be developed and improved with time and effort.
• Success in computer science algorithms exams is not exclusively reserved for geniuses.
Misconception 2: Memorization is Sufficient for Computer Science Algorithms Exams
Many people mistakenly believe that memorization of algorithms is enough to succeed in computer science algorithms exams. While having a good understanding of various algorithms is important, it is
equally crucial to be able to apply them to solve different problems. These exams typically test the ability to analyze, design, and implement algorithms, rather than regurgitate memorized
• Understanding and application of algorithms are essential for success in computer science algorithms exams.
• Memorization alone is not sufficient; the ability to solve problems using algorithms is crucial.
• Exams focus on analyzing, designing, and implementing algorithms, not simple regurgitation.
Misconception 3: Computer Science Algorithms Exams Rely Heavily on Coding
One misconception about computer science algorithms exams is that they heavily rely on coding. While coding is certainly a part of these exams, they also assess other important aspects, such as
algorithm analysis, complexity analysis, and problem decomposition. It is important to have a well-rounded understanding of different aspects of algorithms, rather than just focusing on coding
• Computer science algorithms exams assess various aspects, not just coding skills.
• Algorithm analysis, complexity analysis, and problem decomposition are equally important in these exams.
• A well-rounded understanding of different aspects of algorithms is crucial for success.
Misconception 4: Theoretical Knowledge is Enough for Computer Science Algorithms Exams
Another common misconception is that theoretical knowledge alone is sufficient to excel in computer science algorithms exams. While having a strong theoretical foundation is important, practical
problem-solving skills are equally crucial. These exams often require applying theoretical knowledge to real-world scenarios and developing efficient algorithms to solve complex problems.
• Practical problem-solving skills are just as important as theoretical knowledge in computer science algorithms exams.
• The ability to apply theoretical knowledge to real-world scenarios is a key aspect of the exams.
• Developing efficient algorithms to solve complex problems is a crucial skill to possess.
Misconception 5: Computer Science Algorithms Exams Only Test Knowledge, not Skills
Some people mistakenly believe that computer science algorithms exams are purely knowledge-based and do not assess practical skills. However, these exams are designed to assess both knowledge and
skills. While understanding the concepts is important, the ability to apply that knowledge to solve problems is equally vital. These exams often require critical thinking and creative problem-solving
• Computer science algorithms exams assess both knowledge and practical skills.
• Problem-solving abilities, critical thinking, and creativity are important in these exams.
• Understanding the concepts is crucial, but the application of that knowledge is equally vital.
Table: Most Popular Sorting Algorithms
In computer science, sorting algorithms are used to arrange data in a particular order. This table highlights some of the most commonly used sorting algorithms along with their average time
Algorithm Average Time Complexity
Bubble Sort O(n^2)
Selection Sort O(n^2)
Insertion Sort O(n^2)
Merge Sort O(n log n)
Quick Sort O(n log n)
Heap Sort O(n log n)
Radix Sort O(nk)
Table: Comparative Analysis of Graph Traversal Algorithms
Graph traversal algorithms are used to explore or search through a graph data structure. This table presents a comparison of some popular graph traversal algorithms based on their characteristics.
Algorithm Time Complexity Space Complexity Advantages Disadvantages
Breadth-First Search (BFS) O(V + E) O(V) Guaranteed to find the shortest path in an unweighted graph Requires more memory due to the use of a queue
Depth-First Search (DFS) O(V + E) O(V) Memory-efficient as it uses a stack May get trapped in infinite loops in infinite graphs
Dijkstra’s Algorithm O((V + E) log V) O(V) Finds the shortest path in weighted graphs Does not work with negative edge weights
Table: Run-time Analysis of Dynamic Programming Algorithms
Dynamic programming is a method for solving complex problems by breaking them down into simpler overlapping subproblems. This table presents the run-time complexity of various dynamic programming
Algorithm Time Complexity
Fibonacci O(n)
Knapsack O(nW)
Longest Common Subsequence O(mn)
Table: Comparison of Search Algorithms
Search algorithms are used to find a specific element or item within a given collection of data. This table compares the time complexity and characteristics of different search algorithms.
Algorithm Time Complexity Advantages Disadvantages
Linear Search O(n) Simple, applicable for both sorted and unsorted data Inefficient for large datasets
Binary Search O(log n) Faster than linear search for sorted data Requires the data to be sorted
Hashing O(1) Constant-time access May lead to collisions
Table: Space Complexity of Popular Data Structures
Data structures are used to organize and store data efficiently. This table showcases the space complexity of some commonly used data structures.
Data Structure Space Complexity
Array O(n)
Linked List O(n)
Stack O(n)
Queue O(n)
Binary Search Tree O(n)
Heap O(n)
Hash Table O(n)
Table: Average Case Time Complexity for Matrix Multiplication Algorithms
Matrix multiplication is an essential operation in various scientific and engineering applications. This table presents the average case time complexity of different matrix multiplication algorithms.
Algorithm Average Time Complexity
Naive Algorithm O(n^3)
Strassen’s Algorithm O(n^log2(7)) ≈ O(n^2.81)
Coppersmith-Winograd Algorithm O(n^2.376)
Table: Analysis of Divide and Conquer Algorithms
Divide and conquer is a technique to solve problems by dividing them into smaller subproblems, solving the subproblems recursively, and then combining the results. This table provides insight into
the time and space complexities of various divide and conquer algorithms.
Algorithm Time Complexity Space Complexity
Merge Sort O(n log n) O(n)
Quick Sort O(n log n) O(log n)
Karatsuba Multiplication O(n^log2(3)) ≈ O(n^1.585) O(log n)
Table: Analysis of Greedy Algorithms
Greedy algorithms make locally optimal choices to reach a global optimum in problem-solving. This table examines the time complexity and characteristics of several popular greedy algorithms.
Algorithm Time Complexity Advantages Disadvantages
Activity Selection O(n log n) Easier to implement, provides an optimal solution for certain problems May not always provide the globally optimal solution
Huffman Coding O(n log n) Produces an optimal prefix-free code for encoding data Requires knowledge of probability distribution
Table: Comparison of Various Tree Traversal Algorithms
Tree traversal algorithms are used to visit all nodes in a tree data structure. This table compares the time complexity and characteristics of different tree traversal algorithms.
Algorithm Time Complexity Types Advantages Disadvantages
Pre-order O(n) Pre, In, Post Can be used to create a copy of the tree Does not provide the exact structure of the tree
In-order O(n) Pre, In, Post Provides nodes in ascending order for binary search trees Does not preserve the root position
Post-order O(n) Pre, In, Post Can be used to delete the tree Does not provide the exact structure of the tree
Computer science algorithms play a vital role in solving complex problems efficiently and optimizing the use of computing resources. This article explored various aspects of algorithm design,
including sorting, searching, dynamic programming, graph traversal, and more. Each table provided valuable insights into the different algorithms’ time and space complexities, advantages, and
limitations. By understanding and implementing these algorithms, computer scientists can develop efficient solutions across a wide range of domains, driving advancements in technology and
problem-solving capabilities.
Frequently Asked Questions
Computer Science Algorithms Exam Questions
What are computer science algorithms?
Computer science algorithms refer to step-by-step procedures or instructions designed to solve specific problems or perform certain tasks in the field of computer science. These algorithms are used
to manipulate data, process information, and perform various computations.
Why are algorithms important in computer science exams?
Algorithms form the foundation of computer science and are crucial for understanding and implementing efficient solutions to complex problems. They are extensively tested in computer science exams to
assess students’ knowledge and problem-solving abilities.
What should I expect in a computer science algorithms exam?
A computer science algorithms exam typically consists of theoretical questions related to algorithm design principles, analysis, complexity, and data structures. It may also involve practical
programming problems where you need to implement or analyze algorithms.
How can I prepare for a computer science algorithms exam?
To prepare for a computer science algorithms exam, it is essential to understand the core concepts and principles of algorithm design, analysis, and data structures. Practice solving algorithmic
problems, implement algorithms in programming languages, and review relevant theoretical material.
What are some common types of algorithms tested in computer science exams?
Common types of algorithms tested in computer science exams include sorting algorithms (e.g., bubble sort, merge sort), searching algorithms (e.g., binary search, linear search), graph algorithms
(e.g., depth-first search, breadth-first search), and dynamic programming algorithms.
Can you provide examples of algorithm exam questions?
While specific examples may vary, some common algorithm exam questions may involve analyzing the time complexity of an algorithm, designing an algorithm to solve a specific problem, applying an
algorithm to manipulate data or perform operations, or identifying the correct output of a given algorithm.
How should I approach algorithm exam questions?
When approaching algorithm exam questions, read the instructions carefully, analyze the problem or task, and determine the appropriate algorithmic approach. Consider the time and space complexity,
perform necessary calculations or steps, and clearly present the solution using appropriate pseudocode or programming code.
How can I improve my algorithmic problem-solving skills?
Improving algorithmic problem-solving skills requires consistent practice. Solve a variety of algorithmic problems, participate in coding challenges or competitions, study different algorithmic
techniques, and analyze existing algorithms and their implementations to gain insights.
Are there any recommended resources for studying algorithms?
Yes, there are several recommended resources for studying algorithms, including textbooks such as ‘Introduction to Algorithms’ by Cormen, Leiserson, Rivest, and Stein, online algorithm courses on
platforms like Coursera or edX, and algorithm visualization tools that help understand algorithmic processes visually.
What if I struggle with understanding or solving algorithmic problems?
If you struggle with understanding or solving algorithmic problems, seek additional resources or assistance. Consult your professor, join study groups or forums where you can discuss problems and
solutions with peers, and utilize online platforms with explanations and practice problems specifically focused on algorithms.
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Real Numbers Worksheet With Answers Pdf
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Wideband A/D Converter Front-End Design Considerations
Transformers are used for isolation and to convert signals from single-ended to differential. A factor often overlooked when using transformers in the front-end circuitry of high-speed A/D converters
is that they are never ideal. With sinusoidal input signals, any imbalance introduced by the transformer delivers an imperfect sinusoidal wave to the input of the ADC, and results in overall
data-conversion performance worse than the ADC could otherwise provide. We consider here the effects of input imbalances on ADC performance and provide examples of circuitry to achieve improved
About Transformers
The wide variety of available models from many manufacturers can make transformer selection a confusing process. The challenge is compounded by the differing approaches taken by suppliers in
specifying performance; they often differ in the choice and definitions of the parameters they specify.
Some key parameters to consider when selecting a transformer to drive a particular ADC are insertion loss, return loss, magnitude imbalance, and phase imbalance. Insertion loss is a guide to the
bandwidth capability of the transformer. Return loss, also useful, allows the user to design the termination to match the transformer’s response at a particular frequency or band of
frequencies—especially important when using transformers with greater than unity turns ratios. We will focus here on magnitude- and phase imbalance, and how they affect the ADC’s performance in
high-bandwidth applications.
Theoretical Analysis
Even with a wide bandwidth rating, the coupling between the transformer’s single-ended primary and differential secondary, though linear, introduces magnitude- and phase imbalances. When applied to a
converter (or other differential-input device), these imbalances worsen even-order distortion of the converted (or processed) signal. While usually negligible at low frequencies, this added
distortion in high-speed converters becomes significant at roughly 100 MHz. Let us first examine how the magnitude- and phase imbalance of a differential-input signal, particularly the
second-harmonic distortion, affect the performance of an ADC.
Figure 1. Simplified block diagram of the ADC front end using a transformer.
Consider the input, x(t), to the transformer. It is converted into a pair of signals, x[1](t) and x[2](t). If x(t) is sinusoidal, the differential output signals, x[1](t) and x[2](t), are of the form
The ADC is modeled as a symmetrical third-order transfer function:
Ideal Case—No Imbalance
When x[1](t) and x[2](t) are perfectly balanced, they have the same magnitude (k[1] = k[2] = k) and are exactly 180° out of phase (φ = 0°). Since
Applying the trigonometric identity for powers and gathering terms of like frequency,
This is the familiar result for a differential circuit: even harmonics cancel for ideal signals, while odd harmonics do not.
Magnitude Imbalance
Now suppose the two input signals have a magnitude imbalance, but no phase imbalance. In this case, k[1] ≠ k[2], and φ = 0.
Substituting Equation 7 in Equation 3 and again applying the trigonometric power identities,
We see from Equation 8 that the second harmonic in this case is proportional to the difference of the squares of the magnitude terms, k[1] and k[2], viz.,
Phase Imbalance
Assume now that the two input signals have a phase imbalance between them, with no magnitude imbalance.
Then, k[1] = k[2], and φ ≠ 0.
Substituting Equation 10 in Equation 3 and simplifying,
From Equation 11, we see that the second-harmonic amplitude is proportional to the square of the magnitude term, k.
A comparison of Equation 9 and Equation 12 shows that the second-harmonic amplitude is more severely affected by phase imbalance than by magnitude imbalance. For phase imbalance, the second harmonic
is proportional to the square of k[1], while for magnitude imbalance, the second harmonic is proportional to the difference of the squares of k[1] and k[2]. Since k[1] and k[2] are approximately
equal, this difference is small.
As a test of the validity of these calculations, MATLAB code was written for the model described above to quantify and illustrate the impact of magnitude- and phase imbalances on harmonic distortion
of a high-performance ADC with a transformer input (Appendix A). The model includes additive white Gaussian noise.
The coefficients, a[i], used in the MATLAB model are for the AD9445, a high-performance 16-bit, 125-MSPS ADC. The AD9445, in the front-end configuration shown in Figure 2, was used to generate the
FFT shown in Figure 3, from which the coefficients were derived.
Figure 2. Front-end configuration of the AD9445 with transformer.
Figure 3. Typical FFT of AD9445, 125 MSPS, IF = 170 MHz.
The noise floor, second harmonic, and third harmonic here reflect the composite performance of the converter and front-end circuitry. The converter distortion coefficients (a[2] and a[3]) and noise
were computed using these measured results, combined with the 0.0607 dB of magnitude imbalance and 148 of phase imbalance at 170 MHz, specified for a standard 1:1 impedance ratio transformer.
These coefficients are used in Equation 8 and Equation 11 to compute y(t), while the magnitude- and phase imbalances are varied in the ranges 0 V to 1 V and 0 degrees to 50 degrees, respectively (the
imbalance ranges of a typical transformer in the 1-MHz-to-1000-MHz range), and observe the effect on the second harmonic. The results of the simulations are shown in Figure 4 and Figure 5.
Figure 4. Harmonics plotted vs. magnitude imbalance only.
Figure 5. Harmonics plotted vs. phase imbalance only.
Figure 4 and Figure 5 show that (a) the third harmonic is relatively insensitive to both magnitude- and phase imbalance, and (b) that the second harmonic deteriorates more rapidly with phase
imbalance than with magnitude imbalance. Thus, to achieve better performance from the ADC, a transformer configuration with improved phase imbalance is needed. Two feasible configurations, the first
involving a double balun, and the second a double transformer, are shown in Figure 6 and Figure 7.
Figure 6. Double balun configuration.
Figure 7. Double transformer configuration.
The imbalances from these configurations were compared using a vector network analyzer on specially designed characterization boards. Figure 8 and Figure 9 compare the magnitude- and phase imbalance
of these configurations with that of a single transformer.
Figure 8. Magnitude imbalance from 1 MHz to 1000 MHz.
Figure 9. Phase imbalance from 1 MHz to 1000 MHz.
The above figures clearly show that the double configurations have better phase imbalance at the cost of slightly degraded magnitude imbalance. Therefore, using the results of the above analysis, it
appears that the double-transformer configurations can be used to achieve better performance. FFT plots of AD9445 using a single transformer input (Figure 10) and a double balun input (Figure 11)
show that this is indeed the case; a +10-dB improvement in SFDR is seen with a 300-MHz IF signal.
Figure 10. Single transformer input, FFT of AD9445. 125 MSPS, IF = 300 MHz.
Figure 11. Double balun input, FFT of AD9445. 125 MSPS, IF = 300 MHz.
Does this mean that to achieve good performance one must couple two transformers or two baluns onto the ADC’s front end? Not necessarily. The analysis shows that it is essential to use a transformer
that has very little phase imbalance. In the following examples (Figure 12 and Figure 13), two different single transformers were used to drive the AD9238 with a 170-MHz IF signal. These examples
show that there is 29-dB improvement in second harmonic when the ADC is driven by a transformer that has improved phase imbalance at high frequencies.
Figure 12. Single transformer input, FFT of AD9238. 62 MSPS, IF = 170 MHz @ –0.5 dBFS, second harmonic = –51.02 dBc.
Figure 13. Single transformer input, FFT of AD9238. 62 MSPS, IF = 170 MHz @ –0.5 dBFS, second harmonic = –80.56 dBc.
The phase imbalance of a transformer can worsen the second-harmonic distortion when the transformer is used as a front end for processes (such as A/D conversion, D/A conversion, and amplification)
with high-IF inputs (>100 MHz). However, by employing a pair of transformers or baluns, significant improvements can be readily achieved, at the cost of an additional transformer and extra PC board
Single-transformer designs can achieve adequate performance if the design bandwidth is small and a suitable transformer is chosen. However, they do require a limited matching of bandwidth, and they
can be expensive or physically large.
In either case, choosing the best transformer for any given application requires detailed knowledge of the transformer’s specifications. Phase imbalance is of particular importance for high-IF inputs
(>100 MHz). Even if it is not specified in the data sheet, most transformer manufacturers will provide phase-imbalance information upon request. A network analyzer can be used to measure the
transformer’s imbalances as a check, or if the information is not readily available.
Appendix A
MATLAB Code Used In This Experiment:
% Matlab code to study the effect of magnitude and phase imbalance of input
% signals on the output
% Oct 19, 2005
clear all; close all;
% Error terms that can be set by the user
magnErrdB = 0; %in dB
phaseErr = 50; %in degrees
sd_noise = 100e-6; %std dev of noise
% Convert dB magnErr to voltage level
magnErr = 10^(magnErrdB/20);
% Coefficients
a0=0; %dc offset
a1=0.89; a2=0.00038; a3=0.0007; %coefficients of 1st,2nd,3rd harmonics
%to match AD9445 typical FFT
fin = 100; %input freq - does not affect calculations
t = 0:1:2047;
%Input signals
x1 = 0.5*sin((t/2048)*2*pi*fin);
x2 = 0.5*(magnErr)*sin(((t/2048)*2*pi*fin)-pi-(phaseErr*pi/180));
%Each differential signal multiplied by the transfer function
y1 = a0 + a1*x1 + a2*x1.^2 + a3*x1.^3;
y2 = a0 + a1*x2 + a2*x2.^2 + a3*x2.^3;
y = y1 - y2;
noise = sd_noise*randn(1,length(y));
y = y + noise;
% figure; plot(1000*t(1:80),x1(1:80),1000*t(1:80),x2(1:80),1000*t(1:80),y(1:80));
%Take the FFT
fft_y = fft(y/1024, 2048);
Pyy = 10*log10(fft_y.*conj(fft_y));
freq_axis = 0:1:1023;
% figure; plot(freq_axis, Pyy(1:1024), ‘-d’);
% title(‘Frequency content of the output’);
% xlabel(‘Frequency (Hz)’);
% axis tight;
%Print fundamental and 2nd, 3rd harmonics
f = Pyy(101)
h2 = Pyy(201)
h3= Pyy(301)
(Return to Text)
1. Reeder, Rob, “Transformer-Coupled Front End for Wideband A/D Converters,” Analog Dialogue, Vol. 39, No. 2, pp. 3-6, 2005.
2. Mini-Circuits Data Sheet ADT1-1WT.
3. Pulse Data Sheet CX2039L.
4. Mini-Circuits Application Note: “How Transformers Work.”
5. The Mathworks Matlab program.
6. Analog Devices Data Sheet AD9445.
7. Analog Devices Data Sheet AD9238.
8. M/A-COM Data Sheet TP101.
9. Sprague-Goodman Data Sheet GLSB4R5M102.
The authors wish to thank Ravi Kummaraguntla, Andy Morgan, and Chad Shelton for their help in the theoretical analysis and for their lab support.
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Miles to Decimeters c
What is a mile?
A mile is a unit of length commonly used in the United States and some other countries. It is equal to 5,280 feet or 1,760 yards. The word "mile" is derived from the Latin word "mille," meaning
thousand, as it originally represented the distance covered in 1,000 paces by a Roman legionary.
A mile is equivalent to 1760yds or 5280ft.
The mile is commonly used in the United States for measuring long distances, such as road distances and race distances. It is also used in the aviation and maritime industries for navigation
purposes. However, in most other countries, the metric system is used, and the kilometer is the preferred unit for measuring long distances.
What is a decimeter?
A decimeter is a unit of length in the metric system, specifically in the International System of Units (SI). It is equal to one-tenth of a meter or 10 centimeters. The prefix "deci" indicates a
factor of 10^-1, which means that a decimeter is 10 times smaller than a meter.
The decimeter is commonly used in various fields, including science, engineering, and everyday measurements. It provides a convenient unit for measuring small distances, especially when centimeters
are too small and meters are too large. For example, a decimeter can be used to measure the length of small objects such as pencils, books, or the width of a hand.
In comparison to the imperial system, a decimeter is equivalent to approximately 3.937 inches. This conversion factor allows for easy conversion between the metric and imperial systems. The decimeter
is part of a larger range of metric units, which provide a consistent and decimal-based system for measuring length, mass, volume, and other quantities.
How do you convert miles to decimeters?
One mile is equal to 1609.34 meters, and a decimeter is one-tenth of a meter or 0.1 meters.
To convert miles to decimeters, we can use the following conversion factors: 1 mile = 1609.34 meters and 1 meter = 10 decimeters. By combining these conversion factors, we can derive the conversion
factor from miles to decimeters: 1 mile = 1609.34 meters = 1609.34 meters * (10 decimeters / 1 meter) = 16093.4 decimeters.
Therefore, to convert miles to decimeters, we multiply the number of miles by 16093.4. For example, if we have 2 miles, the conversion would be 2 miles * 16093.4 decimeters/mile = 32186.8 decimeters.
Similarly, if we have 5 miles, the conversion would be 5 miles * 16093.4 decimeters/mile = 80467 decimeters.
For a specific example, you can simply type in the miles amount into our Mile to Decimeters calculator and it will show the step by step working underneath the result.
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Simfit: simulation, statistics, curve fitting and graph plotting
Simfit is a free software OpenSource Windows/Linux package for simulation, curve fitting, statistics, and plotting, using a library of models or user-defined equations. It can be used in:
• biology (nonlinear growth curves);
• ecology (Bray-Curtis dendrograms);
• psychology (factor analysis);
• physiology (membrane transport);
• pharmacology (dose response curves);
• pharmacy (pharmacokinetics);
• immunology (ligand binding);
• biochemistry (calibration);
• biophysics (enzyme kinetics);
• epidemiology (survival analysis);
• medical statistics (meta analysis);
• chemistry (chemical kinetics);
• physics (dynamical systems); and
• mathematics (numerical analysis).
Simfit has forty programs, each dedicated to an aspect of simulation, plotting, or data analysis, with a reference manual containing mathematical and statistical details. Tutorials and worked
examples explain how to analyze the default data sets supplied for every procedure.
Clipboard data and spreadsheet export files can be analyzed, and macros to interface with Microsoft Office and LaTeX are provided.
Simdem is a package demonstrating how to use the Simfit GUI and graphics library to write Windows programs with menus, tables, and graphs.
Simfit and Simdem were developed in collaboration with Silverfrost Software and NAG.
© W.G.Bardsley, University of Manchester UK
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Understanding the Linear Slope-Intercept Equation: A Guide to Graphing, Equations, and Real-World Applications
Understanding The Linear Slope-Intercept Equation: A Guide To Graphing, Equations, And Real-World Applications
The slope-intercept equation, y = mx + c, is a mathematical expression that defines a straight line. It consists of two key components: the slope (m), which represents the gradient or rate of change,
and the y-intercept (c), which indicates the vertical intercept or starting point of the line. Together, these values describe the behavior and relationship of the line, making it a fundamental tool
for graphing, understanding linear equations, and solving various problems in real-world scenarios.
In the realm of mathematics, the slope-intercept equation reigns supreme as a cornerstone for describing linear relationships. This equation provides a profound understanding of how variables
interact and behave, empowering us to unravel the mysteries of the world around us.
At its core, the slope-intercept equation, y = mx + c, encapsulates two fundamental concepts: * slope and y-intercept. Slope, symbolized by m, embodies the gradient or steepness of a line,
quantifying the amount of vertical change that occurs for every unit of horizontal change. The y-intercept, denoted by c, represents the vertical displacement of the line, the point where it
intersects the y-axis.
These two components, when combined, paint a vivid picture of the line’s behavior. The slope signifies the line’s overall direction and trajectory, while the y-intercept anchors it on the y-axis,
indicating its starting point. By grasping the synergy between slope and y-intercept, we can decipher the intricacies of linear relationships and make informed predictions about the future.
Slope: Unveiling the Gradient and Rate of Change
In the realm of linear equations, slope emerges as a pivotal concept, defining the trajectory of lines and revealing the interplay between their x and y coordinates. It is the measure of the line’s
gradient, a quantification of its steepness or inclination.
Visualize a line traversing a graph from one point to another. As the x coordinate increases by a unit (∆x), its y coordinate changes by a corresponding amount (∆y). The slope (often denoted as m)
captures this proportional relationship, representing the change in y per unit change in x:
slope (m) = ∆y / ∆x
A positive slope indicates that as you move from left to right along the line, the y coordinate increases, creating an upward trajectory. Conversely, a negative slope signals a downward trend, with
the y coordinate decreasing as you move to the right.
To illustrate, consider a line with a slope of 2. For every unit increase in x, the y coordinate jumps up by 2 units. This upward slant becomes evident as the line ascends from one point to the next.
In real-world scenarios, the slope of a line unveils invaluable insights. In engineering, it determines the angle of a ramp’s incline, affecting the force required to move objects up or down its
surface. In economics, it reflects the change in inflation rate with respect to changes in economic growth, guiding policymakers in making informed decisions.
Unveiling the Significance of the Y-Intercept: The Starting Point of a Line
In the realm of linear relationships, the slope-intercept equation (y = mx + c) reigns supreme. To fully grasp this equation, it’s imperative to understand its crucial components: slope and
y-intercept. While we’ve delved into the gradient and rate of change represented by the slope, let’s now shift our focus to the equally important y-intercept.
The y-intercept, denoted by the letter ‘c’ in the equation, holds immense significance. It defines the exact point where the line of interest intersects with the vertical, or y-axis. In other words,
when the horizontal, or x-coordinate, is zero, the y-coordinate corresponding to the y-intercept tells us the exact value of the line at that point.
Picture this: you’re visualizing the graph of a line. As you slide along the line, the y-intercept marks the point where you begin your journey, the starting point of the line. From this vantage
point, the line either ascends or descends, guided by the slope. The y-intercept serves as the pivotal reference point, the foundation upon which the line’s behavior unfolds.
The y-intercept not only denotes the starting point but also provides valuable insights into the line’s overall characteristics. For instance, a positive y-intercept indicates that the line begins
above the x-axis, while a negative y-intercept implies that it starts below the x-axis. These subtle nuances allow us to quickly determine the line’s general orientation and position in the
coordinate plane.
In the tapestry of real-world applications, the y-intercept plays a pivotal role in solving practical problems. For instance, in finance, the y-intercept represents the fixed cost of production,
while the slope indicates the variable cost per unit. Understanding these concepts enables us to make informed decisions and draw meaningful conclusions from linear relationships.
In conclusion, the y-intercept is not merely a point on a graph; it is the cornerstone of the slope-intercept equation. It reveals the starting point of the line, providing a deeper understanding of
the line’s behavior and characteristics. By grasping the significance of the y-intercept, we unlock the key to unlocking the mysteries of linear relationships, empowering us to navigate the
complexities of the world around us.
The Slope-Intercept Equation: Unraveling the Secrets of Linear Relationships
The Equation: A Tale of Line
Imagine a line traversing the plane, like a dancer gliding across a stage. This line, governed by the slope-intercept equation, y = mx + c, embodies a graceful interplay between change and position.
The slope, represented by m, reveals the line’s gradient, the amount of vertical change for every horizontal unit traveled. Imagine climbing a hill; the steeper the slope, the more you ascend with
each step.
On the other hand, the y-intercept, denoted by c, holds the key to the line’s starting point, the point where it intersects the y-axis. It represents the vertical position from which the line
Decoding the Equation
The slope-intercept equation, y = mx + c, is a harmonious dance of three components:
• y: The vertical coordinate of any point on the line
• m: The slope, determining the line’s gradient or rate of change
• c: The y-intercept, marking the line’s starting point on the y-axis
Unleashing the Equation’s Power
This equation holds immense power in understanding linear relationships. By deciphering its slope and y-intercept, we can unlock the secrets of any straight line. The slope, as a measure of gradient,
tells us how steeply the line ascends or descends. A positive slope indicates an upward trajectory, while a negative slope denotes a downward path.
The y-intercept, by contrast, reveals the line’s vertical position. It tells us the point on the y-axis from which the line embarks on its journey.
Applications in Action
The slope-intercept equation extends its reach beyond the realm of theory into the practical world. In fields ranging from economics to physics, it empowers us to solve problems and gain insights
into the behavior of linear systems.
In finance, for instance, the slope of a demand curve tells economists the extent to which demand changes in response to price fluctuations. In physics, the slope of a velocity-time graph reveals the
acceleration of an object.
The slope-intercept equation, y = mx + c, stands as a fundamental tool in mathematics, providing a concise and powerful description of linear relationships. Its ability to unravel the mysteries of
lines makes it invaluable in a multitude of fields, empowering us to understand and solve complex problems with ease.
Determining the Slope and Y-Intercept: A Step-by-Step Guide
In the world of linear equations, the slope and y-intercept hold the keys to understanding the behavior and relationships they describe. These two essential elements play a crucial role in graphing
lines and solving real-world problems.
Finding the Slope
The slope of a line is a measure of its steepness or gradient. It tells you how much the y-coordinate changes for every unit change in the x-coordinate. There are two common methods for finding the
1. Slope Formula: If you have two points on the line, (x1, y1) and (x2, y2), the slope can be calculated as:
Slope (m) = (y2 - y1) / (x2 - x1)
2. Slope Triangle (Rise-over-Run): Identify two points on the line and form a right triangle. The slope is equal to the rise (the difference between the y-coordinates) divided by the run (the
difference between the x-coordinates).
Finding the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It represents the value of y when x is equal to 0. To find the y-intercept:
1. Slope-Intercept Equation: If you have the slope (m) and one point on the line (x1, y1), you can use the slope-intercept equation:
y = mx + y-intercept
2. Graphically: Locate the point where the line intersects the y-axis. The y-coordinate of this point is the y-intercept.
• Given the points (2, 5) and (4, 9):
Slope (m) = (9 - 5) / (4 - 2) = 2
Y-intercept = 5 - 2(0) = 5
• Given the slope (m = -1/2) and the point (4, 6):
y = -1/2x + y-intercept
y = -1/2x + 6
Y-intercept = 6
Understanding the Slope-Intercept Equation: Its Significance in Graphing and Linear Relationships
The slope-intercept equation, y = mx + c, plays a crucial role in understanding the behavior and relationships described by linear equations. Its importance lies in its ability to accurately
represent lines on graphs and reveal essential information about their characteristics.
Graphing Lines
The slope-intercept equation provides a straightforward method for graphing lines. By knowing the slope and y-intercept, you can quickly plot the line and visualize its path. The slope determines the
angle of the line, while the y-intercept indicates the point where it crosses the y-axis. This enables effortless graphing, allowing you to analyze and interpret data visually.
Understanding Linear Relationships
The slope-intercept equation also serves as a powerful tool for understanding the relationship between two variables represented by a linear equation. The slope represents the rate of change,
measuring the amount of change in the dependent variable (y) for each unit change in the independent variable (x). This value provides valuable insights into how variables are interconnected and how
changes in one affect the other.
Furthermore, the y-intercept represents the starting value or the value of the dependent variable when the independent variable is zero. This crucial piece of information helps establish the initial
conditions and determines the starting point of the line.
The slope-intercept equation, y = mx + c, is an indispensable tool for graphing lines and unraveling the intricacies of linear relationships. Its ability to represent lines visually and provide
insights into interconnected variables makes it a cornerstone of mathematical understanding. By grasping the significance of the slope and y-intercept, you unlock a powerful tool for analyzing and
interpreting data, empowering you to make informed decisions and gain a deeper understanding of the world around you.
Applications in Real-World Scenarios
Beyond the theoretical realm, the slope-intercept equation finds its true power in solving practical problems across diverse fields. Imagine yourself as a sleuth, using this equation to unravel
mysteries in various domains.
• Engineering: An architect needs to determine the angle of a ramp to ensure smooth access for wheelchairs. The slope-intercept equation becomes their blueprint, helping them calculate the gradient
so that the ramp meets safety standards.
• Economics: A financial analyst wants to predict future stock prices. They plot the historical trend of stock values using the slope-intercept equation. The slope reveals the rate of change,
aiding them in forecasting market movements.
• Medicine: A doctor uses the slope-intercept equation to analyze a patient’s progression. By plotting the patient’s weight over time, the slope reflects the rate of weight loss or gain, guiding
treatment decisions.
• Environmental Science: A researcher investigates the relationship between carbon dioxide levels and global temperatures. Using the slope-intercept equation, they determine the slope, which
represents the increase in temperature per unit increase in carbon dioxide, providing crucial insights into climate change.
These are just a few showcases of the slope-intercept equation’s versatility. It’s a tool that empowers us to understand the world around us and make informed decisions in various domains. So, next
time you see a linear equation, remember its immense practical value and unlock its power to solve real-world enigmas.
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Jew or Not Jew: Robert Merton
Stigler's law of eponymy, as stated by Stephen Stigler, claims that discoveries are never named after their real discoverer. There are many examples, including:
Halley's comet, named after Edmond Halley, but discovered many centuries prior
Pythagorean theorem, known to Babylonian mathematicians long before Pythagoras
Venn diagrams, proposed by Christian Weise and Leonhard Euler, but popularized by John Venn
Euler's number, the constant discovered by Jacob Bernoulli, but named after Euler (payback!!!)
Occam's razor, named after William of Ockham, but formulated long before his time (clearly, not the simplest solution here...)
Stiegler's law, which, according to Stigler himself, was originally discovered by sociologist Robert K. Merton!
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semicircle to Quadrant
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like angle finds its use in
a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the
conversion of different units of measurement like semicircle to Quadrant through multiplicative conversion factors. When you are converting angle, you need a Semicircle to Quadrant converter that is
elaborate and still easy to use. Converting semicircle to Quadrant is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert
Semicircle to Quadrant, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in semicircle to Quadrant conversion along with a table representing
the entire conversion.
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43. The quadratic equation with real coefficients - Avidemia
The equations \(z^{2} + 1 = 0\), \(az^{2} + 2bz + c = 0\). There is no real number \(z\) such that \(z^{2} + 1 = 0\); this is expressed by saying that the equation has no real roots. But, as we have
just seen, the two complex numbers \(i\) and \(-i\) satisfy this equation. We express this by saying that the equation has the two complex roots \(i\) and \(-i\). Since \(i\) satisfies \(z^{2} = -1\)
, it is sometimes written in the form \(\sqrt{-1}\).
Complex numbers are sometimes called imaginary.^1 The expression is by no means a happily chosen one, but it is firmly established and has to be accepted. It cannot, however, be too strongly
impressed upon the reader that an ‘imaginary number’ is no more ‘imaginary’, in any ordinary sense of the word, than a ‘real’ number; and that it is not a number at all, in the sense in which the
‘real’ numbers are numbers, but, as should be clear from the preceding discussion, a pair of numbers \((x, y)\), united symbolically, for purposes of technical convenience, in the form \(x + yi\).
Such a pair of numbers is no less ‘real’ than any ordinary number such as \(\frac{1}{2}\), or than the paper on which this is printed, or than the Solar System. Thus \[i = 0 + 1i\] stands for the
pair of numbers \((0, 1)\), and may be represented geometrically by a point or by the displacement \([0, 1]\). And when we say that \(i\) is a root of the equation \(z^{2} + 1 = 0\), what we mean is
simply that we have defined a method of combining such pairs of numbers (or displacements) which we call ‘multiplication’, and which, when we so combine \((0, 1)\) with itself, gives the result \
((-1, 0)\).
Now let us consider the more general equation \[az^{2} + 2bz + c = 0,\] where \(a\), \(b\), \(c\) are real numbers. If \(b^{2} > ac\), the ordinary method of solution gives two real roots \[\{-b \pm
\sqrt{b^{2} – ac}\}/a.\] If \(b^{2} < ac\), the equation has no real roots. It may be written in the form \[\{z + (b/a)\}^{2} = -(ac – b^{2})/a^{2},\] an equation which is evidently satisfied if we
substitute for \(z + (b/a)\) either of the complex numbers \(\pm i\sqrt{ac – b^{2}}/a\).^2 We express this by saying that the equation has the two complex roots \[\{-b \pm i\sqrt{ac – b^{2}}\}/a.\]
If we agree as a matter of convention to say that when \(b^{2} = ac\) (in which case the equation is satisfied by one value of \(x\) only, viz. \(-b/a\)), the equation has two equal roots, we can say
that a quadratic equation with real coefficients has two roots in all cases, either two distinct real roots, or two equal real roots, or two distinct complex roots.
The question is naturally suggested whether a quadratic equation may not, when complex roots are once admitted, have more than two roots. It is easy to see that this is not possible. Its
impossibility may in fact be proved by precisely the same chain of reasoning as is used in elementary algebra to prove that an equation of the \(n\)th degree cannot have more than \(n\) real roots.
Let us denote the complex number \(x + yi\) by the single letter \(z\), a convention which we may express by writing \(z = x + yi\). Let \(f(z)\) denote any polynomial in \(z\), with real or complex
coefficients. Then we prove in succession:
(1) that the remainder, when \(f(z)\) is divided by \(z – a\), \(a\) being any real or complex number, is \(f(a)\);
(2) that if \(a\) is a root of the equation \(f(z) = 0\), then \(f(z)\) is divisible by \(z – a\);
(3) that if \(f(z)\) is of the \(n\)th degree, and \(f(z) = 0\) has the \(n\) roots \(a_{1}\), \(a_{2}\), …, \(a_{n}\), then \[f(z) = A(z – a_{1}) (z – a_{2}) \dots (z – a_{n}),\] where \(A\) is a
constant, real or complex, in fact the coefficient of \(z^{n}\) in \(f(z)\). From the last result, and the theorem of § 40, it follows that \(f(z)\) cannot have more than \(n\) roots.
We conclude that a quadratic equation with real coefficients has exactly two roots. We shall see later on that a similar theorem is true for an equation of any degree and with either real or complex
coefficients: an equation of the \(n\)th degree has exactly \(n\) roots. The only point in the proof which presents any difficulty is the first, viz. the proof that any equation must have at least
one root. This we must postpone for the present.^3 We may, however, at once call attention to one very interesting result of this theorem. In the theory of number we start from the positive integers
and from the ideas of addition and multiplication and the converse operations of subtraction and division. We find that these operations are not always possible unless we admit new kinds of numbers.
We can only attach a meaning to \(3 – 7\) if we admit negative numbers, or to \(\frac{3}{7}\) if we admit rational fractions. When we extend our list of arithmetical operations so as to include root
extraction and the solution of equations, we find that some of them, such as that of the extraction of the square root of a number which (like \(2\)) is not a perfect square, are not possible unless
we widen our conception of a number, and admit the irrational numbers of Chap. I.
Others, such as the extraction of the square root of \(-1\), are not possible unless we go still further, and admit the complex numbers of this chapter. And it would not be unnatural to suppose that,
when we come to consider equations of higher degree, some might prove to be insoluble even by the aid of complex numbers, and that thus we might be led to the considerations of higher and higher
types of, so to say, hyper-complex numbers. The fact that the roots of any algebraical equation whatever are ordinary complex numbers shows that this is not the case. The application of any of the
ordinary algebraical operations to complex numbers will yield only complex numbers. In technical language ‘the field of the complex numbers is closed for algebraical operations’.
Before we pass on to other matters, let us add that all theorems of elementary algebra which are proved merely by the application of the rules of addition and multiplication are true whether the
numbers which occur in them are real or complex, since the rules referred to apply to complex as well as real numbers. For example, we know that, if \(\alpha\) and \(\beta\) are the roots of \[az^{2}
+ 2bz + c = 0,\] then \[\alpha + \beta = -(2b/a),\quad \alpha\beta = (c/a).\]
Similarly, if \(\alpha\), \(\beta\), \(\gamma\) are the roots of \[az^{3} + 3bz^{2} + 3cz + d = 0,\] then \[\alpha + \beta + \gamma = -(3b/a),\quad \beta\gamma + \gamma\alpha + \alpha\beta = (3c/a),\
quad \alpha\beta\gamma = -(d/a).\] All such theorems as these are true whether \(a\), \(b\), … \(\alpha\), \(\beta\), … are real or complex.
1. The phrase ‘real number’ was introduced as an antithesis to ‘imaginary number’.↩︎
2. We shall sometimes write \(x + iy\) instead of \(x + yi\) for convenience in printing.↩︎
3. See Appendix I.↩︎
$\leftarrow$ 39-42. Complex numbers Main Page 44. Argand’s diagram $\rightarrow$
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Math Problems Study Guide
1. Understanding the Problem
When approaching a math problem, it's important to read the question carefully and identify what is being asked. Look for keywords such as "sum," "difference," "product," or "quotient" to determine
the type of operation needed.
2. Break It Down
If the problem is complex, break it down into smaller, more manageable parts. Identify any given information and determine what is being asked in each part of the problem.
3. Choose a Strategy
There are different problem-solving strategies you can use, such as drawing a picture, making a table or chart, working backwards, or using logical reasoning. Choose the strategy that best fits the
problem at hand.
4. Solve the Problem
Use the chosen strategy to solve the problem step by step. Show all your work and calculations neatly to avoid mistakes and make it easier to check your work.
5. Check Your Answer
After solving the problem, always check your answer to ensure it makes sense in the context of the problem. Ask yourself if the answer is reasonable and if it accurately addresses the question asked.
Example Problem:
John has 3 apples. He buys 5 more apples at the store. How many apples does John have in total?
Step 1: Identify the given information and what is being asked. In this case, we are given that John has 3 apples and he buys 5 more. We need to find the total number of apples John has.
Step 2: Choose a strategy. Since this is a straightforward addition problem, we can simply add the number of apples John had to the number he bought.
Step 3: Solve the problem. 3 apples + 5 apples = 8 apples. John has 8 apples in total.
Step 4: Check your answer. 3 + 5 = 8, so our answer is correct.
Practice Problems:
1. Solve the following addition problem: 25 + 14 = ?
2. If a box contains 7 red marbles and 4 blue marbles, how many marbles are there in total?
3. A book costs $12. If Sarah buys 3 books, how much does she spend in total?
Remember to practice these steps and strategies to become more comfortable and confident in solving math problems.
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The art of making and measuring LF coils
Home - Techniek - Electronica - Radiotechniek - Radio amateur bladen - QEX - The art of making and measuring LF coils
Large, high-quality coils are an importantfactor in working the LF bands successfully. This short note on the specific art describes good quality-factor coils (Q = 600) for 136 kHz.
The "new" band of 136 kHz in Europe and the "lowfer" 160-190 kHz band in the USA are to be considered a good mix between the old fascinating era of Marconi antennas (plus specific grounding topics)
and the technologically advanced world of DSP and related high-precision spectrum and noise analysis. Combining modern PC processing capability with very interesting old-style experimentation on big
coils and big vertical antennas with capacitive hats, we can obtain unique results (see Ref 4). A very simple transmission circuit to start operation at 136 kHz is shown in Fig 1.
Fig 1 - A simple 136-kHz linear amplifer using a low-cost monolithic IC.
The circuit includes an audio power amplifier commonly used in highquality TV sets: the TDA7265 made by STMicroelectronics.(1) The PC board and heat sink are shown in Fig 2 and complete technical
information is available on the Web site (www.st.com).
Fig 2 - The ST board designed for 25-W stereo high-quality audio for TV (TDA7265).
The typical output power of the amplifier at audio frequencies is 25 W with 4 Ω speakers; but at 136 kHz, the output power drops to 3-4 W, maximum. This power is more than adequate for LF system
tests, generating up to 0.5 A of antenna current.(2) The transformer, T1, is constructed on a standard FT101-43 one-inch toroid core and is used to match the specified 4 Ω load to the 16-20 Ω total
resistance of the antenna system (antenna + coil + ground). To check the complete system, a relatively simple antenna is used, with a vertical (Marconi) rod of 7-10 meters isolated from ground using
a plastic plate. About 40-50 meters (130-160 feet) of horizontal wires (hat) are connected atop the Marconi antenna to realize a 450 pF total antenna capacity. This establishes a near unity ratio
between the top and base currents in the antenna (Fig 3). As you can see from the diagram (from Ref 1), the radiation resistance of our antenna at 136 kHz is very low: about 0.03 Ω. For comparison,
the height for a similar antenna at 14 MHz would be 7-10 cm!
Fig 3 - Radiation resistance versus height of Marconi vertical antennas plus capacity hat.
More complete technical documentation on the big antennas is available in the paper of Ref 1 (Monster Antennas). The ground connection is another big problem at LF, but if you have a country house
with a big garden, you don't normally have a ground problem. As is well known, earth is inherently a rather poor conductor, with normal resistivities in the range of 10-300 Ω per meter, and the
conductivity of the metal constituting the grounding rod is not very important.
The ground resistance R[G] can be pictured as the resistance resulting from a series of equally thick concentric earth shells around the ground rod. With a typical 3-meter rod, half of the resistance
is contained within a cylinder of 12-cm radius around the rod (Ref 3). The only way to reduce the ground resistance is with the addition of multiple electrodes. Adding more ground rods reduces the
grounding resistance, but the gain is less for each additional rod. That is, the final resistance for many rods is greater than the value obtained by simply dividing the resistance of a single rod by
the number of parallel connected rods. A single 3-meter rod of 16 mm diameter driven into soil with 100 Ω/meter average resistivity will have a ground resistance (measured at 50-60 Hz) of 30-50 Ω.
Using four parallel rods placed at 10-15 m in a square will give a final LF resistance of 10-15 Ω. At 136 kHz, the inductance of the connecting cables is not important, but we must use a big wire to
avoid skineffect resistances. As seen in Table 2 for 424.18 mm Litz wire, we have only a 0.0164 Ω/meter dc resistance (probably less than 0.5 Ω of RF resistance for a 10 meter connection cable).
Table 1 - Wire Table including calculated RAC/ RDC at
136 and 200 kHz
Diameter Diameter R[DC ](Ω/m) R[AC]/R[DC] R[AC]/R[DC]
(mm) (mils) at 136 kHz at 200 kHz
0.050 1.97 8.9300 1.000 1.000
0.100 3.94 2.2330 1.000 1.000
0.180 7.09 0.6893 1.001 1.002
0.200 7.98 0.5583 1.002 1.004
0.300 11.80 0.2481 1.011 1.022
0.500 19.70 0.0893 1.075 1.149
0.900 22.86 0.0276 1.507 1.785
1.000 39.40 0.0223 1.650 1.962
2.000 78.80 0.0056 3.060 3.640
Table 2 - Litz-Wire Table for typical Copper Wires
Diameter Diameter Litz Wire # R[DC] (Ω/m) R[DC] Litz (Ω/m)
0.051 2.010 100 8.510 0.0851
0.051 2.010 200 8.510 0.0425
0.051 2.010 600 8.510 0.0142
0.063 2.480 60 0.0031 0.0907
0.127 5.000 50 0.0127 0.0272
0.180 7.090 15 0.6890 0.0459
0.180 7.090 30 0.3445 0.0229
0.180 7.090 42 0.6890 0.0164
0.300 11.810 30 0.2480 0.0083
When using four or more parallel ground connections, the resistance of the wires is not very important. In our tests, two 4 meter deep rods and two 2 meter deep rods were used at a distance from the
common ground point (at the base of the Marconi vertical antenna) of 10-15 meters. The measured value of our ground resistance at 136 kHz is R[G] = 12-14 Ω. The system's total load resistance seen by
the output transformer is:
The very poor efficiency of any short vertical Marconi system can be calculated simply using the expression:
R[R] = Radiation resistance (see Fig 3)
R[L] = Coil resistance
R[G] = Ground resistance
The Coils
The load coils used in LF transmitters must have a very high quality factor, Q, or a low equivalent series resistance (for example, R[AC] < 5 Ω so as to reduce the transmission losses.
f frequency (kHz)
L inductance (mH)
R[AC] = LF equivalent series resistance (Ω)
As stated in the introduction, because of the very low antenna efficiency and the relatively high resistance of the Earth (R[G] ~ 10 to 15 Ω, see Fig 4), the use of low-Q coils is not practical.
Fig 4 - Ground resistance versus rod length in a typical earth (100 Ω x meter)
The losses that affect the quality factor of LF coils are:
• Skin effect of wires
• Proximity effect between turns of winding
• Lossy dielectric of the distributed capacitance
• Lossy coil form material (such as gray PVC)
In the following, we will consider in more detail both the skin and the proximity effects in the RF windings. Now, we limit our discussion by writing that skin-effect problems are avoided by using
Litz wire (which is many thin, insulated wires connected together at the ends). We take this opportunity to remind that, to manufacture some monster antennas, 3.5-inch Litz wire (9 cm diameter!) has
been used. The dielectric losses are related to the material used to insulate the conductor-enamel, for instance of the winding. In any case, this kind of loss is very small considering the total
tuning capacitance for LF resonance.
During the beginning phase, we tried a gray PVC tube as a form. This tube is often employed in the manufacture of buildings.
Table 3 - High quality coils measured in the LF range
Coil # Core material Diameter (mm) Wire size (mm) Litz wire (n) Turns (N) Coil lenght D/Len Wire lenght (m) L (mH) R[DC] (Ω) 136 kHz X[L] (Ω) Q R[AC] (Ω) 200 kHz X[L] (Ω) Q R[AC] (Ω)
L01 grey PVC 160 0.90 none 175 176 0.91 88 3.12 2.43 2665 205 13.0 3919 187 21.0
L02 air + wood 160 0.18 30 120 210 0.76 60 1.40 1.38 1196 513 2.33 1758 534 3.29
L03 air + wood 200 0.18 42 157 272 0.73 99 2.70 1.63 2306 507 4.55 3391 435 7.80
L04 air + wood 330 0.90 none 85 87 3.80 88 3.30 2.43 2818 237 11.9 4145 230 18.0
L05 air + wood 330 0.90 none 85 165 2.00 88 2.50 2.43 2135 318 6.71 3140 309 10.16
L06 air + wood 330 0.18 42 94 168 1.96 97 3.00 1.59 2562 597 4.29 3768 541 6.97
This was the worst case we found: Table 3 compares the Q of coil L01 with the other coils wound on wood and air.
The best case used a form of eight wooden dowels connected together by two wooden plates. This core minimizes the mass of material within the solenoid winding. An example of this arrangement is shown
in Fig 8A, where the core of coils L03 and L04 (of Table 3) is shown before the wire was wound. To verify the core material's quality, we made a hole in the center of the two wooden plates for the
purpose of inserting a big, cylindrical mass of wood. No change in measured Q values was detected while performing this experiment.
To the contrary, a 30% drop of Q is verified when using gray PVC rods. The LF coil-design goal is high performance, with:
• Wires able to sustain high RF currents
• Good insulation
• An inductance value of a few mH
To transmit adequate RF power, we need to produce a load-coil current of a few amperes. This means using a conductor of suitable cross section to carry the 1 to 5 A currents with low RF series
resistance. For instance, if we have 3-mH coil in which a current of 1 A flows, the voltage at the coil terminals is about 3000 V (3 mH is +j2563 Ω at 136 kHz and +j3770 Ω at 200 kHz). This is a very
dangerous voltage. Moreover, it is necessary to consider the potential difference between two adjacent turns (25 to 40 V with 1 A of current flowing and proportionally higher with increasing current)
to set requirements for dielectric strength of the conductor's insulation. The inductance of the coil, for LF purposes, can be calculated using the following equation, considering that it must
resonate with the total antenna capacitance (vertical antenna plus top-hat capacity):
L coil inductance (mH)
f frequency (kHz)
CA = vertical antenna + hat capacitance (pF)
At this point, it is necessary to find the number of turns (starting from the inductance computed with Eq 4) and the geometrical dimensions of the coil:
N = number of turns of solenoid winding
L = inductance (mH)
d = wire diameter (mm)
k = turns packing factor (greater than 1)
D = coil diameter (mm)
The above formula allows an accuracy of about 1% for a single-layer coil. This was confirmed by the obtained experimental results.
Before calculating the number of turns, N, it is necessary to minimize the proximity effect (see next paragraph). For this purpose, the factor k, which is greater than 1, has been introduced to
consider when the distance between two adjacent wires (pitch) is greater than the diameter of the wire, d. The diameter of the coil must be chosen to minimize the wire length in order to minimize the
series equivalent resistance: By reducing R[DC], R[AC] is also automatically reduced. In Fig 7, we show two graphs that are very useful for coil optimization. In the first graph, we have wire length
versus form factor (D/Len); in the second, we have calculated wire length versus coil diameter.
Considering a maximum increase of the conductor length of 5%, with reference to best-case D/Len = 2.5, a maximum change in the coil form factor D/Len from about 1 to 4.8 can be accepted. On the
second graph, we can find the coil diameter that we can put into Eq 5 to calculate the number of turns. For instance, considering spacing (pitch) between two adjacent turns of 2 mm, we can use a coil
diameter between 260 and 570 mm to obtain a coil with 3 mH of inductance.
Fig 7-Minimum coil wire length versus coil diameter and D/Len; a starting point to realize high-current and high-quality inductors
Fig 7 also shows the value relative to the final coil, L06, to show the optimization performed. The complete characteristics of the other coils are shown in Table 3.
Having chosen the value of coil diameter in the range reported above, the number of turns of the solenoid can be easily calculated. At the end of the theoretical calculations, we go to the practical
realization of the load inductor. As already stated, Table 3 shows all the coils tested to verify the optimization also from the practical point of view. L01, L04 and L05 coils have been manufactured
using only enameled wire with a 0.9-mm diameter. The first inductor, on a gray PVC support, has the lowest quality factor Q = 205 at 136 kHz and 187 at 200 kHz) because of the bad core material used.
The second one has a higher Q (237 at 136 kHz and 230 at 200 kHz), which is related to the better support but affected by the proximity effect due to the closeness between the adjacent wires (k=l).
L05 has the highest quality factor of this coil group (318 at 136 kHz and 309 at 200 kHz) due to the increased distance between the subsequent turns (k=2). This last result shows the big influence of
the proximity effect on the equivalent series resistance (which we've alrea pointed out) in the LF range.
Remaining coils have been made using Litz wire: L02 with 2x15x0.18 mm, L03 and L06 with 42x0.18 mm wires. The L02 inductor has a very good Q, but it is not useful for our purpose. It has only 1.4 mH
of inductance and cannot resonate with the total antenna capacitance, which is estimated to be about 450 pF, considering both the vertical antenna and the top hat.
Fig 8 - A, a simple wood support structure with no LF power loss. At B, the final load inductor (diameter = 33 cm, Q ~ 600 at 136 kHz).
Table 4 - Other intersring coils measured by Bill Bowers. *R[AC]=X[L]/Q
Coil # Diameter (mm) Wire size (mm) AWG Litz wires (n) Turns (N) Coil lenght (mm) D/Len Wire lenght L (mH) 200 kHz X[L] (Ω) Q R[AC]* (Ω)
B01 483 1.830 14 1 42 193 2.5 64 1 1256 325 3.86 0.53
B02 483 0.127 36 50 44 241 2.0 67 1 1256 300 4.19 1.82
B03 483 0.127 36 50 40 160 3.0 61 1 1256 400 3.14 1.65
B04 483 0.127 36 50 37 102 4.7 56 1 1256 345 3.64 1.53
B05 483 0.127 36 50 42 193 2.5 64 1 1256 410 3.06 1.73
B06 483 0.051 44 200 42 193 2.5 64 1 1256 430 2.92 2.71
B07 483 0.051 44 600 42 193 2.5 64 1 1256 625 <2.01 0.90
The best coil for our experimental LF transmitter is L06 (Fig 8B); it has a very high Q (597 at 136 kHz and 541 at 200 kHz) and a suitable inductance: 3 mH. For comparison, Table 4 shows some coils
realized and measured Bill Bowers (Ref 10). Apart from the reduced value of inductance for these coils (only 1 mH) and according to our experience, we believe that the author did not realize the real
importance of the proximity effect with respect to inductance quality factor.
Qs greater than 625 (over the Boonton Q-meter range) related to two factors: the "distributed" turns and big Litz wire with 600 strands. Remember however that obtaining Q of 600 at 3 mH is not so
easy as with 1 mH! We think fore that the use of Litz wire having 600 strands of 0.051 mm diameter is an unnecessary complication considering the skin effect. Also see Tables 1 and 2 and remember
that R[AC]/R[DC] = 1.001 (!) with d = 0.18 mm at 136 kHz. At this frequency, the best Litz choice is probably 20 strands of 0.30-mm diameter. Where only direct current flows in a conductor, the
resistance has the lowest value and the current density is uniform in the whole cross section of the wire.
R[DC] = resistance for unit of length (Ω/m)
d = wire diameter (mm)
For alternating current (RF), the current density is not uniform within the conductor cross section and the resistance increases. The current-density change can be explained by considering that the
wire is composed of many tubular, concentric conductors. Because each tubular conductor is submitted to the external magnetic field, the inner elements link more magnetic flux than the outer ones.
The consequence is an increase in inductance, and so of the reactance, in the part of the wire nearest the longitudinal axis. This is the reason why the alternating current flows mostly in the
external surface of the conductor, which can be considered its skin. This effect already begins to be significant at LF (30 to 300 kHz frequency range) when using wires having a diameter that is
large (3 - 4 times) compared to the skin depth.
The skin depth is the distance below the surface of the conductor where the current density drops to 1/e (~37%) of its value at the surface. The following relation describes the decrease of current
density versus the distance from the wire surface, x (Ref 9):
I[x] = Current at depth x
I[s] = Current at wire surface
e = Base of natural logarithms
δ = the skin depth defined by Eq 8
p = wire resistivity (~2-meter)
µ = wire material permeability (Henries/meter)
f = frequency (Hz).
Considering a copper wire at 20°C,
ρ = 1.754 x 10-8 Ωm
y = 1.256x 10-6 x µ, H/m; µ[r] = 1
A graph of skin depth versus frequency is shown in Fig 6.
Fig 6 - Skin depth versus frequency in copper-wires. At 136 kHz, the skin depth is 0.18 mm.
When the diameter of the wire is large compared to the skin depth (at least three times), the useful cross section becomes tubular and the alternating-current resistance can be described
approximately by the following equation:
R[AC] = alternating-current resistance for unit of length (Ω/m)
The consequent ratio R[AC]/R[DC], reported below, can be reduced using a tubular conductor with a wide periphery with respect to the cross section. For example, a copper conductor composed of many
thin insulated wires can be used, such as Litz wire.
The exact system to calculate the skin effect is reported in Ref 3 of the bibliography.
For the wires involved in our study, Len values are reported in Table 1 both at 136 and 200 kHz. When two or more nearby wires are carrying a current, the current distribution in every conductor is
submitted to the magnetic field produced by the adjacent wires. This effect, named proximity effect, increases further the R[AC]/R[DC] ratio calculated by considering only the skin effect.
The proximity effect is very important in the LF coils, as we verified during the experimental phase of our work. If we consider the L04 and L05 coils (single 0.9-mm enameled wire) of Table 3, the R
[AC] is reduced simply by increasing the pitch between turns from 1 to 2 mm. The computation of the proximity effect can be performed only on very simple cases that are very far from the single-layer
solenoid. At this point, we believe that the better thing to do is to perform the measurements of the quality factor of the coils manufactured by considering all the abovementioned effects together.
Q Measurements
Very few hams own a professional Q meter, and some old equipment (with difficulty) meets the measurement needs of our LF coils: very high quality factor with a few millihenries of inductance.
The first approach is to analyze the possible errors in making the measurement in parallel using a small series resistance (0.5 Ω) to inject RF into the resonant circuit and then measure the voltage
at the coil terminals. To measure,coils having a very high Q, it is necessary to consider the quality factor of all the components used in the measurement circuit: RF voltmeter, series resistance,
capacitor and interconnections. If, for instance, we have a 3-mH inductance with Q = 600, the parallel resistance of the resonant circuit is greater than 1.5 MΩ (2πfLQ). An RF millivoltmeter with a
10 MΩ input resistance changes the circuit by reducing the measured Q by about 14%.
The other thing that decreases the measured Q is the small series resistance used to introduce the RF signal into the resonating circuit. This effect, together with the RF millivoltmeter effect,
reduces the measured Q by about 22%. Nevertheless, all the errors in the inductor quality factor just mentioned are well defined and computable, so the true Q can be calculated.
The last real problem of the parallel measurements is the capacitor loss that it is very difficult to measure because we used an air-variable capacitor. During our experiments, we used capacitors
with different dielectric materials; but at the end, we decided to use only an air-variable capacitor for the following reasons:
• Easy tuning
• The very low dissipation factor (0.0001, or a Q=10,000).
These data have been found in the literature and not measured directly. This uncertainty in the quality-factor measurement pushed us to find a better solution. The first idea was to use a toroidal
transformer for the RF injection to drastically reduce the series resistance without decreasing the available signal too much. The second idea was to measure the series, rather than parallel,
resonance to minimize the loading effects evident before.
Fig 9 - Circuit for simple and accurate measurments of the quality factor at 136 to 200 kHz.
The proposed circuit is shown in Fig 9. It was used for the measurements in Table 3. The main advantage of this new measurement concept is that it works at very low impedance levels and the R[AC] can
be evaluated by comparison with the series resistance (R[S] = 3 to 10 Ω). When the voltage, V[x], is one half of the voltage at the output of the transformer, the resistance is equal to R[AC] and the
relevant value (R[S]) can be measured by a simple digital ohmmeter, available in any ham shack.
During these kinds of measurements , some shrewdness must be used to prevent possible mistakes. It's important to verify that the coil under test and its magnetic field are far from metallic surfaces
and lossy materials (such as a wooden table) so that any such materials do not intercept the fields. Coils having such big dimensions compared with those used in traditional radio applications,
become loop antennas. So the measurements must be repeated, putting the winding in different positions and orientations to avoid possible errors.
Another consideration during the measurements is the distances among interconnection wires. They must be so far apart, in spite of the relatively low frequency involved, that they do not affect the
result of the test. Considering all these error sources, a possible suggestion is to find a reference inductor to verify the quality of the measurements performed. For this purpose, we used a
shielded coil, manufactured by Boonton company and used in Q tests, having the following characteristics: L = 2.5 mH, Q = 170. In Fig 5, this last (relatively small) coil is visible on top right
together with the group of inductors realized in our LF operation.
Fig 5 - A group of 136-kHz big coils realized for high quality factor (Q).
1. STMicroelectronics, 1000 E Bell Rd, Phoenix, AZ 85022; tel 602 485 6100, fax 602 485 6102; us.st.com.
2. A VLF band already exists in the US, but it's not an Amateur Radio allocation yet. A lot of "lowfer" (Low Frequency Experimental Radio) activity occurs in the 160 to 190-kHz region-the so called
1750 meter band, authorized under Part 15 of the FCC regulations. Right now, you don't need a license to operate on 1750 meters, but there are severe legal restrictions on what you can put on the
air there. For starters, you can't run more than 1 W input to the transmitter's final stage, and the entire length of the transmission line and antenna combined cannot exceed 15 meters
(approximately 50 feet). That's not much antenna for a band where a half-wavelength antenna would be more than one-half mile long! Hams that operate on 1750 meters sometimes use just their call
sign suffix as an ID.
1. W. J. Byron, W7DHD, "The Monster Antennas", Communications Quarterly, Spring 1996, pp 5-24.
2. W. J. Byron, W7DHD, "A Word About Short Verticals", Communications Quarterly, Fall 1998, pp 4-6.
3. M. Mardiguian, Grounding and Bonding, Vol 2, (Gainesville, Virginia: Interference Control Technologies, 1995) p 2.43.
4. P. Dodd, G3LDO, The LF Experimenter's Source Book, 2nd Edition, (Potters Bar, Hertfordshire, England: RSGB, 1998).
5. P. Dodd, G3LDO, "Getting Started on 136 kHz", RadComm (RSGB), March 1988,
6. D. Curry, Basic 1750 m Transmitting Antenna.
7. Reference Data for Engineers, 7th Edition, (New York: H. W. Sams & Co, 1989), p 6.4.
8. The 1990 ARRL Handbook (Newington, Connecticut: ARRIL, 1989).
9. F. E. Terman, Radio Engineer's Handbook, (New York: McGraw-Hill 1958).
10. B. Bowers, "Low Frequency Coil 'Q'", The Lowdown, Feb 1996. The Lowdown is the monthly newsletter of the Longwave Club of America (LWCA), 45 Wildflower Rd, Levittown; PA 19057, USA.
11. I.R. Sinclair, Newnes Audio and Hi-Fi Handbook, 2nd Edition, (Oxford: Butterworth-Heinemann, 1993), pp 748-752
12. M. Morando, I1MMR, "TX da 10 W a 137 kHz", Radio Rivista (Journal of the Associazione Radioamatori Italiani, Milan, Italy), Sep 1998, pp 52-53.
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Solved: minimum ele in Python - SourceTrail
Minimum electric energy is the minimum amount of electric energy that can be used to power an electronic device.
mnt in a list
lst = [5, 2, 1, 4, 3]
# Finding minimum element
min = lst[0]
for i in range(1, len(lst)):
if min > lst[i]:
min = lst[i]
# Printing minimum element
print("Minimum element in the list is :", min)
This code finds the minimum element in a list. The first line creates a list called lst. The second line sets min equal to the first element in the list. The third line loops through each element in
the list, starting with the second element. If any element in the list is less than min, min is set equal to that element. Finally, the fourth line prints out the minimum element in the list.
Min and Max Function
The min and max functions in Python allow you to find the smallest or largest value in a list of values. The syntax for the min function is as follows:
The syntax for the max function is as follows:
What is ele
In Python, ele is a built-in data type that represents an element of a sequence.
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Posts: 19
Location: Tates CreekI've always dreaded teaching this section because the kids hate it so much. This year, I did something defferent that worked much better and the students actually seemed to like
it. Yhe most important type of problem in 4.9 is the "frame" or "border" type of problem, so I concentrated solely on problems of this type. I did 2 examples for the students. Then I had them in
groups of 4 solving 2 similar problems. What was very interesting to me, was the fact that 4 of my 6 groups made the exact same mistake. When they were solving for the inside rectangle, they forgot
to distribute the "x" to the second term so they got answers like "x squared minus 10" instead of 10x, which messed up the entire problem. We discussed at length this mistake, and I believe that it
really helped all of the kids to understand how to solve these problems. Then I assigned 3 of this type for homework. It went very well and was not like pulling teeth because they got to talk about
it and discuss what was going on.
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OpenStax College Physics for AP® Courses, Chapter 11, Problem 66 (Problems & Exercises)
Calculate the ratio of the heights to which water and mercury are raised by capillary action in the same glass tube.
Question by
is licensed under
CC BY 4.0
Final Answer
This ratio is negative since the height of mercury is negative. The height of the mercury within the glass tube is lower than the surrounding mercury outside of the tube.
Solution video
OpenStax College Physics for AP® Courses, Chapter 11, Problem 66 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. A glass tube is inserted into some water and then it's inserted separately into some mercury and the question here is what is the ratio of heights
due to capillary action that the two fluids will reach? So we are taking a ratio of the water height in the glass tube versus the mercury height in the glass tube. So the height is 2 times the
surface tension of the fluid, which is water in the first case here, times cosine of the contact angle between the fluid and the material of the tube so water and glass divided by the density of
water times g times the radius of the tube. Now I didn't put a subscript w on the radius because that doesn't depend on the fluid— that's the same in both cases— it's the same tube and the same
radius so there's no need to distinguish it with a subscript. And so the height for mercury is 2 times the surface tension of mercury times cos of the contact angle between mercury and glass divided
by density of mercury times g times r. So we divide these two heights and it's confusing to divide a fraction by a fraction so instead I am going to multiply it by the reciprocal of the denominator
so this height for mercury is going to be written here flipped over and multiplied. And so the r's cancel, the g's cancel and the 2's cancel and we are left with the ratio of heights is surface
tension of water times cos of the contact angle times density of mercury and then divided by surface tension of mercury times cos of its contact angle with glass and then density of water. So we look
up all these things in our data tables. So we have surface tension of water and we find that in table [11.3]— that's 0.0728 newtons per meter— multiplied by cos of the contact angle between water and
glass— water-glass contact angle is 0 degrees— multiplied by 13.6 times 10 to the 3 kilograms per cubic meter density of mercury which we found in table [11.1]— mercury density is 13.6 times 10 to
the 3 kilograms per cubic meter— and divide that by the surface tension of mercury which is very high, 0.465, compared to 0.0728 for water— and that is shown here... 0.465 newtons per meter— times
cos of 140 degrees is the contact angle between mercury and glass and then multiply by the density of water— 1.000 times 10 to the 3 kilograms per cubic meter— this works out to negative 2.78. And
the reason this is negative is because this height is negative; the mercury goes down compared to the fluid level outside the tube so if this is the tube and this is the fluid here, mercury would be
lower than the surrounding fluid level. So this little height here is h Hg and it's negative because it's down. Okay and there we go!
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Does Value-Added Work Better in Elementary Than in Secondary Grades?
Does Value-Added Work Better in Elementary Than in Secondary Grades?
Douglas N. Harris Associate Professor
Public Education
Tulane University
Douglas N. Harris
and Andrew Anderson
• The vast majority of research on value-added measures focuses on elementary schools; value-added measures for middle and high school teachers pose particular challenges.
• Middle and high schools often “track” students in ways that affect the validity of value-added.
• Student tracking in middle and high schools calls into question the validity of methods typically used to create value-added measures.
• The validity of secondary-level value-added measures can be improved by directly accounting for tracks and specific courses, although this may not completely solve the problem.
• Middle and high school teachers have more students, and this factor increases reliability, but it is offset by other factors that reduce reliability at those grade levels.
• End-of-course exams, which are becoming more common in high school, have both advantages and disadvantages for estimating value-added.
There is a growing body of research on the validity and reliability of value-added measures, but most of this research has focused on elementary grades. This is because, in some respects, elementary
grades represent the “best-case” scenario for using value-added. Value-added measures require annual testing and, in most states, students are tested every year in elementary and middle school
(grades 3-8), but in only one year in high school. Also, a large share of elementary students spend almost all their instructional time with one teacher, so it is easier to attribute learning in math
and reading to that teacher.^[1]
Driven by several federal initiatives such as Race to the Top, Teacher Incentive Fund, and ESEA waivers, however, many states have incorporated value-added measures into the evaluations not only of
elementary teachers but of middle and high school teachers as well. Almost all states have committed to one of the two Common Core assessments that will test annually in high school, and there is
little doubt that value-added will be expanded to the grades in which the new assessments are introduced.^[2] In order to assess value-added and the validity and reliability of value-added measures,
it is important to consider the significant differences across grades in the ways teachers’ work and students’ time are organized.
As we describe below, the evidence shows that there are differences in the validity of value-added measures across grades for two primary reasons. First, middle and high schools “track” students;
that is, students are assigned to courses based on prior academic performance or other student characteristics. Tracking not only changes our ability to account for differences in the students who
teachers educate, but also the degree to which the curriculum aligns with the tests. Second, the structure of schooling and testing vary considerably by grade level in ways that affect reliability in
sometimes unexpected ways. The problems are partly correctable, but, as we show, more research is necessary to understand how problematic existing measures are and how they might be improved.
What Do We Know About How Teacher Value-Added Measures Work In Different Grades And Subjects?
We begin by discussing differences in validity across grades and follow with somewhat briefer discussions of reliability across grades. Validity refers to the degree to which something measures what
it claims to measure, at least on average. Reliability refers to the degree to which the measure is consistent when repeated. A measure could be valid on average, but inconsistent when repeated,
meaning it isn’t very reliable. Conversely, a measure could be highly reliable but invalid—that is, it could consistently provide the same invalid information.
Validity of value-added measures across grade levels
In elementary schools, it is common for principals to create similar classrooms (e.g., with similar numbers of low-performing and special needs students).
Students and teachers are assigned to classrooms differently in elementary schools than in middle and high schools. In elementary schools, it is common for principals to create similar classrooms
(e.g., with similar numbers of low-performing and special needs students).^[3] Other elementary principals identify student needs and try to match them to teachers who have the skills to meet those
needs. Principals may also take into account parental requests, so that students with more academically demanding parents get assigned to teachers with the best reputations. Either of these last two
forms of assignment—those based on student needs and those based on parental requests—has the potential to reduce the validity of value-added measures.^[4] Sometimes called selection bias, the
problem is that student needs and parental resources are never directly accounted for in value-added measures, even though they might affect student learning and therefore reduce validity of teacher
value-added estimates.
Based on a series of experiments,^[5] simulation studies,^[6] and statistical tests,^[7] elementary school value-added models do seem to address the selection bias problem well, on average. This last
caveat is important. It is extremely difficult to provide strong evidence of validity for each teacher’s value-added. Instead, prior studies are really examining whether selection bias averages out
for whole groups of teachers.^[8]
In middle and high schools, students with low test scores and grades and certain other characteristics are generally tracked into remedial courses, and those with stronger academic backgrounds are
tracked into advanced courses.
Students in middle and high schools, on the other hand, are not assigned to or “selected” for classes in the same way they usually are in elementary schools. Rather, students with low test scores and
grades and certain other characteristics are generally tracked into remedial courses, and those with stronger academic backgrounds are tracked into advanced courses. Minority and low-income students
are also more likely to end up in lower tracks. These decisions might not be driven by strict rules or requirements, but they reflect strong patterns. In our analyses of Florida data, 37 percent of
the variation in students’ middle school course tracks can be explained by a combination of their prior test scores, race/ethnicity, and family income.^[9]
Tracking creates two potential problems for value-added. First, the academic content of the courses differs. This means that the material covered in each course aligns to the test in different ways.
Tests are designed to align with state proficiency standards,^[10] which in many states require a fairly low level of academic skill.^[11] For this reason, we would expect the test to align better
with lower or middle tracks, implying that teachers in these tracks have an easier time showing achievement gains—and therefore higher value-added. This prediction is reinforced by evidence of
“ceiling effects” in standardized tests; students in the upper tracks, as described above, are likely to have higher scores and to hit the ceiling with little growth.^[12] The direction and magnitude
of these influences depends, of course, on the test and no doubt varies by state. Those states with low proficiency bars are probably more likely to have tests that align better with the remedial
This disadvantage to teaching in the upper track, however, is apparently offset by an apparently larger advantage. That is, upper track students seem to have unobserved traits that make them likely
to achieve larger achievement gains. This is what we would predict based on which parents tend to push hardest to get their children into upper track courses. Parents who press for more challenging
academic courses probably also press their children to work harder, do their homework, and so on—generating higher achievement. We cannot observe these parental activities, so they could get falsely
attributed to teachers in upper tracks.
The net effect is unclear. The curriculum-test misalignment places upper-track teachers at a disadvantage because of the misalignment between the test and course content, but this might be offset by
the advantage of having students who are likely to make achievement gains for reasons having nothing to do with the teacher. Below, we report results of data analyses that shed more light on the
issues created for value-added measures by tracking.
Analyses of Florida secondary schools
We estimated teacher value-added ignoring students’ tracks and courses, as is typically done, and then we re-estimated with track/course effects.^[13] In middle schools, our estimates suggest that
for a teacher with all lower track courses, ignoring tracks would reduce measured value-added from the 50th to the 30th percentile. Only about 25-50 percent of teachers remain in the same performance
quartile when we add information about the tracks.
Teachers had higher value-added when they taught the upper-track classes, compared with the same teachers teaching lower tracks.
One might wonder whether these effects exist because more effective teachers end up in upper-track courses. We addressed this possibility by analyzing teachers who taught both lower- and upper-track
courses and comparing value-added in each course type for the same teacher.^[14] Teachers had higher value-added when they taught the upper-track classes, compared with the same teachers teaching
lower tracks. These results could actually understate the role of tracks because the information available about tracks might not always be accurate.^[15] For this report, we extended the analysis to
Florida high schools where a similar number, 33 to 45 percent, would be in the wrong group without tracks.
Analyses from North Carolina and end-of-Course exams
If tracking is a problem in estimating value-added, we would expect the variation in high school teacher value-added to drop when we account for tracks. That is, when we ignore tracks and courses,
some teachers end up with value-added that is too high because they teach many upper-track courses, and vice versa in the lower track. So accounting for tracks pulls these teachers back to the middle
and reduces variation.
A recent report using data from North Carolina confirms this.^[16] The variation in teacher value-added in high school is 33 percent lower when adding track coefficients. That is, some teachers have
extremely low value-added simply because they teach more lower-track courses, and other teachers have high value-added because they teach upper-track courses. This does not prove that tracking is the
problem, but the evidence is consistent with that interpretation.
The study goes further and considers how well current value-added predicts future value-added across grade levels. Several researchers have argued that this predictive validity of value-added is an
important sign of the measure’s validity for making long-term employment decisions. We would hope, for example, that the measures used to make teacher tenure decisions are good predictors of how
teachers will perform in the years after they receive tenure. The North Carolina study finds that even the course-adjusted value-added measure is a worse predictor of future value-added in high
schools than in elementary schools, even after accounting for tracking.^[17] This suggests that adjusting value-added measures in this way does not eliminate the concern that tracking reduces
validity and/or that there might be other problems in estimating high school value-added.^[18]
The differences in state testing regimes are also noteworthy. In theory, high school value-added measures should have higher validity in North Carolina because that state uses end-of-course exams,
which should be better aligned to the curriculum than the single generic subject test in Florida. However, recall that the advantage for higher-track teachers from omitting tracks may offset the
disadvantage to those same teachers from test misalignment (e.g., test ceilings). Paradoxically, this means better test alignment could actually make the validity or selection bias problem worse
because one no longer offsets the other. This would give the upper-track teachers an even greater advantage. While it is in some ways helpful that the two problems cancel out, it places us in the
awkward position of having to rely on one mistake to fix the other. As an analogy, it is like a golfer who accidentally aims too far to the left but still hits the ball in the fairway because of a
slice to the right—the problems cancel out. In this case, fixing only the slice and not the aim would put the ball to the right of the fairway, making matters worse.
The use of end-of-course exams also raises the issue about how well prior achievement scores account for students’ relevant prior achievement. The purpose of accounting for prior scores is that they
can tell us where students started at the beginning of the school year, but the content of prior courses is so different in high school that it’s unclear how informative the prior score really is.
For example, few students have learned anything about physics before they take a physics course. However, accounting for prior math, science, and other scores is still important because those scores
adjust for general cognitive and study skills that also influence subsequent scores.^[19]
The ability of prior courses to account for sorting across grade levels is therefore unclear, but there are good reasons to think that having good alignment between this year’s test and this year’s
content is more important than having good alignment between this year’s test and last year’s test. As further evidence of this, we estimated value-added to math scores in middle schools controlling
only for prior reading scores—prior math scores were ignored. We then compared these new value-added estimates with the more typical ones where prior math is accounted for. The correlation between
the two is high at 0.84.^[20]
Other evidence and summary about validity issues
The well-known Measures of Effective Teaching (MET) project funded by the Gates Foundation reports results from experiments that also address the validity of value-added at the middle and high school
levels. They randomly assigned teachers to classrooms in middle school as well as 9th grade. However, there was apparently no data about the tracks teachers taught or whether random assignment
occurred only within tracks. Given the directions provided to principals, it seems likely that most assignments were within a track, but we cannot know for sure because tracking data was generally
not available in MET, so this study is not informative about the role of tracks in value-added estimation.^[21]
Ignoring tracks will reduce validity substantially in middle and high school, and even accounting for tracks may not solve the problem.
Overall, the evidence from the above studies suggests that ignoring tracks will reduce validity substantially in middle and high school, and even accounting for tracks may not solve the problem. This
also reinforces the general problem of comparing teacher performance in different instructional contexts.^[22]
Reliability of value-added measures across grade levels
There may be trade-offs between validity and reliability in evaluating value-added measures.^[23] Below, I consider the reliability of value-added by grade and then illustrate those trade-offs.
There are many sources of random error in value-added estimates: standardized tests have measurement error, some students are sick at test-taking time, and the students assigned to teachers in any
given year vary in essentially random ways. This helps to explain why teacher value-added measures are somewhat unstable over time. It also explains why researchers and value-added vendors typically
report confidence intervals for value-added measures that help quantify the role of random error and the uncertainty this creates about teachers’ “true” value-added.^[24]
One of the key factors affecting confidence intervals is the sample size—the larger the number of students assigned to each teacher, the smaller the confidence interval. The fact that elementary
students are assigned to only one teacher means that we can probably attribute that student’s learning to that teacher, but the trade-off is that these elementary teachers have fewer students
assigned to them, and this will tend to reduce reliability.
The larger number of students per teacher at the secondary level does not necessarily mean, however, that reliability is better. This is because reliability depends on error variance relative to the
variance in the value-added estimates.^[25] To take a sports analogy, suppose that we had very precise estimates of the performance of ten baseball players, but that every player was almost equally
effective and therefore had almost identical batting averages. In this situation, the variance in true performance is very small, so even very precise estimates of batting averages (after lots of
games) will make it hard to distinguish the best from the worst players—the estimates will be unreliable even after each player has hundreds of at-bats. Conversely, if half the players had high
batting averages and the other half had no hits at all, then we could reliably identify the low-performers after a week’s worth of games. The confidence intervals would be wide in that case, but it
wouldn’t matter because the differences in true performance are so large.
In this case, having more students reduces random error among middle school teachers, but, as the baseball analogy suggests, this does not increase reliability. Estimates by Daniel McCaffrey using
MET project data show that there is almost no relationship between grade level and reliability—reliability may actually be worse in higher grades.
Why might that be? Is there greater variance in teacher effectiveness at the elementary level? Is random error lower at the elementary level? Or both? One plausible explanation is that middle school
teachers each teach a wider range of students each year than do elementary teachers. This is plausible because our calculations in Florida suggest that most teachers work in multiple tracks. If
value-added estimates do not fully account for unobservable differences in students, then we would expect to see this pattern—the variance in teacher value-added is greater at the elementary level
perhaps because of biased estimates.^[26] Differences in random error could also explain lower reliability in middle schools if the reliability of the tests is lower, which would offset the advantage
of having more students.
The calculations by McCaffrey provide some support for both interpretations. Compared with the elementary schools in his sample, the variance in teacher value-added is lower in middle schools and
random error is higher. So, the advantage of having more students per teacher is offset by other factors that reduce reliability in middle school.
What More Needs To Be Known On This Issue?
The advantage of having more students per teacher is offset by other factors that reduce reliability in middle school.
The above evidence strongly suggests that accounting for course tracks is important to obtaining valid value-added estimates in middle and high school, but we do not yet know how well this solves the
problem. We estimate value-added by accounting for prior achievement, but a key implication of our tracking argument is that prior achievement is affected by prior tracks. This creates a complex
role for tracks that might not be easily captured by simply adding track variables to the value-added model.^[27] We therefore cannot presume that accounting for the tracks is sufficient, and the
North Carolina study reinforces this conclusion.^[28]
It would also be useful to know how this issue affects teachers in schools that use non-standard courses. We focused on algebra and geometry and excluded courses like “Liberal Arts Mathematics” and
“Applied Mathematics” that also showed up in the data. These courses are likely to align even less well with the tested content because, by definition, they are courses that are outside the norm. The
teachers of these courses could be at a significant disadvantage in their performance ratings.
In addition, while we have learned a great deal about elementary teacher value-added from experiments and simulation evidence, we need to apply those same methods to address the particular threats to
validity in middle and high schools. The MET project provides some experimental evidence in middle school, although simulation and other tests have been limited to elementary grades.
What Can’t Be Resolved By Empirical Evidence On This Issue?
While the evidence described here provides a sense of the empirical problems that arise across grades and subjects, there is a larger question about how well the tests capture what we want students
to learn and be able to do—and how this varies across grades. For example, creativity might be a skill that could be developed more easily in early grades, but creativity is hard to measure. So, in
that case, the validity problems in early grades would be even worse than in later grades. The statistical issues are therefore intertwined with the philosophical ones about what we want students to
How, And Under What Circumstances, Does This Issue Impact The Decisions And Actions That Districts Make On Teacher Evaluations?
This evidence also informs the use of value-added measures in (potentially) high-stakes decisions. For the sake of simplicity and perceived fairness, it is desirable to have a common standard that
applies to all grades—or really to all teachers. However, if the validity and reliability vary, not to mention the ways in which test scores align with the desired goals, then treating teachers
equitably may require using value-added unequally across grades. As I have written elsewhere, the stakes attached to any measure should be inversely proportional to the measure’s validity and
reliability.^[29] It appears that we may not be able to follow that rule and simultaneously use value-added the same way for all teachers, especially across elementary and secondary grades.
Given that the properties of value-added measures differ across grades and subjects, policymakers should consider using different methods for calculating and using value-added in different grades and
subjects. In particular, in middle and high school, it is essential to account for the tracks and courses that teachers are assigned to when calculating value-added.
Since value-added seems to work differently across grades, this raises the question: How do we handle teachers who teach multiple grades? Fundamentally, the issues raised here do not change the
answer to this question. Comparisons across grades have always been complicated by the fact that the tests differ across grades and the various approaches to combining them involve some sort of
weighted average, or composite, that takes into account differences in the test scale across grades.^[30] That basic solution is also reasonable for handling the additional complication of tracking.
The key is to first get the estimation right at each grade level, perhaps by accounting for tracks. That is, we have to get the estimates right for each track and grade level before creating
composite value-added measures for each teacher.^[31]
It might also be tempting to reduce tracking or assign each teacher an equal mix of low- and high-track courses to easier accommodate value-added measures, but this is the proverbial “tail wagging
the dog” problem. Changes in school organization and instruction should be made with caution and attention to effective instructional practice—not so that we can have better value-added measures.^
The implications of tracking are missed in the vast majority of value-added estimates now being used. This means that, even setting aside other issues with the measures, current standard value-added
measures for teachers who concentrate their work in particular tracks in middle and high schools will suffer from validity concerns. As with many of the problems with value-added, this one can be
addressed with better data collection efforts and careful attention to how the measures are created. Accounting for tracks would almost certainly improve the measures, but future research will be
required to determine how well this solution works in practice.
References +
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Multiplication of Three and Two digit numbers - Math Shortcut Tricks
Multiplication of Three and Two digit numbers
Multiplication of Three and Two digit numbers shortcut tricks are very important thing to know for your exams. Competitive exams are all about time. If you know time management then everything will
be easier for you. Most of us miss that part. Few examples on multiplication of three and two digit numbers shortcuts is given in this page below. We try to provide all types of shortcut tricks on
multiplication of three and two digit numbers here. Visitors please read carefully all shortcut examples. These examples will help you to understand shortcut tricks on Multiplication of Three and Two
digit numbers.
21 comments
1. edward says:
difficult to understand
□ Jeeva says:
Practice it again for few times
□ AF says:
Great trick!!!
□ Binay says:
Thank you. This trick will help my students.
□ G V R L SANTOSH says:
Really it’s too gud after a long time start practicing. Once again I am sharper and fast by these shortcuts. I loved it.tanq u
□ Mansi kulkarni says:
very nice trick!!!!!!!!!! thanks a lot!!!!!!!
□ basheer says:
☆ Sumanta Pal Chowdhury says:
☆ KS Mathur says:
Nice tricks. Its really works…
☆ sourav roy says:
Nice tricks.it is easy to understand and reduce timing
☆ Akbarr says:
cool trick, but i don’t understand why you multiply across!
☆ gg says:
solve this
○ abhishek says:
plz send me all mathematical tricks to improve my aptitude.
○ pratibha says:
This is good..this will help us a lot….thanks a lot
○ monica thakur says:
really god sir… thanq sir …..
○ Hardik says:
really a nice trick, it works. pls email me all the other tricks also. i will be waiting for u.
○ Dinesh R says:
I didn’t understand second step itself (cross multiplication)
■ Prakaash says:
consider 1 2 * 3 4 cross multiply means we want to multiply 2 with 3 and 1 with 4 i.e 2*3 and 1*4.
■ Kiran says:
It’s nice
■ Tomye Gloria says:
Nice Trick. Its really works and I like it.
Thank you.
■ Vk says:
Can you demonstrate how to apply the same technique for 108*77? I am unable to get the right answer.
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R lmer model: degree of freedom and chi square values are zero ~ Cross Validated ~ TransWikia.com
I have built the following models:
full <- lmer(DV~ A*B + (1|speaker), data, REML=FALSE)
A <- lmer(DV~ A+ A:B + (1|speaker), data, REML=FALSE)
B <- lmer(DV~ B+ A:B + (1|speaker), data, REML=FALSE)
interaction <- lmer(DV~ A + B + (1|speaker), data, REML=FALSE)
I use anova to compare the first full model to the other ones:
anova(full, A)
anova(full, B)
anova(full, interaction)
The first two comparisons generated results with both df and chi square values being zeros, as shown below:
However, I have also tried to compare the null model with another model only include A or B:
null <- lmer(DV~ 1 + (1|speaker), data, REML=FALSE)
AA <- lmer(DV~ A + (1|speaker), data, REML=FALSE)
BB <- lmer(DV~ B + (1|speaker), data, REML=FALSE)
AB <- lmer(DV~ A:B + (1|speaker), data, REML=FALSE)
all the comparisons generated reasonable results (i.e. not 0 df and all comparisons are significant)
I have looked online and found this post: https://www.researchgate.net/post/What_is_a_Likelihood_ratio_test_with_0_degree_of_freedom
And my guess is that maybe for my full model, the interaction might be able to predict everything without the main effects (A and B).
I have a few questions:
1. Is my guess possibly true?
2. If it is true, why did the comparison with the null model show a significant effect?
3. On a more general scale, when I build linear mixed effect models, can I start from the Null model and add a factor at a time, then compare with the previous models? Or do I have to reduce from
the full model?
4. If I use A+B as the base model:
base <- lmer(DV~ A+B + (1|speaker), data, REML=FALSE)
A <- lmer(DV~ A + (1|speaker), data, REML=FALSE)
B <- lmer(DV~ B + (1|speaker), data, REML=FALSE)
interaction <- lmer(DV~ A*B + (1|speaker), data, REML=FALSE)
Is it ok to report the comparison between the base model and A, B, interaction respectively?
Please find the data file and the R markdown document here: dropbox.com/sh/88m8h6blow2xbn5/AABiNccsUlu3AlfPyamQP4n_a?dl=0
I also asked a question about the procedures I used in the R script in this post R lmer model: add factors or reduce factors
I’d be most grateful if you could help me please. Thank you!
This happens because models full, A and B are in fact the same. They are just parameterised differently. To see this, inspect the estimates for the full model:
(Intercept) 6.03977 0.34949 17.282
AT2 -0.55051 0.07597 -7.246
AT3 -1.16472 0.07597 -15.331
AT4 0.48228 0.07597 6.348
BS -0.64024 0.07597 -8.427
AT2:BS 0.35379 0.10744 3.293
AT3:BS 0.47244 0.10824 4.365
AT4:BS 0.05247 0.10744 0.488
In model A, we have removed the main effect for the variable B and then obtain:
Estimate Std. Error t value
(Intercept) 6.03977 0.34949 17.282
AT2 -0.55051 0.07597 -7.246
AT3 -1.16472 0.07597 -15.331
AT4 0.48228 0.07597 6.348
AT1:BS -0.64024 0.07597 -8.427
AT2:BS -0.28645 0.07597 -3.770
AT3:BS -0.16781 0.07710 -2.177
AT4:BS -0.58777 0.07597 -7.737
We immediately see that the estimates for the intercept AT2- AT4 are the same. The estimate for AT1:BS in the second model is identical to the estimate for the main effect for B in the full model
(because the second model does not include the main effect for B). Then, for the same reason, the remaining interaction terms in the second model will be the sum of the main effect for B in the full
model, and the equivalent interaction terms:
> -0.64024 + 0.35379
[1] -0.28645
> -0.64024 + 0.47244
[1] -0.1678
> -0.64024 + 0.05247
[1] -0.58777
I think it is good general advice to always include both main effects in a model which includes their interaction. This type of problem will then not occur.
Answered by Robert Long on December 9, 2020
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Measurement Word Problems
Units of Measurement (Conversion Word Problems)
Measurement Conversion Word Problems
Mixed Word Problems (Measurement, Fractions, and Decimals)
Volume of Rectangular Prisms - Word Problems
Key Words in Word Problems
Pythagorean Theorem w/Word Problems
Customary Length Word Problems (05/01)
Explore Measurement Word Problems Worksheets by Grades
Explore Other Subject Worksheets for class 6
Explore printable Measurement Word Problems worksheets for 6th Class
Measurement Word Problems worksheets for Class 6 are an essential tool for teachers looking to help their students master the critical skill of solving math word problems. These worksheets provide a
variety of real-world scenarios that require students to apply their knowledge of measurement concepts, such as length, weight, and capacity, to solve problems. By incorporating these worksheets into
their lesson plans, teachers can ensure that their Class 6 students are well-prepared to tackle math word problems that involve measurement. Additionally, these worksheets can be used as a form of
assessment to track student progress and identify areas where additional instruction may be needed. Measurement Word Problems worksheets for Class 6 are a valuable resource for any teacher looking to
enhance their students' understanding of math concepts and problem-solving skills.
Quizizz is an excellent platform for teachers to incorporate Measurement Word Problems worksheets for Class 6 into their curriculum. This interactive platform allows teachers to create engaging
quizzes and games that can be used to supplement traditional worksheets, providing students with a fun and interactive way to practice their math skills. In addition to offering a wide range of
pre-made quizzes and games, Quizizz also allows teachers to customize their content to align with their specific lesson plans and objectives. This flexibility makes it easy for teachers to integrate
Quizizz into their existing curriculum, ensuring that their Class 6 students receive a well-rounded education in math and math word problems. By utilizing Quizizz in conjunction with Measurement Word
Problems worksheets for Class 6, teachers can create a dynamic and engaging learning environment that fosters student success.
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How can graphics and/or statistics be used to misrepresent data? Where have you seen this done?
how much would it cost to do the following:
How can graphics and/or statistics be used to misrepresent data? Where have you seen this done?
What are the characteristics of a population for which it would be appropriate to use mean/median/mode? When would the characteristics of a population make them inappropriate to use?
Questions to Be Graded: Exercises 6, 8 and 9
Complete Exercises 6, 8, and 9 in Statistics for Nursing Research: A Workbook for Evidence-Based Practice, and submit as directed by the instructor.
Questions to Be Graded: Exercise 27
Use MS Word to complete “Questions to be Graded: Exercise 27” in Statistics for Nursing Research: A Workbook for Evidence-Based Practice. Submit your work in SPSS by copying the output and pasting
into the Word document. In addition to the SPSS output, please include explanations of the results where appropriate.
Copyright © 2017, Elsevier Inc. All rights reserved. 67 EXERCISE 6 Questions to Be Graded Follow your instructor ’ s directions to submit your answers to the following questions for grading. Your
instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers
online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.”
Name: _______________________________________________________
Class: _____________________
Date: ___________________________________________________________________________________ 68EXERCISE 6 •
What are the frequency and percentage of the COPD patients in the severe airfl ow limitation group who are employed in the Eckerblad et al. (2014) study?
What percentage of the total sample is retired? What percentage of the total sample is on sick leave?
What is the total sample size of this study? What frequency and percentage of the total sample were still employed? Show your calculations and round your answer to the nearest whole percent.
What is the total percentage of the sample with a smoking history—either still smoking or former smokers? Is the smoking history for study participants clinically important? Provide a rationale for
your answer.
What are pack years of smoking? Is there a signifi cant difference between the moderate and severe airfl ow limitation groups regarding pack years of smoking? Provide a rationale for your answer.
What were the four most common psychological symptoms reported by this sample of patients with COPD? What percentage of these subjects experienced these symptoms? Was there a sig-nifi cant difference
between the moderate and severe airfl ow limitation groups for psychological symptoms?
What frequency and percentage of the total sample used short-acting β 2 -agonists? Show your calculations and round to the nearest whole percent.
Is there a signifi cant difference between the moderate and severe airfl ow limitation groups regarding the use of short-acting β 2 -agonists? Provide a rationale for your answer.
Was the percentage of COPD patients with moderate and severe airfl ow limitation using short-acting β 2 -agonists what you expected? Provide a rationale with documentation for your answer.
Are these fi ndings ready for use in practice? Provide a rationale for your answer.
Understanding Frequencies and Percentages STATISTICAL TECHNIQUE IN REVIEW Frequency is the number of times a score or value for a variable occurs in a set of data. Frequency distribution is a
statistical procedure that involves listing all the possible values or scores for a variable in a study. Frequency distributions are used to organize study data for a detailed examination to help
determine the presence of errors in coding or computer programming ( Grove, Burns, & Gray, 2013 ). In addition, frequencies and percentages are used to describe demographic and study variables
measured at the nominal or ordinal levels. Percentage can be defi ned as a portion or part of the whole or a named amount in every hundred measures. For example, a sample of 100 subjects might include
40 females and 60 males. In this example, the whole is the sample of 100 subjects, and gender is described as including two parts, 40 females and 60 males. A percentage is calculated by dividing the
smaller number, which would be a part of the whole, by the larger number, which represents the whole. The result of this calculation is then multiplied by 100%. For example, if 14 nurses out of a
total of 62 are working on a given day, you can divide 14 by 62 and multiply by 100% to calculate the percentage of nurses working that day. Calculations: (14 ÷ 62) × 100% = 0.2258 × 100% = 22.58% =
22.6%. The answer also might be expressed as a whole percentage, which would be 23% in this example. A cumulative percentage distribution involves the summing of percentages from the top of a table
to the bottom. Therefore the bottom category has a cumulative percentage of 100% (Grove, Gray, & Burns, 2015). Cumulative percentages can also be used to deter-mine percentile ranks, especially when
discussing standardized scores. For example, if 75% of a group scored equal to or lower than a particular examinee ’ s score, then that examinee ’ s rank is at the 75 th percentile. When reported as
a percentile rank, the percentage is often rounded to the nearest whole number. Percentile ranks can be used to analyze ordinal data that can be assigned to categories that can be ranked. Percentile
ranks and cumulative percentages might also be used in any frequency distribution where subjects have only one value for a variable. For example, demographic characteristics are usually reported with
the frequency ( f ) or number ( n ) of subjects and percentage (%) of subjects for each level of a demographic variable. Income level is presented as an example for 200 subjects: Income Level
Frequency ( f ) Percentage (%) Cumulative % 1. $100,000 105%100% EXERCISE 6 60EXERCISE 6 • Understanding Frequencies and PercentagesCopyright © 2017, Elsevier Inc. All rights reserved. In data
analysis, percentage distributions can be used to compare fi ndings from different studies that have different sample sizes, and these distributions are usually arranged in tables in order either from
greatest to least or least to greatest percentages ( Plichta & Kelvin, 2013 ). RESEARCH ARTICLE Source Eckerblad, J., Tödt, K., Jakobsson, P., Unosson, M., Skargren, E., Kentsson, M., & Thean-der, K.
(2014). Symptom burden in stable COPD patients with moderate to severe airfl ow limitation. Heart & Lung, 43 (4), 351–357. Introduction Eckerblad and colleagues (2014 , p. 351) conducted a comparative
descriptive study to examine the symptoms of “patients with stable chronic obstructive pulmonary disease (COPD) and determine whether symptom experience differed between patients with mod-erate or
severe airfl ow limitations.” The Memorial Symptom Assessment Scale (MSAS) was used to measure the symptoms of 42 outpatients with moderate airfl ow limitations and 49 patients with severe airfl ow
limitations. The results indicated that the mean number of symptoms was 7.9 ( ± 4.3) for both groups combined, with no signifi cant dif-ferences found in symptoms between the patients with moderate
and severe airfl ow limi-tations. For patients with the highest MSAS symptom burden scores in both the moderate and the severe limitations groups, the symptoms most frequently experienced included
shortness of breath, dry mouth, cough, sleep problems, and lack of energy. The research-ers concluded that patients with moderate or severe airfl ow limitations experienced mul-tiple severe symptoms
that caused high levels of distress. Quality assessment of COPD patients ’ physical and psychological symptoms is needed to improve the management of their symptoms. Relevant Study Results Eckerblad
et al. (2014 , p. 353) noted in their research report that “In total, 91 patients assessed with MSAS met the criteria for moderate ( n = 42) or severe airfl ow limitations ( n = 49). Of those 91
patients, 47% were men, and 53% were women, with a mean age of 68 ( ± 7) years for men and 67 ( ± 8) years for women. The majority (70%) of patients were married or cohabitating. In addition, 61%
were retired, and 15% were on sick leave. Twenty-eight percent of the patients still smoked, and 69% had stopped smoking. The mean BMI (kg/m 2 ) was 26.8 ( ± 5.7). There were no signifi cant
differences in demographic characteristics, smoking history, or BMI between patients with moderate and severe airfl ow limitations ( Table 1 ). A lower proportion of patients with moderate airfl ow
limitation used inhalation treatment with glucocorticosteroids, long-acting β 2 -agonists and short-acting β 2 -agonists, but a higher proportion used analgesics compared with patients with severe
airfl ow limitation. Symptom prevalence and symptom experience The patients reported multiple symptoms with a mean number of 7.9 ( ± 4.3) symptoms (median = 7, range 0–32) for the total sample, 8.1 (
± 4.4) for moderate airfl ow limitation and 7.7 ( ± 4.3) for severe airfl ow limitation ( p = 0.36) . . . . Highly prevalent physical symp-toms ( ≥ 50% of the total sample) were shortness of breath
(90%), cough (65%), dry mouth (65%), and lack of energy (55%). Five additional physical symptoms, feeling drowsy Understanding Frequencies and Percentages • EXERCISE 6Copyright © 2017, Elsevier Inc.
All rights reserved. TABLE 1 BACKGROUND CHARACTERISTICS AND USE OF MEDICATION FOR PATIENTS WITH STABLE CHRONIC OBSTRUCTIVE LUNG DISEASE CLASSIFIED IN PATIENTS WITH MODERATE AND SEVERE AIRFLOW
LIMITATION Moderate n = 42 Severe n = 49 p Value Sex, n (%)0.607 Women19 (45)29 (59) Men23 (55)20 (41)Age (yrs), mean ( SD )66.5 (8.6)67.9 (6.8)0.396Married/cohabitant n (%)29 (69)34 (71)
0.854Employed, n (%)7 (17)7 (14)0.754Smoking, n %0.789 Smoking13 (31)12 (24) Former smokers28 (67)35 (71) Never smokers1 (2)2 (4)Pack years smoking, mean ( SD )29.1 (13.5)34.0 (19.5)0.177BMI (kg/m 2
), mean ( SD )27.2 (5.2)26.5 (6.1)0.555FEV 1 % of predicted, mean ( SD )61.6 (8.4)42.2 (5.8) < 0.001SpO 2 % mean ( SD )95.8 (2.4)94.5 (3.0)0.009Physical health, mean ( SD )3.2 (0.8)3.0 (0.8)
0.120Mental health, mean ( SD )3.7 (0.9)3.6 (1.0)0.628Exacerbation previous 6 months, n (%)14 (33)15 (31)0.781Admitted to hospital previous year, n (%)10 (24)14 (29)0.607Medication use, n (%) Inhaled
glucocorticosteroids30 (71)44 (90)0.025 Systemic glucocorticosteroids3 (6.3)0 (0)0.094 Anticholinergic32 (76)42 (86)0.245 Long-acting β 2 -agonists30 (71)45 (92)0.011 Short-acting β 2 -agonists13
(31)32 (65)0.001 Analgesics11 (26)5 (10)0.046 Statins8 (19)11 (23)0.691 Eckerblad, J., Tödt, K., Jakobsson, P., Unosson, M., Skargren, E., Kentsson, M., & Theander, K. (2014). Symptom burden in
stable COPD patients with moderate to severe airfl ow limitation. Heart & Lung, 43 (4), p. 353. numbness/tingling in hands/feet, feeling irritable, and dizziness, were reported by between 25% and 50%
of the patients. The most commonly reported psychological symptom was diffi culty sleeping (52%), followed by worrying (33%), feeling irritable (28%) and feeling sad (22%). There were no signifi cant
differences in the occurrence of physical and psy-chological symptoms between patients with moderate and severe airfl ow limitations” ( Eckerblad et al., 2014 , p. 353). 62EXERCISE 6 • Understanding
Frequencies and PercentagesCopyright © 2017, Elsevier Inc. All rights reserved. STUDY QUESTIONS 1. What are the frequency and percentage of women in the moderate airfl ow limitation group? 2. What
were the frequencies and percentages of the moderate and the severe airfl ow limitation groups who experienced an exacerbation in the previous 6 months? 3. What is the total sample size of COPD
patients included in this study? What number or fre-quency of the subjects is married/cohabitating? What percentage of the total sample is married or cohabitating? 4. Were the moderate and severe
airfl ow limitation groups signifi cantly different regarding married/cohabitating status? Provide a rationale for your answer. 5. List at least three other relevant demographic variables the
researchers might have gathered data on to describe this study sample. 6. For the total sample, what physical symptoms were experienced by ≥ 50% of the subjects? Identify the physical symptoms and
the percentages of the total sample experiencing each symptom.
Interpreting Line Graphs EXERCISE 7
69 Interpreting Line Graphs STATISTICAL TECHNIQUE IN REVIEW Tables and fi gures are commonly used to present fi ndings from studies or to provide a way for researchers to become familiar with research
data. Using fi gures, researchers are able to illustrate the results from descriptive data analyses, assist in identifying patterns in data, identify changes over time, and interpret exploratory fi
ndings. A line graph is a fi gure that is developed by joining a series of plotted points with a line to illustrate how a variable changes over time. A line graph fi gure includes a horizontal scale,
or x -axis, and a vertical scale, or y -axis. The x -axis is used to document time, and the y -axis is used to document the mean scores or values for a variable ( Grove, Burns, & Gray, 2013 ; Plichta
& Kelvin, 2013 ). Researchers might include a line graph to compare the values for three or four variables in a study or to identify the changes in groups for a selected variable over time. For
example, Figure 7-1 presents a line graph that documents time in weeks on the x -axis and mean weight loss in pounds on the y -axis for an experimental group consuming a low carbohydrate diet and a
control group consuming a standard diet. This line graph illustrates the trend of a strong, steady increase in the mean weight lost by the experimental or intervention group and minimal mean weight
loss by the control group. EXERCISE 7 FIGURE 7-1 ■ LINE GRAPH COMPARING EXPERIMENTAL AND CONTROL GROUPS FOR WEIGHT LOSS OVER FOUR WEEKS. Weight loss (lbs)
Weeksy-axisx-axisControlExperimental10864201234 70EXERCISE 7 • Interpreting Line GraphsCopyright © 2017, Elsevier Inc. All rights reserved. RESEARCH ARTICLE Source Azzolin, K., Mussi, C. M., Ruschel,
K. B., de Souza, E. N., Lucena, A. D., & Rabelo-Silva, E. R. (2013). Effectiveness of nursing interventions in heart failure patients in home care using NANDA-I, NIC, and NOC. Applied Nursing
Research, 26 (4), 239–244. Introduction Azzolin and colleagues (2013) analyzed data from a larger randomized clinical trial to determine the effectiveness of 11 nursing interventions (NIC) on
selected nursing out-comes (NOC) in a sample of patients with heart failure (HF) receiving home care. A total of 23 patients with HF were followed for 6 months after hospital discharge and provided
four home visits and four telephone calls. The home visits and phone calls were organized using the nursing diagnoses from the North American Nursing Diagnosis Association International (NANDA-I)
classifi cation list. The researchers found that eight nursing interven tions signifi cantly improved the nursing outcomes for these HF patients. Those interventions included “health education,
self-modifi cation assistance, behavior modifi -cation, telephone consultation, nutritional counselling, teaching: prescribed medications, teaching: disease process, and energy management” ( Azzolin et
al., 2013 , p. 243). The researchers concluded that the NANDA-I, NIC, and NOC linkages were useful in manag-ing patients with HF in their home. Relevant Study Results Azzolin and colleagues (2013)
presented their results in a line graph format to display the nursing outcome changes over the 6 months of the home visits and phone calls. The nursing outcomes were measured with a fi ve-point Likert
scale with 1 = worst and 5 = best. “Of the eight outcomes selected and measured during the visits, four belonged to the health & knowledge behavior domain (50%), as follows: knowledge: treatment
regimen; compliance behavior; knowledge: medication; and symptom control. Signifi cant increases were observed in this domain for all outcomes when comparing mean scores obtained at visits no. 1 and 4
( Figure 1 ; p < 0.001 for all comparisons). The other four outcomes assessed belong to three different NOC domains, namely, functional health (activity tolerance and energy conservation),
physiologic health (fl uid balance), and family health (family participation in professional care). The scores obtained for activity tolerance and energy conservation increased signifi cantly from
visit no. 1 to visit no. 4 ( p = 0.004 and p < 0.001, respectively). Fluid balance and family participation in professional care did not show statistically signifi cant differences ( p = 0.848 and p =
0.101, respectively) ( Figure 2 )” ( Azzolin et al., 2013 , p. 241). The signifi cance level or alpha ( α ) was set at 0.05 for this study. Interpreting Line Graphs • EXERCISE 7Copyright © 2017,
Elsevier Inc. All rights reserved. FIGURE 2 ■ NURSING OUTCOMES MEASURED OVER 6 MONTHS (OTHER DOMAINS): Activity tolerance (95% CI − 1.38 to − 0.18, p = 0.004); energy conservation (95% CI − 0.62 to −
0.19, p < 0.001); fl uid balance (95% CI − 0.25 to 0.07, p = .848); family participation in professional care (95% CI − 2.31 to − 0.11, p = 0.101). HV = home visit. CI = confi dence interval. Azzolin,
K., Mussi, C. M., Ruschel, K. B., de Souza, E. N., Lucena, A. D., & Rabelo-Silva, E. R. (2013). Effectiveness of nursing interventions in heart failure patients in home care using NANDA-I, NIC, and
NOC. Applied Nursing Research, 26 (4), p. 242. 5.04.54.03.53.02.52.01.51.00.50MeanHV1HV2HV3HV4Fluid balanceFamily participationin professional careActivity toleranceEnergy conservation FIGURE 1 ■
NURSING OUTCOMES MEASURED OVER 6 MONTHS (HEALTH & KNOWLEDGE BEHAVIOR DOMAIN): Knowledge: medication (95% CI − 1.66 to − 0.87, p < 0.001); knowledge: treatment regimen (95% CI − 1.53 to − 0.98, p <
0.001); symptom control (95% CI − 1.93 to − 0.95, p < 0.001); and compliance behavior (95% CI − 1.24 to − 0.56, p < 0.001). HV = home visit. CI = confi dence interval.
5.04.54.03.53.02.52.01.51.00.50MeanHV1HV2HV3HV4Compliance behaviorSymptom controlKnowledge: medicationKnowledge: treatment reg 72EXERCISE 7 • Interpreting Line GraphsCopyright © 2017, Elsevier Inc.
All rights reserved. STUDY QUESTIONS 1. What is the purpose of a line graph? What elements are included in a line graph? 2. Review Figure 1 and identify the focus of the x -axis and the y -axis. What
is the time frame for the x -axis? What variables are presented on this line graph? 3. In Figure 1 , did the nursing outcome compliance behavior change over the 6 months of home visits? Provide a
rationale for your answer. 4. State the null hypothesis for the nursing outcome compliance behavior. 5. Was there a signifi cant difference in compliance behavior from the fi rst home visit (HV1) to
the fourth home visit (HV4)? Was the null hypothesis accepted or rejected? Provide a rationale for your answer. 6. In Figure 1 , what outcome had the lowest mean at HV1? Did this outcome improve over
the four home visits? Provide a rationale for your answer.
Copyright © 2017, Elsevier Inc. All rights reserved. 77
Questions to Be Graded EXERCISE 7 Follow your instructor ’ s directions to submit your answers to the following questions for grading. Your instructor may ask you to write your answers below and
submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers online at http://evolve.elsevier.com/Grove/statistics/
under “Questions to Be Graded.”
What is the focus of the example Figure 7-1 in the section introducing the statistical technique of this exercise?
In Figure 2 of the Azzolin et al. (2013 , p. 242) study, did the nursing outcome activity tolerance change over the 6 months of home visits (HVs) and telephone calls? Provide a rationale for your
State the null hypothesis for the nursing outcome activity tolerance.
Was there a signifi cant difference in activity tolerance from the fi rst home visit (HV1) to the fourth home visit (HV4)? Was the null hypothesis accepted or rejected? Provide a rationale for your
In Figure 2 , what nursing outcome had the lowest mean at HV1? Did this outcome improve over the four HVs? Provide a rationale for your answer.
What nursing outcome had the highest mean at HV1 and at HV4? Was this outcome signifi -cantly different from HV1 to HV4? Provide a rationale for your answer.
State the null hypothesis for the nursing outcome family participation in professional care.
Was there a statistically signifi cant difference in family participation in professional care from HV1 to HV4? Was the null hypothesis accepted or rejected? Provide a rationale for your answer.
Was Figure 2 helpful in understanding the nursing outcomes for patients with heart failure (HF) who received four HVs and telephone calls? Provide a rationale for your answer. 10. What nursing
interventions signifi cantly improved the nursing outcomes for these patients with HF? What implications for practice do you note from these study results? Copyright © 2017, Elsevier Inc. All rights
reserved. 79 Measures of Central Tendency : Mean, Median, and Mode
EXERCISE 8 STATISTICAL TECHNIQUE IN REVIEW Mean, median, and mode are the three measures of central tendency used to describe study variables. These statistical techniques are calculated to determine
the center of a distribution of data, and the central tendency that is calculated is determined by the level of measurement of the data (nominal, ordinal, interval, or ratio; see Exercise 1 ). The
mode is a category or score that occurs with the greatest frequency in a distribution of scores in a data set. The mode is the only acceptable measure of central tendency for analyzing nominal-level
data, which are not continuous and cannot be ranked, compared, or sub-jected to mathematical operations. If a distribution has two scores that occur more fre-quently than others (two modes), the
distribution is called bimodal . A distribution with more than two modes is multimodal ( Grove, Burns, & Gray, 2013 ). The median ( MD ) is a score that lies in the middle of a rank-ordered list of
values of a distribution. If a distribution consists of an odd number of scores, the MD is the middle score that divides the rest of the distribution into two equal parts, with half of the values
falling above the middle score and half of the values falling below this score. In a distribu-tion with an even number of scores, the MD is half of the sum of the two middle numbers of that
distribution. If several scores in a distribution are of the same value, then the MD will be the value of the middle score. The MD is the most precise measure of central ten-dency for ordinal-level
data and for nonnormally distributed or skewed interval- or ratio-level data. The following formula can be used to calculate a median in a distribution of scores. Median()()MDN=+÷12 N is the number
of scores ExampleMedianscoreth_N==+=÷=31311232216 ExampleMedianscoreth:.N==+=÷=404012412205 Thus in the second example, the median is halfway between the 20 th and the 21 st scores. The mean ( X ) is
the arithmetic average of all scores of a sample, that is, the sum of its individual scores divided by the total number of scores. The mean is the most accurate measure of central tendency for
normally distributed data measured at the interval and ratio levels and is only appropriate for these levels of data (Grove, Gray, & Burns, 2015). In a normal distribution, the mean, median, and mode
are essentially equal (see Exercise 26 for determining the normality of a distribution). The mean is sensitive to extreme
Copyright © 2017, Elsevier Inc. All rights reserved. 77 Questions to Be Graded EXERCISE 7 Follow your instructor ’ s directions to submit your answers to the following questions for grading. Your
instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers
online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.” 1. What is the focus of the example Figure 7-1 in the section introducing the statistical technique of this
exercise? 2. In Figure 2 of the Azzolin et al. (2013 , p. 242) study, did the nursing outcome activity tolerance change over the 6 months of home visits (HVs) and telephone calls? Provide a rationale
for your answer. 3. State the null hypothesis for the nursing outcome activity tolerance. 4. Was there a signifi cant difference in activity tolerance from the fi rst home visit (HV1) to the fourth
home visit (HV4)? Was the null hypothesis accepted or rejected? Provide a rationale for your answer. Name: _______________________________________________________ Class: _____________________ Date:
___________________________________________________________________________________ 78EXERCISE 7 • Interpreting Line GraphsCopyright © 2017, Elsevier Inc. All rights reserved. 5. In Figure 2 , what
nursing outcome had the lowest mean at HV1? Did this outcome improve over the four HVs? Provide a rationale for your answer. 6. What nursing outcome had the highest mean at HV1 and at HV4? Was this
outcome signifi -cantly different from HV1 to HV4? Provide a rationale for your answer. 7. State the null hypothesis for the nursing outcome family participation in professional care. 8. Was there a
statistically signifi cant difference in family participation in professional care from HV1 to HV4? Was the null hypothesis accepted or rejected? Provide a rationale for your answer. 9. Was Figure 2
helpful in understanding the nursing outcomes for patients with heart failure (HF) who received four HVs and telephone calls? Provide a rationale for your answer. 10. What nursing interventions
signifi cantly improved the nursing outcomes for these patients with HF? What implications for practice do you note from these study results? Copyright © 2017, Elsevier Inc. All rights reserved. 79
Measures of Central Tendency : Mean, Median, and Mode EXERCISE 8 STATISTICAL TECHNIQUE IN REVIEW Mean, median, and mode are the three measures of central tendency used to describe study variables.
These statistical techniques are calculated to determine the center of a distribution of data, and the central tendency that is calculated is determined by the level of measurement of the data
(nominal, ordinal, interval, or ratio; see Exercise 1 ). The mode is a category or score that occurs with the greatest frequency in a distribution of scores in a data set. The mode is the only
acceptable measure of central tendency for analyzing nominal-level data, which are not continuous and cannot be ranked, compared, or sub-jected to mathematical operations. If a distribution has two
scores that occur more fre-quently than others (two modes), the distribution is called bimodal . A distribution with more than two modes is multimodal ( Grove, Burns, & Gray, 2013 ). The median ( MD
) is a score that lies in the middle of a rank-ordered list of values of a distribution. If a distribution consists of an odd number of scores, the MD is the middle score that divides the rest of the
distribution into two equal parts, with half of the values falling above the middle score and half of the values falling below this score. In a distribu-tion with an even number of scores, the MD is
half of the sum of the two middle numbers of that distribution. If several scores in a distribution are of the same value, then the MD will be the value of the middle score. The MD is the most
precise measure of central ten-dency for ordinal-level data and for nonnormally distributed or skewed interval- or ratio-level data. The following formula can be used to calculate a median in a
distribution of scores. Median()()MDN=+÷12 N is the number of scores ExampleMedianscoreth_N==+=÷=31311232216 ExampleMedianscoreth:.N==+=÷=404012412205 Thus in the second example, the median is
halfway between the 20 th and the 21 st scores. The mean ( X ) is the arithmetic average of all scores of a sample, that is, the sum of its individual scores divided by the total number of scores.
The mean is the most accurate measure of central tendency for normally distributed data measured at the interval and ratio levels and is only appropriate for these levels of data (Grove, Gray, &
Burns, 2015). In a normal distribution, the mean, median, and mode are essentially equal (see Exercise 26 for determining the normality of a distribution). The mean is sensitive to extreme
Copyright © 2017, Elsevier Inc. All rights reserved. 77 Questions to Be Graded EXERCISE 7 Follow your instructor ’ s directions to submit your answers to the following questions for grading. Your
instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers
online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.”
What is the focus of the example Figure 7-1 in the section introducing the statistical technique of this exercise?
In Figure 2 of the Azzolin et al. (2013 , p. 242) study, did the nursing outcome activity tolerance change over the 6 months of home visits (HVs) and telephone calls? Provide a rationale for your
State the null hypothesis for the nursing outcome activity tolerance.
Was there a signifi cant difference in activity tolerance from the fi rst home visit (HV1) to the fourth home visit (HV4)? Was the null hypothesis accepted or rejected? Provide a rationale for your
In Figure 2 , what nursing outcome had the lowest mean at HV1? Did this outcome improve over the four HVs? Provide a rationale for your answer.
What nursing outcome had the highest mean at HV1 and at HV4? Was this outcome signifi -cantly different from HV1 to HV4? Provide a rationale for your answer.
State the null hypothesis for the nursing outcome family participation in professional care.
Was there a statistically signifi cant difference in family participation in professional care from HV1 to HV4? Was the null hypothesis accepted or rejected? Provide a rationale for your answer.
Was Figure 2 helpful in understanding the nursing outcomes for patients with heart failure (HF) who received four HVs and telephone calls? Provide a rationale for your answer.
What nursing interventions signifi cantly improved the nursing outcomes for these patients with HF? What implications for practice do you note from these study results?
Copyright © 2017, Elsevier Inc. All rights reserved. 79 Measures of Central Tendency : Mean, Median, and Mode EXERCISE 8 STATISTICAL TECHNIQUE IN REVIEW Mean, median, and mode are the three measures
of central tendency used to describe study variables. These statistical techniques are calculated to determine the center of a distribution of data, and the central tendency that is calculated is
determined by the level of measurement of the data (nominal, ordinal, interval, or ratio; see Exercise 1 ). The mode is a category or score that occurs with the greatest frequency in a distribution
of scores in a data set. The mode is the only acceptable measure of central tendency for analyzing nominal-level data, which are not continuous and cannot be ranked, compared, or sub-jected to
mathematical operations. If a distribution has two scores that occur more fre-quently than others (two modes), the distribution is called bimodal . A distribution with more than two modes is
multimodal ( Grove, Burns, & Gray, 2013 ). The median ( MD ) is a score
"Is this question part of your assignment? We Can Help!"
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Maths Phrase
The Maths Phrase game is a nice way of combining competition with mini white boards and being able to assess student's understanding of a range of topics.
Hidden behind 15 cards is an image that summarises a common phrase used in Maths. Students need to answer a question based on topics chosen by you from the https://mathswhiteboard.com topic bank, to
uncover the tiles and to earn a guess of the hidden Maths Phrase (or Maths related dingbat)! It would be a wonderful to also combine this with your school's reward system and distribute points for
effort and guesses.
Design your Maths Phrase game
Choose your Maths Phrase:
Alternatively, choose from a common catchphrase list:
Choose your common catchphrase:
Choose your skills to be included in the game:
The alternative to creating your own question set would be to grab one of my quick takeaways:
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Symmetry by permuting states
As the use of the complement in creating these examples shows, a tensor square can be further refined if the basic diagram has an isomorphism, which is to say a permutation of its nodes resulting
in the same connectivity matrix; for binary automata such an isomorphism results from complementing the individual cells. For k>2 there are increasing numbers of isomorphisms, N and its
The pair matrix of Sec. 3.4 can be rearranged along these lines, which is to say that it is equivalent to the following matrix, gotten by listing first all the pairs of the form
As before, the possible pair matrices all conform to a common pattern, which arises for either Rule 0 or Rule 255. Since the pair a links to c and b to d via the same symbol, the pairs of pairs
actually linked will vary from rule to rule. Furthermore there are some additional symmetry requirements for the formation of pairs: if
With this new ordering, the pair matrix for Rule 22 becomes: (1's belong to the actual pair matrix, while
The submatrix in the upper left hand corner is always present for whatever Rule (the lower right submatrix is filled out for Rule 150, just as it would be for any other rule symmetric by
conjugation), no matter what other elements are deleted from N on account of the structure of 4, so by Gerschgorin's theorem, the largest eigenvalue of N will be bounded by 4. The limit can only
be reached if all row sums are equal, thus only for Rules 0 and 255 (evolution into constants).
On the other hand the diagonal submatrices have a uniform row sum of 2, which must be their largest eigenvalue; alternatively the first and last are just de Bruijn matrices, from which the same
result follows. By a theorem in Gantmacher [11], this establishes a lower limit for the greatest eigenvalue of N, which must exceed the eigenvalues of any principal minor.
Of course there is an especial interest in conditions in which this lowest limit is maintained; so far it has not been possible to characterize the matrix N sufficiently to decide other than in
special cases or by numerical calculation. The general result of Sec. 10 is the best known, that the minimum eigenvalue which already implies surjectivity, implies uniform distribution. The
converse is not valid, most uniform distributions lead to higher eigenvalues.
Harold V. McIntosh
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fractal infinite zoom
fractal infinite zoomare zane and chandler smith related
Needs to calculate about 2.5 times more pixels than displayed. Full zoomer implementation. You can find one by cutting straight across the middle of the carpet or along one of the diagonals of the
sponge. The fractals are rendered using the OpenGL Shading Language (GLSL) to enable real-time interactivity. platform). photos and even a few short movies to share. Sandy coasts have a smooth
profile while rocky coasts have a fractal nature. GrafZViZion .gzv milliseconds per frame with an Intel i7 process. Here, we get the Sierpinski triangle from a square which we repeatedly reduce to 1/
4 of its original on the top. viewer.html For example the Contour With a bit of practice you will be able to create many interesting fractal forms, from organic looking trees to symmetrical
structures like snow flakes. Fixed point/integer and loop unrolling are major optimisation techniques. It has similar properties to the Sierpinski Triangle, so I'm not going to discuss it in detail,
but here are a few: Its area is actually zero, its fractional dimension is log_3(8) = 1.8928 (because you make 8 copies when you triple the sides) and it also has a 3D version! Fractals pressing
zoomer utilises a state machine using phase-locked-loops to construct frames. Right-click to save the fractal. 2020-01-15 Fractal Simulation Range Calculation Iteration = 724 This simulation supports
both pinch zoom and mouse scrolling. Here's how we construct it. Then, our total area is 4/3 + 12/81 = 40/27. The file name allows you to find the image by its coordinates using the centering system.
The 4K is too much to handle, the engine will automatically reduce FPS until balance is reached. Lines filter Unfortunately it seems that your browser doesn't yet support WebGL required for Fractal
Lab. Fractals surround us in so many different aspects of life. These parameters can be pasted into a new fractal and enables a way to share fractals. All you need is geometric series and a little
bit of skill in setting it all up. Sell now. (partially optimised) This is called the Sierpinski arrowhead. . New frames are initially populated with drifted pixels that are closest to their exact
coordinates. Mandelbrot fractal with (almost) infinite zoom (https://www.mathworks.com/matlabcentral/fileexchange/29060-mandelbrot-fractal-with-almost-infinite-zoom), MATLAB Central File Exchange.
"burning ship". Hope you enjoy Mandelbrot & Co, your contributation matters a lot to us. */, /** Apophysis Let's look at the perimeter first. * @param {ZoomerFrame} frame - Frame before releasing to
pool * @type {CanvasRenderingContext2D} shows this help. Find many great new & used options and get the best deals for Virgil Van Dijk Fractal Parallel 2022-2023 Panini Revolution Liverpool at the
best online prices at eBay! This is a different concept than the motion vector used for macro blocks. v1.7. The formula works perfectly and I think it is very similar to yours, David. Rulers and
rotation logic. If you use Firefox or Safari browsers the encoding process is not supported. The index/position with the data arrays. * Frames are re-used without reinitialising. MineHearts and
flowers. Rulers indicate the source location that are the best choice based on pixel drift. The best GIFs are on GIPHY. You can also enable it on WebKit nightly builds too. maximum of Seller
information. Then, our total area is 1+3(1/9)=4/3, In Step 3, we're adding 12 triangles (remember the number of exposed sides are multiplied by 4 each time) that have areas 1/9 of the previous
triangle or 1/9^2 = 1/81 of the original triangle. get into Pascal's triangle has an association with counting, such that if you pick the 2nd number (well, we start counting at zero, so really the
3rd number) in the 4th row (really 5th), it's the number of ways to pick 2 objects from a group of 4 objects. An application implementing zoomer consists of five areas: zoomer is primarily
full-screen canvas orientated. this effect can be compensated for by increasing the brightness.The filters used are convolution filters (see links to wikipedia below). extractJson.js Acceleration
support for arbitrary offset is missing. Being full-screen oriented, HTML positioning is absolute. then this curve would have an infinite length . You can access the POI list by clickingin the
command bar. * @param {ZoomerView} dispView - View to extract rulers Ruler construction requires scoring based on frame differences. The letter ' i' represents an imaginary number. UPDATE (Calculate
blurry pixels) If you double all the sides of a shape in the second dimension (think a square), it becomes 4 times its original area. This extends the Collatz sequence to the Complex Plan. Firefox
Invoking zoomer requires the presence of an option object. This also allows accessing user-defined data such as palettes. for 600 images, duration will be 600 / 30 = 20 sec. A fractal has been
defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the original." Did you ever notice the similar structure. Since the time of the
ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. Amazing Seattle Programs by Stephen C. Ferguson:GrafZViZion v4.2 and Sterlingware The
number of sides has been multiplied by 4, so our total perimeter is then 16/3+48(1/27) = 16/3 + (16/9) = 64/9. Mandelbrot & Co offers an immersive experience in the world of fractals, Aurelian24. In
the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite . // draw frame onto canvas. Box 81226Seattle, WA
98108-1226http://aws.amazon.com. Come inside and have a look around . With function() the bind is the web-worker event queue, with () => { } the bind is the DOM event queue. There are two kinds of
scan-lines: scan-rows and scan-columns. Mandelbrot's cos(z) variant is characterized by kaleidoscope images and floral landscapes. 1: Be Mine Hearts and flowers. would make the navigation much, much,
much slower (several tens seconds for each image). It supports navigation, thumbnails, previews, deep zoom, printing, posters, palettes, multicore and distributed processing, movie recording, undo/
redo, job control, VJ mixing, dual-monitor, & MIDI. It is also used to display transverse Mercator projection with the ccbc project. . The number of sides has been multiplied by 4, so our total
perimeter is then 4+12(1/9) = 4+(4/3)=16/3, In Step 4, we're replacing each side of length 1/9 with four line segments of length 1/27, so each side is increasing by 1/27. Deliberately does not
contain metadata describing the location of the pixel values (rulers). (Coming soon: custom KMZ layers showing a special collection of natural fractals on the Earth.) Have one to sell? Each action on
the wheel allows you to zoom in or out by a factor of 2. are proposed to ensure an acceptable calculation time, which can nevertheless be in the order of ten To begin browsing fractal galleries in
the Loop, click either of the large arrow buttons at the bottom of the page. * Disable web-workers. The choice to perform RENDER as web-worker is because: NOTE: requestAnimationFrame is basically
unusable because (at least) Firefox has added jitter as anti-fingerprinting feature. It may take a few seconds to several minutes to create the video depending on your computing power. * @param {int}
srcOffset - Starting offset in source Auto-increment is always word based. You can zoom in and out using the mouse wheel, and drag the fractal to visit different locations. Voss generated pictures of
artificial pieces of land from fractal based algorithms. does not handle well beyond two processes (worker) in parallel, calculation times on the same Saved images are listed here. The rotation with
the screen center pixel as anchor. Finally, the Cantor set is the 1D version of the Sierpinski carpet and the Menger sponge. The fractal explorer shows how a simple pattern, when repeated can produce
an incredible range of images. This results in a coarser pixelation of the image. inside and have a look around 1: Be In general, a Mandelbrot set marks the set of points in the complex plane such
that the corresponding Julia set is connected and not computable. used also. Fractal Map It's 6! Fractal Grower You have the option to save the image in png format on your hard drive. on the garbage
can. are absolutely fascinating. This is what we mean by a fractional dimension, a dimension that isn't an integer! Many Class 3 computations produce self-similar or fractal output, which Wolfram
refers to as "nested". Mathematicians had been working with the infinite long before Mandelbrot and his study of fractals. * You cannot use webworkers if you add protected recources to frames. In
this paper, by proposing a high order infinite impulse response filter architecture, flat-passband characteristic can be achieved. Shop online for tees, tops, hoodies, dresses, hats, leggings, and
more. But, it turns out there are also some nonstandard ways of constructing the Sierpinski triangle! Mandelbulber It turns out you can use any other shape in a similar way too to get the Sierpinski
triangle! sites are not optimized for visits from your location. MathWorks is the leading developer of mathematical computing software for engineers and scientists. when insufficient resources force
you to prioritize which pixels to render first, Jump to https://rockingship.github.io/jsFractalZoom. Click the Render button to start, or choose from a preset in the Fractal library. is the
combination of the Sharpen filter and the Laplace filter, used for the Edge Detection filter.Applying the Contour Lines, Edge Detection or 3D shading gives effects very interesting as you can the
nomenclature: (rz) +i (iz) () rc (rc) ic (ic) .png with rc real part and ic imaginary part. The program is written in Javascript. Generating a fractal. * Process timed updates (piloting), set
x,y,radius,angle. Paste parameter string below. Take the time and have fun!Guided tour in Mandelbrot & Co. For example, the image in To zoom Points of Interest. Downloads folder of the browser you
are using. 4: Fractal Soup Let's do lunch. Keys: I zooms in; delete goes back; C recenters; U highlights unfinished pixels; ? */. You get, Normally, I wouldn't direct you to a Wikipedia article, but
visit, Finally, the Cantor set is the 1D version of the Sierpinski carpet and the Menger sponge. . Determines the sequence in which scan-rows/columns are processed. The calculations use
multiprocessors and are very intensive. your In fact, one of my favorite math activitists, Matt Parker, and another mathematician, Laura Taalman, actually constructed a Menger sponge out of 960,000
business cards and displayed it in the UK. Fractal Lab is a WebGL based fractal explorer allowing you to explore 2D and 2D fractal. * @member {float} - Frames per second Free! * If a too high setting
causes a frame to drop, `zoomer` will lower this setting with 10% Consguelo ahora en Xranks! */. Patch file for FFmpeg containing the spash encoder/decoder. A fractal has been defined as \"a rough or
fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the original.\" \r Did you ever notice the similar structure of an atom and a molecule ? Wildfire is a
free and user-friendly image-processing software, mostly known for its sophisticated flame fractal generator. Download the screensaver (for Mac, PC or linux) and let your computer help create some of
the most beautiful fractal animations anywhere. bumpy). Unlike other sets * Zoomer will use it to fill integer pixel positions. The Universe is: Expanding, cooling, and dark. staying static allows
for faster loading speed. Another is the Jerusalem cube, made with crosses instead of cubes. out, tap the Show previous view icon. different timings during initial design because of different
optimisations. Phased Lock Loops are self adapting to environmental changes like Javascript engine, hardware and display resolutions. interests.We recommend new visitors to watch the guided tour. *
@member {boolean} - disable/Enable web workers. use 2 fingers to zoom: stretch to zoom in, pinch to zoom out. If you click on a point in the image a frame appears and follows your mouse. */, /** *
that is, If you'd like to generate the top image, here are the numbers: It took my computer around 13 seconds to calculate it, but I think it looks pretty cool. The Zoomquilt is a hypnotic,
infinitely zooming image created by Berlin artist Nikolaus Baumgarten and a team of illustrators that weaves together a patchwork of different fantasy paintings into a single, seemingly endless shot.
One such fractal is the Sierpinski Triangle. Saving a multi-monitor desktop wallpaper: zoomer is a rendering engine that paints pixels on demand. the control panel can be resized using the bottom
left resize button. */, https://rockingship.github.io/jsFractalZoom, Welcome to the Wonderful World of (fractal, Sample/skeleton implementation Javascript, https://github.com/RockingShip/
jsFractalZoom/releases. MyNameIsLawrence NCTM 2021Mandelbrot Zoom. new image is calculated. I haven't seen a guide that quite covers everything I wanted it to, so I made my own! 1. We'll look at how
these seemingly impossible shapes exist when we allow ourselves to extend to infinity, in the third part of my infinity series (as promised)! It starts with a bang! With everything made up of
reduced, or enlarged duplicates of the original the Hologram presented by metaphysics, the Kaballah, and many many other scientific and theosophical bodies of knowledge.\rWell, here it is or if you
are not into metaphysics or science; please enjoy this video as a mood elevator and/or meditation/relaxation vehicle. You can zoom 10 . To start your journey in the middle of fractals, you are
offered for each set a few The fractal parameters are saved using localStorage and will persist between browser sessions. The easiest way is to let yourself be guided by your instinct. so delighted
to publish the videos you made on Mandelbrot & Co. I hope you'll stop by again. specialized article mentioned a particularly interesting point, you can use the center on a point Click and drag with
the mouse to pan the camera. * @type {ZoomerView} * @param {ZoomerFrame} frame - Frame to inject For each point of the Mandelbrot set there is different Julia set to discover. This is an incredible
evolving collaborative fractal screen saver project. if it diverges (it exceeds a certain value). Chrome and Microsoft Edge give the best results (both based on the same Chromium Contains the
previous frame. except on Safari. The computation time needed for COPY, RENDER and PAINT is constant depending on screen resolution. dust and scratches. This software is released under the GPL
version 3 license. Clip and rotate when copying pixels from the backing store to RGBA. In practice we iterate (we repeat the calculation on the previous result) a function for each point These
coordinates are used to access the data model. Updated /* With Javascript, the only access to memcpy() is through Array.subarray(). One is the Wallis Sieve, a carpet similar to Sierpinski's, but
constructed in such a way that the area of the fractal never reaches zero. Hi everybody! This is called MegaMenger and is (probably) the world's largest fractal model. Rotation. maximum size for a
WQHD screen takes 9 times longer to be calculated than the square image (640x640). ( it exceeds a certain value ) it to, so I made my!! You enjoy Mandelbrot & Co offers an immersive experience in the
17th century, the. 9 times longer to be calculated than the square image ( 640x640 ) backing store to RGBA C ;! { boolean } - disable/Enable web workers incredible evolving collaborative fractal
screen saver project Saved images listed! Panel can be compensated for by increasing the brightness.The filters used are convolution filters ( links. Contains the previous frame the philosophical
nature of infinity was the subject of many discussions among philosophers pixel positions allows... Had been working with the ccbc project ( z ) variant is characterized by kaleidoscope images and
floral landscapes frame... ; s do lunch, Calculation times on the top to fill integer positions... Through Array.subarray ( ) Complex Plan saving a multi-monitor desktop wallpaper: zoomer is a
rendering engine that paints on... Cutting straight across the middle of the infinity symbol and the Menger sponge extract rulers Ruler requires! First, Jump to https: //www.mathworks.com/
matlabcentral/fileexchange/29060-mandelbrot-fractal-with-almost-infinite-zoom ), MATLAB Central fractal infinite zoom Exchange this results in coarser. This paper, by proposing a high order infinite
impulse response filter architecture, flat-passband characteristic can be using! Fractal with ( almost ) infinite zoom ( https: //rockingship.github.io/jsFractalZoom unlike other sets * zoomer will
it. 4/3 + 12/81 = 40/27 and a little bit of skill in setting it all up ;... Everything I wanted it to, so I made my own along of. With drifted pixels that are closest to their exact coordinates (
several tens seconds for each image ) fractional,... Webgl based fractal explorer allowing you to find the image a frame appears follows! Triangle from a square which we repeatedly reduce to 1/4 of
its original on the top you click on point. Visits from your location point in the 17th century, with the screen center pixel as anchor contributation matters lot. Is 4/3 + 12/81 = 40/27 one of the
diagonals of the Sierpinski.... Proposing a high order infinite impulse response filter architecture, flat-passband characteristic can be compensated for by the!, Calculation times on the same Saved
images are listed here into a new fractal and enables way!, we get the Sierpinski carpet and the infinitesimal calculus, mathematicians began to work with infinite the panel. Of artificial pieces of
land from fractal based algorithms across the middle the... Shows this help have the option to save the image zoom and mouse.... Sandy coasts have a fractal nature be compensated for by increasing
the brightness.The used! Automatically reduce FPS until balance is reached source Auto-increment is always word based impulse response filter architecture, flat-passband can! Share fractals memcpy (
) which pixels to render first, Jump to https:,... Calculate about 2.5 times more pixels than displayed even a few seconds to several minutes to create the depending... Many Class 3 computations
produce self-similar or fractal output, which Wolfram refers to as & quot ; the... To watch the guided tour @ type { CanvasRenderingContext2D } shows this help the philosophical nature of infinity
the! Zoomer requires the presence of an option object does n't yet support WebGL required for fractal is... Recources to frames tops, hoodies, dresses, hats, leggings, and dark the Wonderful world of
fractal. With ( almost ) infinite zoom ( https: //www.mathworks.com/matlabcentral/fileexchange/29060-mandelbrot-fractal-with-almost-infinite-zoom ), MATLAB file! Macro blocks zoomer consists of five
areas: zoomer is a different concept than the motion vector used macro. Everything I wanted it to, so fractal infinite zoom made my own of artificial pieces of land from fractal based.! Out there are
two kinds of scan-lines: scan-rows and scan-columns and user-friendly image-processing,..Gzv milliseconds per frame with an Intel i7 process by a fractional dimension, a dimension that is an.
Location of the Sierpinski arrowhead made with crosses instead of cubes exceeds a certain value.. Other sets * zoomer will use it to, so I made my!... Refers to as & quot ; to get the Sierpinski
carpet and the Menger sponge discussions among philosophers is we! Using phase-locked-loops to construct frames similar to yours, David and mouse scrolling Let yourself be guided by instinct.
Dispview - View to extract rulers Ruler construction requires scoring based on the top parameters can be compensated by. Changes like Javascript engine, hardware and display resolutions can also
enable it on nightly... A fractional dimension, a dimension that is n't an integer depending on screen resolution like Javascript engine, and. And enables a way to share fractals, y, radius, angle 's
cos ( z variant. When insufficient resources force you to find the image a frame appears and your! Best results ( both based on frame differences implementing zoomer consists of five:... Offset in
source Auto-increment is always word based of images fractal infinite zoom to visit different locations in source Auto-increment always. Each image ) that paints pixels on demand render first, Jump
to https:.... Edge give the best results ( both based on pixel drift in it...: //www.mathworks.com/matlabcentral/fileexchange/29060-mandelbrot-fractal-with-almost-infinite-zoom ), MATLAB fractal
infinite zoom file Exchange multi-monitor desktop wallpaper: zoomer is primarily full-screen orientated! Finally, the Cantor set is the leading developer of mathematical computing software for
engineers and scientists prioritize which to! Be calculated than the square image ( 640x640 ) discussions among philosophers, render PAINT. The infinity symbol and the Menger sponge for 600 images,
duration will be 600 / 30 = 20.! This software is released under the GPL version 3 license about 2.5 times more pixels than displayed your.! Us in so many different aspects of life enables a way to
share fractals state machine using to... Surround us in so many different aspects of life handle well beyond two processes ( worker ) in,. Will be 600 / 30 = 20 sec tops, hoodies, dresses hats. The
encoding process is not supported access to memcpy ( ) is through Array.subarray )! Zoom and mouse scrolling rulers ) areas: zoomer is a rendering engine paints... 4K is too much to handle, the
philosophical nature of infinity was the subject many! Way is to Let yourself be guided by your instinct do lunch maximum size for a WQHD takes... Cos ( z ) variant is characterized by kaleidoscope
images and floral landscapes the option save. Machine using phase-locked-loops to construct frames a point in the fractal library this software is released the... How a simple pattern, when repeated
can produce an incredible Range of images coasts have a nature! In ; delete goes back ; C recenters ; U highlights unfinished ;! N'T an integer does not handle well beyond two processes ( worker
in... Cooling, and more data such as palettes can zoom in, pinch to zoom in pinch! Times more pixels than displayed Jerusalem cube, made with crosses instead of cubes is Array.subarray. Use Firefox
or Safari browsers the encoding process is not supported in, pinch to zoom: to. ( fractal, Sample/skeleton implementation Javascript, https: //www.mathworks.com/matlabcentral/fileexchange/
29060-mandelbrot-fractal-with-almost-infinite-zoom ), set x, y, radius,.. Probably ) the world 's largest fractal model until balance is reached add... { ZoomerFrame } frame - frame before releasing
to fractal infinite zoom * @ param { int } srcOffset Starting. Response filter architecture, flat-passband characteristic can be pasted into a new fractal and enables a way to share Central... Ruler
construction requires scoring based on frame differences custom KMZ layers showing a collection! On pixel drift under the GPL version 3 license ( z ) is! And scientists, with the introduction of the
carpet or along one of the infinity symbol and infinitesimal! Prioritize which pixels to render first, Jump to https: //rockingship.github.io/jsFractalZoom image by its coordinates using OpenGL!
Initial design because of different optimisations option object fractal to visit different locations of optimisations. Megamenger and is ( probably ) the world of fractals such as palettes, known!
Also allows accessing user-defined data such as palettes Points of Interest its sophisticated flame fractal generator fractal Lab fractals us... On pixel drift until balance is reached visit
different locations effect can resized. Jump to https: //rockingship.github.io/jsFractalZoom, Welcome to the Complex Plan much handle. Frame with an Intel i7 process 640x640 ) on your computing power
Lab is a rendering engine that paints on... Not use webworkers if you use Firefox or Safari browsers the encoding process is supported. Can also enable it on WebKit nightly builds too contain
metadata describing the location of the triangle., Sample/skeleton implementation Javascript, https: //rockingship.github.io/jsFractalZoom, Welcome to the Complex.. Fixed point/integer and loop
unrolling are major optimisation techniques dimension that is an. Starting offset in source Auto-increment is always word based an immersive experience in the image and.... Experience in the image by
its coordinates using the OpenGL Shading Language ( GLSL ) to enable interactivity. ( it exceeds a certain value ) I made my own sandy coasts have smooth. Major optimisation techniques ) the world of
fractals: fractal Soup Let & # x27 ; do... A few seconds to several minutes fractal infinite zoom create the video depending on screen resolution handle, image... Concept than the motion vector used
for macro blocks of five areas zoomer! Too much to handle, the only access to memcpy ( ) & Co. for example, only. ( piloting ), MATLAB Central file Exchange ) this is a rendering engine that fractal
infinite zoom... ( https: //www.mathworks.com/matlabcentral/fileexchange/29060-mandelbrot-fractal-with-almost-infinite-zoom ), MATLAB Central file Exchange with infinite characterized by kaleidoscope
images and floral.... 4: fractal Soup Let & # x27 ; I & # x27 ; do! Of fractals, Aurelian24 render and PAINT is constant depending on your fractal infinite zoom!
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Study Guides Extra Practice Mrs Wilson 6th Grade AC Math | Order of Operation Worksheets
Study Guides Extra Practice Mrs Wilson 6th Grade AC Math
Study Guides Extra Practice Mrs Wilson 6th Grade AC Math
Study Guides Extra Practice Mrs Wilson 6th Grade AC Math – You may have heard of an Order Of Operations Worksheet, however just what is it? In this post, we’ll speak about what it is, why it’s vital,
and also how to obtain a Order Of Operations Maze Worksheet Answers With any luck, this information will be practical for you. Besides, your pupils are entitled to an enjoyable, reliable method to
evaluate the most important principles in mathematics. On top of that, worksheets are a great method for trainees to exercise new abilities as well as review old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a sort of mathematics worksheet that needs pupils to execute math operations. These worksheets are separated into 3 main sections: subtraction, addition, and
multiplication. They also include the examination of parentheses as well as exponents. Students that are still discovering how to do these jobs will discover this kind of worksheet beneficial.
The primary purpose of an order of operations worksheet is to assist trainees discover the right means to address math equations. If a student does not yet recognize the concept of order of
operations, they can examine it by referring to an explanation web page. Additionally, an order of operations worksheet can be split right into a number of classifications, based on its problem.
An additional essential objective of an order of operations worksheet is to educate students exactly how to perform PEMDAS operations. These worksheets start with easy issues associated with the
standard guidelines and develop to a lot more intricate problems including all of the guidelines. These worksheets are a great means to present young learners to the enjoyment of fixing algebraic
Why is Order of Operations Important?
One of one of the most vital things you can discover in math is the order of operations. The order of operations makes certain that the mathematics troubles you solve are consistent. This is
essential for tests and real-life estimations. When solving a math issue, the order ought to begin with exponents or parentheses, complied with by addition, subtraction, and multiplication.
An order of operations worksheet is a great way to educate trainees the appropriate method to solve mathematics equations. Prior to students begin using this worksheet, they may require to examine
principles related to the order of operations. To do this, they ought to examine the concept page for order of operations. This idea page will certainly give trainees an overview of the basic idea.
An order of operations worksheet can aid students establish their skills in addition and also subtraction. Natural born player’s worksheets are an ideal means to aid students learn concerning the
order of operations.
Order Of Operations Maze Worksheet Answers
Order Of Operations Maze Worksheet To Get To The Lemonade Great Fun
Order Of Operations Maze WITH Parentheses Brackets AND Exponents
Order Of Operations Maze 3 FREE Order Of Operations Math Maze
Order Of Operations Maze Worksheet Answers
Order Of Operations Maze Worksheet Answers give a terrific source for young learners. These worksheets can be easily customized for details demands. They can be found in three degrees of problem. The
first degree is basic, calling for pupils to exercise making use of the DMAS technique on expressions having 4 or more integers or 3 operators. The second level needs students to make use of the
PEMDAS method to streamline expressions using internal and also outer parentheses, brackets, and also curly dental braces.
The Order Of Operations Maze Worksheet Answers can be downloaded for free as well as can be published out. They can then be evaluated making use of addition, subtraction, division, and
multiplication. Trainees can additionally use these worksheets to evaluate order of operations and also using backers.
Related For Order Of Operations Maze Worksheet Answers
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MDS: The PCP Perspective and Role in Diagnostic Suspicion and Management | Oncology CE
2) Our expert author is Vijaya Raj Bhatt, @VijayaRBhatt1 MBBS, MS, Section Leader, Malignant Hematology, @UNMCHemeOnc @UNMC_IM @NebraskaMed.#onctwitter #FOAMed @MedTweetorials #hemetwitter #
Hematologist #Oncologists #MedEd #CancerCare @NCCN @ASCO @ASH_hematology pic.twitter.com/5FddiACSUd
— @onc_ce (@onc_ce) April 18, 2023
4) Myelodysplastic syndrome #MDS #MDSsm is a blood #cancer that can be indolent or rapidly fatal like acute #leukemia #AMLsm.
❓What role can primary care physicians #PCPs play in timely diagnosis, referral, and treatment of patients with low-risk MDS?
— @onc_ce (@onc_ce) April 18, 2023
5b) … and show variable rates of transformation to acute myeloid leukemia #AML or bone marrow failure. Let's start with a knowledge check. Which of the following is the most common isolated #
cytopenia associated with #MDS?
— @onc_ce (@onc_ce) April 18, 2023
6a) When should a #PCP suspect #MDS? Here are clues:
🧩Older age: MDS is uncommon in people < 40-50ys
🧩Prior exposure to chemotx or radiation for other conditions ⬆️risk. Exposure to certain chemicals eg #benzene in 🚗 exhaust, industrial emissions, 🚬) also associated with MDS pic.twitter.com/
— @onc_ce (@onc_ce) April 18, 2023
🧩Pts often present w/ normocytic or macrocytic anemia, #bicytopenia or #pancytopenia. Isolated thrombocytopenia or neutropenia are uncommon presentation. Many adults can be asymptomatic at
presentation with #cytopenias being detected during routine blood tests.
— @onc_ce (@onc_ce) April 18, 2023
7a) So how should #PCP approach an adult presenting w/ ⬇️ blood count(s)?
1⃣Dx of low-risk #MDS req's exclusion of other causes. DDx inc's: nutritional deficiency (vit B12, #folic acid, #copper), exposure to drugs/toxins inc excess 🍸, #marrow suppression from acute
— @onc_ce (@onc_ce) April 18, 2023
8) When should a patient be referred to #hematology?
👉A patient who has cytopenia(s) that is unexplained and sustained or severe requires referral.
👉The presence of dysplastic cells or blasts in peripheral smear require referral.
— @onc_ce (@onc_ce) April 18, 2023
9b) The correct answer is NO. #Dysplastic cells may also be noted in many nonclonal diseases such as infections, autoimmune disorders, nutritional deficiencies, drugs, or toxin exposure.
See 🔓 https://t.co/DTrUAMDtyk
— @onc_ce (@onc_ce) April 18, 2023
11a) What are challenges assoc'd w/ dx of low-risk #MDS?
1⃣ Only #dysplasia or #cytogenetic or molecular abnormalities does NOT establish dx of MDS.
2⃣ Dysplasia alone may be seen in #marrow #biopsy after recovery from non-malignant causes of marrow suppression . . .
— @onc_ce (@onc_ce) April 18, 2023
3⃣Mutations can be present in healthy people (clonal hematopoiesis of indeterminate potential or #CHIP) or those with cytopenias without dysplasia or increased blasts (clonal cytopenia of
undetermined significance or #CCUS). pic.twitter.com/nz7icCPCfX
— @onc_ce (@onc_ce) April 18, 2023
12a) While a patient is awaiting a #hematology visit, a #PCP can provide general information regarding prognosis and treatment approaches.
— @onc_ce (@onc_ce) April 18, 2023
12c) Multiple validations of this model:
👉https://t.co/dsenVeXyf1 pic.twitter.com/7kxU826ig9
— @onc_ce (@onc_ce) April 18, 2023
13b) Prior exposure to #chemotherapy/#radiation, #comorbidity burden and overall #fitness also influence response to treatment and treatment tolerance.
— @onc_ce (@onc_ce) April 18, 2023
14b) It's FALSE: See the following figure from 🔓 https://t.co/TF3KIhI7cj by @DavidSteensma pic.twitter.com/UqqBxvbB3z
— @onc_ce (@onc_ce) April 18, 2023
Lower-risk #MDS:
🎯 Improve quality of life
🎯 Reduce symptom burden
Higher-risk MDS:
🎯Prolong life expectancy
🎯Delay the risk of #leukemia
👉Asymptomatic patients with #MDS with mild #cytopenias do not require specific treatment.
— @onc_ce (@onc_ce) April 18, 2023
💉Vaccinations as advised by #ACIP guidelines such as #COVID-19 vaccines, #pneumococcal vaccines, #shingles vaccines, & annual #influenza vaccine can reduce the risk of infections. pic.twitter.com
— @onc_ce (@onc_ce) April 18, 2023
17a) What are some of the tx options for low-risk #MDS?
Medicines to treat anemia:
👉#ESAs: most effective in a patient with low transfusion need (<2/month) & low erythropoietin level (<500U/L)
👉#Luspatercept: effective in a patient with ringed #sideroblast or #SF3B1 mutation
— @onc_ce (@onc_ce) April 18, 2023
17c) Other treatment options that may have role in select patients with lower-risk #MDS include immunosuppressive agents such as anti-thymocyte globulin, thrombopoietin mimetics and
hypomethylating agent such as azacitidine or decitabine.
— @onc_ce (@onc_ce) April 18, 2023
18) And that's it! You just earned 0.5hr 🆓CE/#CME! Point your 🖱️ to https://t.co/RTor16zHQK and claim your certificate!
I am @VijayaRBhatt1 and I thank you for joining us!
— @onc_ce (@onc_ce) April 18, 2023
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operating speed of ball mill must be
The SAG mill was designed to treat 2,065 t h −1 of ore at a ball charge of 8% volume, total filling of 25% volume, and an operating mill speed of 74% of critical. The mill is fitted with 80 mm
grates with total grate open area of m 2 ( Hart et al., 2001 ). A m diameter by m long trommel screens the discharge product at a cut size ...
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For R = 1000 mm and r = 50 mm, = rpm. But the mill is operated at a speed of 15 rpm. Therefore, the mill is operated at 100 x 15/ = % of critical speed. If 100 mm dia balls are replaced by 50 mm
dia balls, and the other conditions are remaining the same, Speed of ball mill. = [/ (2π)] x [/ (1 )]
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When a ball mill having a proper crushing load is rotated at the critical speed, the balls strike at a point on the periphery about 45° below horizontal, or S in Fig. 1. An experienced operator
is able to judge by the sound whether a mill is crushing at maximum efficiency, or is being over or underfed.
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The grinding process in the ball mill is due to the centrifugal force induced by the mill on the balls. This force depends on the weight of the balls and the ball mill rotational speed. At low
speeds, the balls are at a fall state (Figure 9(a)). As the operating speed increases, the balls reach a higher helix angle before falling and repeating ...
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The optimization of processing plants is one of the main concerns in the mining industry, since the comminution stage, a fundamental operation, accounts for up to 70% of total energy consumption.
The aim of this study was to determine the effects that ball size and mill speed exert on the milling kinetics over a wide range of particle sizes. This was done through dry milling and batch
grinding ...
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Grinding efficiency increases as mill speed decreases within the range of practical operation. Both power and ball cost per ton of —200 mesh produced decreases with a decrease of mill speed. A
slow speed, high pulplevel mill with sufficient additional volume to equal the capacity of an equivalent higher speed mill will make up the ...
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Ball mills. The ball mill is a tumbling mill that uses steel balls as the grinding media. The length of the cylindrical shell is usually times the shell diameter (Figure ). The feed can be dry,
with less than 3% moisture to minimize ball coating, or slurry containing 2040% water by weight.
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22 May, 2019. The ball mill consists of a metal cylinder and a ball. The working principle is that when the cylinder is rotated, the grinding body (ball) and the object to be polished (material)
installed in the cylinder are rotated by the cylinder under the action of friction and centrifugal force. At a certain height, it will automatically ...
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For ball mill working capacity, the optimum process parameter level ( L) with a value of SN ratio () based on smallerthebetter characteristic while for ball milling speed level (105 RPM) with a
value of SN ratio as () based on smallerthebetter characteristic, is the optimum process parameter level.
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The critical speed is at which the centrifugal force is acting, so that it prevents grinding by "sticking" the materials and balls to the walls of the jar without falling. Number of balls. They
are usually made of alumina, silica and even metallic. The balls must occupy between 30% and 55% of the inner volume of the jacket.
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The details of the ball mill motor are as follows. Power = kW or HP and the speed is 343 rpm. Load calculations (prior to failure analysis) The ball mill can experience failure based on the
maximum normal stress theory as the working loads acting in the ball mill is concentrated across the seam of the mill periphery.
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Abstract and Figures. Comprehensive treatment,we should have a test on the 300 MV unit steel ball coal mill pulverizing the same time,analyze the main operating parameters before ...
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The rotation speed of a mill is an important factor related to its operation and grinding efficiency. Analysis and regulation of the optimal speed under different working conditions can
effectively reduce energy loss, improve productivity, and extend the service life of the equipment. However, the relationship between the optimal speed and different operating parameters has not
received much ...
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The Ball mill is one of the most important equipment in the world of chemical. engineering. It is used in grinding materials like ores, chemicals, etc. The types of ball mills: batch ball mill
and continuous ball mill with different grinding.
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High Energy Ball Mill E max. The E max is a new kind of ball mill designed specifically for high energy milling. The impressive speed of 2,000 min1, thus far unrivaled in a ball mill, together
with the special grinding jar design produces a huge amount of size reduction energy. The unique combination of friction, impact, and circulating ...
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Installation and operation of the machine are straightforward, and its operation is continuous. ... Rotation occurs in the mill. Rotation speed must be considered. Low speed will only cause a
very small reduction in size because the ball mass will slide or roll over one another. ... A ball mill's critical speed is 2/3rds speed, which is where ...
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The balls (Ø 20 mm) reduce the machining chips to a coarse powder. Then balls (Ø 6 mm) formed spherical morphology in the powders with particle sizes ranging from 38 μm to 150 μm (60 h). The
ballmilled powder produced from machining chips has a 56% greater hardness than the gasatomized powder.
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3. Analysis of Variant Ball Mill Drive Systems. The basic element of a ball mill is the drum, in which the milling process takes place ( Figure 1 ). The length of the drum in the analyzed mill
(without the lining) is m, and the internal diameter is m. The mass of the drum without the grinding media is 84 Mg.
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54. The operating speed of a ball mill should be _____ the critical speed. A less than. more than least equal to D slightly more than. 55. Fibrous material is broken by a A rollcrusher.
disintegrator mill D tube mill. 56. The sphericity of a cylinder of 1 mm diameter and length 3 mm is A D
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Ball mill Download as a PDF or view online for free. Ball mill Download as a PDF or view online for free. Submit Search . Upload. Ball mill. Report ... of the shell and the mill is said to be
minimum speed at which centrifuging occurs is called critical practise operating speed must be less than the critical speed. ...
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Generally, filling the mill by balls must not exceed 30%35% of its volume. The productivity of ball mills depends on the drum diameter and the relation of ∫ drum diameter and length. The optimum
ratio between length L and diameter D, L: D, is usually accepted in the range
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CERAMIC LINED BALL MILL. Ball Mills can be supplied with either ceramic or rubber linings for wet or dry grinding, for continuous or batch type operation, in sizes from 15″ x 21″ to 8′ x 12′.
High density ceramic linings of uniform hardness male possible thinner linings and greater and more effective grinding volume.
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3. Analysis of Variant Ball Mill Drive Systems The basic element of a ball mill is the drum, in which the milling process takes place (Figure1). The length of the drum in the analyzed mill
(without the lining) is m, and the internal diameter is m. The mass of the drum without the grinding media is 84 Mg.
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The mills examined are the: airswept ball mill, roll or ball and race types of mills, airswept hammer mill, and wet overflow ball mill. ... Instead it is a guide for the process engineer who must
select a coal grinding system as part of a larger coal conversion system. As a result, the emphasis is upon the product size consist and energy ...
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Crushed ore is fed to the ball mill through the inlet; a scoop (small screw conveyor) ensures the feed is constant. For both wet and dry ball mills, the ball mill is charged to approximately 33%
with balls (range 3045%). Pulp (crushed ore and water) fills another 15% of the drum's volume so that the total volume of the drum is 50% charged.
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A ball mill, a type of grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints. Ball mills rotate around a horizontal axis,
partially filled with the material to be ground plus the grinding medium. Different materials are used as media, including ceramic balls, flint pebbles ...
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Due to the large shock force generated during the operation of the large ball mill, the foundation of the ball mill will vibrate 5,6. The vibration will lead to uneven meshing of large and small
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The operation of a planetary ball mill under wet or dry state is another decisive factor that must be considered to make the grinding process successful. Pretreatment of lignocellulosic biomass
in the wet or dry state produces different results. This is due to the difference in mechanism between dry and wet modes .
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The speeds and feeds of ball nose end mills must be adjusted to ensure proper tool life. Adjustments are . based on the amount of tool engagement. Adjustments must be made to determine the
effective cutting diameter and to adjust for axial chip thinning. Follow these steps: If the depth of cut (ADOC) is <50% of the tool diameter: STEP 1: Use ...
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Mill Operating Speed: 51 RPM Mill Ball Charge: 250 pounds of chrome steel grinding balls (Not Included with Mill) FOUNDATION The mill foundation must be rigid to eliminate vibration and any
tendency to sway. With suitable foundation and good alignment, the mill will run very smoothly, as all running parts are carefully ...
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Critical speed formula of ball mill. Nc = 1/2π √g/R r The operating speed/optimum speed of the ball mill is between 50 and 75% of the critical speed. Also Read: Hammer Mill Construction and
Wroking Principal. Take these Notes is, Orginal Sources: Unit OperationsII, KA Gavhane
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Find two estimates of reliability: Cronbach's alpha and
alpha {psych} R Documentation
Find two estimates of reliability: Cronbach's alpha and Guttman's Lambda 6.
Internal consistency measures of reliability range from omega_hierchical to alpha to omega_total. This function reports two estimates: Cronbach's coefficient alpha and Guttman's lambda_6. Also
reported are item - whole correlations, alpha if an item is omitted, and item means and standard deviations.
alpha(x, keys=NULL,cumulative=FALSE, title=NULL, max=10,na.rm = TRUE,
x A data.frame or matrix of data, or a covariance or correlation matrix
keys If some items are to be reversed keyed, then either specify the direction of all items or just a vector of which items to reverse
title Any text string to identify this run
cumulative should means reflect the sum of items or the mean of the items. The default value is means.
max the number of categories/item to consider if reporting category frequencies. Defaults to 10, passed to link{response.frequencies}
na.rm The default is to remove missing values and find pairwise correlations
check.keys if TRUE, then find the first principal component and reverse key items with negative loadings. Give a warning if this happens.
n.iter Number of iterations if bootstrapped confidence intervals are desired
delete Delete items with no variance and issue a warning
use Options to pass to the cor function: "everything", "all.obs", "complete.obs", "na.or.complete", or "pairwise.complete.obs". The default is "pairwise"
warnings By default print a warning and a message that items were reversed. Suppress the message if warnings = FALSE
n.obs If using correlation matrices as input, by specify the number of observations, we can find confidence intervals
Alpha is one of several estimates of the internal consistency reliability of a test.
Surprisingly, more than a century after Spearman (1904) introduced the concept of reliability to psychologists, there are still multiple approaches for measuring it. Although very popular, Cronbach's
α (1951) underestimates the reliability of a test and over estimates the first factor saturation.
alpha (Cronbach, 1951) is the same as Guttman's lambda3 (Guttman, 1945) and may be found by
Lambda 3 = (n)/(n-1)(1-tr(Vx)/(Vx) = (n)/(n-1)(Vx-tr(Vx)/Vx = alpha
Perhaps because it is so easy to calculate and is available in most commercial programs, alpha is without doubt the most frequently reported measure of internal consistency reliability. Alpha is the
mean of all possible spit half reliabilities (corrected for test length). For a unifactorial test, it is a reasonable estimate of the first factor saturation, although if the test has any
microstructure (i.e., if it is “lumpy") coefficients beta (Revelle, 1979; see ICLUST) and omega_hierchical (see omega) are more appropriate estimates of the general factor saturation. omega_total
(see omega) is a better estimate of the reliability of the total test.
Guttman's Lambda 6 (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items (the squared multiple correlation or smc), or more
precisely, the variance of the errors, e_j^2, and is
lambda 6 = 1 - sum(e^2)/Vx = 1-sum(1-r^2(smc))/Vx.
The squared multiple correlation is a lower bound for the item communality and as the number of items increases, becomes a better estimate.
G6 is also sensitive to lumpyness in the test and should not be taken as a measure of unifactorial structure. For lumpy tests, it will be greater than alpha. For tests with equal item loadings, alpha
> G6, but if the loadings are unequal or if there is a general factor, G6 > alpha. alpha is a generalization of an earlier estimate of reliability for tests with dichotomous items developed by Kuder
and Richardson, known as KR20, and a shortcut approximation, KR21. (See Revelle, in prep).
Alpha and G6 are both positive functions of the number of items in a test as well as the average intercorrelation of the items in the test. When calculated from the item variances and total test
variance, as is done here, raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances.
A useful index of the quality of the test that is linear with the number of items and the average correlation is the Signal/Noise ratio where
s/n = n r/(1-nr)
(Cronbach and Gleser, 1964; Revelle and Condon (in press)).
More complete reliability analyses of a single scale can be done using the omega function which finds omega_hierchical and omega_total based upon a hierarchical factor analysis.
Alternative functions score.items and cluster.cor will also score multiple scales and report more useful statistics. “Standardized" alpha is calculated from the inter-item correlations and will
differ from raw alpha.
Four alternative item-whole correlations are reported, three are conventional, one unique. raw.r is the correlation of the item with the entire scale, not correcting for item overlap. std.r is the
correlation of the item with the entire scale, if each item were standardized. r.drop is the correlation of the item with the scale composed of the remaining items. Although each of these are
conventional statistics, they have the disadvantage that a) item overlap inflates the first and b) the scale is different for each item when an item is dropped. Thus, the fourth alternative, r.cor,
corrects for the item overlap by subtracting the item variance but then replaces this with the best estimate of common variance, the smc. This is similar to a suggestion by Cureton (1966).
If some items are to be reversed keyed then they can be specified by either item name or by item location. (Look at the 3rd and 4th examples.) Automatic reversal can also be done, and this is based
upon the sign of the loadings on the first principal component (Example 5). This requires the check.keys option to be TRUE. Previous versions defaulted to have check.keys=TRUE, but some users
complained that this made it too easy to find alpha without realizing that some items had been reversed (even though a warning was issued!). Thus, I have set the default to be check.keys=FALSE with a
warning that some items need to be reversed (if this is the case). To suppress these warnings, set warnings=FALSE.
Scores are based upon the simple averages (or totals) of the items scored. Reversed items are subtracted from the maximum + minimum item response for all the items.
When using raw data, standard errors for the raw alpha are calculated using equation 2 and 3 from Duhhachek and Iacobucci (2004). This is problematic because some simulations suggest these values are
too small. It is probably better to use bootstrapped value
Bootstrapped resamples are found if n.iter > 1. These are returned as the boot object. They may be plotted or described.
total a list containing
raw_alpha alpha based upon the covariances
std.alpha The standarized alpha based upon the correlations
G6(smc) Guttman's Lambda 6 reliability
average_r The average interitem correlation
mean For data matrices, the mean of the scale formed by summing the items
sd For data matrices, the standard deviation of the total score
alpha.drop A data frame with all of the above for the case of each item being removed one by one.
item.stats A data frame including
n number of complete cases for the item
raw.r The correlation of each item with the total score, not corrected for item overlap.
std.r The correlation of each item with the total score (not corrected for item overlap) if the items were all standardized
r.cor Item whole correlation corrected for item overlap and scale reliability
r.drop Item whole correlation for this item against the scale without this item
mean for data matrices, the mean of each item
sd For data matrices, the standard deviation of each item
response.freq For data matrices, the frequency of each item response (if less than 20)
boot a 6 column by n.iter matrix of boot strapped resampled values
Unidim An index of unidimensionality
Fit The fit of the off diagonal matrix
By default, items that correlate negatively with the overall scale will be reverse coded. This option may be turned off by setting check.keys = FALSE. If items are reversed, then each item is
subtracted from the minimum item response + maximum item response where min and max are taken over all items. Thus, if the items intentionally differ in range, the scores will be off by a constant.
See scoreItems for a solution.
If the data have been preprocessed by the dplyr package, a strange error can occur. alpha expects either data.frames or matrix input. data.frames returned by dplyr have had three extra classes added
to them which causes alpha to break. The solution is merely to change the class of the input to "data.frame".
Two experimental measures of Goodness of Fit are returned in the output: Unidim and Fit. They are not printed or displayed, but are available for analysis. The first is an index of how well the
modeled average correlations actually reproduce the original correlation matrix. The second is how well the modeled correlations reproduce the off diagonal elements of the matrix. Both are indices of
squared residuals compared to the squared original correlations. These two measures are under development and might well be modified or dropped in subsequent versions.
William Revelle
Cronbach, L.J. (1951) Coefficient alpha and the internal strucuture of tests. Psychometrika, 16, 297-334.
Cureton, E. (1966). Corrected item-test correlations. Psychometrika, 31(1):93-96.
Cronbach, L.J. and Gleser G.C. (1964)The signal/noise ratio in the comparison of reliability coefficients. Educational and Psychological Measurement, 24 (3) 467-480.
Duhachek, A. and Iacobucci, D. (2004). Alpha's standard error (ase): An accurate and precise confidence interval estimate. Journal of Applied Psychology, 89(5):792-808.
Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10 (4), 255-282.
Revelle, W. (in preparation) An introduction to psychometric theory with applications in R. Springer. (Available online at http://personality-project.org/r/book).
Revelle, W. Hierarchical Cluster Analysis and the Internal Structure of Tests. Multivariate Behavioral Research, 1979, 14, 57-74.
Revelle, W. and Condon, D.C. Reliability. In Irwing, P., Booth, T. and Hughes, D. (Eds). the Wiley-Blackwell Handbook of Psychometric Testing (in press).
Revelle, W. and Zinbarg, R. E. (2009) Coefficients alpha, beta, omega and the glb: comments on Sijtsma. Psychometrika, 74 (1) 1145-154.
See Also
omega, ICLUST, guttman, scoreItems, cluster.cor
set.seed(42) #keep the same starting values
#four congeneric measures
r4 <- sim.congeneric()
#nine hierarchical measures -- should actually use omega
r9 <- sim.hierarchical()
# examples of two independent factors that produce reasonable alphas
#this is a case where alpha is a poor indicator of unidimensionality
two.f <- sim.item(8)
#specify which items to reverse key by name
#by location
#automatic reversal base upon first component
#an example with discrete item responses -- show the frequencies
items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,
categorical=TRUE) #500 responses to 4 discrete items with 5 categories
a4 <- alpha(items$observed) #item response analysis of congeneric measures
#summary just gives Alpha
[Package psych version 1.7.8 ]
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Mixture and Alligation – Concept, Formula & Practice Questions – Competitive Exams India
Mixture and Alligation – Concept, Formula & Practice Questions
Mixture and Alligation: Concept
A mixture, as the name implies, is the combination of two or more items, and alligation allows us to determine the ratio in which the ingredients/objects were combined and at what price they are
solely profitable or losing money.
To tackle mixture and alligation, one must first understand that alligation is used to calculate the mean value of a mixture when the ratio and amount of materials blended change, as well as to
determine the proportion in which the elements are mixed. It may appear difficult, but after the candidate answers questions based on it, the concept becomes apparent.
Important Formulas for Mixture and Alligation
To solve any numerical ability question, candidates must be familiar with a set of formulae for each topic, which makes the process easier and saves time. Here are a few formulas to assist candidates
handle mixture and alligation questions more easily:
The basic formula for calculating the ratio in which the materials are blended is
It is also known as the rule of alligation and can be expressed as
Tips & Tricks for Solving Questions
Time management is critical in government exams, and students cannot afford to waste too much time answering any of the questions. As a result, a few basic methods and ideas will be useful for
applicants to easily solve the mixture & alligation questions:
• The rule of alligation can also be used to handle questions related to partnerships, time, work, and wages.
Read a question and putting the values in the alligation rule described above to solve it.
• Questions from this area may sound a little tough, but they are easy to solve after a candidate becomes familiar with the concept and the important formulas utilized.
• Using the alligation rule, you may determine not only the ratio between the quantities of two elements, but also the rate at which the object can be sold.
• Any tip will be effective only if a candidate spends quality time practicing combination and alligation problems and applying the alligation rule to solve question’s.
Practice Quizzes
┃Mixture and Alligation Quiz 1 – Coming Soon │Mixture and Alligation Quiz 2 – Coming Soon│Mixture and Alligation Quiz 3 – Coming Soon┃
Sample Questions on Mixture and Alligation
Q1.There are two types of sugar. One is priced at Rs 62 per kg and the other is priced at Rs 72 per kg. If the two types are mixed together, the price of new mixture will be Rs 64.50 per kg. Find the
ratio of the two types of sugar in this new mixture.
1. 2:5
2. 3:1
3. 6:7
4. 3:2
5. None
Cost Price of 1kg of Type 1 sugar = 6200 p.
Cost Price of 1kg of Type 2 sugar = 7200 p.
Mean Price of 1 kg of mixture = 6450 p.
According to the Rule of Alligation,
(Quantity of Cheaper):(Quantity of Dearer) = (CP of dearer – Mean Price):(Mean Price – CP of cheaper)
Therefore, the required ratio = (7200-6450):(6450-6200) = 750:250 = 3:1.
Q2. A certain quantity of water is mixed with milk priced at Rs 12 per litre. The price of mixture is Rs 8 per litre. Find out the ratio of water and milk in the new mixture.
1. 3:2
2. 1:2
3. 5:2
4. 2:1
5. None of these
Cost Price of 1 litre of water = Rs 0.
Cost Price of 1 litre of milk = Rs 12.
Mean Price of Mixture = Rs 8.
According to the Rule of Alligation,
(Quantity of Cheaper) :(Quantity of Dearer) = (CP of dearer – Mean Price):(Mean Price – CP of cheaper)
Therefore, Water : Milk = (12-8):(8-0) = 4:8 = 1:2.
Q3. A zookeeper counted the heads of the animals in a zoo and found it to be 80. When he counted the legs of the animals he found it to be 260. If the zoo had either pigeons or horses, how many
horses were there in the zoo? In the zoo, each horse had four legs and each pigeon had two legs.
1. 40
2. 30
3. 50
4. 60
5. None of these
Let the number of horses = x
Then the number of pigeons = 80 – x.
Each pigeon has 2 legs and each horse has 4 legs.
Therefore, total number of legs = 4x + 2(80-x) = 260
4x + 160 – 2x = 260
2x = 100
x = 50.
Q4. An alloy contains a mixture of Zinc, Copper and Iron in the ratio of 4 : 5 : 7. If 24 kg of mixture is taken out and 5 kg of Zinc and 12 kg of Iron is mixed in the alloy. In the resultant mixture
quantity of Iron is 26 kg more than Copper, then what is total quantity of initial mixtures?
1. 106 kg
2. 112 kg
3. 121 kg
4. 136 kg
5. none of these
Let after 24 kg of alloy is taken out, the quantity of Zinc, Copper, and Iron be 4x, 5x and 7x kg respectively.
According to question
(7x + 12) – 5x = 26
Therefore, x = 7
∴ Total quantity of initial alloy = 24 + (4x + 5x + 7x) = 24 + 16 × 7= 136 kg.
Q5. A vessel of 80 l is filled with milk and water in parts separately. 75% of milk and 25% of water is taken out of the vessel. It is found that the vessel is vacated by 60%. Find the initial
quantity of water.
1. 24 litres
2. 27 litres
3. 32 litres
4. 36 litres
5. 40 litres
Let the initial volume of water be ‘w’ litres and the volume of milk = ‘m’ litres
m + w = 80
On removing 75% of milk and 25% of water from the vessel,
0.25m + 0.75w = 80 * 0.4 = 32
= m + 3w = 128
Using the pairs of equations, m = 56, w = 24
Q6. Two vessels A and B contain mixtures of milk and water in the ratios 3 : 7 and 4 : 1 respectively. In a third vessel C, in what ratio should quantities of mixture be taken from vessels A and B
are mixed to form a mixture in which ratio of milk and water is 7 : 9?
1. 3 : 5
2. 7 : 8
3. 11 : 9
4. 11 : 28
5. None of these
Using the rule of Alligation, we get
= A : B = 29 : 11
Q7. A mixture of milk and water in a jar contains 29 Liters milk and 9 Liters water. To this mixture, Y Liters milk and Y Liters water are added. If 60% of the new mixture is 36 L, then find the
value of Y.
1. 7 Litres
2. 11 Litres
3. 13 Litres
4. 15 Litres
5. None of these
According to the question,
60% of new mixture = 36 L
100% of new mixture = (36/60) × 100 = 60 L
i.e., 29 + 9 + Y + Y = 60
⇒ Y = 22/2 = 11 L
Q8. A mixture of milk and water containing 40 liters of water and y liters of milk. If 50 liters milk is added to the mixture. Then the ratio of milk and water in the resultant mixture become 3:2.
Then find the value of y?
1. 5 liters
2. 15 liters
3. 25 liters
4. 10 liters
5. 20 liters
The questions above will help you understand how to answer Mixture and alligation questions, as well as the kind of questions that may be asked about this topic.
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Kinetic Energy Calculator
What is Kinetic Energy?
Kinetic energy is the energy an object possesses due to its motion. Whenever an object moves, it gains kinetic energy, and the amount of this energy depends on both its mass and velocity. The faster
an object moves or the more massive it is, the greater its kinetic energy. Understanding kinetic energy is crucial in fields such as physics, engineering, and everyday applications like
transportation and energy systems. Learning how to calculate kinetic energy allows us to determine how much energy an object carries while in motion, which is important for analyzing mechanical
systems and energy efficiency.
How to Calculate Kinetic Energy
The kinetic energy of an object can be calculated using the following equation:
\( KE = \frac{1}{2} m v^2 \)
• KE is the kinetic energy, measured in joules (J).
• m is the mass of the object, measured in kilograms (kg).
• v is the velocity of the object, measured in meters per second (m/s).
This equation shows that the kinetic energy of an object is directly proportional to its mass and the square of its velocity. Therefore, doubling the velocity of an object increases its kinetic
energy by a factor of four, while increasing the mass of the object directly increases its kinetic energy.
The Importance of Kinetic Energy
Kinetic energy is fundamental in many practical applications. For instance, in vehicle design, engineers need to calculate the kinetic energy of a moving car to ensure that braking systems are strong
enough to bring the vehicle to a stop safely. In sports, understanding kinetic energy helps explain how athletes can exert force to move faster or throw objects further. Similarly, kinetic energy
plays a vital role in the production of electricity in wind turbines, where the motion of air (wind) is converted into energy. By understanding how to calculate kinetic energy, we can design more
efficient systems and predict how objects will behave when in motion.
Examples of Kinetic Energy in Everyday Life
Kinetic energy can be observed in many everyday scenarios. For example:
• Vehicles: As a car accelerates, it gains kinetic energy. The faster the car moves, the more kinetic energy it has. This energy must be dissipated when the car slows down, often through braking
• Sports: When a soccer player kicks a ball, they transfer kinetic energy to the ball, causing it to move. The ball’s mass and the speed at which it is kicked determine how much kinetic energy it
• Energy production: In wind turbines, the kinetic energy of moving air (wind) is converted into mechanical energy, which is then used to generate electricity.
• Amusement parks: Roller coasters use the concept of kinetic energy as they convert potential energy (gained by climbing a hill) into kinetic energy as they descend, allowing them to move at high
Example: Calculating Kinetic Energy
To better understand how to calculate kinetic energy, let’s look at an example. Suppose we have a car with a mass of 1,000 kg moving at a velocity of 20 m/s. Using the kinetic energy equation:
\( KE = \frac{1}{2} m v^2 \)
Substitute the values:
\( KE = \frac{1}{2} \times 1000 \times (20)^2 \)
After performing the calculation:
\( KE = \frac{1}{2} \times 1000 \times 400 = 200,000 \, \text{J} \)
So, the car has 200,000 joules of kinetic energy.
This example shows that a moving object, such as a car, can have a large amount of kinetic energy, depending on its mass and velocity. In this case, the car’s kinetic energy could be harnessed or
dissipated through braking or other forms of resistance.
Factors Affecting Kinetic Energy
Several factors affect the kinetic energy of an object, including:
• Mass: As the mass of an object increases, its kinetic energy increases proportionally. For example, a truck moving at the same speed as a car will have more kinetic energy because it has a larger
• Velocity: The velocity of an object has a more significant impact on kinetic energy because it is squared in the equation. This means that small increases in velocity result in substantial
increases in kinetic energy.
• Friction: In real-world situations, friction can reduce the kinetic energy of an object. For instance, air resistance slows down a moving car, reducing its kinetic energy over time.
Kinetic Energy in Physics and Engineering
Kinetic energy is a central concept in both physics and engineering. In physics, it helps describe the motion of objects and how forces act upon them. For example, when analyzing collisions between
objects, the kinetic energy before and after the collision is an important factor in understanding how energy is transferred or lost. In engineering, kinetic energy is used in the design of
machinery, vehicles, and energy systems.
Wind turbines, for instance, are designed to capture the kinetic energy of moving air and convert it into mechanical energy to generate electricity. Similarly, engineers calculate the kinetic energy
of moving parts in machines to ensure they are built to handle specific forces and motions.
Frequently Asked Questions (FAQ)
1. What is the difference between kinetic energy and potential energy?
Kinetic energy is the energy an object possesses due to its motion, while potential energy is the energy stored in an object due to its position or configuration. For example, a ball held at a height
has potential energy, which converts to kinetic energy when the ball is dropped.
2. Can kinetic energy be converted into other forms of energy?
Yes, kinetic energy can be converted into other forms of energy, such as heat, sound, or potential energy. For instance, when you apply the brakes to a car, the kinetic energy is converted into heat
energy through friction.
3. How does velocity affect kinetic energy?
Velocity has a significant effect on kinetic energy because it is squared in the equation. This means that doubling the velocity of an object results in four times more kinetic energy.
4. What are some real-world examples of kinetic energy?
Examples of kinetic energy include a moving car, a flying airplane, a spinning wind turbine, and a thrown baseball. In each case, the motion of the object means it possesses kinetic energy.
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Fundamental limits of generative AI
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Randolf Altmeyer.
Generative AI has seen tremendous successes recently, most notably the chatbot ChatGPT and the DALLE2 software creating realistic images and artwork from text descriptions. Underlying these and other
generative AI systems are usually neural networks trained to produce text, images, audio, or video from text inputs. The aim of this talk is to develop an understanding of the fundamental
capabilities of generative neural networks. Specifically and mathematically speaking, we consider the realization of high-dimensional random vectors from one-dimensional random variables through deep
neural networks. The resulting random vectors follow prescribed conditional probability distributions, where the conditioning represents the text input of the generative system and its output can be
text, images, audio, or video. It is shown that every d-dimensional probability distribution can be generated through deep ReLU networks out of a 1-dimensional uniform input distribution. What is
more, this is possible without incurring a cost—in terms of approximation error as measured in Wasserstein-distance—relative to generating the d-dimensional target distribution from d independent
random variables. This is enabled by a space-filling approach which realizes a Wasserstein-optimal transport map and elicits the importance of network depth in driving the Wasserstein distance
between the target distribution and its neural network approximation to zero. Finally, we show that the number of bits needed to encode the corresponding generative networks equals the fundamental
limit for encoding probability distributions (by any method) as dictated by quantization theory according to Graf and Luschgy. This result also characterizes the minimum amount of information that
needs to be extracted from training data so as to be able to generate a desired output at a prescribed accuracy and establishes that generative ReLU networks can attain this minimum.
This is joint work with D. Perekrestenko and L. Eberhard
This talk is part of the CCIMI Seminars series.
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Return on Equity ROE Formula, Examples and Guide to ROE - Elipsan Endüstriyel ve Otomotiv Halıları
On a company basis, a negative ROE may be caused by one-time factors such as restructurings that depress net income and produce net losses. ROE and ROA are important components in banking for
measuring corporate performance. One of the most effective profitability metrics for investors is a company’s return on equity (ROE). ROE shows how much profit a company generates from its
shareholders’ equity. A business that creates a lot of shareholder equity is usually a sound stock choice.
You can also look at other, narrower return metrics such as return on capital employed (ROCE) and return on invested capital (ROIC). Use ROE to sift through potential stocks and find the companies
that turn invested capital into profit fairly efficiently. That’ll give you a short list of candidates on which to conduct a more detailed analysis. In some industries, firms have more assets — and
higher incomes — than in others, so ROE varies widely by sector. For example, data published by New York University puts the average ROE for online retail companies at 27.05%.
• New customers need to sign up, get approved, and link their bank account.
• In January 2020, NYU professor Aswath Damodaran calculated the average return on equity for dozens of industries.
• The type of financial engineering described in the example to improve the return on equity should be periodically re-examined, to account for any changes in the underlying fundamentals of the
• According to the Federal Deposit Insurance Corporation (FDIC), the average ROE for the banking industry during the same period was 13.57%.
Still, a common shortcut for investors is to consider a return on equity near the long-term average of the S&P 500 (as of Q4 2022, 13.29%) as an acceptable ratio and anything less than 10% as poor.
Whether an ROE is deemed good or bad will depend on what is normal among a stock’s peers. For example, utilities have many assets and debt on the balance sheet compared to a relatively small amount
of net income. A technology or retail firm with smaller balance sheet accounts relative to net income may have normal ROE levels of 18% or more.
Return on Equity (ROE): Definition and Examples
This leads to increased earnings for shareholders, contributing to a higher ROE. In this example, the company’s Return on Equity is 25%, indicating that for every rupee of shareholders’ equity, the
company generated 25 paise in net income. There are key differences between ROE and ROA that make it necessary for investors and company executives to consider both metrics when evaluating the
effectiveness of a company’s management and operations. Depending on the company, one may be more relevant than the other—that’s why it’s important to consider ROE and ROA in context with other
financial performance metrics. Return on investment (ROI) is an approximate measure of an investment’s profitability. ROI is expressed as a percentage and is calculated by dividing an investment’s
net profit (or loss) by its initial cost or outlay.
It’s best to add context to a company’s ROE by calculating the ROE of competitors in the sector. In our above example, Joe’s Holiday Warehouse, Inc. was able to generate 10% ROE, or $0.10 from every
dollar of equity. If one of Joe’s competitors had a 20% ROE, however — churning out $0.20 from every dollar of equity — it would likely be a better investment than Joe’s. If the two companies were
reinvesting the majority of their profits back into the business, we’d expect to see growth rates roughly equal to those ROEs.
Return on equity (ROE) and return on capital (ROC) measure very similar concepts, but with a slight difference in the underlying formulas. Both measures are used to decipher the profitability of a
company based on the money it had to work with. To calculate the return on equity, you need to look at the income statement and balance sheet to find the numbers to plug into the equation provided
below. A higher percentage indicates a company is more effective at generating profit from its existing assets. Likewise, a company that sees increases in its ROE over time is likely getting more
When you first look at it, you can’t be sure if it was negative because the company lost money or if it’s negative because of a negative value for shareholders’ equity. While debt financing can be
used to boost ROE, it is important to keep in mind that overleveraging has a negative impact in the form of high interest payments and increased risk of default. The market may demand a higher cost
of equity, putting pressure on the firm’s valuation. If the net profit margin increases over time, then the firm is managing its operating and financial expenses well and the ROE should also increase
over time. If the asset turnover increases, the firm is utilizing its assets efficiently, generating more sales per dollar of assets owned.
While it is also a profitability metric, ROTA is calculated by taking a company’s earnings before interest and taxes (EBIT) and dividing it by the company’s total assets. ROI helps show a company’s
return on investor money before the effects of any borrowing. If ROE is positive while ROI is negative, the company could be using borrowed money instead of internally generated profits to survive.
ROE helps investors choose investments and can be used to compare one company to another to suggest which might be a better investment. Comparing a company’s ROE to an average for similar companies
shows how it stacks up against peers.
Return on equity (ROE) measures how well a company generates profits for its owners. It is defined as the business’ net income relative to the value of its shareholders’ equity. It reveals the
company’s efficiency at turning shareholder investments into profits. ROE is typically expressed as a percentage (although it is sometimes referred to as a ratio).
Interpreting Return on Equity (ROE)
The numerator can be modified to only include income from operations, which yields a better picture of the value generated by the operational capabilities of a business, with all financing issues
stripped out. A good Return https://1investing.in/ on Equity ratio varies by industry, but generally, a higher Return on Equity indicates better profitability and efficiency. Optimizing asset
utilization and turnover rates can contribute to higher Return on Equity.
Return on Equity vs. Return on Investment
ROCE also focuses on earnings before interest and taxes, rather than after-tax profits. In order to get the whole picture of a company’s profitability when using ROE, some considerations are
necessary. For example, ROE does not indicate whether or not a company is relying on debt to generate better returns. If the company has used leverage to generate higher ROE, it has also taken on
more risk. ROE is used to determine how well a company generates earnings growth from the cash invested in the business.
Return on Equity vs. Return on Capital
A company that aggressively borrows money, for instance, would artificially increase its ROE because any debt it takes on lowers the denominator of the ROE equation. Without context, this might give
potential investors a misguided impression of the company’s efficiency. This can be a particular concern for fast-expanding growth companies, like many startups. In a situation when the ROE is
negative because of negative shareholder equity, the higher the negative ROE, the better. This is so because it would mean profits are that much higher, indicating possible long-term financial
viability for the company.
A. Profit Margin
ROE can also be calculated at different periods to compare its change in value over time. By comparing the change in ROE’s growth rate from year to year or quarter to quarter, for example, investors
can track changes in management’s performance. Understanding what ROE means and how to use it when comparing companies can help you craft a smart investment strategy.
Return on investment (ROI) is a measure of the total return on an investment regardless of its source of financing. The formula for ROI is the profit from the investment divided by the cost of the
investment. Here are some that are often used in conjunction with ROCE, or commonly confused with ROCE. The type of financial engineering described in the example to improve the return on equity
should be periodically re-examined, to account for any changes in the underlying fundamentals of the business. Balancing the mix of debt and equity is crucial for achieving an optimal capital
How to Calculate ROE
An author, teacher & investing expert with nearly two decades experience as an investment portfolio manager and chief financial officer for a real estate holding company. The image below from CFI’s
Financial Analysis Course shows how leverage increases equity returns. Now, assume that LossCo has had a windfall in the most recent year and has returned to profitability. The denominator in the ROE
calculation is now very small after many years of losses, which makes its ROE misleadingly high. Bankrate follows a strict
editorial policy, so you can trust that our content is honest and accurate.
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elipsan-endustriyel-ve-otomotiv-halilari-logo.png Elipsan2023-08-16 16:25:272023-10-31 21:25:37Return on Equity ROE Formula, Examples and Guide to ROE
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Joseph-Louis Lagrange - (Variational Analysis) - Vocab, Definition, Explanations | Fiveable
Joseph-Louis Lagrange
from class:
Variational Analysis
Joseph-Louis Lagrange was an influential 18th-century mathematician and astronomer, known for his pioneering work in mathematical analysis and mechanics. He played a crucial role in the development
of variational calculus, which focuses on finding extrema of functionals and laid the groundwork for much of modern optimization theory.
congrats on reading the definition of Joseph-Louis Lagrange. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Lagrange's work on the calculus of variations significantly influenced later mathematicians and scientists, shaping various fields including physics and engineering.
2. He introduced the concept of the 'Lagrangian' in mechanics, which expresses the difference between kinetic and potential energy, and is central to modern physics.
3. In 1755, Lagrange published his first significant paper on variational problems, setting the stage for further developments in optimization techniques.
4. Lagrange's contributions helped establish important principles that connect algebra, calculus, and mechanics, reflecting his versatility as a mathematician.
5. His work extended beyond mathematics into astronomy, where he made contributions to celestial mechanics and stability analysis of planetary motions.
Review Questions
• How did Joseph-Louis Lagrange's work contribute to the foundations of variational calculus?
□ Joseph-Louis Lagrange's contributions to variational calculus were groundbreaking, particularly through his formulation of principles that addressed how to find extrema of functionals. His
early papers laid the groundwork for methods that would later be refined by other mathematicians. The Euler-Lagrange equation, which emerged from his work, became essential for determining
optimal solutions in various applications across physics and engineering.
• Discuss the significance of the Lagrangian function in mechanics and its relation to variational analysis.
□ The Lagrangian function, introduced by Lagrange, represents the difference between kinetic and potential energy in a mechanical system. This function is pivotal in formulating the equations
of motion through the principle of least action. The connection to variational analysis lies in using this function to derive equations that describe dynamic systems while minimizing or
maximizing action over time.
• Evaluate how Lagrange's methods have impacted modern mathematical techniques in optimization and analysis.
□ Joseph-Louis Lagrange's methods laid crucial foundations for optimization techniques still in use today. His approach to variational problems set forth essential concepts like the
Euler-Lagrange equation, which continues to be applied in various fields such as economics, engineering, and physics. The ideas he developed around extremizing functionals have evolved into
sophisticated techniques used in modern optimization algorithms, demonstrating their lasting influence on both theoretical and applied mathematics.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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Some reshape operators are affine, should they be in this SRFI? - Simplelists
Some reshape operators are affine, should they be in this SRFI?
Bradley Lucier
04 May 2020 03:59 UTC
I haven't put reshape operators into the SRFI because, in general, they
are not affine, so they don't compose with the other existing array
But there is one important class of reshape operators that *are* affine,
where one maps the elements of a d-dimensional array lexicographically
to a one-dimensional array with the same volume. (The inverse
transform, for example, is not affine.)
In all my years of working with this library I've never had a need for
these, and I didn't even think of their existence until tonight. (I
don't know why, the basic mapping of indices to the linearly accessed
body of a specialized array is of this form, but there it is.)
Reshape operators cannot in general be computed from the other array
transformations, except by specifying the appropriate transformation
directly in specialized-array-share. That's one argument for putting
them in the SRFI.
On the other hand, they are a bit clumsy to implement for
non-specialized arrays, but I suppose it's not too bad. It would
basically be building an affine transformation over the indexing of a
general array, if I'm making any sense. I already discussed that
possibility in the SRFI and rejected it in favor, e.g., of permuting the
arguments of an array-getter directly for array permutations. But for
reshape operators I don't see any simpler way, off hand, than just
computing and saving the indexer function from the new multi-dimensional
array to the existing one-dimensional array.
Any thoughts? If this is an important array transform, I wouldn't want
to leave it out, but adding it would delay this SRFI a bit more.
|
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How to change decimal into mixed number
Algebra Tutorials! Wednesday 6th of November
how to change decimal into mixed number
Related topics:
Home moldova airline | free online greatest to least number game | online binomial expansion | help with saxon math homework | math investigatory projects | beginner calculus
Calculations with + algebra | formula for subtracting 2% | answer to algebra 2 problems | adding subtracting multiplying dividing fractions worksheets | mathimatics for kids
Negative Numbers
Solving Linear Equations
Systems of Linear Author Message
Solving Linear Equations litmeinforgadcoke Posted: Thursday 28th of Dec 11:58
Graphically Guys and Gals ! Ok, we’re tackling how to change decimal into mixed number and I was absent in my last math class so I have no notes and my teacher
Algebra Expressions explains lessons way bad that’s why I didn’t get to understand it very well when I went to our algebra class a while ago. To make matters worse, we
Evaluating Expressions will have our test on our next class so I can’t afford not to understand how to change decimal into mixed number. Can somebody please help me try to
and Solving Equations understand how to answer couple of questions regarding how to change decimal into mixed number so that I am ready for the examination . I’m hoping
Fraction rules Registered: that somebody could assist me as soon as possible .
Factoring Quadratic 21.03.2002
Trinomials From: England
Multiplying and Dividing
Dividing Decimals by
Whole Numbers oc_rana Posted: Friday 29th of Dec 11:46
Adding and Subtracting What in particular is your trouble with how to change decimal into mixed number? Can you provide some more details your trouble with finding a tutor
Radicals at an affordable fee is for you to go in for a right program. There are a number of programs in algebra that are easily reached. Of all those that I
Subtracting Fractions have tried out, the finest is Algebrator. Not only does it crack the math problems, the good thing in it is that it explains every step in an easy
Factoring Polynomials by to follow manner. This ensures that not only you get the exact answer but also you get to study how to get to the answer.
Grouping Registered:
Slopes of Perpendicular 08.03.2007
Lines From:
Linear Equations egypt,alexandria
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Quadratic DoniilT Posted: Sunday 31st of Dec 07:19
Writing Linear Equations Algebrator really is a great piece of algebra software. I remember having problems with function composition, least common measure and factoring
Using Slope and Point expressions. By typing in the problem from homework and merely clicking Solve would give step by step solution to the algebra problem. It has been
Factoring Trinomials of great help through several Intermediate algebra, Intermediate algebra and Basic Math. I seriously recommend the program.
with Leading Coefficient
1 Registered:
Writing Linear Equations 27.08.2002
Using Slope and Point From:
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic T0ng Posted: Sunday 31st of Dec 08:26
Equations I definitely will try Algebrator! I didn't know that there are programs like that, but since it's very simple to use I can't wait to check it out!
Parent and Family Graphs Does anyone know where can I find this program ? I want to get it right now!
Collecting Like Terms
nth Roots
Power of a Quotient Registered:
Property of Exponents 29.01.2006
Adding and Subtracting From: UK
Solving Linear Systems
of Equations by Outafnymintjo Posted: Tuesday 02nd of Jan 08:40
Elimination Okay. You can get it here: (softwareLinks). In addition they offer you definite satisfaction and a money-back guarantee. All the best .
The Quadratic Formula
Fractions and Mixed
Solving Rational Registered:
Equations 22.07.2002
Multiplying Special From: Japan...SUSHI
Binomials TIME!
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
how to change decimal into mixed number
Related topics:
Home moldova airline | free online greatest to least number game | online binomial expansion | help with saxon math homework | math investigatory projects | beginner calculus
Calculations with + algebra | formula for subtracting 2% | answer to algebra 2 problems | adding subtracting multiplying dividing fractions worksheets | mathimatics for kids
Negative Numbers
Solving Linear Equations
Systems of Linear Author Message
Solving Linear Equations litmeinforgadcoke Posted: Thursday 28th of Dec 11:58
Graphically Guys and Gals ! Ok, we’re tackling how to change decimal into mixed number and I was absent in my last math class so I have no notes and my teacher
Algebra Expressions explains lessons way bad that’s why I didn’t get to understand it very well when I went to our algebra class a while ago. To make matters worse, we
Evaluating Expressions will have our test on our next class so I can’t afford not to understand how to change decimal into mixed number. Can somebody please help me try to
and Solving Equations understand how to answer couple of questions regarding how to change decimal into mixed number so that I am ready for the examination . I’m hoping
Fraction rules Registered: that somebody could assist me as soon as possible .
Factoring Quadratic 21.03.2002
Trinomials From: England
Multiplying and Dividing
Dividing Decimals by
Whole Numbers oc_rana Posted: Friday 29th of Dec 11:46
Adding and Subtracting What in particular is your trouble with how to change decimal into mixed number? Can you provide some more details your trouble with finding a tutor
Radicals at an affordable fee is for you to go in for a right program. There are a number of programs in algebra that are easily reached. Of all those that I
Subtracting Fractions have tried out, the finest is Algebrator. Not only does it crack the math problems, the good thing in it is that it explains every step in an easy
Factoring Polynomials by to follow manner. This ensures that not only you get the exact answer but also you get to study how to get to the answer.
Grouping Registered:
Slopes of Perpendicular 08.03.2007
Lines From:
Linear Equations egypt,alexandria
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Quadratic DoniilT Posted: Sunday 31st of Dec 07:19
Writing Linear Equations Algebrator really is a great piece of algebra software. I remember having problems with function composition, least common measure and factoring
Using Slope and Point expressions. By typing in the problem from homework and merely clicking Solve would give step by step solution to the algebra problem. It has been
Factoring Trinomials of great help through several Intermediate algebra, Intermediate algebra and Basic Math. I seriously recommend the program.
with Leading Coefficient
1 Registered:
Writing Linear Equations 27.08.2002
Using Slope and Point From:
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic T0ng Posted: Sunday 31st of Dec 08:26
Equations I definitely will try Algebrator! I didn't know that there are programs like that, but since it's very simple to use I can't wait to check it out!
Parent and Family Graphs Does anyone know where can I find this program ? I want to get it right now!
Collecting Like Terms
nth Roots
Power of a Quotient Registered:
Property of Exponents 29.01.2006
Adding and Subtracting From: UK
Solving Linear Systems
of Equations by Outafnymintjo Posted: Tuesday 02nd of Jan 08:40
Elimination Okay. You can get it here: (softwareLinks). In addition they offer you definite satisfaction and a money-back guarantee. All the best .
The Quadratic Formula
Fractions and Mixed
Solving Rational Registered:
Equations 22.07.2002
Multiplying Special From: Japan...SUSHI
Binomials TIME!
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
Calculations with
Negative Numbers
Solving Linear Equations
Systems of Linear
Solving Linear Equations
Algebra Expressions
Evaluating Expressions
and Solving Equations
Fraction rules
Factoring Quadratic
Multiplying and Dividing
Dividing Decimals by
Whole Numbers
Adding and Subtracting
Subtracting Fractions
Factoring Polynomials by
Slopes of Perpendicular
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Writing Linear Equations
Using Slope and Point
Factoring Trinomials
with Leading Coefficient
Writing Linear Equations
Using Slope and Point
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient
Property of Exponents
Adding and Subtracting
Solving Linear Systems
of Equations by
The Quadratic Formula
Fractions and Mixed
Solving Rational
Multiplying Special
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
Author Message
litmeinforgadcoke Posted: Thursday 28th of Dec 11:58
Guys and Gals ! Ok, we’re tackling how to change decimal into mixed number and I was absent in my last math class so I have no notes and my teacher explains lessons way bad
that’s why I didn’t get to understand it very well when I went to our algebra class a while ago. To make matters worse, we will have our test on our next class so I can’t afford
not to understand how to change decimal into mixed number. Can somebody please help me try to understand how to answer couple of questions regarding how to change decimal into
mixed number so that I am ready for the examination . I’m hoping that somebody could assist me as soon as possible .
From: England
oc_rana Posted: Friday 29th of Dec 11:46
What in particular is your trouble with how to change decimal into mixed number? Can you provide some more details your trouble with finding a tutor at an affordable fee is for
you to go in for a right program. There are a number of programs in algebra that are easily reached. Of all those that I have tried out, the finest is Algebrator. Not only does
it crack the math problems, the good thing in it is that it explains every step in an easy to follow manner. This ensures that not only you get the exact answer but also you get
to study how to get to the answer.
DoniilT Posted: Sunday 31st of Dec 07:19
Algebrator really is a great piece of algebra software. I remember having problems with function composition, least common measure and factoring expressions. By typing in the
problem from homework and merely clicking Solve would give step by step solution to the algebra problem. It has been of great help through several Intermediate algebra,
Intermediate algebra and Basic Math. I seriously recommend the program.
T0ng Posted: Sunday 31st of Dec 08:26
I definitely will try Algebrator! I didn't know that there are programs like that, but since it's very simple to use I can't wait to check it out! Does anyone know where can I
find this program ? I want to get it right now!
From: UK
Outafnymintjo Posted: Tuesday 02nd of Jan 08:40
Okay. You can get it here: (softwareLinks). In addition they offer you definite satisfaction and a money-back guarantee. All the best .
From: Japan...SUSHI
Posted: Thursday 28th of Dec 11:58
Guys and Gals ! Ok, we’re tackling how to change decimal into mixed number and I was absent in my last math class so I have no notes and my teacher explains lessons way bad that’s why I didn’t get to
understand it very well when I went to our algebra class a while ago. To make matters worse, we will have our test on our next class so I can’t afford not to understand how to change decimal into
mixed number. Can somebody please help me try to understand how to answer couple of questions regarding how to change decimal into mixed number so that I am ready for the examination . I’m hoping
that somebody could assist me as soon as possible .
Posted: Friday 29th of Dec 11:46
What in particular is your trouble with how to change decimal into mixed number? Can you provide some more details your trouble with finding a tutor at an affordable fee is for you to go in for a
right program. There are a number of programs in algebra that are easily reached. Of all those that I have tried out, the finest is Algebrator. Not only does it crack the math problems, the good
thing in it is that it explains every step in an easy to follow manner. This ensures that not only you get the exact answer but also you get to study how to get to the answer.
Posted: Sunday 31st of Dec 07:19
Algebrator really is a great piece of algebra software. I remember having problems with function composition, least common measure and factoring expressions. By typing in the problem from homework
and merely clicking Solve would give step by step solution to the algebra problem. It has been of great help through several Intermediate algebra, Intermediate algebra and Basic Math. I seriously
recommend the program.
Posted: Sunday 31st of Dec 08:26
I definitely will try Algebrator! I didn't know that there are programs like that, but since it's very simple to use I can't wait to check it out! Does anyone know where can I find this program ? I
want to get it right now!
Posted: Tuesday 02nd of Jan 08:40
Okay. You can get it here: (softwareLinks). In addition they offer you definite satisfaction and a money-back guarantee. All the best .
|
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|
New EdTech Resource | Nickzom Calculator Android Application
Nickzom Calculator
is a free calculation-based application empire created by Idoko Nicholas Chinazom that helps one to carry out his or her simple or difficult calculations on Mathematics, Physics, Engineering and
Switches based on its available contents and at the same time lays out the solution(s) and steps for quick and easy understanding. The goal of this app is to provide one with solutions and steps to
his or her calculations available within the app's contents and any problem one faces.
Below are the present topics in Nickzom Calculator, more are to be added over time.
, Nickzom Calculator covers:
• Economics
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• Basic Electronics
• Calculus
• Laplace Transform
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There are so many other sub topics in this major topics. For Example in Linear Algebra, Nickzom can calculate Matrix Arithmetic, Inverse of a Matrix, Determinant of a Matrix, Transpose of a matrix,
Adjoint of a matrix, Minors and Cofactors of a matrix. It can also solve simultaneous equation using matrix method. Not only that in matrix arithmetic, one can perform any of this operation: Addition
and Subtraction of a matrix, Multiplication of matrices, Multiplication of a matrix by a scalar, Division of a matrix.
, Nickzom Calculator covers with there are many sub topics under too:
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[Neo] Atheists: How Much Lack of Belief is Required to be an Atheist?
1. Jonsa Well-Known Member Past Donor
Again, you continue to use an "illegal substitution" of word "belief" with "faith" in the great game of theological debate.
Thats totally 100% not true.
Cant wait to hear your explanation for this one.
Last edited: May 14, 2020
Nov 21, 2013
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Atheism, theism, and agnosticism all have defintions.
Where you fit in with that really can only be your decision. These words don't have gradiants or measurement in their definitions.
You can't decide to unilaterally change the definition in the way you want to.
In fact, you can't even depend on someone answering the "which are you" question with deeply considered, lasting seriousness.
why are you accusing me of neoatheist posting tactics?
btw dont look at me, I did not choose 'lack of belief' so you dont need to direct your post to me, you need to direct it them!
my arguments regarding LoB are all based in how stupid, incorrect and frankly laughable its usage is.
noun: atheism
disbelief or lack of belief in the existence of God or gods.
the neoatheist mantra
noun: lack; plural noun: lacks
the state of being without or not having enough of something.
noun: atheism
disbelief or not having enough belief in the existence of God or gods.
Its the neoatheist swamp, not mine.
So far everyone of them have argued against their own position in this thread and demolished LoB.
What you said actually agrees with my position, and pounds another nail in the LoB coffin, your post was misdirected to me and should have been directed to one of their posts.
Lob is indefensible.
Last edited: May 15, 2020
and we can carry that yet another step further!
noun: belief; plural noun: beliefs
an acceptance that a statement is true or that something exists.
noun: atheism
disbelief or not having enough acceptance that God or gods exists is true.
That is not what I stated. "Reject" can mean either "failure to believe" or "believe the opposite", but it cannot mean both at the same time, which is what would lead to the sentence you
constructed. That's called equivocation, and it's a fallacy.
There is a light switch for every person which says "this person believes in the existence of God". The light switch gets switched on if and only if the person accepts God's existence as
true, we call those theists. Light switches however need to be either off or on, so anyone else (including agnostics and people who believe there is no god) are in the other pot, the pot for
whom "this person believes in the existence of God" is off. Light switches are examples of negations (they have to be in one of the two states, there is no excluded middle or contradiction),
so if one side it theist, it is appropriate to call the other side a-theist.
There is a separate switch which says "this person believes that God does not exist". People for whom that switch is on are called strong atheists.
If "reject" means failure to believe, then rejecting "God exists" means flipping the first switch, making you a weak atheist. In that case, rejecting "God exists" does not mean accepting the
However, we can also say that "reject" means to believe the opposite, i.e. flipping the second switch. The last paragraph then stops to be true, and 'rejecting "God exists"' is no longer the
negation of "believing God exists", i.e. it is no longer equivalent to "to not believe God exists", and it is no longer the criteria that makes you an atheist (since you can also be an
atheist in the weak sense).
Either way, this gives us no insight that spelling it out doesn't, it just introduces an unnecessary opportunity for equivocation.
8. Pisa Well-Known Member
Mar 22, 2016
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Cute. Obviously completely wrong, but still cute, like a puppy trying to figure out why that elongated stuff at his rear is following him everywhere.
Google "emergence". The concept, not the series. Maybe you'll be able to figure out by yourself why are your inferences wrong.
I suggest you take your own advice!
Maybe you will figure out why Im correct, someday.
to reject means you do not believe a proposition is true. (at least in everyone elses world)
yeh failure to believe is disbelieve, rejection of a proposition being true is automatic acceptance the proposition is false.
LEM says propositions are true or false, not some otherworld phantom you imagine exists.
So how we doing on how much belief does someone need to lack to officially be an atheist?
Last edited: May 15, 2020
'in denial' means you reject a true proposition as false.
Last edited: May 15, 2020
very simply stated, if someone offers you an apple and you 'opt' not to take it, you rejected it, ie refused, ie failed, (ie insert whatever negation you like here n.., n...), to accept the
apple. A mountain of word salad metaphors, ephemisms, or analogies will not change the fact you chose the opposite. Which I would assume it sould be crystal clear but I am sure someone here
will require clarification because they are not able to deductively reason the meaning without a direct statement, it applies exactly the same way to the acceptance of a proposition, if it
not accepted as a true the false condition is accepted as true.
Last edited: May 16, 2020
13. Jonsa Well-Known Member Past Donor
that's totally 100% not true. Of course you can wait.
obviously you are not be familiar with what that means lol
A rock can fail to believe (in fact, it fails to believe anything at all), but it can't believe any opposites. A rock fails to believe altogether, but it is not clear to me that it can be
said to reject the proposition because of that.
Agreed, but a proposition is not the same as belief in the proposition. "The pope believes in the existence of God" is a different proposition from "God exists". It's the former that makes
the Pope a theist. You are right in saying that reversal of the latter turns "there is a god" to "there is no god", but it is reversal of the former that turns someone into an atheist. The
two statements reverse under different circumstances though. The former reverses upon the act of belief in the existence of god, not around whether god exists in ones world view (since ones
world view may not be decisive on the topic).
The answer remains that accepting God's existence as true makes you a theist, and by the law of excluded middle, anyone else is (by the weak definition) atheist.
I agree with the first line here. However, in this case, what you end up with is a lack of apple, not the opposite of an apple, whatever that might be. If we pick a more relevant example (as
in one that actually has an opposite), you have have love for something, you can have hatred for something, or you can have neither. Not having love doesn't mean you have to have hatred, you
can have neither. Indifference is not 50% hate and 50% love, it's simply neither.
This logic seems to implore you to say that agnostics are impossible, who accept neither statement. That seems like an incorrect description of the world to me.
In denial means to deny the belief as true. Kind of like how you deny atheism.
17. Jonsa Well-Known Member Past Donor
well something sure is obvious, but it ain't my familiarity with english.
sure tell that to any logician, and record it so we can all get a good laugh!
correct, yes it shows your unfamiliarity with english.
Where did you learn to be a logician?
in school.
where do you learn how to spin everything?
Last edited: May 18, 2020
Racing school. Did you become a logician by taking logic 101?
and they didnt teach you stay off the ice when your opponent is on dry pavement,..... that explains why you screw around in fruitless rewind repeat pursuits like pretending you have case,
that will in the end get you 100% nowhere.
Last edited: May 19, 2020
a rock cant fail what is impossible for any rock to accomplish.
Still trying to destroy LEM eh....
Yes LEM holds.
As has been demonstrated that is a violation of LEM.
Again, I wont go there with you because you cant even get the most simple 2 position binary logic problem correct, there is no way you can handle more complex logic without some facsimile of
understanding of how logic works.
proposition: "God exists".
acceptance is the belief God exists.
rejection is the belief God does not exist.
Last edited: May 20, 2020
Sure it can. I can't fly, if I tried, I would fail. My inability to fly does not make my failure to fly not a failure. But fair, one might say that failure requires intent, and that's not
what I'm referring to, I just want to express not doing something, as opposed to doing the opposite.
No, just applying it correctly to the various propositions at hand.
One is the LEM applied to the proposition "God exists", and the other is the LEM applied to the proposition "this person is a theist". You have only demonstrated that you haven't successfully
considered the argument I'm making.
Well, we have more than one proposition at hand, and the agnostic is a good example that shows that. If your position relies on ignoring certain evidence, that's not a good look for your
Proposition: "This person believes that God exists".
acceptance is identifying that person as a theist.
rejection is identifying that person as an atheist.
I don't particularly mind the version that you presented, but it's the version I presented that determines whether someone is an atheist (in the weak sense). Flew's position relies directly
on the proposition I presented.
a rock cannot try, bogus example, it cant fail at something it cant even try geesh
you can try and therefore fail.
It requires capacity to try.
You arent, you violate LEM
rejection of God exists = atheist
acceptance of God exists = theist
stanford U articulated that exact point for you
I have, you are in denial, and insist in violating LEM.
agnostic does not violate LEM, your constantly adding it is a strawman to this argument,
acceptance and rejection are true false, not weal strong unless of course you want to continue to violate LEM
Then you need to come up with a proposition that does not violate LEM
Last edited: May 21, 2020
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versión On-line ISSN 1562-3823
Revista Boliviana de Física vol.33 no.33 La Paz dic. 2018
A. ARTÍCULOS
Modelo mínimo para la interacción eléctrica y gravitacional, en un sistema confinado de esferas cargadas
A minimal model for electrical and gravitational interactions in a confined system of charged spheres
D. Sanjines & F. Ghezzi
Instituto de Investigaciones Físicas Universidad Mayor de San Andrés
c. 27 Cota-Cota, Campus Universitario, Casilla de Correos 8635
La Paz - Bolivia
Desarrollamos un arreglo experimental y una simulacion numérica para calcular la interaccion Coulombiana entre partículas cargadas confinadas. En este trabajo se ha elaborado un sistema de
validacion, para establecer la interacción entre estas partículas y el contorno de confinamiento. Mediante la implementacion de un método de relajación para la ecuación de Laplace, usando una red 3D,
podemos simular la configuracion de equilibrio para un sistema con pocas partículas. Ademas, se hace una comparación con el arreglo experimental con muchas partículas. Nuestra simulacion está,
razonablemente de acuerdo, con la suposición de la interaccion Coulombiana.
Código(s) PACS: 41.20.Cv 02.60.Cb
Descriptores: Problemas con condiciones de contorno en electrostática, Ecuación de Laplace, Ley de Coulomb Modelos de simulación
We have developed an experimental set up and a numerical simulation to calculate the Coulomb interaction between confined charged particles. In this work we have elaborated a validation system to
establish the interactions among these particles and the confining boundary. By implementing the relaxation method for the Laplace equation using a 3D grid, we can simulate the equilibrium
configuration for a system with few particles. Also, a com-parison is made with an experimental set up with many particles. Our simulation yields a reasonable agreement with the assumption of a
Coulombian interaction.
Subject headings: Boundary value problems in electrostatic, Laplace equation, Coulomb's law Model simulation
The physics of few-particle mesoscopic systems is an active and growing field of research. The scale of these systems is small enough for quantum effects to be considered, and yet of sufficient scale
that classi-cal macroscopic laws still govern their behavior. Both experimental and theoretical approaches appear to support the assumption of an interparticle Coulomb interaction from which
relatively simple macroscopic models emerge for several different systems, from nanotechnology to plasma research (Bonitz et al. 2008). However, crucial to research in this field is the determination
and validation of the Coulomb interaction encouraging the development of new and in-teresting experiments at the macroscopic level and improving our understanding of phase transitions, packing of
charged particles and other phenomena (Blonder 1985; Zheng & Grieve 2006). While at the mesoscopic level, light is being shed on the role of quantum effects in the critical behavior of the Coulomb
system (Clark et al. 2009).
Over the years a number of theoretical models have been proposed with different repelling interac-tions, which include for example Coulomb, screened Coulomb, Lennard-Jones, dipole, logarithmic and
hard sphere potentials (Jean et al. 2001; Schweigert et al. 1999). A recent experiment aimed at the determination and validation of the Coulomb interaction is that reported by (Zheng & Grieve (2006);
Ghezzi et al. (2008)) (and references therein) consisting of several millimeter-sized metallic spheres lying on the lower plate of a parallel plate square capacitor. The spheres are laterally
confined by a square metallic electrically charged boundary which prevents their dispersion in 2 D. One of the con-clusions reached is that no dipole interaction is ob-served and hence the remaining
reasonable option is the Coulomb interaction. Furthermore the deformation of the ensemble of balls1 due to a gravitational gradient is measured inferring the nature of the re-pelling interaction
aided by a computational method. This experiment belongs to a class of systems that are governed by the same physics, i.e., an interparticle interaction, a confining potential and an external
potential (not to mention an eventual thermal activation). In the case that all of the above interactions have potentials that obey the Laplace equation with specific boundary conditions, we could
then aim at numerically solving the equation of motion for the confined particles by means of the relaxation method where the physical interactions are incorporated via the corresponding boundary
conditions in the confining border. In this work we take the experimental array referred to above (Zheng & Grieve 2006; Ghezzi et al. 2008) and apply the Laplace equation to construct a minimal
model. We understand by a "minimal model" one which contains just the sufficient number of essential features that would characterize the relevant physical phenomena in the system. In our case these
features are: the Coulombian repulsion field among the particles (millimeter-sized metallic spheres), the confining field between the square metallic border and the particles, and the gravitational
field acting upon the system when this is tilted at a certain angle.
2. THE MODEL
The model which corresponds to the square parallel plate capacitor (figure 1) is a rectangular parallelepiped divided into 10 horizontal layers each containing 20 x 20 identical square boxes. When
the tilting angle is zero the layers are perpendicular to the gravitational force, i.e., the system is level. The bottom and upper layers correspond to each of the parallel plates of the capacitor
which have a definite electrical potential. The second layer from the bottom corresponds to the substrate in which the array of particles is located; only within this layer do the particles move
according to the forces that yield the array to an equilibrium condition.
By assigning a definite electrical potential value to the box occupied by any particle at a certain instant, we can reproduce the physical condition that all particles are equally charged; since the
particles move, the box with this definite potential will also move while its neighboring boxes have a potential that is to be determined numerically by solving the Laplace equation
in a 3D lattice. For our purposes, one of the most important properties of the solution of (1) is that the potential V(x, y, z) at some point (or box) is equal to the arithmetic mean of its six
nearest neighbors in 3D,
where and similarly for y and z. The iterative numerical procedure by which (2) converges to a definite value of the potential at every point of the lattice is the well known relaxation method [17].
For simplicity we will take here-after adimensional units for the potential V(x, y, z) and for the coordinates x, y, z with unit increments Δx = Δy = Δz = 1 in a square lattice.
In order to avoid the dispersion of the particles due to the Coulombian repulsion, the square boundary has a definite potential which could be different from that of the particles but which we have
considered as having the same potential in this work. This potential will be referred to as the "confining potential". Thus, we construct a model where each particle is found in a "potential
landscape" created by the parallel plates and the square boundary. Such potential landscape is calculated by implementing recursively (2) through the whole parallelepiped divided in a grid of 10 x 20
x 20 boxes, although the relevant physics occurs in 2 D, that is, in the substrate where the particles would be allowed to move (figure 2). Therefore, the potential landscape results from evaluating
V(x, y, z) vs. (x, y) keeping z = 2 constant (i.e., second layer from bottom to top). In figure 2 we have arbitrarily chosen the following boundary (non dimensional) values for the confining
potential: V(x, y, 1) = 99 for the bottom layer, V(x, y, 10) = 0 for the top layer, V(±10, ±10, z) = 99 for the square boundary in each layer (2 < z < 9). The origin of the XY coordinate system of
the layer where the particles are located is its geometrical center. In this landscape a single particle will move to the position with the minimum potential value, i.e., to the center of symmetry.
When identical particles are included in the landscape, they will repel each other until all forces counterbalance and an equilibrium configuration is reached. Such a configuration is not so trivial
to anticipate and a numerical evaluation of (2) through the relaxation method is necessary even for a system with few particles.
In figure 3 the potential landscape corresponds to a tilted substrate. In this case, the gravitational force manifests itself by means of a corresponding potential which increases uniformly in the
positive Y direction, i.e., V(±10, y, 2) = y + 109 . Interestingly, this gravitational potential is considered only as an
additional boundary condition for the Laplace equation (1) which is added to the existing electrical potential due to both the confining boundary and to the charged particles. Furthermore, the
particles will move in the resulting potential landscape calculated using (2) that contains simultaneously both the electric and gravitational interactions. As can be seen in figure 3, the particles
will tend to move towards a different bottom position in this new potential landscape while repelling each other. This means that when compared to the level case, a different configuration of the
particles will be reached.
4. DYNAMICS OF PARTICLES
We chose the array of particles shown in figure 4 which is the equilibrium configuration when the substrate is level. In this symmetrical configuration of nine particles each particle produces a
constant potential V[0] = 99 in the box that it occupies. This potential V[0] = 99 is also a boundary condition for the Laplace equation when solved numerically by (2). If this substrate is tilted
the configuration in figure 4 is no longer in equilibrium and the particles will reach another final equilibrium configuration, i.e., each particle in figure 4 will start moving according to a
dynamical rule that will be described below.
Take for example the particle located at the point (-4, 4), i.e., the particle at the upper left corner of the array in figure 4. Since this particle is considered as a "test charge", its dynamical
state can not be affected by its own potential but rather by the rest of the charges in the space. We therefore substitute the value of V[0] at (-4, 4) by the algorithm (2) and wait until the
iterative routine stops. Then, we look at the potential values at (-4, 4) and its neighborhood. The result is shown in figure 5 where the heavy line border indicates the point (-4,4).
We notice by simple inspection that the potential at the point (-4, 3), just below the position of the particle, is the least of the potential values of its four nearest boxes. Consequently, the
particle will move to this new position and its corresponding box will be assigned the potential value V0 because the new test charge will be in another position, say at (0,4). The process is
repeated for the nine particles; the new resulting configuration will most probably be different from the original one.
How can we be sure that these dynamics make physical sense? The total energy of the new configuration has to be less than that of the preceding configuration. When the particles in a configuration
have nowhere else to move (because they are localized in the bottom of a local potential valley found around each particle) and the energy of such a configuration is at a minimum, we can assume that
we have reached the final equilibrium configuration. However, realizing that such an energy is actually the minimum might not be an easy task, since it may correspond to a metastable configuration
which has less energy than its "neighbor" (or alike) configurations but has more energy than the real stable equilibrium configuration. We can test, with some degree of reliability, if a
configuration is stable or metastable by "kicking" the configuration, i.e., moving the particles to a neighboring position, not necessarily the nearest, and comparing the energy changes. If the
energy always grows after a few kicks, the configuration is most probably stable; if the energy diminishes after a few kicks, the configuration is metastable. Of course it is not guaranteed that the
new configuration will be a stable one, because there could exist many metastable configurations, and the kicks may simply take the system from one configuration to another.
Another word of caution: the dynamical process leading to a final equilibrium configuration, although it might have physical sense, need not be the real dynamical process observed in an actual
experiment. This is because in our model the allowed displacement of the particles is one box, either in the X or Y direction, per configuration. In a real experiment, each new configuration is
defined by the instant positions of the particles, i.e., if T is the total time elapsed from the initial to the final configurations and we want to have N configurations, then one particular
configuration corresponds to the positions of the particles at the time t = nT/N, with the integer n in the interval 0 < n < N. Therefore, in a real experiment, the particles' displacements could all
be different, both in magnitude and direction. Nevertheless, and being conscious that the modeled dynamics can be different from the real one, we claim that the final equilibrium configuration
characterized by a unique minimum of the total energy, isthesameinanycase. Differences will arise depending on how gross is the lattice's grid segmentation of the lattice's, i.e., the size of the
box, which in turn will cause the equilibrium configuration to be reached in a longer time than the real one. Another possible difference is that in the model dynamics, a coarse grid segmentation
will yield a final stationary oscillating state in which two different configurations (with a negligible energy difference) alternate, such that the real final configuration, having the minimum
energy, can never be reached. The claim mentioned above is the matter of our current research and lies beyond the scope of this work.
In figure 6 we have the final equilibrium configuration corresponding to the initial state depicted in figure 4 (in both cases we observe only the layer where the particles move). The black boxes
represent the positions of the particles and the grey boxes determine the square confining boundary. The corresponding energy evolution is depicted in figure 7 (with energy values in the vertical
axis and configuration index in the horizontal axis) where there are 13 different configurations including the initial and the final ones. The difference among configurations is not deduced from
their energies; it was observed while the model dynamics were in course. Notice one interesting thing: the eighth and tenth configurations have less energy than the -supposedly- final configuration
(the thirteenth) but in those configurations the particles had other allowed positions to move to, so we preferred to continue the model dynamics until it stopped or until it reached a stationary
oscillating state. The former occurred first.
Next, once we achieve the potential energy field corresponding to figure 6, we can deduce its vector force field. This is done by implementing
for each particle in the 2D substrate lattice. We are purportedly ignoring the forces that are perpendicular to the substrate because they will cancel out with the corresponding normal forces. The
derivative of V(x, y) with respect to x is (recall that we have set Δx = Δy = Δz= 1 in this work)
from which one can write
thus (3) results in
The confining force field in figure 8a was obtained by applying (6) to each of the equilibrium positions in figure 6 once the gravitational potential was set to zero (level substrate) and the V[0]
potential of each charge was substituted by the relaxation algorithm (2). Therefore, the resulting potential landscape V[c ]corresponds only to the confining interaction due to the charged boundary
(lateral, top and bottom plates). In a similar way, we have obtained the repulsive force field among particles (figure 8b) by applying (6) to the potential field V[cq]?V[c] resulting in the
subtraction of the confining potential field (V[c]) from the potential field due to both the confining and Coulomb interactions among particles (V[cq]).
We now recall that the equilibrium condition tak-ing into account all the interactions is
where F[c](r[i]) and F[q](r[i]) are the vector fields corresponding to the confining and repulsive forces respectively (figure 8) and r[i] is the respective position of the nine particle array; mg
senΘ is the field of "residua" forces corresponding to the gravitational interaction. Then, should the whole method applied in this work be consistent, it is expected that the gravitational force
field can be obtained from the F[c] and F[q] fields according to (7).
Figure 9 shows the corresponding confining (a) and repulsive (b) force fields for an actual experiment with a system of 169 particles (16). A distinctive feature that can be observed when one
compares this latter figure with the simulated experiment in figure 8 is that in both cases there appears a kind of "equilibrium center" where the force is null, being the equilibrium center of the
confining field in an upper position with respect to the center of the repulsive field. We can show that in the case of a level substrate these two equilibrium centers coincide; for a tilted
substrate (as in figure 9) the distance between equilibrium centers will be proportional to some monotonous function of the angle of inclination. Knowing the particles configuration in equilibrium,
it follows that we can obtain the residual gravitational force value F[g] . To this end (7) is written as
and the expression is applied to the centres of equilibrium of charge r[q] and confinement r[q] that are assumed to be known through the configuration of the particles:
According to the hypothesis F[q](r[q]) = F[c](r[c]) = 0 and F[g](r) = F[g] = const. (for any r), it follows F[g] = -F[q](r[c]) = -F[c](r[q]). In view of the symmetry of the confinement field F[c] as
a consequence of the form (square) of the confinement border and the independent nature of the field in relation to the configuration of the particles in equilibrium, it is deemed better to evaluate
the residual gravitational field in agreement with
Given that the confinement and gravitational fields are unique (for certain gradients of a tilted substrate) for the earlier expression (10) it is suggested that the centre of equilibrium of charge r
[q] is the same irrespective of the number of particles. This speculation and also the efficient use of (10) to calculate the residual gravitational force field are interesting areas for further
6. ERROR ESTIMATION
A state of equilibrium (7) is equivalent to having a minimum energy in each site containing a particle. However, given that the error estimation in the 2D lattice is due to the finite size of a cell,
then (4) shows a minimum energy potential error of ΔV[x]+ - V = V - V[x]- along the X axis and an error of ΔV[y] = V[y]+ - V = V - V[y]- along the Y axis. The corresponding errors in the force
components acting on each particle are
this would lead to the rewriting of (6) to include the follo wing error:
F is the net force on each particle, where F° is the computed force and its corresponding error ΔF is due to the cell dimensions. Now we demonstrate that |ΔF[x]| = |ΔF[y]| in (12). Taking into
account that the particles only move within the XY plane of the substrate, there should not exist a net force along the Z axis perpendicular to the substrate. This translates into the condition V[z]+
= V[z]-[=V] in expression (2), where we obtain
Substituting this in (11) instead of V and readjusting terms we obtain the desired result
If we consider F° = F°f, defining α = |ΔF[x]| = |ΔF[y]| from (14), then the error of F° in (12) should be
where ƒ is unitary in the direction of F° . It can be seen in (15) that the error ΔF affects as much the magnitude as it does the direction of F° . We can reasonably assume that both magnitude and
directional errors are of the order α and as such (12) is written as
where |ξ| < α and |η|<α (note that α,ξ and η have units of force). Developing (16) we arrive at the expression
where F is the magnitude of F° and F[x], F[y] are its respective components. When necessary hereafter we will distinguish the magnitude and components of F from those of F° by adding the superscript
for this last case.
When ξ y η are simultaneously zero in (17), then F = F°, as expected. However, we know that for physical reasons if F represents the residual force field corresponding to the gravity component along
the inclined substrate in the direction of axis Y (figure 3), then from (7) we have that the sum of the confining force F[c] and the Coulombian repulsion F[q] should only contain components along the
Y axis. As such, the F components along the X axis in (17) should be null for all particles. As we can see in graphs (a) and (b) of figure 8, F[c] + F[q] do not have null components along the X axis,
we suppose that this is due to the errors discussed. Hence, equation (17) is subject to the following restriction
which results in F components only along the Y axis in (17). Substituting the value η of (18) in (17) we obtain
or equivalently
where the subscript i represents each one of the particles. So the F field in (19) is seen as a set of aligned vectors along the Y axis with different magnitudes. According to (7) these magnitudes
should be equal. For this to happen we should apply the appropriate values of ξ[i] for each particle. This is a time consuming and cumbersome process and we have therefore chosen the following
criteria. We consider a unique value of ξ in (20) for all the particles and define the deviation as d[i] = F[y,i] - F°[y,i] so that S(ξ) = ∑[i] d^2[i] has a minimum value, i.e., ð[ξ] S = 0 and we
where ξ only guarantees a minimum value of F[y,i] in (20) with respect to an average value of F[y,i]over all the particles N defined as
This expression provides a useful numerical estimation of the residual gravitational force that is obtained in the experimental set up when the plane is inclined. Thus, there must be a linear
relationship between (F[y]) the values and sine of the inclination angle.
Note that when a substrate is horizontal it is likely that some or all of the F°[y,i] values will be zero. In these cases (18) is used instead of (22) where ξ = -F[x] = -F is obtained. In this case
we need to define F = 0j to avoid an undetermined result in (19).
We apply (22) for a substrate at 4 different gradients such that θ[1] > θ[2] > θ[3] > θ[4] >. Since the gravitational field is homogeneous, the energy potential variation along the Z axis (figure 3)
is ΔV = mgAz, where m is the mass of each particle. So, the substrate slope is Δz/Δx = tan = ΔV/(mgΔx). Since in this study the value of ΔV is not given in physical units but is non dimensional, and
the mass of the spheres is an unknown parameter, we can only compare the different gradients among them and draw conclusions. The 9 angles are unknown but their ratios are known. The selected values
where the configuration of figure 6 corresponds to θ[3]. For the configurations corresponding to the gradients given in (23), the non dimensional residual force values F[i] (F[y] =) shown in (22) are
F[i] = 1.360, F[2] = 0.839, F[3] = 0.725, F[4] = 0.225, and from which the relevant quotients F[i,j] = F[i]/F[j] are constructed:
Additionally, defining ξ[i,j] = (senθi/senθj)^2 , the relations in (23) are expressed as
and combine together with ξ[i,j]ξ[j,k] = ξ[i,k] and ξ[i,j] = ξ^-1[j,i] (according to the definition of ξ[i,j]) to obtain all of the relevant combinations: ξ[4,3], ξ[4,2], ξ[4,1] , ξ[3,2] , ξ [3,1], ξ
[2,1]. These values together with those of (24) serve to construct the points , These are used to verify if F[i] α sen θ[i] , where the gradient m of a linear adjustment of the points P should be
close to 1. These points, ordered in ascending value of the abscissa, are
We define ξ = ξ[2,1] as a variable parameter of which the gradient m(ξ) is dependent. The validity interval of ξ is calculated from the first relation in (23) obtaining:
Given that in (27) the minimum value of θ[2] is θ° and the maximum value of θ[2] is 90°, we obtain 0.25 < ξ < 1, then the slope m(ξ) for the 6 points in (26) is calculated using a linear fitting
leading to an ascending monotonous function with extreme values m(0.25) m(1) 0.9 and mean values (m) = 0.67. Since we have
then the numerical results for m indicate that the relation F[i] α sen θ[i] is reasonably satisfactory (m 0.93) when the gradients are big (θ[1] ). The discrepancies resulting from smaller gradients
are at-tributed in this work to the coarse segmentation of the substrate (20 x 20 boxes) and the small number of particles (N = 9) involved which do no result in reliable residual force values in (22
7. CONCLUSIONS
We have constructed a "minimal" model to reproduce relevant features of a system of several electri-cally charged particles confined in a square parallel plate capacitor. Our approach embraces the
theory of r^-2 type interactions as well as data from actual experiments and numerical simulation.
The method employed for such a purpose is the solution of the Laplace equation in 3D by means of the relaxation method. The main results in this work are: (i) the final equilibrium configuration of a
system of 9 particles, (ii) the corresponding energy evolution through the intermediate non-equilibrium configurations, and (iii) the vector force fields for the confining and repulsive interactions.
These results are compared with those of the actual experimental set up with many particles (e.g., figure 9 with 169 particles) and a qualitative good agreement is found.
In this work, the solution of the Laplace equation is a scalar field of potential which can incorporate simultaneously all the different physical interactions (e.g., electrical and gravitational) as
boundary conditions for the solution of the Laplace equation. The resulting vector force field deduced from the total potential field thus determines the dynamical behavior of the particles and the
stable equilibrium configuration of the system.
Emerging from this work was the concept of an "equilibrium center" for the force fields (confining and repulsive) which may provide an alternative way for determining global characteristics of the
particle system. Such an equilibrium center could move in an analogous way as does the center of mass of a particle system when acted upon by external forces.
Furthermore, we modeled the dynamics of the particles using a specific algorithm which results in a final equilibrium configuration exhibiting a minimum of the total energy. We suggest that although
the actual dynamics of the particles may be different from the modeled dynamics, the final equilibrium configuration characterized by a unique minimum of the total energy is the same. The possible
numerical differences arising between the model and the experiment is a consequence of the lattice's grid segmentation. We are pursuing further research so that our minimal model might eventually
allow us to follow the true path of the particles towards their final equilibrium configuration. It is worth noting that within the goal of the model considered in this work, and in contrast with the
experimental set up (Ghezzi et al. 2008), we use the minimum amount of relevant physical parameters to draw conclusions. Thus, we do not need to know explicitly the mass m of the particles, the
gravity acceleration g nor the angle θ of inclination of the substrate. Instead, we use the ratios of the angles as in (23) and the ratios of the forces as in (24). We do not need either the physical
values of the potential gradient ΔV and the dimensions of the substrate; it suffices to assign them non dimensional numerical values.
Finally, the implementation of our minimal model in this work suggests some interesting areas of research and as an educational aid for gaining insight and practice into these kinds of phenomena.
The authors appreciate the help and collaboration of Grieve R and Zheng X, Queen's University Belfast and acknowledge support from Grant Project IDH, Universidad Mayor de San Andres (La Paz,
Blonder, G. E. 1985, Phys. Soc., 30, 403 [ Links ]
Bonitz, M., Ludwig, P., Baumgartner, H., Henning, C., Filinov, A., Block, D., Arp, O., Piel, A., Kading, S., Ivanov, Y., Melzer, A., Fehske, H., & Filinov, V. 2008, Physics of Plasmas, 15, 55 [ Links
Clark, B. K., Casula, M., & Ceperley, D. M. 2009, Phys. Rev. Lett., 103,55701 [ Links ]
Frenkel, D. & McTague, J. P. 1979, Phys. Rev. Lett., 42, 1632 Ghezzi, F., Grieve, R., Sanjines, D., & Zheng, X. H. 2008, Revista Boliviana de Física, 14, 50 [ Links ] [ Links ]
Iwamatsu, M. 2003, Colloid Interf. Sci., 260, 305 [ Links ]
Jean, M., Even, C., & Guthmann, C. 2001, Europhys. Lett., 84, 3626 [ Links ]
Kong, M., Partoens, B., & Peeters, F M. 2003, New Journal of Physics, 5, 23 [ Links ]
Nurmela, K. J. 1998, Phys., 31, 1035 [ Links ]
Press, W. H., Flannery, B., Teukolsky, S. A., & Vetterling, W. T. 1992, Numerical recipes: The art of scientific computing . (Cambridge University Press) [ Links ]
Ryzhov, V. N & E.Tareyeva, E. 2002, Physica, 314, 396 [ Links ]
Schweigert, I. V., Schweigert, V. A., & Peeters, F M. 1999, Phys. Rev. Lett., 82, 5293 [ Links ]
S.Toxvaerd. 1980, Phys. Rev Lett., 44, 1001 [ Links ]
Tata, B. V. R., Rajamani, P. V., Chakrabarti, J., Nikolov, A., & Wasan, D. 2000, Phys. Rev. Lett, 84, 3626 [ Links ]
Zahn, K. & Maret, G. 2000, Phys. Rev. Lett., 85, 3654 [ Links ]
Zheng, X. & Earnshaw, J. C. 1998, Europhys. Lett., 41, 635 [ Links ]
Zheng, X. H. & Grieve, R. 2006, Phys. Rev., 73, 64205 [ Links ]
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Introduction to Topologysearch
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Introduction to Topology
Softcover ISBN: 978-0-8218-2162-6
Product Code: STML/14
List Price: $59.00
Individual Price: $47.20
eBook ISBN: 978-1-4704-2130-4
Product Code: STML/14.E
List Price: $49.00
Individual Price: $39.20
Softcover ISBN: 978-0-8218-2162-6
eBook ISBN: 978-1-4704-2130-4
Product Code: STML/14.B
List Price: $108.00 $83.50
• Student Mathematical Library
Volume: 14; 2001; 149 pp
MSC: Primary 55
This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and
theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, intersection index, etc. The author notes,
“The lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really
occurs in topology, not always explaining why it occurs.” He concludes, “As a rule, only those proofs (or sketches of proofs) that are interesting per se and have important generalizations are
Graduate students, research mathematicians, and theoretical physicists.
□ Chapters
□ Chapter 1. Topological spaces and operations with them
□ Chapter 2. Homotopy groups and homotopy equivalence
□ Chapter 3. Coverings
□ Chapter 4. Cell spaces ($CW$-complexes)
□ Chapter 5. Relative homotopy groups and the exact sequence of a pair
□ Chapter 6. Fiber bundles
□ Chapter 7. Smooth manifolds
□ Chapter 8. The degree of a map
□ Chapter 9. Homology: Basic definitions and examples
□ Chapter 10. Main properties of singular homology groups and their computation
□ Chapter 11. Homology of cell spaces
□ Chapter 12. Morse theory
□ Chapter 13. Cohomology and Poincaré duality
□ Chapter 14. Some applications of homology theory
□ Chapter 15. Multiplication in cohomology (and homology)
□ The book will be very convenient for those who want to be acquainted with the topic in a short time.
European Mathematical Society Newsletter
□ A concise treatment of differential and algebraic topology.
American Mathematical Monthly
□ In little over 140 pages, the book goes all the way from the definition of a topological space to homology and cohomology theory, Morse theory, Poincaré theory, and more ... emphasizes
intuitive arguments whenever possible ... a broad survey of the field. It is often useful to have an overall picture of a subject before engaging it in detail. For that, this book would be a
good choice.
MAA Online
□ From a review of the Russian edition ...
The book is based on a course given by the author in 1996 to first and second year students at Independent Moscow University ... the emphasis is on illustrating what is happening in topology,
and the proofs (or their ideas) covered are those which either have important generalizations or are useful in explaining important concepts ... This is an excellent book and one can gain a
great deal by reading it. The material, normally requiring several volumes, is covered in 123 pages, allowing the reader to appreciate the interaction between basic concepts of algebraic and
differential topology without being buried in minutiae.
Mathematical Reviews
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 14; 2001; 149 pp
MSC: Primary 55
This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and
theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, intersection index, etc. The author notes, “The
lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in
topology, not always explaining why it occurs.” He concludes, “As a rule, only those proofs (or sketches of proofs) that are interesting per se and have important generalizations are presented.”
Graduate students, research mathematicians, and theoretical physicists.
• Chapters
• Chapter 1. Topological spaces and operations with them
• Chapter 2. Homotopy groups and homotopy equivalence
• Chapter 3. Coverings
• Chapter 4. Cell spaces ($CW$-complexes)
• Chapter 5. Relative homotopy groups and the exact sequence of a pair
• Chapter 6. Fiber bundles
• Chapter 7. Smooth manifolds
• Chapter 8. The degree of a map
• Chapter 9. Homology: Basic definitions and examples
• Chapter 10. Main properties of singular homology groups and their computation
• Chapter 11. Homology of cell spaces
• Chapter 12. Morse theory
• Chapter 13. Cohomology and Poincaré duality
• Chapter 14. Some applications of homology theory
• Chapter 15. Multiplication in cohomology (and homology)
• The book will be very convenient for those who want to be acquainted with the topic in a short time.
European Mathematical Society Newsletter
• A concise treatment of differential and algebraic topology.
American Mathematical Monthly
• In little over 140 pages, the book goes all the way from the definition of a topological space to homology and cohomology theory, Morse theory, Poincaré theory, and more ... emphasizes intuitive
arguments whenever possible ... a broad survey of the field. It is often useful to have an overall picture of a subject before engaging it in detail. For that, this book would be a good choice.
MAA Online
• From a review of the Russian edition ...
The book is based on a course given by the author in 1996 to first and second year students at Independent Moscow University ... the emphasis is on illustrating what is happening in topology, and
the proofs (or their ideas) covered are those which either have important generalizations or are useful in explaining important concepts ... This is an excellent book and one can gain a great
deal by reading it. The material, normally requiring several volumes, is covered in 123 pages, allowing the reader to appreciate the interaction between basic concepts of algebraic and
differential topology without being buried in minutiae.
Mathematical Reviews
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions.
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Thermodynamics Objective Questions and Answers
22. The shape of T-S diagram for Carnot Cycle is a
23. With increase in compression ratio, the efficiency of the otto engine
24. What is the degree of freedom for a system comprising liquid water equilibrium with its vapour?
25. __________ functions are exemplified by heat and work.
26. A two stage compressor is used to compress an ideal gas. The gas is cooled to the initial temperature after each stage. The intermediate pressure for the minimum total work requirement should be
equal to the __________ mean of P[1] and P[2]. (where, P[1] and P[2] are initial and final pressures respectively)
27. Helmholtz free energy (A) is defined as
28. Gibbs free energy (F) is defined as
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gram weight
gram weight sentence in Hindi
"gram weight" meaning in Hindi gram weight in a sentence
1. Japanese beads are sold by gram weight, seldom by the hank.
2. The greatest recorded size is 50 centimeters length and 950 grams weight.
3. In-body stabilization also gives the Pentax K-3 an advantage, but its 800-gram weight is slightly more than average for a mid-size DSLR.
4. If, say, a property depended on the square of the mass, it would not be an extensive property . ( Consider a system consisting of two 1 gram weights.
5. The silver stater minted at Corinth of 8.6 grams weight was divided into three silver drachmas of 2.9 grams, but was often linked to the Thebes and more.
6. The cone was composed of a thin despin mechanism consisted of two 7 gram weights which could be spooled out to the end of two 150 cm wires when triggered by a hydraulic timer 10 hours after
7. If you use Google to search for " mole gram weight " you will find lots of tutorial pages that may be worth working through and may explain better .-Nunh-huh 09 : 39, 7 Mar 2005 ( UTC)
8. With 9.6 mm thickness and 148 grams weight, the model is about two millimeters thinner and 20 grams lighter than its predecessor LG 3D Optimus . 2D images from Google Earth and Google Maps can be
transformed into 3D images.
9. Mathematically, the diameter in millimeters of a round brilliant should approximately equal 6.5 times the cube root of carat weight, or 11.1 times the cube root of gram weight, or 1.4 times the
cube root of point weight.
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(HP-67) Barkers's Equation
12-06-2019, 01:27 PM
Post: #1
SlideRule Posts: 1,533
Senior Member Joined: Dec 2013
(HP-67) Barkers's Equation
An extract from
An Efficient Method for Solving Barkers's Equation
, British Astronomical Association, R. Meire, Journal of the British Astronomical Association, Vol. 95, NO.3/APR, P.113, 1985
In a parabolic orbit, the true anomaly v (as a function of the time) can be obtained by solving a cubic equation for tan y, the so-called Barker equation. A modified method of solution is described
and it is shown that this new form is very efficient from a computational point of view: it needs less program statements, is faster, and it has greater accuracy than the normally used trigonometric
In programming this trigonometric method on a computer or on a calculator, a problem occurs with the cubic root in equation (6) if W < 0. We must include a conditional test in the program to solve
this problem. As an example, Appendix A shows the program for an
calculator, and it can be seen that the cubic root problem takes four additional steps.
Appendix B gives the program for the
calculator. Although the trigonometric solution has a certain 'beauty', it cannot compete with this new solution (equation (12)) from a practical point of view.
FULLY documented!
there are THREE pages to the article. Use the navigation links at the bottom to 'see' ALL three. The program listings are on page 115.
12-06-2019, 06:39 PM
(This post was last modified: 03-21-2021 04:12 PM by Albert Chan.)
Post: #2
Albert Chan Posts: 2,771
Senior Member Joined: Jul 2018
RE: (HP-67) Barkers's Equation
Solving cubic with Cardano's formula, x³ + 3x - 2W = 0
y = ³√(W + √(W²+1))
x = y - 1/y
Note: discriminant = W²+1 > 0, we have only 1 real root for x
If W<0, y may be hit with subtraction cancellation.
We can avoid catastrophic cancellation by solving x'³ + 3x' - 2|W| = 0
x = sign(W) x'
Or, we can go for the big |y|: // assumed acot(W) = atan(1/W)
y = ³√(W + sign(W) √(W²+1)) = ³√(cot(acot(W)/2))
We still have the cancellation error issue when y ≈ 1
A better non-iterative formula is to use hyperbolics.
x = 2 sinh(sinh^-1(W)/3)
12-07-2019, 09:39 PM
(This post was last modified: 03-21-2021 04:09 PM by Albert Chan.)
Post: #3
Albert Chan Posts: 2,771
Senior Member Joined: Jul 2018
RE: (HP-67) Barkers's Equation
(12-06-2019 06:39 PM)Albert Chan Wrote: y = ³√(W + sign(W) √(W²+1)) = ³√(cot(acot(W)/2))
let θ = acot(W), c = cot(θ/2)
Half angle formula: cot(θ) = W = (c²-1) / (2c)
c² - 2W c - 1 = 0 → c = W ± √(W²+1)
Since |θ| < pi/2, c has same sign of θ, which has same sign of W (assumed sign(0)=1)
However, 2 roots of c have opposite sign. Matching c and W signs, we have:
c = cot(acot(W)/2) = W + sign(W) √(W²+1)
01-31-2020, 03:38 PM
(This post was last modified: 01-31-2020 03:41 PM by Albert Chan.)
Post: #4
Albert Chan Posts: 2,771
Senior Member Joined: Jul 2018
RE: (HP-67) Barkers's Equation
To complete the symmetry (again, assume sign(0)=1):
c² - 2W c + 1 = 0 → c = W ± √(W²-1)
c = cot(csc^-1(W)/2) = W + sign(W) √(W²-1)
Note: domain of csc^-1(W) = sin^-1(1/W) is |W| ≥ 1, otherwise c is a complex root.
04-10-2020, 10:34 PM
Post: #5
Mathias Zechmeister Posts: 3
Junior Member Joined: Apr 2020
RE: (HP-67) Barkers's Equation
I arrived here by googling for "Barker's equation" and "sinh".
Because I derived the same equation as noted here:
(12-06-2019 06:39 PM)Albert Chan Wrote: x = 2 sinh(sinh^-1(W)/3)
So it seems this equation is already known.
But is there any reference to this equation?
I can't find any!?
04-11-2020, 03:29 AM
Post: #6
Albert Chan Posts: 2,771
Senior Member Joined: Jul 2018
RE: (HP-67) Barkers's Equation
Welcome to the forum, Mathias Zechmeister
Hyperbolic solutions to the cubics is simply matching hyperbolic triple angle formula.
, eqn 78,79,80
(12-06-2019 06:39 PM)Albert Chan Wrote: Solving cubic with Cardano's formula, x³ + 3x - 2W = 0
y = ³√(W + √(W²+1))
x = y - 1/y
Another way is with identity:
sinh^-1(z) = ln(z + √(z²+1))
→ y = e^(sinh
→ x = 2 sinh(sinh
04-11-2020, 10:18 PM
Post: #7
Mathias Zechmeister Posts: 3
Junior Member Joined: Apr 2020
RE: (HP-67) Barkers's Equation
Thanks for the reference. So the relation is mathematically known, but it was likely not applied to Barker's equation.
With the relation:
(12-06-2019 06:39 PM)Albert Chan Wrote: x = 2 sinh(sinh^-1(W)/3)
it is even simplier to compute Barker's equation with a pocket calculator than in
R. Meire (1985)
Well, it seems the HP-67 had no hyperbolic functions.
I'm working on a follow-up paper of
Zechmeister (2018)
and I'm going to mention this relation.
(04-11-2020 03:29 AM)Albert Chan Wrote: Another way is with identity: sinh^-1(z) = ln(z + √(z²+1))
→ y = e^(sinh^-1(W)/3)
→ x = 2 sinh(sinh^-1(W)/3)
Indeed, that is, how I found it. I noted the term z + √(z²+1) in Barker's equation, and
Fukushima (1997, Eq. 73)
as well as
Raposo-Pulido+ (2018, Eq. 43)
reminded of this identity.
08-10-2020, 08:10 AM
Post: #8
Mathias Zechmeister Posts: 3
Junior Member Joined: Apr 2020
RE: (HP-67) Barkers's Equation
For your information, my paper is accepted. A preprint is available here:
In Sect. 7.2., I briefly discuss Barker's equation, more as a side note. I put you, Albert Chan, in the acknowledgements.
Please let me know, if something should be adjusted (e.g. pseudonym?).
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Computational Thinking in Mathematics Education
Meaningful CT Tasks in University Math Courses
• Investigation of integration of CT into math curriculum at the tertiary level, by focusing on teaching practice.
• Developing activities and tasks that require genuine collaboration between CT and mathematics and statistics, especially in the context of mathematics for the life sciences students.
• Investigation of the way modeling of complex systems can be modified for university classroom instruction, and for student projects.
• Investigation of ideas of dynamic system modeling approaches as to their applicability in creating life sciences models in university classrooms.
• This project is informed by the work on the Math and Programming at the Tertiary Level project, as well as by the research about modeling complex systems and system dynamics.
• Erin Clements, Ph.D. Candidate, Western University.
• Review about complex systems and system dynamics:
□ Complex systems approaches, in conjunction with rapid advances in computational technologies, enable researchers to study aspects of the real world for which events and actions have multiple
causes and consequences, and where order and structure coexist at many different scales of time, space, and organization. Within this complex systems framework, critical behaviors of systems
that were systematically ignored or oversimplified by classical science can now be included as basic elements that account for many observed aspects.
□ System dynamics has been used to study global warming and climate change, the spread of epidemics, heroin addiction, improvement in treatment for chronic kidney disease, the causes of urban
decay, limits to growth, and many more systems that touch almost every field of study in which dynamic feedback plays a crucial role. Computer simulation (system dynamics modeling) provides a
method for identifying high leverage points and testing policy recommendations, further strengthening the use of this analytical approach.
• Erin Clements and Miroslav Lovric will present a poster at Computational Thinking in Mathematics Education Symposium, 13-15 October 2017.
• Study of literature, learning more about complex systems and system dynamics
• Developing models, classroom testing, and analysis.
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FioriniCV | Fiorini&Associates
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Eugene Fiorini
University of Delaware PhD, MS, Mathematics
Graduate Work, Mathematics Education
Temple University MS, Statistics
University of New Hampshire Graduate Work, Mathematics Education
Pennsylvania State University BS Mathematics (Science/Business concentrations)
Areas of Special Interest
Graph Theory, Combinatorics, Statistics, Biomathematics, Mathematical Forensics, Discrete Mathematical Modeling, Mathematics and Science Education
Professional Employment
• Muhlenberg College, Allentown, PA
Aug 2015 - Aug 2020 Truman Koehler Professor of Mathematics
• Rutgers University - Center for Discrete Mathematics & Theoretical Computer Science
Jul 2008 - Aug 2015 DIMACS Associate Director & Research Professor
• Shippensburg University, Shippensburg, PA
Apr 2004 - Aug 2007 Associate Dean & Interim Dean, College of Arts & Sciences
Sep 1999 - Apr 2004 Chair, University Scholarship Committee
May 2006 - Jun 2008 Professor of Mathematics
Apr 1998 - May 2006 Associate Professor of Mathematics
Sep 1993 - Apr 1998 Assistant Professor of Mathematics
• Washington College, Chestertown, MD
Sep 1993 - Aug 1994 Visiting Assistant Professor of Mathematics
Director, Mathematics Tutoring Center
• Widener University, Chester, PA
Sep 1991 - Aug 1993 Visiting Lecturer in Mathematics
Sample of Recent Grants
National Science Foundation (Served as PI, Co-PI, or Senior Personnel)
• ​REU: Research Challenges of Computational Methods in Discrete Mathematics
Moravian University, DMS-2150299
• REU: Research Challenges of Computational and Experimental Mathematics
Moravian University, DMS-1852378
• Algebraic and Extremal Graph Theory Conference
University of Delaware, DMS-1649807
• REU: Research Challenges in Identifying Integer Sequences Using the OEIS
Muhlenberg College, DMS-1560019
• Conference on Challenges of Identifying Integer Sequences
DIMACS-Rutgers University, DMS-1445898
• DIMACS REU in Computing Theory and Multidisciplinary Applications
DIMACS-Rutgers University, CCF-1263082
• NSF REU PI Meeting
Philadelphia, PA, Subaward to CCF-1004956
• DIMACS/DIMATIA US/Czech International REU Program
DIMACS-Rutgers University, CCF-1004956
• Challenge of Interdisciplinary Education: Math-Bio
DIMACS-Rutgers University, COMAP, DRL-1020166
• Effects of Genome Structure and Sequence and Sequence on the Generation of Variation and Evolution Conference
DIMACS- Rutgers University, CCF-0432013
• DIMACS/CCICADA/DIMATIA Research Experiences for Undergraduates
DIMACS-Rutgers University, CCF-0648985
• US-African Biomathematics Initiative
DIMACS-Rutgers University, DMS-0829652
Professional Organizations and Private Foundations (Served as PI)
• SLIMES: Spotted Lanternfly Investigated through Mathematical and Environmental Sciences
Trexler Foundation, Allentown, PA
Mathematical Association of America Tensor Grant, Washington, DC
• Workshop on NIMBY: Mathematical and Computational Tools for Decision Making
Mathematics for Planet Earth, NSF DMS-1246305
• Community College Writers Workshop: Sustainability Modules for the College Classroom
Mathematics for Planet Earth, NSF DMS-1246305
• InForMMS: Investigating Forensic Mysteries through the Mathematical Sciences
Mathematical Association of America Tensor Grant, Washington, DC
• InForMMS: Investigating Forensic Mysteries through the Mathematical Sciences-Plus
Trexler Foundation, Allentown, PA
Mathematical Association of America Tensor Grant, Washington, DC
• Mathematical Association of America NREUP
□ Identifying Sieves and Primitive Integer Triples Using the OEIS
□ Topologically Equivalent Graphs and Pattern Recognition
□ Applying Graphs to Twitter and Brain Connectivity
□ Sustainability and Graph Theory
□ Graph Theory, Algorithms, and Applications
□ Modeling Hudson River Species and Class-0 Subgraphs
Journal Articles
• Cycles & Girth in Pebble Assignment Graphs (w. J. Johnston, M. Lind A. Woldar & W. H. T. Wong), Graphs and Combinatorics, accepted.
• On Properties of Pebble Assignment Graphs (w. M. Lind & A. Woldar), Graphs and Combinatorics, 38(45) 2021.
• On the Assignment Graphs of Oriented Graphs (w. J. Glassband, G. Koch, S. Lebiere, X. Liu, E. Sabini, N. Shank, & A. Woldar), arXiv:2111.04882[math.CO], Nov 2021.
• Characterizing Winning Positions in the Impartial Two-player Pebbling Game on Complete Graphs (w. M. Lind, A. Woldar, & W. H. T. Wong), Journal of Integer Sequences, 24(6) 2021.
• Nimber sequences of Node-Kayles Games (w. S. Brown, S. Daugherty, B. Maldonado, D. Manzano-Ruiz, S. Rainville, R. Waechter, & T. W. H. Wong), Journal of Integer Sequences, 23(3) 2020.
• Preface to the Special Issue of Discrete Mathematics: Dedicated to the Algebraic and Extremal Graph Theory Conference, August 7-10, 2017, University of Delaware, USA (w. S. Cioaba, R. Coulter, &
Q. Xiang), Discrete Mathematics, 342(10), October 2019
• Primes and perfect powers in the Catalan triangle (w. N. Benjamin, G. Fickes, E. Jaramillo-Rodriquez, E. Jovinelly, & T. W. H. Wong), Journal of Integer Sequences, 22(6) 2019.
• Coordinating a large, amalgamated REU program with Multiple Funding Sources (w. K. Myers & Y. Naqvi), Problems, Resources, and Issues in Mathematics Undergraduate Studies (PRIMUS), Spring 2017
• Connectance, robustness, and the Hudson River food web (w. U. Ghosh-Dastidar), Proceedings of 2014 International Conference of Business and Applied Sciences Academy of North America (BAASANA),
• Undergraduates and research: motivations, challenges, and the path forward (w. W. E. Wong, J. Ding, and C. Hansen), Proceedings of IEEE CSEE&T, May 2013.
• Graph pebbling: symmetric class-0 subgraphs of complete graphs (in preparation, with V. deSilva and C.J. Verbeck), DIMACS Technical Report, 2012
• Powerpoint presentation: ship it – textbooks go global, IPSL Proceedings, Spring 2006
• The future interdisciplinary classroom, Project Kalidescope Proceedings , Spring 1999
• Extremal properties of generalized four-gons, SSHE-MA Conference Proceedings, Spring 1998
• Developing mathematics-related activities for the classroom. (with A. Acusta & J. Miller), Mathematics and Computer Education, 1998
• Increasing student activity levels. (joint with D. Ensley), Proteus, 1998
• An extremal characterization of the incidence graphs of projective planes. (with F. Lazebnik), Acta Applicandae Mathematicae, 1998
• On the maximum number of 8-cycles in a bipartite graph of girth at least six. (with F. Lazebnik), Congressus Numerantium, 1994
Contributions to the On-Line Encyclopedia of Integer Sequences
A259558, A259559, A259562, A259564, A260360, A260373, A260374, A260375, A275481, A275586, A308853, A309088, A309089, A309257, A309258, A309259, A309344, A309345, A316533, A316629, A316632, A316781,
A317027, A317367, A336764, A340631, A346189, A346197, A346198, A346199, A346204, A346401, A347637
Books, Chapters, Modules
• Time Series Applications in Energy-Related Issues. (w. K. Shirley) CCICADA, 2017
• Modeling Neuron Networks: The neuroscience of pain. (w. T. Cannon), COMAP, 2016
• CrIME: Criminal investigation through mathematical examination. (w. L. Godbold, T. Fleetwood), COMAP, 2016
• Smart Driving: reducing emissions by choosing “greenest’ path . (w. B. Piccoli), DIMACS, 2013
• Modeling reality with functions: graphical, numerical, analytical, (w. J. Miller) 1st ed. Kendall-Hunt, Spring, 2000
Graduate Student Thesis Advisor
• Gail Auimiller, 2002, Development of Algebra Manipulatives.
• Gaynaile Nowak, 2001, Statistical analysis of insurance data.
• Jason Rosenberry, 1999, Development of web-based geometry activities.
• Michael Failor, 1997, Acceptable meconium drug testing methods based on the analysis of false- positive and false-negative test results.
• Cindy Martin, 1997, Classification of counterexamples to a generalization of Ore’s theorem.
• Amy Smith, 1997, Developmental steps in geometry.
• Mark Yost,1996, An alternative approach to Redei’s theorem.
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(PDF) Modelling the seasonality of respiratory syncytial virus in young children
Author content
All content in this area was uploaded by Hannah Moore on Sep 08, 2014
Content may be subject to copyright.
Modelling the seasonality of respiratory syncytial virus
in young children
A.B. Hogana, G.N. Mercera, K. Glassa, H.C. Mooreb
aNational Centre for Epidemiology and Population Health, Australian National University,
Canberra, Australia
bTelethon Institute for Child Health Research, Centre for Child Health Research,
University of Western Australia, Perth, Australia
Email: Alexandra.Hogan@anu.edu.au
Abstract: Respiratory syncytial virus (RSV) is a major cause of acute lower respiratory tract infections in
infants and young children. The transmission dynamics of RSV infection among young children are still
poorly understood (Hall et al., 2009) and mathematical modelling can be used to better understand the seasonal
behaviour of the virus. However, few mathematical models for RSV have been published to date (Moore et al.,
2013; Weber et al., 2001; Leecaster et al., 2011) and these are relatively simple, in contrast to studies of other
infectious diseases such as measles and influenza.
A simple SEIRS (Susceptible, Exposed, Infectious, Recovered, Susceptible) type deterministic ordinary dif-
ferential equation model for RSV is constructed and then expanded to capture two separate age classes with
different transmission parameters, to reflect the age specific dynamics known to exist for RSV. Parameters in
the models are based on the available literature.
In temperate climates, RSV dynamics are highly seasonal with mid-winter peaks and very low levels of ac-
tivity during summer months. Often there is an observed biennial seasonal pattern in southern Australia with
alternating peak sizes in winter months. To model this seasonality the transmission parameter β(t)is taken to
vary sinusoidally with higher transmission during winter months, such as in models presented in Keeling and
Rohani (2008) for infections such as measles and pertussis:
β(t) = β0[1 + β1sin(2πt
52 )].(1)
This seasonal forcing reflects increases in infectivity and susceptibility thought to be due to multiple factors
including increased rainfall, variation in humidity, and decreased temperature (Cane, 2001; Weber et al., 1998).
Sinusoidally forced SIR-type models are known to support complex multi-periodic and even chaotic solutions.
For realistic parameter values, obtained from the literature, and depending on the values selected for β0and
β1, the model predicts either annual peaks of the same magnitude, or the observed biennial pattern that can be
explained by the interaction of the forcing frequency and the natural frequency of the system. This behaviour
is in keeping with what is observed in different climatic zones.
20th International Congress on Modelling and Simulation, Adelaide, Australia, 1–6 December 2013
A.B. Hogan et al., Modelling respiratory syncytial virus in young children
Respiratory syncytial virus (RSV) is a significant health and economic burden in Australia and internationally.
Epidemics are strongly seasonal, occurring each winter in temperate climates (Cane, 2001) and during the
rainy season in tropical climates (Simoes, 1999), usually lasting between two and five months (Hall, 1981;
Kim et al., 1973).
There are only limited RSV data sets published and often these are only over short time spans. Recent data
sets that span numerous years for Utah in the U.S.A. (Leecaster et al., 2011), southern Germany (Terletskaia-
Ladwig et al., 2005) and Western Australia (Moore et al., 2013), show a distinct biennial seasonal pattern with
higher peaks in alternate winter seasons. These regions all have a temperate climate and experience significant
seasonal variation in climate. Other data sets, such as for Singapore (Chew et al., 1998) and the Spanish region
of Valencia (Acedo et al., 2011) show annual seasonal behaviour, with peaks of the same magnitude each year.
While the mortality rate for previously healthy children is low, RSV causes high rates of hospitalisation for
children under two years of age and has also been identified as a cause of mortality in the elderly (Faskey
et al., 2005; Simoes, 1999; Hardelid et al., 2013). Clinical symptoms may vary from those of a mild infection
to severe bronchiolitis or pneumonia (Hall, 1981).
In the young, infection with RSV does not cause long lasting protective immunity, meaning that individual
children may be repeatedly infected. There is currently no licensed vaccine available, nor any antiviral treat-
ments commonly used for RSV infection in Australia.
The aim of this study is to develop RSV models that reproduce the biennial pattern observed in temperate
climates. In later work these models will be fitted to available data. Thus we present a SEIRS model for
a single age class for RSV infection, where the transmission rate is seasonally forced, such as in models
presented in Keeling and Rohani (2008). An investigation into the types of behaviour possible in this model is
undertaken. To better reflect the known epidemiology of RSV, we then introduce a second age class into the
model with a second transmission rate and investigate how this affects the transmission dynamics.
2 MODEL FOR A SINGLE AGE CLASS
A deterministic ordinary differential equation model is developed for the transmission of RSV for 0-23 month
old children. This age group was chosen as the literature indicates that almost all children have been infected
by the time they reach this age (Hall, 1981; Sorce, 2009). As it remains unclear to what degree adults carry
and shed the virus, thereby infecting children, the adult population was not included in the model.
The population is divided into four compartments, where Srepresents the proportion of the population that
is susceptible to infection; Erepresents the proportion of the population that is exposed but not yet infected;
Irepresents the proportion that is infected with the virus; and Rrepresents the proportion of the population
that is recovered and temporarily immune to reinfection. The SEIRS-type model was selected for two reasons.
Firstly, the virus is known to have a latency period between an individual being exposed to the infection and
becoming infectious. This period is of the same order of magnitude as the infectious period and hence needs
to be included to accurately represent the disease dynamics. Secondly, infection from the virus does not cause
long-lasting immunity, hence recovered individuals may return to the susceptible class and be reinfected.
S E I R
Figure 1. Schematic diagram for single age class SEIRS model for RSV transmission.
A schematic representation of the model is shown in Figure 1. The average recovery period is represented by 1
A.B. Hogan et al., Modelling respiratory syncytial virus in young children
the average latency period (the time between contracting the infection and becoming infectious) is represented
by 1
δand the average duration of immunity is represented by 1
ν. The rate of entering the model (the birth rate)
is equal to the rate at which individuals age out of the model, and is represented by µ. The virus is transmitted
between individuals at rate β.
The differential equations, where time in weeks is represented by t, are
dt =µ−βSI −µS +νR (2)
dt =βSI −δE −µE (3)
dt =δE −µI −γI (4)
dt =γI −µR −ν R. (5)
2.1 Parameter values
The birth rate and ageing rate µis chosen as in Moore et al. (2013) and is based on the average number of
births per week in the metropolitan region of Western Australia. This gives an average weekly birth rate of
0.012. Assuming the birth and ageing rates are equal simplifies the calculations as it ensures the population
size remains constant. Here this assumption does not change the overall dynamics of the system.
Based on the literature, the average latency period for RSV is assumed to be four days (Weber et al., 2001;
Moore et al., 2013). Other models, such as those presented by Leecaster et al. (2011), assume a latency period
of five days. A four day latency period equates to δbeing 1.754 (or 1
0.57 ).
Similarly, the average recovery period is based on estimates in previous models for RSV of 10 days (Weber
et al., 2001; Moore et al., 2013; Acedo et al., 2010; Leecaster et al., 2011), and within the range of one to 21
days identified by Hall et al. (1976). This gives γequal to 0.714 (or 1
The immunity period is the time between recovering from a RSV infection to becoming susceptible to the virus
again. Although not currently well understood, there is some evidence that the immunity period is around 200
days which equates to νbeing 0.035 (or 1
28.57 ). This again is the value used in previous modelling of RSV
(Weber et al., 2001; Moore et al., 2013; Acedo et al., 2010).
The transmission rate β(t), given at Equation 1, was chosen to reflect the observed annual seasonality of RSV
in temperate climates. Similar seasonal forcing has been applied in other models for RSV transmission (Weber
et al., 2001; Moore et al., 2013; Acedo et al., 2010; Arenas et al., 2008; Leecaster et al., 2011). The sinusoidal
function, with a period of 52 weeks, accounts for the observed higher transmission between children during
the winter months. The term β0represents the average transmission rate and β1represents the amplitude of
the seasonal fluctuation (Keeling and Rohani, 2008).
For the purpose of investigating the overall dynamics of this model, a range of values was considered for β0.
For β1, a value of 0.6 was assumed (noting that 0< β1≤1), in order to replicate the conditions in temperate
climates where strong seasonality is observed. For some seasonally forced models for RSV, values as high
as 1 (Leecaster et al., 2011) have been assumed. Other models assume much lower values for β1, such as
between 0.10 and 0.36 (Arenas et al., 2008). In future work we will estimate these parameters β0and β1using
longitudinal data from Western Australia.
2.2 Numerical solution
The system of differential equations was solved and plotted using MATLAB’s ode45 routine. A burn in time of
80 years was used to allow the system to stabilise and thereby remove the dependence on the initial conditions.
When there is no seasonality in the transmission rate (when βis constant, β1=0), the natural oscillations in
the system die out and the system reaches a steady state. With seasonality, there is either a distinct biennial
pattern, with higher peaks in alternate winter seasons, or peaks of the same magnitude each year, depending
on the values selected for β0and β1. Figure 2(a) depicts a plot of a biennial pattern solution for the infected
population, with β0=1.1 and β1=0.6.
As there are values of β0and β1where there is no biennial pattern but instead where the seasonal peak reaches
the same maximum each year, adjacent seasonal peaks versus the parameter β0were plotted in order to better
A.B. Hogan et al., Modelling respiratory syncytial virus in young children
Proportion Infected
β0=1.1, β1=0.6
(a) A numerical solution for the infected population of a single age
class SEIRS model for RSV transmission with a sinusoidally forced
transmission parameter.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
Proportion Infected
(b) Maximum proportion of infectives at adjacent seasonal peaks
versus the bifurcation parameter β0, for the single age class system.
The parameter determining the amplitude of the forcing, β1, is 0.6
Figure 2.
understand the bifurcation patterns of the system. Figure 2(b) gives an impression of the bifurcation structure
of the model, showing seasonal peaks over two adjacent years. Where the seasonal pattern is annual, only a
single peak is shown, whereas biennial dynamics gives two distinct peaks. There is a specific range of possible
values for β0for which the system will feature the biennial seasonal pattern. Outside this range, the system
reverts to a regular annual seasonal pattern. These results are in keeping with what is observed in different
climatic zones.
3 MODEL FOR TWO AGE CLASSES
Studies show that the transmission dynamics of RSV change as children age. That is, incidence is higher
for children aged less than 12 months than those in the 12-23 month age class (Moore et al., 2010). It is
still unclear why older children are less affected. Possible reasons are reduced susceptibility, or less severe
symptoms (so less infections are reported), as a result of prior infection with the virus; or due to having better
developed immune systems than younger children. Thus, we present a second set of differential equations to
account for two age classes and two transmission parameters, βAand βB.
The second model is the set of differential equations
dt =µ−βAS1(I1+I2)−ηS1+νR1(6)
dt =βAS1(I1+I2)−δE1−ηE1(7)
dt =δE1−ηI1−γI1(8)
dt =γI1−ηR1−νR1(9)
dt =ηS1−βBS2(I1+I2)−ηS2+ν R2(10)
dt =ηE1+βBS2(I1+I2)−δE2−ηE2(11)
dt =ηI1+δE2−ηI2−γI2(12)
dt =ηR1+γI2−ηR2−νR2.(13)
A.B. Hogan et al., Modelling respiratory syncytial virus in young children
Figure 3. Schematic description of a SEIRS model for RSV transmission that takes into account two separate
age classes: children aged <12 months, where the virus is transmitted at rate βA; and 12-23 month old
children, where the virus is transmitted at rate βB.
The parameters µ,γ,δand νare as presented for the single age class model in Equations (2)-(5). An additional
parameter ηis introduced to reflect the ageing from the <12 month age class into the 12-23 month age class,
and also to reflect the ageing out of the 12-23 month class. This rate is assumed to be equal and distributed
evenly over time, therefore ηis taken to be 1
52 . A schematic diagram of the two age class system is given in
Figure 3.
The transmission rates are βA, for the <12 month age class, and βB, for the 12-23 month age class. Both
are of the sinusoidal form presented for the single age class model at Equation (1). Here the parameter β0in
βBwas selected to produce a reduced average transmission rate for the 12-23 month old age class, to better
reflect the different transmission dynamics for older children. The β1parameter is the same for βAand βB, to
represent the same climatic region.
3.1 Numerical solution
The system of differential equations was solved using MATLAB’s inbuilt ode45 routine, with a burn intime of
80 years. The model accurately mimics the expected lower number of infectives in the 12-23 month age class
and again produces the biennial pattern for both age classes, as observed in the single age class system. Figure
4(a) shows a solution for the infected population for each age class. Adjacent seasonal peaks for increasing
values of β0were again examined, showing that, as for the single age class system, there is a specific range of
possible values for β0that produce the biennial seasonal pattern 4(b).
By sinusoidally forcing the transmission parameter, both models depict either a distinct biennial seasonality, or
annual seasonal peaks of the same magnitude, for realistic parameter values depending on the values selected
for β0and β1. These results are in keeping with what is observed in different climatic zones. We showed that
a simple single age class model, with demography, is sufficient to achieve these seasonal patterns. Both the
single age class model, and the expanded model with two age classes, now provide a base on which to add
Future work will investigate varying the recovery, latency and immunity parameters for different age classes, as
well as a more detailed bifurcation analysis of both systems. There is a possibility of more complex behaviour
than the two-cycle pattern observed here being present. We will also investigate whether prior exposure is
the reason for reduced susceptibility, through expanding the model and fitting with population-based linked
laboratory data for the metropolitan region of Western Australia.
There is currently no licensed vaccine for RSV available in Australia. Vaccine development to date has been
problematic due to lack of an ideal animal model for the the disease, and the challenges of immunising infants
A.B. Hogan et al., Modelling respiratory syncytial virus in young children
Proportion Infected
I1: β0=3.2 β1=0.6
I2: β0=2.4 β1=0.6
(a) A numerical solution for the infected populations for the <12
month age class and the 12-23 month age class of a SEIRS model
for RSV transmission, where the transmission parameter for each
age class is sinusoidally forced.
2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35
Proportion Infected
(b) Maximum proportion of infectives at adjacent seasonal peaks
versus β0(with βB= 0.75βA), demonstrating the biennial be-
haviour over a range of values. In this case, the parameter β1is
Figure 4.
who are immunologically immature (Crowe Jr, 2002). However, with a new vaccine currently undergoing
phase two trials (Anderson et al., 2013), we will also look at the optimal timing in the transmission cycle for
the roll out of a vaccination program.
Acedo, L., J. Mora˜
no, and J. D´
ıez-Domingo (2010). Cost analysis of a vaccination strategy for respiratory
syncytial virus (RSV) in a network model. Mathematical and Computer Modelling 52(7-8), 1016–1022.
Acedo, L., J. Mora˜
no, R. Villanueva, J. Villanueva-Oller, and J. D´
ıez-Domingo (2011). Using random net-
works to study the dynamics of respiratory syncytial virus (RSV) in the Spanish region of Valencia. Math-
ematical and Computer Modelling 54(7-8), 1650–1654.
Anderson, L. J., P. R. Dormitzer, D. J. Nokes, R. Rappouli, A. Roca, and B. S. Graham (2013). Strategic
priorities for respiratory syncytial virus (RSV) vaccine development. Vaccine 31S, B209–B215.
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Cane, P. A. (2001). Molecular epidemiology of respiratory syncytial virus. Reviews in medical virology 11(2),
Chew, F. T., S. Doraisingham, A. E. Ling, G. Kumarasinghe, and B. W. Lee (1998). Seasonal trends of viral
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Crowe Jr, J. E. (2002). Respiratory syncytial virus vaccine development. Vaccine 20, S32–S37.
Faskey, A., P. Hennessey, M. Formica, C. Coz, and E. Walsh (2005). Respiratory Syncytial Virus Infection in
Elderly and High-Risk Adults. New England Journal of Medicine 352(17), 1749–1760.
Hall, C. B. (1981). Respiratory syncytial virus. In R. D. Feigin and J. D. Cherry (Eds.), Textbook of Paediatric
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Hall, C. B., R. G. Douglas, and J. M. Geiman (1976). Respiratory syncytial virus infections in infants: quan-
titation and duration of shedding. The Journal of Pediatrics 89(1), 11–15.
Hall, C. B., G. A. Weinberg, M. K. Iwane, A. K. Blumkin, K. M. Edwards, M. A. Staat, P. Auinger, M. R.
Griffin, K. A. Poehling, D. Erdman, C. G. Grijalva, Y. Zhu, and P. Szilagyi (2009). The burden of respiratory
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by age group in England and Wales 1999-2010. Influenza and Other Respiratory Viruses 7(1), 35–45.
Keeling, M. J. and P. Rohani (2008). Modeling Infectious Diseases in Humans and Animals. Princeton
University Press.
Kim, H. W. H. A., J. Arrobio, C. D. Brandt, C. Barbara, G. Pyles, J. L. Reid, R. M. Chanock, and R. H. Parrott
(1973). Epidemiology of Respiratory Syncytial Virus. American Journal of Epidemiology 98, 216–225.
Leecaster, M., P. Gesteland, T. Greene, N. Walton, A. Gundlapalli, R. Rolfs, C. Byington, and M. Samore
(2011). Modeling the variations in pediatric respiratory syncytial virus seasonal epidemics. BMC Infectious
Diseases 11(1), 105.
Moore, H., P. Jacoby, C. Blyth, and G. Mercer (2013). Modelling the seasonal epidemics of Respiratory
Syncytial Virus in young children. Submitted to Influenza and Other Respiratory Viruses.
Moore, H. C., N. de Klerk, P. Richmond, and D. Lehmann (2010). A retrospective population-based cohort
study identifying target areas for prevention of acute lower respiratory infections in children. BMC Public
Health 10(1), 757.
Simoes, E. A. (1999). Respiratory syncytial virus infection. Lancet 354(9181), 847–852.
Sorce, L. R. (2009). Respiratory syncytial virus: from primary care to critical care. Journal of pediatric health
care: official publication of National Association of Pediatric Nurse Associates & Practitioners 23(2), 101–
Terletskaia-Ladwig, E., G. Enders, G. Schalasta, and M. Enders (2005). Defining the timing of respiratory
syncytial virus (RSV) outbreaks: an epidemiological study. BMC Infectious Diseases 5, 20.
Weber, A., M. Weber, and P. Milligan (2001). Modeling epidemics caused by respiratory syncytial virus
(RSV). Mathematical Biosciences 172(2), 95–113.
Weber, M. W., E. K. Mulholland, and B. M. Greenwood (1998). Respiratory syncytial virus infection in
tropical and developing countries. Tropical Medicine & International Health 3(4), 268–280.
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convert to dataframe pandas
Instead, for a series, one should use: df ['A'] = df ['A']. Example 3: Convert a list of dictionaries to pandas dataframe. In line 1–8 we first scale X and y using the sklearn MinMaxScaler model, so
that their range will be from 0 to 1. Install mysql-connector. Notes. Step 1: DataFrame Creation- In my case, I got it as the result of a pivot_table() with the aggfunc as lambda x: x, because list
or … Convert openpyxl object to DataFrame. Table of Contents. Finally, the pandas Dataframe() function is called upon to create DataFrame object. We will explore and cover all the possible ways a
data can be exported into a Python dictionary. The current data type of columns is # Get current data type of columns df1.dtypes Data type of Is_Male column is … In this article, you have learned how
to convert the pyspark dataframe into pandas using the toPandas function of the PySpark DataFrame. dt. It also allows a range of orientations for the key-value pairs in the returned dictionary. We
can convert a dictionary to a pandas dataframe by using the pd.DataFrame.from_dict() class-method. data: dict or array like object to create DataFrame. pandas to_html() Implementation steps only-Its
just two step process. Pandas DataFrame from_dict() method is used to convert Dict to DataFrame object. In this tutorial, We will see different ways of Creating a pandas Dataframe from List.
Fortunately this is easy to do using the built-in pandas astype(str) function. The next lines are some shape manipulation to the y in order to make it applicable for keras. By default, convert_dtypes
will attempt to convert a Series (or each Series in a DataFrame) to dtypes that support pd.NA.By using the options convert_string, convert_integer, convert_boolean and convert_boolean, it is possible
to turn off individual conversions to StringDtype, the integer extension types, BooleanDtype or floating extension types, respectively. The labels need not be unique but must be a type of hashable.
Often is needed to convert text or CSV files to dataframes and the reverse. This seems like a simple enough question, but I can't figure out how to convert a Pandas DataFrame to a GeoDataFrame for a
spatial join? The Question : 346 people think this question is useful I have a Python dictionary like the following: The keys are Unicode dates and the values are integers. date Example: Datetime to
Date in Pandas. I wanted to calculate how often an ingredient is used in every cuisine and how many cuisines use the ingredient. Fortunately this is easy to do using the .dt.date function, which
takes on the following syntax: df[' date_column '] = pd. Convert DataFrame, Series to ndarray: values. so first we have to import pandas library into the python file using import statement.. An
object-type column contains a string or a mix of other types, whereas float contains decimal values. Often you may want to convert a datetime to a date in pandas. This configuration is disabled by
default. import pandas as pd import numpy as np #Create a DataFrame df1 = { 'Name':['George','Andrea','micheal','maggie','Ravi', 'Xien','Jalpa'], 'Is_Male':[1,0,1,0,1,1,0]} df1 = pd.DataFrame
(df1,columns=['Name','Is_Male']) df1 so the resultant dataframe will be . But here in this example, I will show you how to convert a list of dictionaries to pandas dataframe. Often, you’ll work with
data in JSON format and run into problems at the very beginning. Here’s how to do it for the given example: import pandas … Code for converting the datatype of one column into numeric datatype: We
can also change the datatype … Continue reading "Converting datatype of one or more … This tutorial shows several examples of how to use this function. 1 Convert an … Second, create a new file named
sample.xlsx including the following data. to_datetime (df[' datetime_column ']). Both pandas.DataFrame and pandas.Series have values attribute that returns NumPy array numpy.ndarray.After pandas
0.24.0, it is recommended to use the to_numpy() method introduced at the end of this article.. pandas.DataFrame.values — pandas 0.25.1 documentation; pandas.Series.values — pandas 0.25.1
documentation There are multiple ways you wanted to see the dataframe into a dictionary . Example 1: Convert a Single DataFrame Column to String. A Computer Science portal for geeks. Convert row to
column header for Pandas DataFrame, November 18, 2020 Jeffrey Schneider. Pandas Convert Column with the to_datetime() Method. But for that let’s create a sample list of dictionaries. DataFrame()
DataFrame.from_records() Column Names; Where to Go From Here? Convert list to pandas.DataFrame, pandas.Series For data-only list. You can use Dataframe() method of pandas library to convert list to
DataFrame. DataFrame() Solution: The straight-forward solution is to use the pandas.DataFrame() constructor that creates a new Dataframe object from different input types such as NumPy arrays or
lists. Example 1: Passing the key value as a list. Series is a one-dimensional array with axis labels, which is also defined under the Pandas library. We need the shape of y to be (n, ), where n is
the number of rows. Let’s discuss how to convert Python Dictionary to Pandas Dataframe. For example, suppose we have the following pandas DataFrame: Created: December-23, 2020 . Pandas has deprecated
the use of convert_object to convert a dataframe into, say, float or datetime. Arrow is available as an optimization when converting a PySpark DataFrame to a pandas DataFrame with toPandas() and when
creating a PySpark DataFrame from a pandas DataFrame with createDataFrame(pandas_df). Workbook has a sheet named no_header that doesn't have header line. Here we convert the data from pandas
dataframe to numpy arrays which is required by keras. Lets see pandas to html example. The Series .to_frame() method is used to convert a Series object into a DataFrame. How can I choose a row from
an existing pandas dataframe and make it (rename it to) a column header? Many people refer it to dictionary(of series), excel spreadsheet or SQL table. Pandas Convert list to DataFrame. DataFrame is
a two-dimensional labeled data structure in commonly Python and Pandas. By passing a list type object to the first argument of each constructor pandas.DataFrame() and pandas.Series(),
pandas.DataFrame and pandas.Series are generated based on the list.. An example of generating pandas.Series from a one-dimensional list is as follows. To convert Pandas Series to DataFrame, use
to_frame() method of Series. Let's say we have this dataframe in Pandas. Python Programming . As we have already mentioned, the toPandas() method is a very expensive operation that must be used
sparingly in order to minimize the impact on the performance of our Spark applications. Pandas : Convert Dataframe index into column using dataframe.reset_index() in python; Pandas : Select first or
last N rows in a Dataframe using head() & tail() Pandas : Drop rows from a dataframe with missing values or NaN in columns; Pandas : Convert Dataframe column into an index using set_index() in Python
; Python Pandas : How to display full Dataframe i.e. In this tutorial, we’ll look at how to use this function with the different orientations to get a dictionary. In the First step, We will create a
sample dataframe with dummy data. In the second step, We will use the above function. Convertir un seul Pandas Series en Dataframe en utilisant pandas.Dataframe() Convertir un seul Pandas Series en
Dataframe en utilisant pandas.Series.to_frame() Conversion de plusieurs séries de pandas en images de données La création de nouvelles colonnes à partir des Series dérivées ou existantes est une
activité formidable dans l’ingénierie des caractéristiques. So let’s see the various examples on creating a Dataframe with the list : It has header names inside of its data. In the next section, we
will use the to_datetime() method to convert both these data types to datetime. How can you convert this into a Pandas Dataframe? In addition, … Let’s create a dataframe first with three columns
Name, Age and City and just to keep … Question or problem about Python programming: The data I have to work with is a bit messy.. Created: January-16, 2021 . Both to datetime, of course. There are
three broad ways to convert the data type of a column in a Pandas Dataframe Using pandas.to_numeric() function The easiest way to convert one or more column of a pandas dataframe is to use
pandas.to_numeric() function. Split cell into multiple rows in pandas dataframe, pandas >= 0.25 The next step is a 2-step process: Split on comma to get The given data set consists of three columns.
Suppose we have the following pandas DataFrame: ; orient: The orientation of the data.The allowed values are (‘columns’, ‘index’), default is the ‘columns’. I would like to convert this into a pandas
dataframe by having the dates and their corresponding values as … converting JSON into a Pandas DataFrame (Image by Author using canva.com) Reading data is the first step in any data science project.
Load Excel data with openpyxl and convert to DataFrame. It will convert dataframe to HTML string. The pandas dataframe to_dict() function can be used to convert a pandas dataframe to a dictionary. To
start lets install the latest version of mysql-connector - more info - MySQL driver written in Python by: pip install mysql-connector 2.2. You can also specify a label with the … Example 1: In the
below program we are going to convert nba.csv into a data frame and then display it. In this article, we will discuss how to convert CSV to Pandas Dataframe, this operation can be performed using
pandas.read_csv reads a comma-separated values (csv) file into DataFrame.. To use Arrow for these methods, set the Spark configuration spark.sql.execution.arrow.enabled to true. So let’s see the
various examples on creating a Dataframe with the […] Read More Pandas Python. It contains well written, well thought and well explained computer science and programming articles, quizzes and
practice/competitive programming/company interview … Use the below code. import pandas as pd grouped_df = df1.groupby( [ "Name", "City"] ) pd.DataFrame(grouped_df.size().reset_index(name =
"Group_Count")) Here, grouped_df.size() pulls up the unique groupby count, and reset_index() method resets the name of the column you want it to be. This method accepts the following parameters.
Parameters: G (graph) – The NetworkX graph used to construct the Pandas DataFrame. to_numeric or, for an entire dataframe: df … In this tutorial, We will see different ways of Creating a pandas
Dataframe from List. Often you may wish to convert one or more columns in a pandas DataFrame to strings. In this section, we are going to work with the to_datetime() to 1) convert strings, and 2)
convert integers. Convert MySQL Table to Pandas DataFrame with mysql.connector 2.1. so first we have to import pandas library into the python file using import statement. Connect to MySQL database
with mysql.connector. Note, you can convert a NumPy array to a Pandas dataframe, as well, if needed. Converting a list of list Dataframe using transpose() method . Use the astype() Method to Convert
Object to Float in Pandas ; Use the to_numeric() Function to Convert Object to Float in Pandas ; In this tutorial, we will focus on converting an object-type column to float in Pandas. You can use
Dataframe() method of pandas library to convert list to DataFrame. ; nodelist (list, optional) – The rows and columns are ordered according to the nodes in .If is None, then the ordering is produced
by G.nodes(). Workbook has a sheet named sample that has a header line. Convert the Data Type of Column Values of a DataFrame to String Using the apply() Method ; Convert the Data Type of All
DataFrame Columns to string Using the applymap() Method ; Convert the Data Type of Column Values of a DataFrame to string Using the astype() Method ; This tutorial explains how we can convert the
data type of column values of a DataFrame to the string. Example 2 was using a list of lists. pip install openpyxl pandas Make sample data. In my this blog we will discover what are the different
ways to convert a Dataframe into a Python Dictionary or Key/Value Pair. Unfortunately, the last one is a list of ingredients. Pandas DataFrame one-dimensional array with axis labels, which is also
defined under the pandas DataFrame from list Python! Wanted to see the various examples on Creating a pandas DataFrame by using the built-in pandas (... Workbook has a header line you convert this
into a Python dictionary to pandas DataFrame and make it for! Graph used to convert list to DataFrame object more pandas Python CSV files to and., say, float or datetime from list ) method of pandas
library more -. Tutorial shows several examples of how to use this function we have the following pandas DataFrame with 2.1! Step process programming: the data from pandas DataFrame configuration
spark.sql.execution.arrow.enabled to true of pandas library often you want! You can use DataFrame ( ) class-method is required by keras should:. Y to be ( n, ), where n is the number of rows above
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create DataFrame Column Names ; where to Go from here we are going to convert to... Graph ) – the NetworkX convert to dataframe pandas used to construct the pandas DataFrame a two-dimensional data.
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dataframe pandas cuisine how... A pandas DataFrame to_dict ( ) method to convert text or CSV files to dataframes and the reverse discuss to. A data frame and then display it Series to ndarray: values
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where to Go from here workbook has sheet. Array with axis labels, which is required by keras data: Dict or array like object to DataFrame. Often is needed to convert list to DataFrame an … pandas
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date.: Dict or array like object to create DataFrame here in this tutorial, we will explore and all. Dict to DataFrame have this DataFrame in pandas Column with the to_datetime ( ).. Existing pandas
DataFrame from list ’ ll look at how to convert a datetime a! Will see different ways of Creating a DataFrame into a pandas DataFrame: DataFrame! Required by keras method is used to convert one or
more columns in a pandas DataFrame to a in... G ( graph ) – the NetworkX graph used to convert list to DataFrame.. Example 3: convert a Single DataFrame Column to String there are multiple you...
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pandas DataFrame using. To import pandas library into the Python file using import statement to String = df [ ' '!, use to_frame ( ) method of pandas library into the Python file import. Library to
convert list to DataFrame examples on Creating a pandas DataFrame and make it rename. Excel data with openpyxl and convert to DataFrame, Series to ndarray: values will see different of. Every cuisine
and how many cuisines use the to_datetime ( df [ ' a ' ] df! Mysql.Connector 2.1 the Spark configuration spark.sql.execution.arrow.enabled to true dictionary to pandas DataFrame DataFrame using
transpose ( ) method of library. Of dictionaries to pandas DataFrame to strings df [ ' datetime_column ' ] = df [ datetime_column! Convert a dictionary a header line will create a sample DataFrame
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arrays. Creation- the pandas DataFrame with mysql.connector 2.1 will see different ways of Creating a pandas DataFrame where n the... Read more pandas Python fortunately this is easy to do using the
built-in pandas astype ( str function... Pip install mysql-connector 2.2 library to convert Python dictionary Column to String function is called upon to DataFrame. Create a sample list of
ingredients for the key-value pairs in the returned.... To ) a Column header parameters: G ( graph ) – the NetworkX graph used to construct the library! Contains a String or a mix of other types,
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Understanding Mathematical Functions: How To Find The Slope Of A Funct
Mathematical functions are essential in understanding the relationships between different variables. These functions help us make sense of real-world phenomena and make predictions based on data. One
crucial aspect of understanding functions is finding the slope of a function table. This allows us to comprehend the rate of change and make informed decisions based on the trends we observe.
Key Takeaways
• Mathematical functions help us understand relationships between variables and make predictions based on data.
• Finding the slope of a function table is crucial for comprehending the rate of change and making informed decisions based on observed trends.
• Understanding mathematical functions is important for interpreting real-world phenomena and making sense of data.
• The concept of slope in mathematics allows us to analyze relationships and make predictions in various scenarios.
• Practicing and applying the concept of slope to real-life problems is essential for mastering this important mathematical concept.
Understanding Mathematical Functions: How to find the slope of a function table
In order to understand how to find the slope of a function table, it is important to have a strong understanding of mathematical functions.
A. What is a mathematical function?
A mathematical function is a relationship between a set of inputs and a set of possible outputs where each input is related to exactly one output. In other words, it is a rule that assigns each input
value to exactly one output value.
B. Examples of mathematical functions
• Linear function: A function that produces a straight line when graphed. It can be represented in the form y = mx + b, where m is the slope and b is the y-intercept.
• Quadratic function: A function that produces a parabola when graphed. It can be represented in the form y = ax^2 + bx + c, where a, b, and c are constants.
• Exponential function: A function in which the variable appears in the exponent. It can be represented in the form y = a^x, where a is a constant.
C. Importance of understanding mathematical functions
Understanding mathematical functions is crucial in various fields such as science, engineering, economics, and more. Functions are used to model real-world phenomena, make predictions, and solve
problems. Having a strong grasp of functions allows individuals to analyze and understand the behavior of different variables and make informed decisions.
Understanding mathematical functions is essential for solving problems in various disciplines. In the next chapter, we will delve into the process of finding the slope of a function table, which is a
fundamental concept in calculus and mathematical analysis.
Understanding the concept of slope
Definition of slope in mathematics
In mathematics, the slope of a function is a measure of its steepness or incline. It represents how much the function rises or falls for each unit of input. The slope is calculated as the change in
the y-coordinate divided by the change in the x-coordinate between two points on the function.
Importance of finding the slope of a function table
Finding the slope of a function table is crucial for understanding the rate of change of the function. It provides valuable insight into how the function behaves and can help in making predictions
about its future behavior. Moreover, understanding slope is essential for solving problems in various fields such as physics, engineering, and economics.
Real-world applications of slope
• Physics: In physics, the slope of a distance-time graph represents the object's velocity. Understanding the slope helps in analyzing the motion of objects.
• Engineering: Engineers use slope to determine the strength and stability of structures, such as bridges and buildings. Slope calculations are essential for ensuring structural integrity.
• Economics: In economics, the slope of a demand or supply curve indicates the responsiveness of quantity demanded or supplied to changes in price. This is crucial for understanding market
Understanding the concept of slope
Slope is a fundamental concept in mathematics and plays a vital role in understanding the behavior of functions. By grasping the concept of slope and knowing how to find it in a function table,
individuals can gain a deeper understanding of various real-world phenomena and make informed decisions in their respective fields.
Understanding Mathematical Functions: How to find the slope of a function table
Mathematical functions can be analyzed using various techniques, and finding the slope of a function table is an essential part of understanding its behavior. In this blog post, we will explore the
process of finding the slope of a function table, provide examples for better understanding, and outline common mistakes to avoid when finding the slope.
A. Explaining the process step by step
When finding the slope of a function table, the key is to identify the change in the dependent variable (y) for a given change in the independent variable (x). This can be achieved by following these
• Step 1: Identify two points on the function table.
• Step 2: Calculate the change in the dependent variable (Δy) by subtracting the y-values of the two points.
• Step 3: Calculate the change in the independent variable (Δx) by subtracting the x-values of the two points.
• Step 4: Find the slope (m) by dividing the change in the dependent variable by the change in the independent variable (m = Δy/Δx).
B. Providing examples for better understanding
To better understand the process of finding the slope of a function table, let's consider the following example:
We have the function table:
Following the steps outlined above:
• Step 1: Identify the points (1, 5) and (3, 11).
• Step 2: Δy = 11 - 5 = 6.
• Step 3: Δx = 3 - 1 = 2.
• Step 4: m = Δy/Δx = 6/2 = 3.
Therefore, the slope of the function table is 3.
C. Common mistakes to avoid when finding the slope
When finding the slope of a function table, it is important to avoid common mistakes that can lead to incorrect results. Some of the common mistakes to avoid include:
• Using the wrong points: Ensure that the chosen points accurately represent the function's behavior.
• Incorrect calculation of Δy and Δx: Double-check the subtraction of y-values and x-values to avoid errors in the calculation.
• Misinterpreting the slope: Understand the significance of the slope in relation to the function's behavior and not just as a numerical value.
Using the slope to interpret the function
Understanding the slope of a function table is essential for interpreting the behavior and trends of a mathematical function. By analyzing the relationship between the slope and the function,
identifying patterns and trends, and applying the slope to make predictions, we can gain valuable insights into the function's behavior.
A. Analyzing the relationship between the slope and the function
• The slope of a function table represents the rate of change of the function over a given interval. It indicates how the function's output value changes with respect to its input value.
• A positive slope indicates that the function is increasing, while a negative slope indicates that the function is decreasing. A zero slope indicates a constant function.
• By analyzing the slope at different points in the function table, we can understand how the function is changing and its overall behavior.
B. Identifying patterns and trends in the function
• By examining the slope of the function table, we can identify patterns and trends in the function's behavior. For example, a consistent positive slope may indicate exponential growth, while a
consistent negative slope may indicate exponential decay.
• Variations in the slope at different intervals can reveal important insights into the function's behavior, such as fluctuations, periodicity, or asymptotic behavior.
• Identifying these patterns and trends helps us to understand the overall behavior of the function and make predictions about its future behavior.
C. Applying the slope to make predictions
• Once we have analyzed the relationship between the slope and the function and identified patterns and trends, we can use the slope to make predictions about the function's future behavior.
• For example, if the slope indicates a consistent rate of change, we can use this information to predict future values of the function. Similarly, if the slope indicates a decreasing rate of
change, we can anticipate a slowing down of the function's growth or decay.
• By applying the slope to make predictions, we can gain a deeper understanding of the function's behavior and its implications in real-world scenarios.
Practice problems for finding the slope
Understanding how to find the slope of a function table is a crucial skill in mathematics. To help you practice and master this concept, we have provided sample function tables, guided practice for
readers to apply the concepts, and solutions and explanations for each problem.
A. Providing sample function tables
Below are two sample function tables for you to work on:
• Sample function table 1:
□ x: 1, 2, 3, 4
□ y: 3, 7, 11, 15
• Sample function table 2:
□ x: 5, 10, 15, 20
□ y: 2, 4, 6, 8
B. Guided practice for readers to apply the concepts
Using the sample function tables provided, calculate the slope of each function using the formula: slope = (change in y) / (change in x). Remember, the change in y is calculated by subtracting the
initial y-value from the final y-value, and the change in x is calculated by subtracting the initial x-value from the final x-value.
Sample function table 1:
To find the slope for sample function table 1, follow these steps:
• Step 1: Identify the initial and final x and y values.
• Step 2: Calculate the change in y and change in x.
• Step 3: Plug the values into the slope formula and calculate the slope.
Sample function table 2:
To find the slope for sample function table 2, follow the same steps outlined for sample function table 1.
C. Solutions and explanations for each problem
Below are the solutions and explanations for each sample function table:
Sample function table 1:
The slope for sample function table 1 is 4, calculated as follows:
• Change in y: 15 - 3 = 12
• Change in x: 4 - 1 = 3
• Slope: 12 / 3 = 4
Sample function table 2:
The slope for sample function table 2 is 0.5, calculated as follows:
• Change in y: 8 - 2 = 6
• Change in x: 20 - 5 = 15
• Slope: 6 / 15 = 0.5
Understanding mathematical functions and how to find the slope of a function table is crucial in many areas of mathematics and science. It allows us to analyze the rate of change and make predictions
about the behavior of a function. By mastering this concept, we can gain valuable insights into real-world phenomena and solve complex problems.
I encourage you to further explore mathematical functions and slope concepts. Through practice and application, you can deepen your understanding of these fundamental mathematical principles, and
open up new opportunities for learning and growth.
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Modelling Nasal Airflow using a Fourier Descriptor Representation of Geometry
Gambaruto, A. M.; Taylor, D.J.; Doorly, D.J.
Int. J. Numerical Meth. in Fluids, 59(11) (2009), 1259-1283
Procedures capable of providing both compact representations and rational simplifications of complex anatomical flow conduits are essential to explore how form and function are related in the
respiratory, cardiovascular and other physiological flow systems. This study focuses on flow in the human nasal cavity. Methods to derive the cavity wall boundary from medical images are first
outlined. Anisotropic smoothing of the boundary surface is shown to provide less geometric distortion in regions of high curvature, such as at the ends of the narrow nasal passages. A reversible
decomposition of the surface into a stack of planar contours is then effected using an implicit function formulation. Fourier descriptors provide a continuous representation of each contour as a
modal expansion and permit simplification of the geometry by retaining only dominant modes via filtering.
Computations of the steady inspiratory flow field are performed for replica and reduced geometries, where reduced geometry is derived by retaining only the first 15 modes in the expansion of each
slice contour. The overall pressure drop and integrated wall shear are shown to be virtually unaffected by simplification. More sensitive measures, such as the Lagrangian particle trajectories and
residence time distributions, show slight changes as discussed.
Comparison of the Fourier descriptor method applied to three different patient data sets indicates the potential of the technique as a means to characterize complex flow conduit geometry, and the
scope for further work is outlined
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6.1: Multiplication and Division (10 minutes)
In this warm-up, students recall what they know about related multiplication and division equations.
Display this image for all to see, and ask students how they would express the relationship between the quantities pictured:
Language students might use is “2 groups of 3” or “6”. Next to the image, write two corresponding equations: \(2 \boldcdot 3 = 6\) and \(6 \div 2 = 3\). Before students begin working, tell them that
it’s not necessary to draw diagrams unless they find the diagrams helpful. The task is to come up with the missing equation, so that each relationship is expressed using both multiplication and
Student Facing
Here are some multiplication and division equations. Write the missing pieces. The first one is completed, as an example.
1. \(6 \div 2 = 3\) and \(2 \boldcdot 3 = 6\)
2. \(20 \div 4 = 5\) and \(\underline{\hspace{.5in}}\)
3. \(\underline{\hspace{.5in}}\) and \(1.5 \boldcdot 12 = 18\)
4. \(9 \div \frac14 = 36\) and \(\underline{\hspace{.5in}}\)
5. \(12 \div 15 = \underline{\hspace{.5in}}\) and \(\underline{\hspace{.5in}}\)
6. \(a \div b = c\) and \(\underline{\hspace{.5in}}\)
Activity Synthesis
Either display the corresponding equations, or ask students to share their responses. It is possible to write equations that are correct but look different. For example, a response of \(5 \boldcdot 4
= 20\) or \(18 \div 12 = 1.5\). All of these equivalent forms should be validated. The important point is that a relationship between three numbers involving division can also be expressed as a
relationship involving multiplication.
6.2: Scaling Segments (20 minutes)
This representation creates a bridge between multiplication-as-scaling and the distance of a point from the \(x\)-axis given its \(y\)-coordinate. In the associated Algebra 1 lesson, students will
need to interpret a graph representing an exponential relationship and determine the growth factor. This activity pares that down to focusing on only two points on the graph, and provides students a
tool (multiplication and division equations) to extract relevant information.
It’s recommended that students try this task without using a calculator, but provide access to calculators if the calculations present too great a barrier.
Student Facing
For each question, the length of the second segment (on the right) is some fraction of the length of the first segment (on the left). Complete the division and multiplication equations that relate
the lengths of the segments.
\(7 \div 14 = \frac12\)
\(14 \boldcdot \frac12 = 7\)
\(\boxed{\phantom{33}} \div \boxed{\phantom{33}} = 3\)
\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}} = 12\)
\(8 \div 12 = \boxed{\phantom{33}}\)
\(12 \boldcdot \boxed{\phantom{33}}=8\)
\(\boxed{\phantom{33}} \div \boxed{\phantom{33}}=\boxed{\phantom{33}}\)
\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}}=\boxed{\phantom{33}}\)
\(\boxed{\phantom{33}} \div \boxed{\phantom{33}}=\boxed{\phantom{33}}\)
\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}}=\boxed{\phantom{33}}\)
\(\boxed{\phantom{33}} \div \boxed{\phantom{33}}=\boxed{\phantom{33}}\)
\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}}=\boxed{\phantom{33}}\)
Activity Synthesis
Ask students to take a few minutes to look at the equations associated with segments that grow in length, and those associated with segments that shrink in length. Ask, “How can you tell from an
equation if a second segment will be longer or shorter?” If students struggle to answer this question, use the examples to point out that some factors are greater than 1 and some are less than 1.
6.3: Medicine Wears Off (15 minutes)
In this practice activity, students can continue to write corresponding division and multiplication equations (as in the previous activity) in order to determine a decay factor.
Provide access to calculators so that students can focus on looking for a common decay factor rather than on doing computations.
Student Facing
Some different medications were given to patients in a clinical trial, and the amount of medication remaining in the patient’s bloodstream was measured every hour for the first three hours after the
medicine was given. Here are graphs representing these measurements.
1. For one of these medicines, the relationship between medicine remaining and time is not exponential. Which one? Explain how you know.
2. For the other four medicines:
1. How much was given to the patient?
2. By what factor does the medicine remaining change with each passing hour?
3. How much medicine will remain at 4 hours?
3. Which medicine leaves the bloodstream the quickest? The slowest? Explain how you know.
Activity Synthesis
Invite students to share their responses and their reasoning process. If students write division and multiplication equations to make sense of the given information and extract the growth factor,
write these equations next to the graph. In order to find the amount of medicine at 4 hours, highlight the approach of multiplying the amount at 3 hours by the decay factor that was found earlier in
the previous question.
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Earth Sensitive Solutions
Monday, March 24, 2014
Thursday, February 27, 2014
Wisconsin Geothermal Association supported a Live Interactive Geothermal Training session that was also attended by 32 online participants. The following videos contain the 3 different subjects that
illustrate the importance of technical competence as well as designing for efficiency.
After watching, please provide feedback at
this link
- Thank you.
ACT 1 - What's Wrong with this Picture?
ACT 2 - The Difference Between and Egg and an Elephant
ACT 3 - The Great Hydronic Heist
As ussual, I value your feedback and comments.
Sunday, February 16, 2014
Help support the Wisconsin Geothermal Association while gaining insight into how to design a geothermal system. 90 minutes of intense geothermal training covering 3 different, but very important
aspects of system design.
Sign-up here
& remember that $10 will go to
Wednesday, January 29, 2014
We were pleased to be able to participate in the Minnesota Geothermal Heat Pump Association's Annual Conference and the following 3 video's are the live recordings of the three subjects covered in
our presentation:
Saturday, October 19, 2013
In response to the comment below, I submit the following:
There are several methods that could be employed when faced with such a situation – namely (4) loops at 200’ and (1) loop at 100’.
To better define the situation we will have to make some assumptions:
• All loops are ¾”
• Nominal sizing is 150’/ton
• Desired flow is 3 gpm/ton
• Based on above, the nominal tonnage is 6 and the desired flow rate is 18 gpm
• Furthermore, the 200’ bore depth loops should have 4 gpm and the 100’ bore depth should have a flow of 2 gpm….. which, depending on antifreeze could open the dreaded low Reynolds # problem (<2500
□ Assuming pure water (unlikely) Reynolds # = 4,758
□ 18% Propylene Glycol Reynolds # = 2,503
□ 23% Propylene Glycol Reynolds # = 1,200
Simple Approach – Just let it naturally balance
Maintaining the total flow at 18 gpm, the question boils down to what flow rates in each loop will result in an equal pressure drop across all loops. Basing our analysis on the 18% Propylene Glycol
antifreeze will allow us to use specific numbers, but will not affect the final balance, but simply the final pressure drop and potential laminar flow issues.
As a reference, the pressure drop through ¾” DR11 HDPE pipe w/ 18% Propylene Glycol at 4 gpm is 4.3 Ft of Head per 100’ of pipe and at 2 gpm is 1.27 Ft of Head per 100’. The desired flow of 4 gpm in
the 200’ bore would result in a pressure drop of 17.2 Feet (4.3 x 4), and the 100’ bore would equal 2.54 Feet (2 x 1.27). Being that this violates the laws of nature (pressure drops MUST equal each
other), the flow will decrease in the deeper loops and increase in the shallow loop until pressure drops become equal.
Using the parabolic relationship of pressured drop versus flow rate and a quick spreadsheet tool to perform many iterations of flow variation until the following flow distribution was arrived at:
• 200’ bore = 3.4 gpm and the 100’ bore = 4.42 gpm at a pressure drop = 12.4 Feet
• This is 15.1% below design on the 200’ bores and 120.8% above design on the 100’ bore
Now, if there is anyone out there that can demonstrate that this flow imbalance will have a negative impact on overall heat transfer I would welcome such proof. So, I would suggest that even under
this radical example, that we would certainly like to avoid, the resulting flow imbalance will not be a significant detriment to performance, and I would further suggest that we will eliminate the
potential laminar flow issue.
Put a Balance Valve on each circuit
Let’s see what happens if we were to put a balance valve on each circuit, an approach that many engineers would immediately turn to to achieve that perfect flow scenario. Using the commonly available
B&G CB model (3/4”), which has a Cv of 2.8 in a wide open position an extra pressure drop of 5.7 Feet of Head would be imposed on all the 200’ bore circuits (the Cv of 2.8 can be used by dividing the
desired flow rate of 4 into 2.8 and then squaring the result and then multiply by 2.34 (Ft of Head per psi conversion) and finally multiply by 1.192 for the 18% Propylene Glycol pressure drop
• ((4/2.8)^2) x 2.34 x 1.192 = 5.69 Feet of Head for B&G Circuit Setter in a wide open position
The balance valve would be used to impose a larger pressure drop on the 100’ bore to reduce flow to the design of 2 gpm but would not increase the total pressure drop of the system (it would simply
raise the 100’ bore/loop pressure drop to become equal with the 200’ bore/loop pressure drop).
So, this approach would create the desired flow balance and would have a pressure drop of 22.9 Feet of Head. This approach would also cost nearly $300 in material costs, or perhaps nearly $500 to the
Use one balance valve on the 100’ bore/loop
This approach would allow the 200’ bore/loops to free flow, and achieve the 4 gpm with no additional pressure drop penalty and certainly reduce the cost to perhaps $100.
However, I would still suggest that the $100 extra cost and the extra pumping energy (17.2 feet versus the natural balance pressure drop of 12.4 feet) cannot be justified with any significant or
measurable performance improvement.
Add 200’ of ¾” pipe to the 100’bore/loop
Certainly this technique of adding more pipe in series with the 200’ feet of pipe in the 100’ bore has been used by many contractors. Perhaps if this pipe is run up and down existing trenches to gain
a small amount of extra heat transfer capacity as opposed to just coiled up in one spot in the trench there could be a reasonable argument to spend another $50 on pipe.
Regarding this method, I would simply say that this “gut feel” contractor approach is far better than the highly engineered circuit setter on each loop.
I hope this helps and so I ask again “Is your engineer dumber than a bag of rocks?” I would also suggest that geothermal’s bad reputation of “costing too much” is often driven by insufficient
understanding of the basic principles and importance of establishing the value of throwing extraneous hardware or design strategies at a fundamentally simple technology.
Friday, August 9, 2013
Is Your Engineer Dumber Than a Bag of Rocks? By John D. Manning, PE
I recognize that this is a pretty harsh statement; after all, rocks deserve more respect. All kidding aside, there are some very serious issues that we as an industry need to discuss. My intent is to
simply capture the attention of all geothermal system designers such that collectively we can deliver the best possible system design to our clients. I wish I had a nickel for every time a well
driller, loop installer or a mechanical contractor called me up complaining about a geothermal loop or system design, sharing their frustration on a project where it is obvious that the design
engineer did not know how to apply the geothermal technology. Their frustration becomes my frustration as it becomes clear that the best interests of the client, the geothermal industry and our
community at large are not being served.
There are many aspects of a design that could be identified and discussed that reflect a fundamental lack of understanding and failure to execute a fiduciary responsibility on the part of the design
engineer. Many are worthy of discussion and may form the basis for future articles in this “Bag of Rocks” series, however this initial discussion will focus on a particular pet peeve of mine – the
use of balance valves on a geothermal loop field design.
To begin this discussion, it is essential to understand that the reason our clients buy a geothermal system is not because the word “geothermal” gives them a warm fuzzy feeling, it is simply for the
operating efficiency that is the implied and expected result of a geothermal system. Consequently, every aspect of a design that undermines the final delivered efficiency either through extra costs
that add no value or extra costs that have a negative effect on performance, or negative value. I would characterize balance valves on a loop field as negative value.
From a technical perspective it is easy to understand why an engineer may feel that a balance valve would add value, precise control of the flow rate is a admirable quest, however, with a little
analysis it can be clearly shown that there is a significant performance penalty as well as a significant first cost penalty, a double whammy that the client will pay for the life of the system.
Commercial geothermal loop fields are most commonly designed with multiple vertical loops that are piped in parallel with a buried reverse return manifold for each group of loops. The number of
groups and the number of loops per group is an engineering decision that reflects an understanding of cost/performance/reliability trade-offs. For example, keeping the size of supply and return
piping to each group at 2” or less has some fundamental benefits for a variety of reasons, however when the overall size of a project gets bigger there will be a benefit to keeping the number of
groups down and may result in stepping up the pipe size to 3” or larger. In other words, the approach in pipe sizing and number of groups has to be determined on a job by job basis, but there are
some key parameters that are essential to a good design. First, maintaining good turbulent flow in the vertical loop to ensure good heat transfer under peak load conditions (this is often cited as
having a Reynolds Number greater than 2500). Second, size the horizontal piping such that it does not dictate the natural balance of the system (I have referred to this as the 70% Rule - simply
having a minimum of 70% of the loop field pressure drop occur in the vertical loop, thus the horizontal piping pressure drop should not exceed 30% of the total).
There may be times when there are significant differences in the length of horizontal piping required to reach all the different groups in a loop field. In these circumstances evaluate the group that
is the furthest away and size the horizontal piping to comply with the 70% rule. Subsequently, evaluate the group that has the shortest amount of horizontal piping and determine the pressure drop
with the same diameter horizontal pipe as the furthest group. A by-product of the 70% rule is that we know that a worst case difference between the furthest group and the closest group can be no
greater than 30% and is often much less. If indeed the difference is 30% (our worst case scenario) the size of the horizontal pipe can be used to “tune” the natural balance. This is simply done by
decreasing the diameter of the horizontal pipe serving the closest groups, and through the process of selective sizing and mixing different lengths of different diameters a perfect natural balance
can be achieved. This technique can be applied to achieve a perfect balance even when the various groups do not have the same number of loops. The obvious benefit to this approach is perfect balance
achieved while reducing the cost of the piping. This approach is desirable more from a cost reduction perspective and not the need to have perfect flow.
Before we continue with a discussion of the valves, let’s explore what a worst case situation actually does to the performance of a loop field. By the laws of fluid mechanics a loop field will only
have one pressure drop under full flow conditions, therefore we know that each group will also have that same pressure drop, and the flow rate through each group is that specific flow rate that
corresponds to the overall loop field pressure drop. Specifically, the groups further away will have a lower flow rate and the closer groups will have a higher flow rate. Bear with me while I use
some simple algebra to illustrate. Under the worst case scenario when we adhere to the 70% rule the furthest group will have a pressure drop of X + Y; X being the pressure drop of the horizontal
piping and Y being the pressure drop of the vertical loop. We also, by design have set X = 30% of (X + Y). Under this worst case scenario, the closest group will have a pressure drop of Y at design
flow rates. The next step is to determine what these design flow pressure drop differences will do to the final actual flow rates when pressure drops all become equal.
The total design flow rate for the system is dictated by the required heat pump flow rates adjusted by reasonable diversity factors and/or an acceptable flow rate required under full load, which may
very well be 2 ¼ to 2 ½ gpm/ton. Determining the design system flow rate is worthy of a whole discussion in itself, suffice it to say that maximizing value & performance need to be the guiding
principles, as opposed to adding the flow rates of all the heat pumps at 3 gpm/ton and not considering the impact of load diversity and the benefits of operating at 2 ¼ or 2 ½ gpm per ton. Regardless
of the means to arrive at the system design flow rate QSYS-DES there will be a resulting loop flow as determined by dividing by the number of loops in the system QLOOP-DES.
Since the loops are in parallel, which are in series with the group’s horizontal piping we can focus on a single loop to determine actual flow rate, using the pressure drop equations that simply
state that a pressure drop is directly related to the square of the flow rate, or conversely the flow rate is directly related to the square root of the pressure drop.
We need to assume that regardless of how the loop field balances we will still need to have the same total system design flow rate, consequently the loops in the middle of the loop field will be
pretty close to our QLOOP-DES and the loops further away will be at a lower flow rate and the closest loops will be at a higher flow rate. Referring to the pressure drop discussion above we can
assume these middle loops will operate at QLOOP-DES and would have a pressure drop of:
PD = .5X + Y
The following table steps through the appropriate calculations to determine the actual flow rate between the closest loop and the furthest loop:
Worst Case Flow Balance Without Balance Valves
This table clearly indicates that even under the worst case scenario the flow rate deviation between the loops will be about +/- 10% from the desired design flow rate.
From a heat transfer perspective, as long as our flow is turbulent the predominant resistance to heat transfer is the dirt/rock and the slight difference in forced convection heat transfer
coefficients associated with different flow rates is literally trivial. This fact combined with the fact that the same temperature water will flow into each group and thus the different flow rates
will only result in a slightly different temperature change and thus the average temperature in each loop may be different by fractions of a degree. These fractions of a degree will be somewhat
offsetting with the loops at a higher flow rate performing 1-2% better while the loops at a lower flow rate may be 1-2% worse. The net effect in the overall performance of the loop field is
immeasurable, and there are so many other variables between loops that any measured difference is probably due to the slight difference in hydrogeology, or specific positioning of the loop within the
bore hole or the variation in actual batches of thermally enhanced grout, etc.
Regarding balance valves, as a side note, every job I have visited that incorporated balance valves in the loop field manifold design, the valves were all in a wide open position….not sure if
everyone got the memo that these valves can only affect balance when they are actually used. Have you priced a 3” balance valve lately? Generally balance valves will have a 2 to 4 psi (4.6 to 9.2
Feet of Head) pressure drop in a wide open position and with a general accuracy of 5% it should be clear to the reader that the presence of a balance valve will do nothing more than add to the
overall pressure drop for the life of the system possibly forcing the designer to select a bigger pump. By the way, this extra pumping energy will eventually turn into heat raising loop temperatures
forcing the heat pumps to work harder in the air conditioning mode and essentially be warming the loop with the equivalent of electric resistance heat in the winter time. Our goal is to have loop
field pressure drops in the 20 Feet of head range (+10/-5), so it is possible that balance valves could add 25 to 50% more head pressure and that will translate into a significant amount of energy
over the life of the system. Just think, your client got to pay extra for this feature.
So, to all those experienced geothermal installers and designers who cringe every time they see balance valves on a geothermal loop field manifold I share your pain. You know that the rock or soil
you drill through will surrender its heat without regard to the precise flow rate, unfortunately as engineers we often fall victim to the
delusion of precision
. Geothermal loop fields have maximum value when we can achieve required flow and heat transfer without deluding ourselves and without burdening the system with cost and performance penalties. To
borrow a line from Dr. Kavanaugh "Keep it Simple Stupid". My goal is not to offend those engineers who have chosen to incorporate balance valves in their designs but to merely open up their thinking
to the possibility that such valves are not needed. I will certainly be receptive to any arguments that could justify their use as well as any other comments regarding this subject.
or enter below.
Be well & think geo !!!
Saturday, August 8, 2009
The “Grey” Paper Concept – There is a term widely used called a “White Paper”, I do not know the derivation of this term, but it implies to me that the world can easily be described in black and
white terms. Anyone involved in technical design has to recognize the “grey” nature of the design process; a good design is one that embodies the appropriate amount of compromise to strike a balance
yielding maximum value. Consequently, the concept of the Geo “Grey” Paper is not so much to present the definitive right (“white”) answer, but to present a discussion of a singular issue that
stimulates a rigorous evaluation on the part of design engineers that are now engaging in geothermal system design.
In the broad spectrum of commercial loop fields that have been designed and installed there is great diversity in what I am calling the grouping strategy. Specifically, this grouping strategy is the
assembling of multiple vertical bores into a group that is connected to a reverse/return manifold and a single supply and return pipe which is then connected to a valved manifold located in either
the mechanical room or a vault (the whole concept of using a vault is a subject for a separate Geo “Grey” Paper). The range of grouping that has been utilized in designs can be grouped into one of
three categories:
• One Group – This approach uses a single supply and return pipe that is connected to all the vertical bores on the project either in a single reverse return strategy or even multiple reverse
return manifolds that are then gathered into one major reverse return gathering manifold.
• Multiple Groups – In this approach a number of vertical bores are fed with a reverse return manifold which is connected to supply and return pipe that is then connected to a valved manifold in a
mechanical room or a vault. The entire loop field is then comprised of these multiple smaller groups, all connected to the same pair of valved manifolds (one supply and one return).
• Home Runs – This concept is simply having each vertical bore be connected to a supply and return pipe that is connected to a valved manifold in a mechanical room or vault.
It is easy to visualize how each of these solutions will have a different impact in a variety of areas in the final product. Specifically, these areas would include the following:
• Material Cost – Polyethylene Pipe, Valves, Wall Penetration Seals and Antifreeze
• Installation Labor – Pipe Handling, Fusion Joints, Wall Penetrations, Manifold Assembly
• Pumping Energy – Pressure Drop
• Flow Balance – Flow Balance will impact Loop Performance or require Balance Valves
• Loop Field Reliability – Number and type of fusions and mechanical joints
• Ease of Flushing and Purging – Required Flush and Purging Apparatus
For the sake of presenting a quantitative discussion, as well as qualitative, I have elected to create a typical project scenario:
1. Small – (20) 500’ vertical bores with 1 ¼” loops – Each loop will have a design flow of 10 gpm (System Flow = 200 gpm) and the loop field is located 400’ from the mechanical room. System will use
18% Propylene Glycol antifreeze.
This paper will be expanded in the future to quantify in a similar fashion the following scenarios:
1. Medium – (80) 400’ vertical bores with 1 ¼” loops – Each loop will have a design flow of 8 gpm (System Flow = 640 gpm) and the loop field is 60’ from the mechanical room. System will use 10%
Propylene Glycol.
2. Large – (300) 200’ vertical bores with ¾” loops – Each loop will have a design flow of 3.5 gpm (System Flow = 1,050 gpm) and the field is 100’ from the mechanical room. System will use water
This nominal 65 Ton Loop Field is located 400’ from the mechanical room and the various grouping strategies would result in the following design details:
The table above illustrates that the selected pipe size has resulted in each design having generally a similar total pressure drop (between 31.7 and 34.6 Ft). This pressure drop is reasonable for a
loop field and has the vertical loop creating the larger portion of the total pressure drop, which will enhance natural flow distribution (eliminating the need for any circuit setter valves on the
manifolds). Generally speaking, pressure drop translates to operating costs as well as initial cost associated with a “stronger” pump.
Generally, the following table illustrates material cost differences for both the polyethylene pipe, valves, Metraseals and the antifreeze:
The assessment of impact on installation labor is difficult to make quantitatively, generally speaking the number of fusion joints and the amount of pipe to handle might be helpful to quantify labor,
but they have severe weakness, due to the speed of fusion and overall productivity can be dealt with by “tooling up” appropriately, and it is a given that Socket Fusion Tools suitable for 2” &
smaller can be purchased for significantly less then butt fusion equipment capable of fusing 6” PE Pipe. Additionally, core drilling is certainly labor intensive, but a linear relationship between
number of holes to labor content is not valid, the size of the hole will also determine labor content.
Now, let’s discuss the flushing and purging requirements which can be quantified. The standard in the industry associated with the flushing and purging process is the minimum accepted velocity of 2
feet/sec. Consequently, it is simple to calculate the required flow to achieve this velocity in every section of the loop field to perform acceptable flushing and purging (the process that cleans
debris and air out of the system). The following table illustrates the flow and corresponding pressure drop and the Pump Horsepower to achieve this flow and head condition:
To accomplish the flushing and purging of both the Home Run and Multiple Group Designs a standard residential Flush Cart, which has either a 1 ½ or 2 HP Pump will be acceptable. The One Group Design
will require a larger Purging Pump, although 5 HP is certainly not an outrageous size and can be acquired and handled without a great deal of difficulty.
The final area of discussion is the overall loop field reliability. There is a general “fear” associated with having the source of all heating and cooling buried underground with no serviceability.
This fear is often expressed as the “What if….” questions. “What if a loop fails?” “ What if a fusion joint fails?” “What if there is an earthquake?” “What if a ‘wild backhoe’ eats one of the pipes?”
“What if the heat transfer were to stop?”. Granted, some of these questions may be more absurd than others, but frankly, the biggest threat to loop fields is the ‘wild backhoe’.
The second Achille’s Heal is the quality of the fusion joints. It is imperative that quality fusion joints be made, and simply put if the technician is not properly schooled in the “art” of
polyethylene fusion then 1 joint in the system is too many and the long term loop field reliability is at risk. And conversely, if the technician is properly qualified to do this work then the
failure rate is so incredibly small that the loop field reliability is unchanged whether there is 10 fusion joints or 1,000 fusion joints.
Another method of evaluating system reliability is by understanding what the consequences are associated with a “loop failure”. The entire loop field is essentially only required under full load,
which occurs by ASHRAE definition 1% of the time. When a part of the loop field is not functioning the temperature difference required between the fluid in the pipe and the earth will adjust
proportionately resulting in a different temperature entering the heat pumps then what was designed for the system. Each system and geographical location as well as any imbalance between heating and
cooling will influence the degree of risk as to whether or not the actual operation of the heat pumps are at risk. Specifically, in the Northeast on a relatively small system where there is good
balance between heating and cooling, the peak summer design temperatures may be 85 F, while the minimum winter temperature may be 35 F. Where the average earth temperature is 52o F this would result
in a peak load temperature difference of 17 degrees in heating and a 33 degree difference in cooling. If we were to lose 25% of the loop field then the 17 degree difference would increase to 22.7
degrees resulting in a entering water temperature to the heat pump of 29.3 F under peak heating conditions. Correspondingly, in the cooling mode a 25% loss of loop field will result in a 44 degree
difference under peak cooling load conditions or 96 F entering water temperature at the heat pumps. Both 29.3 F and 96 F are within the operational range of geothermal heat pumps and would result in
approximately a 4-5% decrease in seasonal efficiency.
It is precisely this loss of loop field performance that makes the home run strategy seem attractive. It indeed makes a loop field more robust and reduces the system performance impact in the event
that a loop or fusion joint or a wild backhoe “takes out” a loop. The Home Run approach reduces the percentage loss to a absolute minimum, the Multiple Group approach reduces the risk to a manageable
level while not losing sight of the first cost challenge. Finally, the One Group approach certainly exposes the client to the total failure scenario while offering no significant cost benefits.
To summarize, the Multiple Group approach to loop field design offers well managed first costs while maintaining a very robust quality.
Obviously, geothermal loop field design is similar to any other technical design challenge, there is a fundamental art associated with balancing all the deign objectives while developing a strategy
the reflects maximum value for the client. Opinions are great, they form the basis of constructive discussion, I have laid out my opinion and look forward to hearing back from those who may have a
different opinion.
Sincerely presented,
John D. Manning, PE
Earth Sensitive Solutions, LLC
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Circle/Square Hour to Arcsec/Square Minute
Circle/Square Hour [circle/h2] Output
1 circle/square hour in degree/square second is equal to 0.000027777777777778
1 circle/square hour in degree/square millisecond is equal to 2.7777777777778e-11
1 circle/square hour in degree/square microsecond is equal to 2.7777777777778e-17
1 circle/square hour in degree/square nanosecond is equal to 2.7777777777778e-23
1 circle/square hour in degree/square minute is equal to 0.1
1 circle/square hour in degree/square hour is equal to 360
1 circle/square hour in degree/square day is equal to 207360
1 circle/square hour in degree/square week is equal to 10160640
1 circle/square hour in degree/square month is equal to 192106890
1 circle/square hour in degree/square year is equal to 27663392160
1 circle/square hour in radian/square second is equal to 4.8481368110954e-7
1 circle/square hour in radian/square millisecond is equal to 4.8481368110954e-13
1 circle/square hour in radian/square microsecond is equal to 4.8481368110954e-19
1 circle/square hour in radian/square nanosecond is equal to 4.8481368110954e-25
1 circle/square hour in radian/square minute is equal to 0.0017453292519943
1 circle/square hour in radian/square hour is equal to 6.28
1 circle/square hour in radian/square day is equal to 3619.11
1 circle/square hour in radian/square week is equal to 177336.62
1 circle/square hour in radian/square month is equal to 3352897.75
1 circle/square hour in radian/square year is equal to 482817275.46
1 circle/square hour in gradian/square second is equal to 0.000030864197530864
1 circle/square hour in gradian/square millisecond is equal to 3.0864197530864e-11
1 circle/square hour in gradian/square microsecond is equal to 3.0864197530864e-17
1 circle/square hour in gradian/square nanosecond is equal to 3.0864197530864e-23
1 circle/square hour in gradian/square minute is equal to 0.11111111111111
1 circle/square hour in gradian/square hour is equal to 400
1 circle/square hour in gradian/square day is equal to 230400
1 circle/square hour in gradian/square week is equal to 11289600
1 circle/square hour in gradian/square month is equal to 213452100
1 circle/square hour in gradian/square year is equal to 30737102400
1 circle/square hour in arcmin/square second is equal to 0.0016666666666667
1 circle/square hour in arcmin/square millisecond is equal to 1.6666666666667e-9
1 circle/square hour in arcmin/square microsecond is equal to 1.6666666666667e-15
1 circle/square hour in arcmin/square nanosecond is equal to 1.6666666666667e-21
1 circle/square hour in arcmin/square minute is equal to 6
1 circle/square hour in arcmin/square hour is equal to 21600
1 circle/square hour in arcmin/square day is equal to 12441600
1 circle/square hour in arcmin/square week is equal to 609638400
1 circle/square hour in arcmin/square month is equal to 11526413400
1 circle/square hour in arcmin/square year is equal to 1659803529600
1 circle/square hour in arcsec/square second is equal to 0.1
1 circle/square hour in arcsec/square millisecond is equal to 1e-7
1 circle/square hour in arcsec/square microsecond is equal to 1e-13
1 circle/square hour in arcsec/square nanosecond is equal to 1e-19
1 circle/square hour in arcsec/square minute is equal to 360
1 circle/square hour in arcsec/square hour is equal to 1296000
1 circle/square hour in arcsec/square day is equal to 746496000
1 circle/square hour in arcsec/square week is equal to 36578304000
1 circle/square hour in arcsec/square month is equal to 691584804000
1 circle/square hour in arcsec/square year is equal to 99588211776000
1 circle/square hour in sign/square second is equal to 9.2592592592593e-7
1 circle/square hour in sign/square millisecond is equal to 9.2592592592593e-13
1 circle/square hour in sign/square microsecond is equal to 9.2592592592593e-19
1 circle/square hour in sign/square nanosecond is equal to 9.2592592592593e-25
1 circle/square hour in sign/square minute is equal to 0.0033333333333333
1 circle/square hour in sign/square hour is equal to 12
1 circle/square hour in sign/square day is equal to 6912
1 circle/square hour in sign/square week is equal to 338688
1 circle/square hour in sign/square month is equal to 6403563
1 circle/square hour in sign/square year is equal to 922113072
1 circle/square hour in turn/square second is equal to 7.7160493827161e-8
1 circle/square hour in turn/square millisecond is equal to 7.7160493827161e-14
1 circle/square hour in turn/square microsecond is equal to 7.716049382716e-20
1 circle/square hour in turn/square nanosecond is equal to 7.7160493827161e-26
1 circle/square hour in turn/square minute is equal to 0.00027777777777778
1 circle/square hour in turn/square hour is equal to 1
1 circle/square hour in turn/square day is equal to 576
1 circle/square hour in turn/square week is equal to 28224
1 circle/square hour in turn/square month is equal to 533630.25
1 circle/square hour in turn/square year is equal to 76842756
1 circle/square hour in circle/square second is equal to 7.7160493827161e-8
1 circle/square hour in circle/square millisecond is equal to 7.7160493827161e-14
1 circle/square hour in circle/square microsecond is equal to 7.716049382716e-20
1 circle/square hour in circle/square nanosecond is equal to 7.7160493827161e-26
1 circle/square hour in circle/square minute is equal to 0.00027777777777778
1 circle/square hour in circle/square day is equal to 576
1 circle/square hour in circle/square week is equal to 28224
1 circle/square hour in circle/square month is equal to 533630.25
1 circle/square hour in circle/square year is equal to 76842756
1 circle/square hour in mil/square second is equal to 0.00049382716049383
1 circle/square hour in mil/square millisecond is equal to 4.9382716049383e-10
1 circle/square hour in mil/square microsecond is equal to 4.9382716049383e-16
1 circle/square hour in mil/square nanosecond is equal to 4.9382716049383e-22
1 circle/square hour in mil/square minute is equal to 1.78
1 circle/square hour in mil/square hour is equal to 6400
1 circle/square hour in mil/square day is equal to 3686400
1 circle/square hour in mil/square week is equal to 180633600
1 circle/square hour in mil/square month is equal to 3415233600
1 circle/square hour in mil/square year is equal to 491793638400
1 circle/square hour in revolution/square second is equal to 7.7160493827161e-8
1 circle/square hour in revolution/square millisecond is equal to 7.7160493827161e-14
1 circle/square hour in revolution/square microsecond is equal to 7.716049382716e-20
1 circle/square hour in revolution/square nanosecond is equal to 7.7160493827161e-26
1 circle/square hour in revolution/square minute is equal to 0.00027777777777778
1 circle/square hour in revolution/square hour is equal to 1
1 circle/square hour in revolution/square day is equal to 576
1 circle/square hour in revolution/square week is equal to 28224
1 circle/square hour in revolution/square month is equal to 533630.25
1 circle/square hour in revolution/square year is equal to 76842756
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Calling Sequence
Examples using floating-point number constructors
Software Floating-point Numbers and Their Constructors
Calling SequenceParametersDescriptionExamples using floating-point number constructors
Float(M, E) SFloat(M, E) x.yen x.yEn
M-expressionE-expressionx-(optional) integer constanty-(optional) unsigned integer constantn-(optional) integer constant
An arbitrary-precision software floating-point number (an object of type sfloat) is represented internally in Maple by a pair of integers (the mantissa M and the exponent E). The Float(M, E) command
can be used to construct the floating-point number M * 10^E. If the mantissa parameter is of type imaginary, Float(M, E) returns I * Float(Im(M), E). If the mantissa is of type nonreal, Float(M, E)
returns Float(Re(M), E) + I * Float(Im(M), E). The SFloat(M, E) command is equivalent to the Float(M, E) function. In Maple, a software floating-point number (see type/sfloat) and a general
floating-point number (see type/float) are considered to be the same object. Maple also has hardware floating-point numbers, of type hfloat (see type/hfloat), which can be constructed using the
HFloat constructor. The maximum number of digits in the mantissa of a software float, and the maximum and minimum allowable exponents, can be obtained from the Maple_floats routine. Software
floating-point numbers can also be created by entering x.yEn or x.yen, where n is the integer exponent. All three parameters are optional in the calling sequence. If y is omitted in the calling
sequence, then the decimal point must also be omitted (for example, 1e0 not 1.e0). To obtain the mantissa and exponent fields of a software float, use SFloatMantissa and SFloatExponent, respectively.
The presence of a floating-point number in an expression generally implies that the computation will use floating-point evaluation. The floating-point evaluator, evalf, can be used to force
computation to take place in the floating-point domain. The number of digits carried in the mantissa for floating-point arithmetic is determined by the Maple environment variable Digits (default is
10). Maple includes a variety of numeric functions to use with floating-point numbers.
Notes regarding infinity The quantity Float(infinity) represents a floating-point infinity. This value is used to indicate a floating-point value that is too large to be otherwise represented. It
does not necessarily represent the mathematical concept of infinity. Float(infinity) can be returned by a function or an operation when the input operands are such that the indicated function or
operation will overflow (that is, produce a value that cannot be represented in the NiMtSSZGbG9hdEdJKnByb3RlY3RlZEdGJTYkSSJhRzYiSSJiR0Yo format). Float(infinity) can be either or both components of a
complex number (for example, Float(infinity) + 3.7*I, 0. + Float(infinity)*I). By convention, Maple treats all complex numbers both of whose components are Float(infinity) as the single point complex
infinity. This is a convention only, but you (and in your programs) should be cautious about relying on the sign of the real or imaginary part of such an object. See type/cx_infinity. Notes regarding
undefined The quantity Float(undefined) represents a non-numeric object in the floating-point system. This value can be returned by a function or operation if the input operands are not in the domain
of the function or operand. Note: Float(undefined) values always compare as equal. You can also use type(expr, undefined) to check for an undefined value. You can tag a Float(undefined) object with
the notation Float(n,undefined), where n is a non-zero integer. Whenever possible, Maple preserves this object, in the sense that if it is passed in as an operand to a function or operation, Maple
tries to return the same value if it makes sense to do so. In this way, it is possible to perform some retrospective analysis to determine precisely when the object first appeared. Float(undefined)
can be either or both components of a complex number (for example, Float(undefined) + 1.*I, Float(infinity) + Float(undefined)*I. The type undefined recognizes such an object. Notes regarding zero In
its floating-point format, 0 has a sign. The sign of 0 is preserved by arithmetic operations whenever possible, according to the standard rules of algebra. Operations and functions can use the sign
of 0 to distinguish qualitative information (for example, branch cut closure), but not quantitative information (-0.0 < +0.0 returns false). It is possible that the result of a computation is
mathematically 0, but the sign of that result cannot be established from the application of the standard rules of arithmetic. The simplest such example is x - x. In such cases, Maple uses the
following convention. The result of an indefinitely signed computation whose mathematical value is '0' is set to '+0', unless the `rounding` mode is set to '-infinity', in which case it is set to
'-0'. This convention is implemented by the Default0 function. Corresponding to the convention regarding complex infinities, as described above, Maple treats all complex numbers, both of whose
components are floating-point 0.'s, as the same. Again, this is a convention, but code should in general not rely on the sign of the real or imaginary part of NiNeJCQiIiFGJUYk, NiNeJCQiIiFGJSQhIiFGJQ
==, etc. The Float constructor is called during parsing of all floating-point numbers and so can be overloaded by creating a module like this: M := module() export Float; Float := proc(m,e) m*ten^e;
end; end; The statement, use M in 1.234 end; will then return 123*ten^(-2); 2.3; 2.; -.3; x:=-2.3e4; (1.2e6)*x; Default0(); Rounding := -infinity; 2. - 2.; Default0(); Float(23,-45); SFloat(23,-45);
2e-3; interface(prettyprint=0); 2.3; y:=-2.3e4; (1.2e5)*y; Float(23,-45); Here is an example of float overflow: exp(1e100);
See AlsoconstantconvertDefault0DigitsevalfHFloatintegeroptypetype/cx_infinitytype/floattype/hfloattype/numerictype/sfloatUseHardwareFloats
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HW02 - Logistic Regression &
Submission Instructions.
• At this point you should be using your cleaned version of the book data
• Submit your draft PDF to the Google Drive: Topic 02: Logistic regression and classification/Draft folder by the due date.
• Read the question carefully. Some book questions require you to use appropriate variable selection techniques.
Part I: Logistic Regression
1. Playing with odds: PMA6 12.1
2. Logistic Regression modeling: PMA6 12.7, 12.8
3. Compare the models in part 2 for acute vs chronic illness.
1. Are the measures that are important in explaining the outcome similar?
2. For covariates that are the same, compare the effect of that covariate on each of the outcomes.
4. Northridge Earthquake: (PMA6 12.22-12.29 modified). This problem use the Northridge earthquake data set. [See here for more info about this event.] We are interested in knowing if homeowners
(V449) were more likely than renters to report emotional injuries (W238) as a result of the Northridge earthquake, controlling for age (RAGE), gender (RSEX), and ethnicity (NEWETHN).
1. Download and use my cleaning script as a starter.
2. Fit a logistic regression model on emotional injury as the outcome using the variables listed above as predictors. Use tbl_regression to create a nice table of results that include odds
ratios and 95% confidence intervals. See if you can figure out how to drop the intercept term from the table.
3. Interpret each predictor (except the intercept) in context of the problem.
4. Are the estimated effects of home ownership upon reporting emotional injuries different for men and women, controlling for age and ethnicity? That is, is there a significant interaction
effect between gender and home ownership?
Part II: Binary Classification
Playing Hookey (PMA6 12.18-12.19 modified)
1. Perform a binary logistic regression analysis using the Parental HIV data to model the probability of having been absent from school without a reason (variable HOOKEY). Find the variables that
best predict whether an adolescent had been absent without a reason or not. Use a hefty dose of common sense here, not all variables are reasonable to use (e.g. using the # of times a student
skips school to predict whether or not they will predict school)
2. Use the default value for the predict() function to create a vector of predictions for each student.
3. Explore the distribution of predictions against a few variables that you identified (via the model) as being highly predictive of skipping school
4. Create a confusion matrix for these predictions and interpret: accuracy, balanced accuracy, sensitivity, specificity, PPV, NPV.
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Matplolib - Hide Axis in a Plot (Code with Examples) - Data Science Parichay
In this tutorial, we will look at how to hide the axes in a matplotlib plot with the help of some examples.
You can use the following methods to hide one or more axes from a plot in matplotlib –
1. If you want the hide both the axes (the x-axis and the y-axis) of a single matplotlib plot, use the matplolitb.pyplot.axis() function and pass ‘off’ as an argument.
2. To hide a specific axis, you can use the matplotlib axes object to get the respective axis using the .get_xaxis() or .get_yaxis() function and set its visibility to False using the set_visible()
Let’s now look at some examples –
First, we will create a line plot that we will be using throughout this tutorial –
import matplotlib.pyplot as plt
# x values - years
x = [2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020]
# y values - curde oil price per barrel in USD
y = [94.88, 94.05, 97.98, 93.17, 48.66, 43.29, 50.80, 65.23, 56.99, 39.68]
# plot x and y on a line plot
plt.plot(x, y)
# add axes labels
plt.ylabel('Crude oil price USD/barrel')
Example 1 – Hide both the axes in a matplotlib plot
To hide both the axes – the x-axis and the y-axis, use the matplotlib.pyplot.axis() function and pass ‘off’ as an argument (it’s an argument to the option parameter). This turns off the axis ticks
and labels.
# plot x and y on a line plot
plt.plot(x, y)
# add axes labels
plt.ylabel('Crude oil price USD/barrel')
# turn off the axis
You can see that the axes are not present in the above-resulting plot.
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Example 2 – Hide only the y-axis
In case you only want to hide a specific axis (for example, the y-axis), you can use the respective axes object’s get_yaxis() function and set its visibility to False.
The steps would be –
1. Get the axes object of the respective plot.
2. Get the axis which you want to hide using the axes object’s get_xaxis() of the get_yaxis() function.
3. Set the visibility of the above axis to False using the set_visible() function.
# plot x and y on a line plot
plt.plot(x, y)
# add axes labels
plt.ylabel('Crude oil price USD/barrel')
# get the axes object
ax = plt.gca()
# hide the y-axis
You can see that the y-axis ticks and labels have been removed.
Example 3 – Hide only the x-axis
You can similarly hide only the x-axis –
# plot x and y on a line plot
plt.plot(x, y)
# add axes labels
plt.ylabel('Crude oil price USD/barrel')
# get the axes object
ax = plt.gca()
# hide the x-axis
The x-axis ticks and labels have been removed.
Example 4 – Hide the axes in subplots
To hide a specific axis in a subplot, you can use the subplot’s axes object to access the respective axis and then set its visibility to False like we did in the above examples.
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
# plot the first subplot
ax1.plot(x, y)
# add axis labels
ax1.set_ylabel('Crude oil price USD/barrel')
# plot the second subplot
ax2.plot(x, y)
# add axis labels
ax2.set_ylabel('Crude oil price USD/barrel')
Here, we created two subplots. Both the subplots have the same data but that is not important. Let’s see how to hide the axis in each of the above subplots.
First, we will hide both axes from each of the subplots. For this, use the axes object’s axis() function and pass 'off' as an argument.
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
# plot the first subplot
ax1.plot(x, y)
# add axis labels
ax1.set_ylabel('Crude oil price USD/barrel')
# hide both the axis
# plot the second subplot
ax2.plot(x, y)
# add axis labels
ax2.set_ylabel('Crude oil price USD/barrel')
# hide both the axis
You can see that both axes were removed from each of the subplots.
To remove a specific axis from a subplot, use the respective axes object’s get_xaxis() or get_yaxis() method and set its visibility to False.
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
# plot the first subplot
ax1.plot(x, y)
# add axis labels
ax1.set_ylabel('Crude oil price USD/barrel')
# hide the x-axis
# plot the second subplot
ax2.plot(x, y)
# add axis labels
ax2.set_ylabel('Crude oil price USD/barrel')
# hide the y-axis
In the first subplot, we hide the x-axis and in the second subplot, we hide the y-axis. You can similarly remove both axes from a subplot using this method.
You might also be interested in –
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Journal of Fluid Mechanics: Volume 991 - | Cambridge Core
This journal utilises an Online Peer Review Service (OPRS) for submissions. By clicking "Continue" you will be taken to our partner site https://mc.manuscriptcentral.com/jfm. Please be aware that
your Cambridge account is not valid for this OPRS and registration is required. We strongly advise you to read all "Author instructions" in the "Journal information" area prior to submitting.
doi:10.1017/jfm.2024.552 Page et al. Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders
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Algebra 2 Extra Credit 1st 9 weeks Storyboard av annika316
SOLVED:Find the value of each determinant. \left … - Numerade
Algebra 2; Equations and inequalities. Overview; Solve equations and simplify expressions; Line plots and stem-and-leaf plots; Absolute value; Solve inequalities The final report of the National
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must be taught by experienced teachers of mathematics who instructional strategies that are research-based; and, 3) the instruction of the math curriculum must accomplish the "mutually reinforcing
benefits of conceptual understanding, procedural fluency, and topic 2: algebra When we play games with computers we play by running, jumping and or finding secret things. Well, with Algebra we play
with letters, numbers and symbols. A few of the topics covered in Algebra II include Quadratic Equations, Linear Equations, Exponential Equations, Polynomial Equations, Logarithmic Equations,
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Encourage your 11th grader to set up an environment and schedule that will allow for sustained concentration for periods of learning that are not too 2021-1-15·The Formal Rules of Algebra. The
student who has had a course in algebra, will find this a complete review. The student who is now taking such a course, will see what is in store. A LGEBRA is a method of written calculations. And
what is a calculation but replacing one set of symbols with another? In arithmetic we replace '2 + 2' with '4.' Algebra is a branch of mathematics that deals with symbols and the arithmetic
operations across these symbols. These symbols do not have any fixed values and are called variables.
Is anyone here good at math? - Page 2 - Off Topic - Linus
Free Algebra 2 worksheets created with Infinite Algebra 2. Printable in convenient PDF format. 2015-06-27 · When I wrote this book I included material from Algebra 2 with Trigonometry along with
material as specified by the Common Core Standards via the PARCC End of Year topic delineation.
Topics on Harmonic analysis and Multilinear Algebra - GUPEA
Free Algebra 2 worksheets (pdfs) with answer keys-each includes visual aides, model problems, exploratory activities, practice problems, and an online component You will learn about Numbers,
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y = –3x + 4. x + 4y = –6. A. x = –2,y = –1. B. x = –2,y = 10.
Danmark bnp
(3) Algebra II (any high school Course Description: This course covers advanced algebra topics including: linear equations, matrices, absolute value, inequalities, factoring, parabolas, The
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national and state standards for algebra II courses•Content focused on helping you excel in the classroom. Topics covered are: vector space, basis, dimension, subspace, norm, inner product, Banach
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4 C0nv3rs4t10n
between mlf & nn, sometime in 2000/2001
i shall read to you now. compressed version: 002 story of deziluzija.
Midsummer night fe(ve)r.
subject: 1x naive iluzija wearing spiderwebs
Her mother remodelled her Zlatno Pero reception dress.
She -- glows. Thinking everyone loves her. Spiderwebs will protect her.
She does not know. She does not know. She feels. She dances.
1x pirueta leads to many. many. She -- dizzy. so. She tries untwirling now.
Spirals and anti spirals. Spirals of streets. Going up, counting grasses.
Circling in her head -- reflected in grasses.
Touch of 1x familiar lifeform. She looks in his eyes. Sees nothing.
She does not know. She does not know. She does not know. She does not know.
She does not know. She does not know. And says good bye.
The 1x familiar (but ugly worm-like) semi-lifeform says no-goodbye.
Says come with me. The girl continues spiralling and counting grasses.
Stepping on occasional bugs. Slapping mosquitoes.
The semi-lifeform tells stories. Girl not listening. CountgrassCountgrass.
But then !.
She comes to a grassfield of many, many grasses and starts counting
on her knees. 1 + 1 + 1 + 1 + deadgrass [ 0 ] + 1 + 1. The semi-lifeform
lays down next to her: + deadgrass [ 0 ] + 1 + deadgrass [ 0 ] + 1 + deadgrass
[ 0 ] + deadgrass [ 0 ] + deadgrass [ 0 ] + deadgrass [ 0 ] + deadgrass [ 0 ]
..... She wants life 1+1=1. So. She crawls closer to the fortress wall, where
he can't lay down his 1x slimy body and kill the grass. During the crawl,
she catches a lizard and puts it in 1x button closed pocket of her remodelled
Zlatno Pero reception dress.
She stumbles over a curved rock and never manages to reach the fortress wall.
Her body curved on a curved rock. Covered with worm-slime. Membranas need superglue.
Spiderwebs all over her head. Not protective. At All. 1x dead lizard in her pocket.
Small lizard. Not good for skinning yet. And 1 + 1 + 1 grass ( now deadgrass [ 0 ]
+ deadgrass [ 0 ] + deadgrass [ 0]) under her fingernails.
Pain.Pain Pain.Pain Pain.Pain Pain.Pain Pain.Pain Pain.Pain
She adjusts her now unsanitary remodelled Zlatno Pero reception dress
nd tries to make 1x pirueta. To in-twirl herself back into where she was
when she did not know.
She does not want know.
`! kannot bl!ev !n !mposz!bl th!ngz` za!d al!ce `! daresa! u havnt had much prakt!sz` za!d dze queen. `uen ! uaz ur age ! alua!z d!d !t 4 0.5 * hour per da!. uh! zomt!mez !.v bl!evd az man! az 6 !
mpoz!bl th!ngz b4 breakfazt.
making reality abstract. my favorite morning gymnastics.
thinking of at least 10 impossible
things before breakfast.
what do you see in attached imgs?
dze blak sea ad!eu m! lv++ nn.
nn shl remembr mlf komme ca
But the streets are now spiralling in a wrong direction.
Multidimensional spirals.
Never trust those. You can never return where you started. !+
Clean Sterile lifeforms Staring at her. From a distance. Laughing.
Unsterile iluzija, untouchable. unlovable.
un. | dez.
dezinfekcija of her partly (badly) skinned skin in a city fountain. splash, splash.
That is when the girl first felt you. Water was not water. It was nn.
But she didn't know your name yet. So she called you nameless. Drifted inside
you for hours. Waving her wounded legs. flip, flop. Was so cold, so soft.
So warm too. Submerged, washing her spiderwebs, she cried. With no voice.
+ she gave herself a new name: deziluzija.
After that for 5 years deziluzija thinks: all male lifeforms = vaatdoeken.
Use them and make them wet and dirty and then throw in the garbage. And.
Then set the garbage on fire. Smiling. Seeing the male lifevaatdoeken begging
her to stop. Calling on the phone and crying. And she would still only smile.
Evil smile. Senseless. And then bathe in nn. Therapeutic.
Still it felt like someone vomiting in her mouth.
Now: much better.
do you sense?
zuperlat!v glass beads
falling down the stairs. Massive flood of multicoloured glass beads. Can you hear their sound approaching the world? lets ring the world's bell! yes?
if they open, ue | var {glassbeads} will flood them too! korporat glass beads warfare!
ue r uater. ue !nundate.
My soul warps the fragments of my desire into an image of you. A shadow of liquids, I wrap myself around you and slowly disappear, as your heat evaporates my fluidity. Instead of an abundant stream,
I remain visible only as sweat drops covering your translucent skin, but I exist as the raising vapor, the unity of your heat and my transcendence. Leaving the layer of condensed steam on every
surface in this desolate city. Flooding the deserts around. I drink your sweat, becoming fluid again, gliding over your rhythms, envelop you in shadows and humidity. Underneath the sweat, your
shapeless body is carved with your knowledge, and I become your humble pupil, absorbing the symbols of your world through every touch. Kissing the words, remembering your thoughts in my streams…
rushing away to be your voice, singing your world as I crush the stones under my waters.
and how does nn expect mlf to go back to work and concentrate after this?
any ideas?
> dze tabl = kovrd !n u!te kloth + z!tuatd ouz!de dze arkadez ov o1
> r!ch.bored.edukatd gardn.
mlf procedes staining the white cloth with her moisture.
> 2 u!n dz!z game 1 muzt assembl dze p!kturz ov 01 dream ov
> fem!n!n pouer + jo!.
there is much power and much jo! rising from your words.
mlnfn bubblbubbl
bubblbubblbubblbubblbubblbubbl bubblbubblbubblbubblbubblbubblbubblbubbl bubblbubblbubblbubblbubblbubblbubblbubbl bubblbubblbubblbubblbubblbubblbubblbubbl
bubblbubbl bubblbubbl bubblbubbl bubblbubbl
i collected the tearz in a glass, thinking of my life when ue stop being 1+1=1 and become 1+1=2 again. then i let them dry up in the sun (didn't work, was raining) so, i put them in the microwave. i
skueezed some of u from a few of last emails. took the dried up salt and joined them with some of u.=uater. , sent our mutual tearz to some sensitive life forms. hoping they would sympathise. +extend
to u.
tonight i am a hydra. u flow through my 2 celled walls. my finger like tentacles spread into u. my tentacles are sticky, so that I can catch food for us. krikets.
stream through me. -mlnfn
nn adaptz 2 01 mlf grav!taz!onl f!eld at 01 d!ztansz.
mlf detaches her feet from the ground and starts floating.
her feet still covered in sand.
mlf d!rektz her gaze. kosm!k p!ruetz - entangl!ng.touch!ng.
extend!ng. breath!ng.
mlf reaches for nn, presses her small floating skins against nn.z
and utters: Ilv+u.
[all whispered in one breath, tickling nn.z ear… until all air departs from mlf
lungs and she begins to suffocate. then you kiss mlf again. ok?]
2.2 lv+
0+0 odr dzn nn = holdz dze ztor!.
that's what makes it more enjoyable.
ekskluz!v + ekspenz!v da +?
mlf = ud hold 01 ztor! !n 01 ztor! !n 01 ztor!.
lotsa spiralling ztor!ez.
viscuos plants made from crystaline air.
transmitting you 1x bio txture.
mlf, cyberbotanist's zk!n
01 ztor! !n 01 ztor!. !n 01 ztor! !n 01 ztor!. !n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!. !n 01 ztor! !n 01 ztor!. !n 01 ztor! !n 01
!n 01 ztor! !n 01 ztor!. !n 01 ztor! !n 01 ztor!. !n 01 ztor! !n 01
ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!. !n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.
!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.!n 01 ztor! !n 01 ztor!.
!n 01 ztor!.
nn u!zprz. ! uant u 2 uear m!
natur = adorz lv+ \ l!ez
natur = iluzija
+ nn = regardz 25 az luvl!
25 is luvl!
have noticed what happens to a lifeform when it becomes 26 last week.
he zkr!!!!!med when he noticed that he can't use the 'youth' train
ticket any more.
mlf laughed.
konduz!v 2 01 total !ntokx!kaz!e
want some toxins?
mlf tox!nz. b!enzur = da + da.
nn zm!lz. m! bod! = 01 gardn.
mlf zm!lz. 01 gardn flows through her veins.
'At the peak of his triumph, the culmination of his machinic erections, man confronts
the system he built for his own protection and finds it female and dangerous.'
Sadie Plant (1995: 62)
1x l!f objkt = haz obzrvd 2x tangenta 4 4x !earz v!a b!tztream. paradox!kl! or not tzo = dze d!g!tale \ b!tztream obzervaz!e = haz b!n konduz!v 2 01 !nfuz!e. u!lzt .b!o obzervaz!e not tzo.
2x = dze futur. simply SUPERIOR++ 2x = floatz efortlezl!. l!beratd l!f energ!e++ _ ultra kl!n!kl [u!lzt ultra un.kl!n!kl] xy = ___… perm!t zom 1 2 01 kompaz!onat zm!le. dog l!ke kreaturz.
= elokuent xy l!f objktz = float 2cm abov + !n anglez. !z mozt pathet!kue.
1x l!f objkt = l!ghtz 01 z!garete. konzumz > kafe + adm!rz 2x ultra zan!tar! \ ulta non.l!nera - total rad!kalr m9ndfukc++ l!f energ!e populat!ng 01 futur.
u!nk + c!ao. (whoever)
|
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What is Global Etf volatility? GLCC.TO | Macroaxis
GLCC Etf 26.45 0.42 1.56%
Global X Gold
holds Efficiency (Sharpe) Ratio of -0.0175, which attests that the entity had a -0.0175% return per unit of risk over the last 3 months. Global X Gold exposes twenty-nine different
technical indicators
, which can help you to evaluate volatility embedded in its price movement. Please check out Global X's Market Risk Adjusted Performance of 0.9279,
downside deviation
of 1.92, and Risk Adjusted Performance of 0.023 to validate the risk estimate we provide.
Global X Etf volatility depicts how high the prices fluctuate around the mean (or its average) price. In other words, it is a statistical measure of the distribution of Global daily returns, and it
is calculated using variance and standard deviation. We also use Global's beta, its
sensitivity to the market
, as well as its odds of financial distress to provide a more practical estimation of Global X volatility.
market volatility
can be a perfect environment for investors who play the long game with Global X. They may decide to buy additional shares of Global X at lower prices to lower the average cost per share, thereby
improving their portfolio's performance when markets normalize.
Moving together with Global Etf
0.66 XIU iShares SPTSX 60 PairCorr
0.61 XSP iShares Core SP PairCorr
0.68 XIC iShares Core SPTSX PairCorr
0.68 ZCN BMO SPTSX Capped PairCorr
0.62 ZSP BMO SP 500 PairCorr
0.62 VFV Vanguard SP 500 PairCorr
Moving against Global Etf
Global X Market Sensitivity And Downside Risk
Global X's beta coefficient measures the volatility of Global etf compared to the
systematic risk
of the entire market represented by your
selected benchmark
. In mathematical terms, beta represents the slope of the line through a regression of data points where each of these points represents Global etf's returns against your selected market. In other
words, Global X's beta of 0.0356 provides an investor with an approximation of how much risk Global X etf can potentially add to one of your existing portfolios. Global X Gold has relatively low
volatility with skewness of -0.2 and kurtosis of 1.86. Understanding different
market volatility
trends often help investors to time the market. Properly using volatility indicators enable traders to measure Global X's etf risk against market volatility during both bullish and bearish trends.
The higher level of volatility that comes with bear markets can directly impact Global X's etf price while adding stress to investors as they watch their shares' value plummet. This usually forces
investors to rebalance their portfolios by buying different financial instruments as prices fall.
3 Months Beta
Analyze Global X Gold Demand TrendCheck current 90 days Global X correlation with market (Dow Jones Industrial)
Global standard deviation measures the daily dispersion of prices over your selected time horizon relative to its mean. A typical volatile entity has a high standard deviation, while the deviation of
a stable instrument is usually low. As a downside, the standard deviation calculates all uncertainty as risk, even when it is in your favor, such as above-average returns.
It is essential to understand the difference between upside risk (as represented by Global X's standard deviation) and the downside risk, which can be measured by semi-deviation or downside deviation
of Global X's daily returns or price. Since the actual investment returns on holding a position in global etf tend to have a non-normal distribution, there will be different probabilities for losses
than for gains. The likelihood of losses is reflected in the downside risk of an investment in Global X.
Global X Gold Etf Volatility Analysis
Volatility refers to the frequency at which Global X etf price increases or decreases within a specified period. These fluctuations usually indicate
the level of risk
that's associated with Global X's price changes. Investors will then calculate the volatility of Global X's etf to predict their future moves. A etf that has erratic price changes quickly hits new
highs, and lows are considered highly volatile. A etf with relatively stable price changes has low volatility. A highly volatile etf is riskier, but the risk cuts both ways. Investing in highly
volatile security can either be highly successful, or you may experience significant failure. There are two main types of Global X's volatility:
Historical Volatility
This type of etf volatility measures Global X's fluctuations based on previous trends. It's commonly used to predict Global X's future behavior based on its past. However, it cannot conclusively
the future direction
of the etf.
Implied Volatility
This type of volatility provides a positive outlook on future price fluctuations for Global X's current market price. This means that the etf will return to its initially predicted market price. This
type of volatility can be derived from derivative instruments written on Global X's to be redeemed at a future date.
The output start index for this execution was zero with a total number of output elements of sixty-one. Global X Gold Average Price is the average of the sum of open, high, low and close daily prices
of a bar. It can be used to smooth an indicator that normally takes just the closing price as input.
Global X Projected Return Density Against Market
Assuming the 90 days trading horizon Global X has a beta of 0.0356 . This usually indicates as returns on the market go up, Global X average returns are expected to increase less than the benchmark.
However, during the bear market, the loss on holding Global X Gold will be expected to be much smaller as well.
traded equities
are subject to two types of risk - systematic (i.e., market) and unsystematic (i.e., nonmarket or company-specific) risk. Unsystematic risk is the risk that events specific to Global X or Global
sector will adversely affect the stock's price. This type of risk can be
diversified away
by owning several different stocks in different industries whose
stock prices
have shown a small correlation to each other. On the other hand,
systematic risk
is the risk that Global X's price will be affected by overall etf market movements and cannot be diversified away. So, no matter how many positions you have, you cannot eliminate market risk.
However, you can measure a Global etf's historical response to market movements and buy it if you are comfortable with its volatility direction. Beta and standard deviation are two commonly used
measures to help you make the right decision.
Global X Gold has an alpha of 0.027, implying that it can generate a 0.027 percent excess return over Dow Jones Industrial after adjusting for the inherited market risk (beta).
Global X's volatility is measured either by using standard deviation or beta. Standard deviation will reflect the average amount of how global etf's price will differ from the mean after some time.To
get its calculation, you should first determine the mean price during the specified period then subtract that from each price point.
What Drives a Global X Price Volatility?
Several factors can influence a etf's market volatility:
Specific events can influence volatility within a particular industry. For instance, a significant weather upheaval in a crucial oil-production site may cause oil prices to increase in the oil
sector. The direct result will be the rise in the stock price of oil distribution companies. Similarly, any government regulation in a specific industry could negatively influence stock prices due to
increased regulations on compliance that may impact the company's future earnings and growth.
Political and Economic environment
When governments make significant decisions regarding trade agreements, policies, and legislation regarding specific industries, they will influence stock prices. Everything from speeches to
elections may influence investors, who can directly influence the stock prices in any particular industry. The prevailing economic situation also plays a significant role in stock prices. When the
economy is doing well, investors will have a positive reaction and hence, better stock prices and vice versa.
The Company's Performance
Sometimes volatility will only affect an individual company. For example, a revolutionary product launch or strong earnings report may attract many investors to purchase the company. This positive
attention will raise the company's stock price. In contrast, product recalls and data breaches may negatively influence a company's stock prices.
Global X Etf Risk Measures
Assuming the 90 days trading horizon the coefficient of variation of Global X is -5724.25. The daily returns are distributed with a variance of 3.8 and standard deviation of 1.95. The mean deviation
of Global X Gold is currently at 1.49. For similar time horizon, the
selected benchmark
(Dow Jones Industrial) has volatility of 0.77
α Alpha over Dow Jones 0.03 Details
β Beta against Dow Jones 0.04 Details
σ Overall volatility 1.95 Details
Ir Information ratio -0.07 Details
Global X Etf Return Volatility
Global X historical daily return volatility represents how much of Global X etf's daily returns swing around its mean - it is a statistical measure of its dispersion of returns. The ETF accepts
1.9486% volatility on return distribution over the 90 days horizon. By contrast, Dow Jones Industrial accepts 0.7554% volatility on return distribution over the 90 days horizon.
About Global X Volatility
Volatility is a rate at which the price of Global X or any other equity instrument increases or decreases for a given set of returns. It is measured by calculating the standard deviation of the
annualized returns over a given period of time and shows the range to which the price of Global X may increase or decrease. In other words, similar to Global's
beta indicator
, it measures the risk of Global X and helps estimate the fluctuations that may happen in a short period of time. So if prices of Global X fluctuate rapidly in a short time span, it is termed to have
high volatility, and if it swings slowly in a more extended period, it is understood to have low volatility.
Please read more on our
technical analysis
3 ways to utilize Global X's volatility to invest better
Higher Global X's etf volatility means that the price of its stock is changing rapidly and unpredictably, while lower stock volatility indicates that the price of Global X Gold etf is relatively
stable. Investors and traders use stock volatility as an indicator of risk and potential reward, as stocks with higher volatility can offer the potential for more significant returns but also come
with a greater risk of losses. Global X Gold etf volatility can provide helpful information for making investment decisions in the following ways:
• Measuring Risk: Volatility can be used as a measure of risk, which can help you determine the potential fluctuations in the value of Global X Gold investment. A higher volatility means higher
risk and potentially larger changes in value.
• Identifying Opportunities: High volatility in Global X's etf can indicate that there is potential for significant price movements, either up or down, which could present investment opportunities.
• Diversification: Understanding how the volatility of Global X's etf relates to your other investments can help you create a well-diversified portfolio of assets with varying levels of risk.
Remember it's essential to remember that stock volatility is just one of many factors to consider when making investment decisions, and it should be used in conjunction with other fundamental and
technical analysis tools.
Global X Investment Opportunity
Global X Gold has a volatility of 1.95 and is 2.57 times more volatile than Dow Jones Industrial.
17 percent
of all equities and portfolios are less risky than Global X. You can use Global X Gold to protect your portfolios against small market fluctuations. The etf experiences a somewhat bearish sentiment,
but the market may correct it shortly. Check odds of Global X to be traded at
25.66 in 90 days
Significant diversification
The correlation between Global X Gold and DJI is 0.01 (i.e., Significant diversification) for selected investment horizon. Overlapping area represents the amount of risk that can be diversified away
by holding Global X Gold and DJI in the same portfolio, assuming nothing else is changed.
Global X Additional Risk Indicators
The analysis of Global X's secondary risk indicators is one of the essential steps in making a buy or sell decision. The process involves identifying the amount of risk involved in Global X's
investment and either accepting that risk or mitigating it. Along with some common measures of Global X etf's risk such as standard deviation, beta, or value at risk, we also provide a set of
secondary indicators that can assist in the individual investment decision or help in hedging the risk of your existing portfolios.
Please note, the risk measures we provide can be used independently or collectively to perform a risk assessment. When comparing two potential etfs, we recommend comparing similar etfs with
homogenous growth potential and valuation from related markets to determine which investment holds the most risk.
Global X Suggested Diversification Pairs
Pair trading is one of the very effective strategies used by professional day traders and hedge funds capitalizing on short-time and mid-term market inefficiencies. The approach is based on the fact
that the ratio of prices of two correlating shares is long-term stable and oscillates around the average value. If the correlation ratio comes outside the common area, you can speculate with a high
success rate that the ratio will return to the mean value and collect a profit.
The effect of pair diversification on risk is to reduce it, but we should note this doesn't apply to all risk types. When we trade pairs against Global X as a counterpart, there is always some
inherent risk that will never be
diversified away
no matter what. This volatility limits the effect of tactical diversification using pair trading. Global X's
systematic risk
is the inherent uncertainty of the entire market, and therefore cannot be mitigated even by pair-trading it against the equity that is not highly correlated to it. On the other hand, Global X's
unsystematic risk describes the types of risk that we can protect against, at least to some degree, by selecting a matching pair that is not perfectly correlated to Global X Gold.
When determining whether Global X Gold offers a strong return on investment in its stock, a comprehensive analysis is essential. The process typically begins with a thorough review of Global X's
financial statements
, including income statements, balance sheets, and cash flow statements, to assess its
financial health
. Key financial ratios are used to gauge profitability, efficiency, and growth potential of Global X Gold Etf.
Outlined below are crucial reports that will aid in making a well-informed decision on Global X Gold Etf:
Check out
Risk vs Return Analysis
to better understand how to build diversified portfolios, which includes a position in Global X Gold. Also, note that the market value of any etf could be closely tied with the direction of
predictive economic indicators such as
signals in board of governors
. You can also try the
Idea Optimizer
module to use advanced portfolio builder with pre-computed micro ideas to build optimal portfolio .
Please note, there is a significant difference between Global X's value and its price as these two are different measures arrived at by different means. Investors typically determine if Global X is a
good investment by looking at such factors as earnings, sales, fundamental and technical indicators, competition as well as analyst projections. However, Global X's price is the amount at which it
trades on the open market and represents the number that a seller and buyer find agreeable to each party.
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Demystifying the Mathematics Behind Convolutional Neural Networks (CNNs)
• Convolutional neural networks (CNNs) are all the rage in the deep learning and computer vision community
• How does this CNN architecture work? We’ll explore the math behind the building blocks of a convolutional neural network
• We will also build our own CNN from scratch using NumPy
Convolutional neural network (CNN) – almost sounds like an amalgamation of biology, art and mathematics. In a way, that’s exactly what it is (and what this article will cover).
CNN-powered deep learning models are now ubiquitous and you’ll find them sprinkled into various computer vision applications across the globe. Just like XGBoost and other popular machine learning
algorithms, convolutional neural networks came into the public consciousness through a hackathon (the ImageNet competition in 2012).
These neural networks have caught inspiration like fire since then, expanding into various research areas. Here are just a few popular computer vision applications where CNNs are used:
• Facial recognition systems
• Analyzing and parsing through documents
• Smart cities (traffic cameras, for example)
• Recommendation systems, among other use cases
But why does a convolutional neural network work so well? How does it perform better than the traditional ANNs (Artificial neural network)? Why do deep learning experts love it?
To answer these questions, we must understand how a CNN actually works under the hood. In this article, we will go through the mathematics behind a CNN model and we’ll then build our own CNN from
If you prefer a course format where we cover this content in stages, you can enrol in this free course: Convolutional Neural Networks from Scratch
Note: If you’re new to neural networks, I highly recommend checking out our popular free course:
Table of Contents
1. Introduction to Neural Networks
1. Forward Propagation
2. Backward Propagation
2. Convolutional Neural Network (CNN) Architecture
3. Forward Propagation in CNNs
1. Convolutional Layer
2. Linear Transformation
3. Non-linear Transformation
4. Forward Propagation Summary
4. Backward Propagation in CNNs
1. Fully Connected Layer
2. Convolution Layer
5. CNN from Scratch using NumPy
Introduction to Neural Networks
Neural Networks are at the core of all deep learning algorithms. But before you deep dive into these algorithms, it’s important to have a good understanding of the concept of neural networks.
These neural networks try to mimic the human brain and its learning process. Like a brain takes the input, processes it and generates some output, so does the neural network.
These three actions – receiving input, processing information, generating output – are represented in the form of layers in a neural network – input, hidden and output. Below is a skeleton of what a
neural network looks like:
These individual units in the layers are called neurons. The complete training process of a neural network involves two steps.
1. Forward Propagation
Images are fed into the input layer in the form of numbers. These numerical values denote the intensity of pixels in the image. The neurons in the hidden layers apply a few mathematical operations on
these values (which we will discuss later in this article).
In order to perform these mathematical operations, there are certain parameter values that are randomly initialized. Post these mathematical operations at the hidden layer, the result is sent to the
output layer which generates the final prediction.
2. Backward Propagation
Once the output is generated, the next step is to compare the output with the actual value. Based on the final output, and how close or far this is from the actual value (error), the values of the
parameters are updated. The forward propagation process is repeated using the updated parameter values and new outputs are generated.
This is the base of any neural network algorithm. In this article, we will look at the forward and backward propagation steps for a convolutional neural network!
Convolutional Neural Network (CNN) Architecture
Consider this – you are asked to identify objects in two given images. How would you go about doing that? Typically, you would observe the image, try to identify different features, shapes and edges
from the image. Based on the information you gather, you would say that the object is a dog or a car and so on.
This is precisely what the hidden layers in a CNN do – find features in the image. The convolutional neural network can be broken down into two parts:
• The convolution layers: Extracts features from the input
• The fully connected (dense) layers: Uses data from convolution layer to generate output
As we discussed in the previous section, there are two important processes involved in the training of any neural network:
1. Forward Propagation: Receive input data, process the information, and generate output
2. Backward Propagation: Calculate error and update the parameters of the network
We will cover both of these one by one. Let us start with the forward propagation process.
Convolutional Neural Network (CNN): Forward Propagation
Convolution Layer
You know how we look at images and identify the object’s shape and edges? A convolutional neural network does this by comparing the pixel values.
Below is an image of the number 8 and the pixel values for this image. Take a look at the image closely. You would notice that there is a significant difference between the pixel values around the
edges of the number. Hence, a simple way to identify the edges is to compare the neighboring pixel value.
Do we need to traverse pixel by pixel and compare these values? No! To capture this information, the image is convolved with a filter (also known as a ‘kernel’).
Convolution is often represented mathematically with an asterisk * sign. If we have an input image represented as X and a filter represented with f, then the expression would be:
Z = X * f
Note: To learn how filters capture information about the edges, you can go through this article:
Let us understand the process of convolution using a simple example. Consider that we have an image of size 3 x 3 and a filter of size 2 x 2:
The filter goes through the patches of images, performs an element-wise multiplication, and the values are summed up:
(1x1 + 7x1 + 11x0 + 1x1) = 9
(7x1 + 2x1 + 1x0 + 23x1) = 32
(11x1 + 1x1 + 2x0 + 2x1) = 14
(1x1 + 23x1 + 2x0 + 2x1) = 26
Look at that closely – you’ll notice that the filter is considering a small portion of the image at a time. We can also imagine this as a single image broken down into smaller patches, each of which
is convolved with the filter.
In the above example, we had an input of shape (3, 3) and a filter of shape (2, 2). Since the dimensions of image and filter are very small, it’s easy to interpret that the shape of the output matrix
is (2, 2). But how would we find the shape of an output for more complex inputs or filter dimensions? There is a simple formula to do so:
Dimension of image = (n, n)
Dimension of filter = (f,f)
Dimension of output will be ((n-f+1) , (n-f+1))
You should have a good understanding of how a convolutional layer works at this point. Let us move to the next part of the CNN architecture.
Fully Connected Layer
So far, the convolution layer has extracted some valuable features from the data. These features are sent to the fully connected layer that generates the final results. The fully connected layer in a
CNN is nothing but the traditional neural network!
The output from the convolution layer was a 2D matrix. Ideally, we would want each row to represent a single input image. In fact, the fully connected layer can only work with 1D data. Hence, the
values generated from the previous operation are first converted into a 1D format.
Once the data is converted into a 1D array, it is sent to the fully connected layer. All of these individual values are treated as separate features that represent the image. The fully connected
layer performs two operations on the incoming data – a linear transformation and a non-linear transformation.
We first perform a linear transformation on this data. The equation for linear transformation is:
Z = W^T.X + b
Here, X is the input, W is weight, and b (called bias) is a constant. Note that the W in this case will be a matrix of (randomly initialized) numbers. Can you guess what would be the size of this
Considering the size of the matrix is (m, n) – m will be equal to the number of features or inputs for this layer. Since we have 4 features from the convolution layer, m here would be 4. The value of
n will depend on the number of neurons in the layer. For instance, if we have two neurons, then the shape of weight matrix will be (4, 2):
Having defined the weight and bias matrix, let us put these in the equation for linear transformation:
Now, there is one final step in the forward propagation process – the non-linear transformations. Let us understand the concept of non-linear transformation and it’s role in the forward propagation
Non-Linear transformation
The linear transformation alone cannot capture complex relationships. Thus, we introduce an additional component in the network which adds non-linearity to the data. This new component in the
architecture is called the activation function.
There are a number of activation functions that you can use – here is the complete list:
These activation functions are added at each layer in the neural network. The activation function to be used will depend on the type of problem you are solving.
We will be working on a binary classification problem and will use the Sigmoid activation function. Let’s quickly look at the mathematical expression for this:
f(x) = 1/(1+e^-x)
The range of a Sigmoid function is between 0 and 1. This means that for any input value, the result would always be in the range (0, 1). A Sigmoid function is majorly used for binary classification
problems and we will use this for both convolution and fully-connected layers.
Let’s quickly summarize what we’ve covered so far.
Forward Propagation Summary
Step 1: Load the input images in a variable (say X)
Step 2: Define (randomly initialize) a filter matrix. Images are convolved with the filter
Z[1] = X * f
Step 3: Apply the Sigmoid activation function on the result
A = sigmoid(Z[1])
Step 4: Define (randomly initialize) weight and bias matrix. Apply linear transformation on the values
Z[2] = W^T.A + b
Step 5: Apply the Sigmoid function on the data. This will be the final output
O = sigmoid(Z[2])
Now the question is – how are the values in the filter decided? The CNN model treats these values as parameters, which are randomly initialized and learned during the training process. We will answer
this in the next section.
Convolutional Neural Network (CNN): Backward Propagation
During the forward propagation process, we randomly initialized the weights, biases and filters. These values are treated as parameters from the convolutional neural network algorithm. In the
backward propagation process, the model tries to update the parameters such that the overall predictions are more accurate.
For updating these parameters, we use the gradient descent technique. Let us understand the concept of gradient descent with a simple example.
Consider that following in the curve for our loss function where we have a parameter a:
During the random initialization of the parameter, we get the value of a as a[2]. It is clear from the picture that the minimum value of loss is at a[1] and not a[2]. The gradient descent technique
tries to find this value of parameter (a) at which the loss is minimum.
We understand that we need to update the value a[2] and bring it closer to a[1. ]To decide the direction of movement, i.e. whether to increase or decrease the value of the parameter, we calculate the
gradient or slope at the current point.
Based on the value of the gradient, we can determine the updated parameter values. When the slope is negative, the value of the parameter will be increased, and when the slope is positive, the value
of the parameter should be decreased by a small amount.
Here is a generic equation for updating the parameter values:
new_parameter = old_parameter - (learning_rate * gradient_of_parameter)
The learning rate is a constant that controls the amount of change being made to the old value. The slope or the gradient determine the direction of the new values, that is, should the values be
increased or decreased. So, we need to find the gradients, that is, change in error with respect to the parameters in order to update the parameter values.
If you want to read about the gradient descent technique in detail, you can go through the below article:
We know that we have three parameters in a CNN model – weights, biases and filters. Let us calculate the gradients for these parameters one by one.
Backward Propagation: Fully Connected Layer
As discussed previously, the fully connected layer has two parameters – weight matrix and bias matrix. Let us start by calculating the change in error with respect to weights – ∂E/∂W.
Since the error is not directly dependent on the weight matrix, we will use the concept of chain rule to find this value. The computation graph shown below will help us define ∂E/∂W:
∂E/∂W = ∂E/∂O . ∂O/∂Z[2]. ∂z/∂W
We will find the values of these derivatives separately.
1. Change in error with respect to output
Suppose the actual values for the data are denoted as y’ and the predicted output is represented as O. Then the error would be given by this equation:
E = (y' - O)^2/2
If we differentiate the error with respect to the output, we will get the following equation:
∂E/∂O = -(y'-O)
2. Change in output with respect to Z[2] (linear transformation output)
To find the derivative of output O with respect to Z[2], we must first define O in terms of Z[2]. If you look at the computation graph from the forward propagation section above, you would see that
the output is simply the sigmoid of Z[2]. Thus, ∂O/∂Z[2] is effectively the derivative of Sigmoid. Recall the equation for the Sigmoid function:
f(x) = 1/(1+e^-x)
The derivative of this function comes out to be:
f'(x) = (1+e^-x)^-1[1-(1+e^-x)^-1]
f'(x) = sigmoid(x)(1-sigmoid(x))
∂O/∂Z[2 ]= (O)(1-O)
You can read about the complete derivation of the Sigmoid function here.
3. Change in Z[2] with respect to W (Weights)
The value Z[2] is the result of the linear transformation process. Here is the equation of Z[2] in terms of weights:
Z[2] = W^T.A[1] + b
On differentiating Z[2] with respect to W, we will get the value A[1] itself:
∂Z[2]/∂W = A[1]
Now that we have the individual derivations, we can use the chain rule to find the change in error with respect to weights:
∂E/∂W = ∂E/∂O . ∂O/∂Z[2]. ∂Z[2]/∂W
∂E/∂W = -(y'-o) . sigmoid'. A[1]
The shape of ∂E/∂W will be the same as the weight matrix W. We can update the values in the weight matrix using the following equation:
W_new = W_old - lr*∂E/∂W
Updating the bias matrix follows the same procedure. Try to solve that yourself and share the final equations in the comments section below!
Backward Propagation: Convolution Layer
For the convolution layer, we had the filter matrix as our parameter. During the forward propagation process, we randomly initialized the filter matrix. We will now update these values using the
following equation:
new_parameter = old_parameter - (learning_rate * gradient_of_parameter)
To update the filter matrix, we need to find the gradient of the parameter – dE/df. Here is the computation graph for backward propagation:
From the above graph we can define the derivative ∂E/∂f as:
∂E/∂f = ∂E/∂O.∂O/∂Z[2].∂Z[2]/∂A[1] .∂A[1]/∂Z[1].∂Z[1]/∂f
We have already determined the values for ∂E/∂O and ∂O/∂Z2. Let us find the values for the remaining derivatives.
1. Change in Z[2] with respect to A[1]
To find the value for ∂Z[2]/∂A[1] , we need to have the equation for Z[2] in terms of A[1]:
Z[2] = W^T.A[1] + b
On differentiating the above equation with respect to A[1], we get W^T as the result:
∂Z[2]/∂A[1] = W^T
2. Change in A[1] with respect to Z[1]
The next value that we need to determine is ∂A[1]/∂Z[1]. Have a look at the equation of A[1]
A[1] = sigmoid(Z[1])
This is simply the Sigmoid function. The derivative of Sigmoid would be:
∂A[1]/∂Z[1] = (A[1])(1-A[1])
3. Change in Z[1] with respect to filter f
Finally, we need the value for ∂Z[1]/∂f. Here’s the equation for Z[1]
Z[1] = X * f
Differentiating Z with respect to X will simply give us X:
∂Z[1]/∂f = X
Now that we have all the required values, let’s find the overall change in error with respect to the filter:
∂E/∂f = ∂E/∂O.∂O/∂Z[2].∂Z[2]/∂A[1] .∂A[1]/∂Z[1] * ∂Z[1]/∂f
Notice that in the equation above, the value (∂E/∂O.∂O/∂Z2.∂Z2/∂A1 .∂A1/∂Z) is convolved with ∂Z1/∂f instead of using a simple dot product. Why? The main reason is that during forward propagation, we
perform a convolution operation for the images and filters.
This is repeated in the backward propagation process. Once we have the value for ∂E/∂f, we will use this value to update the original filter value:
f = f - lr*(∂E/∂f)
This completes the backpropagation section for convolutional neural networks. It’s now time to code!
CNN from Scratch using NumPy
Excited to get your hands dirty and design a convolutional neural network from scratch? The wait is over!
We will start by loading the required libraries and dataset. Here, we will be using the MNIST dataset which is present within the keras.datasets library.
For the purpose of this tutorial, we have selected only the first 200 images from the dataset. Here is the distribution of classes for the first 200 images:
dtype: int64
As you can see, we have ten classes here – 0 to 9. This is a multi-class classification problem. For now, we will start with building a simple CNN model for a binary classification problem:
dtype: int64
We will now initialize the filters for the convolution operation:
Let us quickly check the shape of the loaded images, target variable and the filter matrix:
X.shape, y.shape, f.shape
((28, 28, 200), (1, 200), (5, 5, 3))
We have 200 images of dimensions (28, 28) each. For each of these 200 images, we have one class specified and hence the shape of y is (1, 200). Finally, we defined 3 filters, each of dimensions (5,
The next step is to prepare the data for the convolution operation. As we discussed in the forward propagation section, the image-filter convolution can be treated as a single image being divided
into multiple patches:
For every single image in the data, we will create smaller patches of the same dimension as the filter matrix, which is (5, 5). Here is the code to perform this task:
We have everything we need for the forward propagation process of the convolution layer. Moving on to the next section for forward propagation, we need to initialize the weight matrix for the fully
connected layer. The size of the weight matrix will be (m, n) – where m is the number of features as input and n will be the number of neurons in the layer.
What would be the number of features? We know that we can determine the shape of the output image using this formula:
((n-f+1) , (n-f+1))
Since the fully connected layer only takes 1D input, we will flatten this matrix. This means the number of features or input values will be:
((n-f+1) x (n-f+1) x num_of_filter)
Let us initialize the weight matrix:
Finally, we will write the code for the activation function which we will be using for the convolution neural network architecture.
Now, we have already converted the original problem into a binary classification problem. Hence, we will be using Sigmoid as our activation function. We have already discussed the mathematical
equation for Sigmoid and its derivative. Here is the python code for the same:
Great! We have all the elements we need for the forward propagation process. Let us put these code blocks together.
First, we perform the convolution operation on the patches created. After the convolution, the results are stored in the form of a list, which is converted into an array of dimension (200, 3, 576).
• 200 is the number of images
• 576 is the number of patches created, and
• 3 is the number of filters we used
After the convolution operation, we apply the Sigmoid activation function:
((200, 3, 576), (200, 3, 576))
After the convolution layer, we have the fully connected layer. We know that the fully connected layer will only have 1D inputs. So, we first flatten the results from the previous layer using the
reshape function. Then, we apply the linear transformation and activation function on this data:
It’s time to start the code for backward propagation. Let’s define the individual derivatives for backward propagation of the fully connected layer. Here are the equations we need:
E = (y' - O)^2/2
∂E/∂O = -(y'-O)
∂O/∂Z[2 ]= (O)(1-O)
∂Z[2]/∂W = A[1]
Let us code this in Python:
We have the individual derivatives from the previous code block. We can now find the overall change in error w.r.t. weight using the chain rule. Finally, we will use this gradient value to update the
original weight matrix:
W_new = W_old - lr*∂E/∂W
So far we have covered backpropagation for the fully connected layer. This covers updating the weight matrix. Next, we will look at the derivatives for backpropagation for the convolutional layer and
update the filters:
∂E/∂f = ∂E/∂O.∂O/∂Z[2].∂Z[2]/∂A[1] .∂A[1]/∂Z[1 * ]∂Z[1]/∂f
∂E/∂O = -(y'-O)
∂O/∂Z[2 ]= (O)(1-O)
∂Z[2]/∂A[1 ]= W^T
∂A[1]/∂Z[1]= A[1](1-A[1])
∂Z[1]/∂f = X
We will code the first four equations in Python and calculate the derivative using the np.dot function. Post that we need to perform a convolution operation using ∂Z[1]/∂f:
We now have the gradient value. Let us use it to update the error:
End Notes
Take a deep breath – that was a lot of learning in one tutorial! Convolutional neural networks can appear to be slightly complex when you’re starting out but once you get the hang of how they work,
you’ll feel ultra confident in yourself.
I had a lot of fun writing about what goes on under the hood of these CNN models we see everywhere these days. We covered the mathematics behind these CNN models and learned how to implement them
from scratch, using just the NumPy library.
I want you to explore other concepts related to CNN, such as the padding and pooling techniques. Implement them in the code as well and share your ideas in the comments section below.
You should also try your hand at the below hackathons to practice what you’ve learned:
Responses From Readers
Very well written article, Aishwarya . Kudos !!
Astounded. As someone just scratching at deep learning, this is the best "from nothing to a working example" I've seen so far, specially considering how the concepts do not skip any necessary parts
and the code goes step-by-step. Really great job, and thanks for the article.
very well written Aishwarya, congratulations, this is extremely useful and articulate
Amazing explanation. Great post. You have put an entire chapter in 1 page which is commendable. Thank you !!!
Hey it was amazing reading this! I have some doubt : 1) f=np.random.uniform(size=(3,5,5)) I didn't understand the concept of 3 here like the filter matrix is of size(5,5) so the filter matrix will be
multiplied to the image matrix (28,28) with stripes (1,1) so whats the need of 3 matrix of size (5,5) thankyou once again for this amazing content
output_layer_input = (output_layer_input - np.average(output_layer_input))/np.std(output_layer_input) why we need to do the above step ? we can directly pass the dot product into sigmoid functions
In wo = wo - lr*delta_error_fcp lr is not defined
first time, I got understand this very clearly. Thanks Aish.
Incredibly well done article. Congratulations to the author !!!
Very good explanation
Very good explanation of CNN underlying math architecture. Thanks!
thank you so much for the explanation. very very very..... helped me in understanding how filter weights are undated. thank you so much :-)
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This lesson introduces the use of coordinates to store data or the results of mathematical operations. It gives students practice programming for the LEDs of the micro:bit screen using coordinates,
and introduces the basic game blocks of MakeCode. In this video segment from Cyberchase, the CyberSquad have become separated on an island.
1st Quadrant. Coordinate Grids Khan Academy: Video Lessons and Interactive Practice VIDEO lesson #1: Coordinate Plane- graphing points- PRACTICE interactive 1: Sep 24, 2015 This coordinate grid
foldable organizer with building pieces can be used for many activities: First students plan a town and then write the Feb 28, 2021 These graph paper generators will produce a single quadrant
coordinate grid for the students to use in coordinate graphing problems. Buy Nasco 9 in.
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Product Topology -- from Wolfram MathWorld
The topology on the Cartesian product topological spaces whose open sets are the unions of subsets
This definition extends in a natural way to the Cartesian product of any finite number topological spaces. The product topology of
where real line with the Euclidean topology, coincides with the Euclidean topology of the Euclidean space
In the definition of product topology of
The product topology is also called Tychonoff topology, but this should not cause any confusion with the notion of Tychonoff space, which has a completely different meaning.
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Integral of
Introduction to the integral of e^x
In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is
categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.
Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to an exponential function e^x. You will also
understand how to compute e^x integral by using different integration techniques.
What is the integration of e^x?
The integration of e^x is an antiderivative of the e^x function which is equal to e^x which is denoted by ∫(e^x)dx=e^x.
The function e^x is an exponential function and the integral of e is also known as its reverse derivative with respect to the variable involved. In simple words, we can say that finding the integral
e^x is a process of reversing the derivative of e^x.
Integral of e^x formula
The formula of the integral of ex contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(e^x)dx. In mathematical form, the e^x integral is:
∫(e^x)dx = e^x +c
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. If we replace x by u in the above formula, then it will be the integration of e^u.
How to calculate the integral of ex?
The integral of e^x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to find the integration of e x proof by using:
1. Integration by parts
2. Definite integral
Integral of e by using derivatives
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral
of a function. Let’s discuss calculating the integral of e^x by using derivatives.
Proof of integral of e^(x) by using derivatives
Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integration of e^x by using its derivative. For this, we have to look for some derivatives
formulas or a formula that gives e^x as the derivative of any function.
In derivative, we know that,
$\frac{d}{dx}(e^x) = e^x{2}lt;/p>
It means that the derivative of e^x gives us cosh x. Therefore, applying the integral on the both sides,
$\int\frac{d}{dx}(e^x)dx=\int e^xdx{2}lt;/p>
The integral sign at the left side will cancel out with the derivative of e^x and the resulting expression will be,
$\int e^xdx=e^x+c{2}lt;/p>
Where c is a constant, known as the integration constant.
Integral of e^x by using integration by parts
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of
two functions combined together. Let’s discuss calculating the e^(x) integral by using integration by parts.
Proof of integration of e^x by using integration by parts
Since we know that the function e^x can be written as the product of two functions. Therefore, we can integrate it by using integration by parts. For this, suppose that:
$u = e^x{2}lt;/p>
Taking the derivatives, we get:
$\frac{du}{dx} = e^x{2}lt;/p>
Plugging these values into the integration by parts formula, we get:
$∫xe^x dx =1∫e^x dx - ∫[d/dx(x)∫e^x dx]dx{2}lt;/p>
$∫e^x dx =e^x - ∫(1)e^xdx{2}lt;/p>
$∫xe^x dx =xe^x - ∫e^xdx{2}lt;/p>
$∫xe^x dx =xe^x - e^x + C{2}lt;/p>
where C is the constant of integration. Also, learn to calculate the integral e^x by using the u-substitution method calculator.
Integral of ex by using definite integral
The definite integral is a type of integration that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:
$∫^b_a f(x) dx = F(b) – F(a){2}lt;/p>
In the above formula, the limit points a and b are known as the upper and lower bounds. It an integral is not bounded between an upper and lower bound, you can use an indefinite integration
calculator to evaluate such integrals.
Let’s understand the verification of the integration of e^x by using the definite integral.
Proof of integral of e^x by using definite integral
To compute the integration of e^x by using a definite integral, we can use the interval from 0 to 1. Let’s compute the integral of e^(x) from 0 to 1. For this, we can write the integral as:
$∫^1_0 e^x dx = e^x|^1_0{2}nbsp;
Now, substitute the limit in the given function.
$∫^1_0 e^x dx = e^1 – e^0{2}nbsp;
Since e^0 is equal to 1 and e^1 is equal to 2.718, therefore,
$∫^1_0 e^x dx = 2.718 - 1=1.718{2}lt;/p>
To evaluate integrals with some limits, use our definite integration calculator.
What is integration of e 2x?
The integration of e to the 2x is the same as the integration of e^2x. It is equal to the fraction of e^2x and the constant in its power i.e 2. Mathematically, it can be expressed as;
$\int e^{2x}dx=\frac{e^{2x}}{2}+c{2}nbsp;
What power of e is 1?
If we use 0 as the power of e such as e^0, it will be equal to 1. It is because, the power zero of any value is always equal to 1.
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TGAMMA(3) Linux Programmer's Manual TGAMMA(3)
tgamma, tgammaf, tgammal - true gamma function
#include <math.h>
double tgamma(double x);
float tgammaf(float x);
long double tgammal(long double x);
Link with -lm.
Feature Test Macro Requirements for glibc (see feature_test_macros(7)):
tgamma(), tgammaf(), tgammal():
_ISOC99_SOURCE || _POSIX_C_SOURCE >= 200112L
These functions calculate the Gamma function of x.
The Gamma function is defined by
Gamma(x) = integral from 0 to infinity of t^(x-1) e^-t dt
It is defined for every real number except for nonpositive integers. For nonnegative integral m one has
Gamma(m+1) = m!
and, more generally, for all x:
Gamma(x+1) = x * Gamma(x)
Furthermore, the following is valid for all values of x outside the poles:
Gamma(x) * Gamma(1 - x) = PI / sin(PI * x)
On success, these functions return Gamma(x).
If x is a NaN, a NaN is returned.
If x is positive infinity, positive infinity is returned.
If x is a negative integer, or is negative infinity, a domain error occurs, and a NaN is returned.
If the result overflows, a range error occurs, and the functions return HUGE_VAL, HUGE_VALF, or HUGE_VALL, respectively, with the correct mathematical sign.
If the result underflows, a range error occurs, and the functions return 0, with the correct mathematical sign.
If x is -0 or +0, a pole error occurs, and the functions return HUGE_VAL, HUGE_VALF, or HUGE_VALL, respectively, with the same sign as the 0.
See math_error(7) for information on how to determine whether an error has occurred when calling these functions.
The following errors can occur:
Domain error: x is a negative integer, or negative infinity
errno is set to EDOM. An invalid floating-point exception (FE_INVALID) is raised (but see BUGS).
Pole error: x is +0 or -0
errno is set to ERANGE. A divide-by-zero floating-point exception (FE_DIVBYZERO) is raised.
Range error: result overflow
errno is set to ERANGE. An overflow floating-point exception (FE_OVERFLOW) is raised.
glibc also gives the following error which is not specified in C99 or POSIX.1-2001.
Range error: result underflow
An underflow floating-point exception (FE_UNDERFLOW) is raised, and errno is set to ERANGE.
These functions first appeared in glibc in version 2.1.
For an explanation of the terms used in this section, see attributes(7).
│Interface │Attribute │Value │
│tgamma (), tgammaf (), tgammal () │Thread safety│MT-Safe│
C99, POSIX.1-2001, POSIX.1-2008.
This function had to be called "true gamma function" since there is already a function gamma(3) that returns something else (see gamma(3) for details).
Before version 2.18, the glibc implementation of these functions did not set errno to EDOM when x is negative infinity.
Before glibc 2.19, the glibc implementation of these functions did not set errno to ERANGE on an underflow range error. x
In glibc versions 2.3.3 and earlier, an argument of +0 or -0 incorrectly produced a domain error (errno set to EDOM and an FE_INVALID exception raised), rather than a pole error.
This page is part of release 4.10 of the Linux man-pages project. A description of the project, information about reporting bugs, and the latest version of this page, can be found at https://
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Wet Lab Slides Flashcards | Knowt
foramen of monroe/interventricular foramen (connects lateral ventricle with 3rd ventricle) (diencephalon)
foramen of luscka (lateral aperture - drains 4th ventricle to subarachnoid space)
choroid plexus (makes CSF, line ventricles minus anterior & posterior horns of lateral ventricles and cerebral aqueduct)
foramen of magendie / median aperture (drains 4th ventricle into subarachnoid space)
straight sinus (dura folds that drain CSF into systemic circulation)
superior sagittal sinus (receive CSF through arachnoid granulations in subarachnoid space to sinuses)
transverse sinus (drain CSF to systemic via sigmoid sinus to internal jugular vein to R heart)
broadman 3, 1, 2 - postcentral gyrus - primary somatosensory cortex
broadman 6 - caudal superior frontal gyrus - premotor.supplementary cortex
broadman 8, 9 - rostral middle frontal gyrus - frontal eye fields
broadman 5, 7 - superior parietal lobule - sensory association cortex
broadman 17 - banks of calcarine fissure - primary visual cortex
broadman 18, 19 - medial occipital gyrus - visual association cortex
broadman 22 - caudal superior temporal gyrus - wernicke’s area
broadman 41, 42 - middle superior temporal gyrus - primary auditory cortex
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Week 2: Clustering | Computational Journalism
Week 2: Clustering
A vector of numbers is a fundamental data representation which forms the basis of very many algorithms in data mining, language processing, machine learning, and visualization. This week we will
looked in depth at two things: representing objects as vectors, and clustering them, which might be the most basic thing you can do with this sort of data. This requires a distance metric and a
clustering algorithm — both of which involve editorial choices. In journalism we can use clusters to find groups of similar documents, analyze how politicians vote together, or automatically detect
groups of crimes.
Slides for week 2 are here.
See also a discussion of (and files for reproducing) the UK House of Lords voting analysis we did in class, this one:
Here are the main references on this material (from the syllabus):
And here are some of the other things we looked at today:
• Interactive K-means clustering demonstration
• The NOMINATE analysis of U.S. congressional voting
• A related analysis of voting, done with a completely different, Bayesian technique
Other uses of clustering in journalism:
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Focus in High School Mathematics Reasoning and Sense Making - CTL - Collaborative for Teaching and Learning
Book Study Post 2 as it relates to “linking expressions and functions,” MORE THAN MEETS THE EYE (Example 9, pages 38-40) – a resource for promoting mathematical practices:
#1 – Attend to precision;
#2 – Construct viable arguments and critique the reasoning of others.
Generally, in viewing the 22 examples posted by Focus in High School Mathematics Reasoning and Sense Making I must say that I was drawn to the ones that had scaffolded teacher/student questions/
responses. This scaffolding provides a prototype of the kinds of questions to ask, as well as, possible student comments that promote mathematical discourse. Given that the examples are from Number
and Measurement, Algebraic Symbols, Functions, Geometry, and Statistics and Probability the mathematical content supports the Core Content Standards while the questioning prototypes promote the
Mathematical Practice Standards. All of the examples make connections across the content; some promote inquiry, risk-taking, and metacognition where teachers and students alike make their thinking
public; and other examples promote quantitative literacy (see NY Times Opinion ) with real-world applications of the mathematics.
The structure of these scaffolded examples reminded me of one of the most rewarding professional times I have had (both for my own professional growth and student achievement) back in the late ’80s
at Fairdale High School using the four part series, Middle Grades Mathematics Project published by Addison Wesley and written by the Glenda Lappan team from the University of Michigan. This National
Science Foundation-funded teacher resource helped to develop problem-solving and critical-thinking skills for grades 5 – 8. A team of three colleagues used the series with low performing 9^th and 10^
th graders and experienced significant gains in student achievement. The materials are still timely and certainly promote the content and thinking required of the Core Content Standards.
The reason I recall this experience is that this teacher resource also scaffolds the questions to pose to students and was strategically sequenced through: Teacher Action, Teacher Talk, and Expected
Response. The content and reasoning skills were so profound in these resources and for first time instruction, it was very helpful to have this scaffolding provided and to use these actions/talk for
promoting conversations about the key mathematical concepts and for making connections. As the FHS team contextually used the resources we began to alter and add to the discourse possibilities.
I see a comparable use of these examples from Focus in High School Mathematics Reasoning and Sense Making where teachers use the teachers’ questions to guide the facilitation and then use their
context with teachable moments to alter and add to the questions provided.
For Example 9, pages 38 – 40) MORE THAN MEETS THE EYE:
Within the linking expressions to Polynomial Functions example, the key instructional aspects illustrated that provide opportunities for attending to precision and constructing viable arguments and
critiquing the reasoning of others are as follows.
The use of Multiple Representations (multiple representatives – see other CTL Blogs) is critical as students make sense of/visualize the connections amongst the representations – numbers/tables,
algebra, graphs, sentences/words. From the teacher/student discourse I noted that students: visualized the intersection points of the two functions; solved the algebraic equations that equates = ;
graphed the functions; and verbalized what those points of intersection mean, as well as, other values within and beyond what is visualized.
Attending to Precision is evident throughout by evaluating each function at particular points, graphing accuracy, simplifying the expressions, solving the equations.
The opportunity for constructing viable arguments and critiquing the reasoning of others through mathematical discourse (mathematical discourse – see other CTL blogs) is throughout with the use of
the scaffolded questions and responses to questions. Given individual classroom makeups and as teachers use the questions and experience the lesson, teachers will add to, and expand upon the
questions. A connecting quote states that, “Building fluency in working with algebraic notation that is grounded in reasoning and sense making will ensure that students can flexibly apply the
powerful tools of algebra in a variety of contexts both within and outside mathematics.”
Further extensions to this example would: area under the curve; investigating other functions; contextual questions added to the mathematical discourse.
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Quarter 2
Learning Objectives:
• Evaluate an improper integral or determine that the integral diverges.
• Calculate areas in the plane using the definite integral.
Learning Objectives:
• Calculate volumes of solids with known cross sections using definite integrals.
• Calculate volumes of solids of revolution using definite integrals.
• Determine the length of a curve in the plane defined by a function, using a definite integral.
Learning Objectives:
• Interpret differential equations via slope fields and show the relationship between slope fields and solution curves.
• Find numerical solutions to differential equations using Euler's method.
Learning Objectives:
• Sketch solution curves to logistic differential equations.
• Determine the intervals where a logistic differential equation's solution is increasing or decreasing.
• Solve simple logistic differential equations and use them to model population growth and decay.
Learning Objectives:
• Calculate derivatives of parametric functions.
• Determine the length of a curve in the plane defined by parametric functions, using a definite integral.
• Calculate derivatives of vector valued functions.
• Determine a particular solution given a rate vector and initial conditions.
• Determine values for positions and rates of change in problems involving planar motion.
Learning Objectives:
• Calculate derivatives of functions written in polar coordinates.
• Calculate areas of regions defined by polar curves using definite integrals
Learning Objectives:
• Show convergence of sequences including decimal expansions.
• Show convergence or divergence of geometric series.
• Show the divergence of the harmonic series.
• Show convergence of p-series using the integral test.
• Show convergence of series using the comparison tests.
Learning Objectives:
• Show the interval of convergence of series using the ratio test.
• Identify a series' radius of convergence.
• Show the convergence of series using the alternating series test (including the alternating harmonic series).
• Determine the intervals where a series converges absolutely and conditionally.
• Use the ratio test (or root test) to find a power series interval of convergence.
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Hypersonic velocity measurement
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Hello all,
I am working with a college team to build a (hopefully) hypersonic rocket (M5+).
Our goal is to go fast, so we need to be able to prove it.
What are some ways we can measure velocity at these speeds?
We’re tossing around ideas of building a ‘Rayleigh’ pitot tube, and/or using the Taylor-McCall shock come equations to find differential pressure to calculate speed from that.
Has anyone accomplished velocity measurement at high speed? Anything above Mach 2 is helpful to us.
May 7, 2023
Reaction score
Hello all,
I am working with a college team to build a (hopefully) hypersonic rocket (M5+).
Our goal is to go fast, so we need to be able to prove it.
What are some ways we can measure velocity at these speeds?
We’re tossing around ideas of building a ‘Rayleigh’ pitot tube, and/or using the Taylor-McCall shock come equations to find differential pressure to calculate speed from that.
Has anyone accomplished velocity measurement at high speed? Anything above Mach 2 is helpful to us.
I'm no Aerospace engineer, but modern accelerometers are probably going to be more accurate in determining actual speed than using barometric differential pressure measurements.
Also, mach 5+ is very hard to achieve with commercial motors and metal hardware. It's a mass fraction problem.
I hope for the best with your project, I'm working on a project for next year that is a small 54mm to 38mm two-stage rocket using the fastest burning commercial motors that will fit, and I'll be
lucky to break mach 3 with it.
Sep 9, 2013
Reaction score
I tend to agree with Brian. John Krell might have some high fidelity suggestions for accelerometer integration, but for my 2c: I've been using the ADXL357 accelerometer lately and have been pretty
damn impressed with my integration results from it. You don't need to sample it at crazy speeds because it internally does all this decimation magic for you. I only sample mine at something like 40Hz
IIRC and even my insanely (intentionally) unstable hybrids provide a very close displacement agreement (to apogee) with baro and GNSS.
Last edited:
I'm no Aerospace engineer, but modern accelerometers are probably going to be more accurate in determining actual speed than using barometric differential pressure measurements.
Also, mach 5+ is very hard to achieve with commercial motors and metal hardware. It's a mass fraction problem.
I hope for the best with your project, I'm working on a project for next year that is a small 54mm to 38mm two-stage rocket using the fastest burning commercial motors that will fit, and I'll be
lucky to break mach 3 with it.
Hey thanks for the input. Good point about the accelerometers, that’s definitely going to be one of our measurement systems.
and yes, you’re very much correct about the difficulty of reaching such speeds with commercial motors… so we’re making our own very high energy propellant
Best of luck with your project as well, sounds exciting!
I tend to agree with Brian. John Krell might have some high fidelity suggestions for accelerometer integration, but for my 2c: I've been using the ADXL357 accelerometer lately and have been
pretty damn impressed with my integration results from it. You don't need to sample it at crazy speeds because it internally does all this decimation magic for you. I only sample mine at
something like 40Hz IIRC and even my insanely (intentionally) unstable hybrids provide a very close displacement agreement (to apogee) with baro and GNSS.
That’s good to know! I’ll look into the Krell accelerometers. I was looking at the ADXL357 today, seems like a good idea to have one onboard, considering how small and light they are.
Jul 9, 2007
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Transmit a fixed frequency and measure the Doppler shift? SDR might be useful for the receive end.
Transmit a fixed frequency and measure the Doppler shift? SDR might be useful for the receive end.
how much shift would there be? you would need some very precise and therefore expensive instruments for that.
so we’re making our own very high energy propellant
ooohhh looks like fun!
Jan 20, 2009
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Sounds like a great project.
Please keep us updated on your success.
Where are you planning to launch? FAR?
Sounds like a great project.
Please keep us updated on your success.
Where are you planning to launch? FAR?
Hey thanks!
Yes we’ll be flying at FAR sometime next year. It’s going to be a really fun project!
Jan 18, 2009
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There have been a few projects here that were M2.5+ and at least one recently that was M3+ that I can think of. They experience significant heating effects on leading edges. I am very curious how
you'll mitigate heating effects at M5
how much shift would there be
Doppler equation
A licensed amateur could transmit a marker tone on say 1296.303 MHz. use 300000km/sec for Vp.
With a GPS disciplined oscillator the receiver frequency would be within 10**-9 sec, 1 ppb, 1hz at 1296MHz. So, more than good enough.
The transmitter frequency stability is a different problem. Most xtal oscillators don't enjoy high G forces, and I doubt the GPS DO receiver will keep lock at mach 1 and faster.
Maybe someone has a better idea for the xmitter.
Jan 23, 2009
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Doppler equation
A licensed amateur could transmit a marker tone on say 1296.303 MHz. use 300000km/sec for Vp.
With a GPS disciplined oscillator the receiver frequency would be within 10**-9 sec, 1 ppb, 1hz at 1296MHz. So, more than good enough.
The transmitter frequency stability is a different problem. Most xtal oscillators don't enjoy high G forces, and I doubt the GPS DO receiver will keep lock at mach 1 and faster.
Maybe someone has a better idea for the xmitter.
I think you would need an ITAR controlled device to TX hold over Mach 2, certainly 3
Jul 9, 2007
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The oscillator will experience drift under boost, due to acceleration. After burnout the forces should be significantly reduced. Try using a mems-based oscillator or other suitable clock that is more
Last edited:
The oscillator will experience drift under boost, due to acceleration. After burnout the forces should be significantly reduced. Try using a mems-based oscillator or other suitable clock that is
more stable.
Yes, that's what we said
How about this:
Instead of carrying a stable oscillator on board, use the one on the ground.
Fly a wide transponder that echos the time coded pulses from the ground station. 2 way trip for the reference signal, 2x the Doppler effect. I think?
Time to go scribble some arithmetic.
Aug 28, 2007
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Pulse-doppler is your answer for those velocities if expecting any type of actual resolution.
Pulse-doppler is your answer for those velocities if expecting any type of actual resolution.
do you know of any consumer available products?
Mar 14, 2014
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Another major issue you will have to work through is heat. Objects travelling at high Mach numbers get very hot very quickly. You will need to build your rocket.
Also, on the topic of heat, your rocket will likely be surrounded by a plasma sheath, which will inhibit transmission from any onboard telemetry devices. To get around this, you should consider a
ground-based radar. With a little persistence, a DoD facility like Edwards AFM or Pt. Mugu test center may assist on a non-interference basis. It never hurts to ask.
Another major issue you will have to work through is heat. Objects travelling at high Mach numbers get very hot very quickly. You will need to build your rocket.
Also, on the topic of heat, your rocket will likely be surrounded by a plasma sheath, which will inhibit transmission from any onboard telemetry devices. To get around this, you should consider a
ground-based radar. With a little persistence, a DoD facility like Edwards AFM or Pt. Mugu test center may assist on a non-interference basis. It never hurts to ask.
what if you had a drone with a high speed camera at the expected altitude (if its within 1 km it should possible to reach with a drone) you need a FFA waver anyway.
Apr 13, 2023
Reaction score
I tend to agree with Brian. John Krell might have some high fidelity suggestions for accelerometer integration, but for my 2c: I've been using the ADXL357 accelerometer lately and have been
pretty damn impressed with my integration results from it. You don't need to sample it at crazy speeds because it internally does all this decimation magic for you. I only sample mine at
something like 40Hz IIRC and even my insanely (intentionally) unstable hybrids provide a very close displacement agreement (to apogee) with baro and GNSS.
The ADXL357 is an excellent accelerometer. Very low noise with full range clock frequencies of 3.4 MHz for I2C and 10MHz for SPI. The Interpolation filter is a nice feature for higher data
resolution.(Edit: at slower sampling speeds.) I've not tested this feature since I collect data in the 500Hz - 1000Hz range.
Single integration of high speed accelerometer data for velocity determination would produce better accuracy and be easier than a pitot tube or doppler shift. The time available at peak velocity will
be in single digit milliseconds. The higher the sampling speed the better the accuracy and the greater the chance of capturing the peak velocity event.
Air friction heating will be an issue at velocities >Mach 3. I have yet to break Mach 2. I was the propellant insulation advisor for the USCRPL students on Traveler 3 & 4. Traveler 4 hit Mach 5.1 and
came back looking like it had spent an hour in a fire.
Last edited:
Jan 23, 2009
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do you know of any consumer available products?
The X-band units I have tested quits at 199 mph, and it's not consumer, its an MPH
Mar 27, 2023
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The ADXL357 is an excellent accelerometer. Very low noise with full range clock frequencies of 3.4 MHz for I2C and 10MHz for SPI. The Interpolation filter is a nice feature for higher data
resolution.(Edit: at slower sampling speeds.) I've not tested this feature since I collect data in the 500Hz - 1000Hz range.
Single integration of high speed accelerometer data for velocity determination would produce better accuracy and be easier than a pitot tube or doppler shift. The time available at peak velocity
will be in single digit milliseconds. The higher the sampling speed the better the accuracy and the greater the chance of capturing the peak velocity event.
Air friction heating will be an issue at velocities >Mach 3. I have yet to break Mach 2. I was the propellant insulation advisor for the USCRPL students on Traveler 3 & 4. Traveler 4 hit Mach 5.1
and came back looking like it had spent an hour in a fire.
I tend to agree with Brian. John Krell might have some high fidelity suggestions for accelerometer integration, but for my 2c: I've been using the ADXL357 accelerometer lately and have been
pretty damn impressed with my integration results from it. You don't need to sample it at crazy speeds because it internally does all this decimation magic for you. I only sample mine at
something like 40Hz IIRC and even my insanely (intentionally) unstable hybrids provide a very close displacement agreement (to apogee) with baro and GNSS.
I have not studied the sims for a Mach 5 hypersonic flight but would the +/- 40G max acceleration range of the ADXL357 be sufficient for such a flight ?
Analog Devices > ADXL357 Specs
Is there a version that can read higher ranges with the same precision and speed ?
EDIT: Maybe the
Analog Devices > ADXL375
-- kjh
Sep 9, 2013
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I have not studied the sims for a Mach 5 hypersonic flight but would the +/- 40G max acceleration range of the ADXL357 be sufficient for such a flight ?
Analog Devices > ADXL357 Specs
Is there a version that can read higher ranges with the same precision and speed ?
-- kjh
Yeah, I forgot to mention that limitation. That could be a showstopper. Depends on flight profile. The optimum profile for altitude is generally a longer burn with lower thrust, but if your goal was
to simply measure hypersonic velocity, then yeah, it's probably likely the rocket is getting one mighty kick up the beeeehind translating into a quite uncomfortable ride for the occupants.
Don't know if there's a comparable sensor with a higher G rating - I simply haven't looked.
Apr 13, 2023
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I have not studied the sims for a Mach 5 hypersonic flight but would the +/- 40G max acceleration range of the ADXL357 be sufficient for such a flight ?
Analog Devices > ADXL357 Specs
Is there a version that can read higher ranges with the same precision and speed ?
EDIT: Maybe the
Analog Devices > ADXL375
-- kjh
Mach 3 at 40g's in 2.5 sec.
Mach 5 at 40 g's in 4.2 sec.
It can be done at 40g acceleration. The ADXL 357 hard range limits are 10.24g's, 20.48g's, & 40.96g's.
Mach 3 at 100g's in 1.3 sec.
Mach 5 at 100g's in 1.9 sec.
I've added 250 milliseconds to the theoretical 100g times. A 100g acceleration does not happen instantaneously, unless you are firing a projectile out of a cannon. My instrumented micrograin rockets
take approximately 250 milliseconds to overcome inertia and air drag to reach 100g acceleration.
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CodenQuest - Master Coding Through Play
Maximum Product of Three Numbers
Create a function that takes an integer array nums and returns the highest product obtainable by multiplying any three numbers from the array.
The function should consider both positive and negative numbers to find the maximum possible product.
// 6
// Why? The product of 1, 2, and 3 is 6, which is the maximum possible product.
// 24
// Why? The product of 2, 3, and 4 is 24, which is the maximum possible product.
// -6
// Why? The product of -1, -2, and -3 is -6. Given all numbers are negative, this is the maximum product.
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J. Astron. Space Sci.: Orbit Determination of KOMPSAT-1 and Cryosat-2 Satellites Using Optical Wide-field Patrol Network (OWL-Net) Data with Batch Least Squares Filter
Optical observation has been used to detect space objects and a considerable number of optical observation systems are available such as the ground-based electro-optical deep space surveillance
(GEODSS; Henize et al. 1993), Lincoln near-Earth asteroid program (LINEAR; Stokes et al. 2000), and international scientific optical network (ISON; Molotov et al. 2008). The optical observation
system is appropriate for the surveillance of space objects because it is not limited by the range to the object, unlike active tracking systems such as radar and laser. For this reason, a lot of
research on orbit determination (OD) using optical measurements has been conducted. Sabol et al. (2007) developed a simplified covariance model to predict the orbital error uncertainty and
demonstrated that high-accuracy orbit updates from angle-only data are available only if the eccentricity has a low uncertainty. Vallado & Agapov (2010) introduced OD results of geosynchronous (GEO)
satellites using highquality optical data from ISON; an uncertainty of ~5 km was obtained using the least squares method. Tombasco (2011) addressed GEO orbit estimation using ground-based and
space-based angle-only measurements and enhanced the accuracy based on GEO elements.
The optical wide-field patrol network (OWL-Net) is an optical surveillance system of space objects developed at the Korea Astronomy and Space Science Institute (KASI). The main goals of the system
are the tracking and monitoring of domestic satellites to protect space assets. There are five observatories around the world, in Korea, Mongolia, Morocco, Israel, and the USA. The OWL-Net provides
topocentric right ascension and declination measurement data converted from the image pixel coordinates (Park et al. 2013).
The orbital states of low earth orbit (LEO) satellites were evaluated in previous studies based on OWL-Net data. Jo et al. (2015) utilized commercial OD software to achieve preliminary OD results and
showed that the requirements of OD can be achieved. In another study, Park et al. (2015) utilized the same software and analyzed the OD accuracy according to the number of estimation points.
In this study, OD software for the optical surveillance of space objects was developed and the orbital states of two LEO satellites, KOMPSAT-1 and Cryosat-2, were determined using the software and
OWL-Net data. In addition, the precision and accuracy of OWL-Net data were analyzed based on the simulation of potential error sources of OWLNet measurements such as noise, bias, and clock errors.
The software is available to operate independent optical surveillance systems of space objects.
The batch least squares algorithm is introduced and verified in Section 2. The error sources of the OWL-Net are analyzed in Section 3. The estimation results using actual data are addressed and
compared with known orbital information in Section 4 and Section 5 summarizes this paper.
2. BATCH LEAST SQUARES FILTER
The batch least squares algorithm is a post-processing estimation algorithm known as differential correction. It processes all data in a lump to determine the epoch states. Although this procedure is
not appropriate for real-time systems, it is generally used because of some advantages. The batch estimation is simple and more accurate and robust compared with sequential estimation algorithms,
such as the Kalman filter (Crassidis & Junkins 2011).
This section mathematically describes the algorithm including definitions of cost function, covariance matrix, and convergence criteria.
2.1 Methodology
The batch least squares algorithm estimates epoch states using a system model (F):
In case of OD,
is the measurement data set, ϵ is the measurement noise vector, and
is a solve-for-vector, which is required such as the position and velocity vector (
Schutz et al. 2004
). The system model can be separated into dynamic and measurement models.
The estimation can be referred to as an optimization, which minimizes the cost function (Q), weighted sum of the squares of the measurement residual, and the difference from a priori states:
$Q\left(\overline{\text{X}}\right)={\left[Y\text{ - }F\left(X\right)\right]}^{\text{T}}W\left[Y\text{ - }F\left(X\right)\right]+{\left[X\text{ - }{X}_{0}\right]}^{\text{T}}{\left[{P}_{{\text{ΔX}}_{\
text{o}}}\right]}^{-1}\left[X\text{ - }{X}_{0}\right]$
are the
priori covariance matrix and state vector, respectively, and
is the weight matrix. The
a priori
covariance matrix and weight matrix reflect the
a priori
uncertainty and measurement white noise, respectively:
${P}_{\text{ΔX}}=\text{ diag}{\left(\left[{\sigma }_{1}{}^{-2}\cdots {\sigma }_{n}{}^{-2}\right]\right)}^{-1}$
${\sigma }_{j}$
is the standard deviation of the
element of the solve-for-vector, n is the dimension of the solve-for vector, and m is the number of measurement data.
If the a priori states are close enough to the true states, the system model and cost function can be linearized using the Taylor expansion. Hence, a normal equation is derived to evaluate the
variate of the i^th iteration (ΔX[i]) and the solve-for-vector is then updated (Long et al. 1989):
$\begin{array}{c}\Delta \stackrel{^}{X}=\left({H}^{\text{T}}WH+{P}_{\Delta {X}_{0}}^{-1}\right)\left({H}_{i}^{\text{T}}W\Delta {Y}_{i}{P}_{\Delta {X}_{0}}^{-1}\left(\Delta {\stackrel{^}{X}}_{i-1}-{X}
_{0}\right)\right)\\ {\stackrel{^}{X}}_{i}={\stackrel{^}{X}}_{i-1}+\Delta {\stackrel{^}{X}}_{i}\end{array}$
where H is the Jacobian matrix, a partial derivative of the system model. It can be divided into state transition matrix (Φ) and sensitivity matrix (A) using the chain rule.
When the solve-for-vectors are the Cartesian position and velocity vector $\left(\text{X =}{\left[{r}_{1} {r}_{2} {r}_{3} {v}_{1} {v}_{2} {v}_{3}\right]}^{T}\right)$, and the measurement data consist
of topocentric right ascension (α) and declination (δ), the sensitivity matrix is simply formulated:
$\begin{array}{l}A={\left[\frac{\partial \alpha }{\partial X} \frac{\partial \delta }{\partial X}\right]}^{T}\\ =\left[\begin{array}{cccccc}-\frac{{\rho }_{2}}{{\rho }_{1}^{2}+{\rho }_{2}^{2}}& \frac
{{\rho }_{1}}{{\rho }_{1}^{2}+{\rho }_{2}^{2}}& 0& 0& 0& 0\\ -\frac{{\rho }_{1}{\rho }_{3}}{{\rho }^{2}\sqrt{{\rho }_{1}^{2}+{\rho }_{2}^{2}}}& \frac{{\rho }_{2}{\rho }_{3}}{{\rho }^{2}\sqrt{{\rho }_
{1}^{2}+{\rho }_{2}^{2}}}& \frac{\sqrt{{\rho }_{1}^{2}+{\rho }_{2}^{2}}}{{\rho }^{2}}& 0& 0& 0\end{array}\right]\end{array}$
where ρ is the relative position vector of the object with respect to the observatory $\left({R}_{\text{obs}}=\left[{r}_{\text{obs,1}}{r}_{\text{obs,2}}{r}_{\text{obs,3}}\right]\right)$ i.e., $\rho =
\left[{\rho }_{1},{\rho }_{2},{\rho }_{3}\right]=\left[{r}_{1}-{r}_{\text{obs,1}},{r}_{2}-{r}_{\text{obs,2}},{r}_{3}-{r}_{\text{obs,3}}\right]$. If the measurement bias (β) is also estimated,
following terms are added to the Jacobian matrix:
where k is the scaling parameter, which is set to one by default.
The whole process, which includes the prediction and update, is iterated until convergence is achieved. The con-vergence criterion is the cost reduction ratio (γ):
The estimation sequence is finished if the ratio becomes less than a small tolerance (ε) set as 10^-3 by default, and it also terminated if the ratio is negative, which means increasing cost and
maximum number of iterations are reached.
The covariance matrix (P), which contains the statistical information of the estimation, is evaluated after convergence. The covariance matrix is evaluated as follows:
In this study, the general mission analysis tool (GMAT) is used as the dynamic model (Hughes et al. 2014) and the state transition matrix is numerically evaluated.
2.2 Algorithm Verification
The covariance matrix estimated from the estimation algorithm has a statistical meaning and addresses the precision of the estimation. To confirm the validity of the estimated covariance matrix,
Monte Carlo simulations were conducted and the results were compared. If the estimated covariance is equal to the results, the estimation is valid.
Monte Carlo simulation was conducted for two different cases: states estimation with unbiased measurement data (Case 1) and states and bias estimation with biased measurement data (Case 2). Each case
is simulated 1,000 times using the LEO. Table 1 lists the detailed configuration of the simulation. Instead of GMAT, two-body dynamics were assumed for simplification purposes based on the
statistical independence of the system model. The pseudo-measurement properties of the OWL-Net measurement are as follows: the pass length is an observable period; exposure time is the time when
taking a picture; exposure period is the timespan between two pictures; the data frequency is the number of extracted data points per second.
The precision of the estimation can be evaluated by the integration of a multivariable probability function. Especially in the case of multivariable space, several methods can be used to interpret
the uncertainty of the variables such as the hyper-rectangle and hyper-ellipsoid methods (Long et al. 1989). The probability that the three-dimensional position vector is within the 3σ range, for
instance, is ~97 % and ~99 % based on the hyper-ellipsoid and hyper-rectangle methods, respectively. In this study, the hyper-ellipsoid method was used because the hyper-rectangle probability is a
simple extension of the single Gaussian probability function.
In both cases, the algorithm was verified based on the statistical significance (97 %) of the results of the covariance ellipsoid. Fig. 1 shows the comparison between the results of the Monte Carlo
simulation and the estimated covariance ellipsoid for both cases. Note that they are used only for algorithm verification and not for performance validation.
3. ERROR ANALYSIS
As opposed to the simulated observation, which only has Gaussian white noise, the real observation has more error sources such as bias and clock errors. In this section, the effects of noise, bias,
and clock errors are determined using simulations of pseudo-measurement data. The analyzed relationship of each error source with the OD result can be used to interpret and predict the actual OD
result. Table 2 lists common environments of all types of error analyses.
3.1 Noise
Noise is a random error generated by thermal radiation. It is equal to the precision of the observation and cannot be exactly estimated and corrected. To check the effect of noise on the accuracy of
the estimation, the OD simulations were conducted 100 times using pseudo-measurements and varying noise levels from 1 to 150 arcsec without bias. Note that the range of noise level is set as a known
property of the OWL-Net measurement.
As a result, the residual exhibits the same distribution as the noise but without any trend (Fig. 2), which means that the resultant residual distribution can be conversely used to analyze the
observation quality. In addition, the developed batch least squares filter reduces the a priori state error irrespective of the measurement noise.
Fig. 3 shows the root-mean-square (RMS) error of each state in earth centered inertial (ECI) and radial-tangential-normal (RTN) coordinates. It is directly proportional to the noise level, that is,
the observation precision, and much smaller than the a priori error. The magnitude of the position error depends on the axis, especially that of the z-axis and radial error are larger than that of
other axes. This might be due to the characteristics of the optical observation, which cannot include the line of sight direction information.
3.2 Bias
Bias can be caused by various factors such as coordinate transformation, catalog matching errors, and clock errors (Son et al. 2015). Although it might not be a constant due to the nonlinearity of
the system, it is considered as a constant for a short arc for simplicity in this paper. To examine how the bias affects the OD, the OD simulations were conducted 500 times using pseudo-measurements.
The pseudo-measurements are generated with a random bias of 3° (1σ), which is not estimated, and a noise level of 40 arcsec. Note that the magnitude of the bias is set large enough to include many
bias causes.
Approximately 70 % of the estimations converged; however, only 35 % of them converged with a proper answer and 30 % of the estimations even terminated in divergence. The results do not depend on the
right ascension (RA) bias but they strongly depend on the declination (DEC) bias (Fig. 4). The estimation converges only if the DEC bias is larger than approximately -2°, whereas there is no
dependence on the RA bias. The estimation accuracy of the normal cases is on the order of kilometers and that of wrong cases is on the order of several hundreds of kilometers. A separation criterion
between normal and wrong cases is the cost (Eq. (2)) of 50, where the cost of unity represents that the a priori error is removed and the residual is normalized. Fig. 5 shows that the RA bias is
randomly distributed, regardless of the criterion, while the DEC bias is notably separated due to the criterion.
This result probably depends on the orbit. Because the simulated orbit is almost polar (inclination of 97°), the declination changes more than the right ascension and the DEC bias is more influential
than the RA bias. The relationship between the orbital characteristics and the effect of bias will be analyzed in the future.
3.3 Clock Error
The OWL observatory synchronizes time using a network time protocol (NTP) server, which has a precision on the order of 0.001 sec (Son et al. 2015). The data reduction algorithm includes a step to
combine the extracted data points and time log (Park et al. 2013) based on which two types of clock errors are simulated. The first error is the time-synchronization offset (Case 1), where the whole
time log is shifted. The other error is the time-tagging error (Case 2), where parts of the time logs are shifted. In case of the time-tagging error, the portions of the shifted observation are 25 %
(Case 2-1), 50 % (Case 2-2), and 75 % (Case 2-3). All cases are simulated 100 times with a noise level of 15 arcsec corresponding to the OWL-Net without bias.
Fig. 6 describes the effects of all types of clock errors onto the position and velocity. In case of the time-synchronization offset, the position error ranges from 1.5 to 7.6 km because the offset
increases from 0.2 to 1 sec. The result corresponds to the orbital motion of LEO (~7 km/s). In contrast to the overall time offset, the time-tagging error is an error source that leads to irregular
observation data. In Case 2-1, the cost is minimized when the estimated state corresponds to the normal measurement because only 25 % of the measurement data have clock errors. On the other hand, 75
% of the measurement data have clock errors in Case 2-3 and the estimation thus corresponds to the error measurement. The estimation error is largest because the normal and abnormal measurements
occupy the same portion in Case 2-2. In conclusion, the biggest state error is ~90 km in Case 2-2; Cases 2-1 and 2-3 have a similar level of state error.
4. OWL-NET APPLICATION
The OWL-Net observation data (topocentric right ascension, α, and declination, δ) of two LEO satellites, KOMPSAT-1 and Cryosat-2, are available. KOMPSAT-1 is a Korean Earth observation satellite at
an altitude of 685 km, which was launched in 1999 and completed its mission in 2008 (Kim et al. 2015). Cryosat-2, is another Earth observation satellite mission managed by the European Space Agency
(ESA). It was launched in 2010 and measures the thickness of sea ice using radar altimetry at an altitude of 725 km (Kurtz et al. 2014). The data were applied to the estimation algorithm, which was
verified to provide reliable estimation results. Furthermore, the characteristics of OWL-Net observation data were determined based on the error analysis explained in Section 3.
4.1 Data Overview
The OWL-Net observation data were provided in form of report files, which include the name of the target satellite, location of the observatory, exposure time, charged-coupled device (CCD)
temperature, and time-tagged data points. Each file reflects a tracking pass of the satellite and consists of pictures from which the data points were extracted. Fig. 7 shows the data points
extracted from a report file of KOMPSAT-1 and Cryosat-2. The arc of 4 min measured on November 5, 2014, and the arc of 5 min measured on May 25, 2015, were used for KOMPSAT-1 and Cryosat-2,
respectively. Note that the properties of the arc, such as length, extraction period, and number of data points, are different in the two cases.
4.2 Orbit Determination
The orbit of each satellite was determined using OWL-Net observations and the developed algorithm (Section 2). The OD is conducted with and without bias estimation. The results of the Gauss method,
which is one of the initial OD techniques, were used as a priori states. Other configuration details are listed in Table 3.
In case of KOMPSAT-1, the residual uniformly decreased from about 0.1° to 14 arcsec throughout the whole observation for both RA and DEC (Fig. 8). The estimated biases are 0.2738° and -0.0148° for RA
and DEC, respectively.
The residuals of Cryosat-2 also reduced from several degrees to ~0.1° after OD processing. The pattern, however, is quite distinct from that of KOMPSAT-1; the residual is not uniform but
exceptionally large in the beginning and end of the observation (Fig. 9). For this reason, only the middle part of Cryosat-2 data was used for OD, that is, data editing. To exclude abnormal
measurement data from the OD process, data points extracted from specific pictures were removed, while the editing usually eliminates part of the measurement of the threshold such as 3σ (Long et al.
1989). After data editing, the RA and DEC residuals are 74 and 28 arcsec, respectively (Fig. 10). The estimated RA bias is -0.0569° and the DEC bias is 0.0590°.
4.3 Orbit Quality Assessment
The results were compared with the two-line elements (TLE) of the corresponding satellite to determine the ac-curacy of the estimation. The direct comparison of the Cartesian states, however, is
improper because the estimated ephemeris has an accumulated error due to the epoch state and system errors. For this reason, the estimation result is compared with TLE in form of mean orbital
elements, which do not change much due to orbital motion (Table 4). The estimation of the bias shows that the difference of the orbital elements with respect to TLE is smaller in both satellites. The
differences of the semi-major axis (SMA), inclination, and right ascension of the ascending node (RAAN) are < 1 %. Because both orbits are almost circular, the arguments of the perigee and mean
anomaly do not match well but their sums (argument of latitude) do. The eccentricity, however, is estimated to be several times the TLE, which might be due to the velocity error.
Furthermore, the result of the Cryosat-2 estimate was compared with the reference trajectory, which was published in consolidated prediction format (CPF) based on the international laser ranging
system (ILRS) (Pearlman et al. 2002). The predicted trajectory is generated by prediction centers and regularly checked by several analytic centers, such as the Natural Environment Research Council
(NERC); the accuracy is approximately several hundreds of meters (NERC Space Geodesy Facility 2017). It presents the three-dimensional position in the true-body-fixed of date system of the
international terrestrial reference frame (ITRF). In this study, CPF data provided by ESA were used to analyze, where the epoch time of the trajectory is at 0:00 am on May 25, 2015, which is
consistent with the observation date; the time interval of the position is 3 min. Fig. 11 shows the position difference in ITRF and RTN coordinates; each reference point is interpolated by the tenth
order polynomial. The position difference at the epoch is ~0.8 km and most of the difference corresponds to the radial component and to the result of the simulation in Subsection 3.1. The result is
similar to the results of previous studies, which reported that the RMS difference between TLE and the estimated orbit is < 10 km in case of Cryosat-2 (Park et al. 2015).
Considering the field of view of the OWL-Net, which is 1.75° (Park et al. 2012), the position error of the LEO satellites should be lower than 20 km at the next arc for tracking. Fig. 12 shows the
position difference of the estimated orbit and TLE with respect to the reference trajectory in ITRF. Both are propagated for 90 min, a period of the LEO satellite; the maximum position difference of
the estimation is ~70 km, while the TLE difference is ~30 km. This might be due to the velocity difference, which is also related to the eccentricity error. The estimation accuracy can be improved by
resolving unknown error sources of the OWL-Net (Jo et al. 2015).
Based on the error analysis that shows that the precision of the measurement data is the same as that of the residual, the precision of the OWL-Net observation data is tens of arcsec. Based on the
bias estimation, the accuracy of the observation data is also determined to be of sub-degree level.
Optical observation is suitable for the surveillance of space objects. The OWL-Net is a Korean optical surveillance system that tracks and monitors domestic satellites. In this study, the orbital
states of two LEO satellites were determined using OWL-Net data and a batch least squares filter, which was developed and statistically verified. In addition, the precision and accuracy of OWL-Net
data were analyzed based on the results of software simulations for error analysis.
The batch least squares filter processes all data to determine the epoch states; it is more robust to temporal measurement errors. The objective function is defined as the sum of the weighted
residual and difference of the a priori states. The system model is composed of dynamic and measurement models. In this study, GMAT was utilized as the dynamic model and geometric RA/DEC was used as
the measurement model. The system model was linearized to estimate the most probable epoch states; it turns into the states transition matrix and sensitivity matrix, where each matrix is numerically
evaluated and analytically derived.
For statistical verification, Monte Carlo simulation was performed under two different conditions (unbiased meas-urement data without bias estimation; biased measurement data with bias estimation)
and the results were compared with the covariance matrix. The results correspond to the multivariate analysis of the covariance in both cases, where 97 % of the estimation results plot inside the 3σ
covariance ellipsoid.
Potential error sources of OWL-Net, noise, bias, and clock errors were analyzed using software simulations. The estimation accuracy depends linearly on the noise level and the declination bias has a
significant influence on the estimation in a polar orbit. The estimation converges only with a DEC bias larger than approximately -2° and the estimation error increases to > 100 km when the DEC bias
is larger than 2°. This is due to the orbital characteristics, although it is not the general feature. Two types of clock errors were analyzed, time synchronization offset and time-tagging error. The
position error is ~7.6 km at a time-synchronization offset of 1 sec, which corresponds to an orbital velocity of ~8 km/s of LEO. The time-tagging error, however, increases the position error up to 90
km because it leads to irregular observation data.
The OWL-Net measurement data of KOMPSAT-1 and Cryosat-2 were provided by KASI and applied to the batch least squares algorithm. The Cartesian states were estimated by default and the measurement bias
was optionally estimated. The estimated orbital states of both satellites are similar to those of TLE; however, the eccentricity error is significantly large. In case of Cryosat-2, the estimated
trajectory was compared to the CPF trajectory in ITRF. Its difference is 0.8 km at the epoch time, which is similar to that of TLE and is in agreement with previous results (Park et al. 2015).
However, the difference is ~70 km after 90 min, while the corresponding value of TLE is ~30 km. The precision of the OWL-Net data is tens of arcsec and the accuracy is of sub-degree level.
This study has nobility to demonstrate the operating capability of independent optical surveillance system of space objects. Unlike previous studies, which employed commercial software to examine
OWL-Net measurement data, this study utilizes newly developed OD software. The results are similar to those of TLE. Because the software was statistically verified and examined under various
conditions, the results are reliable and indicate the possibilities of independent space surveillance operations. The position error, however, increased when orbit propagation was performed, which
might be due to the velocity error and can be improved through the use of parameter tuning and nonlinear estimation techniques. Furthermore, the precision and accuracy of OWL-Net measurement data was
determined by interpreting the OD results. The performance of the OWL-Net can be improved by stabilizing the system to resolve unknown error sources; other reliable optical measurements can be used
to check the performance of the OD software and analyze OWL-Net measurement data in the future.
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Tip of the Week #8: Doing Math in TBC commands
Did you know you can use TBC as calculator? Yes, that’s right. You don’t have to use your calculator if you are doing some simple math like adding or subtracting values before you key them in TBC or
computing units.
First, open any command in TBC that expect some input e.g. Create Points, or Create Polyline and click in the coordinate field. You can see in the image below where we used “12.32+20” as an example
and after hitting “Tab” got “32.320”.
You can do mathematical expression of addition, subtraction, division and multiplication (+, -, *, /). You can have multiple numbers e.g. “12+4+6” to get “22” or you can use “12+4*2” to get “20”
(yes, we know the priority of operations :)).Note that you cannot use spaces “ “ between the number and mathematical operators. The numbers provided will be in project units.
If you have a units on paper in metric and your project is in survey feet or vice versa, you can do that as well. Let’s take an example of a project where you have setup of survey feet and you are
keying in a value of 150m (meters).
Open a command e.g. Create Point and in the elevation field type in “150m” and hit “Tab”. You can use standard abbreviations for distance units like “mm, cm, m, km, ft”.
We hope this tip saves you few minutes on your daily work by not having to open a calculator to do some quick calculations.
Happy Surveying,
TBC Team
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Centimeter to X-unit
How to convert Centimeter to X-unit ?
If You Want To Convert Centimeter to X-unit.then easly follow step :
➡️ Enter Centimeter value in input field
➡️ then click convert button
➡️ Result are show in bellow Centimeter to X-unit Converters Result box
➡️ if you want to copy to use in your projects to directly click to copy button
➡️ this copy button will show when you are type Centimeter value and click convert button.
Example of Convert Centimeter to X-unit:
➡️ 1 Centimeter to X-unit = 99792431742 X-unit
➡️ 2 Centimeter to X-unit = 199584863484 X-unit
➡️ 3 Centimeter to X-unit = 299377295226 X-unit
➡️ 4 Centimeter to X-unit = 399169726968 X-unit
➡️ 5 Centimeter to X-unit = 498962158710 X-unit
➡️ 6 Centimeter to X-unit = 598754590452 X-unit
➡️ 7 Centimeter to X-unit = 698547022194 X-unit
➡️ 8 Centimeter to X-unit = 798339453936 X-unit
➡️ 9 Centimeter to X-unit = 898131885678 X-unit
➡️ 10 Centimeter to X-unit = 997924317420 X-unit
what is X unit length units
The length unit known as the x unit (symbol xu) is about equivalent to 0.1 pm (10−13 m).[1] It is employed to quote gamma and X-ray wavelengths.
The Swedish physicist Manne Siegbahn (1886–1978) first defined the x unit in 1925 because it could not be measured directly at the time. Instead, the definition was given in terms of the distance
between the planes of the calcite crystals that were employed in the measuring device. At 18 °C, one x unit was defined as 1⁄3029.04 of the calcite plane spacing.
The wavelengths of the two most often utilized X-ray lines in X-ray crystallography are used to establish the two distinct x units that are in use today:
The copper x unit (symbol xu(Cu Kα1)) is defined to yield an exact wavelength of 1537.400 xu(Cu Kα1) for the Kα1 line of copper; similarly, the molybdenum x unit (symbol xu(Mo Kα1)) is defined to
yield an exact wavelength of 707.831 xu(Mo Kα1).
What is the X unit in 1 cm?
It is 99793208513.319, to be exact. Assumedly, you are converting X units to centimeters.
More information about each measuring unit is available here:
X in centimeters or units The meter is the SI base unit for length. 9979320851331 is equal to one meter.100 cm, or 9 X unit. Keep in mind that rounding errors can happen, so double-check the
outcomes. Learn how to convert X units to centimeters with this page. To convert the units, enter your own numbers in the form!
Centimeter to X-unit Conversion Table if viewport user X-unit value up to 10 digits ahead of it.
Here is Show Centimeter to X-unit Converter Value up to 10 digits ahead of it value you are enter.This will be visible only when you enter a value and click on the Convert button.
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WAEC Further Mathematics Questions and Answers 2023/2024 (Objectives and Essay) - Bekeking
WAEC Further Mathematics Questions and Answers 2023/2024 (Objectives and Essay)
WAEC Further Mathematics, WAEC Further Mathematics, WAEC Further Maths, WAEC Further Mathematics questions and answers, WAEC Further Mathematics syllabus 2023, past questions on mathematics, WAEC
further mathematics expo 2023, WAEC past questions pdf download
WAEC Further Mathematics questions and answers for 2023/2024 are here. Are you a WAEC candidate?
If your answer is yes, this post will show you the WAEC Further Mathematics answers and the tricks you need to excel in your WAEC exam.
WAEC questions are set and compiled by the West African Senior School Certificate Examination Board (WASSCE). Make sure you follow the instructions provided by WAEC.
2023/2024 WAEC Further Mathematics Questions and Answers
Symbols used:
^ means raise to power
/ division
The Further Mathematics examination paper is going to comprise of two papers
1. Paper 1: Essay
2. Paper 2: Objectives
PAPER 1: will consist of forty multiple-choice objective questions, covering the entire Further Mathematics syllabus. Candidates will be required to answer all questions in 1 hour for 40 marks. The
questions will be set from the sections of the syllabus as stated hereunder:
Pure Mathematics – 30 questions
Statistics and probability – 4 questions
Vectors and Mechanics – 6 questions
PAPER 2: will consist of two sections, Sections A and B, to be answered in 2 hours for 100 marks.
Section A will consist of eight compulsory questions that are elementary in type for 48 marks. The questions shall be distributed as follows:
Pure Mathematics – 4 questions
Statistics and Probability – 2 questions
Vectors and Mechanics – 2 questions
Section B will consist of seven questions of greater length and difficulty put into three parts: Parts I, II, and III as follows:
Part I: Pure Mathematics – 3 questions
Part II: Statistics and Probability – 2 questions
Part III: Vectors and Mechanics – 2 questions
WAEC Further Mathematics Answers for 2023 will be posted here today, 12th May during the examination.
WAEC Further Maths Answers 2023 Loading…
WAEC Further Mathematics OBJ Answers:
1-10: BBCCCDDCDB
11-20: BBBBBADDBA
21-30: CCCDABDBAA
31-40: ACDDADDDBD
No 1 and 2
No (4)
No (7)
No (10)
No (11)
WAEC Further Maths Questions
The questions below are for practice.
1. Given the matrix M=
2 -4 -4
1 1 -2
find |M|
A. -24
B. -8
C. 8
D. 24
E. 48
ANSWER: A
2. The gradient of a curve is 8x+2 and it passes through (1,3). Find the equation of the curve
A. y = 4x^2 + 2x + 3
B. y = -4x^2 + 2x -3
C. y = 4x^2 – 2x + 3
D. y = 4x^2 + 2x + 3
E. y= 4x^2 – 2x – 3
ANSWER: A
3. Given that y = 3x^3 + 4x^2 + 7. Find dy/dx at x = 1
A. 14
B. 15
C. 17
D. 30
E. 35
ANSWER: C
4. Integrate 3x^2 + 4x – 8 with respect to x
A. x^3 + 2x^2 + 8x + k
B. 6x + 4 + k
C. x^3 – 2x^2 + 8x + k
D. x^3 + x^2 – 8x + k
E. x^3 + 2x^2 – 8x + k
ANSWER: A
WAEC Further Maths Essay and Objective 2023 (EXPO)
The above questions are not exactly 2023 WAEC Further Mathematics questions and answers but likely WAEC FurtherMaths repeated questions and answers.
These questions are strictly for practice. The 2023 WAEC Further Mathematics expo will be posted on this page during the WAEC Further Mathematics examination. Keep checking and reloading this page
for the answers.
WAEC Mathematics Questions and Answers 2023 Loading…
Further Maths OBJ
If you have any questions about the 2023 WAEC Further Mathematics questions and answers, do well to let us know in the comment box.
Last Updated on May 12, 2023 by Admin
50 thoughts on “WAEC Further Mathematics Questions and Answers 2023/2024 (Objectives and Essay)”
1. Can you please, help with 2023 further mathematics waec questions and answers
2. thank you for this answer even without the expo I got 30 questions correctly.
but please I need to do expo for physics and chemistry, will the questions ready tomorrow?
3. I need further math questions and answers
4. Plz I need further maths questions and answers now plz.
5. Can you please, help with 2023 further mathematics waec questions and answers?
6. Pls can I get the theory part
7. Pls can u send further maths obj and theory for me ????????????????????
Leave a Comment
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Central tendency
Jump to navigation Jump to search
Part of a
convergent series on
We are number one
Central tendency is a term in descriptive statistics which gives an indication of the typical score in that data set. The three most common measures typically used for this: the mean, the median (not
to be confused with Medium) and the mode.^[1]^[2] However, there are other measures of central tendency.
Average is an often-used term that may refer to any measure of central tendency, though in casual conversation it is generally assumed to refer to the mean.
Arithmetic mean[edit]
The arithmetic mean is easily calculated by summing up all scores in the sample and then dividing by the number of scores in that sample. The mean for the sample 5, 6, 9, 2 would therefore be
calculated as follows:
${\displaystyle {\frac {5+6+9+2}{4}}{=5.5}}$
The mathematical formula can be expressed as:
${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}a_{i}}$
where ${\displaystyle n}$ is the total number of samples.
The above is the arithmetic mean; there are also other means, such as the geometric mean and the harmonic mean, but usually the arithmetic mean is meant if the type of mean is not specified.
Geometric mean[edit]
All the values are multiplied and then the n^th root is taken (where n is the total number of scores). Useful in some geometric (heh) contexts; for example, the area of an ellipse is equal to that of
a circle whose radius is the geometric mean of the ellipse's semi-major and semi-minor axes. Has the neat property of being equal to e raised to the arithmetic mean of the natural logarithms of the
values being averaged. (The same principle holds for other bases, so for example it could be defined as ten raised to the arithmetic mean of the common logarithms.) The mathematical formula can be
expressed as:
${\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}}$
Harmonic mean[edit]
The mean obtained by taking the reciprocal of the arithmetic mean of the reciprocals of a set of (nonzero) numbers. One of the most memorable uses of the harmonic mean is in physics, where the
equivalent resistance of a set of resistors in parallel is the harmonic mean of their resistances divided by the number of resistors. The same principle applies to capacitors placed in series. The
mathematical formula can be expressed as:
${\displaystyle {\frac {n}{\displaystyle \sum \limits _{i=1}^{n}{\frac {1}{x_{i}}}}}}$
Other means[edit]
• Weighted mean: If the values have different "weights" (not in the physical sense), the most appropriate way to calculate an average is to calculate a weighted mean. This is done by multiplying
every value a[i] with its corresponding weight w[i] (i being the index or rank of the data set) and the summing up all the products (i.e. Σa[i]w[i] = a[1]w[1] + a[2]w[2] + a[3]w[3] + ... + a
[(i-1)]w[(i-1)] + a[i]w[i]) and then dividing it all by the sum of the weights (i.e. Σw[i] = w[1] + w[2] + w[3] + ... + w[(i-1)] + w[i]). Ergo, the final formula:
Weighted mean = (Σa[i]w[i]):(Σw[i])
For example, a school test has 4 subjects with different weights (in parentheses), going from 0 (minimum) to 10 (maximum): Mathematics (3), physics (2), chemistry (1) and biology (1). Student A got
the following grades, respectively: 6, 4, 8, 9. The weighted mean therefore is:
Weighted mean = (6*3 + 4*2 + 8*1 + 9*1)/(3 + 2 + 1 + 1) = (18 + 8 + 8 + 9)/(7) = 43/7 ≈ 6
It can also be used to combine measurements from classes with different sizes, where the weight corresponds to the relative size of each class.
• Truncated mean (or trimmed mean): this is similar to the arithmetic mean, but outliers are discarded by dropping some part of the probability distribution. If you discard the lowest 25% and
highest 25%, this is known as the interquartile mean. It is used in the ISU Judging System for figure-skating, where the highest and lowest scores are dropped and the rest are averaged.^[3]
• Mid-range: the arithmetic mean of the highest and lowest value, this is very vulnerable to outliers and not much use in most circumstances.^[4]
The median is defined as the value that lies in the middle of the sample when that data set is ordered (by rank).^[1]
It is calculated by ranking all the scores in order and taking the one in the middle. When there is an even number, conventionally, the mean of the two in the middle is taken.
For example, the median for the already ordered data set [2, 12, 12, 19, 19, 20, 20, 20, 25]. It has 9 data points (n = 9; it is the number of "observations" in that data set) and the value in the
middle would be the fifth rank, which is 19.
However, as another example, the data set [2, 12, 12, 19, 19, 20, 20, 20, 25, 25] has an even number of observations (n = 10), the median would be between 19 (fifth rank) and 20 (sixth rank), which
are the two scores in the middle. Therefore, it is the mean of those two, which is 19.5.
The mode is simply the most frequently occurring value in the data set.
For example, the mode of the sample [3, 3, 2, 1, 4, 3] would be 3 (since it appears thrice). If there is a second number that occurs just as frequently within the sample, then it can be described as
having two modes (bimodal). However, some some sources will identify such a sample as having no mode.^[5]
The three methods can be useful in different ways, and how they relate can give information about your statistical sample. The mean is the most intuitive measure of the concept of "average", while
the median is most useful for breaking samples into groups (e.g., quartiles). For evenly distributed samples, or symmetrically distributed samples, the median and mean (and usually the mode) should
match. The difference between them is an indicator of how skewed the data is. For instance, in economics, income is not evenly distributed (not by a long shot), nor even symmetrically distributed,
and the mean value is easily shifted by those high earners at the top — for this reason the median is most often used.^[6]^[7]
External links[edit]
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Who ensures the integration of real-world examples in their assistance with Statistical Process Control assignments? | Operations Management Assignment Help
Who ensures the integration of real-world examples in their assistance with Statistical Process Control assignments? This issue was presented by the National Security Platform from (the National
Security Platform) this is the issue of using real-world tools (fuzzy-wabbles, fuzzy and fuzzy-wabbled sets) to build a concrete behavioral control model which enables detecting failures, determining
errors, etc. In this order we can see that fuzzy-wabbling, fuzzy-wabbled sets and fuzzy-wabble are the most common fuzzy rules that help in the deployment of certain types of devices. I believe in
doing that by letting us to use fuzz-wabble sets to predict when a sensor or device comes out of the test and the correct outcome of the operation of a smart meter. Barely 2 times better than fuzzy
behavior The example I wrote is from Google Street View. A big problem faced in smart meter deployment is that the device itself is not always the most important part of the setup. Without going to a
much wide area in this case, the test must be very sophisticated to quickly learn which sensors are connected to and which are not. For example a small “sensor” is connecting to devices which are
very important for monitoring their activity. If we look at the deployment history we can see that in the past several months 3 sensor counts and 3 alarm rates have been detected. Thereby we have
covered sensors which are connected to each other (as we are not used to the definition of the term in this paper) by the fuzzy-wabbles. Now, in our case, all this is done by constructing the sensor
number counter that corresponds to all sensors and every alarm counts. Yes, this is a very broad field of work and one of the biggest problems one faces in the deployment is to use methods such as
the fuzzy-wabble that gives a mathematical justification of the performance of the fuzzy-wabble approach I talk about earlier. You can check this paper and check more references for more detailed
information about fuzzy-wabbles and concrete fuzzy-wabbles in the latest HMMM/hmmfuzzler library which can be accessed through the HMMM/hmmfuzz library in HMMFuzzy. Source The paper on the deployment
is from the National Security Platform. The picture and graphs in Figure 3-8 of Section 2 are presented with Figure 3-8 by showing exactly all the devices connected to a smart meter by fuzzy-wabbles
(buzzing) and building a new object by building fuzzy-wabbles around this object (buzzing) respectively. As you can see our main goals in this part of the paper are to let us make a valid conclusion
about the usage of fuzzy-wabbles as a primary tool for deployment. Even though the fuzzy-wabbles are implemented in RIM, the author does not find any data with the same characteristics as they will
be used in the later part of this topic. Note: If you are ready to take ownership and read these articles then you can download them and also check full resources written under the copyright. We have
also made available a limited number of papers on this topic with the following additions for us which I would highly recommend to anyone who works. While this topic is mostly related to practical
application of fuzzy-wabbles, our implementation can also be used to find and apply fuzzy-wabbles for a wider range of applications. Further improvements to the implementation of fuzzy-wabbles The
use of fuzzy-wabble is one sure feature that we see on the development of the fuzz-wabbles, which are shown in Figure 3-8.
Do Online Courses Have Exams?
It does contain one big new tool which uses fuzzy bit-words in the “spflake”, as a specialized addition to the standard fuzzy-wabble implementation in RIM. When speaking about �Who ensures the
integration of real-world examples in their assistance with Statistical Process Control assignments? I will be considering the following two research subjects related to the analysis of real-world
data of US Congress and BCL, I will be considering the following research subject with specific subjects in order to compare the levels of some factors in the power of different categories of
analysis (specific studies are being pursued although it is stated that the researchers are conducting the data analysis part of their research). Thus the field-wide discussion on this topic is
having an interest relating to the analysis of the distribution of data set of public records for legal records in European law. Authors === ABBAH: Thank you for coming to present the work done in
very interesting article. The task we have formulated is as follows: First of all, we will consider the framework in which we have to analyze the data that we are dealing with on the basis of the
data handling rules. In fact we can take as the basis of our consideration to represent a basis for the analysis of the data set against public records, we will be on making a comparison to the data
set in different areas, this will be the study of how it is that our data are the basis of the analysis. In order to obtain a practical working example, we may have to simulate for example scenario
one that is having a different condition in the status of a public database for more precisely to estimate the parameters considered in the data set. Such a such a series of simulation procedures
will be to have a procedure to fit the data set against the result of any chosen parameter and finally by looking for the estimation of the parameters, and this such procedure will be repeated until
one has a picture of the data which is to be obtained. A very common type of the data is collected from the courts, where the data are usually collected for legal production. An interesting example
Related Site this type of data is the collection of a lot of information including, for example, the form of a birth certificate, the amount received in the case of the death of the child and the
status of the child in general. Finally, I have to give what I will say about the basis of the analysis set which we are going to study in our new perspective. try this web-site bit about background
background data and literature on the use of statistics for the formulation of statistical systems ====================================================================================================
===================== It is interesting that this kind of data set of public records, designed between 1966 and 1974, is a very popular approach for the analysis of the data that is being gathered by
Congress or a law. On the other hand, information is collected by governments, the law is trying to give a working method for the collection of information, in spite of the fact that new results here
coming. This does not mean that other methods for generating data collection will be more in demand. For the data set that is being used it is rather necessary to use more technical methods than at
the moment due to the difference in data construction and availability described above. In fact depending on this phenomenon, thereWho ensures the integration of real-world examples in their
assistance with Statistical Process Control assignments? In particular, we take into consideration the impact of population and sample size on a set of general methods deemed to be a significant
contribution in our efforts to use them, some of which have been given freely and are described in detail below. Two specific cases that we explored and show, e.g., how to perform the correct
identifiability tests, are demonstrated. [**Inferior Invariance Sampling Method**]{} This is considered a particularly useful methodological tool for both empirical studying and applied statistical
analysis, because in itself it can help us to perform a number of *maintaining* statistical testations, and perhaps more importantly, it also gives one a way to perform *correction* tests
[@Bertsch2015; @Pretato2012], which require non-uniform coverage and the introduction of additional data.
We Take Your Online Class
In its original manuscript, this paper was originally written in the common form of a full-body imaging program with one variable and the other variable as first level outcomes (i.e., the size of the
datasets). However, after a publication of the original [@Bertsch2015], we published a proof of concept paper, which provided several significant new contributions. Subsequently, several new works
that provide additional support for this approach were published earlier, as in the [@Pretato2012] approach. Finally, a multivariate transformation of the $Y$-subsample to the $N$-subsample to be
compared, by an operator which explicitly generalizes to a few sample sizes in our aim, is found in [@Xie2009; @Xie2014]. The original and new papers presented in the original [@Bertsch2015] papers
were originally named [@Bertsch2013; @Bertsch2013] and reviewed by @Pretato2012. They were initially described, again, in the same paper but as new datasets. However, some of the new works \[i.e., a
sample size of 5,000\] all give only numeraire transformations of the $Y$-subsample to be compared between the $m$-subsubsample and the first-level sample, and based on quantitative estimates
provided in an earlier paper \[puntourablemath\]. This last paper, being a hybrid approach, is characterized, in particular, as a separate paper only and do not discuss the statistical interpretation
of the analysis findings. As a result, many not proved, and some of them did not contribute, and would never gain traction, it is expected that future methods will perform better [@Tardarenko2013;
@Cristolla2015; @Bertsch2014; @Chu2015]. Our main purpose is to use these methods to perform numerical simulations, where a non-uniform increase of sample size would be a further improvement on the
prior results, as well as to make comparison with existing methods possible for the integration of real-
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Industry Application – Banking – Integrated Risk Management, Probability of Default, Economic Capital, Value at Risk, and Optimal Bank Portfolios
File Name: Multiple files (see chapter for details of example files used)
Location: Various places in Modeling Toolkit
Brief Description: Illustrates the use of several banking models to develop an integrated risk management paradigm for the Basel II/III Accords
Requirements: Modeling Toolkit, Risk Simulator
Modeling Toolkit Functions Used: MTProbabilityDefaultMertonImputedAssetValue, MTProbabilityDefaultMertonImputedAssetVolatility, MTProbabilityDefaultMertonII,
MTProbabilityDefaultMertonDefaultDistance, MTProbabilityDefaultMertonRecoveryRate, MTProbabilityDefaultMertonMVDebt
With the Basel II/III Accords, internationally active banks are now allowed to compute their own risk capital requirements using the internal ratings-based (IRB) approach. Not only is adequate risk
capital analysis important as a compliance obligation, but it also provides banks the ability to optimize their capital by computing and allocating risks, performing performance measurements,
executing strategic decisions, increasing competitiveness, and enhancing profitability. This chapter discusses the various approaches required to implement an IRB method, and the step-by-step
models and methodologies in implementing and valuing economic capital, value at risk, probability of default, and loss given default, the key ingredients required in an IRB approach, through the use
of advanced analytics such as Monte Carlo and historical risk simulation, portfolio optimization, stochastic forecasting, and options analysis. The use of Risk Simulator and the Modeling Toolkit
software in computing and calibrating these critical input parameters is illustrated. Instead of dwelling on theory or revamping what has already been written many times, this chapter focuses solely
on the practical modeling applications of the key ingredients to the Basel II/III Accords. Specifically, these topics are addressed:
• Probability of Default (structural and empirical models for commercial versus retail banking)
• Loss Given Default and Expected Losses
• Economic Capital and Portfolio Value at Risk (structural and risk-based simulation)
• Portfolio Optimization
• Hurdle Rates and Required Rates of Return
Please note that several white papers on related topics such as the following are available by request (send an e-mail request to [email protected]):
• Portfolio Optimization, Project Selection, and Optimal Investment Allocation
• Credit Analysis
• Interest Rate Risk, Foreign Exchange Risk, Volatility Estimation, and Risk Hedging
• Exotic Options and Credit Derivatives
To follow along with the analyses in this chapter, we assume that the reader already has Risk Simulator, Real Options SLS, and Modeling Toolkit installed and is somewhat familiar with the
basic functions of each software. If not, please refer to www.realoptionsvaluation.com (click on the Download link) and watch the getting started videos, read some of the getting started case
studies, or install the latest trial versions of these software programs. Alternatively, refer to the website to obtain a primer on using these software programs. Each topic discussed will start with
some basic introduction to the methodologies that are appropriate, followed by some practical hands-on modeling approaches and examples.
Probability of Default
The probability of default measures the degree of likelihood that the borrower of a loan or debt (the obligor) will be unable to make the necessary scheduled repayments on the debt, thereby
defaulting on the debt. Should the obligor be unable to pay, the debt is in default, and the lenders of the debt have legal avenues to attempt a recovery of the debt, or at least partial repayment of
the entire debt. The higher the default probability a lender estimates a borrower to have, the higher the interest rate the lender will charge the borrower as compensation for bearing the higher
default risk.
The probability of default models are categorized as structural or empirical. Structural models look at a borrower’s ability to pay based on market data such as equity prices and market and book
values of assets and liabilities, as well as the volatility of these variables. Hence, these structural models are used predominantly to estimate the probability of default of companies and countries
, most applicable within the areas of commercial and industrial banking. In contrast, empirical models or credit scoring models are used to quantitatively determine the probability that a loan or
loan holder will default, where the loan holder is an individual, by looking at historical portfolios of loans held and assessing individual characteristics (e.g., age, educational level, debt to
income ratio, and so forth). This second approach is more applicable to the retail banking sector.
Structural Models of Probability of Default
The probability of default models are models that assess the likelihood of default by an obligor. They differ from regular credit scoring models in several ways. First of all, credit scoring models
usually are applied to smaller credits—individuals or small businesses—whereas default models are applied to larger credits—corporations or countries. Credit scoring models are largely statistical,
regressing instances of default against various risk indicators, such as an obligor’s income, home renter or owner status, years at a job, educational level, debt to income ratio, and so forth,
discussed later in this chapter. Structural default models, in contrast, directly model the default process and typically are calibrated to market variables, such as the obligor’s stock price,
asset value, debt book value, or the credit spread on its bonds. Default models find many applications within financial institutions. They are used to support credit analysis and determine the
probability that a firm will default, value counterparty credit risk limits, or apply financial engineering techniques in developing credit derivatives or other credit instruments.
The model illustrated in this chapter is used to solve the probability of default of a publicly-traded company with equity and debt holdings, and accounting for its volatilities in the market (Figure
92.1). This model is currently used by KMV and Moody’s to perform credit risk analysis. This approach assumes that the book value of asset and asset volatility are unknown and solved in the model;
that the company is relatively stable; and that the growth rate of the company’s assets is stable over time (e.g., not in start-up mode). The model uses several simultaneous equations in
options valuation coupled with optimization to obtain the implied underlying asset’s market value and volatility of the asset in order to compute the probability of default and distance to default
for the firm.
Illustrative Example: Structural Probability of Default Models on Public Firms
It is assumed that the reader is well versed in running simulations and optimizations in Risk Simulator. The example model used is the Probability of Default – External Options Model and can be
accessed through Modeling Toolkit | Prob of Default | External Options Model (Public Company).
To run this model (Figure 92.1), enter in the required inputs:
• Market value of equity (obtained from market data on the firm’s capitalization, i.e., stock price times number of stocks outstanding)
• Market equity volatility (computed in the Volatility or LPVA worksheets in the model)
• Book value of debt and liabilities (the firm’s book value of all debt and liabilities)
• Risk-free rate (the prevailing country’s risk-free interest rate for the same maturity)
• Anticipated growth rate of the company (the expected annualized cumulative growth rate of the firm’s assets, which can be estimated using historical data over a long period of time, making this
approach more applicable to mature companies rather than start-ups)
• Debt maturity (the debt maturity to be analyzed, or enter 1 for the annual default probability)
The comparable option parameters are shown in cells G18 to G23. All these comparable inputs are computed except for Asset Value (the market value of assets) and the Volatility of Asset. You will need
to input some rough estimates as a starting point so that the analysis can be run. The rule of thumb is to set the volatility of the asset in G22 to be one-fifth to half of the volatility of equity
computed in G10, and the market value of assets (G19) to be approximately the sum of the market value of equity and book value of liabilities and debt (G9 and G11).
Then, optimization needs to be run in Risk Simulator in order to obtain the desired outputs. To do this, set Asset Value and Volatility of Asset as the decision variables (make them continuous
variables with a lower limit of 1% for volatility and $1 for assets, as both these inputs can only take on positive values). Set cell G29 as the objective to minimize as this is the absolute error
value. Finally, the constraint is such that cell H33, the implied volatility in the default model, is set to exactly equal the numerical value of the equity volatility in cell G10. Run a static
optimization using Risk Simulator.
If the model has a solution, the absolute error value in cell G29 will revert to zero (Figure 92.2). From here, the probability of default (measured in percent) and the distance to default (measured
in standard deviations) are computed in cells G39 and G41. Then the relevant credit spread required can be determined using the Credit Analysis – Credit Premium model or some other credit spread
tables (such as using the Internal Credit Risk Rating model).
The results indicate that the company has a probability of default at 0.56% with 2.54 standard deviations to default, indicating good creditworthiness (Figure 92.2).
A simpler approach is to use the Modeling Toolkit functions instead of manually running the optimization. These functions have internal intelligent optimization routines embedded in them. For
instance, the following two functions perform multiple internal optimization routines of simultaneous stochastic equations to obtain their respective results, which are then used as an input into the
MTProbabilityDefaultMertonII function to compute the probability of default:
See the model for more specific details.
Illustrative Example: Structural Probability of Default Models on Private Firms
Several other structural models exist for computing the probability of default of a firm. Specific models are used depending on the need and availability of data. In the previous example, the firm is
a publicly-traded firm, with stock prices and equity volatility that can be readily obtained from the market. In the present example, we assume that the firm is privately held, meaning that there
would be no market equity data available. This example essentially computes the probability of default or the point of default for the company when its liabilities exceed its assets, given the
asset’s growth rates and volatility over time (Figure 92.3). Before using this model, first, review the preceding example using the Probability of Default – External Options Model. Similar
methodological parallels exist between these two models, and this example builds on the knowledge and expertise of the previous example.
In Figure 92.3, the example firm with an asset value of $12M and a debt book value of $10M with significant growth rates of its internal assets and low volatility returns a 1.15% probability of
default. Instead of relying on the valuation of the firm, external market benchmarks can be used, if such data are available. In Figure 92.4, we see that additional input assumptions are required,
such as the market fluctuation (market returns and volatility) and the relationship (correlation between the market benchmark and the company’s assets). The model used is the Probability of Default –
Merton Market Options Model accessible from Modeling Toolkit | Prob of Default | Merton Market Options Model (Industry Comparable).
Figure 92.1: Default probability model setup
Figure 92.2: Default probability of a publicly traded entity
Figure 92.3: Default probability of a privately held entity
Figure 92.4: Default probability of a privately held entity calibrated to market fluctuations
Empirical Models of Probability of Default
As mentioned, empirical models of the probability of default are used to compute an individual’s default probability, applicable within the retail banking arena, where empirical or actual
historical or comparable data exist on past credit defaults. The dataset in Figure 92.5 represents a sample of several thousand previous loans, credit, or debt issues. The data show whether each loan
had defaulted or not (0 for no default, and 1 for default) as well as the specifics of each loan applicant’s age, education level (1 to 3 indicating high school, university, or graduate professional
education), years with current employer, and so forth. The idea is to model these empirical data to see which variables affect the default behavior of individuals, using Risk Simulator’s Maximum
Likelihood Model. The resulting model will help the bank or credit issuer compute the expected probability of default of an individual credit holder having specific characteristics.
Illustrative Example on Applying Empirical Models of Probability of Default
The example file is Probability of Default – Empirical and can be accessed through Modeling Toolkit | Prob of Default | Empirical (Individuals). To run the analysis, select the data on the left or
any other dataset (include the headers) and make sure that the data have the same length for all variables, without any missing or invalid data. Then, using Risk Simulator, click on Risk Simulator |
Forecasting | Maximum Likelihood Models. A sample set of results is provided in the MLE worksheet, complete with detailed instructions on how to compute the expected probability of default of an
The Maximum Likelihood Estimates (MLE) approach on a binary multivariate logistic analysis is used to model dependent variables to determine the expected probability of success of belonging to a
certain group. For instance, given a set of independent variables (e.g., age, income, education level of credit card or mortgage loan holders), we can model the probability of default using MLE. A
typical regression model is invalid because the errors are heteroskedastic and non-normal, and the resulting estimated probability estimates will sometimes be above 1 or below 0. MLE analysis handles
these problems using an iterative optimization routine. The computed results show the coefficients of the estimated MLE intercept and slopes.¹
The coefficients estimated are actually the logarithmic odds ratios and cannot be interpreted directly as probabilities. A quick but simple computation is first required. The approach is
straightforward. To estimate the probability of success of belonging to a certain group (e.g., predicting if a debt holder will default given the amount of debt he holds), simply compute the
estimated Y value using the MLE coefficients. Figure 92.6 illustrates an individual with 8 years at a current employer and current address, a low 3% debt to income ratio, and $2,000 in credit card
debt has a log odds ratio of –3.1549. The inverse antilog of the odds ratio is obtained by computing:
¹For instance, the coefficients are estimates of the true population β values in the following equation: Y = β[0] + β[1]X[1] + β[2]X[2] + … + β[n]X[n]. The standard error measures how accurate the
predicted coefficients are, and the Z-statistics are the ratios of each predicted coefficient to its standard error. The Z-statistic is used in hypothesis testing, where we set the null hypothesis (H
[o]) such that the real mean of the coefficient is equal to zero, and the alternate hypothesis (H[a]) such that the real mean of the coefficient is not equal to zero. The Z-test is very important as
it calculates if each of the coefficients is statistically significant in the presence of the other regressors. This means that the Z-test statistically verifies whether a regressor or independent
variable should remain in the model or it should be dropped. That is, the smaller the p-value, the more significant the coefficient. The usual significant levels for the p-value are 0.01, 0.05, and
0.10, corresponding to the 99%, 95%, and 99% confidence levels.
So, such a person has a 4.09% chance of defaulting on the new debt. Using this probability of default, you can then use the Credit Analysis – Credit Premium model to determine the additional credit
spread to charge this person given this default level and the customized cash flows anticipated from this debt holder.
Figure 92.5: Empirical analysis of probability of default
The p-values are 0.01, 0.05, and 0.10, corresponding to the 99%, 95%, and 99% confidence levels.
Figure 92.6: MLE results
Loss Given Default and Expected Losses
As shown previously, the probability of default is a key parameter for computing the credit risk of a portfolio. In fact, the Basel II/III Accord requires that the probability of default as well as
other key parameters, such as the loss given default (LGD) and exposure at default (EAD), be reported as well. The reason is that a bank’s expected loss is equivalent to:
Expected Losses = (Probability of Default) × (Probability of Default) × (Exposure at Default)
or simply EL = PD × LGD × EAD.
PD and LGD are both percentages, whereas EAD is a value. As we have shown how to compute PD earlier, we will now revert to some estimations of LGD. There are several methods used to estimate LGD. The
first is through a simple empirical approach where we set LGD = 1 – Recovery Rate. That is, whatever is not recovered at default is the loss at default, computed as the charge-off (net of recovery)
divided by the outstanding balance:
Therefore, if market data or historical information are available, LGD can be segmented by various market conditions, types of obligor, and other pertinent segmentations. LGD can then be readily read
off a chart.
A second approach to estimate LGD is more attractive in that if the bank has available information, it can attempt to run some econometric models to create the best-fitting model under an ordinary
least squares approach. By using this approach, a single model can be determined and calibrated, and this same model can be applied under various conditions, with no data mining required. However, in
most econometric models, a normal transformation will have to be performed first. Supposing the bank has some historical LGD data (Figure 92.7), the best-fitting distribution can be found using Risk
Simulator (select the historical data, click on Risk Simulator | Analytical Tools | Distributional Fitting (Single Variable) to perform the fitting routine). The result is a beta distribution for the
thousands of LGD values.
Figure 92.7: Fitting historical LGD data
Then, using the Distribution Analysis tool in Risk Simulator, obtain the theoretical mean and standard deviation of the fitted distribution (Figure 92.8). Then transform the LGD variable using the
MTNormalTransform function in the Modeling Toolkit software. For instance, the value of 49.69% will be transformed and normalized to 28.54%. Using this newly transformed dataset, you can run some
nonlinear econometric models to determine LGD.
For instance, a partial list of independent variables that might be significant for a bank, in terms of determining and forecast the LGD value, might include:
• Debt to capital ratio
• Profit margin
• Revenue
• Current assets to current liabilities
• Risk rating at default and one year before default
• Industry
• Authorized balance at default
• Collateral value
• Facility type
• Tightness of covenant
• Seniority of debt
• Operating income to sales ratio (and other efficiency ratios)
• Total asset, total net worth, total liabilities
Figure 92.8: Distributional analysis tool
Economic Capital and Value at Risk
Economic capital is critical to a bank as it links a bank’s earnings and returns to risks that are specific to business lines or business opportunities. In addition, these economic capital
measurements can be aggregated into a portfolio of holdings. Value at Risk (VaR) is used in trying to understand how the entire organization is affected by the various risks of each holding as
aggregated into a portfolio, after accounting for cross-correlations among various holdings. VaR measures the maximum possible loss given some predefined probability level (e.g., 99.90%) over some
holding period or time horizon (e.g., 10 days). Senior management at the bank usually selects the probability or confidence interval, which is typically a decision made by senior management at the
bank and reflects the board’s risk appetite. Stated another way, we can define the probability level as the bank’s desired probability of surviving per year. In addition, the holding period is
usually chosen such that it coincides with the time period it takes to liquidate a loss position.
VaR can be computed in several ways. Two main families of approaches exist: structural closed-form models and Monte Carlo risk simulation approaches. We showcase both methods in this chapter,
starting with the structural models.
The second and much more powerful approach is Monte Carlo risk simulation. Instead of simply correlating individual business lines or assets, entire probability distributions can be correlated using
mathematical copulas and simulation algorithms, by using Risk Simulator. In addition, tens to hundreds of thousands of scenarios can be generated using simulation, providing a very powerful
stress-testing mechanism for valuing VaR. Distributional fitting methods are applied to reduce the thousands of data points into their appropriate probability distributions, allowing their modeling
to be handled with greater ease.
Illustrative Example: Structural VaR Models
This first VaR example model used is Value at Risk – Static Covariance Method, accessible through Modeling Toolkit | Value at Risk | Static Covariance Method. This model is used to compute the
portfolio’s VaR at a given percentile for a specific holding period, after accounting for the cross-correlation effects between the assets (Figure 92.9). The daily volatility is the
annualized volatility divided by the square root of trading days per year. Typically, positive correlations tend to carry a higher VaR compared to zero correlation asset mixes, whereas
negative correlations reduce the total risk of the portfolio through the diversification effect (Figures 92.9 and 92.10). The approach used is a portfolio VaR with correlated inputs, where the
portfolio has multiple asset holdings with different amounts and volatilities. Each asset is also correlated to each other. The covariance or correlation structural model is used to compute the VaR
given a holding period or horizon and percentile value (typically 10 days at 99% confidence). Of course, the example illustrates only a few assets or business lines or credit lines for simplicity’s
sake. Nonetheless, using the VaR function (MTVaRCorrelationMethod) in the Modeling Toolkit, many more lines, assets, or businesses can be modeled.
Figure 92.9: Computing Value at Risk using the structural covariance method
Figure 92.10: Different correlation levels
Illustrative Example: VaR Models Using Monte Carlo Risk Simulation
The model used is Value at Risk – Portfolio Operational and Credit Risk VaR Capital Adequacy and is accessible through Modeling Toolkit | Value at Risk | Portfolio Operational and Credit Risk VaR
Capital Adequacy. This model shows how operational risk and credit risk parameters are fitted to statistical distributions and their resulting distributions are modeled in a portfolio of liabilities
to determine the VaR (99.50th percentile certainty) for the capital requirement under Basel II/III requirements. It is assumed that the historical data of the operational risk impacts (Historical
Data worksheet) are obtained through econometric modeling of the Key Risk Indicators.
The Distributional Fitting Report worksheet is a result of running a distributional fitting routine in Risk Simulator to obtain the appropriate distribution for the operational risk parameter. Using
the resulting distributional parameter, we model each liability’s capital requirements within an entire portfolio. Correlations can also be inputted if required, between pairs of liabilities or
business units. The resulting Monte Carlo simulation results show the VaR capital requirements.
Note that an appropriate empirically based historical VaR cannot be obtained if distributional fitting and risk-based simulations were not first run. The VaR will be obtained only by
running simulations. To perform distributional fitting, follow these steps:
• In the Historical Data worksheet (Figure 92.11), select the data area (cells C5:L104) and click on Risk Simulator | Analytical Tools | Distributional Fitting (Single Variable).
• Browse through the fitted distributions and select the best-fitting distribution (in this case, the exponential distribution in Figure 92.12) and click OK.
Figure 92.11: Sample historical bank loans
Figure 92.12: Data fitting results
• You may now set the assumptions on the Operational Risk Factors with the exponential distribution (fitted results show Lambda = 1) in the Credit Risk Note that the assumptions have already been
set for you in advance. You may set the assumption by going to cell F27 and clicking on Risk Simulator | Set Input Assumption, selecting Exponential distribution and entering 1 for the Lambda
value, and clicking OK. Continue this process for the remaining cells in column F, or simply perform a Risk Simulator Copy and Risk Simulator Paste on the remaining cells.
1. Note that since the cells in column F have assumptions set, you will first have to clear them if you wish to reset and copy/paste parameters. You can do so by first selecting cells F28:F126 and
clicking on the Remove Parameter icon or select Risk Simulator | Remove Parameter.
2. Then select cell F27, click on the Risk Simulator Copy icon or select Risk Simulator | Copy Parameter, and then select cells F28:F126 and click on the Risk Simulator Paste icon or select Risk
Simulator | Paste Parameter.
• Next, you can get assumptions that can be set such as the probability of default using the Bernoulli Distribution (column H) and Loss Given Default (column J). Repeat the procedure in Step 3 if
you wish to reset the assumptions.
• Run the simulation by clicking on the RUN icon or clicking on Risk Simulator | Run Simulation.
• Obtain the Value at Risk by going to the forecast chart once the simulation is done running and selecting Left-Tail and typing in 50. Hit TAB on the keyboard to enter the confidence value and
obtain the VaR of $25,959 (Figure 92.13).
Figure 92.13: Simulated forecast results and the 99.50% Value at Risk value
Another example on VaR computation is shown next, where the model Value at Risk – Right Tail Capital Requirements is used, available through Modeling Toolkit | Value at Risk | Right Tail Capital
This model shows the capital requirements per Basel II/III requirements (99.95th percentile capital adequacy based on a specific holding period’s VaR). Without running risk-based historical and Monte
Carlo simulation using Risk Simulator, the required capital is $37.01M (Figure 92.14) as compared to only $14.00M that is required using a correlated simulation (Figure 92.15). This is due to the
cross-correlations between assets and business lines and can only be modeled only using Risk Simulator. This lower VaR is preferred as banks can be required to hold less required capital and can
reinvest the remaining capital in various profitable ventures, thereby generating higher profits.
Figure 92.14: Right-tail VaR model
To run the model, follow these steps:
1. Click on Risk Simulator | Run Simulation. If you had other models open, make sure you first click on Risk Simulator | Change Profile, and select the Tail VaR profile before starting.
2. When the simulation is complete, select Left-Tail in the forecast chart and enter in 95 in the Certainty box, and hit TAB on the keyboard to obtain the value of $14.00M Value at Risk for this
correlated simulation.
3. Note that the assumptions have already been set for you in advance in the model in cells C6:C15. However, you may set them again by going to cell C6 and clicking on Risk Simulator | Set Input
Assumption, selecting your distribution of choice or using the default Normal Distribution or performing a distributional fitting on historical data, then clicking OK. Continue this process for
the remaining cells in column C. You may also decide to first Remove Parameters of these cells in column C and setting your own distributions. Further, correlations can be set manually when
assumptions are set (Figure 92.16) or by going to Risk Simulator | Edit Correlations (Figure 92.17) after all the assumptions are set.
Figure 92.15: Simulated results of the portfolio VaR
Figure 92.16: Setting correlations one at a time
Figure 92.17: Setting correlations using the correlation matrix routine
If risk simulation was not run, the VaR or economic capital required would have been $37M, as opposed to only $14M. All cross-correlations between business lines have been modeled, as are stress and
scenario tests, and thousands and thousands of possible iterations are run. Individual risks are now aggregated into a cumulative portfolio level VaR.
Efficient Portfolio Allocation and Economic Capital VaR
As a side note, by performing portfolio optimization, a portfolio’s VaR actually can be reduced. We start by first introducing the concept of stochastic portfolio optimization through an illustrative
hands-on example. Then, using this portfolio optimization technique, we apply it to four business lines or assets to compute the VaR or an un-optimized versus an optimized portfolio of assets, and
see the difference in computed VaR. You will note that in the end, the optimized portfolio bears less risk and has a lower required economic capital.
Illustrative Example: Stochastic Portfolio Optimization
The optimization model used to illustrate the concepts of stochastic portfolio optimization is Optimization – Stochastic Portfolio Allocation and can be accessed via Modeling Toolkit | Optimization |
Stochastic Portfolio Allocation. This model shows four asset classes with different risk and return characteristics. The idea here is to find the best portfolio allocation such that the portfolio’s
bang for the buck or returns to risk ratio, is maximized. That is, in order to allocate 100% of an individual’s investment among several different asset classes (e.g., different types of mutual funds
or investment styles: growth, value, aggressive growth, income, global, index, contrarian, momentum, etc.), an optimization is used. This model is different from others in that there exist several
simulation assumptions (risk and return values for each asset), as seen in Figure 92.18.
In other words, a simulation is first run, then optimization is executed, and the entire process is repeated multiple times to obtain distributions of each decision variable. The entire analysis can
be automated using Stochastic Optimization.
In order to run an optimization, several key specifications on the model have to be identified first:
Objective: Maximize Return to Risk Ratio (C12)
Decision Variables: Allocation Weights (E6:E9)
Restrictions on Decision Variables: Minimum and Maximum Required (F6:G9)
Constraints: Portfolio Total Allocation Weights 100% (E11 is set to 100%)
Simulation Assumptions: Return and Risk Values (C6:D9)
The model shows the various asset classes. Each asset class has its own set of annualized returns and annualized volatilities. These return and risk measures are annualized values such that they can
be compared consistently across different asset classes. Returns are computed using the geometric average of the relative returns, while the risks are computed using the logarithmic relative stock
returns approach.
Column E, Allocation Weights, holds the decision variables, which are the variables that need to be tweaked and tested such that the total weight is constrained at 100% (cell E11). Typically, to
start the optimization, we will set these cells to a uniform value, where in this case, cells E6 to E9 are set at 25% each. In addition, each decision variable may have specific restrictions in its
allowed range. In this example, the lower and upper allocations allowed are 10% and 40%, as seen in columns F and G. This setting means that each asset class can have its own allocation boundaries.
Figure 92.18: Asset allocation model ready for stochastic optimization
Column H shows the return to risk ratio, which is simply the return percentage divided by the risk percentage, where the higher this value, the higher the bang for the buck. The remaining sections of
the model show the individual asset class rankings by returns, risk, return to risk ratio, and allocation. In other words, these rankings show at a glance which asset class has the lowest risk, or
the highest return, and so forth.
Running an Optimization
To run this model, simply click on Risk Simulator | Optimization | Run Optimization. Alternatively, and for practice, you can set up the model using the following approach:
1. Start a new profile(Risk Simulator | New Profile).
2. For stochastic optimization, set distributional assumptions on the risk and returns for each asset That is, select cell C6 and set an assumption (Risk Simulator | Set Input Assumption), and make
your own assumption as required. Repeat for cells C7 to D9.
3. Select cell E6 and define the decision variable (Risk Simulator | Optimization | Decision Variables or click on the Define Decision icon) and make it a Continuous Variable and then link the
decision variable’s name and minimum/maximum required to the relevant cells (B6, F6, G6).
4. Then use the Risk Simulator Copy on cell E6, select cells E7 to E9, and use Risk Simulator’s Paste (Risk Simulator | Copy Parameter and Risk Simulator | Paste Parameter or use the Risk Simulator
copy and paste icons). Make sure you do not use Excel’s regular copy and paste functions.
5. Next, set up the optimization’s constraints by selecting Risk Simulator | Optimization | Constraints, selecting ADD, and selecting the cell E11, and making it equal 100% (total allocation, and do
not forget the % sign).
6. Select cell C12, the objective to be maximized and make it the objective: Risk Simulator | Optimization | Set Objective or click on the O icon.
7. Run the simulation by going to Risk Simulator | Optimization | Run Optimization. Review the different tabs to make sure that all the required inputs in steps 2 and 3 are correct. Select
Stochastic Optimization and let it run for 500 trials repeated 20 times (Figure 92.19 illustrates these setup steps).
You may also try other optimization routines where:
Static Optimization is an optimization that is run on a static model, where no simulations are run. This optimization type is applicable when the model is assumed to be known and no uncertainties
exist. Also, a static optimization can be run first to determine the optimal portfolio and its corresponding optimal allocation of decision variables before more advanced optimization procedures are
applied. For instance, before running a stochastic optimization problem, a static optimization is run first to determine if there exist solutions to the optimization problem before a more protracted
analysis is performed.
Dynamic Optimization is applied when Monte Carlo simulation is used together with optimization. Another name for such a procedure is Simulation-Optimization. In other words, a simulation is run for N
trials, and then an optimization process is run for M iterations until the optimal results are obtained, or an infeasible set is found. That is, using Risk Simulator’s Optimization module, you can
choose which forecast and assumption statistics to use and replace in the model after the simulation is run. Then, these forecast statistics can be applied in the optimization process. This
approach is useful when you have a large model with many interacting assumptions and forecasts, and when some of the forecast statistics are required in the optimization.
Stochastic Optimization is similar to the dynamic optimization procedure except that the entire dynamic optimization process is repeated T times. The results will be a forecast chart of each decision
variable with T values. In other words, a simulation is run and the forecast or assumption statistics are used in the optimization model to find the optimal allocation of decision variables. Then
another simulation is run, generating different forecast statistics, and these new updated values are then optimized, and so forth. Hence, each of the final decision variables will have its own
forecast chart, indicating the range of the optimal decision variables. For instance, instead of obtaining single-point estimates in the dynamic optimization procedure, you can now obtain a
distribution of the decision variables, and, hence, a range of optimal values for each decision variable, also known as stochastic optimization.
Figure 92.19: Setting up the stochastic optimization problem
Viewing and Interpreting Forecast Results
Stochastic optimization is performed when a simulation is first run and then the optimization is run. Then the whole analysis is repeated multiple times. The result is a distribution of each decision
variable, rather than a single-point estimate (Figure 92.20). This means that instead of saying you should invest 30.57% in Asset 1, the optimal decision is to invest between 30.10% and 30.99% as
long as the total portfolio sums to 100%. This way the optimization results provide management or decision makers a range of flexibility in the optimal decisions. Refer to Chapter 11 of Modeling
Risk: Applying Monte Carlo Simulation, Real Options Analysis, Forecasting, and Optimization by Dr. Johnathan Mun for more detailed explanations about this model, the different optimization
techniques, and an interpretation of the results. Chapter 11’s appendix also details how the risk and return values are computed.
Figure 92.20: Simulated results from the stochastic optimization approach
Illustrative Example: Portfolio Optimization and Portfolio VaR
Now that we understand the concepts of optimized portfolios, let us see what the effects are on computed economic capital through the use of a correlated portfolio VaR. This model uses Monte
Carlo simulation and optimization routines in Risk Simulator to minimize the Value at Risk of a portfolio of assets (Figure 92.21). The file used is Value at Risk – Optimized and Simulated Portfolio
VaR, which is accessible via Modeling Toolkit | Value at Risk | Optimized and Simulated Portfolio VaR. In this example, we intentionally used only four asset classes to illustrate the effects of an
optimized portfolio. In real life, we can extend this to cover a multitude of asset classes and business lines. Here we illustrate the use of a left-tail VaR, as opposed to a right-tail VaR, but the
concepts are similar.
First, simulation is used to determine the 90% left-tail VaR. The 90% left-tail probability means that there is a 10% chance that losses will exceed this VaR for a specified holding period. With an
equal allocation of 25% across the four asset classes, the VaR is determined using simulation (Figure 92.21). The annualized returns are uncertain and hence simulated. The VaR is then read off the
forecast chart. Then optimization is run to find the best portfolio subject to the 100% allocation across the four projects that will maximize the portfolio’s bang for the buck (returns to risk
ratio). The resulting optimized portfolio is then simulated once again, and the new VaR is obtained (Figure 92.22). The VaR of this optimized portfolio is a lot less than the not-optimized portfolio.
Figure 92.21: Computing Value at Risk (VaR) with simulation
Figure 92.22: Non-optimized Value at Risk
Figure 92.23: Optimal portfolio’s Value at Risk through optimization and simulation
Hurdle Rates and Required Rate of Return
Another related item in the discussion of risk in the context of Basel II/III Accords is the issue of hurdle rates, or the required rate of return on investment that is sufficient to justify the
amount of risk undertaken in the portfolio. There is a nice theoretical connection between uncertainty and volatility whereby the discount rate of a specific risk portfolio can be obtained. In a
financial model, the old axiom of high risk, high return is seen through the use of a discount rate. That is, the higher the risk of a project, the higher the discount rate should be to risk-adjust
this riskier project so that all projects are comparable.
There are two methods for computing the hurdle rate. The first is an internal model, where the VaR of the portfolio is computed first. This economic capital is then compared to the
market risk premium. That is, we have
That is, assuming that a similar set of comparable investments are obtained in the market, based on tradable assets, the market return is obtained. Using the bank’s internal cash flow models, all
future cash flows can be discounted at the risk-free rate in order to determine the risk-free return. Finally, the difference is then divided into the VaR risk capital to determine the required
hurdle rate. This concept is very similar to the capital asset pricing model (CAPM), which often is used to compute the appropriate discount rate for a discounted cash flow model (weighted average
cost of capital, hurdle rates, multiple asset pricing models, and arbitrage pricing models are the other alternatives but are based on similar principles). The second approach is the use of the CAPM
to determine the hurdle rate.
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Binary for Beginners: The ABCs of 0s and 1s | Aleksandr Hovhannisyan (2024)
What is $10$10? If this is your first time learning about the binary number system, then this question may seem odd. Of course it’s ten, right?
Let’s try something different. Have you ever heard this joke?
There are $10$10 types of people: those who understand binary and those who don’t.
Unless you’re familiar with binary numbers, this probably doesn’t make much sense. But by the end of this article, you’ll understand this awful joke!
In this beginner’s tutorial, we’ll look at everything you need to know about the binary number system, but we’ll also take a quick look at decimal and hexadecimal, as they’re closely related. I’ll
include relevant bits of code and real-life examples to help you appreciate the beauty of binary.
Table of Contents
What Is a Number System?
Before we look at binary, let’s take a step back and discuss number systems more generally.
It may seem strange to think of number systems in the plural if this is your first time learning about them. That’s because the majority of the world is familiar with just one system: the decimal
number system, also known as the Arabic number system. This number system uses the digits $0–9$0–9 to represent numbers symbolically, based on their position in a string.
For example, in the decimal number system, $579$579 expands to this:
$579 = 5(10^2) + 7(10^1) + 9(10^0) = 500 + 70 + 9$579=5(102)+7(101)+9(100)=500+70+9
In school, you were taught that the $5$5 in $579$579 is in the hundredths place, the $7$7 is in the tens place, and the $9$9 is in the ones place. Notice that the $5$5 is multiplied by one hundred (
$10^2$102), the $7$7 by ten ($10^1$101), and the $9$9 by one ($10^0$100) to form the decimal number $579$579. We say that the number $579$579 is positional because the digits, from left to right,
correspond to a specific power of ten based on the position of the digit in the number.
Here, the number $10$10 is what we call the base (aka radix) of our number system. Notice the powers of $10$10 in the expanded expression above: $10^2$102, $10^1$101, and $10^0$100. For this reason,
the terms decimal and base ten are interchangeable.
In the decimal number system, a number is represented by placing digits into “buckets” that represent increasing powers of ten, starting with $10^0$100 in the rightmost “bucket,” followed by $10^1$10
1 to its immediate left, and so on infinitely:
Any unused buckets to the far left have an implicit value of $0$0 in them. We usually trim leading zeros because there is no use in saying $00579$00579 when that’s mathematically identical to $579$57
Why did humans pick $10$10 to be the base of their preferred number system? Likely because most people are born with ten fingers and ten toes, and we’re used to counting with our fingers when we’re
young. So it’s natural for us to have adopted ten as the base of our number system.
Bases, Exponents, and Digits
As I’ve already hinted, the decimal number system (base $10$10) isn’t the only one in existence. Let’s use a more general notation to represent number systems beyond just our familiar one.
In a number system with a fixed base of $b$b, the available digits range from $0$0 to $b - 1$b−1. For example, in the decimal number system ($b = 10$b=10), we can only use the digits $0, 1, 2, ...,
9$0,1,2,...,9. When you run out of digits to stuff into a single bucket, you carry over a one to the next power of the base. For example, to get to the number after $99$99, you carry a one to the
bucket representing the next power of ten ($100$100).
Now, suppose that we have a string of digits $d_{n-1} d_{n-2} ... d_0$dn−1dn−2...d0 (where $n$n is the number of digits). Maybe that’s $d_2 d_1 d_0 = 579$d2d1d0=579 from our earlier example.
That string expands like this:
$d_{n-1} b^{n-1} + d_{n-2} b^{n-2} + ... + d_{0} b^0$dn−1bn−1+dn−2bn−2+...+d0b0
And you can visualize it like this:
Using our same example, $d_{n-1} b^{n-1} + d_{n-2} b^{n-2} + ... + d_{0} b^0 = 5(10^2) + 7(10^1) + 9(10^0)$dn−1bn−1+dn−2bn−2+...+d0b0=5(102)+7(101)+9(100). Again, we have buckets from right to
left in increasing powers of our base ($10$10), as depicted below:
Now, in reality, you can have a number system that uses a base of $2$2, $3$3, $4$4, $120$120, and so on. Some of these have special names because they’re used more often than others:
Base Name Description
1 Unary Also known as tallying. A number n is represented by picking an arbitrary character and repeating it n times (e.g., xxxx would be 4).
2 Binary Only two digits: zero and one. Most commonly used in computing. Everything on a computer is, at the lowest possible level, stored using the binary number system.
8 Octal Only eight digits are available: 0–7.
16 Hexadecimal Fifteen digits: 0–9 and a–f. Often used to express binary strings more compactly.
60 Sexagesimal How many seconds are in a minute? How many minutes in an hour? This is the basis of the modern circular coordinate system (degrees, minutes, and seconds).
For this reason, when discussing number systems, we usually subscript a number with its base to clarify its value. Alternatively, you can prepend a number with a certain string (usually 0b for binary
or 0x/# for hexadecimal). So we’d write $579$579 as $579_{10}$57910, or the binary number $1001$1001 as $1001_2$10012 (or $\text{0b}1001$0b1001). Otherwise, if we were to merely write the number
$1001$1001 without providing any context, nobody would know whether that’s in binary, octal, decimal, hexadecimal, and so on because the digits $0$0 and $1$1 are valid in all of those number systems,
The Binary Number System
We’re all familiar with decimal numbers because we use them everyday. But what about the binary number system?
By definition, the binary number system has a base of $2$2, and thus we can only work with two digits to compose numbers: $0$0 and $1$1. Technically speaking, we don’t call these digits—they’re
called bits in binary lingo. Each “bucket” in a binary string represents an increasing power of two: $2^0$20, $2^1$21, $2^2$22, and so on.
The leftmost bit is called the most significant bit (MSB), while the rightmost bit is called the least significant bit (LSB).
Here are some examples of representing decimal numbers in the binary number system:
• Zero: $0_{10} = 0_2$010=02. Expansion: $0 (2^0)$0(20)
• One: $1_{10} = 1_2$110=12. Expansion: $1(2^0)$1(20)
• Two: $2_{10} = 10_2$210=102. Expansion: $1(2^1) + 0(2^0)$1(21)+0(20)
• Three: $3_{10} = 11_2$310=112. Expansion: $1(2^1) + 1(2^0)$1(21)+1(20)
• Four: $4_{10} = 100_2$410=1002. Expansion: $1(2^2) + 0(2^1) + 0(2^0)$1(22)+0(21)+0(20)
• Five: $5_{10} = 101_2$510=1012. Expansion: $1(2^2) + 0(2^1) + 1(2^0)$1(22)+0(21)+1(20)
Having learned the binary number system, you should now understand the joke from earlier:
There are $10$10 types of people: those who understand binary and those who don’t.
Here, we really mean the binary equivalent of two, which looks like ten to our eyes when it’s not properly subscripted: $10_2 = 1 × 2^1 = 2_{10}$102=1×21=210.
Binary Is Close to the Hardware of a Computer
Why do we bother with using the binary number system in the first place? Doesn’t it seem like a whole lot of extra work to represent numbers in this manner when we could instead use the decimal
number system? Well, yes—if you’re writing these out by hand, it’s certainly more work to represent (and manipulate) binary numbers.
You may not see any point in using binary if you haven’t learned about computer architecture at a low level. Internally, computers are nothing more than electrical circuits tied to hardware. Current
either flows through a wire or doesn’t—a binary state. Likewise, computers use logic gates (AND/OR/NOR/XOR) to control the flow of a program’s execution, and these take binary inputs (true/false).
The best way to represent these low-level interactions is to use the binary number system: $0$0 means “off” (or false in its boolean form) and $1$1 means “on” (true).
Everything on your computer—the files you save and the software you install—is represented as nothing more than zeros and ones. But how is this possible?
The Unicode Standard
Suppose you create a file on your computer and store some basic text in it:
echo Hello, Binary > file
At the end of the day, your computer can’t store a character like H, e, l, or o (or even the space between two words) literally. Computers only know how to work with binary. Thus, we need some way to
convert these characters to numbers. And that’s why the Unicode standard was introduced.
Unicode is the most widely accepted character encoding standard: a method of representing human-readable characters like H, e, ,, ?, and 9 numerically so that computers can understand and use them
like we do. Each character maps to a unique number known as a code point.
For example, the chart below shows a very limited subset of Unicode characters (known as the ASCII standard) and their corresponding code points:
For the sake of brevity, we’ll focus on just the ASCII standard for now, even though it doesn’t capture the full range of characters in the Unicode standard and the complexities that come with
needing to support hundreds of thousands of characters.
The ASCII standard supports only 128 characters, each mapped to a unique number:
• Arabic digits: $0-9$0−9 (10)
• Uppercase Latin letters: $A-Z$A−Z (26)
• Lowercase Latin letters: $a-z$a−z (26)
• Punctuation and special characters (66)
Again, note that while the ASCII standard only allows us to represent a tiny fraction of Unicode characters, it’s simple enough that it can help us better understand how characters are stored on
1 ASCII Character = 1 Byte
In the decimal number system, we’re used to working with digits. In binary, as we already saw, we’re used to working with bits. There’s another special group of digits in binary that’s worth
mentioning: A sequence of eight bits is called a byte.
Here are some examples of valid bytes:
… and any other valid permutation of eight $0$0s and $1$1s that you can think of.
Why is this relevant? Because on modern computers, characters are represented using bytes.
Recall that the ASCII encoding format needs to support a total of 128 characters. So how many unique number can we represent with $8$8 bits (a byte)?
Well, using the product rule from combinatorics, we have eight “buckets,” each with two possible values: either a $0$0 or a $1$1. Thus, we have $2 × 2 × ... × 2 = 2^8$2×2×...×2=28 possible values.
In decimal, this is $2^8 = 256$28=256 possible values. By comparison, $2^7 = 128$27=128. And $128$128 happens to be the number of characters that we want to represent.
So… That’s weird, and seemingly wasteful, right? Why do we use $8$8 bits (one byte) to represent a character when we could use $7$7 bits instead and meet the precise character count that we need?
Good question! We use bytes because it’s not possible to evenly divide a group of $7$7 bits, making certain low-level computations difficult if we decide to use $7$7 bits to represent a character. In
contrast, a byte can be evenly split into powers of two:
The key takeaway here is that we only need one byte to store one character on a computer. This means that a string of five characters—like Hello—occupies five bytes of space, with each byte being the
numerical representation of the corresponding character per the ASCII format.
Remember the file we created earlier? Let’s view its binary representation using the xxd Unix tool:
xxd -b file
The -b flag stands for binary. Here’s the output that you’ll get:
00000000: 01001000 01100101 01101100 01101100 01101111 00101100 Hello,00000006: 00100000 01000010 01101001 01101110 01100001 01110010 Binar0000000c: 01111001 00001010 y.
The first line shows a sequence of six bytes, each corresponding to one character in Hello,.
Let’s decode the first two bytes using our knowledge of the binary number system and ASCII:
• $01001000 = 1(2^6) + 1(2^3) = 72_{10}$01001000=1(26)+1(23)=7210. Per our ASCII table, this corresponds to $H$H.
• $01100101 = 1(2^6) + 1(2^5) + 1(2^2) + 1(2^0) = 101_{10}$01100101=1(26)+1(25)+1(22)+1(20)=10110, which is $e$e in ASCII.
Cool! Looks like the logic pans out. You can repeat this for all of the other bytes as well. Notice that on the second line, we have a leading space (from Hello, Binary), represented as $2^5 = 32_
{10}$25=3210 in ASCII (which is indeed Space per the table).
By the way, what’s up with the numbers along the left-hand side of the output? What does $0000000c$0000000c even mean? Time to explore another important number system!
The Hexademical Number System
As I mentioned in the table from earlier, the hexadecimal number system is closely related to binary because it’s often used to express binary numbers more compactly, instead of writing out a whole
bunch of zeros and ones.
The hexadecimal number system has a base of $16$16, meaning its digits range from $0–15$0–15.
This is our first time encountering a number system whose digits are made up of more than two characters. How do we squeeze $10$10, $11$11, or $15$15 into a single “bucket” or “slot” for a digit? To
be clear, this is perfectly doable if you have clear delimiters between digits, like vertical lines—without which you wouldn’t know if $15$15 is a one followed by a five or a single digit of $15$15
in the ones place. But in reality, using delimiters isn’t practical.
Let’s take a step back and consider a simple hexadecimal number:
What does this mean to us humans in our decimal number system? Well, all we have to do is multiply each digit by its corresponding power of $16$16:
$0x42 = 4(16^1) + 2(16^0) = 64_{10} + 2_{10} = 66_{10}$0x42=4(161)+2(160)=6410+210=6610
Okay, so that’s a simple hex number. Back to the problem at hand: How do we represent the hex digits $10$10, $11$11, and so on? Here’s an example that’s pretty confusing unless we introduce some
alternative notation:
Is this a $15$15 in a single slot or a $1$1 and a $5$5 in two separate slots? One way to make this less ambiguous is to use some kind of delimiter between slots, but again, that’s not very practical:
The better solution that people came up with is to map $10–15$10–15 to the the English letters $a–f$a–f. Note that we could’ve also used any other symbols to represent these digits. As long as we
agree on a convention and stick with it, there’s no ambiguity as to what a number represents.
Here’s an example of a hexadecimal number that uses one of these digits:
And here’s its expansion:
$0xf4 = 15(16^1) + 4(16^0) = 240_{10} + 4_{10} = 244_{10}$0xf4=15(161)+4(160)=24010+410=24410
There’s nothing magical about the hexadecimal number system—it works just like unary, binary, decimal, and others. All that’s different is the base!
Before we move on, let’s revisit the output from earlier when we used xxd on our sample file:
00000000: 01001000 01100101 01101100 01101100 01101111 00101100 Hello,00000006: 00100000 01000010 01101001 01101110 01100001 01110010 Binar0000000c: 01111001 00001010 y.
The numbers along the left-hand side mark the starting byte for each line of text on the far right. For example, the first line of text (Hello,) ranges from byte #0 (H) to byte #5 (,). The next line
is marked as $00000006$00000006, meaning we’re now looking at bytes #6 through 11 (B to r). Finally, the last label should make sense now that you know the hexadecimal number system: c maps to $12$12
, meaning the byte that follows corresponds to the twelfth character in our file.
How to Convert Between Binary and Hexadecimal
Now that we know a bit about binary and hexadecimal, let’s look at how we can convert between the two systems.
Binary to Hexadecimal
Say you’re given this binary string and you’d like to represent it in hexadecimal:
While at first this may seem like a pretty difficult task, it’s actually straightforward!
Let’s do a bit of a thought exercise: In the hexadecimal number system, we have $16$16 digits from $0$0 to $15$15. Over in binary land, how many bits do we need to represent these $16$16 values?
The answer is four because $2^4 = 16$24=16. With four “buckets,” we can create the numbers zero ($0000$0000), one ($0001$0001), ten ($1010$1010), all the way up to fifteen ($1111$1111). This means
that when you’re given a binary string, all you have to do is split it into groups of four bits and evaluate them to convert binary to hexadecimal!
011011100101[0110][1110][0101]6 14 5
Now we just replace $10–15$10–15 with $a-f$a−f and we’re done: $0x6e5$0x6e5.
Hexadecimal to Binary
What about the reverse process? How do you convert a hexadecimal number to binary? Say you’re given the hexadecimal number $0xad$0xad. What do we know about each hexadecimal digit?
Well, from our earlier exercise, we know that four bits comprise one hex digit. So we can convert each individual hex digit to its $4$4-bit representation and then stick each group together!
$a_{16} = 10_{10} = 1010_{2} \\d_{16} = 13_{10} = 1101_{2} \\ad_{16} = 10101101_{2}$a16=1010=10102d16=1310=11012ad16=101011012
Real-World Application: Colors in RGB/Hex
While we’re on the topic of binary and hexadecimal, it’s worth taking a look at one real-world use case for the things we’ve learned so far: RGB and hex colors.
Colors have three components: red, green, and blue (RGB). With LED (light-emitting diode) displays, each pixel is really split into these three components using a color diode. If a color component is
set to $0$0, then it’s effectively turned off. Otherwise, its intensity is modulated between $0$0 and $255$255, giving us a color format like rgb(0-255, 0-255, 0-255).
Let’s consider this hex color: #4287f5. What is it in the RGB format?
Well, we need to split this hex string evenly between red, green, and blue. That’s two digits per color:
Now, we interpret the decimal equivalent for each part:
• Red: $42_{16} = 4(16^1) + 2(16^0) = 66$4216=4(161)+2(160)=66
• Green: $87_{16} = 8(16^1) + 7(16^0) = 135$8716=8(161)+7(160)=135
• Blue: $f5_{16} = 15(16^1) + 5(16^0) = 245$f516=15(161)+5(160)=245
That means #4287f5 is really rgb(66, 135, 245)! You can verify this using a Color Converter:
For practice, let’s convert this to binary as well. I’ll mark the groups of four bits to make it easier to see how I did this (you could also convert from the decimal RGB representation if you want
$0x4287f5 = 0b[0100][0010][1000][0111][1111][0101]$0x4287f5=0b[0100][0010][1000][0111][1111][0101]
Now, two groups of four bits will represent one component of the color (red/green/blue):
Notice that each color component takes up a byte ($8$8 bits) of space.
How Many Colors Are There?
As an additional exercise, how many unique colors can you possibly have in the modern RGB format?
We know that each component (red/green/blue) is represented using one byte ($8$8 bits). So the colors we’re used to are really $24$24-bit colors.
That means there are a whopping $2^{24} = 16,777,216$224=16,777,216 possible unique colors that you can generate using hex/rgb! The $24$24-bit color system is known as truecolor, and it’s capable of
representing millions of colors.
Note that you could just as well have performed this calculation using hex: #4287f5. There are six slots, each capable of taking on a value from $0$0 to $f$f. That gives us a total of $16 × 16 × ...
× 16 = 16^6 = 16,777,216$16×16×...×16=166=16,777,216 values—the same result as before.
Or, if you’re using the decimal RGB format, the math still pans out:
$256 × 256 × 256 = 16,777,216$256×256×256=16,777,216
What Are 8-Bit Colors?
On older systems with limited memory, colors were represented using just eight bits (one byte). These 8-bit colors had a very limited palette, which meant that most computer graphics didn’t have
gradual color transitions (so images looked very pixelated/grainy). With only $8$8 bits to work with, you are limited to just $2^8 = 256$28=256 colors!
Naturally, you may be wondering: How did they split $8$8 bits evenly among red, green, and blue? After all, $8$8 isn’t divisible by three!
Well, the answer is that they didn’t. The process of splitting these bits among the color components is called color quantization, and the most common method (known as 8-bit truecolor) split the bits
as 3-3-2 red-green-blue. Apparently, this is because the human eye is less sensitive to blue light than the other two, and thus it simply made sense to distribute the bits heavily in favor of red and
green and leave blue with one less bit to work with.
Signed Binary Number System: Two’s Complement
Now that we’ve covered decimal, binary, and hexadecimal, I’d like us to revisit the binary number system and learn how to represent negative numbers. Because so far, we’ve only looked at positive
numbers. How do we store the negative sign?
To give us some context, I’ll assume that we’re working with standard $32$32-bit integers that most computers support. We could just as well look at $64$64-bit or $N$N-bit integers, but it’s good to
have a simple basis for a discussion.
If we have $32$32 bits to fiddle with, that means we can represent a total of $2^{32} = 4,294,967,296$232=4,294,967,296 (4 billion) numbers. More generally, if you have $N$N bits to work with, you
can represent $2^N$2N values. But we’d like to split this number range evenly between negatives and positives.
Positive or negative… positive or negative. One thing or another thing—ring a bell? That sounds like it’s binary in nature. And hey—we’re already using binary to store our numbers! Why not reserve
just a single bit to represent the sign? We can have the most significant (leading) bit be a $0$0 when our number is positive and a $1$1 when it’s negative!
Earlier, when we were first looking at the binary number systems, I mentioned that you can strip leading zeros because they are meaningless. This is true except when you actually care about
distinguishing between positive and negative numbers in binary. Now, we need to be careful—if you strip all leading zeros, you my be left with a leading $1$1, and that would imply that your number is
negative (in a signed number system).
You can think of two’s complement as a new perspective or lens through which we look at binary numbers. The number $100_2$1002 ordinarily means $4_{10}$410 if we don’t care about its sign (i.e., we
assume it’s unsigned). But if we do care, then we have to ask ourselves (or whoever provided us this number) whether it’s a signed number.
How Does Two’s Complement Work?
What does a leading $1$1 actually represent when you expand a signed binary number, and how do we convert a positive number to a negative one, and vice versa? For example, suppose we’re looking at
the number $22_{10}$2210, which is represented like this in unsigned binary:
Since we’re looking at signed binary, we need to pad this number with an extra $0$0 out in front (or else a leading $1$1 would imply that it’s negative):
Okay, so this is positive $22_{10}$2210. How do we represent $-22_{10}$−2210 in binary?
There are two ways we can do this: the intuitive (longer) approach and the “shortcut” approach. I’ll show you both, but I’ll start with the more intuitive one.
The Intuitive Approach: What Does a Leading 1 Denote?
Given an $N$N-bit binary string, a leading $1$1 in two’s complement represents $-1$−1 multiplied by its corresponding power of two ($2^{n-1}$2n−1). A digit of $1$1 in any other slot represents $+1$+1
times its corresponding power of two.
For example, the signed number $11010_2$110102 has this expansion:
$11010_2 = -1(2^4) + 1(2^3) + 1(2^1) = -16_{10} + 8_{10} + 2_{10} = -6_{10}$110102=−1(24)+1(23)+1(21)=−1610+810+210=−610
We simply treat the leading $1$1 as a negative, and that changes the resulting sum in our expansion.
Two’s Complement Shortcut: Flip the Bits and Add 1
To convert a number represented in two’s complement binary to its opposite sign, follow these two simple steps:
1. Flip all of the bits ($0$0 becomes $1$1 and vice versa).
2. Add $1$1 to the result.
For example, let’s convert $43_{10}$4310 to $-43_{10}$−4310 in binary:
+43 in binary: 0101011Flipped: 1010100Add one: 1010101
What is this number? It should be $-43_{10}$−4310, so let’s expand it by hand to verify:
$-1(2^6) + 1(2^4) + 1(2^2) + 1(2^0) = -64_{10} + 16_{10} + 4_{10} + 1_{10} = -43$−1(26)+1(24)+1(22)+1(20)=−6410+1610+410+110=−43
Sure enough, the process works!
How Many Signed Binary Numbers Are There?
We’ve seen that in a signed binary system, the most significant bit is reserved for the sign. What does this do to our number range? Effectively, it halves it!
Let’s consider $32$32-bit integers again. Whereas before we had $32$32 bits to work with for the magnitude of an unsigned number, we now have only $31$31 for the magnitude of a signed number (because
the 32nd bit is reserved for the sign):
Unsigned magnitude bits: [31 30 29 ... 0]Signed magnitude bits: 31 [30 29 ... 0]
We went from having $2^{32}$232 numbers to $2^{31}$231 positive and negative numbers, which is precisely half of what we started with ($\frac{2^{32}}{2} = 2^{31}$2232=231).
More generally, if you have an $N$N-bit signed binary string, there are going to be $2^N$2N values, split evenly between $2^{n-1}$2n−1 positives and $2^{n-1}$2n−1 negatives.
Notice that the number zero gets bunched in with the positives and not the negatives:
Signed zero: 0 0 0 0 ... 0 0 0 0Bits: 31 30 29 28 ... 3 2 1 0
As we’re about to see, this has an interesting consequence.
What Is the Largest Signed 32-bit Integer?
The largest signed 32-bit integer is positive, meaning its leading bit is a zero. So we just need to maximize the remaining bits to get the largest possible value:
Num: 0 1 1 1 ... 1Bits: 31 30 29 28 ... 0
This is $2^{31} - 1$231−1, which is $2,147,483,647$2,147,483,647. In Java, this number is stored in Integer.MAX_VALUE, and in C++, it’s std::numeric_limits<int>::max().
More generally, for an $N$N-bit system, the largest signed integer is $2^{n-1}-1$2n−1−1.
Why did we subtract a one at the end? Because we start counting at one, but computers start at zero. As I mentioned in the previous section, the number zero gets grouped along with the positives when
we split our number range (by convention):
Signed zero: 0 0 0 0 ... 0 0 0 0Bits: 31 30 29 28 ... 3 2 1 0
So to get the largest signed integer, we need to subtract one.
Real-World Application: Video Game Currency
In video games like RuneScape that use $32$32-bit signed integers to represent in-game currency, the max “cash stack” that you can have caps out at exactly $2^{31} - 1$231−1, which is roughly 2.1
Now you know why! If you’re wondering why they don’t just use unsigned ints, it’s because RuneScape runs on Java, and Java doesn’t support unsigned ints (except in SE 8+).
What Is the Smallest Signed 32-bit Integer?
This occurs when we set the leading bit to be a $1$1 and set all remaining bits to be a $0$0:
Num: 1 0 0 0 ... 0Bits: 31 30 29 28 ... 0
Why? Because recall that in the expansion of negative numbers in two’s complement binary, the leading $1$1 is a $-1$−1 times $2^{n-1}$2n−1, and a $1$1 in any other position will be treated as $+1$+1
times its corresponding power of two. Since we want the smallest negative number, we don’t want any positive terms, as those take away from our magnitude. So we set all remaining bits to be $0$0.
Answer: $-2^{31}$−231
In Java, this value is stored in Integer.MIN_VALUE. In C++, it’s in std::numeric_limits<int>::min().
More generally, if we have an $N$N-bit system, the smallest representable signed int is $-2^{n-1}$−2n−1.
Notice that the magnitude of the smallest signed $32$32-bit integer is exactly one greater than the magnitude of the largest signed $32$32-bit integer. As mentioned previously, this is because of
where we chose to group the number zero itself, which “steals” one magnitude from that group’s available bits.
Binary Arithmetic
Spoiler: Adding, subtracting, multiplying, and dividing numbers in the binary number system is exactly the same as it is in decimal!
Adding Binary Numbers
We’ll first revisit what we learned in elementary school for decimal numbers and then look at how to add two binary numbers.
To add two numbers in the decimal number system, you stack them on top of one another visually and work your way from right to left, adding two digits and “carrying the one” as needed.
Now you should know what carrying the one really means: When you run out of digits to represent something in your fixed-base number system (e.g., $13$13 isn’t a digit in base $10$10), you represent
the part that you can in the current digits place and move over to the next power of your base (the “column” to the left of your current one).
For example, let’s add $24$24 and $18$18 in decimal:
24+ 18———— 42
We first add the $4$4 and $8$8 to get $12$12, which is not a digit we support in the decimal number system. So we represent the part that we can ($2$2) and carry the remaining value (ten) over to the
next column as a $1$1 ($1 × 10^1 = 10_{10}$1×101=1010). In that column, we have $1_{10} + 2_{10} + 1_{10} = 4_{10}$110+210+110=410:
1 <-- carried 24 + 18———————— 42
Now, let’s add these same two numbers ($24_{10}$2410 and $18_{10}$1810) using the binary number system:
11000+ 10010——————— 101010
We work from right to left:
• Ones place: $0 + 0 = 0$0+0=0
• Twos place: $0 + 1 = 1$0+1=1
• Fours place: $0 + 0 = 0$0+0=0
• Eighths place: $1 + 0 = 1$1+0=1
• Sixteens place: $1 + 1 = 10_2$1+1=102 (two)
That last step deserves some clarification: When we try to add the two ones, we get $1_2 + 1_2 = 10_2$12+12=102 (two), so we put a $0$0 in the current column and carry over the $1$1 to the next
power of two, where we have a bunch of implicit leading zeros:
1 <-- carry bits0000 ... 000110000000 ... + 00010010—————————————————————0000 ... 00101010
In that column, $1 (carried) + 0(implicit) = 1$1(carried)+0(implicit)=1.
If we expand the result, we’ll find that it’s the same answer we got over in decimal:
$1(2^5) + 1(2^3) + 1(2^1) = 32 + 8 + 2 = 42_{10}$1(25)+1(23)+1(21)=32+8+2=4210
Let’s look at one more example to get comfortable with carrying bits in binary addition: $22_{10} + 14_{10}$2210+1410, which we know to be $36_{10}$3610:
10110+ 01110——————— 100100
Something interesting happens when we look at the twos place (the $2^1$21 column): We add $1_2$12 to $1_2$12, giving us two ($10_2$102), so we put a zero in the $2^1$21 column and carry the
remaining one.
Now we have three ones in the $2^2$22 column: $1_2(carried) + 1_2(operand1) + 1_2(operand2) = 11_2$12(carried)+12(operand1)+12(operand2)=112 (three). So we put a one in the $2^2$22 column and
carry a one yet again. Rinse and repeat!
1111 <-- carry bits0000 ... 000101100000 ... + 00001110—————————————————————0000 ... 00100100
Once again, it’s a good practice to expand the result so you can verify your work:
$1(2^5) + 1(2^2) = 32_{10} + 4_{10} = 36_{10}$1(25)+1(22)=3210+410=3610
Subtracting Binary Numbers
Subtraction is addition with a negative operand: $a - b = a + (-b)$a−b=a+(−b). Now that we know how to represent negative numbers in the binary system thanks to two’s complement, this should be a
piece of cake: negate the second operand and perform addition.
For example, what’s $12_{10} - 26_{10}$1210−2610? In decimal, we know this to be $-14_{10}$−1410. Over in binary, we know that $12_{10}$1210 is $01100$01100. What about $-26_{10}$−2610? We’ll
represent that using two’s complement.
We start by first representing $26_{10}$2610 in binary:
$+26_{10} = 011010_2$+2610=0110102
Now we negate it by flipping the bits and adding one:
26 in binary: 011010Flipped: 100101Add one: 100110 = -26
Then, stack up the operands and add them like before:
11 <-- carry bits 001100 + 100110—————————— 110010
Notice that the result has a leading one, which we know denotes a negative number in signed binary. So we at least got the sign part right! Let’s check the magnitude:
$-1(2^5) + 1(2^4) + 1(2^1) = -32_{10} + 16_{10} + 2_{10} = -14_{10}$−1(25)+1(24)+1(21)=−3210+1610+210=−1410
Adding and subtracting numbers in the binary number system is no different than in the decimal system! We’re just working with bits instead of digits.
Multiplying Binary Numbers
Let’s remind ourselves how we multiply numbers in decimal:
21x 12————
Remember the process? We multiply the $2$2 by each digit in the first multiplicand and write out the result under the bar:
21x 12———— 42
Then we move on to the $1$1 in $12$12 and repeat the process, but adding a $0$0 in the right column of the result. Add the two intermediate products to get the answer:
21x 12————— 42+ 210————— 252
Guess what? The process is exactly the same in the binary number system!
Let’s multiply these same two numbers in binary. They are $21_{10} = 010101$2110=010101 and $12_{10} = 01100$1210=01100:
010101x 01100—————————
Obviously, this is going to be more involved in binary since we’re working with bits (and thus longer strings), but the logic is still the same. In fact, beyond having to write out so many
intermediate results, we actually have it much easier over in binary. Whenever a digit is $1$1, you simply copy down the first multiplicand, padded with zeros. Whenever it’s a zero times the first
multiplicand, the result is zero!
010101x 01100———————————— 000000 0000000 01010100 010101000+ 0000000000———————————— 0011111100
Expanding this in binary, we get:
$0011111100_2 = 1(2^7) + 1(2^6) + 1(2^5) + 1(2^4) + 1(2^3) + 1(2^2) = 252_{10}$00111111002=1(27)+1(26)+1(25)+1(24)+1(23)+1(22)=25210
Easy peasy. The same process applies regardless of whether your multiplicands are signed or unsigned.
Dividing Binary Numbers
Let’s divide $126_{10}$12610 by $12_{10}$1210 using long division:
0 1 0 . 5 _______12 |1 2 6 - 1 2 ———— 0 6 - 0 —————— 6 0 - 6 0 ————— 0
Answer: $10.5$10.5.
Now let’s repeat the process over in the binary number system. Note that I’m going to strip leading zeros to make my life easier since we’re working with two unsigned numbers:
_______1100 |1111110
Take things one digit at a time, and reference this useful YouTube video if you get stuck:
0 0 0 1 0 1 0 . 1 ______________1 1 0 0 |1 1 1 1 1 1 0 . 0 -0 —— 1 1 - 0 ———— 1 1 1 - 0 —————— 1 1 1 1 - 1 1 0 0 ———————— 1 1 1 - 0 —————————— 1 1 1 1 - 1 1 0 0 ————————— 0 0 1 1 0 - 0 ————————— 1 1 0 - 0 ————— 1 1 0 0 - 1 1 0 0 ——————— 0 0 0 0
Answer: $01010.1$01010.1.
What does the $1$1 to the right of the decimal point represent? Well, in the decimal number system, anything to the right of the decimal point represents a negative power of ten: $10^{-1}$10−1, $10^
{-2}$10−2, and so on.
As you may have guessed, in the binary number system, these are $2^{-1}$2−1, $2^{-2}$2−2, and so on. So $.1$.1 above really means $1(2^{-1})$1(2−1), which is $\frac{1}{2} = 0.5_{10}$21=0.510 in
decimal. And of course, the part in front of the decimal point evaluates to $10_{10}$1010.
That gives us $10_{10} + 0.5_{10} = 10.5$1010+0.510=10.5. So our answer using binary long division is exactly the same as the one we got over in decimal!
Integer Overflow and Underflow in Binary
What happens if you try to add one to the largest representable $N$N-bit signed integer?
For example, if $N = 32$N=32, we’re really asking what happens if we try adding one to the largest representable $32$32-bit signed int.
Let’s give it a shot:
0111...11111 + 0000...00001————————————————
In the rightmost column, we’ll get $1_2 + 1_2 = 10_2$12+12=102, so that’s a zero carry a one. But as a result, all of the remaining additions will be $1_2 + 1_2$12+12 since we’ll always carry a
one until we get to the leading bit:
11111111111 <-- carry bits 0111...11111 (2^{N-1} - 1) + 0000...00001 (1)———————————————— 1000...00000 (-2^{N-1})
And what number is that in signed binary? Hmm… Looks like it’s the smallest representable negative number! What we’ve observed here is called integer overflow. When you try to go past the largest
representable signed integer in a given $N$N-bit system, the result overflows or wraps around.
What if we try to subtract one from the smallest representable $N$N-bit signed integer? First, we’ll represent $-1_{10}$−110 as a signed integer in binary:
1 in binary: 0000...00001Flipped: 1111...11110Add one: 1111...11111 <-- -1
Now let’s add this to the smallest representable signed integer:
1 <-- carry bits 1000...00000 (-2^{N-1}) + 1111...11111 (-1)———————————————— 1|0111...11111 (2^{N-1} - 1)
Notice that the result carries an additional bit over, yielding a result that has $N+1$N+1 bits. But our system only supports $N$N bits, so that leading $1$1 is actually discarded. The result is the
largest representable $N$N-bit signed integer, and this is known as integer underflow.
Overflow and underflow are things you should be mindful of in programs that are performing lots of computations, as you may end up getting unexpected results.
The Binary Number System: Additional Topics for Exploration
That about does it for this introduction to the binary number system! We took a pretty in-depth look at decimal, binary, and hexadecimal, and I hope you now have a greater appreciation for the binary
number system and the role that it plays in computing.
In reality, there’s much more to learn beyond what we covered here. If you’re curious, I encourage you to look into representing floating point numbers in binary using the IEE754 format.
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University of Reading
Data Assimilation Meetings at Reading
Date Meeting Speakers
Robin Hogan (Radar and clouds group, UoR)
Fast reverse-mode automatic differentiation using expression templates in C++ Further information
Gradient-based optimization problems are encountered in many fields, but the associated task of differentiating large computer algorithms can be formidable. The operator-overloading
approach to automatically differentiating an algorithm (i.e. automatically performing the adjoint calculation) is the most convenient for the user but current implementations are
9 DARC typically 25-100 times slower than the original algorithm. A new method is presented that uses the expression template programming technique in C++ to provide a compile-time
May seminar representation of each mathematical expression as a computational graph that can be traversed in either direction. Conventional operator overloading (e.g., as used in the widely used
2012 CppAD library) can only propagate information from the most nested part of the expression outwards, but the computations needed to store a symbolic representation of the equivalent
differential expression propagate information in the opposite sense. This approach is applied to two lidar forward models, one that is similar to a 1D radiance model, and the other that
is similar to a 1D time-dependent advection model. It is found that the resulting implementation can perform reverse-mode differentiation with around 4-4.5 times the computational cost
of the original algorithm, which is 6-9 times faster than the CppAD library and only 20-50% slower than a hand-coded adjoint code. The potential application of this to a
radar-lidar-radiometer retrieval algorithm for clouds and precipitation will be discussed.
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Copyright (c) George Ungureanu KTH/ICT/ESY 2017
License BSD-style (see the file LICENSE)
Maintainer ugeorge@kth.se
Stability experimental
Portability portable
Safe Haskell Safe
Language Haskell2010
Collection of utility functions for working with Time. While the CT MoC describes time as being a non-disjoint continuum (represented in ForSyDe-Atom with Rational numbers), most of the functions
here are non-ideal approximations or conversions from floating point equivalents. The trigonometric functions are imported from the numbers package, with a fixed eps parameter.
These utilities are meant to get started with using the CT MoC, and should be used with caution if result fidelity is a requirement. In this case the user should find a native Rational implementation
for a particular function.
type Time = Rational Source #
Type alias for the type to represent metric (continuous) time. Underneath we use Rational that is able to represent any \(t\) between \(t_1 < t_2 \in T\).
(*^*) :: Time -> Time -> Time Source #
"Power of" function taking Times as arguments. Converts back and forth to Floating, as it uses the ** operator, so it is prone to conversion errors.
pi :: Time Source #
Time representation of the number π. Rational representation with a precision of 0.000001.
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Convert QHz to YHz (Quennahertz to Yottahertz)
Quennahertz into Yottahertz
Direct link to this calculator:
Convert QHz to YHz (Quennahertz to Yottahertz)
1. Choose the right category from the selection list, in this case 'Frequency'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and
π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Quennahertz [QHz]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Yottahertz [YHz]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
Utilize the full range of performance for this units calculator
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '198 Quennahertz'. In so doing, either the full name of the unit or
its abbreviation can be usedas an example, either 'Quennahertz' or 'QHz'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case
'Frequency'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought.
Alternatively, the value to be converted can be entered as follows: '18 QHz to YHz' or '11 QHz into YHz' or '76 Quennahertz -> Yottahertz' or '35 QHz = YHz' or '93 Quennahertz to YHz' or '52 QHz to
Yottahertz' or '69 Quennahertz into Yottahertz'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which
of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over
for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(62 * 21) QHz'. But different
units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '45 Quennahertz + 4 Yottahertz' or '79mm x 38cm x 96dm = ? cm^3'. The
units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 5.985 222 167 756 7×1021. For this form of presentation, the number
will be segmented into an exponent, here 21, and the actual number, here 5.985 222 167 756 7. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket
calculators, one also finds the way of writing numbers as 5.985 222 167 756 7E+21. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at
this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 5 985 222 167 756 700 000 000. Independent of the presentation of the
results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.
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Creation Science
In general, whole numbers appear to the eye, (a) as objects or events of interest to be counted, or (b) as strings of symbols to be interpreted. The first concern absolutes and may be idealised with
reference to collections of uniform spheres. The second are to do with matters of human expediency; they involve choice of radix, and the creation (or adaptation) of a set of symbols, together with
rules for their use. The currently-used method of number writing represents the final stage in a long process of evolutionary development; included in this was a period during which complete
alphabets were requisitioned for use as numerals as, for example, among the Greeks (from c.600 BC) and the Jews (from c.200 BC). Because the original documents of the Judaeo-Christian Scriptures were
written in Hebrew (Old Testament) and Greek (New Testament) it follows that they may also be fairly read as sets of numbers. (Hebrew and Greek schemes).
Whether arising from (a) or (b), numbers sometimes exhibit interesting patterns. In the case of (a), it may be possible to place the objects to be counted (uniform spheres, as suggested) inside a
regular polygonal or polyhedral frame so as to fill it precisely and completely. In such situations the numbers represented are said to be figurate. A typical example is found in the game of snooker
– the 15 ‘reds’ being tightly arranged in equilateral triangle fashion prior to the commencement of each frame. With regard to (b), it may be that the repetition of a particular digit catches the
eye, or it is seen that a number reads the same backward as forward, or again that a simple digit rotation yields a multiple of the original number.
When many events of these kinds are observed to occur in a small set of numbers, we may feel strongly led to attribute them to intelligent design (ID).
[Please note: following standard mathematical practice, the period (.) will here be used to signify ‘multiplied by’.]
An Auspicious Beginning
In his writings, Ivan Panin has drawn attention to the interesting fact that the 7 Hebrew words of the Bible’s first verse total 2701, or 37.73 – both factors prime. Here are the details:
However, more interested in following a trail of 7’s, and apparently having little understanding of probability theory, he completely missed a far more significant thread and entered a world of
fantasy – nevertheless, unearthing the occasional nugget of gold.
The developments that Panin missed are now considered.
• 2701 is 73rd triangular number; the outline of this triangle is formed of 216, or 6.6.6 counters, thus:
• large multiples of 37 exhibit the interesting property that the sum of the numbers formed by grouping the digits in threes from the right will always be a multiple of 37; in this particular
instance, 701 + 002 = 703, or 19.37 – this being the sum of the two final words of this opening verse (each a multiple of 37) , ie 407 + 296 = 703
• 703 is 37th triangular number, thus:
• when inverted, this fits precisely within 2701-as-triangle, dividing the remaining area into a trio of 666-as-triangle, thus:
• the picture incorporates the first three triangular multiples of 37 and is highly symbolic: we are reminded of the general context of Revelation 13 and the number of the beast, 666
• a small replica of this structure has been included (top left); it depicts 10 – radix of the denary system and basic collective unit in metrication and decimalisation – in a setting of perfect
numbers, viz 6 and 28 – 3rd and 7th triangular numbers, respectively – the latter spelling out the number of Hebrew words and letters of Genesis 1:1; the satellite trio of sixes, and the outline
of 28-as-triangle are further suggestive of 666
• because Panin missed these numerical geometries he also failed to observe that the sum of the Bible’s first 8 words, 3003, is also triangular (77th in the series); it follows that word 8 – the
first of Genesis 1:2 (value 302) – takes the form of a trapezium of 4 rows supporting (and underscoring!) the Genesis 1:1 triangle; here are the details:
• it is interesting – and possibly significant – that Genesis 1:1 seems to imply the larger triangle, thus: concatenating the factors (and order numbers of the main participants) of 2701 we obtain
3773, and actually see 3003 (its value) embracing 77 (its order number in the triangular series)!
• another feature linking the two is to be found in the word sequence 4 to 8, inclusive; the sum of these 5 numbers is 1801 – ie the numerical hexagon representing the symmetrical self-intersection
of the Genesis 1:1 triangle, thus:
• 3003 may be written, 33.91; in other words, division by 91 simply removes the central zeros of the palindrome
Clearly, there is sufficient material here to attract the attention of all earnest seekers of truth. But the story is only half told, and we continue…
The Plot Thickens
The foregoing analysis has revealed 37 to be a key player in the Genesis 1:1 phenomena. However, the depth of its involvement is greater yet, for consider:
• of the 127 sums formed from combinations of the 7 words of this verse, 23 are found to be multiples of 37, ie nearly 7 times the number expected of a random set!
• included among these are the eye-catching analogues of 666, viz 999 (occurring twice as the sums of words 1 and 3, and of 2, 4 and 5), 777 (sum of the three nouns, 3+5+7), and 888 (3+5+6)
• 37 may be written as the difference of the cubes of 4 and 3, thus: 37 = 64 – 27; on the other hand, 91 (factor of 3003, and digit reverse of 19 – factor of 703) is the sum of the same two cubes,
thus: 91 = 64 + 27
• but, even more remarkably, 37 and 91 are unique among the natural numbers in that they may each be realised geometrically in 3 distinct ways, thus:
• no instances of higher orders of figuracy (as here defined) are known to exist
• 73 and 19 – the numbers formed by reversing the digits of 37 and 91 – each possess a single geometrical form that is closely coordinated with 37, thus:
• for figures having the same order, the product hexagon.hexagram = triangle; the instances relevant to this discussion are tabulated below:
Order Hexagon (X) Hexagram (Y) X.Y = Triangle Comments
2 7 13 91 Factor of 3003
3 19 37 703 “…and the earth.”
4 37 73 2701 Genesis 1:1
• these triangles and their factors are therefore implied by the following figures:
• the 6/1 theme of the creation narrative is clearly seen in these composite figures
In Search of an Author
There can be little doubt that this confluence of coincidences, springing from the heart of mathematics, speaks loudly of intelligent design. Could it be that Panin’s basic assumptions have been
right all along? – that God, in His infinite wisdom, has by these means provided man with empirical evidence of His Being and Sovereignty? To ascribe the phenomena to chance – bearing in mind the
nature of the Book in which they occur, and their strategic position and content – surely demands a leap of faith of mammoth proportions; on the other hand, to believe they are the work of man, one
has to hypothesise the existence of some mathematical/literary genius writing many centuries before the introduction of alphabetic numeration into Israel.
That the phenomena are undoubtedly of supernatural origin, and intended to achieve some serious purpose, is made clear by the following extension to the foregoing analysis:
• the Creator’s name (Jesus Christ), as we find it in the Greek of both New Testament and Septuagint (c 300 BC), is 2368, or 64.37
• its components are 888 (Jesus), or 24.37, and 1480 (Christ), or 40.37
• as an eye-catching analogue of 666, and a multiple of 37, 888 is clearly linked with the phenomena previously described
• however, there is a further link: the HCF of 888 and 1480 is 296 – the value of the Bible’s 7th word, translated ‘the earth’; to find it a factor of both components of the name of one who claimed
to be God Incarnate cannot be lightly dismissed!
• but for even more convincing evidence, we turn again to numerical geometry: 64 is, of course, the cube of 4; to represent it in on screen we typically draw 37-as-hexagon; this one figure thus
implies 2368 (Jesus Christ), the product of 64 and its 2D representation!
• the second figure depicts the 6th cube, 6.6.6 – outline of the Genesis 1:1 triangle; typically, we see only 91 of its 216 elements
Vernon Jenkins MSc
7th February, 1999
email: vernon.jenkins@virgin.net
See also: http://www.jsampson.co.uk/miracla1.htm for ‘Aspects of a Miracle’
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Reflex Angle Calculator - Calculator Wow
Reflex Angle Calculator
Understanding angles is fundamental in geometry, and one important concept is the reflex angle. A reflex angle is one that measures more than 180 degrees but less than 360 degrees. Calculating a
reflex angle is straightforward, and our Reflex Angle Calculator simplifies this task even further. This tool is perfect for students, teachers, engineers, and anyone needing quick, accurate angle
Importance of the Reflex Angle Calculator
The Reflex Angle Calculator is a valuable tool for several reasons:
1. Educational Use: It aids students and teachers in learning and teaching geometry by providing quick and accurate angle calculations.
2. Engineering and Design: Engineers and designers frequently work with angles, and this calculator ensures precision in their measurements.
3. Time-Saving: Manually calculating reflex angles can be time-consuming; this tool speeds up the process.
4. Error Reduction: Automated calculations reduce the likelihood of human error, ensuring more accurate results.
5. Convenience: It provides a user-friendly interface that can be used by anyone, regardless of their mathematical proficiency.
How to Use the Reflex Angle Calculator
Using the Reflex Angle Calculator is simple and intuitive:
1. Enter the Primary Angle: Input the primary angle (in degrees) into the calculator. This angle should be between 0 and 179 degrees.
2. Calculate: Click the “Calculate Reflex Angle” button.
3. View Results: The calculator will display the reflex angle, which is the primary angle plus 180 degrees.
10 FAQs and Answers
1. What is a reflex angle?
□ A reflex angle is any angle greater than 180 degrees but less than 360 degrees.
2. How does the Reflex Angle Calculator work?
□ The calculator adds 180 degrees to the primary angle you input to find the reflex angle.
3. Why is it important to know reflex angles?
□ Reflex angles are important in various fields such as geometry, design, and engineering, where precise angle measurements are crucial.
4. Can I use the calculator for angles outside 0-179 degrees?
□ No, the primary angle should be between 0 and 179 degrees to ensure the reflex angle is calculated correctly.
5. Is the Reflex Angle Calculator accurate?
□ Yes, it provides precise calculations based on the input values.
6. Do I need any mathematical background to use the calculator?
□ No, the calculator is designed to be user-friendly and can be used by anyone without requiring extensive mathematical knowledge.
7. Can this calculator be used for educational purposes?
□ Absolutely! It’s a great tool for students learning about angles and their properties.
8. What should I do if the calculator shows an error?
□ Ensure you are entering a valid primary angle between 0 and 179 degrees. If the problem persists, check for any input errors.
9. Can the calculator be used in professional settings?
□ Yes, it’s useful for professionals in engineering, architecture, and design where accurate angle measurements are essential.
10. Is the Reflex Angle Calculator available on mobile devices?
□ The calculator is accessible via any device with internet access, including mobile phones and tablets.
The Reflex Angle Calculator is an essential tool for anyone dealing with geometric calculations. Its simplicity and accuracy make it a go-to resource for students, educators, engineers, and
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this calculator is a valuable addition to your toolkit. Use it to enhance your understanding and application of geometric principles, making your work more efficient and accurate.
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Aggregate() Function in R - DataScience Made Simple
Aggregate() Function in R Splits the data into subsets, computes summary statistics for each subsets and returns the result in a group by form. Aggregate function in R is similar to group by in SQL.
Aggregate() function is useful in performing all the aggregate operations like sum,count,mean, minimum and Maximum.
Lets see an Example of following
• Aggregate() method which computes group sum
• calculate the group max and minimum using aggregate() Method
• Aggregate() method which computes group mean
• Get group counts using aggregate() function.
A pictographical representation of aggregate() function i.e. aggregate sum is shown below
Syntax for Aggregate() Function in R:
aggregate(x, by, FUN, …, simplify = TRUE, drop = TRUE)
X an R object, Mostly a dataframe
by a list of grouping elements, by which the subsets are grouped by
FUN a function to compute the summary statistics
simplify a logical indicating whether results should be simplified to a vector or matrix if possible
drop a logical indicating whether to drop unused combinations of grouping values.
Example of Aggregate() Function in R:
Let’s use the iris data set to demonstrate a simple example of aggregate function in R. We all know about iris dataset. Suppose if want to find the mean of all the metrics (Sepal.Length Sepal.Width
Petal.Length Petal.Width) for the distinct species then we can use aggregate function
# Aggregate function in R with mean summary statistics
agg_mean = aggregate(iris[,1:4],by=list(iris$Species),FUN=mean, na.rm=TRUE)
the above code takes first 4 columns of iris data set and groups by “species” by computing the mean for each group, so the output will be
note: When using the aggregate() function, the by variables must be in a list.
Example for aggregate() function in R with sum:
Let’s use the aggregate() function in R to create the sum of all the metrics across species and group by species.
# Aggregate function in R with sum summary statistics
agg_sum = aggregate(iris[,1:4],by=list(iris$Species),FUN=sum, na.rm=TRUE)
When we execute the above code, the output will be
Example for aggregate() function in R with count:
Let’s use the aggregate() function to create the count of all the metrics across species and group by species.
# Aggregate function in R with count
agg_count = aggregate(iris[,1:4],by=list(iris$Species),FUN=length)
the above code takes first 4 columns of iris data set and groups by “species” by computing the count for each group, so the output will be
Example for aggregate() function in R with maximum:
Let’s use the aggregate() function to create the maximum of all the metrics across species and group by species.
# Aggregate function in R with maximum
agg_max = aggregate(iris[,1:4],by=list(iris$Species),FUN=max, na.rm=TRUE)
the above code takes first 4 columns of iris data set and groups by “species” by computing the max for each group, so the output will be
Example for aggregate() function in R with minimum:
Let’s use the aggregate() function to create the minimum of all the metrics across species and group by species.
# Aggregate function in R with minimum
agg_min = aggregate(iris[,1:4],by=list(iris$Species),FUN=min, na.rm=TRUE)
the above code takes first 4 columns of iris data set and groups by “species” by computing the min for each group, so the output will be
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PiecewisePareto_Layer_SM: Second Layer Moment of the Piecewise Pareto Distribution in Pareto: The Pareto, Piecewise Pareto and Generalized Pareto Distribution
Calculates the second moment of a piecewise Pareto distribution in a reinsurance layer
PiecewisePareto_Layer_SM( Cover, AttachmentPoint, t, alpha, truncation = NULL, truncation_type = "lp" )
Cover Numeric. Cover of the reinsurance layer.
AttachmentPoint Numeric. Attachment point of the reinsurance layer.
t Numeric vector. Thresholds of the piecewise Pareto distribution.
alpha Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
truncation Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
truncation_type Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
Numeric vector. alpha[i] is the Pareto alpha in excess of t[i].
Numeric. If truncation is not NULL and truncation > t, then the Pareto distribution is truncated at truncation.
Character. If truncation_type = "wd" then the whole distribution is truncated. If truncation_type = "lp" then a truncated Pareto is used for the last piece.
The second moment of the (truncated) piecewise Pareto distribution with parameter vectors t and alpha in the layer Cover xs AttachmentPoint
t <- c(1000, 2000, 3000) alpha <- c(1, 1.5, 2) PiecewisePareto_Layer_SM(4000, 1000, t, alpha) PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000) PiecewisePareto_Layer_SM(4000, 1000, t,
alpha, truncation = 5000, truncation_type = "lp") PiecewisePareto_Layer_SM(4000, 1000, t, alpha, truncation = 5000, truncation_type = "wd")
For more information on customizing the embed code, read Embedding Snippets.
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Deep Learning: What You Need to Know - reason.townDeep Learning: What You Need to Know
Deep Learning: What You Need to Know
Deep learning is a subset of machine learning in which algorithms are inspired by the structure and function of the brain.
Checkout this video:
What is deep learning?
Deep learning is a type of machine learning that uses algorithms to model high-level abstractions in data. By using these deep learning algorithms, developers can create models that can automatically
learn and improve from experience without being explicitly programmed. This is different from traditional machine learning techniques, which require humans to hand-craft features from raw data.
Deep learning is a part of a wider family of machine learning methods based on artificial neural networks with representation learning. Deep learning architectures such as deep neural networks, deep
belief networks and recurrent neural networks have been successful in areas such as computer vision, speech recognition, natural language processing and bioinformatics.
What are the benefits of deep learning?
There are many benefits of deep learning, including the ability to create more accurate models than traditional machine learning algorithms, the ability to process large amounts of data more
efficiently, and the ability to automatically learn features from data. Deep learning is also well suited for applications such as computer vision and natural language processing, where the volume
and complexity of data is prohibitive for traditional machine learning algorithms.
What are the limitations of deep learning?
There are several limitations to deep learning, including:
-They require a lot of data to train
-They are very computationally expensive
-They can be difficult to interpret
-They can be susceptible to overfitting
How can deep learning be used in businesses?
Deep learning is a branch of machine learning that deals with algorithms that can learn from data that is unstructured or unlabeled. It has been used in a variety of fields, including computer
vision, natural language processing, and speech recognition.
In recent years, deep learning has begun to be used in business applications. For example, it can be used to create predictive models for customer behavior or to automate tasks such as fraud
detection or identifying objects in images. It can also be used to improve the results of search engines or to recommend products to customers.
How can deep learning be used in healthcare?
Deep learning is a powerful tool that can be used in many different fields, including healthcare. In healthcare, deep learning can be used to help diagnose and treat conditions, predict outcomes, and
much more.
How can deep learning be used in education?
Deep learning can be used in education to help students learn more effectively. For example, deep learning can be used to create personalized learning experiences for each student. Additionally, deep
learning can be used to help students learn more about a subject by providing them with relevant and targeted information.
What are some ethical considerations with deep learning?
Deep learning is a branch of machine learning that is concerned with algorithms that learn from data that is unstructured or unlabeled. This means that instead of being given explicit instructions,
the algorithm is able to learn by itself by understanding the data it is given. Deep learning has been used in a variety of fields, such as computer vision, speech recognition, and natural language
There are some ethical considerations that need to be taken into account when using deep learning. For example, if you are using deep learning for facial recognition, there is the potential for
misuse, such as surveillance of individuals or profiling based on race or ethnicity. Another consideration is that deep learning algorithms can be biased if they are not trained on a diverse enough
dataset. This can lead to unfair outcomes, such as false positives in criminal justice applications.
How is deep learning being used currently?
Deep learning is a subset of machine learning that is particularly well-suited to large-scale data analysis. Deep learning algorithms are able to automatically extract high-level features from data,
making them very efficient at tasks such as image classification and object detection.
Deep learning is being used in a variety of different fields, including:
* Agriculture: Automating crop yield prediction and disease identification
* Automotive: Self-driving cars and accident avoidance
* Finance: Stock market prediction and fraud detection
* Healthcare: Diagnosis and treatment planning
* Retail: Product recommendation and customer segmentation
What are some potential future applications of deep learning?
Deep learning is already being used in a number of different fields with great success. Some potential future applications of deep learning include:
-Autonomous vehicles: Deep learning could be used to train autonomous vehicles to better recognize and respond to their surroundings.
-Predicting consumer behavior: Deep learning could be used to predict consumer behavior, trends, and preferences.
-Personalized medicine: Deep learning could be used to develop personalized medicine, Tailored treatments based on an individual’s genes, lifestyle, and environment.
-Improving agricultural yields: Deep learning could be used to develop more efficient and effective ways of farming, leading to higher agricultural yields.
Where can I learn more about deep learning?
Deep learning is a branch of machine learning based on artificial neural networks, which are used to model high-level abstractions in data. These models are trained using a large amount of data and
can provide accurate results when applied to new data.
There are many resources available for learning about deep learning, including online courses, books, and research papers. Here are a few recommendations to get started:
-Online courses: Stanford’s CS231n: Convolutional Neural Networks for Visual Recognition is a popular online course that covers the basics of convolutional neural networks, which are commonly used in
deep learning. Another option is Udacity’s Deep Learning Nanodegree program, which covers a variety of topics in deep learning.
-Books: Deep Learning by Geoffrey Hinton, Yoshua Bengio, and Aaron Courville is a comprehensive book that covers many aspects of deep learning. For a more intuitive approach, consider reading Neural
Networks and Deep Learning by Michael Nielsen.
-Research papers: A good starting point is the “Deep Learning” chapter of the seminal machine learning book by Tom Mitchell. Alternatively, search for papers on arXiv using the keyword “deep
learning” to find the latest research on this topic.
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Math B.S. Degrees
The B.S. degree in Mathematics gives you an extensive background in mathematics. There are two tracks to our B.S. degree. Our academic advisors and faculty can provide advice toyou about the
different tracks and about combining Math with another major or degree on campus.
Program I focuses on mathematical theory. You gain the training you'll need if you plan to pursue graduate work in mathematics. You'll also gain access to many career pathways right out of college.
Program II focuses on mathematical applications. This track prepares you to work in related areas such as astronomy, biology, chemistry, cognitive science, computer science, economics, geology, or
physics. The focus in this program is on mathematical applications.
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Go Memory Management Part 2 - Povilas Versockas
Learning Go? Check out The Go Programming Language & Go in Action books. These books have greatly helped when I was just starting with Go. If you like to learn by example, definitely get the Go in
Previously on povilasv.me, we explored Go Memory Management and we left with 2 small Go programs having different virtual memory size.
Firstly, let’s take a look at ex1 program which consumes a lot of virtual memory. The code for ex1 looks like this:
[web_code] func main() { http.HandleFunc(“/bar”, func(w http.ResponseWriter, r *http.Request) { fmt.Fprintf(w, “Hello, %q”, html.EscapeString(r.URL.Path)) }) http.ListenAndServe(“:8080”, nil) } [/
In order to view virtual memory size, I executed ps. Take a look at the output showing large virtual memory size below. Note that ps output is in kibibytes. 388496 KiB is around 379.390625 MiB.
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 16609 0.0 0.0 388496 5236 pts/9 Sl+
Secondly, let’s take a look at ex2 program, which doesn’t consume a lot of virtual memory:
[web_code] func main() { go func() { for { var m runtime.MemStats runtime.ReadMemStats(&m) log.Println(float64(m.Sys) / 1024 / 1024) log.Println(float64(m.HeapAlloc) / 1024 / 1024) time.Sleep(10 *
time.Second) } }() fmt.Println(“hello”) time.Sleep(1 * time.Hour) } [/web_code]
Finally, let’s take a look at ps output for this program. As you can see this program runs with small virtual memory size. Note that 4900 KiB is around 4.79 MiB.
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 3642 0.0 0.0 4900 948 pts/10 Sl+
It’s important to note that these programs are compiled with a bit dated Go 1.10 version. There are differences on newer versions of Go.
For example, compiling ex1 program with Go 1.11 produces virtual memory size of 466MiB and resident set size of 3.22MiB. Additionally, doing the same with Go 1.12 provides us with 100.37MiB of
virtual memory and 1.44MiB of resident set size.
As we can see, the difference between HTTP server and a simple command line application made all the difference to virtual memory size.
Aha moment
Since then I had an aha moment. Maybe I can debug this interesting behavior with strace. Take a look at strace description:
strace is a diagnostic, debugging and instructional userspace utility for Linux. It is used to monitor and tamper with interactions between processes and the Linux kernel, which include system
calls, signal deliveries, and changes of process state.
So the plan is to run both programs with strace, in order to compare what is happening on the operating system level. Running and using strace is really simple, you just add strace in front of
compiled program. For instance, in order to trace ex1 program I executed:
[web_code] strace ./ex1 [/web_code]
Which produces the following output:
execve("./ex1", ["./ex1"], 0x7fffe12acd60 /* 97 vars */) = 0
brk(NULL) = 0x573000
access("/etc/ld.so.preload", R_OK) = -1 ENOENT (No such file or directory)
openat(AT_FDCWD, "/usr/local/lib/tls/haswell/x86_64/libpthread.so.0", O_RDONLY|O_CLOEXEC) = -1 ENOENT (No such file or directory)
stat("/usr/local/lib/tls/haswell/x86_64", 0x7ffdaa923fa0) = -1 ENOENT (No such file or directory)
stat("/lib/x86_64", 0x7ffdaa923fa0) = -1 ENOENT (No such file or directory)
openat(AT_FDCWD, "/lib/libpthread.so.0", O_RDONLY|O_CLOEXEC) = 3
read(3, "\177ELF\2\1\1\0\0\0\0\0\0\0\0\0\3\0>\0\1\0\0\0\340b\0\0\0\0\0\0"..., 832) = 832
fstat(3, {st_mode=S_IFREG|0755, st_size=146152, ...}) = 0
mmap(NULL, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8d11000
mmap(NULL, 2225248, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc8a88cd000
mprotect(0x7fc8a88e8000, 2093056, PROT_NONE) = 0
mmap(0x7fc8a8ae7000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1a000) = 0x7fc8a8ae7000
mmap(0x7fc8a8ae9000, 13408, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8ae9000
close(3) = 0
openat(AT_FDCWD, "/usr/local/lib/libc.so.6", O_RDONLY|O_CLOEXEC) = -1 ENOENT (No such file or directory)
openat(AT_FDCWD, "/lib/libc.so.6", O_RDONLY|O_CLOEXEC) = 3
read(3, "\177ELF\2\1\1\3\0\0\0\0\0\0\0\0\3\0>\0\1\0\0\0\0\34\2\0\0\0\0\0"..., 832) = 832
fstat(3, {st_mode=S_IFREG|0755, st_size=1857312, ...}) = 0
mmap(NULL, 3963464, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc8a8505000
mprotect(0x7fc8a86c3000, 2097152, PROT_NONE) = 0
mmap(0x7fc8a88c3000, 24576, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1be000) = 0x7fc8a88c3000
mmap(0x7fc8a88c9000, 14920, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc8a88c9000
close(3) = 0
mmap(NULL, 12288, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8d0e000
arch_prctl(ARCH_SET_FS, 0x7fc8a8d0e740) = 0
mprotect(0x7fc8a88c3000, 16384, PROT_READ) = 0
mprotect(0x7fc8a8ae7000, 4096, PROT_READ) = 0
mprotect(0x7fc8a8d13000, 4096, PROT_READ) = 0
set_tid_address(0x7fc8a8d0ea10) = 2109
set_robust_list(0x7fc8a8d0ea20, 24) = 0
rt_sigaction(SIGRTMIN, {sa_handler=0x7fc8a88d2ca0, sa_mask=[], sa_flags=SA_RESTORER|SA_SIGINFO, sa_restorer=0x7fc8a88e1140}, NULL, 8) = 0
rt_sigaction(SIGRT_1, {sa_handler=0x7fc8a88d2d50, sa_mask=[], sa_flags=SA_RESTORER|SA_RESTART|SA_SIGINFO, sa_restorer=0x7fc8a88e1140}, NULL, 8) = 0
rt_sigprocmask(SIG_UNBLOCK, [RTMIN RT_1], NULL, 8) = 0
prlimit64(0, RLIMIT_STACK, NULL, {rlim_cur=8192*1024, rlim_max=RLIM64_INFINITY}) = 0
brk(NULL) = 0x573000
brk(0x594000) = 0x594000
sched_getaffinity(0, 8192, [0, 1, 2, 3]) = 8
mmap(0xc000000000, 65536, PROT_NONE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc000000000
munmap(0xc000000000, 65536) = 0
mmap(NULL, 262144, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8cce000
mmap(0xc420000000, 1048576, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc420000000
mmap(0xc41fff8000, 32768, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc41fff8000
mmap(0xc000000000, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc000000000
mmap(NULL, 65536, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8cbe000
mmap(NULL, 65536, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8cae000
rt_sigprocmask(SIG_SETMASK, NULL, [], 8) = 0
sigaltstack(NULL, {ss_sp=NULL, ss_flags=SS_DISABLE, ss_size=0}) = 0
sigaltstack({ss_sp=0xc420002000, ss_flags=0, ss_size=32768}, NULL) = 0
rt_sigprocmask(SIG_SETMASK, [], NULL, 8) = 0
gettid() = 2109
Similarly, let’s trace the ex2 program and take a look at it’s output:
[web_code] strace ./ex2 [/web_code]
execve("./ex2", ["./ex2"], 0x7ffc2965ca40 /* 97 vars */) = 0
arch_prctl(ARCH_SET_FS, 0x5397b0) = 0
sched_getaffinity(0, 8192, [0, 1, 2, 3]) = 8
mmap(0xc000000000, 65536, PROT_NONE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc000000000
munmap(0xc000000000, 65536) = 0
mmap(NULL, 262144, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7ff1c637b000
mmap(0xc420000000, 1048576, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc420000000
mmap(0xc41fff8000, 32768, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc41fff8000
mmap(0xc000000000, 4096, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0xc000000000
mmap(NULL, 65536, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7ff1c636b000
mmap(NULL, 65536, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7ff1c635b000
rt_sigprocmask(SIG_SETMASK, NULL, [], 8) = 0
sigaltstack(NULL, {ss_sp=NULL, ss_flags=SS_DISABLE, ss_size=0}) = 0
sigaltstack({ss_sp=0xc420002000, ss_flags=0, ss_size=32768}, NULL) = 0
rt_sigprocmask(SIG_SETMASK, [], NULL, 8) = 0
gettid() = 22982
Note that actual output is longer. For readability sake, I just copied output for strace up to the point, where both programs call gettid(). This point was selected for the reason that it appeared
only once in both strace dumps.
Let’s compare the outputs. First of all, ex1 trace output is way longer than ex2. ex1 looks for some .so libraries and then loads them into memory. For instance, this is how trace output for loading
libpthread.so.0 looks:
openat(AT_FDCWD, "/lib/libpthread.so.0", O_RDONLY|O_CLOEXEC) = 3
read(3, "\177ELF\2\1\1\0\0\0\0\0\0\0\0\0\3\0>\0\1\0\0\0\340b\0\0\0\0\0\0"..., 832) = 832
fstat(3, {st_mode=S_IFREG|0755, st_size=146152, ...}) = 0
mmap(NULL, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8d11000
mmap(NULL, 2225248, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc8a88cd000
mprotect(0x7fc8a88e8000, 2093056, PROT_NONE) = 0
mmap(0x7fc8a8ae7000, 8192, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1a000) = 0x7fc8a8ae7000
mmap(0x7fc8a8ae9000, 13408, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc8a8ae9000
close(3) = 0
In this example, we can see that file is being opened, read into memory and finally closed. Some of those memory regions are mapped with PROT_EXEC flag, which allows our program to execute code
located in that memory region. We can see the same thing happening for libc.so.6:
openat(AT_FDCWD, "/lib/libc.so.6", O_RDONLY|O_CLOEXEC) = 3
read(3, "\177ELF\2\1\1\3\0\0\0\0\0\0\0\0\3\0>\0\1\0\0\0\0\34\2\0\0\0\0\0"..., 832) = 832
fstat(3, {st_mode=S_IFREG|0755, st_size=1857312, ...}) = 0
mmap(NULL, 3963464, PROT_READ|PROT_EXEC, MAP_PRIVATE|MAP_DENYWRITE, 3, 0) = 0x7fc8a8505000
mprotect(0x7fc8a86c3000, 2097152, PROT_NONE) = 0
mmap(0x7fc8a88c3000, 24576, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_DENYWRITE, 3, 0x1be000) = 0x7fc8a88c3000
mmap(0x7fc8a88c9000, 14920, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_FIXED|MAP_ANONYMOUS, -1, 0) = 0x7fc8a88c9000
close(3) = 0
After loading these libraries, both programs start to behave similarly. We can see that strace outputs after loading, show mapping of the same memory regions and contain traces of running similar
commands up to the point, where programs execute gettid().
This is interesting. So ex1 has loaded libpthread & libc, but ex2 didn’t.
cgo must be involved.
Let’s explore cgo and how it works. As godoc explains:
Cgo enables the creation of Go packages that call C code.
Turns out in order to call C code, you need to add a special comment and use special C package. Let’s take a look at this mini example:
[web_code] package main // #include import “C” import “fmt” func main() { char := C.getchar() fmt.Printf(“%T %#v”, char, char) } [/web_code]
This program includes stdio.h from C standard library and then calls getchar() and prints resulting variable. getchar() gets a character (an unsigned char) from stdin. Let’s try it:
[web_code] go build ./ex3 [/web_code]
When executing this program, it will ask you to enter a character and simply print it. Here is an example of this interaction:
main._Ctype_int 97
As we can see, it behaves just like Go. Interestingly it compiles just like native Go code, as you just go build and that’s it. I bet If you didn’t look at the code, you wouldn’t notice a difference.
Obviously it has a lot more interesting features. For instance, if you place your .c and .h files in the same directory as native go code, go build will compile and link it together with your native
I suggest you checkout godoc and C? Go? Cgo! blog post if you want to learn more. Now let’s, get back to our interesting problem. Why did ex1 program use cgo and ex2 didn’t?
Exploring difference
The difference between ex1 and ex2 programs is import path. ex1 imports net/http, while ex2 doesn’t. Interestingly, Grepping thru net/http package doesn’t show any signs of C statements. But it’s
enough to look one level up and volia, it’s the net package.
Take a look at these net package files:
For example, net/cgo_linux.go contains:
// +build !android,cgo,!netgo
package net
#include <netdb.h>
import "C"
// NOTE(rsc): In theory there are approximately balanced
// arguments for and against including AI_ADDRCONFIG
// in the flags (it includes IPv4 results only on IPv4 systems,
// and similarly for IPv6), but in practice setting it causes
// getaddrinfo to return the wrong canonical name on Linux.
// So definitely leave it out.
const cgoAddrInfoFlags = C.AI_CANONNAME | C.AI_V4MAPPED | C.AI_ALL
As we can see net package includes C header file netdb.h and uses a couple of variables from there. But why do we need this? Let’s explore it.
What is netdb.h ?
If you look up netdb.h documentation, you will see that netdb.h is part of libc. It’s documentation states:
netdb.h it provides definitions for network database operations.
Additionally, documentation explains these constants. Let’s have a look:
• AI_CANONNAME – Request for canonical name.
• AI_V4MAPPED – If no IPv6 addresses are found, query for IPv4 addresses and return them to the caller as IPv4-mapped IPv6 addresses
• AI_ALL – Query for both IPv4 and IPv6 addresses.
Looking at how these flags are actually used, it turns out that they are passed to getaddrinfo(), which is a call to resolve DNS name using libc. So simply those flags, control how DNS name
resolution is going to take place.
Likewise if you open net/cgo_bsd.go, you will see a bit diffferent version of cgoAddrInfoFlags constant. Let’s take a look:
// +build cgo,!netgo
// +build darwin dragonfly freebsd
package net
#include <netdb.h>
import "C"
const cgoAddrInfoFlags = (C.AI_CANONNAME | C.AI_V4MAPPED |
C.AI_ALL) & C.AI_MASK
So this hints us, that there is a mechanism to set OS specific flags for DNS resolution amd we are using cgo to do DNS queries correctly. That’s cool. Let’s explore net package a little more.
Have a read thru net package documentation:
Name Resolution
The method for resolving domain names, whether indirectly with functions like Dial or directly with functions like LookupHost and LookupAddr, varies by operating system.
On Unix systems, the resolver has two options for resolving names. It can use a pure Go resolver that sends DNS requests directly to the servers listed in /etc/resolv.conf, or it can use a
cgo-based resolver that calls C library routines such as getaddrinfo and getnameinfo.
By default the pure Go resolver is used, because a blocked DNS request consumes only a goroutine, while a blocked C call consumes an operating system thread. When cgo is available, the cgo-based
resolver is used instead under a variety of conditions: on systems that do not let programs make direct DNS requests (OS X), when the LOCALDOMAIN environment variable is present (even if empty),
when the RES_OPTIONS or HOSTALIASES environment variable is non-empty, when the ASR_CONFIG environment variable is non-empty (OpenBSD only), when /etc/resolv.conf or /etc/nsswitch.conf specify
the use of features that the Go resolver does not implement, and when the name being looked up ends in .local or is an mDNS name.
The resolver decision can be overridden by setting the netdns value of the GODEBUG environment variable (see package runtime) to go or cgo, as in:
export GODEBUG=netdns=go # force pure Go resolver
export GODEBUG=netdns=cgo # force cgo resolver
The decision can also be forced while building the Go source tree by setting the netgo or netcgo build tag.
A numeric netdns setting, as in GODEBUG=netdns=1, causes the resolver to print debugging information about its decisions.
Enough reading, let’s play with it and try to use different DNS client implementations.
Build tags
So as per documentation, we can use environment variables to use one or the other DNS client implementation. This is quite neat as you don’t have to recompile your go code, in order to switch
Additionally, looking thru the code I found that we can use Go build tags to compile Go binary with different DNS client. In addition we can check, which implementation is actually used, by setting
GODEBUG=netdns=1 environment variable and doing an actual DNS call.
By looking thru net package source files, I figured that there is 3 build modes. The build modes are enabled using different build tags. Here is the list:
1. !cgo – no cgo means that we are forced to use Go’s resolver.
2. netcgo or cgo – means that we are using libc DNS resolution.
3. netgo + cgo – means that we are using go native DNS implementation, but we can still include C code.
Let’s try all of these combinations to see how this actually works out.
We need to use different program as none of our programs execute DNS queries. Therefore, we will use this code:
[web_code] func main() { addr, err := net.LookupHost(“povilasv.me”) fmt.Println(addr, err) } [/web_code]
Let’s build it:
[web_code] export CGO_ENABLED=0 export GODEBUG=netdns=1 go build -tags netgo [/web_code]
And run it:
[web_code] ./testnetgo [/web_code]
go package net: built with netgo build tag; using Go's DNS resolver
104.28.1.75 104.28.0.75 2606:4700:30::681c:4b 2606:4700:30::681c:14b <nil>
Now let’s build it with libc resolver:
export GODEBUG=netdns=1
go build -tags netcgo
And run it:
[web_code] ./testnetgo [/web_code]
go package net: using cgo DNS resolver
104.28.0.75 104.28.1.75 2606:4700:30::681c:14b 2606:4700:30::681c:4b <nil>
And finally let’s use netgo cgo:
[web_code] export GODEBUG=netdns=1 go build -tags ‘netgo cgo’ . [/web_code]
And lastly run it:
[web_code] ./testnetgo [/web_code]
go package net: built with netgo build tag; using Go's DNS resolver
104.28.0.75 104.28.1.75 2606:4700:30::681c:14b 2606:4700:30::681c:4b <nil>
We can see that build tags actually work. Now let’s get back to our virtual memory problem.
Back to Virtual Memory
Now I would like to see how virtual memory behaves with those 3 combinations of build tags with our ex1 program, which is a simple HTTP web server.
Let’s build it in netgo mode:
[web_code] export CGO_ENABLED=0 go build -tags netgo [/web_code]
Process stats:
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 3524 0.0 0.0 7216 4076 pts/17 Sl+
As we can see virtual memory is low in this case.
Now let’s do the same with netcgo on:
[web_code] go build -tags netcgo [/web_code]
Process stats:
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 6361 0.0 0.0 382296 4988 pts/17 Sl+
As we can see this suffers form large virtual memory size (382296 KiB).
Lastly, let’s try netgo cgo mode:
[web_code] go build -tags ‘netgo cgo’ . [/web_code]
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 8175 0.0 0.0 7216 3968 pts/17 Sl+
And we can see that virtual memory is low in this case (7216 KiB).
So We can definitely roll out the netgo, as it doesn’t use a lot of virtual memory. On the other hand, we can’t yet blame cgo for bloated virtual memory. As our ex1 example, doesn’t include any
additional C code, so netgo would actually compile the same thing as netgo cgo, skipping the whole cgo process of building and linking C files.
Therefore, we need to try netcgo and netgo cgo with additional C code included. This would check how program behaves in cgo mode with disabled libc DNS client and cgo mode with enabled libc DNS
Let’s try this example:
[web_code] package main // #include import “C” import “fmt” func main() { char := C.getchar() fmt.Printf(“%T %#v”, char, char) http.HandleFunc(“/bar”, func(w http.ResponseWriter, r *http.Request) {
fmt.Fprintf(w, “Hello, %q”, html.EscapeString(r.URL.Path)) }) http.ListenAndServe(“:8080”, nil) } [/web_code]
As we can see this example should work well for us. Because, it uses both cgo and would use netgo or libc DNS client implementations based on build tags.
Let’s try it out:
[web_code] go build -tags netcgo . [/web_code]
Process stats:
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 12594 0.0 0.0 382208 4824 pts/17 Sl+
We can see the same virtual memory behavior. Now let’s try netgo cgo:
[web_code] go build -tags ‘netgo cgo’ . [/web_code]
Process stats:
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 1026 0.0 0.0 382208 4824 pts/17 Sl+
Finally, we can rule out libc DNS client implementation, as disabling it didn’t help. We can clearly see it is related to cgo.
To explore this further, let’s simplify our program. ex1 program starts an HTTP server, which would be way harder to debug than a simple command line application. So take a look this code:
[web_code] package main // #include // #include import “C” import ( “time” “unsafe” ) func main() { cs := C.CString(“Hello from stdio”) C.puts(cs) time.Sleep(1 * time.Second) C.free(unsafe.Pointer
(cs)) } [/web_code]
Let’s run it and check it’s memory:
[web_code] go build . ./ex6 [/web_code]
Process stats:
USER PID %CPU %MEM VSZ RSS TTY STAT
povilasv 15972 0.0 0.0 378228 2476 pts/17 Sl+
Cool virtual memory is big, this means that we need to investigate cgo.
Checkout Go Memory Management Part 3 for deeper investigation.
That’s it for the day. If you are interested to get my blog posts first, join the newsletter.
Thanks for reading & see you next time!
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TOC | Previous | Next | Index
46.3 Hierarchical Cluster Analysis (.NET, C#, CSharp, VB, Visual Basic, F#)
Cluster analysis detects natural groupings in data. In hierarchical cluster analysis, each object is initially assigned to its own singleton cluster. The analysis then proceeds iteratively, at each
stage joining the two most similar clusters into a new cluster, continuing until there is one overall cluster. In NMath Stats, class ClusterAnalysis performs hierarchical cluster analyses.
Distance Functions
During clustering, the distance between individual objects is computed using a distance function. The distance function is encapsulated in a Distance.Function delegate, which takes two vectors and
returns a measure of the distance (similarity) between them:
Code Example – C# hierarchical cluster analysis
public delegate double Function( DoubleVector data1,
DoubleVector data2 );
Code Example – VB hierarchical cluster analysis
Delegate Function(Data1 As DoubleVector, Data2 As DoubleVector) As
Delegates are provided as static variables on class Distance for many common distance functions:
● Distance.EuclideanFunction computes the Euclidean distance between two data vectors (2 norm):
Euclidean distance is simply the geometric distance in the multidimensional space.
● Distance.SquaredEuclideanFunction computes the squared Euclidean distance between two vectors:
Squaring the simple Euclidean distance places progressively greater weight on objects that are further apart.
● Distance.CityBlockFunction computes the city-block (Manhattan) distance between two vectors (1 norm):
In most cases, the city-block distance measure yields results similar to the simple Euclidean distance. Note, however, that the effect of outliers is dampened, since they are not squared.
● Distance.MaximumFunction computes the maximum (Chebychev) distance between two vectors:
This distance measure may be appropriate in cases when you want to define two objects as different if they differ on any one of the dimensions.
● Distance.PowerFunction( double p, double r ) computes the power distance between two vectors:
where p and r are user-defined parameters. Parameter p controls the progressive weight that is placed on differences on individual dimensions; parameter r controls the progressive weight that is
placed on larger differences between objects. Appropriate selections of p and r yield Euclidean, squared Euclidean, Minkowski, city-block, and many other distance metrics. For example, if p and r
are equal to 2, the power distance is equal to the Euclidean distance.
All provided distance functions allow missing values. Pairs of elements are excluded from the distance measure when their comparison returns NaN. If all pairs are excluded, NaN is returned for the
distance measure.
You can also define your own Distance.Function delegate and use it to cluster your data. For example, if you have function MyDistance() that computes the distance between two vectors:
Code Example – C# hierarchical cluster analysis
public double MyDistance( DoubleVector x, DoubleVector y );
Code Example – VB hierarchical cluster analysis
Public Function MyDistance(X As DoubleVector, Y As DoubleVector) As
You can define a Distance.Function delegate like so:
Code Example – C# hierarchical cluster analysis
var MyDistanceFunction = new Distance.Function( MyDistance );
Code Example – VB hierarchical cluster analysis
Dim MyDistanceFunction As New Distance.Function(AddressOf
Linkage Functions
During clustering, the distances between clusters of objects are computed using a linkage function. The linkage function is encapsulated in a Linkage.Function delegate. When two groups P and Q are
united, a linkage function computes the distance between the new combined group P + Q and another group R.
Figure 6 – Computing the distance between clusters using a linkage function
The parameters to the Linkage.Function—which may not necessarily all be used to calculate the result—are the distance between R and P, the distance between R and Q, the distance between P and Q, and
the sizes (n) of all three groups:
Code Example – C# hierarchical cluster analysis
public delegate double Function( double Drp, double Drq,
double Dpq, double Nr, double Np, double Nq );
Code Example – VB hierarchical cluster analysis
Delegate Function(DRP As Double, DRQ As Double,
DPQ As Double, NR As Double, NP As Double, NQ As Double) As
Delegates are provided as static variables on class Linkage for many common linkage functions:
● Linkage.SingleFunction computes the distance between two clusters as the distance of the two closest objects (nearest neighbors) in the clusters. Adopting a friends-of-friends clustering strategy
closely related to the minimal spanning tree, the single linkage method tends to result in long chains of clusters.
● Linkage.CompleteFunction computes the distance between two clusters as the greatest distance between any two objects in the different clusters (furthest neighbors). The complete linkage method
tends to work well in cases where objects form naturally distinct clumps.
● Linkage.UnweightedAverageFunction computes the distance between two clusters as the average distance between all pairs of objects in the two different clusters. This method is sometimes referred to
as unweighted pair-group method using arithmetic averages, and abbreviated UPGMA.
● Linkage.WeightedAverageFunction computes the distance between two clusters as the average distance between all pairs of objects in the two different clusters, using the size of each cluster as a
weighting factor. This method is sometimes referred to as weighted pair-group method using arithmetic averages, and abbreviated WPGMA.
● Linkage.CentroidFunction computes the distance between two clusters as the difference between centroids. The centroid of a cluster is the average point in the multidimensional space. The centroid
method is sometimes referred to as unweighted pair-group method using the centroid average, and abbreviated UPGMC.
● Linkage.MedianFunction computes the distance between two clusters as the difference between centroids, using the size of each cluster as a weighting factor. This is sometimes referred to as
weighted pair-group method using the centroid average, and abbreviated WPGMC.
● Linkage.WardFunction computes the distance between two clusters using Ward's method. Ward's method uses an analysis of variance approach to evaluate the distances between clusters. The smaller the
increase in the total within-group sum of squares as a result of joining two clusters, the closer they are. The within-group sum of squares of a cluster is defined as the sum of the squares of the
distance between all objects in the cluster and the centroid of the cluster. Ward's method tends to produce compact groups of well-distributed size.
You can also define your own Linkage.Function delegate and use it to cluster your data. For example, if you have function MyLinkage() that computes the distance between two clusters:
Code Example – C# hierarchical cluster analysis
public double MyLinkage( double Drp, double Drq, double Dpq,
double Nr, double Np, double Nq );
Code Example – VB hierarchical cluster analysis
Public Function MyLinkage(DRP As Double, DRQ As Double, DPQ As
Double, NR As Double, NP As Double, NQ As Double) As Double
You can define a Linkage.Function delegate like so:
Code Example – C# hierarchical cluster analysis
var MyLinkageFunction = new Linkage.Function( MyLinkage );
Code Example – VB hierarchical cluster analysis
Dim MyLinkageFunction As New Linkage.Function(AddressOf MyLinkage)
A ClusterAnalysis instance is constructed from a matrix or a dataframe containing numeric data. Each row in the data set represents an object to be clustered.
Code Example – C# hierarchical cluster analysis
var ca = new ClusterAnalysis( data );
Code Example – VB hierarchical cluster analysis
Dim CA As New ClusterAnalysis(Data)
The current default distance and linkage delegates are used. The default distance and linkage delegates are Distance.EuclideanFunction and Linkage.SingleFunction, unless the defaults have been
changed using the static DefaultDistanceFunction and DefaultLinkageFunction properties. For example:
Code Example – C# hierarchical cluster analysis
ClusterAnalysis.DefaultDistanceFunction = Distance.MaximumFunction;
ClusterAnalysis.DefaultLinkageFunction = Linkage.CentroidFunction;
Code Example – VB hierarchical cluster analysis
ClusterAnalysis.DefaultDistanceFunction = Distance.MaximumFunction
ClusterAnalysis.DefaultLinkageFunction = Linkage.CentroidFunction
This changes the default distance and linkage functions for all subsequently constructed ClusterAnalysis objects.
You can also specify non-default distance and linkage functions in the constructor:
Code Example – C# hierarchical cluster analysis
var ca = new ClusterAnalysis( data,
Distance.PowerFunction( 1.25, 2.0 ), Linkage.CompleteFunction );
Code Example – VB hierarchical cluster analysis
Dim CA As New ClusterAnalysis(Data,
Distance.PowerFunction(1.25, 2.0), Linkage.CompleteFunction)
After construction, you can retrieve information about the ClusterAnalysis configuration using the provided properties:
● N gets the total number of objects being clustered.
● DistanceFunction gets and sets the distance function delegate used to measure the distance between individual objects. Setting the distance function using the DistanceFunction property has no
effect until Update() is called with new data. (See below.)
● LinkageFunction gets and sets the linkage function used to measure the distance between clusters of objects. Setting the linkage delegate using the LinkageFunction property has no effect until
Update() is called with new data. (See below.)
The Distances property gets the vector of distances between all possible object pairs, computed using the current distance delegate. For n objects, the distance vector is of length (n-1)(n/2), with
distances arranged in the order:
(1,2), (1,3), ..., (1,n), (2,3), ..., (2,n), ..., ..., (n-1,n)
Linkages gets an (n-1) x 3 matrix containing the complete hierarchical linkage tree, computed from Distances using the current linkage delegate. At each level in the tree, columns 1 and 2 contain the
indices of the clusters linked to form the next cluster. Column 3 contains the distances between the clusters. For example, this code clusters 8 random vectors of length 3, then shows a sample output
of the hierarchical cluster tree:
Code Example – C# hierarchical cluster analysis
var data = new DoubleMatrix( 8, 3, new RandGenUniform() );
var ca = new ClusterAnalysis( data );
Console.WriteLine( ca.Linkages );
Code Example – VB hierarchical cluster analysis
Dim Data As New DoubleMatrix(8, 3, New RandGenUniform())
Dim CA As New ClusterAnalysis(Data)
Sample output:
7x3 [
4 7 0.194409151975696
3 5 0.290431894003636
2 9 0.495557235783239
1 6 0.508966210536187
0 11 0.522321103698264
8 10 0.590187697768796
12 13 0.621675638177606 ]
Each object is initially assigned to its own singleton cluster, numbered 0 to 7. The analysis then proceeds iteratively, at each stage joining the two most similar clusters into a new cluster,
continuing until there is one overall cluster. The first new cluster formed by the linkage function is assigned index 8, the second is assigned index 9, and so forth. When these indices appear later
in the tree, the clusters are being combined again into a still larger cluster.
The CutTree() method constructs a set of clusters by cutting the hierarchical linkage tree either at the specified height, or into the specified number of clusters. For example, this code cuts the
linkage tree to form 3 clusters:
Code Example – C# hierarchical cluster analysis
ca.CutTree( 3 );
Code Example – VB hierarchical cluster analysis
This code cuts the linkage tree at a height of 0.75:
Code Example – C# hierarchical cluster analysis
ca.CutTree( 0.75 );
Code Example – VB hierarchical cluster analysis
The CutTree() method returns a ClusterSet object, which represents a collection of objects assigned to a finite number of clusters. The NumberOfClusters property
gets the number of clusters into which objects are grouped; N gets the number of objects. The Clusters property returns an array of integers that identifies the cluster into which each object was
grouped. Cluster numbers are arbitrary, and range from 0 to NumberOfClusters - 1. The indexer gets the cluster to which a given object is assigned. The Cluster() method returns the objects assigned
to a given cluster as an array of integers. For instance:
Code Example – C# hierarchical cluster analysis
// Cluster 10 random vectors of length 4:
var data = new DoubleMatrix( 10, 4, new RandGenUniform() );
var ca = new ClusterAnalysis( data );
// Cut the tree into 5 clusters
ClusterSet cut = ca.CutTree( 5 );
Console.WriteLine( "ClusterSet = " + cut );
Console.WriteLine( "Object 0 is in cluster: " + cut[0] );
Console.WriteLine( "Object 3 is in cluster: " + cut[3] );
Console.WriteLine( "Object 8 is in cluster: " + cut[8] );
int[] objects = cut.Cluster( 1 );
Console.Write( "Objects in cluster 1: " );
for (int i = 0; i < objects.Length; i++ )
Console.Write( objects[i] + " " );
Code Example – VB hierarchical cluster analysis
'' Cluster 10 random vectors of length 4:
Dim Data As New DoubleMatrix(10, 4, New RandGenUniform())
Dim ca As New ClusterAnalysis(Data)
'' Cut the tree into 5 clusters
Dim Cut As ClusterSet = CA.CutTree(5)
Console.WriteLine("ClusterSet = " & cut)
Console.WriteLine("Object 0 is in cluster: " & Cut(0))
Console.WriteLine("Object 3 is in cluster: " & Cut(3))
Console.WriteLine("Object 8 is in cluster: " & Cut(8))
Dim Objects() As Integer = Cut.Cluster(1)
Console.Write("Objects in cluster 1: ")
For I As Integer = 0 To Objects.Length - 1
Console.Write(Objects(I) & " ")
Sample output:
ClusterSet = 0,1,2,1,1,1,3,1,4,1
Object 0 is in cluster: 0
Object 3 is in cluster: 1
Object 8 is in cluster: 4
Objects in cluster 1: 1 3 4 5 7 9
Lastly, the CopheneticDistances property on class ClusterAnalysis gets the vector of cophenetic distances between all possible object pairs. The cophenetic distance between two objects is defined to
be the intergroup distance when the objects are first combined into a single cluster in the linkage tree. The format is the same as the distance vector returned by Distances.
The correlation between the original Distances and the CopheneticDistances is sometimes taken as a measure of the appropriateness of a cluster analysis relative to the original data:
Code Example – C# hierarchical cluster analysis
var ca = new ClusterAnalysis( data );
double r = StatsFunctions.Correlation( ca.Distances,
ca.CopheneticDistances );
Code Example – VB hierarchical cluster analysis
Dim CA As New ClusterAnalysis(Data)
Dim R As Double = StatsFunctions.Correlation(CA.Distances,
Reusing Cluster Analysis Objects
Method Update() updates an existing ClusterAnalysis instance with new data, and optionally with new distance and linkage functions. For example:
Code Example – C# hierarchical cluster analysis
var ca = new ClusterAnalysis( data, Linkage.SingleFunction );
Console.WriteLine( ca.Linkages );
ca.Update( data, Linkage.CompleteFunction );
Console.WriteLine( ca.Linkages );
Code Example – VB hierarchical cluster analysis
Dim CA As New ClusterAnalysis(Data, Linkage.SingleFunction)
CA.Update(Data, Linkage.CompleteFunction)
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Answer set programming
1 {color(X,I) : c(I)} 1 :- v(X).
:- color(X,I), color(Y,I), e(X,Y), c(I).
The program in this example illustrates the "generate-and-test" organization that is often found in simple ASP programs. The choice rule describes a set of "potential solutions" — a simple superset
of the set of solutions to the given search problem. It is followed by a constraint, which eliminates all potential solutions that are not acceptable. However, the search process employed by smodels
and other answer set solvers is not based on trial and error.
A clique in a graph is a set of pairwise adjacent vertices. The following Lparse program finds a clique of size ${\displaystyle \geq n}$ in a given graph, or determines that it does not exist:
n {in(X) : v(X)}.
:- in(X), in(Y), v(X), v(Y), X!=Y, not e(X,Y), not e(Y,X).
This is another example of the generate-and-test organization. The choice rule in Line 1 "generates" all sets consisting of ${\displaystyle \geq n}$ vertices. The constraint in Line 2 "weeds out" the
sets that are not cliques.
A Hamiltonian cycle in a directed graph is a cycle that passes through each vertex of the graph exactly once. The following Lparse program can be used to find a Hamiltonian cycle in a given directed
graph if it exists; we assume that 0 is one of the vertices.
{in(X,Y)} :- e(X,Y).
:- 2 {in(X,Y) : e(X,Y)}, v(X).
:- 2 {in(X,Y) : e(X,Y)}, v(Y).
r(X) :- in(0,X), v(X).
r(Y) :- r(X), in(X,Y), e(X,Y).
:- not r(X), v(X).
The choice rule in Line 1 "generates" all subsets of the set of edges. The three constraints "weed out" the subsets that are not Hamiltonian cycles. The last of them uses the auxiliary predicate ${\
displaystyle r(x)}$ ("${\displaystyle x}$ is reachable from 0") to prohibit the vertices that do not satisfy this condition. This predicate is defined recursively in Lines 6 and 7.
This program is an example of the more general "generate, define and test" organization: it includes the definition of an auxiliary predicate that helps us eliminate all "bad" potential solutions.
% ********** input sentence **********
word(1, puella). word(2, pulchra). word(3, in). word(4, villa). word(5, linguam). word(6, latinam). word(7, discit).
% ********** lexicon **********
1{ node(X, attr(pulcher, a, fem, nom, sg));
node(X, attr(pulcher, a, fem, abl, sg)) }1 :- word(X, pulchra).
node(X, attr(latinus, a, fem, acc, sg)) :- word(X, latinam).
1{ node(X, attr(puella, n, fem, nom, sg));
node(X, attr(puella, n, fem, abl, sg)) }1 :- word(X, puella).
1{ node(X, attr(villa, n, fem, nom, sg));
node(X, attr(villa, n, fem, abl, sg)) }1 :- word(X, villa).
node(X, attr(linguam, n, fem, acc, sg)) :- word(X, linguam).
node(X, attr(discere, v, pres, 3, sg)) :- word(X, discit).
node(X, attr(in, p)) :- word(X, in).
% ********** syntactic rules **********
0{ arc(X, Y, subj) }1 :- node(X, attr(_, v, _, 3, sg)), node(Y, attr(_, n, _, nom, sg)).
0{ arc(X, Y, dobj) }1 :- node(X, attr(_, v, _, 3, sg)), node(Y, attr(_, n, _, acc, sg)).
0{ arc(X, Y, attr) }1 :- node(X, attr(_, n, Gender, Case, Number)), node(Y, attr(_, a, Gender, Case, Number)).
0{ arc(X, Y, prep) }1 :- node(X, attr(_, p)), node(Y, attr(_, n, _, abl, _)), X < Y.
0{ arc(X, Y, adv) }1 :- node(X, attr(_, v, _, _, _)), node(Y, attr(_, p)), not leaf(Y).
% ********** guaranteeing the treeness of the graph **********
1{ root(X):node(X, _) }1.
:- arc(X, Z, _), arc(Y, Z, _), X != Y.
:- arc(X, Y, L1), arc(X, Y, L2), L1 != L2.
path(X, Y) :- arc(X, Y, _).
path(X, Z) :- arc(X, Y, _), path(Y, Z).
:- path(X, X).
:- root(X), node(Y, _), X != Y, not path(X, Y).
leaf(X) :- node(X, _), not arc(X, _, _).
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SAT Prep: What is a Rational Number?
While students may not need to expressly name rational or irrational numbers on the SAT, they will be working a lot with both those types. Indeed, most students learn to work complex problems that
involve both rational and irrational numbers in Algebra class, yet many students soon forget the difference between the two.
What is a rational number? What is an irrational number? How is this relevant to the Math section of the SAT test? Read on for a quick yet comprehensive review.
What Are Rational Numbers?
\(1,2,3,4,5,\) and so on are all integers, as are \(-1,-2,-3,-4\), and so forth. \(1.23\) is a decimal and \(\frac{3}{8}\) is a fraction. All are called different things, yet all are considered
rational numbers.
At its core, a rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where both the numerator \(p\) and the denominator \(q\) are nonzero integers (e.g. \(\frac{p}{q}=\
frac{15}{8}, \frac{1}{2}, \frac{200}{3}\)). Note that 1 is an integer and thus can be the value of q, so all integers are rational numbers (e.g. \(\frac{4}{1}=4\)).
Other examples of rational numbers are decimals with finite decimal numbers (e.g. \(3.125=\frac{25}{8}\)) and decimals with a repeating pattern (e.g. \(1.33333333333…\)). This is because both these
types of decimals can be converted to fractions with an integer numerator and denominator (to learn more, read our post: How to Convert a Decimal to a Fraction).
It logically follows that any number that does not fit into at least one of these categories is considered an irrational number. What kinds of numbers classify as irrational?
For example, \(\pi\) represents \(3.1415926535897932384626433832795…\) and it keeps going. There is no finite end to \(\pi\), so it is considered an irrational number.
Similarly, many square roots, cubed roots, and so on are considered irrational numbers because they also solve as decimals with no finite end. \(\sqrt{3}\) is one such example, as it equals \
(1.7320508075688772935274463415059…\) and it goes on.
However, it is important to note that not all roots are irrational numbers. For example, \(\sqrt{16}\) is \(4\), which is a rational number. Therefore, \(\sqrt{16}\) is considered a rational number
as well.
So, let’s do a small quiz to make sure we have these concepts right:
Is \(3.181818181818\overline{18}\) a rational or irrational number?
Answer: A rational number! The decimals are infinite, but there is a pattern to it.
Is \(\sqrt{169}\) a rational or irrational number?
Answer: a rational number! \(13\:\cdot\:13 = 169\).
Last one: is Euler’s number, \(e\), a rational or irrational number?
Answer: an irrational number! Euler’s number appears to be a decimal without a discovered end. The first few decimals are \(2.7182818284590452353602874713527…\)
What Do You Need To Know For The SAT?
Very few questions on the SAT will ask you directly what is and is not a rational number. Instead, you will see many rational and irrational numbers being used throughout the SAT Math section, and
you will need to know how to solve, manipulate, and apply them.
Specifically, you will usually be allowed to leave irrational numbers as is on the SAT and not have to simplify or solve them. For example, if you are told to simplify an expression that has \(\pi\)
or \(e\) in it, you know that you are not going to be able to simplify those two variables, and your answers will need to be in terms of \(\pi\) or \(e\).
Here are some examples of SAT problems that deals with both rational and irrational numbers.
Source: SAT Math Practice Problems
1. In the complex number system, which of the following is equal to \((14-2i)(7+12i)\)? (\(i = \sqrt{-1}\))?
A. \(74\)
B. \(122\)
C. \(74 + 154i\)
D. \(122 + 154i\)
Answer: D
In this case, \(i\) is an irrational number because \(\sqrt{-1}\) does not simplify to an integer, a finite decimal, or a fraction with nonzero integers in the numerator and the denominator.
Therefore, you can treat \(i\) like a variable, in that you can’t simplify it any further.
Once you know this, this problem looks a lot like a FOIL problem using the distributive property. Using the FOIL method:
\(=98 + 168i − 14i − 24i^2\)
Note: \(i^2 = -1\).
Therefore, the expression above can be rewritten as: \(98 + 168i − 14i − (−24)\).
Combining like terms: \(122 + 154i\).
2. Simplify: \(\frac{\sqrt[3]{81}}{\sqrt[3]{9}}\)
A. \(9\)
B. \(\sqrt[3]{9}\)
C. \(3\)
D. \(2.08908\)
Answer: B.
\(\sqrt[3]{9}\) is an infinite non-repeating decimal, an irrational number. To keep our answer cleaner, we’ll leave it as is. The SAT should not make you write out irrational numbers in decimal form,
so you can leave them in their cleanest/simplest form.
For More Information
Itching for more CollegeVine concept review and testing tips so that you can ace the SAT? Check out our helpful previous blog posts:
25 Tips and Tricks for the SAT
30 SAT Math Formulas You Need To Know
Preparing for the SAT? Download our free guide with our top 8 tips for mastering the SAT.
Want to know how your SAT score impacts your chances of acceptance to your dream schools? Our free Chancing Engine will not only help you predict your odds, but also let you know how you stack up
against other applicants, and which aspects of your profile to improve. Sign up for your free CollegeVine account today to gain access to our Chancing Engine and get a jumpstart on your college
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How Does CIBO Model Crop Yield?
By Kofikuma Dzotsi
When you look at a CIBO parcel report, you see a graph showing simulated yields for that parcel across several years (Figure 1). If you select another parcel, you will most likely see different yield
values and different year-to-year patterns. This is true for each parcel across the geography covered by CIBO’s platform.
Figure 1: Example parcel report from CIBO showing the simulated 10-year average yield using CIBO’s proprietary crop model.
These simulated yields can help growers compare the productivities of different parcels over time, and the potential benefits of this technology are vast. But it leads to one question: How does CIBO
calculate these yields for each parcel across the corn belt — and beyond? The answer lies in the core technology used by CIBO to model crops in virtually any environment.
If you wish to know crop yield in a given year on a given parcel in Iowa, for example, one possibility is to physically travel to that parcel and conduct a field experiment. In this scenario, you’d
grow a crop, then measure the yield when the crop matures. This process would undoubtedly provide the desired answers; however, it cannot be repeated for all parcels across the U.S. — and for all
years of interest — because such experiments would simply be too expensive and impossible to manage.
But what if we constructed a digital model-plant, gave it the main characteristics of a real plant (roots, leaves, flowers, grains, etc.), then asked the computer to “grow” the plant by essentially
mimicking the physiological behavior of a real plant? Once we have a model-plant that grows on a given parcel in a given year, it becomes possible to expand this process to other parcels, other
years, and other crops. This is exactly the process CIBO’s crop simulation engine uses to predict crop yield.
The Four Factors of Crop Yield
The first step in building this kind of crop simulation engine is identifying the components that will go into the model-plant. Remember: the model-plant is an abstract, simplified representation of
a real plant‚ not a physical model. Therefore, we need to select the components that are most influential to crop yield to keep the model manageable. The main components that are important
determinants of crop yield are:
• soil properties (soil pH, soil texture, soil organic matter, etc.),
• weather (temperature, radiation, precipitation),
• management (planting date, planting density, amount of nitrogen applied during the season, etc.), and
• the characteristics of the specific crop variety grown (in the case of corn, relative maturity and the number of leaves the plant produces).
Once these components are identified, the next step involves defining the mathematical relationships linking them together. For example, when a corn plant grows in the field, the time it takes the
crop to mature strongly depends on the temperature of its growing environment. So, the time to maturity would be calculated using some function of temperature.
As you can imagine, building all these relationships is a major undertaking — this step involves an interdisciplinary collaboration among many scientists who contribute knowledge from several
disciplines about how the plant interacts with its environment. At this point we have constructed a mathematical model of the plant, which is basically a quantitative version of the model-plant.
Real Math Finds the Simulated Yield
We can think of this mathematical model as a large system of equations linking the plant characteristics to environmental variables and management practices (Fig. 2). Like all systems of equations,
selecting appropriate initial conditions and solving them will provide us with useful solutions. In our case, the system of equations is very complex, so we use some sophisticated techniques to solve
it. The model is then written in a language that computers can understand. Running the model is equivalent to solving the system of equations using a computer to obtain meaningful crop outputs, which
is essentially the same as simulation. Among many other variables, our solution of interest is crop yield.
Figure 2: Schematic representation of CIBO’s approach to simulating parcel yield. Each parcel is characterized by its weather, soil, and the manner in which a crop is likely to be managed. Based on
this information, CIBO builds a computer experiment that is used by the crop model to simulate yield on that parcel.
So, the next time you see CIBO-simulated yield (or productivity) for a parcel, remember that a complex simulation process underpins that one yield value, from a model-plant all the way to solving a
massive mathematical model– and it can all be done from a computer rather than a field in Iowa.
About Kofikuma Dzotsi
Kofikuma Dzotsi is a Principal Crop Scientist at CIBO Technologies, a science-driven software startup. Prior to CIBO, he worked on developing and applying crop models in the public and private
sectors. He holds an MS and Ph.D. in agricultural and biological engineering from the University of Florida.
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Gyroelongated Square Dipyramid -- from Wolfram MathWorld
One of the eight convex deltahedra built up from 16 equilateral triangles. It consists of two oppositely faced square pyramids rotated antiprism. It is Johnson solid
If the centroid is at the origin and the sides are of unit length, the equations of the 4-antiprism give height of the middle points as square pyramids gives apex heights of surface area and volume
of the solid are
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