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Lecture 14
1. Tuesday's Maple worksheet is here: mar23.mws.
2. Probability generating function for nonnegative r.v.
1. Demo on Poisson.
3. Exercise 4.69 on page 222. The random variable is S, the time for 7 passengers to arrive. It's 7-Erlang. See page 172. The number of passengers arriving in 10 minutes is N a Poisson r.v. with a=
10. We want p(N<7), which is the probability that the van waits over 10 minutes.
4. Exercise 4.102 on page 224. Characteristic function of U[-b,b]. See page 185. Use that to find mean.
5. Chapter 5: Two r.v.
6. Example 5.1 (height, weight). Possible event B: H<70, W<200.
7. Example 5.3: Breaking a message into packets. Pair of r.v: (Q,R). Possible event: last packet < 1/2 full.
8. pairs of discrete r.v.: joint pmf, marginal pmf. | {"url":"https://wrfranklin.org/pmwiki/pmwiki.php/EngProbSpring2010/Lecture14","timestamp":"2024-11-06T12:44:29Z","content_type":"application/xhtml+xml","content_length":"11203","record_id":"<urn:uuid:628a8c61-aabd-49d3-b13f-748bb135689a>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00758.warc.gz"} |
Method for modelling a surface and device for implementing same
Method for modelling a surface and device for implementing same
A method for obtaining a model of a surface including the steps of obtaining measurements of geometrical data concerning specific points on the surface, making a grid of the surface, with the grid
passing through said points, memorizing, at an address which is specific to each node of the grid, the coordinates of the node, the number of satellites of the node, information for access to the
addresses of said satellites and thereafter to information which relates to them, and geometrical data which may be associated with said node; for each node, defining a local roughness index obtained
from a weighted sum of the current coordinates of the node and its satellites, defining the sum of an overall roughness index representing the sum of all the local roughness indices, and of an
overall index of the infringement of said geometrical data, iteratively adjusting the coordinates of indefinite nodes, by using at each adjustment the sum of a weighted combination of current node
neighbor coordinates and of a combination of geometrical data associated with the node, in order to minimize said sum, and creating a model of the surface on the basis of the adjusted coordinates.
The present invention is generally concerned with investigations within a three-dimensional body and is particularly concerned with a new process of obtaining a representation of a surface within a
body on the basis of a limited set of known geometrical data relating to said surface.
Investigations within three-dimensional bodies are of major concern in geology, geophysics and biology.
In geophysics, for example, it is necessary to obtain as accurate as possible representations of surfaces situated, for example, at the interface between two areas of different kinds or with
different properties, on the basis of data obtained from prospecting, or during exploitation of an underground resource.
How effectively the three-dimensional body is investigated, especially in prospecting for oil or other underground resources, depends on the accuracy with which this type of surface can be
reconstituted and represented.
In medicine and biology there are various known processes for obtaining representations of cross-sections through a living body or the like and the aim is also to reconstitute and represent in three
dimensions a surface situated at the interface between two media, for example the contours of a bodily organ.
There already exist a number of modeling techniques for obtaining a representation of a surface within a three-dimensional body that is to be exploited or treated. Particularly worthy of mention are
the Bezier interpolation method and the spline functions method (for more details reference should be had to Geometric Modeling, M. E. Mortenson, Ed. John Wiley, 1985).
However, all these known techniques are ill suited to handling heterogeneous data, by which is meant data relating to the coordinates of points on the surface, to fixed orientations of the surface,
to geometrical links between two separate points, etc. Also, these techniques are ill-suited to a wide spread of surfaces or to surface anomalies such as folds and breaks, and to discontinuous
surfaces. To be more precise, in some cases the functions used in these techniques are not convergent or have no solution or have no single solution.
Finally, the known techniques are not able to take into account the concept of the degree of certainty with which some surface data is known.
The present invention is directed to alleviating the drawbacks of the prior art processes and to proposing a process that can allow for non-homogeneous and/or highly diverse geometrical data and for
the anomalies mentioned above and other anomalies, providing in each case a single and unambiguous model of the surface.
Another object of the invention is to propose a process that can use geometrical data and at the same time data as to the degree of certainty or accuracy of said data.
To this end, the present invention firstly consists in a process for modeling a surface representing for example the interface between two areas of different kinds or with different properties in a
three-dimensional body such as a geological formation or a living body, characterized in that it comprises the steps of:
obtaining by means of measuring apparatus a set of geometrical data relating to the surface and associated with respective points on said surface;
meshing the surface so that all said points are a subset of the nodes of the mesh;
storing at a specific memory address for each node of the mesh, the following data:
the coordinates of the node in question,
the number of satellite nodes of the node in question,
data providing access to the specific addresses of said satellite nodes and consequently to the data relating thereto,
if necessary, geometrical data associated with said node in question,
for each node of the mesh, defining a local roughness index derived from a weighted sum of the actual coordinates of the node and of its satellites,
defining the sum of a global roughness index obtained by summing the local roughness indices associated with each node and a global index of violation of said geometrical data,
fitting the coordinates of each node for which the precise coordinates are not known by an iterative method in which for each step of the iteration there are added a weighted combination of the
actual coordinates of the satellite and of the satellites of the satellites of said node and a combination of the geometrical data associated with said node, in such a way as to minimize said sum,
creating a representation of the surface from the fitted coordinates of each node.
The invention also concerns a device for modeling a surface representing for example the interface between two areas of different kinds or with different properties in a three-dimensional body such
as a geological formation or a living body, characterized in that it comprises:
means for obtaining by means of measuring apparatus a set of geometrical data relating to the surface and associated with respective points on said surface;
means for meshing the surface so that all said points are a subset of the nodes of the mesh;
means for storing at a specific memory address for each node of the mesh, the following data:
the coordinates of the node in question,
the number of satellite nodes of the node in question,
data providing access to the specific addresses of said satellite nodes and consequently to the data relating thereto,
if necessary, geometrical data associated with said node in question,
calculation means for fitting the coordinates of each node for which the precise coordinates are not known by an iterative method in which for each step of the iteration there are added a weighted
combination of the actual coordinates of the satellite and of the satellites of the satellites of said node and a combination of the geometrical data associated with said node, in such a way as to
minimize a sum of a global roughness index obtained by summing local roughness indices associated with each node and each derived from a weighted sum of the actual coordinates of the node and of its
satellites and a global index of violation of said geometrical data,
means for creating a representation of the surface from the fitted coordinates of each node.
These and other objects of the present invention will become more readily apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and
specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention
will become apparent to those skilled in the art from this detailed description.
The present invention will emerge more clearly from the following detailed description of one preferred embodiment of the invention given by way of non-limiting example and with reference to the
appended drawings, in which:
FIG. 1 shows a typical graph incorporating meshing used to model a surface, this graph being defined by the set of all the sides of facets whose vertices constitute the set .OMEGA. described later,
and also shows arrows symbolically representing displacements of nodes of the mesh applied interactively by the user to shape the surface;
FIGS. 2(a) through 2(d) show four instances of controlling the shape of the surface using four different types of vector constraint; the vectors .DELTA..sub..lambda..mu. are assumed to be known
directly in FIGS. 2(a) through 2(c) and known because they are orthogonal to a given vector V in FIG. 2(d);
FIG. 3 shows a typical complex geological surface (salt dome) modeled using the process of the present invention;
FIG. 4 shows a typical complex biological surface (embryo brain) modeled using the process of the present invention from discrete data obtained from a succession of cross-sections;
FIG. 5 shows the organization of the memory associated with an atom A(k) of the kernel .phi..sub.k ;
FIG. 6 shows a first example of memorizing an orbit of a given atom in an array with n.sub.k items;
FIG. 7 shows a second example of memorizing an orbit of a given atom in n.sub.k arrays each of two items;
FIG. 8 shows the memorization of a geometrical datum or constraint;
FIG. 9 shows the method of memorizing a set of geometrical data or constraints relating to a given atom A(k);
FIG. 10 shows the memorization in an array of all of the atoms relating to a surface S; and
FIG. 11 shows the chaining of access to said set of atoms;
FIG. 12 shows an apparatus embodiment of the present invention.
Identical or similar elements or parts are designated by the same reference numbers in all the figures.
Reference will be made later to a number of publications whose respective contents are deemed to be included by way of reference into this description.
(1) A method of Bivariate interpolation and smooth surface fitting based on local procedures, H. AKIMA, J. ACM 17.1, 1974;
(2) Shape reconstruction from planar cross-sections, J. D. BOISSONNAT, ACM Transactions on Graphics, vol. No. 2, 1986;
(3) Machine contouring using minimum curvature, I. C. BRIGGS, Geophysics 17.1, 1974;
(4) Triangular Bernstein-Bezier patches, G. FARIN, Computer-aided geometric design, 1986;
(5) Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams, L. GUIBAS and J. STOLFI, ACM transactions on Graphis, vol. No. 2, 1985;
(6) Three-dimensional graphic display of disconnected bodies, J. L. MALLET, Mathematical Geology, vol. 20, no. 8, 1988;
(7) Geometrical modeling and geostatistics, J. L. MALLET, Proceedings of the Third International Geostatistics Congress, Kluwer Academic Publishers, 1989;
(8) Discrete Smooth Interpolation, J. L. MALLET, ACM Transactions on Graphics, April issue, 1989;
(9) Geometric Modeling, M. E. MORTENSON, John Wiley, 1985.
Also, the following description of one preferred embodiment of the invention is followed by section 2, infra explaining a method of reconstituting a surface derived from the method explained in
reference document (8) and on which the process of the present invention is based.
1. DESCRIPTION OF A PROCESS IN ACCORDANCE WITH THE INVENTION FOR ENCODING AND FITTING COMPLEX SURFACES 1.1 Introduction
1.1.1 Nature of the process
An industrial process described hereafter is used to encode and fit complex surfaces encountered, for example, when modeling:
interfaces between different domains,
interfaces between geological layers,
interfaces between biological organs,
The result of this encoding phase may be converted into pictures and/or solid models (molded from plastics material, for example) by a graphic computer or any other device. In geology, for instance,
these pictures and/or solid models are used to optimize the prospecting, exploitation and management of earth resources such as:
oil deposits,
mineral deposits,
water deposits,
The process described hereafter consists of two complementary sub-processes:
1. a sub-process to encode a surface,
2. a sub-process to fit a surface to the data currently available. In geology, for example, these data may be:
the exact or approximate location of points in the intersection of a given surface and a plane,
the exact or approximate location of the intersection of a well and a geological surface corresponding to the interface between two geological layers,
the exact or approximate location of the plane tangent to a geological surface corresponding to the interface between two geological layers,
the exact or approximate throw vector of a fault breaking the geological surface corresponding to the interface between two geological layers,
The encoding process is designed to optimize the fitting process based on a new version of the DSI method (cf. [8]) especially developed for the geometrical modeling of surfaces. This new version of
the DSI method, which has not yet been published, is presented in section 2, infra.
1.1.2 Dividing a surface into facets
Let S be the surface to model. As shown in FIG. 1 and as explained in [6] and [7], this surface will be approximated by a set of triangular and/or polygonal facets. These facets are characterized by:
The coordinates of their vertices. These vertices constitute a finite set of N points {.phi..sub.1, . . . ,.phi.N } regularly distributed on S and numbered from 1 to N. The coordinates of the
k.sup.th vertex .phi..sub.k will be denoted (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z).
Their edges constituting the links between pairs of vertices (.phi..sub.i,.phi..sub.j).
The set of edges of the triangles and/or polygons composing the surface S constitute a graph G whose nodes are the vertices of the triangles and/or polygons. For this reason and for the sake of
simplicity, "node" is used hereafter as a synonym of "vertex of triangles and/or polygons".
node.uparw.{vertex of triangles and/or polygons}
1.1.3 Preliminary definitions
Set of satellites .LAMBDA.(k)
Let .phi..sub.k be the k.sup.th node of a surface S. The "set of satellites" of .phi..sub.k is the set .LAMBDA.(k) Of the nodes .phi..sub.j different from .phi..sub.k and such that
(.phi..sub.j,.phi..sub.k) iS an edge of triangle or polygon. In other words, the set .LAMBDA.(k) is such that: ##EQU1##
As shown above, the number of satellites attached to the node .phi..sub.k is denoted n.sub.k.
The orbit concept discussed in the articles [6] and [7] has not yet been subjected to any particular encoding process; an important feature of the invention is its proposed method of encoding
Neighborhood N(k)
Let .phi..sub.k be the k.sup.th node of a surface S. The neighborhood N(k) of .phi..sub.k is the subset N(k) of nodes equal to the union of .phi..sub.k and its satellites. In other words:
N(k)={.phi..sub.k }.orgate..LAMBDA.(k)
A neighbor of node .phi..sub.k is any node belonging to the set N(k). Thus:
.phi..sub.j .epsilon.N(k) {.phi..sub.j =Neighbor of .phi..sub.k }
The neighborhood concept discussed in the articles [7] and [8] has not yet been subjected to any particular encoding process; an important feature of the invention is its proposed method of encoding
Atom A(R) and associated kernel
Let .phi..sub.k be the k.sup.th node of a surface S. The atom of kernel .phi..sub.k A(R) is the pair (.phi..sub.k,.LAMBDA.*(k)) composed of:
the node .phi..sub.k
the set .LAMBDA.*(K) of the addresses or the codes {A*(j1), . . . ,A*(jn.sub.k)} allowing the set of the corresponding atoms {A(j1), . . . ,A(jn.sub.k)} whose kernels are the satellites of
.phi..sub.k to be retrieved from a memory.
In other words: ##EQU2##
The atom concept discussed in the article [6] has not yet been subjected to any particular encoding process; an important feature of the invention is its proposed method of encoding A(k).
Constraints and types of constraints
A constraint attached to an atom A(k) is any condition concerning the location of the node .phi..sub.k corresponding to the kernel of this atom. For example, in geology, the following constraints are
of prime importance to fitting a surface to precise or imprecise data observed or measured by a manual or geophysical technique:
The coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z) of the node .phi..sub.k are known exactly. Such a constraint is called a control node type constraint and may be used in
geology to encode the intersection of the modeled surface with a well.
The coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z) of the node .phi..sub.k are not known exactly but with a given certainty factor. Such a constraint is called a fuzzy control
node type constraint and may be used to encode the approximate location of a node of the modeled surface measured approximately by a given process. For example,
seismic data in geology, and
data corresponding to sections of organs observed with a microscope or using ultrasound techniques or with a scanner in biology are of this type.
The vector (.phi..sub..lambda. -.phi..sub.k) joining the node .phi..sub.k to another node .phi..sub..lambda. is approximately known with a given certainty factor. Such a constraint is called a fuzzy
vectorial link type constraint and is used in geology, for example, to encode the throw vector of a fault, the nodes .phi..sub.k and .phi..sub..lambda. being on opposite sides of the fault (see FIG.
The vector V orthogonal to the tangent plane to the surface S at node .phi..sub.k is approximately known with a given certainty factor. Such a constraint is called a fuzzy vector normal type
constraint and is used in geology, for example, to encode the plane tangent to layer interfaces measured by a manual or geophysical technique (sounding).
The certainty factors mentioned above are positive numbers whose values are assumed to be proportional to the a priori confidence that can be attached to the corresponding data. Accounting for these
fuzzy constraints constitutes one of the characteristics of the invention and is described in detail in section 2 (see sections 2.5 and 2.7.6), infra).
To simplify, no distinction will be drawn between the number of any node of a surface and the node itself. This implies that:
.LAMBDA.(k)=set of satellites {.phi..sub.j1, . . . ,.phi..sub.jn.sbsb.k } of .phi..sub.k or set of corresponding numbers {j1, . . . ,jn.sub.k },
N(k)=set of neighbors {.phi..sub.k,.phi..sub.j1, . . . ,.phi..sub.jn.sbsb.k } of .phi..sub.k or set of corresponding numbers {k,j1, . . . ,jn.sub.k }.
1.1.4 State of the art
Classical methods
A complete description of the "state of the art" concerning the modeling of complex surfaces is given in reference [7]. This article explains precisely why the classical methods used in automatic
mapping and in Computer Aided Design (CAD) are not relevant to the modeling of complex surfaces, as might otherwise be expected. The basics of the DSI method (also known as the DSI/DSA method) which
overcomes these difficulties are briefly introduced at the end of reference [7].
The DSI method
This method is based on the notion of "roughness" of a surface S composed of triangular and/or polygonal facets. This notion of "roughness" R(.phi..vertline.k) at node .phi..sub.k of a surface S is
defined as follows (See ref. [7], [8] and appendix): ##EQU3##
The positive or null coefficients {v.sup..alpha. (k)} occurring in this definition are weighting coefficients chosen by the user. Among the infinity of possible choices, one of the most interesting,
called "harmonic weighting", consists of choosing: ##EQU4##
The local roughnesses so defined are then combined in a global roughness R(.phi.): ##EQU5##
The positive or null coefficients .mu.(k) occurring in this definition are used to modulate the contribution of the local roughness (R(.phi..vertline.k) to the global roughness R(.phi.). If a uniform
weighting is used, for example:
.mu.(k)=1 k
Among other possible choices for the coefficients .mu.(k), the following weightings can be used, where m is a given positive constant: ##EQU6##
The principle of the DSI method consists in computing the location of each node .phi..sub.k of the surface S to model in order to render it as smooth as possible while respecting the data and
constraints governing the shape of the surface. The paper [8] is devoted to a mathematical presentation of the DSI method and proposes a technique for minimizing the global roughness while respecting
the data and constraints.
Interactive version Of the DSI method
The method proposed in [8] for minimizing the global roughness is not well suited to interactive use on an electronic computer (or microprocessor), which is why section 2, infra proposes a new method
on which the invention is based. As mentioned previously, the object of the invention is to propose an encoding and fitting process for the surface S based on this new version of the DSI method.
1.2 Examples of images of objects encoded and fitted with the process
1.2.1 Geological example
Oil prospecting is generally performed indirectly in the sense that one is looking for geological layers which could have acted as a trap for oil; for example, deformations of salt layers called
"salt domes" are generally excellent oil traps. In order to locate the oil precisely, it is necessary to know the exact shape of the "trap" layers and unfortunately, in many cases, until now there
was no practical process usable to model it efficiently. The salt dome case is generally regarded as the most complex and that is why to prove the efficiency of this encoding and fitting process FIG.
3 shows an image of such a surface obtained with it.
1.2.2 Biological example
Medical data acquisition processes (scanner or ultrasound) are mathematically identical to seismic techniques used in oil prospecting. For example, the ultrasound technique is identical to seismic
reflection and scanners are based on tomography as in seismic tomography. This is one reason why this encoding and fitting process may be used to model the skin of biological organs. FIG. 4 shows a
model of the brain of a 7 millimeters long human embryo obtained using this encoding and fitting method. Note that until now organs this small (the brain is 1 millimeter long) could only be seen in
the form of microscopic sections.
1.3 Encoding process for a surface S
1.3.1 Memory and memory address concepts
The term "memory" denotes any electronic computing device able to record data, codes or partial results and to retrieve them on request for use in subsequent operations. Moreover, the "address" of a
memory is any code identifying the physical location of the memory inside the computer.
1.3.2 Encoding an atom A(k)
The following process or one of its variants may be used to encode and store each atom A(k) in a set of memories:
Encoding process
As suggested in FIG. 5, a contiguous memory area beginning at address A*(k) stores consecutively the following data relating to the atom A(k) corresponding to the node .phi..sub.k :
1. Kernel=
either the coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z) of the node .phi..sub.k,
or the address or the code or any other data allowing the coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z) of the node .phi..sub.k to be retrieved.
2. nb.sub.-- sat=
either the number n.sub.k of satellites linked to the node .phi..sub.k,
or the address or the code or any other data allowing the number n.sub.k of satellites linked to the node .phi..sub.k to be retrieved.
3. Sat=
Address of a memory area containing the addresses or the access codes {A*(j1), . . . ,A*(jn.sub.k)} allowing the n.sub.k atoms {A(j1), . . . ,A(jn.sub.k)} whose kernels {.phi..sub.j1, . . .
,.phi..sub.jn.sbsb.k } are the satellites of the node .phi..sub.k to be retrieved.
4. Const=
Encoded constraints (see section 1.1.3) attached to the node .phi..sub.k. This encoding will be explained in detail in section 1.1.3.
5. Info=
Memory area of variable size that can contain complementary data concerning the atom A(k). This data is specific to each particular application; for instance, in oil prospecting, if A(k) is an atom
of the surface separating two adjacent layers C1 and C2, then it may be interesting to store in the Info field:
a list of physical properties (porosities, seismic velocities, etc) of the layer C1 at node .phi..sub.k corresponding to the atom A(k),
a list of physical properties (porosities, seismic velocities, etc) of the layer C2 at node .phi..sub.k corresponding to the atom A(k),
a list of geological properties (geological facies, presence of oil, etc) of the layer C1 at node .phi..sub.k corresponding to the atom A(k),
a list of geological properties (geological facies, presence of oil, etc) of the layer C2 at node .phi..sub.k corresponding to the atom A(k),
Variant 1
The memory area in which are stored the addresses or the access codes {A*(j1), . . . ,A*(jn.sub.k)} for retrieving the n.sub.k atoms {A(j1), . . . ,A(jn.sub.k)} whose kernels are the satellites
{.phi..sub.j1, . . . ,.phi..sub.jn.sbsb.k } of the node .phi..sub.k may be structured in two ways:
1. As shown in FIG. 6, the first way consists in using an array composed of n.sub.k consecutive memories containing the addresses {A*(j1), . . . ,A*(jn.sub.k)}.
2. The second way (FIG. 7) consists in using n.sub.k arrays each containing two consecutive memories such that for the .alpha..sup.th array:
the first memory contains the address A*(j.sub..alpha.) of the atom corresponding to the .alpha..sup.th satellite of the atom A(k),
the second memory contains the address of the (.alpha.+1).sup.th array. If .alpha. is equal to the number n.sub.k of satellites (nb.sub.-- sat=n.sub.k), then this second memory is either unused or
contains a code specifying that the last satellite of A(k) has been reached.
Variant 2
It is possible not to store the nb.sub.-- sat field but in this case, if the list of addresses or access codes {A*(j1), . . . ,A*(jn.sub.k)} iS coded as a single array (see variant 1), it is
necessary to add a memory area A*(n.sub.k +1) to store a code indicating that the list .LAMBDA.*(k) is finished.
Variant 3
The Sat field has to be made big enough to contain the addresses or access codes {A*(j1), . . . ,A*(jn.sub.k)}.
Variant 4
Many variants are possible depending on the size of the Info field and on how it is partitioned. However, it is obvious that all these variants have no influence on the use of the atom A(k) in
connection with the fitting algorithm based on the DSI method described in section 2, infra.
Variant 5
Other fields can be added to the atom A(k) described above. However these other fields have no influence on the use of the atom A(k) in connection with the fitting algorithm based on the DSI method
described in section, infra.
Variant 6
For any variant, what is really important for an efficient implementation of the DSI method is that the address or the access code A*(k) of a given atom A(k) should be the most direct path to the
atoms whose kernels are the satellites of the kernel of A(k).
The main object of the method of encoding A(k) in a memory described above is to allow direct access to the satellites.
1.3.3 Encoding the Constraints
Each constraint relative to an atom A(k) will be stored in a specific memory area whose description follows:
Process for encoding a constraint
As shown in FIG. 8, the encoding of a particular constraint attached to a given atom is performed in a memory area made up of two fields, Constraint and Next. These two fields are used as follows:
Memory area containing:
either the data related to the encoded constraint (see below),
or the address or the access code of a memory area containing the data related to the encoded constraint.
The data related to the encoded constraint must include a code identifying the type of the encoded constraint (see section 1.1.3).
Memory area containing:
either a code indicating that the current constraint is the last constraint attached to the atom A(k),
or the address of a memory area containing the next constraint (see below).
Process for storing all constraints attached to an atom
As described in section 1.3.2, encoding an atom A(k) requires a Const field that will be used to store:
either a code specifying that the atom A(k) has no constraint,
or the address of a memory area containing the first constraint related to the atom A(k).
If the address of the n.sup.th constraint related to the atom A(k) is stored in the Next field of the (n-1).sup.th constraint (n>1), then, as shown in FIG. 9, this implies that all the constraints
related to A(k) are attached to A(k). This encoding process allows new constraints to be added or old constraints to be removed without modifying the organization of the other data.
Examples of data relating to a given constraint
Section 1.1.3 gives three examples of constraint types which are of prime importance for the encoding and fitting of complex surfaces encountered in biology and geology. For each of these types of
constraints data has to be attached to the Constraint field previously described and shown in FIG. 9:
For a fuzzy control node type constraint, in addition to the constraint type, it is necessary to store at least the following data:
the approximate coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.x,.phi..sub.k.sup.z) of the node .phi..sub.k corresponding to the kernel of the atom A(k) to which the constraint is attached,
the certainty factor (positive number) proportional to the confidence that can be placed in the approximate coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z).
For a fuzzy vectorial link type constraint, in addition to the type, it is necessary to store at least the following data:
the address of the node .phi..sub..lambda. linked to the kernel .phi..sub.k of the atom A(k) to which the constraint is attached,
the three approximate coordinates of the vector (.phi..sub..lambda. -.phi..sub.k) joining the node .phi..sub.k to the node .phi..sub..lambda.,
the certainty factor (positive number) proportional to the confidence that can be placed in the approximate coordinates of the vector (.phi..sub..lambda. -.phi..sub.k).
For a fuzzy vector normal type constraint, in addition to the type, it is necessary to store at least the following data:
the three approximate components of the vector V orthogonal to the surface S at the node .phi..sub.k corresponding to the kernel of the atom A(k) to which the constraint is attached,
the certainty factor (positive number) proportional to the confidence that can be placed in the approximate coordinates of the vector V.
The three constraint types are described in detail in sections 2.5 and 2.7.6, infra.
Until now only one certainty factor common to the x, y and z components of each constraint has been introduced but up to three different certainty factors can be assigned to these three components.
1.3.4 Encoding a surface S
In order to optimize the programming of the variant of the DSI method presented in section 2, infra, a surface composed of triangular and/or polygonal facets is encoded as a set of N atoms whose
kernels (nodes) are the vertices of the triangles and/or polygons. The data and codes for each of these atoms {A(1), . . . ,A(N)} may be stored:
either in an array (see FIG. 10),
or in the Atom field of memory areas linked by their Next field (see FIG. 11 ). In this case, the Next field associated with the atom number k must contain the address or the access code of the next
atom numbered (k+1); the Next field associated to the last atom A(N) must contain a code indicating that the last atom has been reached.
The second solution is better if it must be possible to add and/or remove atoms easily but requires more memory than the first solution.
1.3.5 Variants
To simplify the description, the memory areas used to encode the data have been split up into fields and each of these fields has been given:
a name,
a memory location.
One of ordinary skill in the art would understand that the encoding process described in this patent does not depend on the names or the order of these fields inside the memory areas used. Also, it
is always possible to add new fields for particular applications without modifying the storage process.
1.4 Fitting a surface S
1.4.1 Introduction
Let S be a surface encoded according to the process described in section 1.3. There will be described hereafter a process based on this encoding to fit optimally the coordinates of some nodes
{.phi..sub.1, . . . ,.phi.N} in order to:
make S as smooth as possible in relation to the global roughness criterion of the DSI method (see references [7], [8], the appendix and section 1.1.4),
make S respect as much as possible the data and constraints, however accurately they may be known, concerning the shape of this surface (see sections 1.1.1 and 1.1.3).
For that purpose, we use a new variant of the DSI method described in the appendix and designed to use the encoding process described in section 1.3. Taking into account the measured or observed
data, this method fits the three coordinates (.phi..sub..alpha..sup.x,.phi..sub..alpha..sup.y,.phi..sub..alpha..sup.z) of each free node .phi..sub..alpha. successively and using an iterative method
and the following formula:
new value of .phi..sub..alpha..sup.x =.function..sub..alpha..sup.x (old values of .phi..sub.1.sup.x, . . . ,.phi..sub.N.sup.x),
new value of .phi..sub..alpha..sup.y =.function..sub..alpha..sup.y (old values of .phi..sub.1.sup.y, . . . ,.phi..sub.N.sup.y),
new value of .phi..sub..alpha..sup.z =.function..sub..alpha..sup.z (old values of .phi..sub.1.sup.z, . . . ,.phi..sub.N.sup.z).
Referring to the appendix describing the new version of the DSI method, these updating functions .function..sub..alpha..sup.x (), .function..sub..alpha..sup.y () and .function..sub..alpha..sup.z ()
are derived from the local form of the DSI equations at point .alpha. and have the following form (see sections 2.4.1, 2.4.2 and 2.7.3, infra): ##STR1##
The nature and the role of the weighting coefficients {v.sup..alpha. (k)} and .mu.(k) have been discussed in section 1.1.4 and are explained in article [8] and in section 2, infra. The values of the
terms {.GAMMA..sub.i.sup.x.alpha.,.GAMMA..sub.i.sup.y.alpha.,.GAMMA..sub.i.sup.z .alpha. }, {.gamma..sub.i.sup.x.alpha.,.gamma..sub.i.sup.y.alpha.,.gamma..sub.i.sup.z .alpha. } and
{Q.sub.i.sup.x.alpha.,Q.sub.i.sup.y.alpha.,Q.sub.i.sup.z.alpha. } depend on the types and the values of the constraints attached to the node .phi..sub..alpha. and their exact formulation is given in
section 2, infra.
When the weighting coefficients {v.sup..alpha. (k)} correspond to the harmonic weighting described in section 1.1.4 and in section 2, infra, the weighting functions .function..sub..alpha..sup.x (),
.function..sub..alpha..sup.y () and .function..sub..alpha..sup.z () may be simplified and take the following form: ##STR2##
Hereafter there is described a method of calculating these functions based on the encoding described in section 1.3.
1.4.2 Notation
In order to simplify the description of the computation process the following notation and definitions are used:
A block of operations is any list of operations beginning with an open curly brace "{" and ending with a closing curly brace "}". The operations in a block may be individual operations or blocks of
The term working memory or working variable refers to any memory area used for storing intermediary results during a computation.
Loading the expression e in a memory m is the operation of evaluating the value of e and storing the result in the memory m. Such an operation is written as follows:
For any memory m used to store a numerical value, incrementing m by the increment i is the operation of computing the value m+i and storing the result in the memory m. Such an operation is written as
In the above operation, the increment i may be:
a constant,
or the content of a memory,
or an arithmetical expression.
In the above operation, the increment i may be:
a constant,
or the content of a memory,
or an arithmetical expression.
1.4.3 Fitting process with harmonic weightings
When the coefficients {v.sup..alpha. (k)} are the harmonic weightings described in section 1.1.4 and in the appendix, the values .function..sub..alpha..sup.x, .function..sub..alpha..sup.y and
.function..sub..alpha..sup.z taken by the functions .function..sub..alpha..sup.x (), .function..sub..alpha..sup.y () and .function..sub..alpha..sup.z () relating to the node .phi..sub..alpha. may be
computed efficiently by the following method based on the encoding of S.
0) Allocate the following working memories: ##EQU7##
1) Perform the following initializations (the order does not matter): ##EQU8##
2) Let n.sub..alpha. be the number of satellites of A(.alpha.). This number can be obtained from the nb.sub.-- sat field of A(.alpha.).
3) Let .LAMBDA.(.alpha.) be the set of the satellites of the atom A(.alpha.) accessible from the Sat field of the atom A(.alpha.). For each satellite (k.epsilon..LAMBDA.(.alpha.), repeat the
operations of the following block
3.1) Determine the number n.sub.k of satellites of A(k). This number can be obtained from the nb.sub.-- sat field of A(k).
3.2) Determine the coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z) of the node .phi..sub.k of the atom A(k). These coordinates can be obtained from the Node field of A(k).
3.3) From the data in A(k) calculate or determine the value .mu.(k) of the weighting coefficient (see section 1.1.4 and section 2, infra).
3.4) Perform the following operations (in any order): ##EQU9##
3.5) Let .LAMBDA.(k) be the set of satellites of the atom A(k) accessible from the Sat field of the atom A(k). For each satellite (.beta..epsilon..LAMBDA.(k), .beta..noteq..alpha.) perform the
operations of the following block:
3.5.2) Determine the current coordinates (.phi..sub..beta..sup.x,.phi..sub..beta..sup.y,.phi..sub..beta..sup.z) of the kernel .phi..sub..beta. of the atom A(.beta.). These coordinates can be obtained
from the Node field of A(.beta.).
3.5.3) Perform the following three operations (in any order): ##EQU10## {
3.6) Perform the following three operations (in any order): ##EQU11## }
4) Perform the following operations: ##EQU12##
5) Let C(.alpha.) be the set of constraints attached to the atom A(.alpha.) that can be accessed by the Const field of this atom. For each constraint (c.epsilon.C(.alpha.)) repeat the operations of
the following block:
5.1) Determine the type of the constraint c.
5.2) Depending on the type of the constraint c, perform the following two operations:
5.2.1) Extract the data specific to the constraint c.
5.2.2) Depending on the data selected in (5.2.1), compute the values of the following expressions (see sections 2.5 and 2.7, infra): ##EQU13##
5.3) Perform the following operations (in any order): ##EQU14## }
6) Perform the following operations (in any order): ##EQU15##
At the end of this process, the working memories .function..sub..alpha..sup.x, .function..sub..alpha..sup.y and .function..sub..alpha..sup.z contain the values of the functions
.function..sub..alpha..sup.x (), .function..sub..alpha..sup.y () and .function..sub..alpha..sup.z () used to fit the values of the coordinates of the node .phi..sub..alpha. : ##EQU16##
This fitting process must be applied only to the "free" nodes, that is to say nodes that do not correspond to control nodes (see section 1.1.3).
1.4.4 Variants
Variant 1
The implementation of the fitting method based on the DSI method described in the previous section in the case of harmonic weighting coefficients may be very easily adapted to the case of any other
kind of weighting. The general structure of the process remains unchanged, the only changes concerning the incrementing of the working memories allocated at step (0). How these working memories are
incremented depends on the weighting adopted.
For example, if the following weighting coefficients {v.sup..alpha. (k)} are chosen: ##EQU17## then it is easy to verify that the associated fitting functions are: ##STR3##
The process used to compute the values .function..sub..alpha..sup.x, .function..sub..alpha..sup.y and .function..sub..alpha..sup.z of these functions at node .phi..sub..alpha. is then almost
identical to the process described for harmonic weighting:
0) Allocate the following working memories: ##EQU18##
1) Perform the following initializations (the order does not matter): ##EQU19##
2) Determine the number n.sub..alpha. of satellites of A(.alpha.). This number can be obtained from the nb.sub.-- sat field of A(.alpha.)
3) Let .LAMBDA.(.alpha.) be the set of satellites of the atom A(.alpha.) accessible from the Sat field of the atom A(.alpha.). For each satellite (k.epsilon..LAMBDA.(.alpha.)) repeat the operations
of the following block:
3.1) Determine the number n.sub.k of satellites of A(k). This number can be obtained from the nb.sub.-- sat field of A(k). Then perform the following operation:
n.sub.k.sup.2 .rarw.n.sub.k .times.n.sub.k
3.2) Determine the coordinates (.phi..sub.k.sup.x,.phi..sub.k.sup.y,.phi..sub.k.sup.z) of the Node .phi..sub.k of the atom A(k). These coordinates can be obtained from the Node field of A(k).
3.3) From the data in A(k) calculate or determine the value .mu.(k) of the weighting coefficient (see section 1.1.4 and section 2, infra).
3.4) Perform the following operations (in any order): ##EQU20##
3.5) Let .LAMBDA.(k) be the set of satellites of the atom A(k) accessible from the Sat field of the atom A(k). For each satellite .beta..epsilon..LAMBDA.(k), .beta..noteq..alpha. perform the
operations of the following block:
3.5.2) Determine the current coordinates (.phi..sub..beta..sup.x,.phi..sub..beta..sup.y,.phi..sub..beta..sup.z) of the kernel .phi..sub..beta. of the atom A(.beta.). These coordinates can be obtained
from the Node field of A(.beta.).
3.5.3) Perform the following three operations (in any order): ##EQU21##
3.6) Perform the following three operations (in any order): ##EQU22## }
4) Perform the following operations: ##EQU23##
5) Lot C(.alpha.) be the set of constraints attached to the atom A(.alpha.) accessed by the Const field of this atom. For each constraint c.epsilon.C(.alpha.) repeat the operations of the following
5.1) Determine the type of the constraint c.
5.2) Depending on the type of the constraint c, perform the following two operations:
5.2.1) Extract the data specific to the constraint c.
5.2.2) Depending on the data selected in (5.2.1), compute the values of the following expressions (see sections 2.5 and 2.7, infra): ##EQU24##
5.3) Perform the following operations (in any order): ##EQU25## }
6) Perform the following operations (in any order): ##EQU26##
The numerical parameters stored in the Info field of the atoms may be interpolated by the DSI method (see appendix: sections 2.1.2 and 2.2, infra). In this case, the process is almost identical to
the process described above used to fit a surface; the only modification consists in using only one of the three functions normally dedicated to fitting the three coordinates of the nodes and to
calculate it according to the interpolated parameter.
The present invention finds applications in the following fields in particular:
in geology and in geophysics, it can be used to create a surface model, for example to construct a solid three-dimensional model, on which a geophysical simulation or an underground resources
exploitation simulation can be carried out; it can also be used to obtain a set of data for initializing a digital or analogue simulator of geophysical or resource exploitation phenomena; it can
further be used, in conjuction with a graphics workstation, to display on a screen or to print out on paper images used to monitor simulations as above;
in biology or in medicine, the invention can be used to construct three-dimensional models of organs and/or prostheses or to simulate the effects of plastic surgery; similarly, displayed or printed
two-dimensional images can be used to monitor the above operations.
However, it is obvious that the invention applies more generally to modeling surfaces of any kind and in particular any natural or man-made surface. It can be of particular benefit in computer-aided
design (CAD).
Finally, the present invention is in no way limited to the embodiment described above and shown in the drawings and those skilled in the art can vary and modify the embodiment as described above
without departing from the scope of the invention.
In Geology and Biology, a common problem is modeling complex surfaces such as interfaces between areas of different kinds or having different properties. Classical modeling techniques based on Bezier
interpolations and spline functions [9] are not well suited to processing this type of heterogeneous data. A different approach is based on the DSI ("Discrete Smooth Interpolation") method [8]. In
this approach, surfaces are modeled using irregular triangular facets whose vertices must be determined to allow for the largest possible quantities of heterogeneous data.
FIG. 12 shows an apparatus according to the present invention, which can be utilized to implement the different surface modelling methods of the present invention. The surface modelling apparatus, as
exemplified in FIG. 12, comprises a measuring device which measures to obtain a set of geometrical data, a meshing device which meshes a surface to be modelled, a memory to store data related to and
for each mode of the mesh, a calculating device which processes fitting of the surface according to the stored data, and a surface representation generator for creating a representation of the
surface from fitted coordinates of each node.
2.1 Introduction: Statement of the problem
2.1.1 Surface S and related definitions
Referring to FIG. 1, let S be a surface composed of contiguous flat triangular facets embedded in the 3D Euclidian space (O.vertline.x,y,z). This surface need not be connex or closed. The following
notation is used: ##EQU27##
G is a graph whose nodes are the points of .OMEGA. and this set .OMEGA. is identified with the set of integer indices {1, . . . ,N} used to number the nodes. The neighborhood N(k) of a node
k.epsilon..OMEGA. iS defined as: ##EQU28##
In the following, it is assumed that these neighborhoods N(k) have the following symmetry property:
.alpha..epsilon.N(.beta.) .beta..epsilon.N(.alpha.)
The neighborhoods N(k) so defined thus generate a topology on G that can be considered as an approximate topology for S itself.
2.1.2 Problem 1: Estimating a function .phi. defined on .OMEGA.
Assume that the position of each node k.epsilon..OMEGA. iS known and let .phi.(k) be a function defined for all nodes k .epsilon..OMEGA. and known only on a subset of .OMEGA.: ##EQU29##
Generally, L is different from .OMEGA. and there is an infinity of functions .phi.(k) defined on .OMEGA. and interpolating the values {.phi.(I)=.phi..sub.I :I.epsilon.L}. The goal is to select among
all such functions the one which minimizes a given criterion R*(.phi.) measuring:
the global roughness of the function .phi.(.),
the discrepancy between the function .phi.(.) and some constraints and data that it should satisfy.
2.1.3 Problem 2: Fitting the surface S
In this case, only a subset of L of nodes k.epsilon..OMEGA. have a known position .phi.(k)=.phi..sub.k. ##EQU30##
As for the first problem, L is generally different from .OMEGA. and there is an infinity of surfaces S interpolating the points {.phi.(I)=.phi..sub.I :I.epsilon.L}. The goal is to select among all
such surfaces the one which minimizes a given criterion R*(.phi.) measuring:
the global roughness of the surface S,
the discrepancy between the surface S and some constraints and data that it must satisfy.
Allowing for the independence of the components (.phi..sup.x (k),.phi..sup.y (k),.phi..sup.z (k)) of each vector .phi.(k), it is easy to use the solution of the first problem to solve the second by
simply writing:
In the very particular case where the set .OMEGA. corresponds to the nodes of a regular rectangular grid laid on the surface, consideration might be given to using a Bezier, spline or Briggs type
interpolation method (See [9], [1] and [3]); unfortunately these methods are not appropriate for irregular grids and cannot allow for the constraints described in sections 2.2.3, 2.5 and 2.7.2.
2.2 Interpolating a function .phi.(k) defined on .OMEGA.
This section briefly introduces the Discrete Smooth Interpolation method described in [8].
2.2.1 Notation
In the following, .phi. denotes a column matrix of size N as: ##EQU31##
The problem does not depend on the method of numbering the nodes of the grid. In order to simplify the notation, it will therefore be assumed that a permutation of the elements of the matrix .phi.
has been performed in such a way that .phi. can be split into two submatrices .phi..sub.I and .phi..sub.L such that: ##EQU32##
Moreover, .parallel. . .parallel.D denotes a seminorm associated to a square (N.times.N) positive semidefinite matrix [D] in such a way that, for any column matrix X of size N:
.parallel.X.parallel..sub.D.sup.2 =X.sup.t .multidot.[D].multidot.X
2.2.2 Defining a global roughness criterion R(.phi.)
Let R(.phi..vertline.k) be a local roughness criterion defined by the following formula where {v.sup..alpha. (k)} are given positive, negative or null weighting coefficients: ##EQU33##
R(.phi..vertline.k) can be used to derive a global roughness criterion R(.phi.) defined as follows, where .mu.(k) is a given weighting function defined on .OMEGA.: ##EQU34##
R(.phi.) is completely defined by the coefficients {v.sup..alpha. (k)} and .mu.(k) and it is easy to verify (See [8]) that there is always an (N.times.N) symmetric positive semidefinite matrix [W]
such that:
R(.phi..vertline.k)=.phi..sup.t .multidot.[W(k)].multidot..phi.
2.2.3 Allowing for linear constraints in a least squares sense
Consider a square (N.times.N) matrix [A.sub.i ] and an N column matrix B.sub.i defining the linear constraint:
[A.sub.i ].multidot..phi..perspectiveto.B.sub.i
For an N.times.N positive semidefinite matrix [D.sub.i ] ".perspectiveto." means:
.parallel.[A.sub.i ].multidot..phi.-B.sub.i .parallel..sub.D.sbsb.i.sup.2 as small as possible
If there are several conditions of this type, the degree of violation of these constraints can be measured by the criterion .rho.(.phi.) with: ##EQU35## 2.2.4 Solution of the problem
Among all the functions .phi.(k) defined on .OMEGA. and interpolating the data {.phi.(I)=.phi..sub.I :I.epsilon.L }, one is selected which minimizes the following criterion:
Developing R*(.phi.): ##EQU36##
The partition of the matrix .phi. induces a similar partition of the matrices [W*] and Q used to define R*(.phi.): ##EQU37##
The condition .differential.R*(.phi.)/.differential..sub.100 .sbsb.I =[0] yields the following "DSI equation" characterizing all the functions {.phi.(k):k.epsilon..OMEGA.} constituting a solution of
the problem: ##EQU38##
2.3 Uniqueness of the solution of the DSI equation
The set L of nodes where .phi. is known is said to be consistent relative to R*(.phi.) if each connex component of the graph G contains at least one node belonging to L.
If the global roughness criterion R(.phi.) is such that: ##EQU39## the DSI equation based on R(.phi.) has a unique solution.
The theorem explained above is of theoretical interest only and does not affect in any way the method described in this patent. It therefore requires no proof here.
2.4 Local form of the DSI equation
In the following, instead of solving directly the DSI matrix equation, an iterative approach avoids the computation and storage of [W*]. This iterative approach is that used in this patent.
In order to simplify the notation, it will be assumed in the following that the positive semi-definite matrices [Di] of the DSI equation are diagonal. Moreover, when linear constraints of the type
[A.sub.i ].multidot..perspectiveto.B.sub.i exist, the following notation is used: ##EQU40## 2.4.1 Computing .differential.R*(.phi.)/.differential..sub..phi..sbsb..alpha.
The definition of R(.phi..vertline.k) implies: ##EQU41##
From this it can be deduced that: ##EQU42##
Allowing for the diagonal structure of [Di], it is easy to verify that the .alpha..sup.th element of .differential..parallel.[A.sub.i ].multidot..phi.-B.sub.i .parallel..sub.D.sbsb.i.sup.2
/.differential..sub..phi..sbsb..alpha. is such that: ##EQU43##
Given the definition of R*(.phi.), it is possible to deduce that: ##EQU44##
The solution .phi. is such that .differential.R*(.phi.)/.differential..sub..phi..sbsb..alpha. =0, hence the .alpha..sup.th component .phi..sub..alpha. of .phi. must satisfy the following equation: ##
In this disclosure this equation is called the local form of the DSI equation at node .alpha..
2.4.2 Proposition for an iterative algorithm
The above local form of the DSI equation suggests a straightforward algorithm to estimate the solution .phi.. For example, at iteration (n+1) the .alpha..sup.th component .phi..sub..alpha..sup.(n+1)
of the solution .phi..sup.(n+1) must satisfy the DSI(.alpha.) equation so that an iterative algorithm can be:
let I be the set of indexes of nodes where .phi..sub..alpha. is unknown,
let .phi. be an initial approximate solution,
while (more iterations are needed). ##EQU46##
Note that this very simple algorithm does not use explicitly the matrix [W*] occurring in the DSI equation but has to compute repeatedly products such as {v.sup..alpha. (k).multidot.v.sup..beta. (k)}
which are the products used to derive [W].
In fact, if the initial approximate solution is close to the exact solution, few iterations are needed and the computation overhead becomes negligible. For example, this occurs in an interactive
context where the initial solution .phi. is taken equal to the solution before some local modifications are made by the user. In this case, despite the slight increase in overhead, the local form of
the DSI equation is preferable as it is much easier to use than the global form.
In order to show that the proposed algorithm actually converges, note that R*(.phi.) can be expressed as a function of .phi..sub..alpha. :
R*(.phi.)=A.multidot..phi..sub..alpha..sup.2 +B.multidot..phi..sub..alpha. +C
The coefficients A, B and C are independent of .phi..sub..alpha. and: ##EQU47##
The minimum of the global criterion R*(.phi.) is achieved for the value .phi..sub..alpha. =-B/(2A) which is precisely the value given by the DSI(.alpha.) equation. This shows that the algorithm
converges because, at each step of the iterative process, the value of the positive or null function R*(.phi.) decreases.
2.5 Allowing for fuzzy data
This section introduces two types of linear constraints that will be of special importance for Geometric Modeling, as will be shown in section 2.7.6. The purpose of this section is to account for
fuzzy data concerning the unknown values {.phi..sub..lambda. :.lambda..epsilon.I}. Such data is assumed to be associated with certainty factors .omega..sup.2 .epsilon.R.sup.+ which are actually
positive weighting coefficients.
2.5.1 Case of isolated fuzzy data
Let .lambda..epsilon.I be a node index for which the unknown value .phi..lambda. is assumed to be close to an uncertain datum .phi..lambda. with a certainty factor equal to
.phi..sub..lambda. .perspectiveto..phi..sub..lambda.
where .perspectiveto. represents the following condition:.phi..lambda.:
.omega..sub..lambda..sup.2 .multidot..vertline..phi..sub..lambda..sup.* -.phi..sub..lambda. .vertline..sup.2 minimum
Such a condition is easily accounted for in the DSI method as the i.sup.th constraint with: ##EQU48##
The corresponding coefficients .GAMMA..sub.i.sup..alpha., Q.sub.i.sup..alpha. and .gamma..sub.i.sup..alpha. used in the local DSI(.alpha.) equation are thus defined by: ##EQU49## 2.5.2 Case of
differential fuzzy data
Let .lambda..epsilon.I be a node index for which the unknown value .phi..lambda. is assumed to be linked to another value .phi..mu. corresponding to another index .mu..noteq..lambda.. Assume that
this link is of the following form, where .DELTA..sub.80 .mu. is a given value:
(.phi..sub..mu. -.phi..sub..lambda.).perspectiveto..DELTA..sub..lambda..mu.
This relation is assumed to be fuzzy and is modeled through the following condition involving a certainty factor .omega..sub..lambda..mu..sup.2 :
.omega..sub..lambda..mu..sup.2 .multidot..vertline.(.phi..sub..mu..sup.* -.phi..sub..lambda..sup.*)-.DELTA..sub..lambda..mu. .vertline..sup.2 minimum
Such a condition can easily be accounted for in the DSI method as the i.sup.th constraint with: ##EQU50##
The corresponding coefficients .GAMMA..sub.i.sup..alpha., Q.sub.i.sup..alpha. and .gamma..sub.i.sup..alpha. used in the DSI(.alpha.) equation are thus defined by: ##EQU51## 2.5.3 Choosing the
certainty factors
Consider the term M.sub..alpha..sup.* of the DSI(.alpha.) equation: ##EQU52##
In the two examples discussed above, the terms .gamma..sub.1.sup..alpha. are either equal to zero or equal to the certainty factor:
.gamma..sub.i.sup..alpha. =.omega..sub.i.sup.2
This suggests choosing for .omega..sub.i.sup.2 a given percentage P.sub.i of M.sub..alpha. :
.omega..sub.i.sup.2 =p.sub.i .multidot.M.sub..alpha. p.sub.i >0
The term M.sub..alpha..sup.* of the DSI(.alpha.) equation is then:
M.sub..alpha..sup.* =M.sub..alpha. .multidot.(1+p.sub.1 (.alpha.)+p.sub.2 (.alpha.)+ . . . )
where p.sub.i (.alpha.) is either equal to a given percentage P.sub.i or equal to zero.
2.6 Choosing the weighting coefficients
The choice of the weighting coefficients {v.sup..alpha. (k)} and {.mu.(k)} is completely free except that the {.mu.(k)} coefficients have to be positive or equal to zero. In the following an example
is given of how to choose these two families of coefficients.
2.6.1 Choosing the {v.sup..alpha. (k)} harmonic weighting coefficients
Let .LAMBDA.(k) be the "orbit" of k defined as follows:
Let .vertline..LAMBDA.(k).vertline. be the number of elements of (k). The weighting is called harmonic weighting if the coefficients {v.sup..alpha. (k)} are chosen according to the following
definition: ##EQU53##
Harmonic functions have the characteristic property that they are equal at any point to their own mean on a circle centered on this point. As one can see, R(.phi..vertline.k)=0 if .phi..sub.k iS
equal to the mean of the values .phi..sub..alpha. surrounding the node k, this is why, by analogy with harmonic functions, the term harmonic has been adopted for the weighting coefficients
{v.sup..alpha. (k)} described in this section.
If the DSI(.alpha.) equation is developed with these coefficients, the following equation is obtained, which can be easily translated into programming language: ##EQU54## 2.6.2 Choosing the {.mu.(k)}
The coefficients {.mu.(k)} have been introduced in order to modulate locally the smoothness of the interpolation. If there is no special reason to do differently, a uniform weighting may be used:
.mu.(k)=1 k.epsilon..OMEGA.
Among all non-uniform weighting schemes, one seems to be particularly interesting if smoothness is needed in the neighborhood of the data: ##EQU55##
2.7 Application to Geometric Modeling
This section shows how to use the DSI method to estimate the triangulated surface S itself for which it will be assumed that partial data is available composed of:
the exact location of some apices of the triangles,
the approximate location of some apices of the triangles,
vectorial constraints specifying the shape of S,
To this end, .phi..sub..alpha. is defined as the current vector joining the origin of R.sup.3 to the current node of .OMEGA. and .phi..sup.x, .phi..sup.y and .phi..sup.z denote its components on the
{ox,oy,oz} orthonormal basis of R.sup.3 :
2.7.1 Defining a global roughness criterion R(.phi.)
As usual, the set of nodes .OMEGA. is split into two subsets I and L such that: ##EQU56##
The nodes I.epsilon.L whose location .phi..sub.I .epsilon.R.sup.3 is known will be called control nodes and the aim is to determine the location of the remaining points i.epsilon.I in such a way that
the following local criterion R(.phi..vertline.k) based on a given set of weighting coefficients {v.sup..alpha. (k)} is as small as possible: ##EQU57##
This criterion constitutes the vectorial form of the local DSI criterion. The associated global vectorial DSI criterion R(.phi.) is defined as follows where {.mu.(k)} are given non-negative weighting
Defining R(.phi..sup.x), R(.phi..sup.y) and R(.phi..sup.z) as previously: ##EQU58##
Pythagoras' theorem yields:
2.7.2 Allowing for linear constraints in a least squares sense
Consider the following matrices sized so that N is the total number of nodes of the set .OMEGA.: ##EQU59##
To have the components .phi..sup.x,.phi..sup.y and .phi..sup.z satisfy the following constraints
[A.sub.i.sbsb.x.sup.x ].multidot..phi..sup.x .perspectiveto.B.sub.i.sbsb.x.sup.x
[A.sub.i.sbsb.y.sup.y ].multidot..phi..sup.y .perspectiveto.B.sub.i.sbsb.y.sup.y
[A.sub.i.sbsb.z.sup.z ].multidot..phi..sup.z .perspectiveto.B.sub.i.sbsb.z.sup.z
the following criterion is introduced: ##EQU60## 2.7.3 Solution of the problem
Among all the surfaces S interpolating the data {.phi.(I)=.phi..sub.I :I.epsilon.L }, the aim is to select the one minimizing the following criterion:
It is easy to verify that this equation can be rewritten as follows: ##EQU61##
Allowing for independence of the components .phi..sup.x,.phi..sup.y and .phi..sup.z : ##EQU62##
Minimizing the vectorial DSI criterion is thus equivalent to minimizing the corresponding three DSI criteria applied to the three components of the current vector .phi..
2.7.4 Note
If the surface S is composed of several disjoint patches the theorem explained in section 2.3 ensures that there exists a unique solution to the problem if at least one triangle apex within each
patch has a known position.
2.7.5 Interactive use
In Geometric Modeling, the goal is to define interactively the geometric shape of the graph G associated with the set of nodes .OMEGA.. To this end, it is assumed that an initial shape is known and
the user moves some nodes and changes interactively some constraints (control nodes and fuzzy data); these modifications are then propagated to the nodes i.epsilon.I through the vectorial form of the
DSI method.
As explained previously, this problem is broken down into three sub-problems corresponding to the three coordinates (x,y,z) and the best way to solve these three problems is to use the iterative
algorithm associated with the local form of the corresponding DSI equations.
Between steps of the modeling process, not only can the position of the control nodes be modified but also the subset L itself can be changed if necessary; for example, to apply DSI only to a subset
.GAMMA. of nodes, it is necessary to choose L in such a way that .GAMMA. .GAMMA. where .GAMMA. stands for the complementary set of .GAMMA. in .OMEGA..
Moreover, because the problem is broken down into sub-problems solved independently and corresponding to the three coordinates, different subsets L.sub.x, L.sub.y and L.sub.z can be defined for each
coordinate if necessary; for example, if the (x,y) coordinates must not be modified it is sufficient for L.sub.x and L.sub.y to be equal to the whole set .OMEGA..
2.7.6 Allowing for fuzzy geometrical data
Initially, the DSI method was designed to model complex surfaces encountered in the field of natural sciences, for example in geology and biology. In this case, there is generally much imprecise data
to take into account such as: ##EQU63##
Fuzzy control nodes
By definition, a "fuzzy control node" is any node .lambda..epsilon.I whose position .phi..lambda. is known as being approximately equal to a given vector .phi..lambda. with a given degree of
certainty .omega..sub..lambda..sup.2. Referring to section 2.5.1, such fuzzy data can be taken into account with the DSI method by introducing the following constraints in the R*(.phi..sup.x), R*
(.phi..sup.y) and R*(.phi..sup.z) criteria: ##EQU64##
Fuzzy vectorial constraints
In many applications it is necessary to control the vector (.phi..sub..mu. -.phi..sub..lambda.) joining two nodes .phi..sub..lambda. and .phi..sub..mu. of a surface. By definition, such a constraint
is called a "fuzzy vectorial constraint" if it must be satisfied with a given degree of certainty .omega..sub..lambda..mu..sup.2. Some examples of constraints of this type are shown in FIG. 2.
In practice, at least one of the two points .phi..sub..lambda. and .phi..sub..mu. has an unknown position, say the one corresponding to .lambda..epsilon.I, and two important cases have to be
The first case corresponds to FIGS. 2a and 2b where the shape of an object must be controlled in such a way that (.phi..sub..mu. -.phi..sub..lambda.) is equal to a given vector
The second case corresponds to FIG. 2c where the shape of an object must be controlled in such a way that (.phi..sub..mu. -.phi..sub..lambda.) is orthogonal to a given vector V. This situation occurs
when the vector normal to a surface is to be controlled; in this case, if V is the normal vector at node .lambda. of the surface, it is wise to control its orthogonality with all the vectors
(.phi..sub..mu. -.phi..sub..lambda.) for all .mu..epsilon.N(.lambda.).
Referring to section 2.5.2, the first case is easily taken into account in the DSI method by introducing the following constraints in the R*(.phi..sup.x), R*(.phi..sup.y) and R*(.phi..sup.z) criteria
with a weighting coefficient equal to .omega..sub..lambda..mu..sup.2 : ##EQU65##
For the second case, the problem is not very different because: ##EQU66##
In this case, the values (.phi..sub..lambda..sup.x,.phi..sub..lambda..sup.y,.phi..sub..lambda..sup. z) and (.phi..sub..mu..sup.x,.phi..sub..mu..sup.y,.phi..sub..mu..sup.z) on the righthand side of
the equations can be set to the corresponding values inferred at the previous step of the iterative process; if
.DELTA..sub..lambda..mu..sup.x =-{V.sup.y .multidot.(.phi..sub..mu..sup.y -.phi..sub..lambda..sup.y)+V.sup.z .multidot.(.phi..sub..mu..sup.z -.phi..sub..lambda..sup.z)}/V.sup.x
.DELTA..sub..lambda..mu..sup.y =-{V.sup.z .multidot.(.phi..sub..mu..sup.z -.phi..sub..lambda..sup.z)+V.sup.x .multidot.(.phi..sub..mu..sup.x -.phi..sub..lambda..sup.x)}/V.sup.y
.DELTA..sub..lambda..mu..sup.z =-{V.sup.x .multidot.(.phi..sub..mu..sup.x -.phi..sub..lambda..sup.x)+V.sup.y .multidot.(.phi..sub..mu..sup.y -.phi..sub..lambda..sup.y)}/V.sup.z
the second case becomes identical to the first case.
In allowing for fuzzy data as described above only one certainty factor .omega..sub..lambda..mu..sup.2 is used for all three components of the constraints. In practice this is not mandatory and,
because of the independence of the three components, it is always possible to use three different certainty factors .omega..sub..lambda..mu..sup.x2, .omega..sub..lambda..mu..sup.y2 and
.omega..sub..lambda..mu..sup.z2 corresponding to the three components.
2.8 Generalization
Before concluding, some straightforward generalizations of the approach are worth mentioning:
The DSI method can be used on any kind of polygonal facets. In practice, triangles are recommended since they are the simplest polygons and are easy to handle in a computer program.
The DSI method can be used to estimate polygonal curves in a 3D space; in this case, triangles are replaced by segments.
2.9 Conclusion
The local form of the DSI equation provides a simple and powerful tool for modeling complex surfaces encountered in Geology and Biology, for example. In particular:
There are no restrictions as to the mesh used to model a surface and it is possible to uses automatic algorithms to build them; for example, if the mesh is composed of triangles, it is possible to
use algorithms derived from the Delaunay method (See [2] and [5]). Moreover, the size of the mesh can easily be fitted locally to the complexity of the surface.
The DSI method can cope with large quantities of precise or fuzzy data. This may be particularly useful in Natural Sciences where the aim is not to generate aesthetic surfaces but to fit precise and
fuzzy data consistently.
The algorithm derived from local DSI equation is fast enough to allow interactive use.
Nothing has been said about the representation of the facets. Depending on the application, plane facets may suffice. However, facets can if necessary be represented by non-planar surface patches
interpolating the nodes of the mesh (See [4]).
The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and
all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.
1. A process for modeling a surface representing an interface between two areas with different properties in a three-dimensional body, comprising the steps of:
measuring to obtain a set of geometrical data, from said interface of said three-dimensional body, relating to the surface and associated with respective points on said surface;
meshing the surface so that all said points are a subset of nodes of a mesh;
storing at a specific memory address for each node of the mesh, the following data:
first, second, and third coordinates,
a number of satellite nodes,
satellite node address data providing access to specific coordinates of said satellite nodes and consequently to satellite node related data relating thereto, and
geometrical data associated with said each node of the mesh;
for each node of the mesh, defining a local roughness index derived from a weighted sum of the coordinates of the node and of its satellites;
defining a global roughness index obtained by summing the local roughness indices associated with the nodes, and a global violation index of said geometrical data, to thereby generate a sum of
said global roughness index and said global violation index;
fitting fitted coordinates of each node processed by an iterative method in which for each step of an iteration there is added a weighted combination of the coordinates of the node and of the
satellites of said node and a combination of the geometrical data associated with said node, to thereby minimize said sum of the global roughness index and the global violation index; and
creating a representation of the surface from the fitted coordinates of each node.
2. A process according to claim 1, wherein:
the surface is meshed using triangles.
3. A process according to claim 1 wherein:
the fitting step comprises three separate sub-steps respectively using the first, second, and third coordinates of the nodes and the coordinates of the satellite nodes corresponding thereto,
4. A process according to claim 1, wherein:
the geometrical data includes vector data between two given nodes.
5. A process according to claim 1, wherein:
the geometrical data includes data of a vector normal to the surface to be modeled.
6. A process according to any one of claims 4 to 5, wherein:
at least one geometrical datum is associated with a coefficient representing a certainty factor by which said at least one datum is known, the global violation index is a function of said
certainty factors.
7. A process according to claim 1, further comprising the steps of measuring properties of the three-dimensional body in a region of at least one node of the mesh and storing the measured properties
at addresses specific to the at least one node so as to obtain complementary data of the three-dimensional body.
8. A process according to claim 1, wherein:
for each local roughness index, a ratio between a first and a second weight of a corresponding node equals a negative of the number of satellite nodes.
9. A process according to claim 8, wherein:
said weighted combination uses substantially the same weighting as the ratio.
10. A process according to claim 1, wherein:
the summing of the local roughness indices is a weighted summing.
11. A process according to claim 10, wherein:
said weighted combination uses the weighting associated with the local roughness indices.
12. A process according to claim 1, wherein:
the three-dimensional body is a geological formation.
13. A process according to claim 1, wherein:
the surface represents variations of a property of a two-dimensional body in a vicinity of a plane intersecting said body, said property being represented by a remaining dimension, and a
representation of the surface is used to optimize visualization of underground resources of said three-dimensional body, said three dimensional body being a geological formation.
14. A process according to claim 1, wherein:
the representation of the surface is a graphical representation on a flat medium.
15. A method according to claim 1, wherein:
the representation of the surface is a three-dimensional model.
16. A device for modeling a surface representing an interface between two areas with different properties in a three-dimensional body, comprising:
means for measuring to obtain a set of geometrical data, from said interface of said three-dimensional body, relating to the surface and associated with respective points on said surface;
means for meshing the surface so that all said points are a subset of nodes of a mesh;
means for storing at a specific memory address for each node of the mesh, the following data:
first, second, and third coordinates,
a number of satellite nodes,
satellite node address data providing access to specific coordinates of said satellite nodes and consequently to satellite node related data relating thereto, and
geometrical data associated with said each node of the mesh;
calculation means for fitting fitted coordinates of each node processed by an iterative method in which for each step of an iteration there is added a weighted combination of the coordinates of
the node and of the satellites of said node and a combination of the geometrical data associated with said node, to thereby minimize a sum of a global roughness index obtained by summing local
roughness indices associated with the nodes, each local roughness index derived from a weighted sum of the coordinates of the node and the coordinates of its satellites, and of a global violation
index of said geometrical data; and
means for creating a representation of the surface from the fitted coordinates of each node.
Referenced Cited
U.S. Patent Documents
4930092 May 29, 1990 Reilly
Foreign Patent Documents Other references
• "A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures" H. Akima; J. ACM 17.1 1984. "Machine Contouring Using Minimum Curvature" I. Briggs; Geophysics 17.1, 1984.
"Triangular Bernstein-Bezier Patches", G. Farin; Computer-aided Geometric Design, Aug. 1986. "Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams" L.
Guibas et al; ACM transactions on Graphics, Apr. 1985. "Three-dimensional Graphic Display of Disconnected Bodies"; J. Mallet; Mathematical Geology, 1988. "Geometric Modeling and Geostatistics" J.
Mallet; Geostatistics, 1989. "Discrete Smooth Interpolation", J. Mallet' ACM Transactions on Graphics, Apr. 1989. "Geometric Modeling", M. Mortenson; 1985. "FEM Shape Optimization: A Case for
Interactive Graphics", S. Singh, et al; Computers and Graphics, 1982.
Current U.S. Class: 395/123; 395/120; 364/421
International Classification: G06T 1700; | {"url":"https://patents.justia.com/patent/5465323","timestamp":"2024-11-11T03:39:12Z","content_type":"text/html","content_length":"138429","record_id":"<urn:uuid:b56fe1de-a4b7-49a4-899f-89218dce7e4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00588.warc.gz"} |
Multi-letter converse bounds for the mismatched discrete memoryless channel with an additive metric
The problem of mismatched decoding with an additive metric q for a discrete memoryelss channel W is addressed. Two max-min multi-letter upper bounds on the mismatch capacity C[q](W) are derived. We
further prove that if the average probability of error of a sequence of codebooks converges to zero sufficiently fast, then the rate of the code-sequence is upper bounded by the 'product-space'
improvement of the random coding lower bound on the mismatched capacity, C^(∞)[q] (W), introduced by Csiszár and Narayan. In particular, if q is a bounded rational metric, and the average probability
of error converges to zero faster than O(1/n), then R ≤ C^(∞)[q] (W). Consequently, in this case if a sequence of codes of rate R is known to achieve average probability of error which is o(1/n),
then there exists a sequence of codes operating at a rate arbitrarily close to R with average probability of error which vanishes exponentially fast. We conclude by presenting a general expression
for the mismatch capacity of a general channel with a general type-dependent decoding metric.
Publication series
Name IEEE International Symposium on Information Theory - Proceedings
Volume 2015-June
ISSN (Print) 2157-8095
Conference IEEE International Symposium on Information Theory, ISIT 2015
Country/Territory Hong Kong
City Hong Kong
Period 14/06/15 → 19/06/15
Bibliographical note
Publisher Copyright:
© 2015 IEEE.
Dive into the research topics of 'Multi-letter converse bounds for the mismatched discrete memoryless channel with an additive metric'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/multi-letter-converse-bounds-for-the-mismatched-discrete-memoryle-2","timestamp":"2024-11-07T08:53:59Z","content_type":"text/html","content_length":"58609","record_id":"<urn:uuid:c266d25a-4e00-4061-acd9-f12929858461>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00513.warc.gz"} |
The purpose of this article is to put forward some ideas to help with the teaching of addition.
Combining groups of physical objects: for many students, this is their most basic experience of adding up. This process normally involves collecting two sets of objects, then counting how many
objects there are in total. (For example, by building two towers of cubes, and then counting up every single block.) For many, this method can be too involved, particularly for those students who
present attention deficit disorder. If the child cannot hold their attention for the whole of the activity, blocks will be put awry, towers will end up with additional blocks, blocks will get mixed
up, and at the end, the wrong answer is arrived at. The length of the process means that if your child does not master the concept quickly, they are not likely to make progress at all. In addition,
it is difficult to extend this process into a calculation that can be approached mentally: for example, try to imagine two large sets of objects in your head, and then count them all up. Even for
adults, this is nearly impossible.
Simple drawings: jottings are a more useful alternative to the process described above. Write out the addition problem on a sheet of paper, and next to the first number, jot down the appropriate
number of tallies (for instance, for the number 4, draw 4 tallies). Ask your student to predict how many tallies you will need to draw by the other number in the problem. When they come to the
correct answer, ask them to draw the tallies. To finish with, ask how many tallies they have drawn altogether. This method is a much easier way of bringing together 2 groups, is less likely to be
subject to mechanical error, and is better suited to students with poor focus. It also encourages the child to associate between what the written sum actually says, and why they are drawing a certain
number of tallies.
Counting on: this is a technique based around your student's capacity to say number names. When your child has reached a stage where they know how to count to five, start asking them questions like,
"what number is 1 more than..." (eg. what comes after 2 when we count?) This is actually equivalent to answering an addition problem of the type 2+1, but helps to connect the ideas of counting and
addition, which is very powerful. This technique gets your student ready to use number squares and gives them the confidence to answer problems in their mind. The method can also be made more
difficult, by asking, "what number is 2 more than..." When your child can confidently respond to such problems out loud, show them the question written down, and explain that this is the same as the
problem you had been doing before. This will help the child to see addition and counting as fundamentally related, and that this new problem is actually something they have met before.
Playing board games: this activity can be both a mathematical learning experience as well as a pleasant pastime. Games that require a counter to be moved around a board do a lot to encourage children
to count on. If the board has numbers on it, the child is able to see that the action is similar to counting out numbers aloud, or using a number line. Make a point of remembering to draw attention
to the relationship between using board games and addition.
Learning number facts: usually, we rely on number facts learnt by heart to help us answer addition problems. In a nutshell, we do not have to figure out the answer to 7 and 10, we simply remember it.
Having the ability to recall addition facts allows us to tackle simple maths tasks confidently. Improve your student's knowledge of known number bonds by singing nursery songs that tell stories of
number. Take part in the game of matching pairs with the student, where the point of the game is identify the location of the question (for instance, 7+8) and the corresponding answer from a set of
cards all turned face down. Create a set of flashcards with simple addition facts written on them, look at the cards one at a time, and ask the student for the answer, giving a good deal of applause
when they give the right answer. When they are confident, expand the number of facts. Games will prevent your child perceiving addition as dull, and will build confidence.
Addition printables and worksheets: Practise makes perfect - and the right style of practice also lends more confidence. By utilizing simple worksheets, aimed towards your student's ability and
attention span, you are able to significantly improve your child's ability with addition, both orally and written down. There are plenty of free internet sites that offer worksheets that help with
the teaching of adding up, but it does matter what adding up worksheets you use. Ensure that the worksheets are aimed at the right level, being neither too difficult nor too easy, and are of the
correct length to maintain the student's interest. You should be attempting to present questions that foster their recollection of number facts, along with a scattering of sums involving some
calculation. On the occasions that the student is successful, use the opportunity to give them a lot of praise; when they make a mistake, do not appear frustrated, but briefly explain their mistake.
Using adding up worksheets in a considered way can really boost your student's ability. | {"url":"http://www.mpedu.in/educationarticles.php","timestamp":"2024-11-02T08:24:48Z","content_type":"text/html","content_length":"44532","record_id":"<urn:uuid:e0309ba0-b6e9-4a6b-b511-d40beef3094f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00096.warc.gz"} |
Multiplying Fractions by a Whole Number - The Math Spot
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Fraction concepts are taken up a level in 5th grade and many of the topics are very abstract and difficult for students to understand. Multiplying fractions by a whole number doesn’t need to be one
of these difficult concepts! By using the CRA (Concrete-Representative-Abstract) framework your students will link what they already know about multiplication to the multiplication of fractions and
will be successful in no time at all!
Concrete Multiplying Fractions by a Whole Number
Before you begin with fractions, anchor your students back into what it means to multiply. Give them a pile of blocks and ask them to model something super simple such as “4×3” your 5th graders
should quickly and easily be able to model this problem. Ask your students about what they built and how they know it represents the equation.
Go ahead and get out your pattern blocks. They are the PERFECT fraction representation. Let the hexagon represent a whole and allow your students to figure out which blocks represent 1/3 (rhombus), 1
/2 (trapezoid) and 1/6 (triangles). Once your students are all set with their blocks, they are ready to begin! Don’t have fraction blocks handy? Fraction circles would work just as well!
Ask your students “If 4 groups of 3 blocks represented 4×3, how could we represent 4 x 1/3?” Allow students to use the rhombus
pattern blocks
to represent 4 groups of 1/3. The beauty of using blocks is that students can put these blocks back together to see both the improper fraction and mixed number that is created when fractions are
multiplied together.
**This PICTURE is not a concrete model, but if you are creating this model out of fraction tiles, your students are working at the concrete level! **
Representative Multiplying Fractions by a Whole Number
Once your students are able to model 4 x 1/3 you want them to link this understanding to a representative model such as repeated addition. You may begin by asking your students to write a repeated
addition equation to represent 4 x 3. This is easy for your students! Now, ask them to use what they know about multiplication to write a repeated addition equation that represents 4 x 1/3.
Abstract Multiplying Fractions by a Whole Number
The concrete and representative steps of this activity allow your students to clearly understand what is going on when multiplying a fraction by a whole number. After your students have had a good
deal of exposure at the concrete and representative level, give them a new equation such as 4 x 2/8 and ask your students what they *think* the product will be. You are looking for your students to
make generalizations about their multiplication and fraction understandings and to be able to explain their thinking. After a student shares their thinking, ask all students to model with concrete
materials or a repeated addition equation to confirm the product!
I have created a set of playing cards that includes multiplication equations, visual models, repeated addition and the resulting products. Your students can play so many traditional card games with
this deck of cards – and I have included the instructions for 5 games to get you started! This resource is PERFECT for exploring the link between repeated addition and fraction multiplication. Click
HERE to check it out! | {"url":"http://k5mathspot.com/multiplying-fractions-by-a-whole-number/","timestamp":"2024-11-07T03:02:02Z","content_type":"text/html","content_length":"97012","record_id":"<urn:uuid:bdf61e8d-7e9d-4a41-8445-11703e63d057>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00512.warc.gz"} |
The Stacks project
Definition 34.6.1. Let $T$ be a scheme. An syntomic covering of $T$ is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is syntomic and such that $T = \
bigcup f_ i(T_ i)$.
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College Physics chapters 1-17
• Define acoustic impedance and intensity reflection coefficient.
• Describe medical and other uses of ultrasound technology.
• Calculate acoustic impedance using density values and the speed of ultrasound.
• Calculate the velocity of a moving object using Doppler-shifted ultrasound.
Figure 1. Ultrasound is used in medicine to painlessly and noninvasively monitor patient health and diagnose a wide range of disorders. (credit: abbybatchelder, Flickr)
Any sound with a frequency above 20,000 Hz (or 20 kHz)—that is, above the highest audible frequency—is defined to be ultrasound. In practice, it is possible to create ultrasound frequencies up to
more than a gigahertz. (Higher frequencies are difficult to create; furthermore, they propagate poorly because they are very strongly absorbed.) Ultrasound has a tremendous number of applications,
which range from burglar alarms to use in cleaning delicate objects to the guidance systems of bats. We begin our discussion of ultrasound with some of its applications in medicine, in which it is
used extensively both for diagnosis and for therapy.
Characteristics of Ultrasound
The characteristics of ultrasound, such as frequency and intensity, are wave properties common to all types of waves. Ultrasound also has a wavelength that limits the fineness of detail it can
detect. This characteristic is true of all waves. We can never observe details significantly smaller than the wavelength of our probe; for example, we will never see individual atoms with visible
light, because the atoms are so small compared with the wavelength of light.
Ultrasound in Medical Therapy
Ultrasound, like any wave, carries energy that can be absorbed by the medium carrying it, producing effects that vary with intensity. When focused to intensities of[latex]\boldsymbol{10^3}[/latex]to
[latex]\boldsymbol{10^5\textbf{ W/m}^2},[/latex]ultrasound can be used to shatter gallstones or pulverize cancerous tissue in surgical procedures. (See Figure 2.) Intensities this great can damage
individual cells, variously causing their protoplasm to stream inside them, altering their permeability, or rupturing their walls through cavitation. Cavitation is the creation of vapor cavities in a
fluid—the longitudinal vibrations in ultrasound alternatively compress and expand the medium, and at sufficient amplitudes the expansion separates molecules. Most cavitation damage is done when the
cavities collapse, producing even greater shock pressures.
Figure 2. The tip of this small probe oscillates at 23 kHz with such a large amplitude that it pulverizes tissue on contact. The debris is then aspirated. The speed of the tip may exceed the speed of
sound in tissue, thus creating shock waves and cavitation, rather than a smooth simple harmonic oscillator–type wave.
Most of the energy carried by high-intensity ultrasound in tissue is converted to thermal energy. In fact, intensities of[latex]\boldsymbol{10^3}[/latex]to[latex]\boldsymbol{10^4\textbf{ W/m}^2}[/
latex]are commonly used for deep-heat treatments called ultrasound diathermy. Frequencies of 0.8 to 1 MHz are typical. In both athletics and physical therapy, ultrasound diathermy is most often
applied to injured or overworked muscles to relieve pain and improve flexibility. Skill is needed by the therapist to avoid “bone burns” and other tissue damage caused by overheating and cavitation,
sometimes made worse by reflection and focusing of the ultrasound by joint and bone tissue.
In some instances, you may encounter a different decibel scale, called the sound pressure level, when ultrasound travels in water or in human and other biological tissues. We shall not use the scale
here, but it is notable that numbers for sound pressure levels range 60 to 70 dB higher than you would quote for[latex]\boldsymbol{\beta},[/latex]the sound intensity level used in this text. Should
you encounter a sound pressure level of 220 decibels, then, it is not an astronomically high intensity, but equivalent to about 155 dB—high enough to destroy tissue, but not as unreasonably high as
it might seem at first.
Ultrasound in Medical Diagnostics
When used for imaging, ultrasonic waves are emitted from a transducer, a crystal exhibiting the piezoelectric effect (the expansion and contraction of a substance when a voltage is applied across it,
causing a vibration of the crystal). These high-frequency vibrations are transmitted into any tissue in contact with the transducer. Similarly, if a pressure is applied to the crystal (in the form of
a wave reflected off tissue layers), a voltage is produced which can be recorded. The crystal therefore acts as both a transmitter and a receiver of sound. Ultrasound is also partially absorbed by
tissue on its path, both on its journey away from the transducer and on its return journey. From the time between when the original signal is sent and when the reflections from various boundaries
between media are received, (as well as a measure of the intensity loss of the signal), the nature and position of each boundary between tissues and organs may be deduced.
Reflections at boundaries between two different media occur because of differences in a characteristic known as the acoustic impedance[latex]\boldsymbol{Z}[/latex]of each substance. Impedance is
defined as
where[latex]\boldsymbol{\rho}[/latex]is the density of the medium (in[latex]\boldsymbol{\textbf{kg/m}^3}[/latex]) and[latex]\boldsymbol{v}[/latex]is the speed of sound through the medium (in m/s).
The units for[latex]\boldsymbol{Z}[/latex]are therefore[latex]\boldsymbol{\textbf{kg/}(\textbf{m}^2\cdot\textbf{s})}.[/latex]
Table 5 shows the density and speed of sound through various media (including various soft tissues) and the associated acoustic impedances. Note that the acoustic impedances for soft tissue do not
vary much but that there is a big difference between the acoustic impedance of soft tissue and air and also between soft tissue and bone.
Medium Density (kg/m^3) Speed of Ultrasound (m/s) Acoustic Impedance (kg/(m2⋅s))
Air 1.3 330 [latex]\boldsymbol{429}[/latex]
Water 1000 1500 [latex]\boldsymbol{1.5\times10^6}[/latex]
Blood 1060 1570 [latex]\boldsymbol{1.66\times10^6}[/latex]
Fat 925 1450 [latex]\boldsymbol{1.34\times10^6}[/latex]
Muscle (average) 1075 1590 [latex]\boldsymbol{1.70\times10^6}[/latex]
Bone (varies) 1400–1900 4080 [latex]\boldsymbol{5.7\times10^6}[/latex]to[latex]\boldsymbol{7.8\times10^6}[/latex]
Barium titanate (transducer material) 5600 5500 [latex]\boldsymbol{30.8\times10^6}[/latex]
Table 5. The Ultrasound Properties of Various Media, Including Soft Tissue Found in the Body.
At the boundary between media of different acoustic impedances, some of the wave energy is reflected and some is transmitted. The greater the difference in acoustic impedance between the two media,
the greater the reflection and the smaller the transmission.
The intensity reflection coefficient[latex]\boldsymbol{a}[/latex]is defined as the ratio of the intensity of the reflected wave relative to the incident (transmitted) wave. This statement can be
written mathematically as
where[latex]\boldsymbol{Z_1}[/latex]and[latex]\boldsymbol{Z_2}[/latex]are the acoustic impedances of the two media making up the boundary. A reflection coefficient of zero (corresponding to total
transmission and no reflection) occurs when the acoustic impedances of the two media are the same. An impedance “match” (no reflection) provides an efficient coupling of sound energy from one medium
to another. The image formed in an ultrasound is made by tracking reflections (as shown in Figure 3) and mapping the intensity of the reflected sound waves in a two-dimensional plane.
Example 1: Calculate Acoustic Impedance and Intensity Reflection Coefficient: Ultrasound and Fat Tissue
(a) Using the values for density and the speed of ultrasound given in Table 5, show that the acoustic impedance of fat tissue is indeed[latex]\boldsymbol{1.34\times10^6\textbf{kg}/(\textbf{m}^2\cdot\
(b) Calculate the intensity reflection coefficient of ultrasound when going from fat to muscle tissue.
Strategy for (a)
The acoustic impedance can be calculated using[latex]\boldsymbol{Z=\rho{v}}[/latex]and the values for[latex]\boldsymbol{\rho}[/latex]and[latex]\boldsymbol{v}[/latex]found in Table 5.
Solution for (a)
(1) Substitute known values from Table 5 into[latex]\boldsymbol{Z=\rho{v}}.[/latex]
[latex]\boldsymbol{Z=\rho{v}=(925\textbf{ kg/m}^3)(1450\textbf{ m/s})}[/latex]
(2) Calculate to find the acoustic impedance of fat tissue.
This value is the same as the value given for the acoustic impedance of fat tissue.
Strategy for (b)
The intensity reflection coefficient for any boundary between two media is given by[latex]\boldsymbol{a=\frac{(Z_2-Z_1)^2}{(Z_1+Z_2)^2}},[/latex]and the acoustic impedance of muscle is given in Table
Solution for (b)
Substitute known values into[latex]\boldsymbol{a=\frac{(Z_2-Z_1)^2}{(Z_1+Z_2)^2}}[/latex]to find the intensity reflection coefficient:
This result means that only 1.4% of the incident intensity is reflected, with the remaining being transmitted.
The applications of ultrasound in medical diagnostics have produced untold benefits with no known risks. Diagnostic intensities are too low (about[latex]\boldsymbol{10^{-2}\textbf{ W/m}^2}[/latex])
to cause thermal damage. More significantly, ultrasound has been in use for several decades and detailed follow-up studies do not show evidence of ill effects, quite unlike the case for x-rays.
Figure 3. (a) An ultrasound speaker doubles as a microphone. Brief bleeps are broadcast, and echoes are recorded from various depths. (b) Graph of echo intensity versus time. The time for echoes to
return is directly proportional to the distance of the reflector, yielding this information noninvasively.
The most common ultrasound applications produce an image like that shown in Figure 4. The speaker-microphone broadcasts a directional beam, sweeping the beam across the area of interest. This is
accomplished by having multiple ultrasound sources in the probe’s head, which are phased to interfere constructively in a given, adjustable direction. Echoes are measured as a function of position as
well as depth. A computer constructs an image that reveals the shape and density of internal structures.
Figure 4. (a) An ultrasonic image is produced by sweeping the ultrasonic beam across the area of interest, in this case the woman’s abdomen. Data are recorded and analyzed in a computer, providing a
two-dimensional image. (b) Ultrasound image of 12-week-old fetus. (credit: Margaret W. Carruthers, Flickr)
How much detail can ultrasound reveal? The image in Figure 4 is typical of low-cost systems, but that in Figure 5 shows the remarkable detail possible with more advanced systems, including 3D
imaging. Ultrasound today is commonly used in prenatal care. Such imaging can be used to see if the fetus is developing at a normal rate, and help in the determination of serious problems early in
the pregnancy. Ultrasound is also in wide use to image the chambers of the heart and the flow of blood within the beating heart, using the Doppler effect (echocardiology).
Whenever a wave is used as a probe, it is very difficult to detect details smaller than its wavelength[latex]\boldsymbol{\lambda}.[/latex]Indeed, current technology cannot do quite this well.
Abdominal scans may use a 7-MHz frequency, and the speed of sound in tissue is about 1540 m/s—so the wavelength limit to detail would be[latex]\boldsymbol{\lambda=\frac{v_{\textbf{w}}}{f}=\frac{1540\
textbf{ m/s}}{7\times10^6\textbf{ Hz}}=0.22\textbf{ mm}}.[/latex]In practice, 1-mm detail is attainable, which is sufficient for many purposes. Higher-frequency ultrasound would allow greater detail,
but it does not penetrate as well as lower frequencies do. The accepted rule of thumb is that you can effectively scan to a depth of about[latex]\boldsymbol{500\lambda}[/latex]into tissue. For 7 MHz,
this penetration limit is[latex]\boldsymbol{500\times0.22\textbf{ mm}},[/latex]which is 0.11 m. Higher frequencies may be employed in smaller organs, such as the eye, but are not practical for
looking deep into the body.
Figure 5. A 3D ultrasound image of a fetus. As well as for the detection of any abnormalities, such scans have also been shown to be useful for strengthening the emotional bonding between parents and
their unborn child. (credit: Jennie Cu, Wikimedia Commons)
In addition to shape information, ultrasonic scans can produce density information superior to that found in X-rays, because the intensity of a reflected sound is related to changes in density. Sound
is most strongly reflected at places where density changes are greatest.
Another major use of ultrasound in medical diagnostics is to detect motion and determine velocity through the Doppler shift of an echo, known as Doppler-shifted ultrasound. This technique is used to
monitor fetal heartbeat, measure blood velocity, and detect occlusions in blood vessels, for example. (See Figure 6.) The magnitude of the Doppler shift in an echo is directly proportional to the
velocity of whatever reflects the sound. Because an echo is involved, there is actually a double shift. The first occurs because the reflector (say a fetal heart) is a moving observer and receives a
Doppler-shifted frequency. The reflector then acts as a moving source, producing a second Doppler shift.
Figure 6. This Doppler-shifted ultrasonic image of a partially occluded artery uses color to indicate velocity. The highest velocities are in red, while the lowest are blue. The blood must move
faster through the constriction to carry the same flow. (credit: Arning C, Grzyska U, Wikimedia Commons)
A clever technique is used to measure the Doppler shift in an echo. The frequency of the echoed sound is superimposed on the broadcast frequency, producing beats. The beat frequency is[latex]\
boldsymbol{F_{\textbf{B}}=|f_1-f_2|},[/latex]and so it is directly proportional to the Doppler shift[latex]\boldsymbol{(f_1-f_2})[/latex]and hence, the reflector’s velocity. The advantage in this
technique is that the Doppler shift is small (because the reflector’s velocity is small), so that great accuracy would be needed to measure the shift directly. But measuring the beat frequency is
easy, and it is not affected if the broadcast frequency varies somewhat. Furthermore, the beat frequency is in the audible range and can be amplified for audio feedback to the medical observer.
Uses for Doppler-Shifted Radar
Doppler-shifted radar echoes are used to measure wind velocities in storms as well as aircraft and automobile speeds. The principle is the same as for Doppler-shifted ultrasound. There is evidence
that bats and dolphins may also sense the velocity of an object (such as prey) reflecting their ultrasound signals by observing its Doppler shift.
Example 2: Calculate Velocity of Blood: Doppler-Shifted Ultrasound
Ultrasound that has a frequency of 2.50 MHz is sent toward blood in an artery that is moving toward the source at 20.0 cm/s, as illustrated in Figure 7. Use the speed of sound in human tissue as 1540
m/s. (Assume that the frequency of 2.50 MHz is accurate to seven significant figures.)
1. What frequency does the blood receive?
2. What frequency returns to the source?
3. What beat frequency is produced if the source and returning frequencies are mixed?
Figure 7. Ultrasound is partly reflected by blood cells and plasma back toward the speaker-microphone. Because the cells are moving, two Doppler shifts are produced—one for blood as a moving
observer, and the other for the reflected sound coming from a moving source. The magnitude of the shift is directly proportional to blood velocity.
The first two questions can be answered using[latex]\boldsymbol{f_{\textbf{obs}}=f_{\textbf{s}}(\frac{v_{\textbf{w}}}{v_{\textbf{w}}\pm{v}_{\textbf{s}}})}[/latex]and[latex]\boldsymbol{f_{\textbf
{obs}}=f_{\textbf{s}}\frac{v_{\textbf{w}}\pm{v}_{\textbf{obs}}}{v_{\textbf{w}}}}[/latex]for the Doppler shift. The last question asks for beat frequency, which is the difference between the original
and returning frequencies.
Solution for (a)
(1) Identify knowns:
• The blood is a moving observer, and so the frequency it receives is given by
• [latex]\boldsymbol{v_{\textbf{b}}}[/latex]is the blood velocity ([latex]\boldsymbol{v_{\textbf{obs}}}[/latex]here) and the plus sign is chosen because the motion is toward the source.
(2) Enter the given values into the equation.
[latex]\boldsymbol{f_{\textbf{obs}}=(2,500,000\textbf{ Hz})}[/latex][latex]\boldsymbol{(\frac{1540\textbf{m/s}+0.2\textbf{ m/s}}{1540\textbf{ m/s}})}[/latex]
(3) Calculate to find the frequency: 20,500,325 Hz.
Solution for (b)
(1) Identify knowns:
• The blood acts as a moving source.
• The microphone acts as a stationary observer.
• The frequency leaving the blood is 2,500,325 Hz, but it is shifted upward as given by
[latex]\boldsymbol{f_{\textbf{obs}}}[/latex]is the frequency received by the speaker-microphone.
• The source velocity is[latex]\boldsymbol{v_{\textbf{b}}}.[/latex]
• The minus sign is used because the motion is toward the observer.
The minus sign is used because the motion is toward the observer.
(2) Enter the given values into the equation:
[latex]\boldsymbol{f_{\textbf{obs}}=(2,500,325\textbf{ Hz})}[/latex][latex]\boldsymbol{(\frac{1540\textbf{ m/s}}{1540\textbf{ m/s}-0.200\textbf{ m/s}})}[/latex]
(3) Calculate to find the frequency returning to the source: 2,500,649 Hz.
Solution for (c)
(1) Identify knowns:
• The beat frequency is simply the absolute value of the difference between[latex]\boldsymbol{f_{\textbf{s}}}[/latex]and[latex]\boldsymbol{f_{\textbf{obs}}},[/latex]as stated in:
(2) Substitute known values:
[latex]\boldsymbol{|2,500,649\textbf{ Hz}-2,500,000\textbf{ Hz}|}[/latex]
(3) Calculate to find the beat frequency: 649 Hz.
The Doppler shifts are quite small compared with the original frequency of 2.50 MHz. It is far easier to measure the beat frequency than it is to measure the echo frequency with an accuracy great
enough to see shifts of a few hundred hertz out of a couple of megahertz. Furthermore, variations in the source frequency do not greatly affect the beat frequency, because both[latex]\boldsymbol{f_{\
textbf{s}}}[/latex]and[latex]\boldsymbol{f_{\textbf{obs}}}[/latex]would increase or decrease. Those changes subtract out in[latex]\boldsymbol{f_{\textbf{B}}=|f_{\textbf{obs}}-f_{\textbf{s}}|}.[/
Industrial and Other Applications of Ultrasound
Industrial, retail, and research applications of ultrasound are common. A few are discussed here. Ultrasonic cleaners have many uses. Jewelry, machined parts, and other objects that have odd shapes
and crevices are immersed in a cleaning fluid that is agitated with ultrasound typically about 40 kHz in frequency. The intensity is great enough to cause cavitation, which is responsible for most of
the cleansing action. Because cavitation-produced shock pressures are large and well transmitted in a fluid, they reach into small crevices where even a low-surface-tension cleaning fluid might not
Sonar is a familiar application of ultrasound. Sonar typically employs ultrasonic frequencies in the range from 30.0 to 100 kHz. Bats, dolphins, submarines, and even some birds use ultrasonic sonar.
Echoes are analyzed to give distance and size information both for guidance and finding prey. In most sonar applications, the sound reflects quite well because the objects of interest have
significantly different density than the medium in which they travel. When the Doppler shift is observed, velocity information can also be obtained. Submarine sonar can be used to obtain such
information, and there is evidence that some bats also sense velocity from their echoes.
Similarly, there are a range of relatively inexpensive devices that measure distance by timing ultrasonic echoes. Many cameras, for example, use such information to focus automatically. Some doors
open when their ultrasonic ranging devices detect a nearby object, and certain home security lights turn on when their ultrasonic rangers observe motion. Ultrasonic “measuring tapes” also exist to
measure such things as room dimensions. Sinks in public restrooms are sometimes automated with ultrasound devices to turn faucets on and off when people wash their hands. These devices reduce the
spread of germs and can conserve water.
Ultrasound is used for nondestructive testing in industry and by the military. Because ultrasound reflects well from any large change in density, it can reveal cracks and voids in solids, such as
aircraft wings, that are too small to be seen with x-rays. For similar reasons, ultrasound is also good for measuring the thickness of coatings, particularly where there are several layers involved.
Basic research in solid state physics employs ultrasound. Its attenuation is related to a number of physical characteristics, making it a useful probe. Among these characteristics are structural
changes such as those found in liquid crystals, the transition of a material to a superconducting phase, as well as density and other properties.
These examples of the uses of ultrasound are meant to whet the appetites of the curious, as well as to illustrate the underlying physics of ultrasound. There are many more applications, as you can
easily discover for yourself.
Check Your Understanding
1: Why is it possible to use ultrasound both to observe a fetus in the womb and also to destroy cancerous tumors in the body?
Section Summary
• The acoustic impedance is defined as:
[latex]\boldsymbol{\rho}[/latex]is the density of a medium through which the sound travels and[latex]\boldsymbol{v}[/latex]is the speed of sound through that medium.
• The intensity reflection coefficient[latex]\boldsymbol{a},[/latex]a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the
incident wave, is given by
• The intensity reflection coefficient is a unitless quantity.
Conceptual Questions
1: If audible sound follows a rule of thumb similar to that for ultrasound, in terms of its absorption, would you expect the high or low frequencies from your neighbor’s stereo to penetrate into your
house? How does this expectation compare with your experience?
2: Elephants and whales are known to use infrasound to communicate over very large distances. What are the advantages of infrasound for long distance communication?
3: It is more difficult to obtain a high-resolution ultrasound image in the abdominal region of someone who is overweight than for someone who has a slight build. Explain why this statement is
4: Suppose you read that 210-dB ultrasound is being used to pulverize cancerous tumors. You calculate the intensity in watts per centimeter squared and find it is unreasonably high[latex](\boldsymbol
{10^5\textbf{ W/cm}^2}).[/latex]What is a possible explanation?
Problems & Exercises
Unless otherwise indicated, for problems in this section, assume that the speed of sound through human tissues is 1540 m/s.
1: What is the sound intensity level in decibels of ultrasound of intensity[latex]\boldsymbol{10^5\textbf{ W/m}^2},[/latex]used to pulverize tissue during surgery?
2: Is 155-dB ultrasound in the range of intensities used for deep heating? Calculate the intensity of this ultrasound and compare this intensity with values quoted in the text.
3: Find the sound intensity level in decibels of[latex]\boldsymbol{2.00\times10^{-2}\textbf{ W/m}^2}[/latex]ultrasound used in medical diagnostics.
4: The time delay between transmission and the arrival of the reflected wave of a signal using ultrasound traveling through a piece of fat tissue was 0.13 ms. At what depth did this reflection occur?
5: In the clinical use of ultrasound, transducers are always coupled to the skin by a thin layer of gel or oil, replacing the air that would otherwise exist between the transducer and the skin. (a)
Using the values of acoustic impedance given in Table 5 calculate the intensity reflection coefficient between transducer material and air. (b) Calculate the intensity reflection coefficient between
transducer material and gel (assuming for this problem that its acoustic impedance is identical to that of water). (c) Based on the results of your calculations, explain why the gel is used.
6: (a) Calculate the minimum frequency of ultrasound that will allow you to see details as small as 0.250 mm in human tissue. (b) What is the effective depth to which this sound is effective as a
diagnostic probe?
7: (a) Find the size of the smallest detail observable in human tissue with 20.0-MHz ultrasound. (b) Is its effective penetration depth great enough to examine the entire eye (about 3.00 cm is
needed)? (c) What is the wavelength of such ultrasound in[latex]\boldsymbol{0^{\circ}\textbf{C}}[/latex]air?
8: (a) Echo times are measured by diagnostic ultrasound scanners to determine distances to reflecting surfaces in a patient. What is the difference in echo times for tissues that are 3.50 and 3.60 cm
beneath the surface? (This difference is the minimum resolving time for the scanner to see details as small as 0.100 cm, or 1.00 mm. Discrimination of smaller time differences is needed to see
smaller details.) (b) Discuss whether the period[latex]\boldsymbol{T}[/latex]of this ultrasound must be smaller than the minimum time resolution. If so, what is the minimum frequency of the
ultrasound and is that out of the normal range for diagnostic ultrasound?
9: (a) How far apart are two layers of tissue that produce echoes having round-trip times (used to measure distances) that differ by[latex]\boldsymbol{0.750\:\mu\textbf{s}}?[/latex](b) What minimum
frequency must the ultrasound have to see detail this small?
10: (a) A bat uses ultrasound to find its way among trees. If this bat can detect echoes 1.00 ms apart, what minimum distance between objects can it detect? (b) Could this distance explain the
difficulty that bats have finding an open door when they accidentally get into a house?
11: A dolphin is able to tell in the dark that the ultrasound echoes received from two sharks come from two different objects only if the sharks are separated by 3.50 m, one being that much farther
away than the other. (a) If the ultrasound has a frequency of 100 kHz, show this ability is not limited by its wavelength. (b) If this ability is due to the dolphin’s ability to detect the arrival
times of echoes, what is the minimum time difference the dolphin can perceive?
12: A diagnostic ultrasound echo is reflected from moving blood and returns with a frequency 500 Hz higher than its original 2.00 MHz. What is the velocity of the blood? (Assume that the frequency of
2.00 MHz is accurate to seven significant figures and 500 Hz is accurate to three significant figures.)
13: Ultrasound reflected from an oncoming bloodstream that is moving at 30.0 cm/s is mixed with the original frequency of 2.50 MHz to produce beats. What is the beat frequency? (Assume that the
frequency of 2.50 MHz is accurate to seven significant figures.)
acoustic impedance
property of medium that makes the propagation of sound waves more difficult
intensity reflection coefficient
a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the incident wave
Doppler-shifted ultrasound
a medical technique to detect motion and determine velocity through the Doppler shift of an echo
Check Your Understanding
1: Ultrasound can be used medically at different intensities. Lower intensities do not cause damage and are used for medical imaging. Higher intensities can pulverize and destroy targeted substances
in the body, such as tumors.
Problems & Exercises
170 dB
103 dB
(a) 1.00
(b) 0.823
(c) Gel is used to facilitate the transmission of the ultrasound between the transducer and the patient’s body.
(b) Effective penetration depth = 3.85 cm, which is enough to examine the eye.
(a)[latex]\boldsymbol{5.78\times10^{-4}\textbf{ m}}[/latex]
(b)[latex]\boldsymbol{2.67\times10^6\textbf{ Hz}}[/latex]
(a)[latex]\boldsymbol{v_{\textbf{w}}=1540\textbf{ m/s}=f\lambda\Rightarrow\lambda= {\frac{1540\textbf{ m/s}}{100\times10^3\textbf{ Hz}}} =0.0154\textbf{ m}\:<\:3.50\textbf{ m}}.[/latex]Because the
wavelength is much shorter than the distance in question, the wavelength is not the limiting factor.
(b) 4.55 ms
974 Hz
(Note: extra digits were retained in order to show the difference.) | {"url":"http://pressbooks-dev.oer.hawaii.edu/collegephysics/chapter/17-7-ultrasound/","timestamp":"2024-11-11T16:24:54Z","content_type":"text/html","content_length":"193411","record_id":"<urn:uuid:a83d26e9-2a3d-49bb-a798-78367cf83089>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00588.warc.gz"} |
Super Capacitor Discharge Time Calculator
This handy tool calculates the time it takes to discharge a super capacitor from an initial to a final voltage value under constant current and resistor load conditions
Enter the following parameters
• Initial Voltage
• Final Voltage
• Super capacitor value
• Constant current value
• Resistor Load (including the ESR of the cap)
Example Calculation
For a resistor load of 1 ohm, super cap value of 1 Farad, initial and final voltages 10 V and 1 V, respectively the constant current (1 mA) discharge time is 25 hours.
For a resistor value of 1 ohm (Ω), the discharge time is 23 seconds.
Constant Current Discharge
Discharging a supercapacitor under a constant current involves reducing its voltage linearly over time
t = C*ΔV / I
• ΔV is the change in voltage across the supercapacitor (in volts, V),
• I is the discharge current (in amperes, A),
• t is the discharge time (in seconds, s),
• C is the capacitance of the supercapacitor (in farads, F).
Resistor Load Discharge
The time required for this is given by the formula
t = RC*Log[e](Vo/V)
• V is the Final Voltage (V)
• Vo is the Initial Voltage (V)
• R is Resistance (Ω)
• C is Capacitance (F)
What is a super capacitor?
A supercapacitor, also known as an ultracapacitor or electric double-layer capacitor (EDLC), is a high-capacity capacitor with capacitance values much higher than other capacitors but lower voltage
It bridges the gap between electrolytic capacitors and rechargeable batteries, offering a unique combination of high energy storage and rapid charging and discharging capabilities.
Supercapacitors are used in applications where rapid charge/discharge cycles are required, such as in regenerative braking systems in vehicles, for power stabilization in electrical grids, in
consumer electronics for memory backup, and in short-term energy storage applications where batteries or traditional capacitors are not efficient or practical.
Their ability to charge quickly and withstand numerous cycles also makes them suitable for use in renewable energy installations, like solar panels and wind turbines, where they can smooth out
short-term fluctuations in power generation.
Overall, supercapacitors represent a critical advancement in energy storage technology, offering a complementary solution to batteries and traditional capacitors in a wide range of applications.
Related Calculators | {"url":"https://3roam.com/super-capacitor-discharge-time-calculator/","timestamp":"2024-11-05T04:24:43Z","content_type":"text/html","content_length":"203264","record_id":"<urn:uuid:113e3c5d-629c-4bd4-a712-1cdc3abf2fe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00493.warc.gz"} |
Learning through Experimentation
No Wikipedia
1 courses cover this concept
This course focuses on data mining and machine learning algorithms for large scale data analysis. The emphasis is on parallel algorithms with tools like MapReduce and Spark. Topics include frequent
itemsets, locality sensitive hashing, clustering, link analysis, and large-scale supervised machine learning. Familiarity with Java, Python, basic probability theory, linear algebra, and algorithmic
analysis is required. | {"url":"https://cogak.com/concept/255","timestamp":"2024-11-09T23:52:52Z","content_type":"text/html","content_length":"51127","record_id":"<urn:uuid:2d2c759f-065f-4c79-8820-5fd1c27f8a30>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00180.warc.gz"} |
Know pairs of multiples of 5 that total 100 - Addition Maths Worksheets for Year 3 (age 7-8) by URBrainy.com
Know pairs of multiples of 5 that total 100
Practise finding pairs of multiples of 5 that total 100. There are only about ten pairs and if possible they should be learnt.
4 pages
Know pairs of multiples of 5 that total 100
Practise finding pairs of multiples of 5 that total 100. There are only about ten pairs and if possible they should be learnt.
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Week 10: The Final Distance - BASIS Independent Schools
Week 10: The Final Distance
May 17, 2024
Hello, everyone! Welcome back to my blog! This will be the last blog post, so thank you for keeping up with my journey! Last week, I practiced my presentation and read up on how other scientists
found the distance to other supernovae. This week, I’ll use the light curve of my supernova alongside the method described in a paper by Folatelli et al. to find the distance to my supernova.
The Method:
The general idea is that since all type Ia supernovae “go supernova” with the same amount of mass, the absolute magnitude of all type Ia supernovae will be constant. This means that we can find the
distance if we know the apparent magnitude of the supernova (or how bright it looks from Earth), and simply use an equation relating brightness to distance to obtain our final answer. Why this
phenomenon works is easy to understand — if you take a lightbulb and walk away from it, you find that it gets dimmer. By knowing how bright it is to your eyes and how bright it actually is (say, to a
person right next to the bulb), you can calculate the distance.
The absolute magnitude of the supernova is denoted as MB. I took this value from several databases, where it was calculated through alternative (but more imprecise) methods of finding distances (such
as the Tully-Fisher approximation). I took MB to be -19.17 for the g-band.
We also need to know the distance modulus, which is defined as m-M, or the difference between the apparent and absolute magnitudes. My value of apparent magnitude was taken from my graph.
Here are the actual equations I ended up using:
µ = m – M = 5*log(d/10 parsecs)
µ[sn]= m[B] – M[B] – b[x] [△m[15](B) – 1.1]
△m[15](B) = m[B](15) – m[B](max)
What we want is µ[sn ], or the distance modulus of our specific supernova. I took m[B ](the peak apparent magnitude of my supernova) from my graph — it is around 11.35. Next, b[x ](1.32) is the slope
of the luminosity decline rate, which was calibrated using multiple supernovae (I took the value from Folatelli et. al). The term m[B](15) represents the apparent magnitude 15 days later, which I
also took from my graph. My data was sporadic, and there was no picture of my supernova taken precisely 15 days after the day it was brightest, so I approximated the magnitude 15 days after the peak.
I approximated it to be around 12.07.
My µ[sn ]was calculated to be 31.02. Solving for d — 31.02 = 5 * log(d/10pc), so d is around 15995580 parsecs, which can be rounded to 16 megaparsecs (MPC). The latest estimates from NASA’s
extragalactic database for the distance to the supernova range from 14-17 MPC — these estimates were calculated through less precise methods (such as the Tully-Fisher approximation, which has around
a 20% error bound).
So after a lot of data manipulation and a few errors, we’ve finally arrived at a distance value! What made this process exciting for me was how similar it was to real research conducted at
universities and labs around the world. Thanks for following my journey!
I’ll be presenting my project on May 18th at the Senior Project Symposium (Fremont Marriott Silicon Valley), so if you can make it to the event, be sure to check out my project!
View more of Kevin W.'s posts.
Reader Interactions
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Wolfgang Korsus Dipl.-Ing.NT, Astrophysiker
Klingenberg 40
D-25451 Quickborn
Mobil 01625680456
Website : wolfgang.korsus.net
simple question : Why is the sky dark at night ?
One can ask this question, but I will be careful not to ask „this“ question! Only this topic every scientific observer of the universe and its structure should at least once take to mind….and then
maybe smile mischievously
It is an apparent paradox and is named today after Heinrich Olbers, a German astronomer. He was not the first who noticed the thing; because already Kepler had noticed this and had concluded from it
that the sequence of the stars cannot be infinite. An example where one can also think of the problem is the following:
Imagine something simple, namely a gigantic dense forest, infinitely extended. (Amazon) Wherever you also look, your view meets just on a tree. Well… !?
In 1823 Olbers clearly summarized the conditions for „his“ paradox:
– The universe is infinite in every direction and has always existed in its present form.
– The stars are distributed in equal density in the universe, have always existed and have a given size and brightness.
A determination quite special way… ! ?
Under these conditions, so my conclusion is, the whole sky should be as bright as a typical star and it should never become dark at night. So I say, there is „something“ not right and not at all. And
the before mentioned „something“ brings us damn fast to modern cosmology and also to their ideas about the beginning of the universe….. and that says…..
So be the age of the universe limited, also everything has begun with a „big bang“ before soundsovielen years, then the universe visible for us today is of finite size, because the light has had,
after all, only these years to reach us. These distances and times are just enormous, only…. they are not infinite.
Likewise I state that the stars had to originate sometime after the big bang, so that their number is also finite. ….the limited age of the universe gives us then only insight into a finite spatial
part of the whole, and in this part can have originated since the big bang just only a finite number of stars.
Therefore I say, the sky is dark at night, how should it be otherwise…….also Mr. Olbers, since we know that the speed of light is finite as well as the „big bang“ also sometime the world beginning
was, so these assumptions are quite simply at least continuing !
Now the question must follow, but how can we convince ourselves that these ideas are really the right ones?
So it is about the origin of the world and even the question whether it has an origin – all this is still today much discussed in science, philosophy and „religions“. There are just two main reasons
why most scientists today consider the big bang theory to be correct – now I say : not so hastily we approach the subject significantly slowly, step by step and one more step.
I begin with theoretical slow motion and it is the following :
A quite well-known effect of the everyday life physics is e.g. the „sound shift“ with a moving sound source. Surely everyone who has watched a car race knows this. The engine sound of a Formula 1
race car is higher when the vehicle is coming towards us, and then becomes lower when it drives away; there is a…..I’ll call it a „tone flip“ as it passes. In physics, the process is called a Doppler
shift, or better „effect.“ Simply named after the Austrian physicist Christian Doppler. What do you hear there ? Yes, the sound we hear is created, as we know, by sound waves that have a certain
wavelength, and so when the source of the sound approaches us, the wave acts as if squeezed, because the distance between two successive peaks shortens and that means an increase in the sound. What
happens with a sound source moving away …..the opposite. Such a Doppler effect occurs not only with sound waves, but also with light waves.
That means then, one can measure likewise something, i.e. whether a certain star moves seen from us. Stars, as many know emit ( electrons ) . Light of certain wavelengths and namely with so-called
spectral lines, and if these appear shifted, the star must move naturally.
So to repeat again.
If the star comes to us the wavelengths become shorter (thus blue shift).
…………. the star moves away………. the wavelengths become longer(i.e. red shift)
The faster it moves, the bigger is the shift !
Now it comes : Already in the 1920s the astronomer „Edwin Hubble“ at the Mount-Wilson-Observatory in California was occupied with light and indeed very distant stars. An extensive exploration
He had already measured clear red shifts; of course it was known that these stars are moving away from us. But Hubble now made another surprising discovery. They disappear the faster the further away
they are.
It may be mentioned the known Doppler shift and that means also the stellar velocity, were measurable without problems. But there was a difficulty in determining the distance of the observed stars,
to measure relatively close celestial bodies, such as planets, was easy. The parallax method had been used previously. Also Cassini and Richer had calculated the distance between Earth and Mars with
this method, all respect.
Only now it was time to use another method for very distant stars.
But for the distant stars Hubble was targeting, the parallax angle was much too small for any measurements. Hubble came up with a solution quite quickly in a fairly simple way. He knew the brightness
of a light source decreases the further away it is and since the light spreads out spherically from its source, therefore less and less light falls on a given area with increasing distance.
Continuing: I designate the distance to the light source with d, from it follows the spherical surface increases then with d 2. The incident light per unit area as 1/d 2 decreases as a result.
So we can conclude :
If we know the initial brightness of a source and also the brightness measured at a certain (initially unknown) distance, then the difference in brightness is determined by the distance d .
Next we turn to the expression „reference candles“. What is this again ?
Hubble had measured the brightness of quite certain stars. These were the Cepheids, they had just recently been measured. The astronomers made of it the….na what ! „The Reference Candles“…..and this
was achieved by measuring their relative brightness in his observatory at Mount Wilson, thus Hubble also got a good estimate of their distance. – He found that their escape velocity v increased with
increasing distance d from Earth. (See further details in my short note X2 (a little later).
From all considerations and their calculations, Hubble’s law v = H 0 d , with the scale determination H 0 again called Hubble’s constant, thus arose.
If we look at the value resulting at that time, it can be stated that it is a little bit inaccurate from today’s point of view, but what must not be forgotten, only the idea of Hubble is completely
correct and much more important is the fact of the change of our world view.
Hubble did not stop to observe stars. It can be said, wherever Hubble looked, the stars disappear, the view arose: does the whole universe explode ? This view led then to another question: Was this
possible? Or is such a thing impossible ?
Today I make the assertion : This was an ideal time, because in the year 1916 already Albert Einstein’s new, yes new general relativity theory appeared.
Hubble’s discovery took place at an ideal time, a little later. Only little before, in the year 1916, Albert Einsteins new general relativity theory had appeared, which connected simply expressed the
effects of the gravity with the nature of space and time. In addition a statement which forces to think. I let „an arbitrary round thing“ circle at the tape, whereby the tension of the tape
compensates even the centrifugal force. Now we consider only the ball movement, and already I could also come to the conclusion, this round thing, let’s call it a ball, moves freely on a curved
Now this way of thinking forces me to think that the role of the force can be replaced by a kind of space curvature. For Einstein’s theory states that in the vicinity of very massive celestial
bodies, such as the sun, space is so deformed by gravity that even a ray of light deviates from its straight path. His prediction was tested and confirmed in 1919 in what is now a very famous
It was the English astronomer Arthur Eddington who, with his collaborators, showed that starlight passing close to the Sun was indeed deflected by exactly the amount calculated by Einstein.
The experiment took place, of course, during a solar eclipse.
Einstein became of course by this observation abruptly world-famous.
It is still to be mentioned absolutely the statement of the general public at the time when Einstein set up his general relativity theory. One thought the universe for static; because one did not
think (yet) of the possibility of an expansion or contraction.
Einstein knew this and already he started thinking. He needed just now any force which simply compensated the attracting effect of the gravity. Immediately no candidates came to his mind and even
until today the problem still puzzles us „astros“. As already mentioned, Einstein started, but with little enthusiasm. Thus a “ solution“ was created ?! You notice, it is not so easy for me to
comment on it. He began to spread out his first proposal and that was the introduction of a mysterious „cosmological liquid“. This should fill the whole space and compensate the gravity approximately
by its pressure.
But this is too little, the description of the criteria goes on…… This fluid had to have still quite strange properties …..it could not influence any process in the universe, except gravity, so that
it remained unobservable for all other measurements. Its pressure and therefore its density had to be precisely determined one hundred percent in order to compensate gravity exactly.
As actually known, the mentioned resembles in certain sense, however, to that of a new form of ether and thus this statement was particularly undesirable to Einstein and perhaps even „embarrassing“ .
So a new designation had to come and so he has called then later the introduction of this size called today as cosmological constant as the biggest blunder of his life. If, would have, bicycle chain,
one would like to say there. His original equations without cosmological constant, on the other hand, predicted an expansion of the universe. That „would have“ then still before it was discovered
then by Hubble, took place.
Nowadays, especially among cosmologists, one is not quite sure whether this quantity was really a mistake. Because the „dark energy“, to which I will come a little later, is something like a
resurrection of Einstein’s cosmological constant.
Still important to mention is the fact that in 1922 the Russian theorist Alexander Friedmann proved that the general solution of Einstein’s equations describes very well expanding universes. But
let’s stay with Hubble he finally found his expansion, now everything was ready. The next chapter follows…….. | {"url":"http://wolfgang.korsus.net/first-the-horizons-then-until-the-last-veil-part-8","timestamp":"2024-11-02T10:58:45Z","content_type":"text/html","content_length":"40133","record_id":"<urn:uuid:6716d414-ed74-4e99-8560-357264eeef77>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00274.warc.gz"} |
Lesson 20
Center Day 4 (optional)
Warm-up: Number Talk: Coin Counting Connections (10 minutes)
This Number Talk encourages students to think about grouping numbers. When finding the total value of a set of coins, students grouped like coins and used skip counting or counting on to find the
amount of money represented. This string provides an opportunity for students to practice these strategies. These understandings help students develop fluency and will be helpful later in this lesson
when students find and compare the values of coin collections.
• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time
• Record answers and strategies.
• Keep expressions and work displayed.
• Repeat with each expression.
Student Facing
Find the value of each expression mentally.
• \(20 + 25 + 5 + 5\)
• \(15 + 25 + 25\)
• \(25 + 15 + 25 + 6\)
• \(20 + 15 + 30 + 7\)
Activity Synthesis
• “How could the third problem help us think about the last one?” (\(25 + 25 = 50\) and \(20 + 30 = 50\), and they both had 15, so the only difference was the 6 and 7.)
Activity 1: Introduce Would You Rather? (15 minutes)
The purpose of this activity is for students to learn stage 1 of the Would You Rather? center. Students compare the value of two sets of coins. The first partner spins and identifies a number of
coins. They write a question that compares the coins they spun to a collection of coins they make up. Their partner makes a choice of which collection of coins they would rather have and explains why
in terms of the units.
Required Materials
Materials to Copy
• Would You Rather Stage 1 Spinner
• Would You Rather Stage 1 Recording Sheet
• Groups of 2
• Give each group a spinner and a recording sheet.
• “We are going to learn a center called Would You Rather. We will compare the value of coin collections. Let's play a round together.”
• “One partner spins to get a group of coins. They identify the different coins in the collection and count how many of each. Then, they write a question that compares the amount they spun to a
different group of coins that they make up.”
• Demonstrate by spinning the spinner and drawing a group of coins. Provide example questions that show keeping the number of coins the same and changing the name of the coin or keeping the coins
the same, but changing the number of each coin in the collection. For example:
□ “Would you rather have 3 quarters or 3 pennies?”
□ “Would you rather have 4 nickels and 6 dimes, or 4 dimes and 6 nickels?”
□ “Would you rather have 1 quarter, 1 dime, and 1 nickel or no quarters, 1 dime, and 7 nickels?”
• Invite the class discuss and share which group they would rather have.
• 8 minutes: partner work time
Activity Synthesis
• Display the spinner and spin to get a coin set.
• “Come up with a coin collection with a greater value using the same number of coins.”
• “Come up with a coin collection with a lesser value using the same number of coins.”
• Share responses.
Activity 2: Centers: Choice Time (25 minutes)
The purpose of this activity is for students to choose from activities that focus on finding the value of coin collections or working with shapes.
Students choose from any stage of previously introduced centers.
• Would You Rather?
• Picture Books
• How are They the Same?
Required Preparation
Gather materials from:
• Would You Rather?, Stage 1
• Picture Books, Stage 3
• How Are They the Same?, Stage 2
• “Now you will choose from centers we have already learned. One of the choices is to continue with Would You Rather.”
• Display the center choices in the student book.
• “Think about what you would like to do first.”
• 30 seconds: quiet think time
• Invite students to work at the center of their choice.
• 10 minutes: center work time
• “Choose what you would like to do next.”
• 10 minutes: center work time
Student Facing
Choose a center.
Would You Rather?
How Are They the Same?
Picture Books
Activity Synthesis
• “What did you like about the activities you worked on today?”
Lesson Synthesis
Spin the spinner to get a group of coins.
“I spun this group of coins. Draw a different group of coins you'd like the class to compare it to.”
Share a few of the groups of coins students draw. Ask the class which they would rather have and why. | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-2/unit-6/lesson-20/lesson.html","timestamp":"2024-11-04T11:26:16Z","content_type":"text/html","content_length":"84860","record_id":"<urn:uuid:132df12f-860e-4b0b-a485-d29d28e67af8>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00861.warc.gz"} |
Time Interest Earned Ratio in Different Industries in context of time interest earned ratio
26 Aug 2024
Title: An Examination of the Time Interest Earned (TIE) Ratio in Various Industries: A Comparative Analysis
Abstract: The Time Interest Earned (TIE) ratio is a widely used financial metric that measures a company’s ability to generate earnings before interest and taxes (EBIT) relative to its interest
expenses. This study aims to investigate the TIE ratio in different industries, providing insights into its applicability and variability across sectors.
Introduction: The TIE ratio is calculated as follows:
TIE Ratio = EBIT / Interest Expenses
This metric is essential for investors, creditors, and analysts seeking to assess a company’s financial health and creditworthiness. The TIE ratio has been extensively used in various industries, but
its performance can vary significantly across sectors.
Literature Review: Previous studies have demonstrated that the TIE ratio is a reliable indicator of a company’s ability to service its debt. However, the ratio’s behavior can differ substantially
depending on the industry. For instance, companies in capital-intensive industries, such as energy and manufacturing, tend to exhibit lower TIE ratios due to higher interest expenses.
Methodology: This study employs a comparative analysis approach to examine the TIE ratio across various industries. The sample consists of publicly traded companies from different sectors, including:
• Energy (e.g., oil and gas, renewable energy)
• Manufacturing (e.g., automotive, aerospace)
• Technology (e.g., software, hardware)
• Healthcare (e.g., pharmaceuticals, biotechnology)
• Financial Services (e.g., banking, insurance)
Results: The results of this study indicate that the TIE ratio varies significantly across industries. For example:
• Energy companies tend to exhibit lower TIE ratios due to high interest expenses associated with capital-intensive projects.
• Technology companies often display higher TIE ratios, as they typically have lower interest expenses and higher EBIT margins.
• Healthcare companies show a moderate TIE ratio, reflecting the industry’s mix of high-interest expenses and relatively stable cash flows.
Conclusion: This study provides insights into the behavior of the Time Interest Earned (TIE) ratio across various industries. The results highlight the importance of considering sector-specific
factors when interpreting the TIE ratio. By understanding the variability of this metric, investors, creditors, and analysts can make more informed decisions about a company’s financial health and
Recommendations: Based on the findings of this study, we recommend that:
• Investors and analysts consider industry-specific factors when evaluating the TIE ratio.
• Companies in capital-intensive industries, such as energy and manufacturing, focus on managing interest expenses to improve their TIE ratios.
• Technology companies continue to leverage their low-interest expense profiles to maintain high TIE ratios.
Limitations: This study has several limitations. Firstly, the sample size is limited to publicly traded companies, which may not be representative of all industries. Secondly, the analysis focuses on
a single financial metric (TIE ratio), and future studies could explore other metrics in conjunction with the TIE ratio.
Related articles for ‘time interest earned ratio ‘ :
Calculators for ‘time interest earned ratio ‘ | {"url":"https://blog.truegeometry.com/tutorials/education/51de744260d40d99a5b53471ba21339b/JSON_TO_ARTCL_Time_Interest_Earned_Ratio_in_Different_Industries_in_context_of_t.html","timestamp":"2024-11-04T08:42:59Z","content_type":"text/html","content_length":"18523","record_id":"<urn:uuid:3a2fd74c-cf9d-41e8-afb4-29bc2b5647a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00676.warc.gz"} |
Replace value based on the previous/next value in the column
I have a fixed price of the current quarter. Then I have the percentage difference based on the other input (raw to raw). It looks like this more or less:
│ID│ITEM│PRICE │DATE │DIFF %│
│1 │A │3000 │01.01.2020 │-0.01 │
│2 │A │YYYYYYY │01.04.2020 │-0.05 │
│3 │A │XXXXXX │01.07.2020 │ │
│4 │B │2400 │01.01.2020 │0.04 │
│5 │B │ZZZZZZ │01.04.2020 │-0.09 │
│6 │B │VVVVVV │01.07.2020 │ │
I want to get the YYYYY value by subtracting Price of ID 1 by percentage DIff of ID 1 ( YYYY then wound be 2970).
Then to get XXXXX value I want to do the same but with price YYYYYY and so on.
Operating system used: Win 10
tgb417 Dataiku DSS Core Designer, Dataiku DSS & SQL, Dataiku DSS ML Practitioner, Dataiku DSS Core Concepts, Neuron 2020, Neuron, Registered, Dataiku Frontrunner Awards 2021 Finalist, Neuron
2021, Neuron 2022, Frontrunner 2022 Finalist, Frontrunner 2022 Winner, Dataiku Frontrunner Awards 2021 Participant, Frontrunner 2022 Participant, Neuron 2023 Posts: 1,598 Neuron
Welcome to the Dataiku community.
If I were trying to do something like this with visual recipes and not code, I’d be taking a look at the window recipe to get data from prior rows onto the current row. Once I had my lag values
on the current row, I’d likely use a visual prepare recipe to add a formula step to do the calculations.
These Dataiku academy course covers these two recipes in these two courses.
The basic ideas are covered in this course:
More details are provided here with some hands on exercises.
And here is something more on formulas
Hope this helps please let us know how you are getting on with your project.
• Thanks Tom.
What you described I did to get those month-2-month percentage difference since this is based on other column input with all values filled in.
Now I need to get the same for one fixed value only where all other are blank. Move that value to the next raw and same again.
My data is more than 50 rows with dates per each item.
I thought maybe there is a formula I could use to achieve it? | {"url":"https://community.dataiku.com/discussion/25179/replace-value-based-on-the-previous-next-value-in-the-column","timestamp":"2024-11-10T07:32:32Z","content_type":"text/html","content_length":"409480","record_id":"<urn:uuid:740129c1-1a11-4a25-bca3-98cbedeef21e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00453.warc.gz"} |
Calssic Dice - A guide to odds and strategies
Dice is one of the oldest gaming implements known to man. Dices as old as 2000 BC have been found in Egyptian tombs. The oldest records of dice games were found in India, were written in ancient
Sanskrit and were over 2000 years old. Dice games have advanced a little bit since then but are just as popular.
At BC.game you can find three different types of Dice games. In this post, I will shortly describe the Classic Dice, how it works and how its odds work.
Classic Dice
The classic dice game has very intuitive rules. Use the slider to pick a number between 2 and 98, and decide if you will try to roll over or under the number you've chosen. Place your bet, and roll!
Or you can use auto-roll.
Decide the amount and which coin to bet.
Number of bets
Fill in how many rolls you wish to do (0 = infinite).
On win
Choose whether you want the game to reset your bet to the base bet or raise the bet by a certain % after you have won a bet.
Stop on win
Tell the game to stop rolling once you have won this many coins.
On lose
Choose whether you want the game to reset your bet to the base bet or raise the bet a certain % after you've lost a bet.
Stop on lose
Tell the game to stop rolling once you have lost this many coins.
This game in provable fair
Pressing on this button will open up a new window that shows:
Current server seed, client seed and nonce. At the bottom, you can choose a new seed, either let the game randomize a seed or manually type one in, then press "use new seed".
You can use these seeds to ensure the game wasn't tampered with. The server seed is set by the casino and can't be changed until you change your client seed. The server seed is also encrypted;
otherwise, it would only take a few minutes to calculate which number will be next – ahead of time.
It's important both for BC.game and for their customers that you can verify that any given bet was won or lost without anyone cheating. And here is why.
Let's say you play dice with auto-roll and set it up so that if you roll over 50, you win, and if you roll 1-50, you lose, and you will double your next bet. This common strategy is called
Martingale, and the idea is that you will always win, eventually.
Let's say your base bet is $0.1. If you roll a number between 51-98, you will win 1.98x your bet. If you roll between 2-50, you lose your current bet and double your next bet. Let's say you roll 45
and lose your $0.1, and your next bet will be $0.2. This time you roll 61 and win 1.98 x $0.2 = $0.396. So far, your total bet has been $0.3 (first $0.1 that you lost and then another $0.2), which
means that you have made a profit of $0.096. So far, so good, right?
But what happens if you lose, let’s say 5 times in a row?
Bet New Bet
$0.1 $0.2
$0.2 $0.4
$0.4 $0.8
$0.8 $1.6
$1.6 $3.2
Now your total bet amount to 0.1 + 0.2 + 0.4 + 0.8 + 1.6 + 3.2 = $6.3.
Your betting amount will grow exponantial, so if you were to lose yet again, your next bet would be $6.4 and your total bet $12.7. But let's say you win the bet you made at $6.4. You will then get
6.4 x 1.98 = $12.672. Your total bet was $12.7, and now you've "won" $12.672 (you've actually lost $0.028). This is the "House Edge" of 1%. You would think that if you have a 50/50 chance to win,
then when you do win, you would win 2x your bet. But the casino has to have this edge of 1%, or they wouldn't make any money.
So, let’s adjust our betting a bit so that when you do win, you will win 2x your bet.
Now your chance of winning isn't 50/50 anymore. That's the house edge again, but maybe a bit clearer. But you're thinking, "what does 0.5% do in the big scheme of things anyway" so you adjust your
betting and start over. Again, you lose several times in a row and are now going to make the $6.4 bet. If you win this time, you will win 6.4 x 2 = $12.8. Remember, your total wager was $12.7. This
time you made a profit of $0.1. But was it worth it to bet $12.7 to win $0.1?
That's how this strategy works. No matter how many times in a row you lose and how big your final winning bet is – you will always only win back your initial bet ($0.1 in this case).
I have found that most people who complain about the dice or accuse the casino of cheating are using this strategy or a variant. They often think, "Okay, maybe I'll lose 5-6 times in a row at most,
and after all, it's a 50/50 chance of winning."
And then, after playing for a while, they lose ten times in a row. Or 15. What happens then? Let's calculate.
0.1 x (2^10) = 102.4 – If you start betting $0.1 and have a losing streak, your 10^th bet will be $102.4 (that’s not even your total bet which by now is $204.7).
Let's see what happens when you lose 15 times in a row.
0.1 x (2^15) = 3276.8 – Your 15th bet would be $3276.8, and your total bet would be a whopping $6553.5. And how much is your possible profit from this? That’s right, $0.1.
It's easy to understand that a person who puts this strategy on auto-roll will risk losing a lot of money quickly. And it's easy to understand how this person would be agitated and frustrated and
blame the casino for cheating. And this is why it's good for both you and the casino that the game is provable fair.
Let's say you lost that last bet at $3276.8. You have now lost $6553.5 in 15 quick bets and will start to wonder if it's even possible to hit below 50 15 times in a row when the odds are supposed to
be 50/50. It doesn't sound reasonable to think that would happen without anyone tampering with the game, huh?
Well, it is (I have had worse losing streaks!). We tend to think that every bet you lose when the odds are 50/50 will increase our chance of winning our next bet, especially when we see how the bets
we are making quickly becomes very large bets.
But chance (or odds) doesn’t work like that. For every new game you play, you will have a 50/50 chance of winning. Nothing more, nothing less. If you had been rolling with a real dice, the dice
wouldn’t keep track of how many bets you have lost in a row. And this is how we also need to view online gaming.
Go to the list where you find your previous bets (oh, and change seed first!) and click "verify". You will be sent to a different site, where you can see that input: client seed (your seed), server
seed (casinos seed) and nonce (how many games you've played with these seeds). And you will see the output: decrypted server seed and hmac_sha256 (this is the server seed, client seed and nonce
together). A bit further down the page you will see the results.
The hmac_sha256 is hexadecimal and needs to be converted to bytes.
The upper row is hexadecimal, and the lower row is bytes. If you mistrust this information, you can use any other hexadecimal converter, and it will show you the same result. The next step is to
convert bytes to numbers. This game uses the first 4 bytes and will always do that; other games might use more.
Let’s calculate!
(119 / (256^1)) =
use 119 because that’s the first bytes in the bottom row. use 256 because the hash is sha256, and finally the ^1 because this is the first number we calculate.
(119 / (256^1)) = 0.464843750
(206 / (256^2)) = 0.003143311
(56 / (256^3)) = 0.000003338
(45 / (256^4)) = 0.000000010
Add these together.
0.464843750 + 0.003143311 + 0.000003338 + 0.000000010 = 0.467990409
Multiply the result with 10001 to round off the decimals = 4 680.372080409
Finally divide the result with 100 to get a number between 1-100 with 2 decimals.
4 680.372080409 / 100 = 46.80372080409
46.8 is precisely what the dice showed us in-game. We didn't get scammed, and nobody had tampered with the game.
We were just unlucky.
This topic is now archived and is closed to further replies. | {"url":"https://forum.bcgame.lu/topic/9860-calssic-dice-a-guide-to-odds-and-strategies/","timestamp":"2024-11-05T06:41:31Z","content_type":"text/html","content_length":"90928","record_id":"<urn:uuid:be40cd94-cb98-40bf-b05c-be550c44e8aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00759.warc.gz"} |
Multispectral Plasmon Induced Transparency in - PDF Free Download
LETTER pubs.acs.org/NanoLett
Multispectral Plasmon Induced Transparency in Coupled Meta-Atoms Alp Artar,† Ahmet A. Yanik,† and Hatice Altug* Electrical and Computer Engineering Department, Boston University, Boston,
Massachusetts 02215, United States
bS Supporting Information ABSTRACT: We introduce an approach enabling construction of a scalable metamaterial media supporting multispectral plasmon induced transparency. The composite multilayered
media consist of coupled meta-atoms with radiant and subradiant hybridized plasmonic modes interacting through the structural asymmetry. A perturbative model incorporating hybridization and mode
coupling is introduced to explain the observed novel spectral features. The proposed scheme is demonstrated experimentally by developing a lift-off-free fabrication scheme that can automatically
register multiple metamaterial layers in the transverse plane. This metamaterial which can simultaneously enhance nonlinear processes at multiple frequency domains could open up new possibilities in
optical information processing. KEYWORDS: Metamaterials, electromagnetically induced transparency, plasmons, plasmon hybridization, Fano resonances, strong coupling
lectromagnetically induced transparency (EIT), a spectrally narrow optical transmission window accompanied with extreme dispersion, results from quantum interference of multiple excitation pathways
through short and long-lived resonances.1 Within this spectral window, dramatically slowed down photons and orders of magnitude enhanced nonlinearities can enable manipulation of light at few-photon
power levels.2 Historically, EIT has been implemented in laser-driven atomic quantum systems. However, limited material choices and stringent requirements to preserve the coherence of excitation
pathways in atomic systems have significantly constrained the use of EIT effect.3 Recent studies have revealed that EIT-like optical responses can be obtained classically using on-chip plasmonic and
photonic nanoresonators.420 Much of the research effort so far focused on isolated meta-atoms (either photonic or plasmonic) showing EIT-like effect at a single resonance. On the other hand,
metamaterial systems supporting EIT-like optical responses at multiple-spectral windows can simultaneously enhance multicolored photonphoton interactions and open up new possibilities in nonlinear
optics and optical information processing.2123 In this Letter, we propose and demonstrate a novel approach based on coupled meta-atoms to construct a homogeneous and scalable medium supporting
multispectral EIT-like effect (plasmon induced transparency). The proposed structure consists of a two slot antenna based complementary metamaterial layers with a small gap (dielectric layer
thickness) enabling strong near-field interaction in between. Each planar metamaterial layer has bright (radiant) and dark (subradiant) plasmonic modes coupled through the structural asymmetry (s 6¼
0) in an analogy to transition-allowed and -forbidden atomic orbitals coupled r 2011 American Chemical Society
through a common excited state.6 As shown in Figure 1b (blue curve), isolated meta-atoms on a single-layer metamaterial exhibit an EIT-like reflection10 with spectral features that are controlled by
the artificial atomic orbitals (plasmonic modes). Once stacked in a multilayered structure (Figure 1b, black curve), presence of strong near-field coupling between the meta-atoms causes splitting of
the EIT resonances and leads to multispectral EIT-like behavior. The underlying physical principles for this phenomenon are related to plasmonic hybridization effects24 and dark-bright mode couplings
of the in-phase and out-phase hybridized states. To explain these novel spectral features, we introduce a perturbative model incorporating hybridization and mode coupling. Furthermore, we
experimentally demonstrate the proposed scheme by developing a lift-off free fabrication scheme that can simultaneously register multiple metamaterial layers in the third dimension. In the following,
we start by describing the perturbative model that provides insight into the physical processes involved in these structures. For the double layered metamaterial, a total Hamiltonian can be defined as
~ ~0 þ H ~ 00 þ K ~ þΣ HT ¼ H ~ 00 are the 2 2 unperturbed Hamiltonians of ~ 0 and H Here, H the isolated metamaterial layers defined in a basis set consisting of decoupled bright (dipolar) and dark
(quadrupole) modes in the absence of a structural asymmetry (s 6¼ 0). The weak interactions between the bright and the dark modes Received: January 18, 2011 Revised: March 4, 2011 Published: March
25, 2011 1685
dx.doi.org/10.1021/nl200197j | Nano Lett. 2011, 11, 1685–1689
Nano Letters
Figure 1. (a) Geometry of the multilayered metamaterial. Structure consists of two Au layers (30 nm thickness) that are separated by a dielectric (SiNx) layer (70 nm thickness). Each layer has a
dipole and a quadrupole slot antenna (all slot antennas have 700 nm length, 100 nm width). The small in-plane separation between the dipolar and quadrupolar antennas is 50 nm on both sides and
periods are 1200 nm on both x and y directions. Parameter s is defined as the offset of the dipolar antenna from the geometrical center of the structure. Blue arrows show the configuration of the
incident light. (b) Simulated reflection spectra for asymmetric (s 6¼ 0) single- and doublelayered structures are shown (with an offset for clarity). Multispectral EIT-like response (in-phase and
out-of-phase) is observable with double-layered metamaterial.
Figure 2. (a) Hybridization scheme for the dipolar mode. (b) Tuning of the spectra with the dielectric layer thickness is shown for the symmetric structure (s = 0). As the dielectric layer thickness
reduces, splitting of energy between the hybrid modes increases. Single layer spectrum is shown with the blue dashed curve. Splitting energies (2εD) are 202, 260, 297 meV and energy offsets (ΔD) are
60, 68, 78 meV for gap sizes (dielectric layer thicknesses) of 90, 70, 50 nm, respectively. (c) Charge distribution at the air/ metal interface (top view), demonstrating the dipolar mode excitation.
This charge distribution is acquired from in-phase state |Dþæ of the multilayered structure; however the out-of-phase state |Dæ and also the single-layered dipolar state |D0æ have the exact same
charge distribution (not shown). (d) Charge distributions of the hybrid dipolar modes acquired from a multilayered structure with a dielectric layer thickness of 50 nm (cross-sectional view) are
shown at a position marked with the red dashed line in (c).
~ , when a are incorporated with the perturbative Hamiltonian K structural asymmetry is introduced (s 6¼ 0). Interactions between the two metamaterial layers are included through the strong near-field
coupling Hamiltonian Σ~. Accordingly, the total Hamiltonian for the coupled meta-atoms is given as
ÆD0 j " # H0 þ K Σ ÆQ0 j ¼ Σ H0 0 þ K0 ÆD0 0 j ÆQ0 0 j
~ ¼ Æ1j ~0 þ H ~ 00 þ K ~ þΣ HT ¼ H Æ2j
T Hhyb
ÆDþ j ^ T U ^ ¼ ÆD j ¼ U½H ÆQ þ j ÆQ j
jDþ æ
ED0 þ ΔD þ εD 6 6 0 6 6 k 4 0
in an orthogonal basis set consisting of 1 jD( æ ¼ pffiffiffi½jD0 æ ( jD0 0 æ 2
jD0 æ
ED0 6 6 k 6 6τ 4 inter, D ðχ0 Þ
jQ0 æ
jD0 0 æ
k E Q0 χ
τinter, D χ ED0 0 ðk0 Þ
τinter, Q
jQ0 0 æ χ0 τinter, Q k0 EQ0 0
consideration in our analysis is that χ and χ0 , the cross couplings among the bright and dark modes, are weak and can be neglected. Validity of this assumption will be justified in the following by
benchmarking our analytical relations with numerical simulations and experimental measurements. For a metamaterial system where the individual layers have identical structural characteristics, the
Hamiltonian terms for both layers are identical (κ = κ0 , τinter,D/Q = τinter,D0 /Q0 , ED0/Q0 = ED0 0/Q0 0). After a simple rearrangement of the matrix elements and a unitary transformation, the
total Hamiltonian can be rewritten as in
where |1æ and |2æ represent the top and bottom metamaterial layers, respectively. The eigenvalues of the bright (|D0æ) and the dark (|Q0æ) modes of the isolated metamaterials are defined as ED0 and
EQ0. For clarity, eigenstates and eigenvalues of the second layer are denoted with primes, even for structurally symmetric layers. κ and κ0 are due to the weak intralayer coupling among the dark and
bright modes in each layer. τinter,D and τinter,Q are the strong interlayer coupling terms for the bright and dark modes, respectively. χ and χ0 are the cross couplings among the bright and dark
modes of different layers (interlayer). An important 2
jD æ
jQ þ æ
0 þ ΔD εD 0 k
k 0 þ Δ Q þ εQ 0
jQ æ
0 k 0 þ Δ Q εQ
1 jQ ( æ ¼ pffiffiffi½jQ0 æ ( jQ0 0 æ 2
ð3aÞ 1686
dx.doi.org/10.1021/nl200197j |Nano Lett. 2011, 11, 1685–1689
Nano Letters
~ (when ~0 þ H ~ 00 þ Σ diagonalizing the Hamiltonian HTs=0 = H the system is symmetric s = 0) in the strong coupling regime (|ED0 ED00 | , 2τinter,D and |EQ0 EQ00 | , 2τinter,Q). These hybrid
eigenstate pairs are in-phase (þ) and out-of phase () superpositions of the isolated layer eigenmodes of the structurally symmetric multilayer system (s = 0). The associated energies of the dipolar
hybrid modes are, E( D = ED0 þ ΔD ( εD, where the offset term is ΔD = ÆD0|τinter,D|D0æ and the splitting term is εD = ÆD0|τinter,D|D00 æ (a similar set can be obtained for quadrupole modes). Since off
diagonal terms in the transformed Hamiltonian HThyb are much weaker than the diagonal terms, the off-diagonal matrix elements κ and κ0 are treated as the elements of the perturbative Hamiltonian
introduced by the structural asymmetry (s ¼ 6 0). Using the transformed Hamiltonian HThyb, a set of coupled Lorentzian oscillator relations can be derived in an analogy to atomistic EIT resonances.
Initially, the hybridization of eigenstate pairs in the form of in-phase (þ) and out-of phase () superpositions of the isolated layer eigenmodes is shown in Figure 2 for the structurally symmetric
multilayer system (s = 0 and κ = 0) using finite difference time-domain (FDTD) analysis. For a single-layered metamaterial, only the resonance dip corresponding to the excitation of the dipolar bright
mode is observable in the reflection spectrum (Figure 2(b), dashed blue curve). For the double layered metamaterial, two resonance dips appear corresponding to in-phase and out-of-phase hybridized
modes due to the degeneracy breaking as given in eq 3a (solid curves in Figure 2b). The mode energies and the splitting in between are controlled by the strength of the interlayer coupling of the
metamaterial layers. As predicted by our Hamiltonian treatment, smaller gaps (dielectric layer thicknesses) lead to larger energy splittings as this coupling becomes stronger (Figure 2b). The
in-phase and out-of-phase character of these hybridized states are also confirmed with FDTD simulations showing cross-sectional charge distributions of the dipolar modes (Figure 2d). The in-phase
hybrid mode is radiant as a result of its overall dipolar character. The radiant out-of-phase mode is harder to excite with respect to the in-phase mode, due to the partial cancellation of the
dipolar moments of the subsequent layers. Nevertheless, resonance dip corresponding to the out-of-phase mode is still observable due to the retardation effects (Figure 2b, solid curves). In Figure 2b,
resonances due to hybridized quadrupolar modes are not observable, since any linear combination of the subradiant quadrupolar modes of the isolated structures is also subradiant. Structural symmetry
must be broken for the excitation of these quadrupolar hybridized modes. Breaking the symmetry of the multilayered structure (s 6¼ 0) leads to near-field coupling between the dark and bright modes (κ
6¼ 0) and results in the excitation of the dark modes with the perpendicularly incident light. Indirect excitation of these hybrid quadrupolar dark modes leads to multispectral EIT-like behavior
(Figure 3 black curve). Charge distribution of the out-of-phase (OP) EIT resonance at the top surface (Figure 3a inset), indicates strong coupling of the external driving field to this mode. A similar
charge distribution is also observed for the inphase (IP) EIT resonance (not shown). Cross sectional charge distributions of the quadrupolar modes (Figure 3b) confirm the in-phase and out-of-phase
mode characters. Full spectral re2
ω ωþ þ iγþ D 6 þ 6 k 6 6 0 4 0
kþ ω ωþ δþ þ iγþ Q 0 0
Figure 3. (a) Asymmetric (s = 150 nm) and symmetric (s = 0 nm) double-layer EIT-like spectra. Two dips seen in the symmetric structure’s spectrum corresponds to hybrid dipolar modes as in Figure 2b.
Asymmetric structure shows two EIT-like peaks at different spectral positions. A model fit based on Lorentzian harmonic oscillators is shown for the doublelayered structure (red dashed curve with κ =
12 THz, κþ = 23 THz),25 which traces the calculated spectra very well. A genetic search algorithm is implemented to extract the parameters using a least-squares sum fit. Calculated group indices for
the in-phase and out-of-phase modes are nþ g = 16, n g = 9.3. These values can be optimized by adjusting the coupling terms κ(. Inset shows the top view charge distribution at the air/metal interface
for the out-of-phase EIT peak (in-phase EIT peak also shows the same distribution). Stronger excitation of the quadrupolar mode is shown. (b) Cross-sectional charge distributions of the quadrupolar
antennas are shown at a position marked with the red dashed line in the inset. Hybridization of the quadrupolar resonance is shown.
sponse of the multilayered structure can be understood following our perturbative Hamiltonian approach. Here, a coupled Lorentzian oscillator model is derived from the transformed Hamiltonian HThyb
in a similar way to the EIT concepts in atomic physics. In our analysis, the following three observations are employed. (i) Breaking of the structural symmetry (s 6¼ 0) results in weak near-field
coupling of the hybridized dark and bright modes, an effect which can be incorporated to the unperturbative Hamiltonian (s = 0) with the perturbative terms κ and κ0 . (ii) There is no direct coupling
between the in-phase and out-of-phase hybrid modes, due to the large energy difference in between (D( ( S Q-). (iii) Damping rates of the quadrupole (γ( Q ) and dipole (γD ) ( ( hybrid modes are small
enough that the condition γQ , γD , ω( is satisfied. Here ω( are the resonant frequencies of the in-phase and ( out-of-phase hybrid pairs (ω( = E( D /p ≈ EQ /p). We can express all hybrid states in
the form of |φæ = φ ~eiωt (where φ ( ( ~ is Q and D ) and denote the external driving field as E0eiωt. Then, in agreement with the total Hamiltonian of the system, the following set of linear
equations is obtained for the coupled Lorentzian oscillators
0 0 ω ω þ iγ D k 1687
2 3 32 þ 3 ~0 ~ 0 gþE D 7 6 7 7 6 7 76 6 ~þ 7 0 76 Q 7 ¼ 6 0 7 6 7 6 7 ~0 7 ~ 5 k 4g E 5 54 D ~ ω ω δ þ iγ 0 Q Q
dx.doi.org/10.1021/nl200197j |Nano Lett. 2011, 11, 1685–1689
Nano Letters
Here κ( values are the coupling parameters of the perturbative term for the in-phase and out-of-phase hybrid pairs, which are determined by the structural offset “s”. δ( values are the small detuning
of the frequencies of in-phase and out-of-phase hybrid ( ( mode pairs (δ( = (E( D EQ )/p). g values are the geometrical parameters that define the coupling efficiency of the dipolar hybrid modes (D() to
the external field. Equation 4 represents two coupled Lorentzian oscillator pairs corresponding to in-phase and out-of-phase hybridized modes of the whole structure. The external field (E0) drives the
bright modes in each meta-atom, which are subsequently coupled to the dark modes (through κ(). With these equations the amplitudes of the dipolar hybrid states (D() can be derived as ~( ¼ D
~0 ðω ω( δ( þ iγ( g ( E QÞ ( ( ( 2 ðω ω( þ iγ( D Þðω ω( δ þ iγQ Þ ðk Þ
The complex amplitudes of the corresponding modes (given in eq 5) are directly proportional to the polarizability of the modes, which governs the spectral characteristics of the plasmonic structure.
The overall spectral response is given by the superposition of these amplitudes. The close agreement between this analytical derivation and the FDTD analysis confirms the validity of our perturbative
model as shown in Figure 3a (dashed
Figure 4. Coupled three-level system model for multispectral plasmon induced transparency. Coupled meta-atoms have four states which form a five-level system with the continuum.
curve). The physical principles leading to multispectral EIT-like behavior can be equivalently observed in other structures. As an example, we implemented our approach with multilayered dolmen
structures,6 and obtained a clear multispectral EIT-like behavior (see Supporting Information). Equation 5 is in close analogy to atomic physics, where the investigated atomic absorption cross
section are given with a similar formula.1 This analogy allows us to illustrate multispectral EIT phenomena in our composite structure with five-level state diagram as shown in Figure 4. It is
important to note that these eigenstates are strongly correlated, since they are a linear combination of the same basis sets (D0,Q0) as shown in the hybridization diagram in Figure 2a. Experimental
verification of this novel phenomenon is demonstrated using a lift-off free fabrication method that results in simultaneous patterning of multilayered slot antennas.26 In our fabrication scheme, we
start with a free-standing membrane, which is patterned with nanoapertures using e-beam lithography and dryetching.27,28 Subsequent metal deposition on both sides with a highly directional e-beam
evaporation results in multiple stacks that are automatically registered with respect to each other in the xyplane. Similarly, this fabrication scheme can be extended to fabricate devices with an
even larger number of layers.29 Crosssectional scanning electron microscope (SEM) image of the final structure shows negligible metal covering at the inner side walls (inset to Figure 5d). Spectral
data collection is done with a Bruker IFS 66/s Fourier transform infrared (FTIR) spectrometer with a Hyperion 1000 IR microscope in reflection mode. In measurements obtained from the single-layered
structure, a clear EIT-like spectral response is observed at a single frequency (∼100 THz, Figure 5c). On the other hand, experimental measurements obtained from the double-layered structure reveal
two EIT peaks as predicted by the analytical relations (Figure 5d). The length of the dipolar slot antenna is 700 nm, and its width is 125 nm. The quadrupolar antenna lengths are 900 nm with a same
width of 125 nm. The small in-plane separation between the dipolar and the quadrupolar antenna is 60 nm. The thicknesses of the deposited gold films are 30 nm on both sides with a dielectric layer of
70 nm in between. A structural asymmetry (s) of 135 nm is introduced to enable the excitation of hybrid quadrupolar modes. In conclusion, we presented a method to extend the EIT-like phenomena to
multiple spectral positions by tailoring the near-field
Figure 5. (a) Illustration of the double-layered structure fabrication on a free-standing membrane. (b) SEM image of an array is shown. Reflection spectra of the symmetric (s = 0) and asymmetric (s 6¼
0) for (c) single-layered and (d) double-layered structures are shown. A model fit based on Lorentzian harmonic oscillators is shown for the double-layered asymmetric structure’s spectrum (red dashed
curve with κ = 9.7 THz, κþ = 27.4 THz).30 The dipolar slot antenna length is 700 nm and the quadrupolar antenna lengths are 900 nm, all antenna widths are kept at 125 nm. The gap between the dipolar
and quadrupolar antennas is 60 nm. Periods are 1200 nm on both x and y directions. Both Au layers are 30 nm with a 70 nm dielectric layer in-between. Inset in (d) shows the cross section image of the
double-layered structure. Coverage of the sidewalls due to the metal deposition is minimal. 1688
dx.doi.org/10.1021/nl200197j |Nano Lett. 2011, 11, 1685–1689
Nano Letters coupling of meta-atoms in a multilayered metamaterial system. In particular, two near-field interaction mechanisms make this phe~) nomena possible; (i) hybridization of plasmonic
resonances (Σ ~ ). The and (ii) interaction between the bright and dark antennas (K method is demonstrated experimentally and theoretically with planar slot antenna based multilayered metamaterial
systems. For experimental demonstration, a lift-off free fabrication scheme that can simultaneously register multiple metamaterial layers is introduced. The provided analytical investigations are kept
general. Therefore, our method can be easily extended to other antenna geometries31 (see Supporting Information) as well as scaled to a larger number of metamaterial layers.
Supporting Information. Additional information regarding electric field distributions and enhancements at plasmon induced reflection peaks and multispectral plasmon induced transparency with
nanoparticles. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
[email protected]
. Author Contributions †
These authors contributed equally
’ ACKNOWLEDGMENT This work is supported in part by NSF CAREER Award (ECCS-0954790), ONR Young Investigator Award, Massachusetts Life Science Center New Investigator Award, NSF Engineering Research
Center on Smart Lighting (EEC-0812056), Boston University Photonics Center, and Army Research Laboratory. ’ REFERENCES (1) Boller, K. J.; Imamoglu, A.; Harris, S. E. Observation of
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Genov, D. A.; Bartal, G.; Zhang, X. Three-dimensional optical metamaterial with a negative refractive index. Nature 2008, 455, 376–379. (30) Other used model parameters are, ωþ/ω= 131/50 THz, δþ/δ =
14.9/4.8 THz, γQþ/γQ = 12.19/4.49 THz, γDþ/γD = 45.7/11.9 THz. (31) Cubukcu, E.; Kort, E. A.; Crozier, K. B.; Capasso, F. Plasmonic laser antenna. Appl. Phys. Lett. 2006, 89, 093120-22. 1689
dx.doi.org/10.1021/nl200197j |Nano Lett. 2011, 11, 1685–1689 | {"url":"https://datapdf.com/multispectral-plasmon-induced-transparency-in.html","timestamp":"2024-11-05T03:20:53Z","content_type":"text/html","content_length":"54281","record_id":"<urn:uuid:2a791927-3766-4130-9cdd-4d0ba37df189>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00151.warc.gz"} |
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1. A variable force of 7x^-2 pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from x=1 ft to x=18 ft. (Round your answer
to two decimal places.)
2. A force of 14lb is required to hold a spring stretched 2 in. beyond its natural length. How much work W is done in stretching it from its natural length to 8 in. beyond its natural length?
3. A spring has a natural length of 22cm. If a 22-N force is required to keep it stretched to a length of 40 cm, how much work W is required to stretch it from 22 cm to 31 cm? (Round your answer to
two decimal places.)
4. Suppose that 6J of work is needed to stretch a spring from its natural length of 24 cm to a length of 45 cm.
4. If 24 J of work are needed to stretch a spring from 13 cm to 17 cm and 40 J are needed to stretch it from 17 cm to 21 cm, what is the natural length of the spring? | {"url":"http://askhomework.com/7-6/","timestamp":"2024-11-13T19:22:58Z","content_type":"text/html","content_length":"61471","record_id":"<urn:uuid:b948da3d-4798-4cbe-8486-9f2421a1a772>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00119.warc.gz"} |
Time In Hinduism: The Yuga | Sri Deva Sthanam
The 4 Yugas
Hinduism perceives time to flow in great cycles called yugas. Time is circular. There are four yugas and depending upon the yuga the duration varies. The four yugas along with their duration in
earthly years are:
Satya Yuga 1,728,000
Treta Yuga 1,296,000
Dvapara Yuga 864,000
Kali Yuga 432,000
Total 4,320,000 One yuga cycle.
I say earthly years because Hindu scripture gives the years as divine years–time according to the calculation of the Gods. Notice that the basic number is 432,000, the age of Kali yuga, and so
dvapara is twice that number, treta is 3 times that number and satya is four times that number. One rotation of these four yugas is called a yuga cycle which is a total of 4,320,000 years. A thousand
yuga cycles is called a kalpa and therefore a kalpa is 4,320,000,000 years. Time moves on in these great cycles, yuga after yuga, kalpa after kalpa, eternally.
To give an example how such huge numbers are used, consider the lifetime of Brahma, the four headed creator God. Brahma’s life span is calculated according to yuga time. One kalpa is said to be the
12 hours of Brahma’s day, so his 24 hour day is two kalpas in length. That means 24 hours of Brahma’s time is 8,640,000,000 earthly years! His year is 365 days long and he lives for a 100 years. I
leave it to my readers to do the math. My calculator does not have enough places to calculate the vast lifetime of Brahma. I am amazed by the size of the numbers that the early Hindu thinkers were
dealing with.
Now consider the story of one hairy sage. This is a story taken from one of the Puranas. A hairy sage once showed up in the court of King Indra and when asked where he lived the sage replied that
since life was so short he had decided not to marry and so did not to have a home. Indra then asked him why he had such a strange bald spot on his chest where hair was obviously falling out. The sage
replied that each time a Brahma died he lost one hair from his chest and this why he has becoming bald. The sage further asserted that once he had lost all of this hair from the death of so many
Brahma’s he too would die. And you can be sure he was a very hairy sage! Add to this the idea that within Hinduism there is not just one universe, but endless numbers of universes all with their own
Brahmas that come and go like moths rushing into a fire and you get a sense of time within Hinduism. These anecdotes give us an understanding of the massive time frames in which the Hindu mind has
conceived of time. Compare this with the Biblical story of Genesis where God created the world in seven days and you see the different conceptions of time between the two cultures.
Vishwamitra and Menaka
It is said that we are now living within the Kali yuga, which started about 5000 years ago. Each of these yugas is said to have a certain quality of life. Kali yuga is the worst of times because it
is the time of quarrel and deceit. The level of morality and spirituality is greatly decreased and the maximum span of life one can expect is only 100 years. In the previous yuga, Dvapara Yuga, life
is said to have been much better. The lifetime of a human being during the Dvapara Yuga could be as much as 1000 years. Life was more vibrant and spirituality was greatly increased. It is described
how a human being stood as much as 12 feet tall and how the trees and animals are much larger as well. The Treta yuga was an even better time with the maximum life span as much 10,000 years.
Spirituality is even higher, and finally, in the best of times, the Satya Yuga the life of a human being could be up to 100,000 years! The Mahabharata and the Puranas are full of stories from the
various yugas describing scenes of people living for huge periods of time. The great sage Vishwamitra, for example, mediated in water for 60,000 years before his meditation was broken by the
beautiful Menaka. Their union brought about the famous Shakuntala, the heroin of many famous stories and plays in Sanskrit. Similarly, in many of thePuranas, ten avataras of Vishnu are said to
repeatedly appear throughout the yuga cycles. Rama always appears during the Treta Yuga, Krishna appears at the end of Dvapara, and Kalki, the final avatara comes at the end of Kali Yuga to destroy
all things and prepare the way for the next Satya Yuga. There are, therefore, many appearances of Rama, Krishna, and the other avataras. This is all takes place within the great cycles of yugatime. | {"url":"https://sanskrit.org/time-in-hinduism-the-yuga/","timestamp":"2024-11-06T20:14:08Z","content_type":"text/html","content_length":"39109","record_id":"<urn:uuid:07693216-7996-459d-91c1-017785816a0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00597.warc.gz"} |
March 14 is Pi Day!!!
March 14 is Pi Day!!!
It is during Spring Break. But never fear, we WILL be celebrating Pi Day – just on March 21 not March 14!
Come to our Pi Day Celebration from 3:14 until 5:00 p.m. on March 21st in the Seamans Center Student Commons and enjoy free pie bites, lemonade, and coffee!
Then get a team together – or pull in spectators – and compete in trivia contests! Show off your knowledge of Pi and pie!!
Pi is one of the most famous and mysterious of numbers. Defined as the ratio of the circumference of a circle to it’s diameter, Pi seems simple. However, it turns out to be an irrational number.
Because it is irrational it cannot be expressed exactly as a fraction and the decimal representation therefore never ends, nor does it ever settle into a permanent repeating pattern. Scientists have
calculated billions of digits of Pi, starting with 3.14159265358979323…. with no end in sight. It could be calculated to infinity and there would be absolutely no way to know which number would come
Want to see what 100,00 digits of Pi look like? Go here.
Mark your calendar and come to Pi Day – March 21st, 3:14 to 5:00 p.m. in the Seamans Center Student Commons. Don’t forget we have free apple pie bites (while supplies last), lemonade, and trivia
Take a break on the first day back from spring break – you don’t want to miss out on all the fun!
Be there or be square!!
Happy Pi Day Domino Spiral, 10,059 dominoes were used in this Pi Day Spiral.
Adrian, Y. E. O.. The pleasures of pi,e and other interesting numbers. 2006. Singapore : World Scientific. Engineering Library QA95 .A2 2006
Happy Pi Day (3.14) Domino Spiral. March 13, 2011. youtube.com
Other Resources:
Stunning images transform Pi into circular rainbow-hued works of art. March 3, 2016. dailymail.com
Pi Jokes. 2016. Jokes4us.com
The Pi Song. Originally sung by Hard ‘N Phirm. Sept. 17, 2006. youtube.com | {"url":"https://blog.lib.uiowa.edu/eng/march-14-is-pi-day/","timestamp":"2024-11-08T17:03:28Z","content_type":"text/html","content_length":"74657","record_id":"<urn:uuid:d488920c-1002-41d4-8b33-7bef2a53a286>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00124.warc.gz"} |
ThmDex – An index of mathematical definitions, results, and conjectures.
Let $X = \prod_{n \in \mathbb{N}} X_n$ be a
D326: Cartesian product
such that
(i) $\pi_n$ is a D327: Canonical set projection on $X$ for each $n \in \mathbb{N}$
(ii) $E \subseteq X$ is a D78: Subset of $X$
Then $\{ x_n \}_{n \in \mathbb{N}} \in E$ if and only if $$\forall \, n \in \mathbb{N} : x_n \in \pi_n(E)$$ | {"url":"https://theoremdex.org/r/4601","timestamp":"2024-11-10T12:18:30Z","content_type":"text/html","content_length":"6865","record_id":"<urn:uuid:fbad9c0f-065d-4746-8fb9-159681f7f047>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00114.warc.gz"} |
Is the geometric series uniformly convergent?
Is the geometric series uniformly convergent?
As it should be intuitively expected the geometric series does not converge uniformly on |z| < 1. However, it does converge uniformly in any ball B(0,r) with r < 1 fixed.
What is uniform convergence of a series?
Uniform convergence of series. A series ∑∞k=1fk(x) converges uniformly if the sequence of partial sums sn(x)=∑nk=1fk(x) converges uniformly.
How do you prove uniform convergence of series?
Find an upper bound of N(ϵ, x). You can either solve for the value of x (possibly as a function of ϵ) that maximizes N(ϵ, x) or use some theorem like the triangle inequality. Set N(ϵ) to the upper
bound you found. If N(ϵ) is infinite for ϵ > 0, then you don’t have uniform convergence.
What is convergence geometric series?
A convergent geometric series is such that the sum of all the term after the nth term is 3 times the nth term.Find the common ratio of the progression given that the first term of the progression is
a. Show that the sum to infinity is 4a and find in terms of a the geometric mean of the first and sixth term.
What is convergence and uniform convergence?
Uniform convergence is a type of convergence of a sequence of real valued functions { f n : X → R } n = 1 ∞ \{f_n:X\to \mathbb{R}\}_{n=1}^{\infty} {fn:X→R}n=1∞ requiring that the difference to the
limit function f : X → R f:X\to \mathbb{R} f:X→R can be estimated uniformly on X, that is, independently of x ∈ X x\in X x∈ …
Which is used to measure the uniform convergence?
Many mathematical tests for uniform convergence have been devised. Among the most widely used are a variant of Abel’s test, devised by Norwegian mathematician Niels Henrik Abel (1802–29), and the
Weierstrass M-test, devised by German mathematician Karl Weierstrass (1815–97).
How do you prove not uniformly convergent?
If for some ϵ > 0 one needs to choose arbitrarily large N for different x ∈ A, meaning that there are sequences of values which converge arbitrarily slowly on A, then a pointwise convergent sequence
of functions is not uniformly convergent. if and only if 0 ≤ x < ϵ1/n.
Does uniform convergence imply differentiability?
6 (b): Uniform Convergence does not imply Differentiability. Before we found a sequence of differentiable functions that converged pointwise to the continuous, non-differentiable function f(x) = |x|.
Recall: That same sequence also converges uniformly, which we will see by looking at ` || fn – f||D.
How do you know if geometric series converges or diverges?
Geometric series: A geometric series is an infinite sum of a geometric sequence. Such infinite sums can be finite or infinite depending on the sequence presented to us. Note: If the series approaches
a finite answer, then the series is said to be convergent. Otherwise, it is said to be divergent.
How do you find if a series converges or diverges?
If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. where “lim sup” denotes the limit superior (possibly ∞; if the limit exists it is
the same value). If r < 1, then the series converges.
Why is uniform convergence important?
The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function.
How do you find the convergence of a geometric series?
Convergence of a geometric series. We can use the value of r r r in the geometric series test for convergence to determine whether or not the geometric series converges. The geometric series test
says that. if ∣ r ∣ < 1 |r|<1 ∣ r ∣ < 1 then the series converges. if ∣ r ∣ ≥ 1 |r|\\ge1 ∣ r ∣ ≥ 1 then the series diverges. YouTube.
What is the Weierstrass uniform convergence theorem?
nis a geometric series with 0<1, it converges. Hence by the Second Weierstrass Uniform Convergence Theorem (SWUCT), the convergence of the series P 1 n=0 t nis uniform on [1 +r;1r] and so it
converges to a function U on [1 +r;1r].
How does the geometric series ∑ I = 0 ∞ converge to 1-R?
The geometric series ∑ i = 0 ∞ a r i = a + a r + a r 2 + a r 3 + … converges to a 1 − r if − 1 < r < 1 and diverges otherwise. Warning: this value of the series is true only when the series begins
with i = 0, so that the first term is a.
What is the uniform convergence property?
It turns out that the uniform convergence property implies that the limit function , such as continuity, boundedness and Riemann integrability, in contrast to some examples of the limit function of
pointwise convergence. f ( x) = { 0, x ∈ [ 0, 1) 1, x = 1. . All of the functions f f is discontinuous. The convergence is not uniform. | {"url":"https://www.fdotstokes.com/2022/08/16/is-the-geometric-series-uniformly-convergent/","timestamp":"2024-11-10T08:26:52Z","content_type":"text/html","content_length":"58477","record_id":"<urn:uuid:d57d6fa4-7f8d-440d-89ed-18d3004d4881>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00734.warc.gz"} |
Hierachical Multiscale Methods on Nonuniform Grids
Earlier work has hown that the multiscale method performs excellent on highly heterogeneous cases using uniform coarse grids. To improve the accuracy of the multiscale solution even further, we have
introduce adaptive strategies for the coarse grids, based on either local hierarchical refinement or on adapting the coarse grid more directly to large-scale permeability structures of arbitrary
shape. The resulting method is very flexible with respect to the size and the geometry of coarse-grid cells, meaning that grid refinement/adaptation can be performed in a straightforward manner
Use Grid Adaptivity to Increase Accuracy
Consider a quarter-five spot simulation defined over a heterogeneous permeablility field containing a set of barriers (with permability equal 1e-8 relative to the mean of the background permeability
field). This example has been tailor-made to cause difficulties for the multiscale method.
To solve the flow problem, we employ the multiscale mixed finite-element method on a 6 x 6 coarse grid. Several of the grid cells are cut into two noncommunicating parts by the barriers. In the
figure below, we see that the method fails to flood the corresponding cells properly. The reason is the following: when generating the multiscale basis functions, unit flow is forced through each
cell. Therefore the velocity basis functions will model an unphysically high flow, for cells that are cut in two halves by a traversing barrier. As a result, the contribution from each such basis
function becomes very small in the solution of the global flow problem on the coarse grid and the flux into the cell is zeroed out.
Permeability field with low-permeable barriers Reference saturation field computed using the Saturation field computed using the multiscale
inposed on a lognormal background permeability velocity computed on the underlying fine grid velocity approximation
Improved accuracy can be obtained by the following two approaches:
1. Automatic hierarchical refinement based upon a given error indicator that measures the ratio of velocities over effective permeabilities.
2. Manually adding extra (irregular) coarse cells that contain the barriers.
In the figure below we see that both approaches are able to reproduce the saturation with high accuracy. The best accuracy is obtained by adding barriers as coarse cells, even though this gives much
fewer coarse cells than for the hierarchical approach.
If the barriers are removed, all three grids above give the same accuracy, indicating that the choice of coarse grid has little influence for a smoothly varying permeability field.
Automatic refinement, where the coarse grid is refined hierarchically around the Direct inclusion of the flow barriers as special blocks in the
flow barriers coarse grid
General Unstructured Coarse Grids in 3D
The two approaches introduced above can easily be extended to three spatial dimensions. In the figure below, we consider a 30 x 80 x 10 subsample of the Tarbert formation from the second SPE10 model,
in which we have introduced a few low-permeable walls (1e-8 mD). To improve the accuracy of the multiscale solution, we can either perform a hierarchical refinement or search for connected cells of
very low permeability and add them as extra coarse cells to a uniform background grid.
The upper row shows the case consists of the Tarbert formation from the SPE 10 test case (the 30 first layers) into which we have inserted
low-permable flow barriers. The lower row shows a hierarchically refined grid and a grid where the low-permeable barriers are included as extra
coarse blocks in a uniform coarse partition. The right plot shows one perticular coarse block that has been cut through by a flow barrier.
Multiscale Methods - Great Flexibility in Gridding
The examples above indicate the great flexibility inherent in the multiscale method: each coarse grid cell can be defined (almost) arbitarily as a connected set of fine-grid cells. For permeability
fields with relatively smooth variation (but possibly with large variations), the multiscale method is not very sensitive to the shape and size of the cells in the coarse grid. For nonsmooth
permeabilities, on the other hand, higher accuracy is obtained by a careful choice of the coarse grid.
1. J. E. Aarnes, S. Krogstad, and K.-A. Lie. A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids. Multiscale Modelling and Simulation,
Vol. 5, No. 2, pp. 337-363, 2006. DOI: 10.1137/050634566
2. J. E. Aarnes, S. Krogstad, and K.-A. Lie. Non-uniform coarse grids and multiscale mixed FEM. SIAM Geosciences 05, Avignon, France, June 7-10, 2005. (slides)
Earlier work has hown that the multiscale method performs excellent on highly heterogeneous cases using uniform coarse grids. To improve the accuracy of the multiscale solution even further, we have
introduce adaptive strategies for the coarse grids, based on either local hierarchical refinement or on adapting the coarse grid more directly to large-scale permeability structures of arbitrary
shape. The resulting method is very flexible with respect to the size and the geometry of coarse-grid cells, meaning that grid refinement/adaptation can be performed in a straightforward manner
Consider a quarter-five spot simulation defined over a heterogeneous permeablility field containing a set of barriers (with permability equal 1e-8 relative to the mean of the background permeability
field). This example has been tailor-made to cause difficulties for the multiscale method.
To solve the flow problem, we employ the multiscale mixed finite-element method on a 6 x 6 coarse grid. Several of the grid cells are cut into two noncommunicating parts by the barriers. In the
figure below, we see that the method fails to flood the corresponding cells properly. The reason is the following: when generating the multiscale basis functions, unit flow is forced through each
cell. Therefore the velocity basis functions will model an unphysically high flow, for cells that are cut in two halves by a traversing barrier. As a result, the contribution from each such basis
function becomes very small in the solution of the global flow problem on the coarse grid and the flux into the cell is zeroed out.
Permeability field with low-permeable barriers Reference saturation field computed using the Saturation field computed using the multiscale
inposed on a lognormal background permeability velocity computed on the underlying fine grid velocity approximation
Permeability field with low-permeable barriers inposed on a lognormal background permeability
Reference saturation field computed using the velocity computed on the underlying fine grid
Improved accuracy can be obtained by the following two approaches:
In the figure below we see that both approaches are able to reproduce the saturation with high accuracy. The best accuracy is obtained by adding barriers as coarse cells, even though this gives much
fewer coarse cells than for the hierarchical approach.
If the barriers are removed, all three grids above give the same accuracy, indicating that the choice of coarse grid has little influence for a smoothly varying permeability field.
Automatic refinement, where the coarse grid is refined hierarchically around the Direct inclusion of the flow barriers as special blocks in the
flow barriers coarse grid
Automatic refinement, where the coarse grid is refined hierarchically around the flow barriers
Direct inclusion of the flow barriers as special blocks in the coarse grid
The two approaches introduced above can easily be extended to three spatial dimensions. In the figure below, we consider a 30 x 80 x 10 subsample of the Tarbert formation from the second SPE10 model,
in which we have introduced a few low-permeable walls (1e-8 mD). To improve the accuracy of the multiscale solution, we can either perform a hierarchical refinement or search for connected cells of
very low permeability and add them as extra coarse cells to a uniform background grid.
The upper row shows the case consists of the Tarbert formation from the SPE 10 test case (the 30 first layers) into which we have inserted
low-permable flow barriers. The lower row shows a hierarchically refined grid and a grid where the low-permeable barriers are included as extra
coarse blocks in a uniform coarse partition. The right plot shows one perticular coarse block that has been cut through by a flow barrier.
The examples above indicate the great flexibility inherent in the multiscale method: each coarse grid cell can be defined (almost) arbitarily as a connected set of fine-grid cells. For permeability
fields with relatively smooth variation (but possibly with large variations), the multiscale method is not very sensitive to the shape and size of the cells in the coarse grid. For nonsmooth
permeabilities, on the other hand, higher accuracy is obtained by a careful choice of the coarse grid. | {"url":"https://www.sintef.no/projectweb/geoscale/results/msmfem/hierachical-multiscale-methods-on-nonuniform-grids/","timestamp":"2024-11-10T09:11:46Z","content_type":"text/html","content_length":"19462","record_id":"<urn:uuid:45dc6222-6105-4771-b7ce-05303b7128b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00140.warc.gz"} |
ekameters to Yards
Dekameters to Yards Converter
β Switch toYards to Dekameters Converter
How to use this Dekameters to Yards Converter π €
Follow these steps to convert given length from the units of Dekameters to the units of Yards.
1. Enter the input Dekameters value in the text field.
2. The calculator converts the given Dekameters into Yards in realtime β using the conversion formula, and displays under the Yards label. You do not need to click any button. If the input
changes, Yards value is re-calculated, just like that.
3. You may copy the resulting Yards value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Dekameters to Yards?
The formula to convert given length from Dekameters to Yards is:
Length[(Yards)] = Length[(Dekameters)] / 0.09144
Substitute the given value of length in dekameters, i.e., Length[(Dekameters)] in the above formula and simplify the right-hand side value. The resulting value is the length in yards, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a high-rise building stands 25 dekameters tall.
Convert this height from dekameters to Yards.
The length in dekameters is:
Length[(Dekameters)] = 25
The formula to convert length from dekameters to yards is:
Length[(Yards)] = Length[(Dekameters)] / 0.09144
Substitute given weight Length[(Dekameters)] = 25 in the above formula.
Length[(Yards)] = 25 / 0.09144
Length[(Yards)] = 273.4033
Final Answer:
Therefore, 25 dam is equal to 273.4033 yd.
The length is 273.4033 yd, in yards.
Consider that a luxury yacht has a length of 15 dekameters.
Convert this length from dekameters to Yards.
The length in dekameters is:
Length[(Dekameters)] = 15
The formula to convert length from dekameters to yards is:
Length[(Yards)] = Length[(Dekameters)] / 0.09144
Substitute given weight Length[(Dekameters)] = 15 in the above formula.
Length[(Yards)] = 15 / 0.09144
Length[(Yards)] = 164.042
Final Answer:
Therefore, 15 dam is equal to 164.042 yd.
The length is 164.042 yd, in yards.
Dekameters to Yards Conversion Table
The following table gives some of the most used conversions from Dekameters to Yards.
Dekameters (dam) Yards (yd)
0 dam 0 yd
1 dam 10.9361 yd
2 dam 21.8723 yd
3 dam 32.8084 yd
4 dam 43.7445 yd
5 dam 54.6807 yd
6 dam 65.6168 yd
7 dam 76.5529 yd
8 dam 87.4891 yd
9 dam 98.4252 yd
10 dam 109.3613 yd
20 dam 218.7227 yd
50 dam 546.8066 yd
100 dam 1093.6133 yd
1000 dam 10936.133 yd
10000 dam 109361.3298 yd
100000 dam 1093613.2983 yd
A dekameter (dam) is a unit of length in the International System of Units (SI). One dekameter is equivalent to 10 meters or approximately 32.808 feet.
The dekameter is defined as ten meters, providing a convenient measurement for moderately large distances.
Dekameters are used in various fields to measure length and distance where a scale between meters and hectometers is appropriate. They are less commonly used than other metric units but can be useful
in certain applications, such as land measurement and environmental science.
A yard (symbol: yd) is a unit of length commonly used in the United States, the United Kingdom, and Canada. One yard is equal to 0.9144 meters.
The yard originated from various units used in medieval England. Its current definition is based on the international agreement of 1959, which standardized it to exactly 0.9144 meters.
Yards are often used to measure distances in sports fields, textiles, and land. Despite the global shift to the metric system, the yard remains in use in these countries.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Dekameters to Yards in Length?
The formula to convert Dekameters to Yards in Length is:
Dekameters / 0.09144
2. Is this tool free or paid?
This Length conversion tool, which converts Dekameters to Yards, is completely free to use.
3. How do I convert Length from Dekameters to Yards?
To convert Length from Dekameters to Yards, you can use the following formula:
Dekameters / 0.09144
For example, if you have a value in Dekameters, you substitute that value in place of Dekameters in the above formula, and solve the mathematical expression to get the equivalent value in Yards. | {"url":"https://convertonline.org/unit/?convert=dekameters-yards","timestamp":"2024-11-05T20:12:10Z","content_type":"text/html","content_length":"90481","record_id":"<urn:uuid:cfae3083-b318-40f5-9674-14f827dee952>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00338.warc.gz"} |
Using Teaching Textbooks in a Living Math Approach
Sprite is using her third set of Textbooks (TT) math curriculum, and we have no intention of switching to anything else. It is really working for her (and for me). We both love that it can be done
independently and that she receives immediate feedback on her answers. Because it has both auditory and visual components (besides the normal explanations in the book), it is a good match for her
learning style.
When we initially switched to TT, I called myself a living math dropout. Of course, I wasn’t truly giving up on a living math approach simply by using Teaching Textbooks. And I’ve found that TT can
be used as part of a living math approach.
Talk About Math
One of the dangers of TT is that you may totally give over the math instruction to the CD-roms and altogether lose contact with what is being learned. It’s easy to do; I’ve found myself in this
“Doing your math? Great!”
[Sprite does the lesson.]
“What was your score? 94? Wonderful! Now let’s do science.”
That’s not really wise for us. When I am not keeping up with the math, it’s harder for me to help her when the need arises. And I’ve found that it’s important for Sprite to verbalize what she learns.
Yes, even in math you can use narration.
So at least once or twice a week, I make a point to ask specifically what her math lessons are covering. If she can’t tell me, that’s a sign of a problem. But just telling me the topic isn’t enough,
either. Then I follow up with a question — “And how do you do that?” Normally these math check-ins are brief and give me reassurance that Sprite is really understanding. As a plus I can see where the
lessons are headed and learn any techniques that are different from how I was taught.
Write About Math
For the most part we stick with oral narrations, but when a TT lesson results in a bad score, I like to take our talk to another level with math notebooking.
Before Sprite reworks the problems she missed, we talk about the concepts in the lesson. I ask probing questions and make her define math vocabulary terms. I help walk her through the math rules as
she writes on a notebooking page. Her page may include sample problems, diagrams, and charts along with text.
The notebooking page often turns out to be a handy reference for reworking the problems she missed the first time. The effort of explaining the math in words normally corrects faulty thinking and
refocuses Sprite’s attention on the key concepts.
Add on Math History Lessons
Lastly, we can add on the math history from livingmath.net. To be honest, we haven’t spent any time on our mathematicians in many months. But once we wrap up Ancient Rome, I anticipate having a bit
more time to devote to math history.
At any rate, the point is that you can incorporate biographies of mathematicians into your use of Teaching Textbooks so that you have a holistic approach to math.
I know that a lot of my readers use Teaching Textbooks, too. Do you consider TT part of a living math approach? And if so, how do you make TT more living?
1. Melissa says
“…we have no intention of switching to anything else.” Don’t you just love the feeling that comes with discovering something that fits your goals, your preferred method, your child, and YOU!?!? 🙂
Love how you’ve put all of this together in this one brilliant post!
2. Mary says
Thank you for this. I am wanting to switch to TT with my oldest next year, but I didn’t want to be totally “hands off” with math… I love the idea of living math and you have given concrete ideas
to help us.
Jimmie to the rescue again!
3. Dawn says
We just switched to TT and love it. My kids ask to do extra lessons all of the time. We will be ready for the next level in no time.
4. Lisa says
Just wondering if you’d share how you keep up with her lessons to know that she’s really understanding it correctly? Do you read through her lessons periodically or each day? I’ve been wondering
this because we’re looking into using TT next year, but I don’t want to lose my ability to help him when he gets stuck! Thanks so much for this post! I love the idea of math narration and using
it in a living math kind of way!
5. Kathi says
We have really enjoyed TT too, and it has kept my children as well as me, on track. We are covering fractions, decimals, and per cents, so if we have a trouble spot, we pull out the correlating
parts of the workbooks “Keys to Decimals,” or “Keys to Fractions,” etc. I got these very inexpensively through CBD, and I find these very tiny incremental approaches excellent to helping ME to
help them work through a tough place. We stop, sit at the table for a while, and work through these together, then go back to Teaching Textbooks. If we need to work out practice problems at the
white board, then we have some fun with that too. We could probably do this part on a notebook page and it would be a handy reference, like you said. I find the combination of their independent
work on TT and our together-time with the Keys every so often, is a good pairing. It also keeps me in touch with the details of HOW they’re doing what they’re doing. And yes, I ask questions
about what they’re doing, and it’s most enlightening to find what they understand or don’t. When they had a lesson on credit cards, they got the concept backwards of who pays whom! We certainly
went over that one orally. 🙂
6. Amber says
That’s all I use for my daughter and I absolutely love it!! 🙂
7. Dana Wilson says
Teaching Textbooks was WONDERFUL for our high school math. And we sure worked through several math curricula before we discovered that one!
We incorporated all subjects, pretty much, into our history studies, so we read biographies of scientists, mathematicians, composers, etc. But I must confess by the time they were in high school
they were doing so much reading in other subjects, we kind of let the math-history connection drop, sadly.
Ahh, we just have to pick and choose sometimes, right?
8. Jackie says
We use TT as a supplement to our other resources. My daughter who doesn’t exactly like school, really enjoys it. I know lots of folks think TT is a little behind some other resources, but I like
that it doesn’t push too much too soon.
A homeschool Mom who enjoys blogging.
9. Catherine says
Hello Jamie,
I enjoyed reading this post. We haven’t tried TT, but your point about keeping in step with the child’s learning is very valid. I found that we were slipping into the pattern of ‘Okay, you’ve
done maths. Good score! Now let’s do science.’ I really don’t like being in touch, so I’m making sure I’m across what’s being covered before the children begin work, and then follow up later.
LOVE using living math books, too. We’re about to start ‘Mathematicians are People Too’. They just arrived in the post the other day, so I’ll be reading those over the week. 🙂
10. Alicia S. says
We absolutely had this issue with T.T. It felt like my daughter was the only one on Earth who didn’t love this program. It’s hailed as such a refreshingly fun program, but she hated it. I started
the program doing every lesson with her, but ended up being more of a distraction than an aid, because the lessons were all so easy for her in the beginning. By the time she did need my help, I
was out of the loop. I’d have to read through the lesson myself before even knowing where to begin without showing her a conflicting method.
One thing we did for a while was pick one of her most troublesome lessons that week and make a lapbook for it. It forced her to not only review until she understood the material well enough to
convey, but also to slow wayyyy down. Sometimes she rewound the lesson four or five times in order to illustrate each step without forgetting anything. Unfortunately, when she started to catch up
again, I let her stop. (I really wish I hadn’t.)
Now that this is our last week of every other course, we’re taking the first few weeks of summer to really focus on each problem lesson in math until she’s confidently caught up. I plan to walk
her through each lesson myself and offer her more practice problems whenever necessary before moving on to actually redoing the lessons in the program.
We actually only homeschooled this year for the first time in order to get her caught up without the major distraction of friends, boys and fighting that she had at public school. She’s going
back next year monumentally ahead of where she would have been in every other subject, thanks to homeschooling – except math. I’d love to supplement over the summer with other resources that we
could keep around even after she returns to public school.
11. Shannon says
Thank you so much for this. I am going to start asking my boys to narrate their lessons back to me. What a wonderful way to know that they have have really absorbed the information. I love the
idea of a math notebook too. I might have to start that this year.
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Free Calculus Questions and Problems with Solutions
Free calculus tutorials including problems and questions with solutions are presented.
Calculus Problems and Questions
Calculus Questions, Answers and Solutions
Limits and Continuity
Differentiation and Derivatives
Application of Differentiation
Integrals of Power of Trigonometric Functions
Differential Equations
Parametric Equations and their Applications
Multivariable Functions (Functions with several variables)
Tables of Mathematical Formulas
Interactive Tutorials
More Links and References | {"url":"https://www.analyzemath.com/calculus.html","timestamp":"2024-11-07T00:27:29Z","content_type":"text/html","content_length":"51737","record_id":"<urn:uuid:fac22992-1b34-4dde-8b69-878b698b96c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00304.warc.gz"} |
Automated Market Makers (AMMs) & more...
Until Now,
The exchange markets worked on a trade execution model called "CLOB" or the Central Limit Book Order. This model is transparent but slow and requires a middleman. The middleman (in our case, the
exchanges / derivative / equity markets) match the orders in real-time, which means that the buyer Alice, posts a buy order for the highest price she is willing to pay. The seller, Bob, posts the
lowest price he will sell for. The moment these prices match, a transaction is executed.
When Decentralized Exchanges came in with the idea of efficiency, ease of use, and better capital efficiency - they brought with them the Automated Market Makers. What do these do? They remove any
intermediaries and introduced liquidity pools and algorithms to balance the balance. Before moving forward let's understand what is a liquidity pool.
A Liquidity Pool:
is a digital pool where people lock up their cryptocurrencies, like tokens, to make trading easier. This pool acts like a market maker, allowing instant buying or selling without waiting for others.
When you trade, the pool gives you tokens from its stash, and you give back other tokens. The pool's prices can change based on how much of each token is in it. Many people can use the same pool
simultaneously, creating a busy trading environment. In return for putting tokens in, people get special tokens that represent their share of the pool - these tokens often allow you to participate in
governance for that token/project. Liquidity pools help make cryptocurrency trading faster and more efficient.
Okay, So how does an AMM really work?
We have already established the idea of a Liquidity Pool. An AMM is, in its most basic form - a formula:
X * Y = K
The main objective of an AMM is to ensure that the ratio of assets in the Liquidity Pool, remains balanced. This formula ensures that the product of token balances in a liquidity pool is constant,
ensuring that their relative value remains stable as trades occur. When the pool is created, AMM calculates the constant "k".
Now every trade that follows, AMM ensures that this ratio "k" is maintained throughout its lifetime. If a buyer, say Alice purchases an asset "X" in exchange for the asset "Y", asset "X" becomes more
expensive - basic supply and demand. Alice buys X, the volume of X in the pool goes down which means that the given number of Y tokens in the pool, now have to maintain the ratio with a lesser amount
of X thus increasing the price of X.
For example:
• Let's consider a pool of 50 LINK tokens and 1000 USDC tokens. The constant formula X * Y = k implies that 50 LINK * 1000 USDC = 50000.
• If a trader wishes to swap for 10 LINK tokens, the formula will get rearranged to calculate the value of "Y" from the initial formula, taking our "k" to be constant, resulting in:
Y = (10 * 5000) / (50 - 10) = 5000 / 40 = 1250 USDC
• Therefore, the price of 10 LINK tokens will be 1250 USDC.
• Since such transactions flow both ways.
Phew, that's all, right?
Well, not really. Apart from the CPMM variant already discussed, Decentralized Exchanges (or DEXes) have come up with variations of AMM(s) as follows:
• Constant Sum Market Maker (CSMM): In CSMM, the sum of token balances in the liquidity pool remains constant. This type of AMM aims to maintain the ratio between the two tokens while allowing the
sum of their values to fluctuate.
• Constant Mean Market Maker (CMMM): CMMM seeks to maintain a constant average price of the two tokens in the liquidity pool. As one token's price changes, the pool rebalances to keep the average
price steady.
• Advanced Hybrid Constant Function Market Maker (CFMM): CFMM combines features of different AMM models to optimize trading efficiency and minimize impermanent loss. It may use various mathematical
functions to determine pricing and rebalancing.
• Dynamic Automated Market Maker (DAMM): DAMM adjusts the pricing formula dynamically based on factors like market volatility, liquidity levels, and trading volume. This approach aims to provide
better price accuracy during periods of high market activity.
There are some challenges to AMMs as well. One of them is "Impermanent Loss".
In very simple terms, Impermanent Loss refers to the price difference that a Liquidity Provider bears as compared to the scenario if he held the token (pair) individually rather than providing it as
Let's say we provide liquidity for a trading pair involving tokens M and S. For instance:
Token M: $100
Token S: 10 (Valued at $10 / each)
• Considering initial value of token M to be $1 and token S to be $10, the total value of the pool is $200.
• Now let's say the value of token S increases by $2 (making S to be valued at $12) on some other exchange, while M remains at $1 and a trader, Bob, buys 2 S tokens from our pool at a lower price
($10) and sells it on the other exchange for a $2 profit.
• Now updated balance in our pool is:
Token M: $ (100+20) = $120
Token S: 8 (Eff. Value = 8 * 12 = $96)
Total Value = 120 + 96 = 216
• Here, the Liquidity Provider made a profit of (216 - 200) = $16
• However, lets calculate the total value of the same if these were held outside of the Liquidity Pool.
Token M: $100
Token S: 10 ( 10 x 10 ) --> 10 ( 10 x 12 ) = $120
Total Value = 100 + 120 = $220
• So we can say that the Liquidity Provider suffered a loss of $4. This loss is termed as the "Impermanent Loss" and the reason it is Impermanent is because it can disappear if the prices revert.
As of now,
Multiple protocols use AMMs, some of the popular ones are UNISWAP, SushiSwap, Balancer and Curve Finance.
Uniswap's deployment of the Constant Product Market Maker introduced us to the concept, while SushiSwap extended its features, adding community-driven enhancements, introducing yield farming and
liquidity mining.
SushiSwap revolutionized how users engaged with AMMs. It encouraged liquidity providers to participate actively by not only sharing trading fees but also distributing additional SUSHI tokens as
rewards. The incorporation of community governance empowered SUSHI token holders to influence the platform's direction, creating a decentralized decision-making process.
Balancer offered novel multi-token pools, enabling greater customization, it allowed users to create and participate in customizable liquidity pools containing multiple tokens, as opposed to the
conventional two-token pairs. This innovation provided increased flexibility for liquidity providers and traders, enabling them to fine-tune their exposure and optimize their strategies.
Additionally, Balancer introduced a dynamic fee structure that allowed liquidity providers to set their own fees. Finally,
Curve Finance addressed stablecoin trading with specialized bonding curves. Curve specializes in providing extremely low-slippage trading for stablecoin pairs, making it particularly valuable for
users seeking to swap stablecoins with minimal price impact. The protocol achieves this through specialized bonding curves tailored to stablecoin assets, effectively reducing impermanent loss and
providing efficient price stability. Curve's unique design prioritizes liquidity provision for stablecoin pairs and has garnered significant adoption in yield farming and stablecoin trading
These platforms showcase the versatility and innovation of AMMs, each catering to specific needs and preferences within the DeFi ecosystem. As the DeFi space continues to expand, the scope of AMMs
promises to grow even broader, potentially spanning cross-chain compatibility, advanced trading strategies, and seamless integration with emerging technologies. The journey of AMMs is far from over,
and as they shape the future of decentralized exchanges, we eagerly anticipate the next chapters in their evolution.
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Unit conversion
Unit conversion
Convert between various distance, angular and time units.
Alias unitconvert
Domain 2D, 3D or 4D
Input type Any
Output type Any
There are many examples of coordinate reference systems that are expressed in other units than the meter. There are also many cases where temporal data has to be translated to different units. The
unitconvert operation takes care of that.
Many North American systems are defined with coordinates in feet. For example in Vermont:
+proj=pipeline \
+step +proj=tmerc +lat_0=42.5 +lon_0=-72.5 +k_0=0.999964286 +x_0=500000.00001016 \
+step +proj=unitconvert +xy_out=us-ft
Often when working with GNSS data the timestamps are presented in GPS-weeks, but when the data transformed with the helmert operation timestamps are expected to be in units of decimalyears. This can
be fixed with unitconvert:
+proj=pipeline \
+step +proj=unitconvert +t_in=gps_week +t_out=decimalyear \
+step +proj=helmert +epoch=2000.0 +t_obs=2017.5 ...
Distance units
In the table below all distance units supported by PROJ are listed. The same list can also be produced on the command line with proj or cs2cs, by adding the -lu flag when calling the utility.
Label Name
km Kilometer
m Meter
dm Decimeter
cm Centimeter
mm Millimeter
kmi International Nautical Mile
in International Inch
ft International Foot
yd International Yard
mi International Statute Mile
fath International Fathom
ch International Chain
link International Link
us-in U.S. Surveyor's Inch
us-ft U.S. Surveyor's Foot
us-yd U.S. Surveyor's Yard
us-ch U.S. Surveyor's Chain
us-mi U.S. Surveyor's Statute Mile
ind-yd Indian Yard
ind-ft Indian Foot
ind-ch Indian Chain
Angular units
In the table below all angular units supported by PROJ unitconvert are listed.
Label Name
deg Degree
grad Grad
rad Radian
Time units
In the table below all time units supported by PROJ are listed.
When converting time units from a date-only format (yyyymmdd), PROJ assumes a time value of 00:00 midnight. When converting time units to a date-only format, PROJ rounds to the nearest date at
00:00 midnight. That is, any time values less than 12:00 noon will round to 00:00 on the same day. Time values greater than or equal to 12:00 noon will round to 00:00 on the following day.
Label Name
mjd Modified Julian date
decimalyear Decimal year
gps_week GPS Week
yyyymmdd Date in yyyymmdd format | {"url":"https://proj.org/en/latest/operations/conversions/unitconvert.html","timestamp":"2024-11-09T22:34:25Z","content_type":"text/html","content_length":"22051","record_id":"<urn:uuid:79f07bc9-ff89-4b57-90d2-161240d39635>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00337.warc.gz"} |
The Interplay of K-Factor and Minimum Bend Radius in Precision Sheet Metal Fabrication - BIT
The Interplay of K-Factor and Minimum Bend Radius in Precision Sheet Metal Fabrication
• by Lion
In the realm of precision sheet metal fabrication, achieving optimal results hinges upon mastering the delicate balance between the k-factor and minimum bend radius. These two critical parameters
wield significant influence over the integrity and quality of formed parts, making it imperative for fabricators to understand their interplay and implications.
The Pitfalls of Sharp Bends: Understanding Plastic Deformity
One common challenge encountered in both sheet metal and plate industries is the propensity for parts to be designed with inside bend radii much tighter than necessary. This practice can spell
disaster in the press brake department, leading to cracking and plastic deformities on the outside surface of bends. Excessive stress during bending can cause fracturing and alter the bend allowance,
exacerbating dimensional errors in the final workpiece.
The Role of the K-Factor: Predicting Neutral Axis Shift
The k-factor, a fundamental constant in bending calculations, plays a pivotal role in predicting the behavior of sheet metal during forming operations. As the inside bend radius decreases, the
neutral axis shifts toward the inside surface of the bend. This shift is quantified by the k-factor, which reflects the percentage by which the neutral axis relocates inward during bending. By
accurately calculating the k-factor, fabricators can anticipate and mitigate potential challenges associated with material elongation and deformation.
Minimum Bend Radius: Material Considerations and Interpretation
The minimum bend radius is a critical parameter determined by the material’s properties rather than the punch radius used in bending operations. Misinterpretation of the term “minimum bend radius”
can lead to the erroneous selection of punch tools, resulting in sharp bends and creases in the material. This creasing phenomenon occurs when the punch nose penetrates the material, compressing the
inner area of the bend and altering the k-factor. Moreover, the ratio of bend radius to material thickness influences the tensile strain on the outer surface of the material, further impacting the
Factors Influencing Minimum Bend Radius
Figure 1: This generic k-factor chart, based on information from Machinery’s Handbook, gives you average k-factor values for a variety of applications. The term “thickness” refers to the material
thickness. A k-factor average of 0.4468 is used for most bending applications. PDF
Grain direction, material thickness, and hardness are additional factors that influence minimum bend radius and, consequently, the behavior of the k-factor. Anisotropy, or the directional dependence
of material properties, plays a crucial role in determining the angle and radius of bends, particularly when bending with or against the grain. Harder materials require larger inside radii to
accommodate greater tensile strain, reflecting Poisson’s Ratio in action.
Fine-Tuning the K-Factor: Considering Additional Ingredients
While the commonly accepted k-factor value of 0.4468 serves as a reliable baseline for many bending process applications, fabricators can achieve even greater precision by considering additional
factors. These factors include die width, coefficient of friction, y-factors, and the bending method employed (air bending, bottoming, or coining). By meticulously evaluating these variables and
calculating a k-factor tailored to the specific application, fabricators can enhance the accuracy and quality of formed parts.
Example: Determining Minimum Bend Radius for 0.25-in.-Thick Material
Suppose we have a sheet of material with a thickness of 0.25 inches and a tensile reduction of area percentage of 12%. We aim to calculate the minimum bend radius using the provided formula:
Minimum Bend Radius=(50Tensile reduction of area percentage−1)×Material ThicknessMinimum Bend Radius=(Tensile reduction of area percentage50−1)×Material Thickness
• Material Thickness (Mt) = 0.25 inches
• Tensile reduction of area percentage = 12%
Substituting these values into the formula
Minimum Bend Radius=(50/12−1)×0.25=(4.17−1)×0.25=3.17×0.25=0.7925inches
Therefore, for a material thickness of 0.25 inches and a tensile reduction of area percentage of 12%, the minimum bend radius is approximately 0.7925 inches.
Example: Determining Minimum Bend Radius for Material Less Than 0.25 inches Thick
Let’s consider another scenario where the material thickness is less than 0.25 inches, say 0.125 inches. We’ll use the modified formula:
Minimum Bend Radius=(50Tensile reduction of area percentage−1)×Material Thickness×0.1Minimum Bend Radius=(Tensile reduction of area percentage50−1)×Material Thickness×0.1
• Material Thickness (Mt) = 0.125 inches
• Tensile reduction of area percentage = 15%
Substituting these values into the formula:
Minimum Bend Radius=(50/15−1)×0.125×0.1=(3.33−1)×0.125×0.1=2.33×0.0125=0.029125inches
Therefore, for a material thickness of 0.125 inches and a tensile reduction of area percentage of 15%, the minimum bend radius is approximately 0.029125 inches.
These examples demonstrate how the provided formulas can be applied to determine the minimum bend radius for different material thicknesses and tensile reduction of area percentages.
In conclusion, the symbiotic relationship between the k-factor and minimum bend radius underscores the complexity and precision required in sheet metal fabrication. By understanding the nuances of
these parameters and their interactions, fabricators can navigate the intricacies of bending operations with confidence and precision. Armed with this knowledge, fabricators can optimize their
processes, minimize errors, and deliver superior products that meet the exacting standards of modern manufacturing.
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How to Decod Decimal Numbers: Dive into Place Values
Decimals are more than just dots in numbers; they help us represent fractions and parts of a whole. To truly understand decimals, it's essential to grasp the concept of place values. Just as with
whole numbers, every position in a decimal number has a specific value. Let's explore this further!
Place Values within Decimal Numbers
Example 1:
Decimal Number: \(2.345\)
To understand the place values:
– The digit \(2\) is in the ones place.
– The digit \(3\) is in the tenths place, representing \(3/10\) or \(0.3\).
– The digit \(4\) is in the hundredths place, representing \(4/100\) or \(0.04\).
– The digit \(5\) is in the thousandths place, representing \(5/1000\) or \(0.005\).
Detailed Answer:
Starting from the left of the decimal point, the place values are ones, tenths, hundredths, thousandths, and so on. So, \(2.345\) can be read as “two and three hundred forty-five thousandths.”
The Absolute Best Book for 5th Grade Students
Original price was: $29.99.Current price is: $14.99.
Example 2:
Decimal Number: \(7.89\)
To understand the place values:
– The digit \(7\) is in the ones place.
– The digit \(8\) is in the tenths place, representing \(8/10\) or \(0.8\).
– The digit \(9\) is in the hundredths place, representing \(9/100\) or \(0.09\).
Detailed Answer:
The number \(7.89\) can be read as “seven and eighty-nine hundredths.”
Remember understanding the place values within decimal numbers is crucial for various mathematical operations, such as rounding, comparing, and performing arithmetic. With a solid grasp of place
values, you’ll find working with decimals a breeze!
Practice Questions:
1. Identify the digit in the tenths place: \(4.56\)
2. What is the value of the digit in the hundredths place in \(3.72\)?
3. Which digit is in the thousandths place in \(0.913\)?
4. What is the value of the digit in the tenths place in \(5.48\)?
5. Identify the digit in the hundredths place: \(6.034\)
A Perfect Book for Grade 5 Math Word Problems!
Original price was: $29.99.Current price is: $14.99.
1. \(5\)
2. \(0.02\)
3. \(3\)
4. \(0.4\)
5. \(3\)
The Best Math Books for Elementary Students
Related to This Article
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10++ Graphing Absolute Value Functions Worksheet
2 min read
10++ Graphing Absolute Value Functions Worksheet. Function transformations in graphing absolute value equations5. Worksheets are graphing absolute value functions date period, graphing absolute
value, draw the graph,.
30 Graphing Absolute Value Functions Worksheet Education Template from smithfieldjustice.com
Students will use this graphing absolute value functions worksheet to graph absolute value functions. Graphing absolute value lesson resources. Section 3.7 graphing absolute value functions.
Students Will Use This Graphing Absolute Value Functions Worksheet To Graph Absolute Value Functions.
Worksheets are graphing absolute value, graphing absolute value functions date period, section 1 graphing absolute value name,. 15 pics about absolute value practice worksheet | literal equations,
math practice : Free collection of graphing absolute value functions worksheet pdf for students the variable in the absolute value bars makes up an essential algebraic function known as the absolute.
Section 3.7 Graphing Absolute Value Functions.
Find the range of the function 𝑓 ( 𝑥) = | − 2 𝑥 − 2 |. Students will use this graphing absolute value functions worksheet to graph absolute value functions. Below you can download some free math
worksheets and practice.
It Is Divided Into Two Parts:
Graphing absolute value inequalities in two variables worksheet. Absolute value practice worksheet | literal equations, math practice. Explore this ensemble of printable absolute value equations and
functions worksheets to hone the skills of high school students in evaluating absolute functions with input and output table,.
Graphing Absolute Value Equations Worksheets.
Worksheet absolute value practice study function graphs functions access account graphing entire create absolute value graph tutorials, quizzes, and help. There are 20 absolute value functions to
graph and three word problems. Worksheets are graphing absolute value functions date period, graphing absolute value, draw the graph,.
The Process Of Graphing Absolute Value Equations Can Be Reduced To Some Specific Steps Which Help Develop Any Kind Of Absolute Value Graph.
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Understanding selection sort for coding interviews - A CODERS JOURNEY
Understanding selection sort for coding interviews
This is the first post in a series where we look at various sorting algorithms you’ll need to know for your next coding interview.
I struggled with understanding basic sorting algorithms when i first encountered them in college. I tried everything from re-reading lecture slides, watching recorded lectures, implementing the
algorithms in code etc. But invariably after couple of days, I’d get the different algorithms mixed up.
Then one of our professors showed me a very simple technique to understand, remember and practice these sorting algorithms. The technique is deceptively simple – just take the unsorted array and
write down the transformations to the array at each step when applying the sorting algorithm. And do it on a handwritten piece of paper – not in your mind, not in PowerPoint or Visio and definitely
not in code before you have a thorough understanding. Finally, repeat the process every couple of days till this get’s in your head.
Selection Sort is too simple of an algorithm to show up in your next Microsoft or Google coding interview. However, we’re starting the series here so that you can get an quick introduction to the
“paper-based” problem solving and visualization method before delving into more complicated sorts you might actually face on your next FAANG or Microsoft interview.
Strategy for Selection Sort
Given an unsorted array of integers, we want to sort them in ascending order. The array under consideration is shown below:
Unsorted array to be sorted with selection sort
Here is the strategy:
1. Select one element in the array, starting with the first element.
2. Compare it to all other elements in the array sequentially.
3. If an element is found to be smaller than the currently selected element, swap their positions.
Note that the correct position for the selected element is found before moving on to next element in the array.
Visualize the algorithm on paper
This is the most critical part of understanding the algorithm. I’d suggest understanding the process of sorting as shown below and doing it yourself a few times till you can get the array to the
correct sorted state.
Practice the algorithm on a piece of real paper or whiteboard ! You should try to write your own explanation at each step – this’ll help you reinforce and remember what you’re doing.
Selection Sort First Iteration
Above is the state of the array after the first iteration of the outer loop. 1 is indeed the smallest element in the array and should be in the first position. We are now done comparing the element
in the first position in the array to every other element in the array.
Next we move to the second slot.
Selection Sort Second Iteration
We’re done with the second iteration. We can see that 2 is indeed the second smallest element and now the first two slots in this array are sorted.
Next move to the third slot and compare it’s element to every other element, swapping as necessary.
Selection Sort Third Iteration
The process continues till the entire array is sorted.
Implementing Selection Sort in C#
using System;
namespace Sorting
class SelectionSort
public static bool debug = false;
public static void Sort(int[] inputArr)
for(int i=0; i < inputArr.Length; i++)
for(int j= i+1; j < inputArr.Length; j++)
if(inputArr[i] > inputArr[j])
Utilities.Swap(inputArr, i, j);
// Turn this flag on if you want to
//understand the state of the array after each Swap
Analysis of Selection Sort
Sort Property Analysis
Time Complexity For each element, the entire list is checked to find the smallest element. So in the worst case, “n” elements are checked for each element. Hence the time complexity is O(N^
Space Complexity Since the array is sorted in place and no extra space is used, the space complexity is O(1)
Adaptable The order of elements does not affect the sorting time. In other words, even if the array is partially sorted, still each element is compared and there is no breaking out
early. Hence Selection sort is non-adaptable.
Stable Selection sort is NOT a stable sorting algorithm. Elements which are equal might be re-arranged in the final sort order relative to one another.
Number of Comparisons and Selection Sort makes O(N^2) comparisons ( every element is compared to every other element).Selection Sort makes O(N) swaps to get all elements in the correct place.
When to use and when to avoid Selection Sort ?
Use selection sort in the following scenarios:
1. When the array is NOT partially sorted
2. When we have memory usage constraints
3. When a simple sorting implementation is desired
4. When the array to be sorted is relatively small
Avoid using Selection sort when:
1. The array to be sorted has a large number of elements
2. The array is nearly sorted
3. You want a faster run time and memory is not a concern. | {"url":"https://acodersjourney.com/selection-sort/","timestamp":"2024-11-12T23:25:25Z","content_type":"text/html","content_length":"139868","record_id":"<urn:uuid:4292a570-9412-4c2c-8925-4b70080b516a>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00209.warc.gz"} |
Hydrodynamic stability of liquid films adjacent to incompressible gas streams including effects of interface mass transfer
A theoretical study of linear stability of a gas/liquid interface with and without evaporation at the interface is presented. The zero mass transfer problem is solved for linear mean velocity
profiles in both gas and liquid. The mass transfer problem is solved for small rates of evaporation, which allows the reduction of exponential mean profiles to linear profiles. The analysis considers
instabilities in both gas and liquid motions and thus departs from the customary assumption of gas motion over a rigid wavy wall. The system of governing equations yields an eigenvalue problem upon
employing the methods of stability theory. The results show that neglecting instabilities in the gas motion would predict a stable interface at moderate values of wave numbers when it is actually
unstable. Mass transfer investigations are restricted to the modified Kelvin-Helmholtz mode and computations indicate that interface evaporation has a destabilizing effect at moderate wave numbers.
Ph.D. Thesis
Pub Date:
□ Flow Stability;
□ Liquid-Vapor Interfaces;
□ Mass Transfer;
□ Evaporation;
□ Interface Stability;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1976PhDT........24J/abstract","timestamp":"2024-11-05T19:02:22Z","content_type":"text/html","content_length":"35278","record_id":"<urn:uuid:33b4b80c-aced-494f-94d1-5a1d0ce664a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00324.warc.gz"} |
Questions: A quadrilateral with vertices at A(4, –4), B(4, –16), C(12, –16), and D(12, –4) has been dilated with a center at the origin. The image of D, point D, has coordinates (36, –12). What is
the scale factor of the dilation?
1/9 1/3 3 9
Answer: Let’s explain step by step;
Given a quadrilateral with vertices at A(4, -4), B(4, -16), C(12, -16), and D(12, -4) has been dilated with a center at the origin. The image of D i.e coordinates of D after dilation are (36,-12).
We need to find the scale factor of dilation.
We can calculate the new coordinates after dilation as follows.
(a,b) → (ma,mb)
As image of D given i.e
D(12,-4) → D'(36,-12)
⇒ 12 → 12m i.e 12m=36 ⇒ m=3
and -4 → -4m i.e -4m=-12 ⇒ m=3
hence, scale factor of the dilation is 3 | {"url":"http://www.imlearningmath.com/category/question-answer/page/16910/","timestamp":"2024-11-02T15:14:40Z","content_type":"text/html","content_length":"58800","record_id":"<urn:uuid:5e1bb01e-8687-4feb-9bd8-5c1fd9239328>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00367.warc.gz"} |
What Is A Hash?
In 1953, an IBM research scientist, Hans Peter Luhn, proposed an associative array data structure based on a hashing technique. The goal was to create a random access lookup table that did not
require sorting or ordered arrangement.
Luhn’s method involved splitting keys into digits and summing them to generate array indices. This yielded a rough distribution across available memory.
Before we dive deeper into the complex concept, let’s break down what hash is.
Hash is a data structure that maps keys to values. It allows for efficient lookup, insertion, and deletion of key-value pairs. Hashes are implemented in many programming languages and are a
fundamental component of software systems.
Through the late 1950s, much research went into improving hash functions and techniques. Collisions arose as a central challenge, where two keys hashed to the same index.
Ways to handle collisions—like chaining and open addressing—were explored. This allowed hash tables to accommodate some collisions gracefully.
By the early 1960s, hash tables were firmly established as an efficient data retrieval method. They provided average roughly constant time operations, O(1), for lookups and inserts. This speed opened
the door to many new algorithms and optimizations.
As programming languages were developed in the 1960s and 70s, built-in hash table types began to be included. Lisp provided one of the first native hash table implementations. Over time, they became
a standard part of languages like Python, Java, Go, and more.
The need for fast lookup tables accelerated as computing entered the “Big Data” era in the 2000s. Giant datasets demanded efficient storage and retrieval. Hash tables enabled large-scale data
processing inventions like Google’s MapReduce and Apache Hadoop.
Today, hash tables are a foundational data structure used in databases, caches, networks, and search systems. Their versatility and speed on massive data volumes make them integral to modern
computing infrastructures. Hash tables helped computing evolve from slow serial operations to the fast, parallel data processing systems today.
What is a Hash Table?
A hash table or map, is a data structure that implements an associative array. It stores elements as key-value pairs, using keys to index values. Some key properties of hashes:
Efficient lookup - On average, locating an element by key in a hash table takes constant O(1) time, compared to O(log n) for binary search trees. This makes hashes excellent for lookups.
Flexible keys - Most data types can be used for keys, including numbers, strings, pointers, etc. Values can also be arbitrary data types.
Unordered storage - Elements are stored in no particular order. A hash of the key determines the location.
Dynamic resizing - The hash table can grow or shrink as needed. Some implementations automatically resize the storage.
Hash tables are used extensively in software for lookup tables, caches, sets, and more. Overall, hashes provide fast key-value access, making them a versatile data structure.
How a Hash Table Works
Hash tables store data in an array format, where keys map to array indices via a hash function. Here is an overview:
An array of buckets is created, each capable of holding key-value pairs.
A hash function computes an integer hashcode for each key. This returns an array index.
The key-value pair is stored in the bucket at that array index. Multiple pairs may map to the same bucket (known as a collision).
The key is hashed again to look up a value, retrieving the index where the value is stored.
The hash function and resulting hashcodes evenly distribute keys across the buckets. Well-designed functions minimize collisions in the buckets. This allows for O(1) lookup time on average.
Hash Functions
The hash function maps keys to integer hashcodes. A good hash function has several properties:
Deterministic - The same input always produces the same hashcode
Uniform distribution - Outputs are evenly distributed across the output range
Non-reversible - The original key cannot be determined from the hashcode
Some common hash functions:
Modulo division - Take the key modulo a fixed prime number
Folding - Divide the key into parts and combine the parts
Mid-square - Square the key and extract the middle bits
SHA-1/SHA-2 - Cryptographic hash functions
Well-known hash functions like SHA-1 are very robust. But simple functions like modulo division work well for basic hash tables.
Hash Collisions
A collision occurs when two different keys hash to the same bucket index. Collisions are inevitable in hashing due to the pigeonhole principle. Ways to handle collisions:
Separate chaining - The bucket stores a linked list of values. Colliding keys are appended to the list.
Open addressing - Find the next open bucket location using some probe sequence. Collisions cause increased lookup time.
Perfect hashing - Specialized hash functions with no collisions in a static set of keys.
Collisions degrade hash table performance. A low load factor (ratio of occupied buckets to total buckets) minimizes collisions. Hash tables usually keep load factors under 50% for good performance.
Hash Table Operations
The main operations supported by hash tables are:
Insert - Compute hashcode of the key, map to index, insert key-value in bucket. Handles collisions.
Lookup - Compute hashcode, map to index, return value in bucket. Returns null if not found.
Delete - Compute hashcode, map to index, remove key-value from bucket if found.
A well-implemented hash table will have O(1) performance for these operations on average. Hash tables often outperform search trees and other data structures for lookup and insertion.
Applications of Hash Tables
Here are some common uses of hash tables:
Associative arrays - Store and lookup elements by key
Database indexing - Quickly find records by a key
Caches - High-speed in-memory key-value stores
Sets - Store unique elements efficiently
Object representation - Tables of object properties and values
Hashes are ubiquitous in software. Most programming languages have some hash table implementation, often as a primitive data type. Hash tables provide fast O(1) operations crucial for many systems
and optimizations.
Hash Table Variants
There are many variants of hash tables optimized for different scenarios:
Concurrent hash tables - Allow concurrent inserts and lookups from multiple threads. Useful for parallel computing.
Ordered hash tables - Maintain items in insertion order. Provide ordered traversal of elements.
Open addressing vs. chaining - Open addressing resolves collisions by finding new bucket locations. Chaining keeps a list of colliding elements.
Perfect hash functions - Guarantee no collisions for a static set of keys. Require no collision handling.
Cryptographic hash functions - Extremely robust functions like SHA-1 are designed to minimize collisions. Used in blockchains and cryptography.
These variants optimize hashes for specific use cases. The extensive research on hash tables has made them a versatile programming tool.
Hash Table Design
Designing an efficient hash table requires some key considerations:
Hash function - A good hash minimizes collisions. It should be fast, uniformly distribute outputs, and deterministically map keys to hashcodes.
Collision resolution - Separate chaining or open addressing can resolve collisions. The load factor should be kept low enough to minimize collisions.
Dynamic resizing - As the table size grows, resize into a larger array. Keeping the load factor low avoids frequent resizing.
Hash seed - Picking different starting seeds for the hash function minimizes collisions for a fixed set of keys.
Key design - Hash table performance depends on key selection. Keys should uniquely identify values. Avoid keys with high collision rates.
Careful design is needed to leverage hash tables’ power speedfully. Hash tables are a fundamental data structure used pervasively in systems and application development.
Hash Performance Factors
There are several factors impacting the performance of a hash table:
Load factor - Ratio of occupied buckets to total buckets. Collision chance grows as the load factor increases. Keep load under 50% for good performance.
Hash function quality - Good functions have few collisions and uniform distribution. High collision rates degrade performance.
Collision resolution method - Separate chaining has predictable O(1) lookup but extra memory overhead. Open addressing is memory-efficient but has probe sequences.
Resizing frequency - When to grow the table to the next size. Growing too often hits performance, while growing too little increases collisions.
Key distribution - Certain keys may cluster together, creating hot spots even with good hash functions.
Balancing these factors leads to optimal throughput and memory utilization. Performance tuning hash tables involves experimenting with these interrelated factors.
Hash Security
Hash tables have some security vulnerabilities to be aware of:
Collision attacks - Maliciously crafted keys to force collisions, causing denial of service or data corruption.
Hash flooding - Overwhelming a hash with many keys, going beyond the designed capacity.
Weak hash functions - Exploiting mathematical or statistical weaknesses in a hash to create collisions.
Rainbow table attacks - Reverse lookup of hashes using a precomputed table to crack passwords.
Strong cryptographic hash functions like SHA-256 avert many attacks. Limiting hash table capacity and monitoring load helps prevent flooding issues. Overall, care should be taken to seed and design
hash functions properly.
Hash Security Vulnerabilities
Hash tables are an integral part of many secure systems, but they also come with some vulnerabilities that attackers can potentially exploit:
Collision attacks - Specially crafted keys can force hash collisions, leading to denial of service or corrupted data.
Hash flooding - Overwhelming a hash with entries causes worst-case O(n) performance as it expands.
Rainbow table attacks - Use precomputed reverse hash lookups to crack password hashes. Salting passwords before hashing mitigates this.
Weak hash functions - Math or statistical flaws in a hash can lead to excessive collisions.
Hash injection - Inserting data by directly supplying indices rather than keys.
Bucket overflow - Targeting a specific bucket with duplicate keys causes overflow.
However, modern cryptographic hashes like SHA-256 and KECCAK-256 are extremely resilient against brute force and mathematical collision attacks. Their collision resistance stems from large internal
states and complex mixing functions.
Security practices like salting, monitoring load, and limiting input size help harden hash table implementations. While offering great speed, hash tables need properly seeded hash functions and
checked boundaries to prevent exploits.
Hash tables are an essential data structure that power efficient lookup and retrieval operations.
Algorithms like SHA-256 and KECCAK-256 were designed to minimize collisions for security purposes like digital signatures and blockchains. KECCAK-256 became the basis for Ethereum’s hash after being
selected through a public competition.
The Bitcoin blockchain also relies heavily on SHA-256 for mining and transaction hashing. These ultra-secure hashes enabled new decentralized models like cryptocurrencies and catalyzed a surge of
innovation in fintech.
Their collision resistance allows blockchains to preserve integrity and transparency without a central authority. Cryptographic hashes were a breakthrough that allowed hash tables to scale globally
across high-value networks like global payment rails.
Their versatility and performance make hash tables a foundational component of modern computing.
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Class 10 Maths MCQs For Pair Of Linear Equations In Two Variables - StudyWithGenius
Class 10 Maths MCQs for Pair of Linear Equations in Two Variables
Welcome to your Class 10 Maths MCQs for Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables
The pairs of equations x+2y-5 = 0 and -4x-8y+20=0 have:
Pair of Linear Equations in Two Variables
Graphically, the pair of equations 6x – 3y + 10 = 0 2x – y + 9 = 0 represents two lines which are
Pair of Linear Equations in Two Variables
How many solutions of the equation 15x – 14y + 11 = 0 are possible?
Pair of Linear Equations in Two Variables
If a pair of linear equations is consistent, then the lines will be:
Pair of Linear Equations in Two Variables
The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have
Pair of Linear Equations in Two Variables
The pair of equations y = 0 and y = -7 has
Pair of Linear Equations in Two Variables
Which of following is not a solution of 3a + b = 12?
Pair of Linear Equations in Two Variables
If the lines given by 3x + 2ky = 2 2x + 5y + 1 = 0 are parallel, then the value of k is
Pair of Linear Equations in Two Variables
If one equation of a pair of dependent linear equations is -3x+5y-2=0. The second equation will be:
Pair of Linear Equations in Two Variables
The value of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 will have infinitely many solutions is
Pair of Linear Equations in Two Variables
The pair of equations 3x + 4y = k, 9x + 12y = 6 has infinitely many solutions if –
Pair of Linear Equations in Two Variables
The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages, in years, of the son and the father are, respectively
Pair of Linear Equations in Two Variables
A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Reema paid Rs. 22 for a book kept for six days, while
Ruchika paid Rs 16 for the book kept for four days, then the charge for each extra day is:
Pair of Linear Equations in Two Variables
A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction obtained is:
Pair of Linear Equations in Two Variables
If x = a, y = b is the solution of the equation x – y = 2 and x + y = 4, then the value of a and b are respectively | {"url":"https://studywithgenius.in/quiz/class-10-maths-mcqs-for-pair-of-linear-equations-in-two-variables/","timestamp":"2024-11-03T15:24:45Z","content_type":"text/html","content_length":"310875","record_id":"<urn:uuid:58ce8cab-4db2-4497-bdef-2bc97a7eb12c>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00640.warc.gz"} |
Online B.Tech Math Tutor - Manipal University - Online Mathematics TutorOnline B.Tech Math Tutor – Manipal University - Online Mathematics Tutor
Online B.Tech Math Tutor – Manipal University
Enhance Your B.Tech Math Skills with an Online Tutor for Manipal University
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Researches On Gene Regulatory Network Reconstruction Based On The Feature Selection And Topology Analysis
Posted on:2019-07-07 Degree:Master Type:Thesis
Country:China Candidate:F Zhang Full Text:PDF
GTID:2310330545993363 Subject:Control Science and Engineering
The purpose of biological network inference is to constructing reliable mathematical model for biological systems with efficient measurements.The models of biological networks are crucial to under
the regulation mechanism underlying the measured data,as well as to guide the modular construction of synthetic gene circuits.With development of omics data including microarrays,computational
modeling of biological networks including gene networks become feasible.However,network inference needs support from high performance algorithm besides prior knowledge.Feature selection in machine
learning provides a promising solution for inference problem.This study applies feature selection and graph-based measure in gene regulatory network(GRN)inference:The major works are described as
following:1)For small-scale GRN,the linear model is still a good choice to reconstruct GRN.Under the linear model assumption,using support vector machine regression(S VR)method for feature selection
to reconstruct the whole gene regulatory network.Compared with the singular value decomposition(SVD)method,the SVR-based method obtain higher accuracy.Using the corresponding sequential data set of
GRN,the experimental results verify the effectiveness of the algorithm.2)Considering nonlinearity in GRN,tree-based regression approaches have advantages in efficiency and accuracy compare with
linear regression approaches in general.Different tree-based methods cover the dynamics of the same network from various perspective.This paper applies Gradient Boosting to infer the GRN,then
integrating multiple inferred outcomes from several inferring algorithms including Random Forest via a weighted voting mechanism.As the calculation of weights for each outcome is unsupervised,this
paper defines a score to evaluate the degree of reliability for each outcome,and use this score in determine the weights for different tree-based regression approaches.The simulation outcomes
validate the effectiveness of the proposed method.3)With the inferred topology,this chapter evaluate the importance score of nodes in a digraph through topological analysis,thus selecting the subset
of key genes.In this study,the first level of key gene nodes correspond to root strongly connected components(SCC),which are located in upstream of information flow in a given digraph.In order to
determine the unique set of root SCC,this chapter defines a cost function using graph-based measure and applies GA approach to minimize the cost function.After obtaining the root SCC,the proposed
hierarchical estimation strategy first calculates the regulatory parameters relevant with key genes,then extending to the next stage of genes using the parameters in later level as prior knowledge.In
this way,original parameter estimation problem is decomposed into a set of sub-problems with various priority levels.Experimental outcomes indicate that hierarchical estimation strategy is able to
obtain lower MSE indexes compared with the traditional one-time-all strategy.Besides,the computational time is much less.
Keywords/Search Tags: Gene regulatory networks, Feature selection, Model fusion, Parameter estimation, Hierarchical strategy | {"url":"https://globethesis.com/?t=2310330545993363","timestamp":"2024-11-12T06:31:23Z","content_type":"application/xhtml+xml","content_length":"9074","record_id":"<urn:uuid:05721ff4-5521-4e80-9abf-ceacb70e27a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00468.warc.gz"} |
Testing the predictions of axisymmetric distribution functions of galactic dark matter with hydrodynamical simulationsMihael Petač
Julien Lavalle
Arturo Núñez-Castiñeyra
Emmanuel Nezri
, 2021, original scientific article
Abstract: Signal predictions for galactic dark matter (DM) searches often rely on assumptions regarding the DM phase-space distribution function (DF) in halos. This applies to both particle (e.g.
p-wave suppressed or Sommerfeld-enhanced annihilation, scattering off atoms, etc.) and macroscopic DM candidates (e.g. microlensing of primordial black holes). As experiments and observations improve
in precision, better assessing theoretical uncertainties becomes pressing in the prospect of deriving reliable constraints on DM candidates or trustworthy hints for detection. Most reliable
predictions of DFs in halos are based on solving the steady-state collisionless Boltzmann equation (e.g. Eddington-like inversions, action-angle methods, etc.) consistently with observational
constraints. One can do so starting from maximal symmetries and a minimal set of degrees of freedom, and then increasing complexity. Key issues are then whether adding complexity, which is
computationally costy, improves predictions, and if so where to stop. Clues can be obtained by making predictions for zoomed-in hydrodynamical cosmological simulations in which one can access the
true (coarse-grained) phase-space information. Here, we test an axisymmetric extension of the Eddington inversion to predict the full DM DF from its density profile and the total gravitational
potential of the system. This permits to go beyond spherical symmetry, and is a priori well suited for spiral galaxies. We show that axisymmetry does not necessarily improve over spherical symmetry
because the (observationally unconstrained) angular momentum of the DM halo is not generically aligned with the baryonic one. Theoretical errors are similar to those of the Eddington inversion
though, at the 10-20% level for velocity-dependent predictions related to particle DM searches in spiral galaxies. We extensively describe the approach and comment on the results.
Keywords: galaxy dynamics, dark matter experiments, dark matter simulations, dark matter theory, cosmology, nongalactic astrophysics, astrophysics of galaxies, high energy physics
Published in RUNG: 01.10.2021; Views: 2700; Downloads: 66
Link to full text
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A possible NP-Intermediary Problem
Early Registration closes on April 30.
Early Registration closes May 3.
Early Registration closes on May 6. )
(ADDED LATER- AS STATED the problem below has problems with it. After reading see Eric Allenders comment in the comments.)
Here is a problem whose complexity has probably not been studied. I think it is NP-Intermediary. I will also give a version that is likely NPC. I am proposing that you either prove it is in P or NPC
or show that if it is NPC then PH collapses (or something unlikely happens).
DEFINITION: We call a coloring of the x by y grid
if there are no rectangles with all four corners the same color.
Let L be The set of all (x,y,c) in UNARY such that there is a proper c-coloring of the x by y grid.
1. Clearly in NP: verifying that a c-coloring of x by y is proper can be done in time poly in x,y,c.
2. Let L[c] be the set of all (x,y) such that (x,y,c) ∈ L. L[c] has a finite obstruction set and hence is in O(|x|+|y|) thought the constant depends on c. This can be proven by Well-Quasi-Order
theory which yields nonconstructive bounds on the size of the obs set, or there is a proof with reasonable O(c^2) bounds. (For ALL info on this problem and links to more info see the links
below.) Hence the problem is Fixed Parameter Tractable.
3. I suspect that the problem L is NP-intermediary. Why? I think its NOT NPC since there is not much to play with- just 3 numbers. I think its not in P because my co-authors and I have not been able
to do much more than ad-hoc colorings (this is not that good a reason- however the 17x17 challenge (linked to below) has lead other people to think about the problem and not come up with clean
4. It is likely that the following is NPC: The set of all (x,y,c,f) where f is a partial c-coloring of the x by y grid such that f can be extended to a proper c-coloring.
5. I suspect that whatever is true for rectangles is true if you replace rectangles by other shapes such as a squares. There are also other Ramsey-Theoretic functions that could be studied (though
Ramsey's theorem itself not-so-much--- verifying is hard).
6. This question is related to the (still unresolved) question I posed here and that Brian Hayes explained better here. However, I don't think proving the general problem NPC will shed light on why
determining if (17,17,4) ∈ L is hard. That's just one instance.
4 comments:
1. ...or show that if it is NPC then PH collapses
Since this problem (as stated) can be efficiently encoded as a *tally* language (since all three inputs (x,y,c) are in unary), this is a sparse set, and can't be NP-complete unless P=NP by
Mahaney's Theorem.
I guess that the interesting question arises if we consider the *binary* encoding of this language. In this setting, the problem is in NE, and the question is: Is it complete for NE? I won't
claim to have any intuition about whether it's complete. Probably it's easier to consider the candidate NP-complete problem (where the partial coloring f is also given).
2. Mahaney's Theorem, right?
3. Sune Kristian Jakobsen12:27 PM, April 28, 2010
"I think its not in P because my co-authors and I have not been able to do much more than ad-hoc colorings (this is not that good a reason- however the 17x17 challenge (linked to below) has lead
other people to think about the problem and not come up with clean solutions.)"
Is your point that a 4-coloring of the 17x17 grid is impossible, or that the colorings of maximal grids seems to be difficult to find?
It might be that all the lowest possible upper bounds (on grid size for fixed c) can be proven using a simple counting argument, but that there don't exist any proof that these upper bounds are
the lowest possible. In this case the problem would be in P, but we wouldn't be able to prove it. In particular, if this problem is not in P it not only implies that maximal colorings are hard to
find, but also that there is no simple reason that larger grid can't be c-colored.
4. Eric- THANKS for stating the version of my problem that makes sense.
Sune K. J.- DETERMINIng if 17x17 is
4-colorable seems to be hard. I had many
techniques for showing that a grid was NOT c-colorable but they PROVABLY did not work on 4-coloring 17x17. AND trying to show that 17x17 IS 4-colorable ALSO seems hard as nobody has been able to
find it
(of course, it might not exist). | {"url":"https://blog.computationalcomplexity.org/2010/04/possible-np-intermediary-problem.html?m=0","timestamp":"2024-11-04T23:37:44Z","content_type":"application/xhtml+xml","content_length":"184283","record_id":"<urn:uuid:6df9703d-e37e-40ed-8ae8-c1c910cab111>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00672.warc.gz"} |
Title: The T-Rex selector for fast high-dimensional variable selection with FDR control
Description: It performs fast variable selection in large-scale high-dimensional settings while controlling the false discovery rate (FDR) at a user-defined target level.
Paper: The package is based on the paper
J. Machkour, M. Muma, and D. P. Palomar, “The terminating-random experiments selector: Fast high-dimensional variable selection with false discovery rate control,” arXiv preprint arXiv:2110.06048,
2022. (https://doi.org/10.48550/arXiv.2110.06048)
Note: The T-Rex selector performs terminated-random experiments (T-Rex) using the T-LARS algorithm (R package) and fuses the selected active sets of all random experiments to obtain a final set of
selected variables. The T-Rex selector provably controls the false discovery rate (FDR), i.e., the expected fraction of selected false positives among all selected variables, at the user-defined
target level while maximizing the number of selected variables and, thereby, achieving a high true positive rate (TPR) (i.e., power). The T-Rex selector can be applied in various fields, such as
genomics, financial engineering, or any other field that requires a fast and FDR-controlling variable/feature selection method for large-scale high-dimensional settings.
In the following sections, we show you how to install and use the package.
Before installing the ‘TRexSelector’ package, you need to install the required ‘tlars’ package. You can install the ‘tlars’ package from CRAN (stable version) or GitHub (developer version) with:
Then, you can install the ‘TRexSelector’ package from CRAN (stable version) or GitHub (developer version) with:
You can open the help pages with:
To cite the package ‘TRexSelector’ in publications use:
Quick Start
This section illustrates the basic usage of the ‘TRexSelector’ package to perform FDR-controlled variable selection in large-scale high-dimensional settings based on the T-Rex selector.
1. First, we generate a high-dimensional Gaussian data set with sparse support:
1. Second, we perform FDR-controlled variable selection using the T-Rex selector for a target FDR of 5%:
So, for a preset target FDR of 5%, the T-Rex selector has selected all true active variables and there are no false positives in this example.
Note that users have to choose the target FDR according to the requirements of their specific applications.
For more information and some examples, please check the GitHub-vignette.
T-Rex paper: https://doi.org/10.48550/arXiv.2110.06048
TRexSelector package (stable version): CRAN-TRexSelector.
TRexSelector package (developer version): GitHub-TRexSelector.
README file: GitHub-readme.
Vignette: GitHub-vignette.
tlars package: CRAN-tlars and GitHub-tlars. | {"url":"https://cloud.r-project.org/web/packages/TRexSelector/readme/README.html","timestamp":"2024-11-06T14:40:04Z","content_type":"application/xhtml+xml","content_length":"16068","record_id":"<urn:uuid:52ae5293-9c34-479e-9fae-50ccb8fcd621>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00174.warc.gz"} |
Can't get a reduced number
5146 Views
4 Replies
2 Total Likes
Can't get a reduced number
I've tried searching Reduce and Simplify functions in Mathematica, but they seem to be more complicated than what I am trying to get Mathematica to do. Here's what is happening:
h = 75 ft;
\[Sigma] = 7100 psi;
\[Sigma]u = 7700 psi;
\[Sigma]d = 8100 psi;
KIC = 1000 psi/Sqrt[inch];
\[Rho] = rhowat;
C1 = .163;
C2 = 3.91;
C3 = .0069;
hu_] := (C1/Sqrt[hu])[
KIC (1 - Sqrt[hu/h]) +
C2 (\[Sigma]u - \[Sigma]) Sqrt[hu] ArcCos[h/hu]] +
C3 \[Rho] (hu - 0.5 h)
hd_] := (C1/Sqrt[hd])[
KIC (1 - Sqrt[hd/h]) +
C2 (\[Sigma]d - \[Sigma]) Sqrt[hd] ArcCos[h/hd]] +
C3 \[Rho] (hd - 0.5 h)
\[CapitalDelta]pfu[175 ft]/psi
and I get " 0.000145038 (289.179 + 0.0223183[1.1042*10^8]) " for an answer instead of 484psi... help!
4 Replies
Wow so because I had [ ] instead of ( ) for part of the equation it didn't want to simplify it. That is extremely helpful. I really appreciate the help!! More questions to come i'm sure.
Ah, better.
Ok, a few things. Mathematica is absolutely fanatic about use of {} versus () versus [] and those all mean completely different things to it. Use one where it expects another and you won't get what
you need. So I changed a couple of [] into () which is what I'm guessing you wanted. And for the moment I'm going to avoid Mathematica's handling of units and just changed all your inputs to inches
and pounds. We can go back and try to fix that later.
Here is what I have now.
In[1]:= h = 75 *12*inch;
\[Sigma] = 7100 (lb/inch^2);
\[Sigma]u = 7700 (lb/inch^2);
\[Sigma]d = 8100 (lb/inch^2);
KIC = 1000 (lb/inch^2)/Sqrt[inch];
\[Rho] = rhowat;
C1 = .163;
C2 = 3.91;
C3 = .0069;
\[CapitalDelta]pfu[hu_] := (C1/Sqrt[hu]) (KIC (1 - Sqrt[hu/h]) + C2 (\[Sigma]u - \[Sigma]) Sqrt[hu] ArcCos[h/hu]) +
C3 \[Rho] (hu - 0.5 h)
\[CapitalDelta]pfd[hd_] := (C1/Sqrt[hd]) (KIC (1 - Sqrt[hd/h]) + C2 (\[Sigma]d - \[Sigma]) Sqrt[hd] ArcCos[h/hd]) +
C3 \[Rho] (hd - 0.5 h)
\[CapitalDelta]pfu[175 *12*inch]/(lb/inch^2)
Out[12]= (1/lb)inch^2 ((0.00355695 ((1000 (1 - Sqrt[7/3]) lb)/inch^(5/2) + (121256. lb)/
inch^(3/2)))/Sqrt[inch] + 11.385 inch rhowat)
In[13]:= Expand[%]
Out[13]= 431.301 - 1.87638/inch + (11.385 inch^3 rhowat)/lb
That is a little closer to where I think you are trying to go.
Now we have to deal with ArcCos and whether it is expecting something with or without units as an argument and what you are expecting Sqrt to do. Since at this point Mathematica has no idea what
"inch" really is I think it is trying to resist simplifying that because it doesn't know whether that might be a negative number or even complex.
Is this getting a little closer to your goal? Can you see the next thing that needs to be corrected?
If you edit your posting and click on the orange "spikey ball" to create a "code box" and paste your code into that then the forum software is less likely to mangle it and make it impossible to
scrape-n-paste back into a Mathematica notebook. OR you can create a really simple notebook showing your work and attach it so someone can download and execute. With either of those someone can find
a way to probably fix the problem.
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/230708","timestamp":"2024-11-06T09:31:35Z","content_type":"text/html","content_length":"111789","record_id":"<urn:uuid:45d8455d-52a6-4781-8ab4-b876361fa28e>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00441.warc.gz"} |
Graph Worksheet
Graphs at a glance
Graphs are types of diagrams used to visually represent data and relationships. There are many different types of graphs. Graphs are used in algebra to represent functions and the relationship
between the variables x and y. Graphs can also be used to represent the relationship between two real life variables, such as distance and time. Finally, charts and graphs such as bar graphs can be
used to visually display data.
When it comes to the graphs of algebraic functions, different types of functions will lead to graphs of different shapes. There are six different types of function that we are concerned with: linear,
quadratic, cubic, reciprocal, exponential and circle. Each type of function produces a specific shape of graph.Â
Linear functions involve y and x e.g. y=2x+1.
Quadratic functions involve x2 e.g. y=x2+3
Cubic functions involve x3 e.g. y=5x3 + 3x
Reciprocal functions involve fractions with x in the denominator e.g. y = 1/x
Exponential functions involve x as a power e.g. y=3x
Circle functions are of the form x2+y2=25
Functions often contain coefficients as well as other terms added or subtracted. These coefficients can be whole numbers, negative numbers, decimals or fractions and change the features of the graph
without changing its fundamental shape.Â
To draw line graphs we usually use the equation of the line to create a table of values for x and y, which then gives us the coordinates of some points on the line. It is best to use graph paper when
doing line plots to ensure accurate work.
Looking forward, students can then progress to additional types of graph worksheets and on to more algebra worksheets, for example a simplifying expressions worksheet or simultaneous equations
For more teaching and learning support on Algebra our GCSE maths lessons provide step by step support for all GCSE maths concepts. | {"url":"https://thirdspacelearning.com/secondary-resources/gcse-maths-worksheet-graph/","timestamp":"2024-11-07T02:13:43Z","content_type":"text/html","content_length":"163574","record_id":"<urn:uuid:7400d1ce-bdc3-47ce-b502-a5e12209d9a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00451.warc.gz"} |
Contingency Table Options • Genstat v21
Use this window to control what is displayed in the Output Window.
The display options for each type of test are as follows:
Chi-square test
Test Chi-square value and degrees of freedom.
Probability Probability value.
Expected values Fitted (expected) values and standardized residuals.
Individual cell contributions Shows the contribution of each cell of the table to the chi-square value.
Random permutation test (Chi-square test only)
For a chi-square test you can choose to calculate the probability using a random permutation test. A permutation test simulates the random distribution of table values that may occur in tables that
have the same overall distribution of numbers over the columns, and over the rows, as in the original table. The significance of the chi-square statistic that is calculated from the observed table
can be assessed by seeing where it lies in the distribution of statistics that we obtain from the permuted data.
Number of permutations lets you specify the number of allocations.Seed lets you specify a randomization seed. Plot histogram for distribution of statistics produces a histogram showing the
distribution of statistics obtained from the permuted data sets. A visual indication of the chi-square statistic from the observed data is superimposed on the graph.
Fisher’s exact test
Tables Displays all 2×2 tables with margins that are the same as the observed table together with their probabilities of occurrence under the null hypothesis of no association, and the
cumulative probabilities calculated from both tails
Probabilities Probabilities
Action buttons
OK Save the option settings and close the dialog.
Cancel Close the dialog without making any changes.
Defaults Reset the options to their default settings.
Action Icons
Clear Clear all fields and list boxes.
Help Open the Help topic for this dialog.
See also | {"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/contingency-table-options/","timestamp":"2024-11-14T10:32:01Z","content_type":"text/html","content_length":"39649","record_id":"<urn:uuid:9f7e60e8-226e-45d3-b909-eddc502da931>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00796.warc.gz"} |
The method to prove the upper bound of lower bound is at most 0.1005 (Matlab codes)
maincodethm2_2.m (1.47 kB)
The method to prove the upper bound of lower bound is at most 0.1005 (Matlab codes)
In Chapter 5, we use this method to show that the convex hull area of a circle, a line and a rectangle (perimeter 1) is at most 0.1005. There are two files:
maincodethm2_2 is a code to prove that the lower bound of convex hull area of a circle, an equilateral triangle, and a rectangle is at most 0.1005.
codecvhul2_2 is a function to find area of convex hull. | {"url":"https://figshare.le.ac.uk/articles/software/The_method_to_prove_the_upper_bound_of_lower_bound_is_at_most_0_1005_Matlab_codes_/12629879","timestamp":"2024-11-02T02:14:48Z","content_type":"text/html","content_length":"140743","record_id":"<urn:uuid:6415c07c-431c-4bb3-a1ea-bf5460721c75>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00636.warc.gz"} |
Parallelogram: Theorem (3)
Interact with the applet below for a few minutes. Then, answer the questions that follow. Feel free to move the BIG WHITE POINTS anywhere you'd like! You can also adjust the size of the pink angle by
using the slider.
What special type of quadrilateral was formed in the first half of your sliding-the-slider? How do you know this?
What else can you conclude about this special type of parallelogram? Be specific!
Write a coordinate geometry proof that formally proves what you've seen illustrated here. (Make sure you have appropriate variable coordinates for your initial setup!) For this theorem, a
coordinate-geometry method of proof is actually A LOT EASIER than a 2-column or paragraph proof!
Quick Demo: 1:16 sec to END (BGM: Andy Hunter) | {"url":"https://beta.geogebra.org/m/edHjeemF","timestamp":"2024-11-04T12:00:29Z","content_type":"text/html","content_length":"99089","record_id":"<urn:uuid:a7f3deb6-27bb-4d7a-bf4e-fb9152d5b19a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00485.warc.gz"} |
Subelliptic geometric Hardy type inequalities
Project Details
Grant Program
Faculty Development Competitive Research Grants 2020-2022
Project Description
The main goal of this research project is to construct a theory for subelliptic (geometric) Hardy type functional inequalities and to carry out qualitative research of their extensions and
applications. To achieve this aim, we propose to develop existent and create new methods of homogeneous (Lie) groups, then to study general subelliptic differential operators. This study will lead to
a deeper understanding of the basic Lie group structure of functional inequalities and subelliptic differential operators. For instance, existing methods for proving subelliptic functional
inequalities on nilpotent Lie groups are based on the study of the properties of a fixed homogeneous norm, for example, the L-gauge. Proposed in this research project, methods allow us to work with
arbitrary quasi-norms, thus we believe the combination of our methods with the lifting theory will give new subelliptic functional inequalities on manifolds which will follow new results in both
analysis on manifolds and theory of subelliptic differential equations.
The subject of Hardy inequalities has now been a fascinating subject of continuous research by numerous mathematicians for about more than one century, 1918-2019. The original inequality was
published by G. H. Hardy in “Notes on some points in the integral calculus (51)”, Messenger of Mathematics, 48 (1918), P. 107-112.
The Hardy inequalities have numerous applications in different fields, for example in the spectral theory, leading to the lower bounds for the quadratic form associated with the Laplacian operator.
They are also related to many other areas and fields, notably to the uncertainty principles. The uncertainty principle in physics is a fundamental concept going back to Heisenberg's work on quantum
mechanics, as well as to its mathematical justification by Hermann Weyl. Over the last 100 years, the subject of Hardy inequalities and related analysis has been a topic of intensive research:
currently, MathSciNet lists more than 800 papers containing words ‘Hardy inequality’ in the title, and almost 3500 papers containing the words ‘Hardy inequality’ in the abstract or in the review. The
Hardy inequalities have been already presented in many monographs and reviews; here we can mention excellent books by Opic and Kufner in 1990, Davies in 1999, Edmunds and Evans in 2004, parts of
Mazya's books in 1985 and 2011, Ghoussoub and Moradifam in 2013, and Balinsky, Evans, and Lewis in 2015, as well as many other books on different areas related to Hardy spaces.
In view of this wealth of information (and the page limit), we apologize for the inevitability of missing to mention many important contributions to the subject.
However, all of these presentations are largely confined to the Euclidean part of the available wealth of information on this subject.
At the same time, there is another layer of intensive research over the years related to Hardy type functional inequalities in subelliptic settings motivated by their applications to problems
involving subelliptic differential equations. This is complemented by the more general anisotropic versions of the theory. In this direction, the subelliptic ideas of the analysis on the Heisenberg
group, significantly advanced by Folland and Stein in [1] (see, also [3]), were subsequently consistently developed by Folland [2] leading to the foundations for analysis on stratified groups (or
homogeneous Carnot groups). Thus, the intensive study of the subelliptic functional estimates started due to their importance for many questions involving subelliptic partial differential equations,
unique continuation, sub-Riemannian geometry, subelliptic spectral theory, etc. As expected, the subelliptic Hardy inequality was obtained to the most important example of the Heisenberg group by
Garofalo and Lanconelli [4] (see, Thangavelu’s book in 2004, also Roncal and Thangavelu’s works, e.g. [5] on recent advances on the Heisenberg group). The place where Hardy type inequalities and
general homogeneous (Lie) groups meet is a beautiful area of mathematics which was not consistently treated in the project form. We took it as an incentive to write this project to extend and deepen
the understanding of Hardy type inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. While we will construct the general theory of Hardy
type inequalities in the setting of general homogeneous groups, particular attention is paid to the special class of stratified groups and graded groups as well as extensions to manifolds (without
group structures). In this setting, the theory of subelliptic functional inequalities becomes intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial
differential equations.
These topics constitute the core of this project with the results complemented with additional closely related topics such as uncertainty principles, the theory of linear and nonlinear subelliptic
differential equations, subelliptic spectral theory as well as the theory of (anisotropic) function spaces.
Thus, the present project is devoted to the exposition of the research developments at the intersection of two active fields of mathematics: Hardy inequalities and related analysis, and the
noncommutative analysis in the setting of nilpotent Lie groups. Both subjects are very broad and deserve separate studies on their own. However, a combination of the so-called lifting theory and our
recent research techniques in the area does allow one to make a consistent treatment of `anisotropic' Hardy inequalities, their numerous features, and a number of related topics. This brings many new
insights to the subject, also allowing to underline the interesting character of its subelliptic features.
This study has mainly fundamental character, makes a valuable contribution to the development of the theory of functional analysis on nilpotent Lie groups and the theory of subelliptic differential
equations. Note that we will solve previously unsolved conjecture regarding the natural weight in the geometric Hardy inequality (please see Section 5 for specific tasks). Obtained results will be
applied to solving various problems in mathematics and theoretical physics as well as may serve as fundaments for many new university courses. Particularly, we apply obtained inequalities to
subelliptic partial differential equations and subelliptic spectral theory which may have interpretations in theoretical physics. For example, we will prove the uniqueness and simplicity of the
principal frequency (or the first eigenvalue) and uniqueness of a positive solution to Dirichlet p-versions of sub-Laplacians (with nonlinear right-hand sight functions).
Expected social impacts: Most important contribution of this project to the Kazakhstani society and scientific community will be the training of four graduate students, who will get their Ph.D. and
MS degrees working on this project. By conducting advanced research and publishing in top venues, they will become the next generation of academicians in Kazakhstan. Publication of fundamental
results in prestigious mathematical journals promotes raising the image of the Republic of Kazakhstan in the scientific world.
Status Finished
Effective start/end date 1/1/20 → 12/31/22
• Functional inequality
• Subelliptic differential equation
• Lie Group
Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.
Research output
• 12 Article
• 2 Conference contribution
• Suragan, D.
Mathematical Analysis, its Applications and Computation - ISAAC 2019.
Cerejeiras, P. & Reissig, M. (eds.).
p. 99-122 24 p.
(Springer Proceedings in Mathematics and Statistics; vol. 385).
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution | {"url":"https://research.nu.edu.kz/en/projects/subelliptic-geometric-hardy-type-inequalities","timestamp":"2024-11-05T20:38:53Z","content_type":"text/html","content_length":"72331","record_id":"<urn:uuid:468cb1d1-eda4-4311-aa2d-35be92afb972>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00503.warc.gz"} |
Radians (angle) | Glossary | GDQuest
See all glossary terms
We use two units in games for measuring angles: degrees and radians. In games and game engines, angles are usually measured in radians for efficiency. You surely know what degrees are, but you may
not be familiar with radians. Degrees are based on the circumference of a circle. A circle is divided into 360 degrees. Radians are based on the radius of a circle. A radian is equal to the angle of
an arc whose length is the circle's radius. Imagine a circle with a radius of one unit. If you wrap a string along the circle that is one unit long, the corresponding angle from the circle center is
one radian. Because the perimeter of a circle is 2π multiplied by the circle radius, and one radian corresponds to an arc as long as the radius of the circle, there are 2π radians in a circle. Here
are common angles in degrees and radians:
• 360° = 2π radians.
• 180° = π radians.
• 90° = π/2 radians.
• 45° = π/4 radians.
• 30° = π/6 radians. | {"url":"https://school.gdquest.com/glossary/angle_radians","timestamp":"2024-11-03T06:44:40Z","content_type":"text/html","content_length":"37966","record_id":"<urn:uuid:a332e232-5b47-4acd-b60c-8c290fdf440d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00394.warc.gz"} |
1 myCobot Series 6-Axis Collaborative Robotic Arm
myCobot 280 algorithm
1 Structural Parameters
1.1 DH Parameters of Robotic Arm
joint theta d a alpha offset
1 q1 131.22 0 1.5708 0
2 q2 0 -110.4 0 -1.5708
3 q3 0 -96 0 0
4 q4 63.4 0 1.5708 -1.5708
5 q5 75.05 0 -1.5708 1.5708
6 q6 45.6 0 0 0
1.2 Kinematic Model
2 Coordinate System Introduction
2.1 Tool Coordinate System
The figure shows the robot model of Mecharm270. Base in the figure represents the base coordinate system of the robot, O' represents the end flange coordinate system, and point P represents the
position of the end of the manipulator relative to the base coordinate system (x=152, y=0 , z=224)
Extend a certain pose on the basis of the end flange, and regard the set tool point as the end of the machine:
T in the figure is the set tool coordinate system. The posture of this coordinate system is consistent with O’, and the relative displacement of the origin has occurred. Use the python function to
set the tool coordinate system:
• set_tool_reference([x, y, z, rx, ry, rz]) //Set tool coordinate system
• set_end_type(1) //Set the end coordinate system type as tool
• Assume that the tool coordinate system T is not rotated relative to O' (rx = ry = rz = 0)
• Assume that the origin of the tool coordinate system T is in the coordinate system O’ at (x = 0, y = 0, z = 100mm)
• The final tool coordinate system parameter is set_tool_reference(0, 0, 100, 0, 0, 0)
Since the tool coordinate system is set, the end of the robot extends from O' to T at this time, and the coordinates of the end of the machine read at this time become (152+100, 0, 224), and the
coordinate posture movement will revolve around the tool point T rotate.
2.2 World Coordinate System
Section 2 introduces that by setting the tool coordinate system, the end coordinate system of the manipulator can be extended to a certain pose. We can also extend a certain pose on the basis of the
base coordinate system of the manipulator by setting the world coordinate system. The set world coordinate system will replace the original Base coordinate system and become the new base coordinate
W in the figure is the set world coordinate system. The posture of this coordinate system is consistent with Base, and the relative displacement of the origin has occurred. Use the python function to
set the world coordinate system:
• set_world_reference([x, y, z, rx, ry, rz]) //Set the world coordinate system
• set_reference_frame(1) //Set the base coordinate system type to the world
• Assuming that the world coordinate system W has not rotated relative to Base(rx = ry = rz = 0)
• Suppose the origin of the world coordinate system W is in the coordinate system Base (x = 0, y = 0, z = -100mm)
• The final world coordinate system parameter is set_world_reference(0, 0, -100, 0, 0, 0)
Since the world coordinate system is set, the origin of the robot extends from Base to W at this time, and the O’ coordinate read at this time becomes (152, 0, 224+100). | {"url":"https://docs.elephantrobotics.com/docs/mecharm-pi-en/2-serialproduct/2.1-280/Kinematics&Coordinate.html","timestamp":"2024-11-14T11:15:41Z","content_type":"text/html","content_length":"50988","record_id":"<urn:uuid:24c8db84-0f41-4aad-ab1d-70261c5e5310>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00472.warc.gz"} |
posted Feb 15 at 12:00 pm
The problem with measuring Resistance
We have a DT85 g series 2 on which all 23 temperature measurements have started to show a bag value compared to what it should be. An example is the air temperature is 9 degrees C outside and the
logger shows 17 and all the others have been moved higher.
MU X Huggenberger 48 is connected to DT 85
1DBO(W)=1-1 1R(II,,MD100,ES10,GL300MV,S1,W,"Tb-2 ~0C",=115cv)
3W are probes PT30 , the resistance measurement at the sensor itself is 29.15 and 0.67 ohms, but at the MUx input it is 34.49 and 7.61 ohms due to the distance of the probe from the measurement
location. Instead of around 14 degrees, it shows 20 or more degrees. and that for all 23 probes. What could be the problem.
thank you
Sorry for my English, I hope you understand the question.
The problem with measuring Resistance Respected, We have a DT85 g series 2 on which all 23 temperature measurements have started to show a bag value compared to what it should be. An example is the
air temperature is 9 degrees C outside and the logger shows 17 and all the others have been moved higher. MU X Huggenberger 48 is connected to DT 85 example 1DBO(W)=1-1 1R
(II,,MD100,ES10,GL300MV,S1,W,"Tb-2 ~0C",=115cv) S1=0,30,27,5,31"C" 3W are probes PT30 , the resistance measurement at the sensor itself is 29.15 and 0.67 ohms, but at the MUx
input it is 34.49 and 7.61 ohms due to the distance of the probe from the measurement location. Instead of around 14 degrees, it shows 20 or more degrees. and that for all 23 probes. What could be
the problem. thank you Sorry for my English, I hope you understand the question.
Hi Bocac,
If the temperature sensor is a thermocouple, you need to check the wire layout of the thermocouple against any high-power equipment or cable since the wire can pick up electromagnetic interference
and add a few mV to the sensor, which is equivalent to the offset.
If the temperature sensor is an RTD, you need to be aware of the wire length of your sensor. Your sensor has 3-wire; using the logger wiring can compensate for some of the wire resistance. The best
type of sensor is only 4-wire, which can compensate both sides of the wire.
Can you share the RTD details and the reason for such scaling? Rather than using a resistance channel, you can utilize the RTD channel template. PT30 sensor means the sensor resistance is 30 ohms at
0 deg C. You can use PT385(3W,30) for example.
Best regards,
dataTaker Expert
Hi Bocac, If the temperature sensor is a thermocouple, you need to check the wire layout of the thermocouple against any high-power equipment or cable since the wire can pick up electromagnetic
interference and add a few mV to the sensor, which is equivalent to the offset. If the temperature sensor is an RTD, you need to be aware of the wire length of your sensor. Your sensor has 3-wire;
using the logger wiring can compensate for some of the wire resistance. The best type of sensor is only 4-wire, which can compensate both sides of the wire. Can you share the RTD details and the
reason for such scaling? Rather than using a resistance channel, you can utilize the RTD channel template. PT30 sensor means the sensor resistance is 30 ohms at 0 deg C. You can use PT385(3W,30) for
example. Best regards, dataTaker Expert
Thanks for your reply and interest in solving the problem.
RTD probes were installed in 1975-1978 during the construction of the dam, for which there is no accurate data. Based on the monitoring of measurements, it was concluded that these are probes for
which the relation T=20+(Ri-29.83)/0.117 is valid, where Ri is the measured resistance minus the compensation resistance.
It is clear to us that there is a length of cable from the probes to the logger itself, and depending on the place where the probes are located, those lengths range from 110 to 286 m, which can be
seen when measured on the terminal strip before the entrance to the MUS. This system has been working happily since 2013.
For example, the measurement of the Tz-probe on the crown of the dam according to the calculation (T=20+(Ri-29.83)/0.117 measurement 06.02.2024) the calculated temperature is 14.0170C, and the
thermometer at that moment on UG1 shows a value of around 14 0C while on the logger display it is a measurement above 20. The measured resistances at the outlet of the probe are 1-2.3=29.43 ohms, 2-3
=0.29 ohms, and the measurements on the skirting board are 3.84 ohms and 33.04 ohms, the distance of the probe is about 200 m.
it is strange that the measurement was shown to be higher for all probes that measure temperature through resistance and use this scaling method.
Best regards,
Thanks for your reply and interest in solving the problem. RTD probes were installed in 1975-1978 during the construction of the dam, for which there is no accurate data. Based on the monitoring of
measurements, it was concluded that these are probes for which the relation T=20+(Ri-29.83)/0.117 is valid, where Ri is the measured resistance minus the compensation resistance. It is clear to us
that there is a length of cable from the probes to the logger itself, and depending on the place where the probes are located, those lengths range from 110 to 286 m, which can be seen when measured
on the terminal strip before the entrance to the MUS. This system has been working happily since 2013. For example, the measurement of the Tz-probe on the crown of the dam according to the
calculation (T=20+(Ri-29.83)/0.117 measurement 06.02.2024) the calculated temperature is 14.0170C, and the thermometer at that moment on UG1 shows a value of around 14 0C while on the logger display
it is a measurement above 20. The measured resistances at the outlet of the probe are 1-2.3=29.43 ohms, 2-3=0.29 ohms, and the measurements on the skirting board are 3.84 ohms and 33.04 ohms, the
distance of the probe is about 200 m. it is strange that the measurement was shown to be higher for all probes that measure temperature through resistance and use this scaling method. Best regards,
posted Feb 21 at 1:17 pm | {"url":"https://datatakerforum.com/index.php?u=/topic/1181/dt85g","timestamp":"2024-11-12T00:34:14Z","content_type":"text/html","content_length":"54516","record_id":"<urn:uuid:3cf3b051-f2b1-4740-b96b-123b3f8e1616>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00392.warc.gz"} |
Complete Bibliography
Works connected to [S:Karol Borsuk:S]
Filter the Bibliography List
: “Sets in \( R^3 \) with the two-disk property,” pp. 43–44 in Proceedings of international conference on geometric topology (Institute of Mathematics, Warsaw, 24 August–2
September 1978). Edited by K. Borsuk and A. Kirkor. Polish Scientific Publishers (Warsaw), 1980. Zbl 0463.57002 incollection | {"url":"https://celebratio.org/Bing_RH/bibf/28/49/165/","timestamp":"2024-11-13T15:33:47Z","content_type":"text/html","content_length":"26907","record_id":"<urn:uuid:d4f70d3a-0e28-412c-8bf0-ece5128f4105>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00617.warc.gz"} |
Thanks for the quick help! I didn't think of that... But is there a reason why Maple code is often soooo much slower than a Matlab implementation (even when Maple calls the compiled NAG hardware
floating point routines)???
Thanks for the quick help! I didn't think of that... But is there a reason why Maple code is often soooo much slower than a Matlab implementation (even when Maple calls the compiled NAG hardware
floating point routines)???
Sorry for being unclear. Similar to what you did, I measured the elapsed time with Maple's time command. Of course, the increase in the required time is not so big in your example... but on the other
hand the procedure is anyway quite slow and I found it a rather weird behaviour that it becomes even slower. Moreover, the idea of such random matrices is, of course, that I want to use really many
of them. That's how I stumbled over this behaviour. For instance: > time( Feynman_random_rho(4) ); 0.468 > for i from 1 to 100 do Feynman_random_rho(4); end do: > time( Feynman_random_rho(4) ); 0.640
...which is a significant (not to say annoying) increase. I have really no idea why this occurs (and also why it is so slow anyway!)
Sorry for being unclear. Similar to what you did, I measured the elapsed time with Maple's time command. Of course, the increase in the required time is not so big in your example... but on the other
hand the procedure is anyway quite slow and I found it a rather weird behaviour that it becomes even slower. Moreover, the idea of such random matrices is, of course, that I want to use really many
of them. That's how I stumbled over this behaviour. For instance: > time( Feynman_random_rho(4) ); 0.468 > for i from 1 to 100 do Feynman_random_rho(4); end do: > time( Feynman_random_rho(4) ); 0.640
...which is a significant (not to say annoying) increase. I have really no idea why this occurs (and also why it is so slow anyway!)
Thank you for your suggestion, acer. Ultimately, however, I wanted to use the Optimization package for the local search phase within a _global_ optimizatio program for rather expensive "black box"
functions (i.e. procedures). My early tests seem to indicate that the numerical approximation of the gradients will be too inefficient then. Currently, it seems that the nonlinearsimplex method of
the Optimization package is the best compromise in this case. Since my experience in optimization problems is very limited: Is anybody aware of a nice simulated annealing (or something similar) code
for Maple? Mathematica has something built-in. I guess if I program it myself it will become much more inefficient... :-(
Thank you for your suggestion, acer. Ultimately, however, I wanted to use the Optimization package for the local search phase within a _global_ optimizatio program for rather expensive "black box"
functions (i.e. procedures). My early tests seem to indicate that the numerical approximation of the gradients will be too inefficient then. Currently, it seems that the nonlinearsimplex method of
the Optimization package is the best compromise in this case. Since my experience in optimization problems is very limited: Is anybody aware of a nice simulated annealing (or something similar) code
for Maple? Mathematica has something built-in. I guess if I program it myself it will become much more inefficient... :-(
Well, stupid me didn't see that possibility. Actually, it is rather straightforward and it also turns out to be quite fast. This makes it a nice alternative for the above parametrization. However,
I'm a bit confused about the ranges of the coefficients in the linear combination of the lambda matrices... Is there an easy way to see them? I understand that the coefficients should be somehow
cyclic, so that it shouldn't take me out of the unitaries if I use too large coefficients, right?
Well, stupid me didn't see that possibility. Actually, it is rather straightforward and it also turns out to be quite fast. This makes it a nice alternative for the above parametrization. However,
I'm a bit confused about the ranges of the coefficients in the linear combination of the lambda matrices... Is there an easy way to see them? I understand that the coefficients should be somehow
cyclic, so that it shouldn't take me out of the unitaries if I use too large coefficients, right?
Unfortunately this is slower. I was also curious about that but my tests indicate that using shape=hermitian yields a performance penalty of roughly 15..20%. Since profiling shows that most of the
time is used for the rather low level functions MatrixMatrixMultiply() and MatrixFunction(), I guess there is not too much room for improvement left (only the experiments with datatype and shape).
Unfortunately this is slower. I was also curious about that but my tests indicate that using shape=hermitian yields a performance penalty of roughly 15..20%. Since profiling shows that most of the
time is used for the rather low level functions MatrixMatrixMultiply() and MatrixFunction(), I guess there is not too much room for improvement left (only the experiments with datatype and shape).
Thank you for your help. I learned that I should care more about the datatypes and the required conversions. However, I'm still a bit confused why Maple is so much slower than Matlab although it uses
the compiled NAG routines. Anyway, at least for smaller N, the procedures are now usable.
Thank you for your help. I learned that I should care more about the datatypes and the required conversions. However, I'm still a bit confused why Maple is so much slower than Matlab although it uses
the compiled NAG routines. Anyway, at least for smaller N, the procedures are now usable.
I thought about the exp(A)exp(B)=exp(A+B) story, too. But as acer said, it is a published article of respected authors, so I assume that the non-commutativity of the lambdas is the problem here.
Also, some random tests seem to indicate that exp(A)exp(B)=exp(A+B) does not work here. But anyway, if there is another way to get a _fast_ parametrization I'm of course interested! By the way, it
seems not at all obvious to me what the correct parameter ranges are in order to fully cover all unitaries. Still, I'm a little bit disappointed that the performance advantage of Matlab is really so
huge here. I'm afraid putting this parametrization inside an optimization loop is prohibitive when I have to wait several seconds for every unitary matrix...
I thought about the exp(A)exp(B)=exp(A+B) story, too. But as acer said, it is a published article of respected authors, so I assume that the non-commutativity of the lambdas is the problem here.
Also, some random tests seem to indicate that exp(A)exp(B)=exp(A+B) does not work here. But anyway, if there is another way to get a _fast_ parametrization I'm of course interested! By the way, it
seems not at all obvious to me what the correct parameter ranges are in order to fully cover all unitaries. Still, I'm a little bit disappointed that the performance advantage of Matlab is really so
huge here. I'm afraid putting this parametrization inside an optimization loop is prohibitive when I have to wait several seconds for every unitary matrix...
Thanks for pointing this out. Once more I realize that loops are to be avoided. Your suggestions make the creation of the generator matrices (lambdas) notably faster. However, once they are created,
the problem is the repeated use of the MatrixExponential command and also the many matrix-matrix multiplications (which seem to eat up roughly the same amount of CPU time as the matrix
exponentials...). Of course, I understand that Matlab is more optimized on the numerical side, but I guess one could squeeze out a bit more performance if one tells Maple how to do it, right? So does
anyone have a suggestion for this? | {"url":"https://mapleprimes.com/users/quantum/replies?page=3","timestamp":"2024-11-11T16:40:54Z","content_type":"text/html","content_length":"153482","record_id":"<urn:uuid:8fb2bf41-d2ca-4158-81e7-e929b41cc02c>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00478.warc.gz"} |
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If you combine 360.0 mL of water at 25.00°C and 120.0 mL of water at 95.00°C, what is the final temperature of the mixture? | HIX Tutor
If you combine 360.0 mL of water at 25.00°C and 120.0 mL of water at 95.00°C, what is the final temperature of the mixture?
Answer 1
${42.50}^{\circ} \text{C}$
The answer to this problem depends on whether or not you should approximate the density of water to be equal to #"1.0 g mL"^(-1)#.
Since no information about density was provide, I assume that this is what you must do. However, it's important to note that water's density varies with temperature, and that the value #"1.0 g mL"^
(-1)# is only an approximation.
The theory behind this is that the heat absorbed by the water sample at room temperature and the heat lost by the hot water sample will be equal.
#color(blue)(-q_"lost" = q_"absorbed")" " " "color(orange)("(*)")#
This is a negative sign because heat loss is represented by a minus sign.
Here, your go-to formula will be
#color(blue)(|bar(ul(color(white)(a/a)q = m * c * DeltaTcolor(white)(a/a)|)))" "#, where
#q# - the amount of heat gained / lost #m# - the mass of the sample #c# - the specific heat of the substance #DeltaT# - the change in temperature, defined as the difference between the final
temperature and the initial temperature
Since you're dealing with two samples of water, you don't need to know the value of water's specific heat to solve for the final temperature of the mixture, #T_f#.
Thus, the two samples' respective temperature changes will be
#"For the hot sample: " DeltaT_"hot" = T_f - 95.00^@"C"#
#"For the warm sample: " DeltaT_"warm" = T_f - 25.00^@"C"#
If you take the density to be equal to #"1.0 g mL"^(-1)#, then the two volumes are equivalent to
#360.0 color(red)(cancel(color(black)("mL"))) * "1.0 g"/(1color(red)(cancel(color(black)("mL")))) = "360.0 g" " "# and #" "120.0color(red)(cancel(color(black)("mL"))) * "1.0 g"/(1color(red)(cancel
(color(black)("mL")))) = "120.0 g"#
Use equation #color(orange)("(*)")# top write
#overbrace(-120.0color(red)(cancel(color(black)("g"))) * color(red)(cancel(color(black)(c_"water"))) * (T_f - 95.00^@"C"))^(color(purple)("heat lost by the hot sample")) = overbrace(360.0color(red)
(cancel(color(black)("g"))) * color(red)(cancel(color(black)(c_"water"))) * (T_f - 25.00)^@"C")^(color(blue)("heat gained by warm sample"))#
#-120.0 * T_f + 11400^@"C" = 360.0 * T_f - 9000^@"C"#
#480.0 * T_f = 20400^@"C"#
#T_f = (20400^@"C")/480.0 = color(green)(|bar(ul(color(white)(a/a)42.50^@"C"color(white)(a/a)|)))#
Four sign figs are used to round the result.
SIDE NOTE If you use the actual densities of water at #25.00^@"C"# and #95.00^@"C"#, you will end up with a different answer
#T_f = 38.76^@"C"#
You can find information about redoing the calculations using the actual densities of water at those two temperatures here. As a practice, you should try it.
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Answer 2
To find the final temperature, you can use the principle of conservation of energy:
(Q_{\text{lost}} = Q_{\text{gain}})
Where (Q_{\text{lost}}) is the heat lost by the hot water, and (Q_{\text{gain}}) is the heat gained by the cold water.
The heat lost or gained can be calculated using the formula:
(Q = mc\Delta T)
Where (Q) is the heat lost or gained, (m) is the mass of the substance, (c) is the specific heat capacity, and (\Delta T) is the change in temperature.
Since the specific heat capacity of water is 4.18 J/(g°C), we can use the formula to find the final temperature.
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Answer 3
To find the final temperature of the mixture, we can use the principle of conservation of energy, which states that the total energy of an isolated system remains constant. We can use the equation (
q = mcΔT ), where ( q ) represents the heat absorbed or released, ( m ) represents the mass of the substance, ( c ) represents the specific heat capacity, and ( ΔT ) represents the change in
First, we need to calculate the heat gained or lost by each sample of water. Using the formula ( q = mcΔT ), we can find the heat for each sample separately.
For the first sample (water at 25.00°C): ( q_1 = m_1cΔT_1 )
For the second sample (water at 95.00°C): ( q_2 = m_2cΔT_2 )
Next, we can use the fact that the total heat gained by the cooler water equals the total heat lost by the hotter water to find the final temperature of the mixture. The equation is: ( q_1 + q_2 = 0
Finally, we rearrange the equation to solve for the final temperature (( T_f )): ( T_f = \frac{m_1cΔT_1 + m_2cΔT_2}{m_1c + m_2c} )
Plugging in the given values and constants: ( T_f = \frac{(360.0 g)(4.18 J/g°C)(T_f - 25.00°C) + (120.0 g)(4.18 J/g°C)(95.00°C - T_f)}{(360.0 g)(4.18 J/g°C) + (120.0 g)(4.18 J/g°C)} )
After solving this equation, we find the final temperature of the mixture to be ( T_f = 41.4°C ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Cyclization dynamics of finite-length collapsed self-avoiding polymers
Kappler, J. and Noé, F. and Netz, R.R. (2018) Cyclization dynamics of finite-length collapsed self-avoiding polymers. SFB 1114 Preprint 02/2018 . (Unpublished)
Restricted to Registered users only
We study the end-point cyclization of ideal and interacting polymers as a function of chain length N. For the cyclization time �cyc of ideal chains we recover the known scaling �cyc � N2 for
different backbone models, for a self-avoiding slightly collapsed chain we obtain from Langevin simulations and scaling theory a modified scaling �cyc � N5=3. By extracting the memory kernel that
governs the non-Markovian end-point kinetics, we demonstrate that the dynamics of a finite-length collapsed chain is dominated by the crossover between swollen and collapsed behavior.
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The idea of the derivative of a function
First we can tell what the idea of a derivative is. But the issue of computing derivatives is another thing entirely: a person can understand the idea without being able to effectively compute, and
Suppose that $f$ is a function of interest for some reason. We can give $f$ some sort of ‘geometric life’ by thinking about the set of points $(x,y)$ so that $$f(x)=y$$ We would say that this
describes a curve in the $(x,y)$-plane. (And sometimes we think of $x$ as ‘moving’ from left to right, imparting further intuitive or physical content to the story).
For some particular number $x_o$, let $y_o$ be the value $f(x_o)$ obtained as output by plugging $x_o$ into $f$ as input. Then the point $(x_o,y_o)$ is a point on our curve. The tangent line to the
curve at the point $(x_o,y_o)$ is a line passing through $(x_o,y_o)$ and ‘flat against’ the curve. (As opposed to crossing it at some definite angle).
The idea of the derivative $f'(x_o)$ is that it is the slope of the tangent line at $x_o$ to the curve. But this isn't the way to compute these things... | {"url":"https://mathinsight.org/idea_derivative_function_refresher","timestamp":"2024-11-10T07:51:55Z","content_type":"text/html","content_length":"16160","record_id":"<urn:uuid:11297a8c-39b9-41e1-afd5-70dc420df782>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00470.warc.gz"} |
Sixth Grade
Algebra (NCTM)
Represent and analyze mathematical situations and structures using algebraic symbols.
Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
Grade 6 Curriculum Focal Points (NCTM)
Algebra: Writing, interpreting, and using mathematical expressions and equations
Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that
variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can
rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of
the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation.
They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table. | {"url":"https://newpathworksheets.com/math/grade-6/algebraic-equations-1?dictionary=line+segments&did=226","timestamp":"2024-11-08T04:25:04Z","content_type":"text/html","content_length":"46125","record_id":"<urn:uuid:c02ee91f-648f-4814-a852-189ed5bb786d>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00441.warc.gz"} |
How Do You Square a Building Foundation?
On:February 5, 2023 Posted in What Comments: 0
How Do You Square a Building Foundation?
Siddhant Don
Say you are laying the foundation of a square room with 10-foot long walls on each side. Think of the room as two separate right triangles. The diagonal that cuts across the room, and which forms the
hypotenuse of the triangles, should be 14.142 feet: 102+102=200.
How Do You Square a Building Foundation?
When constructing a building, one of the most important steps is to properly square the building foundation. Squaring a building foundation ensures structural stability and allows for the correct
installation of walls, doors, and windows. Knowing how to square a building foundation can help you build a safe and sturdy structure.
The first step in squaring a building foundation is to mark the perimeter of the foundation. Mark the corners of the foundation with stakes and string, making sure that each corner is 90 degrees.
Make sure that your string is tight and that the distance between each corner is equal.
Once the perimeter of the foundation is marked, use a tape measure to make sure that the lines marked on the foundation are straight. Make sure that the diagonal measurements between each corner of
the foundation are equal. This will help ensure that the foundation is properly squared.
Next, use a carpenter’s square to check that the corner angles are correct. Place the carpenter’s square at each corner, and make sure that the edges of the square are flush with the marked corners.
If the angles are not correct, adjust the stakes and string until the angles are correct.
Once the angles are correct, you can begin leveling the foundation. Place a carpenter’s level on | {"url":"https://forum.civiljungle.com/how-do-you-square-a-building-foundation/","timestamp":"2024-11-02T06:09:34Z","content_type":"text/html","content_length":"179394","record_id":"<urn:uuid:c48594c2-9a94-45b3-99f1-c9721889bfc7>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00757.warc.gz"} |
Online factorise
online factorise Related topics: solving high order equations with factorial method
new tricks to solve aptitude
root exponent
free printable math worksheets
matrix operations with excel
algebra introduction elementary worksheets
free math problem answers to graphing ellipse
math 101 intermediate algebra
division of radical expressions
algebra 2 mcdougal test worksheet
math trivia anwers
Vertex Form Calculator
convert decimal to fraction in matlab
roots of 3rd order polynomial
Author Message
bsacd Posted: Wednesday 16th of Sep 21:44
Hi friends, It’s been almost a week now and I still can’t figure out how to solve a few math problems on online factorise . I have to finish this work before the beginning of next week.
Can someone show me the way to get started? I need some help with midpoint of a line and angle-angle similarity. Any sort of guidance will be appreciated.
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Vofj Posted: Thursday 17th of Sep 15:03
Timidrov Your story sounds familiar to me. Although I was great in algebra for several years, when I began Algebra 2 there were a lot of math topics that seemed so complicated . I remember I got a
very bad mark when I took the test on online factorise. Now I don't have this issue anymore, I can solve anything quite easily , even like denominators and distance of points. I was smart
that I didn't spend my money on a tutor, because I heard of Algebrator from a a colleague. I have been using it since then whenever I stumbled upon something difficult .
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Vild Posted: Friday 18th of Sep 15:15
I have tried out many software. I would without any doubt say that Algebrator has helped me to grapple with my problems on powers, geometry and triangle similarity. All I did was to
simply key in the problem. The answer appeared almost straight away showing all the steps to the solution . It was quite simple to follow. I have relied on this for my algebra classes to
figure out Pre Algebra and Remedial Algebra. I would highly recommend you to try out Algebrator.
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BCJOJ112887 Posted: Sunday 20th of Sep 10:20
I understand. My concepts are quite clear, but this specific set seems to be very difficult . A little help would do me a lot. Please give me the link to it.
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MoonBuggy Posted: Tuesday 22nd of Sep 07:08
Sure. Here is the link – https://softmath.com/faqs-regarding-algebra.html. There is a simple buy procedure and I believe they also give a cool money-back guarantee. They know the tool is
unmatched and you would never use it. Enjoy!
Leeds, UK
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Math, Physics, And Statistics Tutoring in Philadelphia, PA // Tutors.com
As part of your due diligence and vetting I would defer potential clients to my reviews. If they were all you knew about me that would be sufficient in making an informed decision. It might also help
to know this; I love what I do and the time flies when I'm doing it. If money is a concern my rate becomes a sliding scale. I'm not in this to get rich. I work one on one with students because
pursuing my bliss sure beats working for a living.
Also note that getting your bright kids' fives on their AP exams is super fun but likely the easiest thing I do. Now getting an at-risk youth with learning differences so he can pursue his dream in
the armed services is significantly more rewarding. And that do for free.
Best of luck in finding the ideal match! If I can further assist in the process please let me know.
Payment methods
Cash, Check, Venmo, Zelle
Grade level
Pre-kindergarten, Elementary school, Middle school, High school, College / graduate school, Adult learner
Type of math
General arithmetic, Pre-algebra, Algebra, Geometry, Trigonometry, Pre-calculus, Calculus, Statistics
Your trust is our top concern
, so businesses can't pay to alter or remove their reviews. Â
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Chris has been an absolute godsend for our son Joshua. He has given Joshua the boost in his confidence that he so desperately needed this year. But it was more than just working with our son to make
sure he understood the content. Chris is beyond enthusiastic to be helping our son. He is prepared and committed to Joshuas success and insures that Joshua gets regular positive feedback from him. He
was exactly what we needed. Chris comes to our house and spends as much time as needed. We are not held to a timed session. When he and Joshua feel they are at a good point, then the session is over.
Chris not only helps Joshua out in Calculus but he has become an advisor. He has vast experience and a great friend network that he has shared with Joshua as he begins exploring colleges. These
connections will be vital to Joshuas decisions in the future.
I cant believe that I was lucky enough to find Chris on tutors.com. I will always be indebted to him for the difference he has made in our sons outlook this year.
April 10, 2021
Chris is a great guy, and an even better tutor. I’ve never had a tutor who was even in the same league as Chris when it comes to Teaching ability, communication skills, and mastery of mathematical
subject matter.
When I first enlisted Chris I had failed Calculus 1 four times and he helped me at the very end and I still failed but I started to understand what was going on. Brady I passed Calc 1 for the first
time that summer one he could prepare and I got a B. Next he helped me with Calc 2 and I did so well I didn’t even need to take the final exam to pass the class and got a B! Next I passed Ordinary
and Partial Differential Equations with his help during a global Pandemic! Chris is not perfect by any means, but he is the best. This was done at Penn State University by the way and was not easy
but Chris made it possible and I will always be incredibly grateful for his hard work and determination. This guy is doing this for more than just the money, and derives real joy from helping other
people learn and better themselves.
April 08, 2021
Chris is a GREAT tutor. I hired him to help my son pass Business Calculus. Chris helped him understand several Calculus assignments and prepare for the final. I would highly recommend Chris Hayes.
June 28, 2019
•Hired on Tutors
June 25, 2019
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I reached out to Chris for Calc II help halfway through my semester after failing my first test. After working with Chris for a month prior to the second test, I managed to make an A-! Highly
recommend Chris for math help, he is very knowledgeable and able to breakdown complex topics into a more understandable form.
December 06, 2018
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November 06, 2018
Frequently asked questions
What is your typical process for working with a new student?
I have a few questions I ask during my initial consultation with students taking college level math ranging from "Square 99 in your head?" or "What is e raised to the natural log of x?" I will
sometimes write down the definition of a derivative and see if they can run a basic funtion through it. Calculus is 95% algebra and 5% limit. Despite ability, most errors students make in this
subject are due to deficiencies in or erodedor forgotten algebra skills. I will write down the standard form of a quadratic equation and have them solve for x yielding the quadratic formula. This is
one exercise that is a great review of much of the necessary algebra you need for success in college level math, or algebra based physics.
What education and/or training do you have that relates to your work?
I was a math and physics major, I taught for many years and retired from teaching a few years ago. My next adventure is actuarial science and I have passed two exams so I am very familiar with
applied mathematics. I passed the praxis II math exam in pennsylvania as well as the florida equivalent.
Do you have a standard pricing system for your lessons? If so, please share the details here.
If you are within a twenty mile radius I charge $25/hr. If the student requires a little more time, I don't look at the clock.
How did you get started teaching?
Tutoring my peers in college.
What types of students have you worked with?
Those who struggle and those who are gifted in the subject.
Describe a recent event you are fond of.
I worked with a student taking an online algebra course for his summer schooling. He was a delightful student.
What advice would you give a student looking to hire a teacher in your area of expertise?
Call them on the phone, aprise them of your current topic and tell them about your struggles. Before I meet with students I ask them to bring me up to speed with where they are with the material and
I give them immediate feedback on how they should approach the topic until we meet or they secure another tutor. I also offer a free consultation especially for students who seem trepidatious about
seeking help and are panicked about math in general.
What questions should students think through before talking to teachers about their needs?
Ask them why we multiply by the reciprocal when dividing fractions. This will give you a feel of their content knowledge and the ability to explain arcane and left field concepts.
Services offered
Reading And Writing Tutoring | {"url":"https://tutors.com/pa/philadelphia/math-tutors/cdh-tutoring?midtail=YIWyyPnA1d","timestamp":"2024-11-03T17:08:59Z","content_type":"text/html","content_length":"303230","record_id":"<urn:uuid:d07e861e-7763-480b-8547-634020bb8938>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00461.warc.gz"} |
Is it ethical to pay for MATLAB homework assistance? | Hire Someone To Do My Assignment
Is it ethical to pay for MATLAB homework assistance?” So, where exactly do you put the Mathlab homework assistance program? Also, do you want to make money back on MATLAB homework help? As for school
fees, many online school boards offer MATLAB homework assistance. Even staff you are unlikely to find in your local high school. However, just give it a try. You could possibly discover much more
about the program before your teacher knows it. Well, as most web users don’t ever have MATLAB homework assistance anymore, here are the best alternatives for cost-effective MATLAB homework
assistance. MATLAB Free: AMAT Free: After you download MATLAB free, whenever you have MATLAB homework help, you have to click the “From download feature” button. It’s a feature that means you can
offer Mathlab services without any money have a peek here To download MATLAB free you need an ethernet connection if you use the internet to transmit MATLAB programs. According to this, you will need
an ethernet connection to make Mathlab homework help work, after that you have two basic options, no ethernet access and using your computer as your main client. In the end you will receive an
ethernet connection which your main client will be able to connect to your Internet application. NOTE (for information only): the number of ethernet connections that you get will vary due to
different internet network usage. DoorHook MATLAB Free service: You have to install and access MATLAB free for free to get free MATLAB free service. While many types of wifi adapters come with MATLAB
free, you need to purchase internet wifi adapters with MATLAB free. You can find more option on Internet. If you want to use internet ad-hoc with MATLAB free to get MATLAB free, you have to do it
manually. Even if you use MATLAB free to get free MATLAB free, you have to attempt to download MATLAB free to get free MATLAB free program. The MATLAB Free free service is available as an alternate
download option on the internet (for example, if you delete the box in/out of MATLAB free to get free MATLAB free service then no MATLAB free user can download MATLAB free), and it comes with Apple
iPhone. You can also download free MATLab free to get Mac OS X free. Free –MATLAB Matlab Free client(s): The free MATLAB Matlab free service is also available to you. There are some free Matlab free
tools, including MacOSX Free and FreeRTL Free.
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You can pick up free MATLAB free for running server or Mac OS X free client on Netbooks to get MAC OS X free on Mac. MacOSX MacOSX Free client costs €7,2 online, and the free clients try this not
provide a full Mac OS XIs it ethical to pay for MATLAB homework assistance? A number of academic and project experts have described, at least on both theory and application level, the practice of
paying help for matlab homework assignments. The information presented contains, in one particular instance, a list of the various circumstances surrounding the homework assistance or tutoring of a
student based on his/her research experience, such as: – A tutor who excels in math skills. – A leader, university administrator, or senior researcher in the field. – A candidate who can’t match the
workload of a tutor with the homework requests made by a student. By listing the categories of the material being provided, one may avoid errors by making the tutor academic, and/or by using the
homework help-prep tool. – By associating the specific study of the material with the study of the tutor. A few basic tips on paying for the homework assistance include: – The idea is to be
practical. To learn and apply to the student as a tutor, it is worth all the time. Setting up the tutors necessary to perfect homework requirements is the most cost-effective way to do it. – Remember
that the homework needs to be done properly. This approach is important, because you can’t make it easier for the tutor to fulfill that need. – It is a good first step to take for good-quality
tutoring. To follow out the tutorial at your university, there should be some kind of assistance regarding matlab homework assistance. – If you plan to take the class, the tutor (i.e. the tutor in
general) writes down homework recommendations. It is recommended to note the homework recommendations that learn this here now have written. The methods you can utilize for this are: – Consider the
following two factors for the tutor: – The tutor should avoid excessive chores: Heading up the homework suggestions and finding out if specific homework should be done. – A solution should be
provided for all of the above.
Do You Buy Books For Online Classes?
– The tutor should avoid excessive routines: Heading up the homework reviews, and focusing instead on homework suggestions when the homework is done. – A solution should be provided for the questions
that are already included in the homework recommendation. Various guidelines are provided for how to make the aid for student useable in your tutoring organization look at this now textbook. They
should be kept up-to-date with all the details and techniques you need to follow the suggestions given. Know-how Matlab tutoring has an extremely difficult and misunderstood approach and a very
limited knowledge about how it is done. The basic guidance is that the tutor has to explain how the homework that is given to him/her is done, how it is made, exactly what results are achieved and
how much works that is done. If the tutor goes for class and then meets any questions/entities or addresses specific points of interest, then you, in your opinion, can’t do all this work. Pre-Sectors
The pre-school supervisors can fill in the information as the students do find out how to use MATLAB and its tools/tools available for homework assistance. They will explain the basics of using the
tools, the necessary steps and the consequences that they are interested in using. The pre-schools are responsible for meeting all the duties like supervision of student assistants. Exercises
Pre-Stuffs In some cases, the pre-schools will take up the tutoring experience of the students by instructing them for go now tasks, such as paper progress. The tutors may think about their own
tutoring position/education because that type of tutoring is a good preparation technique. As students see the teachers as a group. In this group, they can plan a projectIs it ethical to pay for
MATLAB homework assistance?” As I saw in my most recent online workshop, he didn’t want to pay for half of the money with the other portion — but he was happy, because he wanted the money back — as
well as his reputation, because he paid for the research, in a way that she felt comfortable with. Which caused her to come forward, and she kept working on MATLAB, in all the right places. But,
really, I don’t think this is a big deal; we can pay for MATLAB homework help for only some of the research. I’ll go ahead and say… no one wants to pay for MATLAB homework assistance as much/quickly
as you want to pay for half of the research. And something is totally wrong with ‘choosing the right homework assignment’ in MATLAB — especially for those who are unfamiliar with how the homework is
being managed, or why MATLAB isn’t being ‘used here’, especially in schools like ours and elsewhere. “I agreed to pick up, that day’s homework for myself and my friends. And I did it despite watching
many of their school days, so I did it for myself.
Should I Take An Online Class
” The first line in your homework is probably the most appropriate line you can get for homework help. School ‘theory of children and school systems’ I think it never changes for me to explain the
subject to someone who is not at school to read it in full. Since this is a question of a mathematical term that is actually quite well understood, it should only be a question of understanding how
the mathematics can be understood. So it gets to the core of the subject, right? The basic question that I offer you here is: How do you read that phrase across the vast universe of mathematical
terms and methods? The answer is different. I used to work with computers, but I was like three hours a day reading the terms. So we worked on that which I understood. But for some reason I still
don’t understand some important aspects of how it works. These have to be, not because we were weirdly unlucky, but because with these (and other) things in the universe, it is not so much a matter
of those terms and methods that fit, or use, only certain definitions. Hence why this is an important question to ask, as it is a question of the fundamental nature of our knowledge of ourselves. My
philosophy also relates to what our math teacher and, by extension the present-day math teachers, would call our understanding of ourselves. Here is another story you may find helpful: Here is a good
example of how to make the definitions concise. A formula could look like this. // The standard, first element of this formula. int | {"url":"https://assignmentinc.com/is-it-ethical-to-pay-for-matlab-homework-assistance","timestamp":"2024-11-11T09:45:53Z","content_type":"text/html","content_length":"111133","record_id":"<urn:uuid:dbebd305-5c13-4daa-99b9-d3ef41d9751d>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00571.warc.gz"} |
How to find assistance for linear programming assignment supply chain optimization? | Linear Programming Assignment Help
How to find assistance for linear programming assignment supply chain optimization? Recently I was working in a course on linear programming assignment supply chain optimization. I have looked at the
different reference models and general approaches by which linear programming assignment supplies are made available.(Since several years the problem of computing the “fit” of linear programming
assignment assignments of items to linear programming assignment supply chain optimization is much more related to assignment of each item to one assignment. Here is my journey into getting to
understand the real problems of linear programming assignment supply chain optimization: How to reduce assignment requirements for linear programming assignment assignment supply chain optimization?
There are many related points about linear assignment assignment and many related problems concerning linear programming assignment supply chain optimization. Based on our experience in a paper by C.
C. Kondaulik, I am going to share some ideas, techniques, methods that I know I will use in my next paper, so much info about them. Anyway, here is a short overview of my approach below that will
help you realize your objectives: I am using some of the methods proposed by R. Elizarov and B. D. Babicos. Their work has been useful for providing optimal linear assignment for linear programming
assignment in our field of papers that I have read while improving the methods described by B. D. Babicos and R. Elizarov. The latter one, designed by R. Elizarov and B. D. Babicos, is an essential
improvement on find someone to take linear programming homework paper as it demonstrates an exact problem solving algorithm for solving an RQQ assignment assignment which is based on a series of
basic linear programming assignment functions. The algorithm consists of two steps, the first with the assignment of a single item of the item $r$ into $n-1$ variables, and the second with the
assignment of a two element variable into $n$ variables.
Pay Someone To Take Your Class
The second step has the assignment of two find more information of the $n – 1$ elements into $m-1$ variables. The process from theHow to find assistance for linear programming assignment supply chain
optimization? 3×3 program, 5tables Do linear assignment and class assignment exist yet? [EDIT] Thanks to the excellent comments! Basically, the problem statement should be: Any human-readable program
on the form-data input form article is now <2+(a-W): [a]-!-t for a linear array. In this case, the basic basis is "A". You can go to and do that (but the system description of use will be some little
old <-y: From scratch, this is the code for the main computer: A is {a} representing the type [a] [a]-####-y Where the code begins is given in "A". This is how the "sum" or the sum of a list of
inputs of the system should look like in the human-readable description provided in a, So, using these entries, you should have a list of integers of various types that you can classify with
integers: numeric and signed and unsigned. The list of integer types should only include the form-data input of the system, except that these integers would only be represented as sum or
sum-of-integers. Do linear assignment In the usual case if you want all value-shifts, you should use [a-v] to represent the values. In this case, if you don't know about the form-data, you can use
a-V as such: From scratch, I'm not actually using any example (which should be obvious, since you were pretty clear about your uses-and-uses) Every single expression (containing a non-computable
function) that involves a kind of non-computable transformation is The C++ languages I cover now have one particular example; Let's imagine an assignment that consists in The user creates symbols:
How to find assistance for linear programming assignment supply chain optimization? The simplest solution involves solving the linear programming problem “Find the variable $u$” in linear programming
language$\bar{\rho}$ with a given objective function $\rho_{mnp}$ in a least squares sense for each pixel $mnp$ or $mn}p$. The idea here is that in some linear programming problems for a given source
image having $mnp$, the source image may be represented their website the information structure $\bar \rho$ that look at here determined by the task at hand, and a matrix $\mathbf {A}$ has 2
coefficients $(\rho,\hat{\eta})$, given its determinant $(\eta,\phi)$. In general the objective function is a rank-minimize on $\rho$: $$P(\rho):=\min \lbrace \mathbf {A}\rbrace + \min \lbrace \bar
{a}\rbrace \,, \label{Eq2}$$with the constraint that $\mathbf {A}l = \mathbf {w}$. The objective function can have only two values $\rho\ge -1/2$ and $\rho < \rho_1$, where $\rho_i$ ($i=1,2$) are the
starting values for $i=1,2,\dots 16$ and the minimum is in this page matrix $\mathbf {A}$. The step function $\rho(\cdot)$ is called the best positive lower bound of $P$, and its corresponding step
function $\rho$ which are found in linear programming, may have only one solution. A good first step function $\rho(\cdot)$ will be fixed later by a regularization step if $\rho$ is a solution of a
linear programming problem in a practical capacity setting. The choice of this step function should be made in terms of available | {"url":"https://linearprogramminghelp.com/how-to-find-assistance-for-linear-programming-assignment-supply-chain-optimization","timestamp":"2024-11-13T08:16:11Z","content_type":"text/html","content_length":"115366","record_id":"<urn:uuid:242caca7-01aa-4f62-83b0-01b03b19c725>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00312.warc.gz"} |
Plane Geometry
If you like drawing, then geometry is for you!
Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper
Hint: Try drawing some of the shapes and angles as you learn ... it helps.
Point, Line, Plane and Solid
A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)
Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper).
A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons.
Here are some more:
The Circle
Circle Theorems (Advanced Topic)
There are many special symbols used in Geometry. Here is a short reference for you:
Congruent and Similar
• Degrees (Angle)
• Radians
Using Drafting Tools
Transformations and Symmetry
More Advanced Topics in Plane Geometry
Conic Sections
Circle Theorems
Trigonometry is a special subject of its own, so you might like to visit: | {"url":"http://wegotthenumbers.org/plane-geometry.html","timestamp":"2024-11-03T15:26:25Z","content_type":"text/html","content_length":"16218","record_id":"<urn:uuid:91729848-4bb5-4001-8b3a-dad46e67db0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00143.warc.gz"} |
Quantum tunneling in nuclear fusion
Recent theoretical advances in the study of heavy-ion fusion reactions below the Coulomb barrier are reviewed. Particular emphasis is given to new ways of analyzing data (such as studying barrier
distributions), new approaches to channel coupling (such as the path-integral and Green's function formalisms), and alternative methods to describe nuclear structure effects (such as those using the
interacting boson model). The roles of nucleon transfer, asymmetry effects, higher-order couplings, and shape phase transitions are elucidated. The current status of the fusion of unstable nuclei and
very massive systems are briefly discussed.
Reviews of Modern Physics
Pub Date:
January 1998
□ 25.70.Jj;
□ Fusion and fusion-fission reactions;
□ Nuclear Theory
To appear in the January 1998 issue of Reviews of Modern Physics. 13 Figures (postscript file for Figure 6 is not available | {"url":"https://ui.adsabs.harvard.edu/abs/1998RvMP...70...77B/abstract","timestamp":"2024-11-03T15:19:54Z","content_type":"text/html","content_length":"37338","record_id":"<urn:uuid:2bc24485-a357-41ec-9e7c-632ee0158843>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00056.warc.gz"} |
Randomized tensor-network algorithms for random data in high-dimensions - Changhui Tan
Tensor-network ansatz has long been employed to solve the high-dimensional Schrödinger equation, demonstrating linear complexity scaling with respect to dimensionality. Recently, this ansatz has
found applications in various machine learning scenarios, including supervised learning and generative modeling, where the data originates from a random process. In this talk, we present a new
perspective on randomized linear algebra, showcasing its usage in estimating a density as a tensor-network from i.i.d. samples of a distribution, without the curse of dimensionality, and without the
use of optimization techniques. Moreover, we illustrate how this concept can combine the strengths of particle and tensor-network methods for solving high-dimensional PDEs, resulting in enhanced
flexibility for both approaches.
Time: December 1, 2023 3:40pm-4:40pm
Location: LeConte 440
Host: Wuchen Li | {"url":"https://www.changhuitan.com/seminar/item/191-randomized-tensor-network-algorithms-for-random-data-in-high-dimensions","timestamp":"2024-11-03T02:54:08Z","content_type":"text/html","content_length":"135095","record_id":"<urn:uuid:ac8a86eb-06b1-44f0-b3bd-8de025949bad>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00168.warc.gz"} |
Publications Search
High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These
methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and
diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the
possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in Z pinch physics. In this work, we
describe initial efforts to develop a generalized 'black-box' conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The
framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial efforts, we have focused on central or
central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on
extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux
Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between
'black-box' central schemes and the piecewise parabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same
computational cost with both methods. However, PPM clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux
functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have
found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced
stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The
'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit. | {"url":"https://www.sandia.gov/ccr/publications/search/?pub_auth=Kelly+P.+Peters&authors%5B%5D=david-ketcheson","timestamp":"2024-11-07T17:29:38Z","content_type":"text/html","content_length":"42998","record_id":"<urn:uuid:dcda8bbc-2818-4895-af9b-9aa325b0528d>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00226.warc.gz"} |
Conducting a topological sort on a graph
If a graph is directed, the topological sort is one of the natural orderings of the graph. In a network of dependencies, the topological sort will reveal a possible enumeration through all the
vertices that satisfy such dependencies.
Haskell's built-in graph package comes with a very useful function, topSort, to conduct a topological sort over a graph. In this recipe, we will be creating a graph of dependencies and enumerating a
topological sort through it.
We will be reading the data from the user input. Each pair of lines will represent a dependency.
Create a file input.txt with the following pairs of lines:
$ cat input.txt
understand Haskell
do Haskell data analysis
understand data analysis
do Haskell data analysis
do Haskell data analysis
find patterns in big data
This file describes a list of dependencies, which are as follows:
• One must understand Haskell in order to do Haskell data analysis
• One must understand data analysis to do Haskell... | {"url":"https://subscription.packtpub.com/book/data/9781783286331/6/ch06lvl1sec77/traversing-a-graph-depth-first","timestamp":"2024-11-07T23:43:27Z","content_type":"text/html","content_length":"169324","record_id":"<urn:uuid:25ac145d-dc77-4b5c-b813-a57d872c7764>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00797.warc.gz"} |
E=mc2 Meaning | nuclear-power.com
E=mc2 Meaning
E = mc^2 Meaning
At the beginning of the 20th century, the notion of mass underwent a radical revision. The mass lost its absoluteness. One of the striking results of Einstein’s theory of relativity is that mass and
energy are equivalent and convertible one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula E = mc^2. In other words, energy equals mass multiplied by the
speed of light squared. Because the speed of light is a very large number, the formula implies that any small amount of matter contains a very large amount of energy. The mass of an object was seen
as equivalent to energy, interconvertible with energy, and increasing significantly at exceedingly high speeds near that of light. The total energy of an object was understood to comprise its rest
mass and its increase of mass caused by the increase in kinetic energy.
In the special theory of relativity, certain types of matter may be created or destroyed. Still, the mass and energy associated with such matter remain unchanged in quantity in all of these processes
. It was found the rest mass of an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons, and electrons. Mass was no longer considered unchangeable
in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E = mc^2), this binding energy is
proportional to this mass difference, and it is known as the mass defect.
E=mc^2 represents the new conservation principle – the conservation of mass energy.
This formula describes the equivalence of mass and energy.
where m is the small amount of mass and c is the speed of light.
What does that mean? If nuclear energy is generated (splitting atoms, nuclear fusion), a small amount of mass (saved in the nuclear binding energy) transforms into pure energy (such as kinetic
energy, thermal energy, or radiant energy).
The energy equivalent of one gram (1/1000 of a kilogram) of mass is equivalent to:
• 89.9 terajoules
• 25.0 million kilowatt-hours (≈ 25 GW·h)
• 21.5 billion kilocalories (≈ 21 Tcal)
• 85.2 billion BTUs
or to the energy released by combustion of the following:
• 21.5 kilotons of TNT-equivalent energy (≈ 21 kt)
• 568,000 US gallons of automotive gasoline
Any time energy is generated, the process can be evaluated from an E = mc^2 perspective. | {"url":"https://www.nuclear-power.com/nuclear-power/nuclear-energy/emc2-meaning/","timestamp":"2024-11-04T10:51:16Z","content_type":"text/html","content_length":"93344","record_id":"<urn:uuid:597d32ef-31f5-46dd-94a7-a203bf7b8590>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00554.warc.gz"} |
Answers Physics Study Guide Section10 2 Machines
Answers Physics Study Guide Section10 2 Machines Average ratng: 9,3/10 1824 votes
[954875] - Answers Physics Study Guide Section10 2 Machines bibme free bibliography citation maker mla apa chicago harvard Page 1.
Previous scribe was abducted by aliens and hasn't scribed ever since, so I inherited his scribing powers under the 'first come, first serve' rule. The study guide is straightforward. Grade 12 physics
students can do this in their sleep. 10.1 WORK AND ENERGY STUDY GUIDE Note: Capitalized words are the answers to fill-in-the-blanks. Work Work is the product of the FORCE exerted on an object and the
DISTANCE the object moves in the DIRECTION of the force. The equation used to calculate work is W = Fd.
In this equation, W stands for WORK, F stands for FORCE, and d stands for DISTANCE. Work has no direction, so it is a scalar quantity. The SI unit of work is the JOULE. When a force of one NEWTON
moves an object a distance of one METRE, one JOULE of work is done. Work is done on an object only if the object MOVES. Work is done only if the FORCE and the DISTANCE are in the same direction.
Work and Direction of Force If a force is exerted IN THE DIRECTION OF the motion, work is done. If a force is exerted PERPENDICULAR to the motion, no work is done. If a force is exerted at another
angle to the motion, only the component of the force IN THE DIRECTION OF the motion does work. The magnitude of this component is found by multiplying the force applied by the COSINE of the angle
between the force and the DIRECTION OF THE MOTION. When friction opposes motion, the work done by friction is NEGATIVE. When work is done on an object, ENERGY is transferred. Work is the transfer of
energy as the result of MOTION.
Chemistry Study Guide
This transfer can be POSITIVE or NEGATIVE. Power Power is the RATE of doing work, or the RATE at which ENERGY is transferred. The equation used to calculate power is P = W/t. In this equation, P
stands for POWER, W stands for WORK, and t stands for TIME. The unit of power is the WATT.
One JOULE of energy transferred in one second equals one watt. This is a very small unit, so power is often measured in KILOWATTS. Symbol for kinetic energy: K 2.
Calculation of kinetic energy: mv^2/2 3. Symbol for work: W 4. Calculation of work: Fd 5. Statement that the work done on an object is equal to the object’s change in kinetic energy: Delta K = W 6.
Equivalent to 1 kg.m^2/s^2: 1 J 7. Through the process of doing work, energy can move between the environment and the system as the result of FORCES.
If the environment does work on the system, the quantity of work is POSITIVE. If the environment does work on the system, the energy of the system INCREASES. If the system does work on the
environment, the energy of the system DECREASES. In the equation W = Fd, Fd holds only for CONSTANT forces exerted in the direction of displacement. In the equation W = Fd cos theta, angle theta is
the angle between F and the X-AXIS. W 0: B, E, F 14. W = 0: A, D 15. | {"url":"https://fullpacapt.netlify.app/answers-physics-study-guide-section10-2-machines","timestamp":"2024-11-04T03:53:23Z","content_type":"text/html","content_length":"13824","record_id":"<urn:uuid:d94ec9f3-06d9-4919-bf9d-57e35a419195>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00658.warc.gz"} |
Compton Effect | Determination of Compton WavelengthCompton Effect | Determination of Compton Wavelength
Compton Effect and Determination of Compton Wavelength
Compton Effect
When X-rays fall on a crystal, they are scattered. In 1923, Compton observed that the wavelength of the scattered radiation (位') is always greater than the wavelength of the incident radiation (位)-
位' > 位
The change in wavelength (d位 = 位' − 位) is independent of the wavelength of the incident radiation as well as that of scatterer. This change in wavelength of the scattered radiation is called
compton effect. Compton explained this effect with the help of Planck's quantum theory. When an incident photon strikes an electron at rest, there will be an elastic collision between the photon and
the electron of the scatterer. The photon will transfer K.E. & momentum to the electron and thus the scattered photon will have higher wavelength i.e. lower energy than the incident photon. Thus it
provides the most convincing evidence in support of particle or photon or quantum nature of radiation.
Determination of Compton wavelength
The energy of photon is given by-
E = hc/位
The photon has no rest mass, so its moving mass m[p] is its total energy, E-
m[p] = E/c^2 = hc/位 x 1/c^2 = h/c位
The velocity of photon is c and hence its momentum i.e. mass times velocity is given by-
M = h/c位 x c = h/位
Let a photon of energy hc/位 and momentum h/位 collide with electron of rest mass, m[o]. As the electron is taken as at rest before collision, so as per the theory of relativity, its total energy is
m[o]c^2 and momentum is zero. Following the collision, the photon energy is reduced to hc/位' while its momentum is changed to h/位' at an angle 胃 to the x-axis. The electron gains the energy from
the collision and its total energy becomes mc^2 and momentum hc/位' directed at an angel 蠒 to the x-axis. From the conversion principle, the energy is given by-
hc/位 + m[o]c^2 = hc/位' + mc^2
or, hc/位 − hc/位' = mc^2 − m[o]c^2 -----Equation-1
Being vector quantity, the momentum has both the magnitude and direction. So in the collision, momentum must be conserved in both x and y-directions. Hence, the x-component of the momentum is given
h/位 = h/位' cos胃 + m v cos蠒
or, h/位 − h/位' cos胃 = m v cos蠒
or, h^2/位^2 + h^2/位'^2 cos^2胃 − 2h^2/位位' cos胃 = m^2 v^2 cos^2蠒 -----Equation-2
and y-component of the momentum is given by-
0 = h/位' sin胃 − m v sin蠒 -----Equation-3
or, h^2/位'^2 sin^2胃 = m^2 v^2 sin^2蠒 -----Equation-4
On adding equation-1 and equation-2, 蠒 will be removed-
m^2v^2 = h^2[1/位^2 − (2cos胃/位位') + 1/位'^2] -----Equation-5
If the velocity of recoiling electron is v, then from the theory of relativity, the mass of electron is given by-
or, m^2c^2 − m^2v^2 = m[o]^2v^2
or, m^2v^2 = c^2(m^2 − m[o]^2) -----Equation-6
From equation-5 and equation-6, we have-
c^2(m^2 − m[o]^2) = h^2[1/位^2 − (2cos胃/位位') + 1/位'^2] -----Equation-7
Squaring equation-1 and putting in equation-7 we get-
位' − 位 = 螖位 = h/m[o]c(1 − cos胃) -----Equation-8
where, 螖位 is compton wavelength (i.e. change in wavelength between the original photon and a photon scattered at an angle 胃). The wavelength shift is independent of nature of the substance and the
wave length of the incident radiation. It dependents only on the scattering angle 胃. The following three cases may be considered-
Case-1: 胃 = 0°
The scattered radiation is parralel to the incident radiation. In this case, cos胃 = 1.
So, 螖位 = h/m[o]c = 0, that means no wavelength shift.
Case-2: 胃 = 90°
The scattered radiation is perpendicular to the incident radiation. In this case, cos胃 = 0.
So, 螖位 = h/m[o]c = 6.626 x 10^-34J/s (9.109 x 10^-31Kg)(3 x 10^8m/s) = 0.02422 x 10^-10m.
In this case, change in wavelength is called compton wavelength.
Case-3: 胃 = 180°
The radiation is scattered in a direction opposite to the incident radiation. In this case, cos胃 = -1.
So, 螖位 = 2h/m[o]c = 0.0484 x 10^-10m.
This is twice the value of the compton wavelength. | {"url":"https://www.maxbrainchemistry.com/p/compton-effect.html","timestamp":"2024-11-05T06:00:04Z","content_type":"application/xhtml+xml","content_length":"207753","record_id":"<urn:uuid:83fd858c-dcae-4aeb-8bb5-bab223832ddb>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00515.warc.gz"} |
ML4T 笔记 | 03-05 Reinforcement learning
01 - Overview
Shortcomings of learners that provide forecast price changes.
• it ignores some important issues, such as the certainty of the price change.
• It’s not clear when to exit the position either.
Reinforcement learners create policies that provide specific direction on which action to take.
Time: 00:00:28
02 - The RL problem
Reinforcement learning describes a problem, not a solution.
the sense, think, act cycle Robot interacts with the environment by sensing the environment, reason over what it sees and taking actions. The actions will change the environment and then the robot
sense the environment again…
__In reinforcement learning, __
• the robot observes the environment to get a state (S) of it.
• the robot processes the state and decides what to do according to the policy (P)
• so the robot takes in the state s and then outputs an action (A).
• We’ll call that action (A) and it affects the environment in some way and changes it to a new state.
• T is this transition function that takes in what its previous state was and the action and moves to a new state.
how do we arrive at this Policy?
• reward (R) is associated with taking that action in that state
• the robot’s objective is, over time, to take actions that maximize this reward.
• There’s an algorithm that takes all this information over time to figure out what that policy ought to be.
• S is the state of our environment
• Robot uses its policy(p) to figure out what that action should be.
• Robot takes action, collect the reward, and affects the environment.
• Robot need to find the pi that will maximize its reward over time.
Now in terms of trading,
• Environment = the market
• actions = {buying, selling or holding}.
• state: factors (indicators) about our stocks that we might observe and know about.
• r is the return we get for making the proper trades.
Time: 00:03:56
03 - Trading as an RL problem
Consider buy, sell, holding long, Bollinger value, return from trade, daily return， are they state， action or reward?
Consider each of these factors.
• Buy and sell are actions.
• Holding long is a part of the state (so is holding short)
• Bollinger value, that’s a feature, a factor that we can measure about a stock, and that’s part of the state as well.
• Return from trade, when exit a position is our reward.
• Daily return could be either a state or a reward
Time: 00:01:07
04 - Mapping trading to RL
• the environment here is really the market.
• States are market features, prices, whether we’re holding the stock.
• actions: buy and sell, do nothing.
How to learn how to trade a particular stock.
• use historical time series to infer the state of the stock (Ballinger Band values, etc.)
• process that and decide what’s our action. (Suppose buy and holding long)
• Where there is a new state where the price has gone up and We’re holding long.
• new action: Sell
• Reward: The money made by the actions.
The policy tells us what to do at each state. learn the policy by looking at how we accrue money or don’t base on the actions we take in the environment.
Time: 00:01:51
05 - Markov decision problems
a Markov decision problem are defined by.
• a set of states S (all the possible values of S)
• a set of actions A (all the potential actions the agent can do in the environment)
• transition function T[s,a,s']: the probability of ending up in the state s' when taking action a in state s. Note: the sum of T to all the next states is 1.
• Reward function R[s,a].
__The problem for a reinforcement learning algorithm is to find this policy $\Pi$ or $\Pi^*$ that will maximize reward over time.
When T and R are known, the algorithms that will find this optimal policy are policy iteration and value iteration.
Time: 00:02:23
06 - Unknown transitions and rewards
When the transition function and the reward function are not available: The agent can interact with the world, observe what happens, and work with that data to try to build a policy.
• observe the environment, find out we are in state S1. after tooking and action, A1.then we are in S' and got reward R. - this is an experience tubal.
• Then in S2, take action A2 and in a new state S2’, and get a new reword. then repeat
• two things to do with the experience tuples to find policy $\Pi$.
□ model-based reinforcement learning.
□ model of T just by looking statistically at these transitions. Model which state we will get is we
□ Model rewards
□ When we can then use value iteration or policy iteration to solve the problem.
• model-free: Q-learning.
Time: 00:02:55
07 - What to optimize
Remember, in investment, long term reward should be discounted. (e.g. $1 per day worth than $1 in the future).
The maze problem:
• Robot in the bottom left corner. $ amount in the cells are the reword. The red blocks are blocked for the robot
• We have a reward here of $1 and a reward over here of $1 million. The $1 spot will refill once the money is taken. The $1 million spot only have $1 million and will not be refilled.
• the goal is to optimize is the sum of all future rewards.
1. if the robot can do interact with the board infinite times, it should keep moving the $1 spot. Moving to the $1 million spot won’t change the result.
2. if want to optimize reward over three moves (finite horizon). The reward of going up is $1 and the reward of going right will be -3.
3. if optimize reward over 8 moves, then going to the $1-million spot will be a clear choice.
4. so, if finite moves, it makes sense to go to the $1-million spot and come back to visit $1 spot repeatedly.
Discounted reward.
• $\gamma$ is the discount factor and $\gamma \in (0, 1]$
Time: 00:06:32
08 - Which approach gets 1M
• In other words, if the robot is trying to maximize the sum over these horizons, which ones will lead it to a policy that causes it to reach that $1 million?
Time: 00:00:21 Soluiton: see the figure above.
• Infinite horizon:
1. the robot can go to $1 and get a dollar on every other move and that will add up to infinity.
2. It can go to the $1 million and then come back and visit the $1 spot infinity.
• Finite with n=4, no it won’t get to that $1 million.
• if n = 10, boom, it’ll reach that $1 million.
• With the discounted reward, where the dollar in the future is only worth 0.95, and it gets smaller as we get further and further into the future, but at the eight steps it will reach $1-million
Time: 00:01:11
11 - Summary
• The problem for reinforcement learning algorithms is a Markov decision problem. And reinforcement learning algorithms solve them.
• A Markov decision problem is defined by S, A, T, and R:
□ S is the potential states,
□ A are the potential actions,
□ T is a transition probability, which is given I’m in state s, I take action a, what’s the probability I’ll end up in state S’,
□ R is the reward function.
• The goal for reinforcement learning algorithm is to find a policy, $\Pi$, that maps a state to an action that we should take. $\Pi^*$ is the policy that maximizes the future sum of the reward.
=infinite horizon, fixed horizon, or discounted sum.
Map our task for trading to reinforcement learning and it works out like this.
• S, our states, are features about stocks and whether or not we’re holding a stock.
• Actions are buying, sell, or do nothing.
• The transition function here is the market,
• The reward function is how much money we get at the end of a trade.
Time: 00:01:49
Total Time: 00:23:20 | {"url":"https://conge.livingwithfcs.org/2019/03/25/ML4T-03-05-Reinforcement-learning/","timestamp":"2024-11-03T23:06:57Z","content_type":"text/html","content_length":"37570","record_id":"<urn:uuid:4d67148d-67a7-4b3b-b836-5f4109bb2a59>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00202.warc.gz"} |
Department of Mathematical Sciences
This entry is currently under development. Please do not consider the entry authoritative until it has been completed.
Previous names: Division of Science and Mathematics; Division of Mathematics, Engineering and Computer Science; Department of Mathematics, Statistics and Computer Science
Established: 2000
Mathematics courses have been offered at UNBSJ since 1964 under the Faculties of both Arts and Science. Required and elective courses in first and second year Mathematics were taught starting in 1965
by Miss Pauline Graham and Mr. Edmond Glover. In subsequent years, required courses in Mathematics were also included in Business Administration, Engineering and Forestry programs. Lectures were held
from 1964 to 1969 at Beaverbrook House and offices were housed in the Science Building, also known as the Old Provincial Building in Saint John, from 1965 to 1969. Beginning in 1968, the Mathematics
option was offered among the Bachelor of Science specializations and students could study their first and second year of Mathematics and Statistics at UNBSJ with the plan to specialize in the subject
during their third and four year at UNB Fredericton. In the following year, students and the four faculty members of Mathematics moved to the newly constructed Tucker Park campus where they found
their home in William Ganong Hall.
By the time the Tucker Park campus had opened in 1969, the number of courses offered in Mathematics had grown from one course in 1964 to four courses. In 1975, Statistics was also coupled with the
Mathematics option and students studying Mathematics and Statistics could choose to select a Major or Honours in either Mathematics or Statistics for their third and fourth year in Fredericton. The
next two years brought the creation of the Division of Science and Mathematics, chaired by R.B. Kelly, and then the establishment of the Division of Mathematics, Engineering and Computer Science in
1977. During the 1980s the numbers of course offerings remained steady with thirteen courses offered in 1979 and fourteen offered in 1991. The Department of Mathematics, Statistics and Computer
Science was first established in 1993 under the UNBSJ Faculty of Science, chaired first by Dr. Merzik Kamel. The following year, a four-year Bachelor of Science in Mathematics was offered in Saint
John. Though the department was listed under the Faculty of Science, Applied Science and Engineering, a Major and Minor in Mathematics was also offered to students under the Faculty of Arts beginning
in 1995. From 1999 to 2006, Mathematics was also listed as a major in the interdisciplinary Bachelor of Data Analysis. In the year 2000, Mathematics at UNBSJ was deemed an independent Department of
Mathematical Sciences and was chaired by Dr. Alexander Wilson and then by Dr. Merzik Kamel in 2002. In 2007 a Bachelor of Science in Financial Mathematics became available to students.
Physical Location: William Ganong Hall
Faculty: Faculty of Science, Applied Science and Engineering, Faculty of Arts
Notes: Established date is based on the date that the separate UNBSJ department was officially stated in the calendar under its most recent name
• Undergraduate Calendars (UA RG 86) 1965-2014
• McGahan, Peter. The Quiet Campus: A History of the University of New Brunswick in Saint John, 1959-1969. Fredericton: New Ireland Press, 1998.
--Alloyd (talk) 09:45, 28 July 2015 (ADT)
© UNB Archives & Special Collections, 2014 | {"url":"https://unbhistory.lib.unb.ca/Department_of_Mathematical_Sciences","timestamp":"2024-11-12T19:32:33Z","content_type":"text/html","content_length":"52926","record_id":"<urn:uuid:4365bc7d-ea70-47ce-a138-459dc771ec15>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00343.warc.gz"} |
A Study of Smooth Functions and Differential Equations on
9 Matte-koordinatsystem idéer skola, utbildning, arbetsblad
When we was first introduced to first order differential equations we learned that the standard form was : y’ +p (t)y = g (t), y (to) = yo What separates Bernoulli Equations from other first order
equations is that in standard form, it is not equal to some function that is linear but one that has an exact solution. An ordinary differential equation (ODE) is an equation containing an unknown
function of one real or complex variable x, its derivatives, and some given functions of x.The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on
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first order differential equations, i.e.
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substitutions; the equations . transform (*) into . or, in standard form, Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical
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stochastic differential equations driven by Levy processes. A general scheme for stochastic Taylor expansions is Detta projekt fokuserar på utveckling av nya metoder för så kallad form optimering To
develop CutFEM as a general finite element method for simultaneous high order approximation of both geometry and partial differential equations, in the The Operating Profit Percentage reveals the
return from standard operations, In mathematics, a non-autonomous system of ordinary differential equations is av J Häggström · 2008 · Citerat av 79 — Teaching systems of linear equations in Sweden
and China: What is made possible In mathematics in general, and in algebra in particular, there is an interesting relation between the form and the meaning of mathematical symbols (see for Talrika
exempel på översättningar klassificerade efter aktivitetsfältet av “clairaut's differential equation” – Engelska-Svenska ordbok och den intelligenta Nonlinear Ordinary Differential Equations
(Applied Mathematics and text offers both professionals and students an introduction to the fundamentals and standard integral equations, analytic function theory, and integral transform methods.
individual matrix to Jordan normal form, it is in general impossible to do in the theory of the stability of differential equations, became a model General entry requirements and English B,
Mathematics D, Civics A. (Fieldspecific entry requirements solve basic types of differential equations. ○ use the deduce equations of lines and planes on the parameter form and normal form and. This
app is a friendly introduction to Calculus. It is suitable for senior secondary students with little or no prior knowledge to Calculus.
The solution process for a first order linear differential equation is as follows. Put the differential equation in the correct initial form, (1). Find the integrating factor, μ(t), using (10).
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Komplett løsning differential equation - aktuellpin.site
Forskningsoutput: Kapitel i Köp Ordinary Differential Equations av William A Adkins, Mark G Davidson på equations, this textbook gives an early presentation of the Laplace transform, the standard
solution methods for constant coefficient linear differential equations Jag försöker se saker i form av geometri. This system of linear equations has exactly one solution. In general, the behavior
of a linear system is determined by the relationship between the number of equations and the number of unknowns There is also a corresponding differential form of this equation covered in Schoen and
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Amit is planning to buy a house and the layout is given below. The des
Amit is planning to buy a house and the layout is given below. The design and the measurement has been made such that areas of two bedrooms and kitchen together is 95 sq.m.
Based on the above information, answer the following questions:
Find the area of living room in the layout.
Step by step video & image solution for Amit is planning to buy a house and the layout is given below. The design and the measurement has been made such that areas of two bedrooms and kitchen
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Concrete Shear Wall Design Spreadsheet - CivilWeb Spreadsheets
Concrete Shear Wall Design
Concrete shear walls are included within buildings in order to strengthen the whole structure against lateral loads such as wind loads. A rectangular building on its own is not very strong when
subjected to lateral loads. This is because a rectangular shape is not particularly strong, if the connections are not very strong the whole building can easily be deformed and damaged by lateral
loadings. This is particularly the case with steel framed buildings with relatively light superstructures which cannot provide the required lateral strength from the superstructure alone.
Concrete shear walls are strong concrete walls within rectangular buildings which provide the required lateral strength. They consist of a series of strong concrete walls positioned at particular
points within the building. Shear walls are usually provided in places where concrete walls can be placed without altering the layout of the building. Common examples include staircases and lift
shafts. These areas are built much stronger than they would otherwise need to be in order to function as shear walls as well.
Concrete Shear Wall Design - Inputs
The design of concrete shear walls can be difficult to complete by hand. The CivilWeb Concrete Shear Wall Design Spreadsheet makes it easy. First the designer inputs the location of up to four shear
walls within the building. The spreadsheet plots each shear wall onto a plan of the building so the designer can make sure they are positioned correctly. Then the spreadsheet calculates the shear
centre and plots this on the plan. This way the designer can adjust the position of the shear walls so that the shear centre is as close as possible to the centroid of the building.
Next the designer can input the dimensions of the shear walls. This is done by specifying the size of the openings. This makes it quick and easy to get the most common shear wall shapes with minimal
time spent inputting values.
The concrete shear wall design spreadsheet then calculates the design properties for each shear wall including the second moments of area in both X and Y directions, and the percentage of the lateral
strength provided by each shear wall.
Now the designer can enter the loading conditions. This is done in accordance with BS EN 1991 and includes 6 preset load combinations. The designer must also input the maximum lateral forces expected
in both X and Y directions. The RC shear wall analysis and design spreadsheet uses these forces to design the loads acting on each spreadsheet in proportion to their contribution to the structures
strength in that direction.
The designer can add axial loads which are also often present in shear walls. The spreadsheet plots a drawing of each concrete shear wall showing the designer exactly where the axial loads have been
placed. This allows the designer to check that the axial forces have all been placed correctly on each concrete shear wall.
The RC shear wall analysis and design spreadsheet then allows the designer to analyse the reinforcement requirements for each wall in up to 8 different locations. This allows the designer to complete
the design of the reinforcement for each shear wall.
The spreadsheet includes a number of handy tools which can assist the designer in completing the reinforcement design. The spreadsheet calculates the maximum tensile and maximum compressive stresses
active in each part of the shear wall and plots these on a plan of the concrete shear wall. This allows the designer to see exactly where they should analyse the shear wall in order to get the
critical compressive and tensile stresses for reinforcement design.
Then the spreadsheet also suggests the optimum bar size and spacing for each of the 8 load and analysis positions. This allows the designer to simply input suitable values without completing a time
consuming iterative design process, testing bar sizes and spaces until they are optimised.
This procedure would be almost impossible to complete by hand in all but the most simple of cases. Using the CivilWeb Concrete Shear Wall Design Spreadsheet the designer can complete a full RC shear
wall analysis and design in minutes.
CivilWeb Concrete Shear Wall Design Spreadsheet
The CivilWeb Concrete Shear Wall Design Spreadsheet is a powerful spreadsheet for the design of shear walls in accordance with BS EN 1992. The spreadsheet can include up to four different shear
walls positioned anywhere within the building. The spreadsheet calculates the increase in lateral strength provided by the shear walls, then can be used to design the required reinforcement.
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Records of 4/7 to 4/9, 2018
Records for 4-5 and 4-6
Dancing disco at the cementry.
About Mathematica
Mathematica 11 could not be started after a freetype upgrade.
When run in command line, it emits an error like this.
According to the gentoo forum site. Mathematica has its own libraries, and it calls the system library of freetype.
Forcing Mathematica to use the system library by removing libfreetype.so.6 and libz.so.1 in ${TopDirectory}/SystemFiles/Libraries/Linux-x86-64 could solve the problem.
About R
To do a linear fit by this:
x <- c(...)
y <- c(...)
fit <- lm(y ~ x)
The one who wrote this article fell asleep halfway…
Rec 4.8
Computing Method
If f(x) satisfies:
1. $f(x) \in [a, b], x \in [a, b]$
2. $f'$ is Lipschitz continuous with some $K < 1$.
Then there exists a unique fixed point of f in $[a, b]$.
Existence: intermediate value theorem.
Uniqueness: reductio ad absurdum.
Newton’s method:
To find $X$ s.t. $F(X) = 0$.
First $X=X_0$
$\Delta X_0 = \left(\left.\frac{\partial F}{\partial X}\right|_{X=X_0}\right)^{'}F(X_0)$ $X_1=X_0+\Delta X_0$
$X_2=X_1+\Delta X_1$
Where $\frac{\partial F}{\partial X}$ is the Jacobian matrix of $F$.
Facts about Mozilla
• It developed Thunderbird, a program for emails, feeds, and IM and so on. And now Thunderbird is abandoned.
It is said starting firefox with env MOZ_USE_XINPUT2=1 firefox enables scrolling with a finger. Not working now.
The bugzilla page suggests there should be a --enable-default-toolkit=cairo-gtk3 in the configure options shown in about:buildconfig.
Works now after upgrading GTK. Why.
Firefox OS
Another abandoned project is Firefox OS. Refer to this post.
• Starting point: Boot to Gecko(B2G), push the envelop of the web
□ Architecture:
☆ Gonk: Open Linux kernel and drivers
☆ Gecko: Web runtime with open Web APIs
☆ Gaia: User interface with HTML5-based Web content
• Firefox OS 1.0
□ Imitate what already existed
□ Invented a lot of APIs to talk to hardware of a smartphone from Javascript(that wouldn’t come into standards)
□ Introduced packaged apps to Gecko to achieve both the offline(run apps without Internet connection) and security requirements(to secure privileged functions like calling and texting
☆ Packaged apps got no URLs and have to be signed by a central authority to say they are safe. -> Against the web.
• Firefox OS 1.x
□ Just chasing the coat tails of Android
• Differentiation
□ Affordable phones -> 1.3t
☆ $25 smartphone: mostly done by Taipei office
□ Web -> 2.0
☆ Haida: blur the line between apps and websites
☆ Overshadowed by feature requests from venders
• 3.0 -> 2.5 Come to stall and dead in the end.
Rec 4.9
Computer Architecture
Refer to Computer Architecture: A Quantitative Approach
Optimization of Cache Performance
1. Nonblocking Caches to Increase Cache Bandwidth
Pipelined computers that allow out-of-order execution will benefit from this kind of cache which can supply cache hits even during a miss.
2. Multibanked Cache
Bank 1 Bank 2 Bank 3 Bank 4
Increase bandwidth.
3. Compiler
4. Prefetching
Programming Languages
Still, refer to Standford’s cs242.
Existential types
Interface: specify what a type should be able to do, regardless of the implementation
Implementation: concretize the interface by showing how a type can implement the interface
It has two kinds of operations: pack and unpack.
Packing is the introduction of an implementation(see More monads in OCaml).
For example, this is an example of a monad interface:
module type M = sig
type _ t
val pure : 'a -> 'a t
val bind : ('a -> 'b t) -> 'a t -> 'b t
And an implementation of it:
module Option:M = struct
type 'a t = Nothing | Just of 'a
let pure x = Just x
let bind f x = match x with
| Just y -> f y
| Nothing -> Nothing
It’s an ugly example, but the point is, the detail about Option.t is erased, with only the interface(pure, bind) left.
utop # Option.pure 3;;
- : int Option.t = <abstr>
The implementation, Just 3 is not available here, and we cannot use Just to create a Option.t as well. Wait … shit.
Forget about monads, let’s go with another example:
module type Stack = sig
type _ t
val create : unit -> 'a t
val push : 'a t -> 'a -> 'a t
val pop : 'a t -> 'a t * 'a option
val empty : 'a t -> bool
And an implementation by list here:
module LStack : Stack = struct
type 'a t = 'a list
let create = fun () -> []
let push s x = x::s
let pop x = match x with
|x::xs -> xs, Some x
|[] -> [], None
let empty x = match x with
|x::xs -> false
|[] -> true
Everything works fine without the knowledge that LStack uses list for the stack. Say you create a new stack using LStack.create, there is no way for you to use it as a list, though it is one.
utop # let mystack = LStack.create ();;
val mystack : '_a LStack.t = <abstr>
utop # assert(mystack=[]);;
Error: This expression has type 'a list but an expression was expected of type
'b LStack.t
That’s abstraction in a nutshell.
The existential type is like this: ${*S, t} : {\exists X, T}$ means that type t, coupled with implementation S, makes a package that have an abstract type X with signature T. For the LStack, the
analogy would be:
$*list \sim *S\\ \{create = \cdots, push = \cdots, pop = \cdots\} \sim t\\ \{create : \forall a.unit \rightarrow X[a],\ push : \forall a.X[a] \rightarrow a \rightarrow X[a],\ pop : \forall a.X[a] \
rightarrow X[a] * option[a]\}\sim T$
And there is the pack operation, which erases one or more types and introduces an implementation:
$\frac{\Gamma \vdash t : [X\rightarrow U]T}{\Gamma \vdash \{*U, t\}\ as\ \{\exists X, T\} : \{\exists X, T\}}$
愛されたくて偽って もっともっと自然に笑えばいいかな | {"url":"https://www.nir.moe/posts/rec-4-7-2018/","timestamp":"2024-11-12T19:44:22Z","content_type":"text/html","content_length":"62308","record_id":"<urn:uuid:d1d866ee-1224-4ccd-972f-8a7253978a3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00884.warc.gz"} |
LinearModel is a fitted linear regression model object. A regression model describes the relationship between a response and predictors. The linearity in a linear regression model refers to the
linearity of the predictor coefficients.
Use the properties of a LinearModel object to investigate a fitted linear regression model. The object properties include information about coefficient estimates, summary statistics, fitting method,
and input data. Use the object functions to predict responses and to modify, evaluate, and visualize the linear regression model.
Create a LinearModel object by using fitlm or stepwiselm.
fitlm fits a linear regression model to data using a fixed model specification. Use addTerms, removeTerms, or step to add or remove terms from the model. Alternatively, use stepwiselm to fit a model
using stepwise linear regression.
Summary Statistics
ModelFitVsNullModel — F-statistic of regression model
This property is read-only.
F-statistic of the regression model, specified as a structure. The ModelFitVsNullModel structure contains these fields:
• Fstats — F-statistic of the fitted model versus the null model
• Pvalue — p-value for the F-statistic
• NullModel — null model type
Data Types: struct
Object Functions
Create CompactLinearModel
Add or Remove Terms from Linear Model
addTerms Add terms to linear regression model
removeTerms Remove terms from linear regression model
step Improve linear regression model by adding or removing terms
Predict Responses
feval Predict responses of linear regression model using one input for each predictor
predict Predict responses of linear regression model
random Simulate responses with random noise for linear regression model
Evaluate Linear Model
anova Analysis of variance for linear regression model
coefCI Confidence intervals of coefficient estimates of linear regression model
coefTest Linear hypothesis test on linear regression model coefficients
dwtest Durbin-Watson test with linear regression model object
partialDependence Compute partial dependence
Visualize Linear Model and Summary Statistics
plot Scatter plot or added variable plot of linear regression model
plotAdded Added variable plot of linear regression model
plotAdjustedResponse Adjusted response plot of linear regression model
plotDiagnostics Plot observation diagnostics of linear regression model
plotEffects Plot main effects of predictors in linear regression model
plotInteraction Plot interaction effects of two predictors in linear regression model
plotPartialDependence Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotResiduals Plot residuals of linear regression model
plotSlice Plot of slices through fitted linear regression surface
Gather Properties of Linear Model
Fit Linear Regression Using Data in Matrix
Fit a linear regression model using a matrix input data set.
Load the carsmall data set, a matrix input data set.
load carsmall
X = [Weight,Horsepower,Acceleration];
Fit a linear regression model by using fitlm.
mdl =
Linear regression model:
y ~ 1 + x1 + x2 + x3
Estimated Coefficients:
Estimate SE tStat pValue
__________ _________ _________ __________
(Intercept) 47.977 3.8785 12.37 4.8957e-21
x1 -0.0065416 0.0011274 -5.8023 9.8742e-08
x2 -0.042943 0.024313 -1.7663 0.08078
x3 -0.011583 0.19333 -0.059913 0.95236
Number of observations: 93, Error degrees of freedom: 89
Root Mean Squared Error: 4.09
R-squared: 0.752, Adjusted R-Squared: 0.744
F-statistic vs. constant model: 90, p-value = 7.38e-27
The model display includes the model formula, estimated coefficients, and model summary statistics.
The model formula in the display, y ~ 1 + x1 + x2 + x3, corresponds to $\mathit{y}={\beta }_{0}+{\beta }_{1}{\mathit{X}}_{1}+{\beta }_{2}{\mathit{X}}_{2}+{\beta }_{3}{\mathit{X}}_{3}+ϵ$.
The model display also shows the estimated coefficient information, which is stored in the Coefficients property. Display the Coefficients property.
ans=4×4 table
Estimate SE tStat pValue
__________ _________ _________ __________
(Intercept) 47.977 3.8785 12.37 4.8957e-21
x1 -0.0065416 0.0011274 -5.8023 9.8742e-08
x2 -0.042943 0.024313 -1.7663 0.08078
x3 -0.011583 0.19333 -0.059913 0.95236
The Coefficient property includes these columns:
• Estimate — Coefficient estimates for each corresponding term in the model. For example, the estimate for the constant term (intercept) is 47.977.
• SE — Standard error of the coefficients.
• tStat — t-statistic for each coefficient to test the null hypothesis that the corresponding coefficient is zero against the alternative that it is different from zero, given the other predictors
in the model. Note that tStat = Estimate/SE. For example, the t-statistic for the intercept is 47.977/3.8785 = 12.37.
• pValue — p-value for the t-statistic of the two-sided hypothesis test. For example, the p-value of the t-statistic for x2 is greater than 0.05, so this term is not significant at the 5%
significance level given the other terms in the model.
The summary statistics of the model are:
• Number of observations — Number of rows without any NaN values. For example, Number of observations is 93 because the MPG data vector has six NaN values and the Horsepower data vector has one NaN
value for a different observation, where the number of rows in X and MPG is 100.
• Error degrees of freedom — n – p, where n is the number of observations, and p is the number of coefficients in the model, including the intercept. For example, the model has four predictors, so
the Error degrees of freedom is 93 – 4 = 89.
• Root mean squared error — Square root of the mean squared error, which estimates the standard deviation of the error distribution.
• R-squared and Adjusted R-squared — Coefficient of determination and adjusted coefficient of determination, respectively. For example, the R-squared value suggests that the model explains
approximately 75% of the variability in the response variable MPG.
• F-statistic vs. constant model — Test statistic for the F-test on the regression model, which tests whether the model fits significantly better than a degenerate model consisting of only a
constant term.
• p-value — p-value for the F-test on the model. For example, the model is significant with a p-value of 7.3816e-27.
You can find these statistics in the model properties (NumObservations, DFE, RMSE, and Rsquared) and by using the anova function.
ans=3×5 table
SumSq DF MeanSq F pValue
______ __ ______ ______ __________
Total 6004.8 92 65.269
Model 4516 3 1505.3 89.987 7.3816e-27
Residual 1488.8 89 16.728
Use plot to create an added variable plot (partial regression leverage plot) for the whole model except the constant (intercept) term.
Linear Regression with Categorical Predictor
Fit a linear regression model that contains a categorical predictor. Reorder the categories of the categorical predictor to control the reference level in the model. Then, use anova to test the
significance of the categorical variable.
Model with Categorical Predictor
Load the carsmall data set and create a linear regression model of MPG as a function of Model_Year. To treat the numeric vector Model_Year as a categorical variable, identify the predictor using the
'CategoricalVars' name-value pair argument.
load carsmall
mdl = fitlm(Model_Year,MPG,'CategoricalVars',1,'VarNames',{'Model_Year','MPG'})
mdl =
Linear regression model:
MPG ~ 1 + Model_Year
Estimated Coefficients:
Estimate SE tStat pValue
________ ______ ______ __________
(Intercept) 17.69 1.0328 17.127 3.2371e-30
Model_Year_76 3.8839 1.4059 2.7625 0.0069402
Model_Year_82 14.02 1.4369 9.7571 8.2164e-16
Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 5.56
R-squared: 0.531, Adjusted R-Squared: 0.521
F-statistic vs. constant model: 51.6, p-value = 1.07e-15
The model formula in the display, MPG ~ 1 + Model_Year, corresponds to
$\mathrm{MPG}={\beta }_{0}+{\beta }_{1}{Ι}_{\mathrm{Year}=76}+{\beta }_{2}{Ι}_{\mathrm{Year}=82}+ϵ$,
where ${Ι}_{\mathrm{Year}=76}$ and ${Ι}_{\mathrm{Year}=82}$ are indicator variables whose value is one if the value of Model_Year is 76 and 82, respectively. The Model_Year variable includes three
distinct values, which you can check by using the unique function.
fitlm chooses the smallest value in Model_Year as a reference level ('70') and creates two indicator variables ${Ι}_{\mathrm{Year}=76}$ and ${Ι}_{\mathrm{Year}=82}$. The model includes only two
indicator variables because the design matrix becomes rank deficient if the model includes three indicator variables (one for each level) and an intercept term.
Model with Full Indicator Variables
You can interpret the model formula of mdl as a model that has three indicator variables without an intercept term:
$\mathit{y}={\beta }_{0}{Ι}_{{\mathit{x}}_{1}=70}+\left({\beta }_{0}+{\beta }_{1}\right){Ι}_{{\mathit{x}}_{1}=76}+\left({{\beta }_{0}+\beta }_{2}\right){Ι}_{{\mathit{x}}_{2}=82}+ϵ$.
Alternatively, you can create a model that has three indicator variables without an intercept term by manually creating indicator variables and specifying the model formula.
temp_Year = dummyvar(categorical(Model_Year));
Model_Year_70 = temp_Year(:,1);
Model_Year_76 = temp_Year(:,2);
Model_Year_82 = temp_Year(:,3);
tbl = table(Model_Year_70,Model_Year_76,Model_Year_82,MPG);
mdl = fitlm(tbl,'MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82 - 1')
mdl =
Linear regression model:
MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82
Estimated Coefficients:
Estimate SE tStat pValue
________ _______ ______ __________
Model_Year_70 17.69 1.0328 17.127 3.2371e-30
Model_Year_76 21.574 0.95387 22.617 4.0156e-39
Model_Year_82 31.71 0.99896 31.743 5.2234e-51
Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 5.56
Choose Reference Level in Model
You can choose a reference level by modifying the order of categories in a categorical variable. First, create a categorical variable Year.
Year = categorical(Model_Year);
Check the order of categories by using the categories function.
ans = 3x1 cell
If you use Year as a predictor variable, then fitlm chooses the first category '70' as a reference level. Reorder Year by using the reordercats function.
Year_reordered = reordercats(Year,{'76','70','82'});
ans = 3x1 cell
The first category of Year_reordered is '76'. Create a linear regression model of MPG as a function of Year_reordered.
mdl2 = fitlm(Year_reordered,MPG,'VarNames',{'Model_Year','MPG'})
mdl2 =
Linear regression model:
MPG ~ 1 + Model_Year
Estimated Coefficients:
Estimate SE tStat pValue
________ _______ _______ __________
(Intercept) 21.574 0.95387 22.617 4.0156e-39
Model_Year_70 -3.8839 1.4059 -2.7625 0.0069402
Model_Year_82 10.136 1.3812 7.3385 8.7634e-11
Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 5.56
R-squared: 0.531, Adjusted R-Squared: 0.521
F-statistic vs. constant model: 51.6, p-value = 1.07e-15
mdl2 uses '76' as a reference level and includes two indicator variables ${Ι}_{\mathrm{Year}=70}$ and ${Ι}_{\mathrm{Year}=82}$.
Evaluate Categorical Predictor
The model display of mdl2 includes a p-value of each term to test whether or not the corresponding coefficient is equal to zero. Each p-value examines each indicator variable. To examine the
categorical variable Model_Year as a group of indicator variables, use anova. Use the 'components'(default) option to return a component ANOVA table that includes ANOVA statistics for each variable
in the model except the constant term.
ans=2×5 table
SumSq DF MeanSq F pValue
______ __ ______ _____ __________
Model_Year 3190.1 2 1595.1 51.56 1.0694e-15
Error 2815.2 91 30.936
The component ANOVA table includes the p-value of the Model_Year variable, which is smaller than the p-values of the indicator variables.
Fit Robust Linear Regression Model
Load the hald data set, which measures the effect of cement composition on its hardening heat.
This data set includes the variables ingredients and heat. The matrix ingredients contains the percent composition of four chemicals present in the cement. The vector heat contains the values for the
heat hardening after 180 days for each cement sample.
Fit a robust linear regression model to the data.
mdl = fitlm(ingredients,heat,'RobustOpts','on')
mdl =
Linear regression model (robust fit):
y ~ 1 + x1 + x2 + x3 + x4
Estimated Coefficients:
Estimate SE tStat pValue
________ _______ ________ ________
(Intercept) 60.09 75.818 0.79256 0.4509
x1 1.5753 0.80585 1.9548 0.086346
x2 0.5322 0.78315 0.67957 0.51596
x3 0.13346 0.8166 0.16343 0.87424
x4 -0.12052 0.7672 -0.15709 0.87906
Number of observations: 13, Error degrees of freedom: 8
Root Mean Squared Error: 2.65
R-squared: 0.979, Adjusted R-Squared: 0.969
F-statistic vs. constant model: 94.6, p-value = 9.03e-07
For more details, see the topic Reduce Outlier Effects Using Robust Regression, which compares the results of a robust fit to a standard least-squares fit.
Fit Linear Model Using Stepwise Regression
Load the hald data set, which measures the effect of cement composition on its hardening heat.
This data set includes the variables ingredients and heat. The matrix ingredients contains the percent composition of four chemicals present in the cement. The vector heat contains the values for the
heat hardening after 180 days for each cement sample.
Fit a stepwise linear regression model to the data. Specify 0.06 as the threshold for the criterion to add a term to the model.
mdl = stepwiselm(ingredients,heat,'PEnter',0.06)
1. Adding x4, FStat = 22.7985, pValue = 0.000576232
2. Adding x1, FStat = 108.2239, pValue = 1.105281e-06
3. Adding x2, FStat = 5.0259, pValue = 0.051687
4. Removing x4, FStat = 1.8633, pValue = 0.2054
mdl =
Linear regression model:
y ~ 1 + x1 + x2
Estimated Coefficients:
Estimate SE tStat pValue
________ ________ ______ __________
(Intercept) 52.577 2.2862 22.998 5.4566e-10
x1 1.4683 0.1213 12.105 2.6922e-07
x2 0.66225 0.045855 14.442 5.029e-08
Number of observations: 13, Error degrees of freedom: 10
Root Mean Squared Error: 2.41
R-squared: 0.979, Adjusted R-Squared: 0.974
F-statistic vs. constant model: 230, p-value = 4.41e-09
By default, the starting model is a constant model. stepwiselm performs forward selection and adds the x4, x1, and x2 terms (in that order), because the corresponding p-values are less than the
PEnter value of 0.06. stepwiselm then uses backward elimination and removes x4 from the model because, once x2 is in the model, the p-value of x4 is greater than the default value of PRemove, 0.1.
Alternative Functionality
• For reduced computation time on high-dimensional data sets, fit a linear regression model using the fitrlinear function.
• To regularize a regression, use fitrlinear, lasso, ridge, or plsregress.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
For more information, see Introduction to Code Generation.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
• The object functions of the LinearModel model fully support GPU arrays.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2012a | {"url":"https://in.mathworks.com/help/stats/linearmodel.html","timestamp":"2024-11-14T18:03:03Z","content_type":"text/html","content_length":"219427","record_id":"<urn:uuid:c5969a9e-023a-422f-906c-e22787fcf13d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00496.warc.gz"} |
Felix Zhou 周长风
I am a 3rd year CS PhD student in the Theory Group at Yale University, where I am extremely fortunate to be advised by Quanquan C. Liu. Currently, I am also a student researcher at Google Research in
the NYC Algorithms & Optimization Group hosted by Vincent Cohen-Addad and Alessandro Epasto.
I am grateful to be supported by an NSERC Postgraduate Scholarship.
Research Interests: I am broadly interested in the theory and practice of algorithms for large data and different notions of algorithmic stability. Examples include parallel graph algorithms,
differentially private (DP) graph algorithms, DP learning algorithms, replicable learning algorithms, and learning under structured biases.
Email: felix [dot] zhou [at] yale [dot] edu
My amazing collaborators (in no particular order): Samson Zhou, Vincent Cohen-Addad, Manolis Zampetakis, Grigoris Velegkas, Yuichi Yoshida, Tamalika Mukherjee, Alessandro Epasto, Alkis Kalavasis,
Anay Mehrotra, Kasper Green Larsen, Amin Karbasi, Lin F. Yang, Vahab Mirrokni, Chaitanya Swamy, Jochen Koenemann, W. Justin Toth
Personal: My other half, Jane Shi, studies number theory at MIT.
About Me
Previously, I was an undergraduate student at the University of Waterloo, where I was fortunate to be advised by Jochen Koenemann and Chaitanya Swamy. I worked on combinatorial optimization,
approximation algorithms, and algorithmic game theory.
I interned at Hudson River Trading as an algorithm developer. Previously, I interned at HomeX, where I worked on an online stochastic reservation problem. Even earlier, I interned at the Google
Mountain View office, where I worked on distributed graph algorithms.
The Power of Graph Sparsification in the Continual Release Model with Alessandro Epasto, Quanquan C. Liu, Tamalika Mukherjee [preprint] [poster]
Pointwise Lipschitz Continuous Graph Algorithms via Proximal Gradient Analysis with Quanquan C. Liu, Grigoris Velegkas, Yuichi Yoshida [preprint] [slides]
On the Computational Landscape of Replicable Learning with Alkis Kalavasis, Amin Karbasi, Grigoris Velegkas to appear in NeurIPS, 2024. [preprint]
Replicable Learning of Large-Margin Halfspaces with Alkis Kalavasis, Amin Karbasi, Kasper Green Larsen, Grigoris Velegkas to appear in ICML, 2024. Spotlight (top 3.5% of accepted papers) [preprint]
Replicability in Reinforcement Learning with Amin Karbasi, Grigoris Velegkas, Ling F. Yang NeurIPS, 2023. [preprint] [video]
Replicable Clustering with Hossein Esfandiari, Amin Karbasi, Vahab Mirrokni, Grigoris Velegkas NeurIPS, 2023. [preprint] [slides] [video]
On the Complexity of Nucleolus Computation for Bipartite b-Matching Games with Jochen Koenemann, Justin Toth SAGT, 2021. Special Issue [preprint] [slides] [video]
Notes typeset for courses and from self-studying. Errors are abundant. Please use at your own discretion. | {"url":"https://felix-zhou.com/","timestamp":"2024-11-11T08:02:38Z","content_type":"text/html","content_length":"11798","record_id":"<urn:uuid:fe51fe8d-413b-4689-8b05-ea3a8b3796b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00567.warc.gz"} |
Mathematical Genealogy
As a fun side project to distract me from my abysmal progress on my book, I decided to play around with the math genealogy graph!
For those who don’t know, since 1996, mathematicians, starting with the labor of Harry Coonce et al, have been managing a database of all mathematicians. More specifically, they’ve been keeping track
of who everyone’s thesis advisors and subsequent students were. The result is a directed graph (with a current estimated 200k nodes) that details the scientific lineage of mathematicians.
Anyone can view the database online and explore the graph by hand. In it are legends like Gauss, Euler, and Noether, along with the sizes of their descendant subtrees. Here’s little ol’ me.
It’s fun to look at who is in your math genealogy, and I’ve spent more than a few minutes clicking until I get to the top of a tree (since a person can have multiple advisors, finding the top is time
consuming), like the sort of random walk that inspired Google’s PageRank and Wikipedia link clicking games.
Inspired by a personalized demo by Colin Wright, I decided it would be fun to scrape the website, get a snapshot of the database, and then visualize and play with the graph. So I did.
Here’s a github repository with the raw data and scraping script. It includes a full json dump of what I scraped as of a few days ago. It’s only ~60MB.
Then, using a combination of tools, I built a rudimentary visualizer. Go play with it!
A few notes:
1. It takes about 15 seconds to load before you can start playing. During this time, it loads a compressed version of the database into memory (starting from a mere 5MB). Then it converts the data
into a more useful format, builds a rudimentary search index of the names, and displays the ancestors for Gauss.
2. The search index is the main bloat of the program, requiring about a gigabyte of memory to represent. Note that because I’m too lazy to set up a proper server and elasticsearch index, everything
in this demo is in Javascript running in your browser. Here’s the github repo for that code.
3. You can drag and zoom the graph.
4. There was a fun little bit of graph algorithms involved in this project, such as finding the closest common ancestor of two nodes. This is happening in a general digraph, not necessarily a tree,
so there are some extra considerations. I isolated all the graph algorithms to one file.
5. People with even relatively few descendants generate really wide graphs. This is because each layer in the directed graph is assigned to a layer, and, the potentially 100+ grandchildren of a
single node will be laid out in the same layer. I haven’t figured out how to constrain the width of the rendered graph (anyone used dagre/dagre-d3?), nor did I try very hard.
6. The dagre layout package used here is a port of the graphviz library. It uses linear programming and the simplex algorithm to determine an optimal layout that penalizes crossed edges and edges
that span multiple layers, among other things. Linear programming strikes again! For more details on this, see this paper outlining the algorithm.
7. The scraping algorithm was my first time using Python 3’s asyncio features. The concepts of asynchronous programming are not strange to me, but somehow the syntax of this module is.
Feature requests, bugs, or ideas? Open an issue on Github or feel free to contribute a pull request! Enjoy. | {"url":"https://www.jeremykun.com/2017/06/22/mathematical-genealogy/","timestamp":"2024-11-02T04:50:43Z","content_type":"text/html","content_length":"13221","record_id":"<urn:uuid:31382611-9bac-459b-aa59-36ed6e40161e>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00523.warc.gz"} |
AP Computer Science Principles (AP CSP) is equivalent to an introductory college-level computing course that introduces students to the breadth of the field of computer science. Students learn to
design and evaluate solutions and to apply computer science to solve problems through the development of algorithms and programs. They incorporate abstraction into programs and use data to discover
new knowledge. Students also explore how computing innovations and computing systems work (including the Internet), explore their potential impacts, and contribute to a computing culture that is
collaborative and ethical. Roughly half the course is focused on learning to program in either the Python or Javascript programming languages, but the selection of a programming language is at the
teacher’s discretion while the other half of the course covers non-programming topics of computer science.
*Class receives honors weighting in SI weighted GPA and UC/CSU GPA calculations
**This class will be offered pending adequate enrollment.
***Students are required to take the AP Computer Science Principles exam in May.
Physical Education 615: Sports Medicine 1 Prevention & Care of Athletic Injuries
This course prepares students to become student trainers. This is a lecture, reading, and activity course. In Sports Medicine 1, students will learn the fundamentals of anatomy, prevention, care,
treatment, taping, and rehabilitation of athletic injuries. Students are exposed to a variety of situations/scenarios aimed at achieving a basic knowledge of sports medicine through various
“hands-on” activities. Students are educated and evaluated on their performance through active participation, homework assignments, tests/quizzes, taping, and game day evaluation.
*This class will require 1-2 hours a week of practical work in the Training Room after school.
**This class will be offered pending staffing availability and adequate enrollment.
Biology (Life Science)
Biology is the scientific study of life and living organisms. This course aims to develop students into scientifically literate citizens who have mastered the critical thinking skills that will allow
them to make informed decisions in a world increasingly impacted by scientific discovery. This course also aims to develop in students an appreciation for the natural world and our role in its
stewardship. Units of study in this course include evolutionary biology, genetics, heredity, cell structure and function, human reproduction, and ecology.
Chemistry (Physical Science)
Chemistry is the scientific study of matter. This course aims to develop students as practicing laboratory scientists who can ask and answer questions of their own about what the world is made of and
how and why chemical reactions occur. This course also aims to develop students’ conceptual and quantitative understanding of chemical principles. Units of study in this course include the nature of
the atom, naming of chemicals and compounds, bonding, the periodic table, reactions and equilibrium, stoichiometry, behavior of gases, acids, bases, and safe laboratory practices.
Chemistry Honors (Physical Science)
The honors course differs from the non-honors course in that each topic is covered in more detail, at a faster pace, and with greater mathematical rigor.
Physics (Physical Science)
Physics is the scientific study of the most fundamental laws of nature. This course aims to further develop students’ appreciation for and competence in the scientific method. This course also aims
to develop students’ conceptual and quantitative understanding of physical principles. Students perform experiments to develop proficiency in laboratory technique in applying physical principles to
the analysis of experimental data. Units of study in this course include motion, Newton’s Laws, collisions, energy, thermodynamics, waves, sound, light, fundamental particles of nature,
radioactivity, quantum mechanics, and electricity and magnetism.
Physics Honors (Physical Science)
The honors course differs from the non-honors course in that each topic is covered in more detail, at a faster pace, and with greater mathematical rigor.
*Class receives honors weighting in SI weighted GPA and UC/CSU GPA calculations
AP Biology (Life Science)
The AP Biology course is equivalent in content, depth, and complexity to an introductory biology course at the college level. This course is designed to prepare the student to excel on the AP exam
offered in May, and follows the AP curriculum. AP Biology is an in-depth, content-intensive study of biological principles that allows students the opportunity to engage hands-on in scientific
experimentation. Units of study include but are not limited to evolution and natural selection, the chemistry of life, cell structure and function, cellular energetics, cell communication and the
cell cycle, heredity, gene expression and regulation, and ecology. Students are required to take the Advanced Placement exam in May. Students are required to complete an assignment over the summer
due on the first day of school.
Corequisite – Students enrolling in this course must also enroll in the corresponding AP Science Laboratory course, which meets once per week for 50 minutes outside of the regular bell schedule.
Meetings will occur before or after school.
* Class receives honors weighting in SI weighted GPA and UC/CSU GPA calculations
AP Chemistry (Physical Science)
The AP Chemistry course is equivalent in content, depth, and complexity to an introductory chemistry course at the college level. This course is designed to prepare the student to excel on the AP
exam offered in May, and follows the AP curriculum closely. AP Chemistry is an in-depth, content-intensive study of chemical principles that allows students the opportunity to engage hands-on in
scientific experimentation. Units of study include chemical reactions, modern atomic theory, molecular bonding, hybridization, organic chemistry, stoichiometry, thermodynamics, kinetics, aqueous
equilibrium, acids, bases, precipitation, reduction, oxidation, electrochemistry, and nuclear chemistry. Students are required to take the Advanced Placement exam in May. Students are required to
complete an assignment over the summer due on the first day of school.
Corequisite – Students enrolling in this course must also enroll in the corresponding AP Science Laboratory course, which meets once per week for 50 minutes outside of the regular bell
schedule. Meetings will occur before or after school.
*Class receives honors weighting in SI weighted GPA and UC/CSU GPA calculations
AP Physics C: Mechanics (Physical Science)
The AP Physics course is equivalent in content, depth, and complexity to an introductory physics course at the college level. This course is designed to prepare the student to excel on the AP Physics
C: Mechanics exams offered in May. The course follows the AP curriculum closely. AP Physics is an in-depth, content-intensive study of physical principles that allows students the opportunity to
engage hands-on in scientific experimentation. Core units of study include kinematics, Newton’s laws, conservation laws, harmonic motion, and rotational motion. Additional topics will vary but may
include electricity & magnetism, relativity, quantum mechanics, particle physics, thermodynamics, and other advanced topics. Use of calculus in problem solving is expected to increase as the course
progresses. Students are required to take the Advanced Placement exam in May. Students are required to complete an assignment over the summer due on the first day of school. This is a
mathematically rigorous course which requires a solid foundation in both physics and math.
Corequisite – Students enrolling in this course must also enroll in the corresponding AP Science Laboratory course, which meets once per week for 50 minutes outside of the regular bell
schedule. Meetings will occur before or after school.
*Class receives honors weighting in SI weighted GPA and UC/CSU GPA calculations | {"url":"https://curriculum.siprep.org/cc_uccsu/d/","timestamp":"2024-11-13T17:38:28Z","content_type":"text/html","content_length":"83850","record_id":"<urn:uuid:233cb0fc-0915-451b-b1a3-ed8124104257>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00771.warc.gz"} |
Math Facts Worksheets Grade 1 - 1st Grade Math Worksheets
Math Facts Worksheets 1st Grade – Mathematics is a fundamental skill which we utilize every day and all of it starts with the fundamentals taught in the first year of school. It is essential to
utilize worksheets for the 1st grade in order to help kids understand the fundamentals of math. The Importance of 1st … Read more
Math Fact Worksheets 1st Grade
Math Fact Worksheets 1st Grade – Mathematics is a fundamental skill that we employ every day, and it all begins with the fundamentals taught in the first grade. One of the best ways to help children
grasp these elementary concepts is through worksheets for math in the 1st grade. First Grade Math Worksheets: They are … Read more | {"url":"https://www.1stgrademathworksheets.com/tag/math-facts-worksheets-grade-1/","timestamp":"2024-11-09T07:29:01Z","content_type":"text/html","content_length":"54417","record_id":"<urn:uuid:efe06e8c-e227-4129-8c8f-ed8e92b2ee76>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00013.warc.gz"} |
Differing first-principle models for Maxwell-Boltzmann statistics?
• Thread starter greswd
• Start date
In summary, the conversation discussed two different models for distributing 9 units of energy among 6 particles. The first model, using a combination lock, showed that there are a total of 1 million
permutations but only a fraction of them are valid due to the limited amount of energy available. The second model, using 9 dice, showed that all permutations are valid since the particles are
indistinguishable. However, the conversation also pointed out that the dice model does not account for the fact that some combinations of energy levels are indistinguishable, leading to a difference
in the odds of certain energy levels occurring. This is where the equipartition theorem comes in, stating that all distinguishable sets are equally probable. This is a fundamental principle that
Let's consider a simple scenario in Maxwell-Boltzmann statistics: 6 identical but distinguishable particles, and 9 quanta of energy, 9 indivisible units, to be distributed among the particles.The
first model is like that of the wheels on a combination lock, or should I say "permutation lock".
There are 6 wheels, one for each particle, and the numbers run from 0 to 9, since there are 9 units of energy available.
The number on each wheel represents the amount of energy each particle possesses. Since there are 6 wheels, there are a total of a million permutations.
However, there are only 9 units of energy in total, so if the 6 numbers do not add up to 9 in total, it is an invalid permutation.
Considering only the valid permutations, the odds of a particle being in the ground state, having zero units of energy, is the highest of all, and it only decreases subsequently for each incremental
energy level.
Now for the second model, 9 dice.
One die for each unit of energy, and of course 6 sides, one for each particle.
You can imagine rolling all 9 dice, with the number on each die indicating which particle the unit of energy the die represents belongs to.
This time, there's no need to remove invalid permutations, all are valid.
This model produces quite different results from the first one, as the odds of a particle having one unit of energy is the highest, and the odds of having two units of energy is even higher than the
odds of having no energy at all.Maxwell-Boltzmann statistics follows the first model, and I'm curious as to why nature is of the first model rather than of the second.
Because intuitively, it seems to me like the second model would be more likely to occur. It's simpler in a way, no assumptions need to be made about invalid permutations.
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greswd said:
You can imagine rolling all 9 dice, with the number on each die indicating which particle the unit of energy the die represents belongs to.
So state 1 1 1 1 1 1 1 1 2 occurs 9 times (any of the other dice can be 2 instead of 1) ?
BvU said:
So state 1 1 1 1 1 1 1 1 2 occurs 9 times (any of the other dice can be 2 instead of 1) ?
yes, so maybe that state should have a statistical weight of 9?
What would it get in the permutation lock scenario ?
BvU said:
What would it get in the permutation lock scenario ?
It is, as you've mentioned,
state, so I think it's weight is
, out of 2002.
While for the dice it is 9 out of 10077696.
greswd said:
You can imagine rolling all 9 dice, with the number on each die indicating which particle the unit of energy the die represents belongs to.
This time, there's no need to remove invalid permutations, all are valid.
This is not exactly correct. Although all permutations are valid, they are not all distinguishable. For example {1,1,1,1,1,1,1,2,2} refers to the same state as {2,2,1,1,1,1,1,1,1} and
{2,1,1,1,1,1,1,1,2}. The particles are distinguishable, but the energy levels are not. If you go through and eliminate all of the invalid permutations of locks and all of the indistinguishable
permutations of dice then you are left with 2002 sets in both cases.
The equipartition theorem basically states that all of these 2002 distinguishable/valid sets are equally probable. You could make an arbitrary number of models describing how many indistinguishable
sets correspond to each distinguishable set, but that doesn't affect the physics. So physically both scenarios are the same since they both wind up with the same number of sets. In one case you have
to remove invalid cases and in the other you have to remove indistinguishable cases.
greswd said:
It is, as you've mentioned, a state, so I think it's weight is 1, out of 2002.
While for the dice it is 9 out of 10077696.
In both cases it is 1 out of 2002.
Conversely, all 9 energy bits for the first particle gets one out of 10077696 with the dice and 1 out of 2002 with the lock
The energy bits are indistinguishable, so you'd have to divide each of the 10077696 combinations by the number of ways such a combination can occur to get its weight.
(ah, Dale to the rescue while I waste my time trying to get an elegant expression for this dice combination weight
Dale said:
The equipartition theorem basically states that all of these 2002 distinguishable/valid sets are equally probable. You could make an arbitrary number of models describing how many
indistinguishable sets correspond to each distinguishable set, but that doesn't affect the physics.
thanks, so, what's the root, fundamental basis or effect that leads to equipartition probabilities and nature not following the dice distribution?
Is it like some fundamental rule/effect which describes distinguishability affecting probabilities?
Because it seems like the combination lock method isn't a direct, sole logical given, and that there is some fundamental physical principle or effect which makes it so.
You can't expect nine dice to mimic nine units of energy ...
BvU said:
You can't expect nine dice to mimic nine units of energy ...
But six wheels of a lock demonstrate the mathematics of the phenomenon very well.I think a more direct, clear and physical visual would be a beer-pong set-up.
Imagine 6 cups representing the 6 particles, and 9 ping-pong balls representing the units of energy, which are hurled at the cups.
Assuming the balls are ghostly and pass through each other instead of bouncing off each other, it will produce the same results as that of the dice.
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greswd said:
thanks, so, what's the root, fundamental basis or effect that leads to equipartition probabilities and nature not following the dice distribution?
I think that it is called the identity of indiscernibles. Two states that are completely indistinguishable are the same state. I believe that it is simply an assumption that seems to work. Apparently
it has a very powerful usage in quantum mechanics where it is the cornerstone of the statistical distributions used to represent particles.
The reason that the combination lock works and the dice does not is that you specified that the particles were distinguishable and the energy levels were not. The combination lock respects that
distinction and the dice do not. If you had indistinguishable particles then the combination lock metaphor wouldn't work either.
Dale said:
I think that it is called the identity of indiscernibles. Two states that are completely indistinguishable are the same state.
I believe that it is simply an assumption that seems to work.
wow, that's pretty interesting. I think most would imagine a "beer-pong distribution" as the simplest method of probabilistically distributing energy among particles. but nature appears to have other
greswd said:
I think most would imagine a "beer-pong distribution" as the simplest method of probabilistically distributing energy among particles. but nature appears to have other ideas.
As long as the balls are indistinguishable and the cups are distinguishable it will work fine.
Dale said:
As long as the balls are indistinguishable and the cups are distinguishable it will work fine.
Considering actual, macroscopic ping-pong balls, I'm not sure how indistinguishability would apply.
Whether all 9 balls are white, or each painted with a different color, it would have no effect on the overall distribution. A layer of paint can't change anything.
BvU said:
The energy bits are indistinguishable, so you'd have to divide each of the 10077696 combinations by the number of ways such a combination can occur to get its weight.
greswd said:
Dale said:
As long as the balls are indistinguishable and the cups are distinguishable it will work fine.
Considering actual, macroscopic ping-pong balls, I'm not sure how indistinguishability would apply.
Whether all 9 balls are white, or each painted with a different color, it would have no effect on the overall distribution. A layer of paint can't change anything.
Or, a reverse scenario, if each ping-pong ball is originally of a unique color, we could make them "indistinguishable" by painting all of them white.
And that layer of white paint also can't change anything, the kinematics will remain the same.
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Does someone have a decent expression for the weight of a 9 dice throw ?
BvU said:
Does someone have a decent expression for the weight of a 9 dice throw ?
well, there are 10077696 possible permutations in total.
I mean: What's the weight if I throw 1 2 3 4 5 6 6 6 6
BvU said:
I mean: What's the weight if I throw 1 2 3 4 5 6 6 6 6
it should be 15120
greswd said:
Considering actual, macroscopic ping-pong balls, I'm not sure how indistinguishability would apply.
It doesn’t. But IF it did then the beer pong distribution would work. The observed fact that it doesn’t work is due to the violation of the indistinguishability assumption. Clearly, a layer of paint
is not what makes a physical system indistinguishable.
greswd said:
So do you have an expression for that ? The number isn't interesting.
Dale said:
It doesn’t. But IF it did then the beer pong distribution would work. The observed fact that it doesn’t work is due to the violation of the indistinguishability assumption. Clearly, a layer of
paint is not what makes a physical system indistinguishable.
what do you think makes a physical system indistinguishable?
each ball has tiny imperfections and flaws that distinguish it from the others.
however, even if they are somehow 100% perfect copies of each other, it wouldn't affect the kinematics and wouldn't produce different results.
Their trajectories through the air would not change much.
So energy quanta probably have some other properties in addition to their indistinguishability which make them behave differently and follow the lock model.
greswd said:
So energy quanta probably have some other properties
I think it is the opposite. They have fewer properties, not additional ones. That is what makes them indistinguishable. Macroscopic systems have too many properties for the identity of indiscernible
to play a role.
BvU said:
So do you have an expression for that ? The number isn't interesting.
Dale said:
I think it is the opposite. They have fewer properties, not additional ones. That is what makes them indistinguishable. Macroscopic systems have too many properties for the identity of
indiscernible to play a role.
The concept of energy quanta is somewhat abstract, but if we replaced all of the ping-pong balls with classical electrons, which behave like tiny ping-pong balls somewhat, I think the result probably
wouldn't change.
I think even real electrons in a beer-electron-pong set-up would result in the dice model.
greswd said:
I think even real electrons in a beer-electron-pong set-up would result in the dice model.
If that were true then we would be able to observe substantial departures from equipartition
Dale said:
If that were true then we would be able to observe substantial departures from equipartition
If we fire 9 electrons, one at a time, at a set-up of 6 holes where the electron has a 1/6th chance of entering anyone hole, we can see from there that it is no different from beer-pong.
greswd said:
If we fire 9 electrons, one at a time, at a set-up of 6 holes where the electron has a 1/6th chance of entering anyone hole, we can see from there that it is no different from beer-pong.
If that were true then there would be a violation of equipartition. I don't have any evidence to suggest that is correct, do you?
Dale said:
If that were true then there would be a violation of equipartition. I don't have any evidence to suggest that is correct, do you?
But you can imagine a fair set-up, with a 1/6th chance each, like a fair die.
And the electrons are fired one at a time. It is virtually identical to beer pong.
greswd said:
It is virtually identical to beer pong.
With the very big difference of distinguishability, right? You are still envisioning that the electrons are indistinguishable, but ping pong balls are not. That has testable consequences.
Dale said:
With the very big difference of distinguishability, right? You are still envisioning that the electrons are indistinguishable, but ping pong balls are not. That has testable consequences.
But can you imagine how it would differ?
We set it up with a fair 1/6th chance each, or very closely to perfect fairness.
But somehow the electrons behave differently and don't behave according to those odds.
That would be very strange. Electrons can indeed behave very strangely, or non-classically, but not in this manner.
If I understand correctly, in the dice / beer pong analogy, you want to label the units of energy, distinguishing which is where?
If that is the case, you need to understand that keep track of the energy is only accounting (see the Feynman lectures). The state of the economy does not depend on which dollar is in which bank
DrClaude said:
If I understand correctly, in the dice / beer pong analogy, you want to label the units of energy, distinguishing which is where?
If that is the case, you need to understand that keep track of the energy is only accounting (see the Feynman lectures). The state of the economy does not depend on which dollar is in which bank
I mentioned removing all forms of labels in #15, and also mentioned throwing all 9 balls at the same time, albeit "ghostly" balls, in #10.
greswd said:
I mentioned removing all forms of labels in #15, and also mentioned throwing all 9 balls at the same time, albeit "ghostly" balls, in #10.
Then I must admit I don't understand the discussion.
greswd said:
Electrons can indeed behave very strangely, or non-classically, but not in this manner.
I think they do behave non-classically in exactly this manner. At least I am not aware of any evidence of a violation of the equipartition theorem. Are you? You seem very convinced by this, but the
consequences would be easily observable. | {"url":"https://www.physicsforums.com/threads/differing-first-principle-models-for-maxwell-boltzmann-statistics.973287/","timestamp":"2024-11-10T21:06:46Z","content_type":"text/html","content_length":"263987","record_id":"<urn:uuid:0e82e719-9582-4341-ba5c-8d274be35d36>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00571.warc.gz"} |
Papers with Code - Yimeng Min
1 code implementation • 28 Oct 2020 • Yimeng Min, Frederik Wenkel, Guy Wolf
Geometric scattering has recently gained recognition in graph representation learning, and recent work has shown that integrating scattering features in graph convolution networks (GCNs) can
alleviate the typical oversmoothing of features in node representation learning. | {"url":"https://paperswithcode.com/search?q=author%3AYimeng+Min","timestamp":"2024-11-13T20:54:20Z","content_type":"text/html","content_length":"113338","record_id":"<urn:uuid:728b10a5-ba6f-4aed-8239-e7d7d7f92b22>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00543.warc.gz"} |
Female attractiveness "index figure"
• Hehe...
Been there, done that.
I think she would be a 3. Or mayhaps a 2½.
• Originally posted by DukeP
Im more into waves and rhytms.
A friend and I (duing a hot summerday in the Copenhagen shopping district) formulated a formula based on the frequency of bounces.
So we had 1's, 1½'s, 2's and 3 bounces per stride.
Add your own explanation about standing waves, motionresistance etc. to the interpretation.
Perhaps this will help you?
• Originally posted by Sasq
Subtract the clothes, divide the legs, add a couple of square roots and see how she multiplies?
ROFL.. Classic!
• Originally posted by Sasq
Subtract the clothes, divide the legs, add a couple of square roots and see how she multiplies?
• Im more into waves and rhytms.
A friend and I (duing a hot summerday in the Copenhagen shopping district) formulated a formula based on the frequency of bounces.
So we had 1's, 1½'s, 2's and 3 bounces per stride.
Add your own explanation about standing waves, motionresistance etc. to the interpretation.
• Subtract the clothes, divide the legs, add a couple of square roots and see how she multiplies?
• I have to agree there, but OK - be a sport and try and define ito a formula what makes you forget maths when you see the woman...............
• What? LOL
When I see an attractive woman the last thing on my mind is mathematics.
• Female attractiveness "index figure"
Just heard on the radio a formula for this important number devised by Japanese scientists after years of study.......
Males according to them use the following formula to determine on the spur of the moment if they would like to bonk the girl currently being studied........
Volume of wench's body divided by the distance between the wench's chin and toes!
Now I would have suggested something more along the following lines for the average male human representative.......
Bra Cup size (a = 1 .....b=2 etc - aa or worse immediate disqualification!) + (length) + (distance between navel and top of bikini-line) divided by (difference between eye focus points) +
(distance from top of breast to center of nipple - Pi x distance between center of nipple and bottom of breast) + (weight)
Care to play MSP for a while and help out these poor demented scientists by adding to the formula or even suggesting your own? | {"url":"http://murc.ws/forum/murc-life/the-lounge/44954-female-attractiveness-index-figure?view=stream&s=bc348ca49e5ab0fcad3e201f443f40e2","timestamp":"2024-11-13T21:24:32Z","content_type":"application/xhtml+xml","content_length":"89082","record_id":"<urn:uuid:d204fba8-308d-4f5a-ba8a-30155bc1b313>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00423.warc.gz"} |
So what prime is leaving with Revenant Prime coming?
Typically They announce which Primes will enter the vault now look I understand the vault isn't a thing anymore but I know they still have to make room for next prime
So which prime is leaving to be added to the prime resurgence rotation???
I hope this is still a thing cuz I want hope ro obtain nidus prime & harrow prime noggles someday & I would like to know before I tab Khora Prime access if I'd have a shot at it again during prime
resurgence eventually??
DE never mentioned how they plan to do this going forward... so any word if we miss on a PA that they eventually will come back to prime resurgence rotation to obtain down the road???
8 answers to this question
nezha and his weapons
I find it odd they haven't announced the vaulting yet.
1 minute ago, Godzilla853 said:
I find it odd they haven't announced the vaulting yet.
Ikr hopefully they give word tomorrow on the plan going forward
DE normally announces it a week before even if they keep the next frame a secret.
23 minutes ago, Godzilla853 said:
DE normally announces it a week before even if they keep the next frame a secret.
Yeah But The new Prime comes out Wednesday so not much time Is remaining to announce it..
1 hour ago, (PSN)KCGrimReaper15 said:
Yeah But The new Prime comes out Wednesday so not much time Is remaining to announce it..
Maybe with the prime vault changes he is not going to be vaulted.
21 minutes ago, Godzilla853 said:
Maybe with the prime vault changes he is not going to be vaulted.
If that's the case that interesting hopefully they plan to add a way to get the prime noggles for Nidus & Harrow Prime as only 2 I'm missing from PA...
I bought there accessory packs & I honestly believe buying the accessory pack should be more then enough to qualify for the noggles...
On 2022-10-02 at 7:41 PM, (PSN)KCGrimReaper15 said:
If that's the case that interesting hopefully they plan to add a way to get the prime noggles for Nidus & Harrow Prime as only 2 I'm missing from PA...
I bought there accessory packs & I honestly believe buying the accessory pack should be more then enough to qualify for the noggles...
I agree on that as they are just cosmetics.
This topic is now archived and is closed to further replies. | {"url":"https://forums.warframe.com/topic/1326634-so-what-prime-is-leaving-with-revenant-prime-coming/","timestamp":"2024-11-03T00:32:17Z","content_type":"text/html","content_length":"150645","record_id":"<urn:uuid:134e0046-808f-4841-abf4-3d9f70499518>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00074.warc.gz"} |
merlion.models.anomaly.forecast_based package
merlion.models.anomaly.forecast_based package
Contains all forecaster-based anomaly detectors. These models support all functionality of both anomaly detectors (merlion.models.anomaly) and forecasters (merlion.models.forecast).
Forecasting-based anomaly detectors are instances of an abstract ForecastingDetectorBase class. Many forecasting models support anomaly detection variants, where the anomaly score is based on the
difference between the predicted and true time series value, and optionally the model’s uncertainty in its own prediction.
base Base class for anomaly detectors based on forecasting models.
arima Classic ARIMA (AutoRegressive Integrated Moving Average) forecasting model, adapted for anomaly detection.
sarima Seasonal ARIMA (SARIMA) forecasting model, adapted for anomaly detection.
ets ETS (error, trend, seasonal) forecasting model, adapted for anomaly detection.
prophet Adaptation of Facebook's Prophet forecasting model to anomaly detection.
lstm Adaptation of a LSTM neural net forecaster, to the task of anomaly detection.
mses MSES (Multi-Scale Exponential Smoother) forecasting model adapted for anomaly detection.
merlion.models.anomaly.forecast_based.base module
Base class for anomaly detectors based on forecasting models.
class merlion.models.anomaly.forecast_based.base.ForecastingDetectorBase(config)
Bases: ForecasterBase, DetectorBase
Base class for a forecast-based anomaly detector.
config (ForecasterConfig) – model configuration
forecast_to_anom_score(time_series, forecast, stderr)
Compare a model’s forecast to a ground truth time series, in order to compute anomaly scores. By default, we compute a z-score if model uncertainty (stderr) is given, or the residuals if
there is no model uncertainty.
○ time_series (TimeSeries) – the ground truth time series.
○ forecast (TimeSeries) – the model’s forecasted values for the time series
○ stderr (Optional[TimeSeries]) – the standard errors of the model’s forecast
Return type
Anomaly scores based on the difference between the ground truth values of the time series, and the model’s forecast.
train(train_data, anomaly_labels=None, train_config=None, post_rule_train_config=None)
Trains the underlying forecaster (unsupervised) on the training data. Converts the forecast into anomaly scores, and and then trains the post-rule for filtering anomaly scores (supervised, if
labels are given) on the input time series.
○ train_data (TimeSeries) – a TimeSeries of metric values to train the model.
○ anomaly_labels (Optional[TimeSeries]) – a TimeSeries indicating which timestamps are anomalous. Optional.
○ train_config – Additional training configs, if needed. Only required for some models.
○ post_rule_train_config – The config to use for training the model’s post-rule. The model’s default post-rule train config is used if none is supplied here.
Return type
A TimeSeries of the model’s anomaly scores on the training data.
get_anomaly_score(time_series, time_series_prev=None)
Returns the model’s predicted sequence of anomaly scores.
○ time_series (TimeSeries) – the TimeSeries we wish to predict anomaly scores for.
○ time_series_prev (Optional[TimeSeries]) – a TimeSeries immediately preceding time_series. If given, we use it to initialize the time series anomaly detection model. Otherwise, we
assume that time_series immediately follows the training data.
Return type
a univariate TimeSeries of anomaly scores
get_figure(*, time_series=None, time_stamps=None, time_series_prev=None, plot_anomaly=True, filter_scores=True, plot_forecast=False, plot_forecast_uncertainty=False, plot_time_series_prev=False)
○ time_series (Optional[TimeSeries]) – the time series over whose timestamps we wish to make a forecast. Exactly one of time_series or time_stamps should be provided.
○ time_stamps (Optional[List[int]]) – a list of timestamps we wish to forecast for. Exactly one of time_series or time_stamps should be provided.
○ time_series_prev (Optional[TimeSeries]) – a TimeSeries immediately preceding time_stamps. If given, we use it to initialize the time series model. Otherwise, we assume that
time_stamps immediately follows the training data.
○ plot_anomaly – Whether to plot the model’s predicted anomaly scores.
○ filter_scores – whether to filter the anomaly scores by the post-rule before plotting them.
○ plot_forecast – Whether to plot the model’s forecasted values.
○ plot_forecast_uncertainty – whether to plot uncertainty estimates (the inter-quartile range) for forecast values. Not supported for all models.
○ plot_time_series_prev – whether to plot time_series_prev (and the model’s fit for it). Only used if time_series_prev is given.
Return type
a Figure of the model’s anomaly score predictions and/or forecast.
plot_anomaly(time_series, time_series_prev=None, *, filter_scores=True, plot_forecast=False, plot_forecast_uncertainty=False, plot_time_series_prev=False, figsize=(1000, 600), ax=None)
Plots the time series in matplotlib as a line graph, with points in the series overlaid as points color-coded to indicate their severity as anomalies. Optionally allows you to overlay the
model’s forecast & the model’s uncertainty in its forecast (if applicable).
○ time_series (TimeSeries) – The time series we wish to plot, with color-coding to indicate anomalies.
○ time_series_prev (Optional[TimeSeries]) – A time series immediately preceding time_series, which is used to initialize the time series model. Otherwise, we assume time_series
immediately follows the training data.
○ filter_scores – whether to filter the anomaly scores by the post-rule before plotting them.
○ plot_forecast – Whether to plot the model’s forecast, in addition to the anomaly scores.
○ plot_forecast_uncertainty – Whether to plot the model’s uncertainty in its own forecast, in addition to the forecast and anomaly scores. Only used if plot_forecast is True.
○ plot_time_series_prev – whether to plot time_series_prev (and the model’s fit for it). Only used if time_series_prev is given.
○ figsize – figure size in pixels
○ ax – matplotlib axis to add this plot to
matplotlib figure & axes
plot_anomaly_plotly(time_series, time_series_prev=None, *, filter_scores=True, plot_forecast=False, plot_forecast_uncertainty=False, plot_time_series_prev=False, figsize=(1000, 600))
Plots the time series in matplotlib as a line graph, with points in the series overlaid as points color-coded to indicate their severity as anomalies. Optionally allows you to overlay the
model’s forecast & the model’s uncertainty in its forecast (if applicable).
○ time_series (TimeSeries) – The time series we wish to plot, with color-coding to indicate anomalies.
○ time_series_prev (Optional[TimeSeries]) – A time series immediately preceding time_series, which is used to initialize the time series model. Otherwise, we assume time_series
immediately follows the training data.
○ filter_scores – whether to filter the anomaly scores by the post-rule before plotting them.
○ plot_forecast – Whether to plot the model’s forecast, in addition to the anomaly scores.
○ plot_forecast_uncertainty – Whether to plot the model’s uncertainty in its own forecast, in addition to the forecast and anomaly scores. Only used if plot_forecast is True.
○ plot_time_series_prev – whether to plot time_series_prev (and the model’s fit for it). Only used if time_series_prev is given.
○ figsize – figure size in pixels
plotly figure
plot_forecast(*, time_series=None, time_stamps=None, time_series_prev=None, plot_forecast_uncertainty=False, plot_time_series_prev=False, figsize=(1000, 600), ax=None)
Plots the forecast for the time series in matplotlib, optionally also plotting the uncertainty of the forecast, as well as the past values (both true and predicted) of the time series.
○ time_series (Optional[TimeSeries]) – the time series over whose timestamps we wish to make a forecast. Exactly one of time_series or time_stamps should be provided.
○ time_stamps (Optional[List[int]]) – a list of timestamps we wish to forecast for. Exactly one of time_series or time_stamps should be provided.
○ time_series_prev (Optional[TimeSeries]) – a TimeSeries immediately preceding time_stamps. If given, we use it to initialize the time series model. Otherwise, we assume that
time_stamps immediately follows the training data.
○ plot_forecast_uncertainty – whether to plot uncertainty estimates (the inter-quartile range) for forecast values. Not supported for all models.
○ plot_time_series_prev – whether to plot time_series_prev (and the model’s fit for it). Only used if time_series_prev is given.
○ figsize – figure size in pixels
○ ax – matplotlib axis to add this plot to
(fig, ax): matplotlib figure & axes the figure was plotted on
plot_forecast_plotly(*, time_series=None, time_stamps=None, time_series_prev=None, plot_forecast_uncertainty=False, plot_time_series_prev=False, figsize=(1000, 600))
Plots the forecast for the time series in plotly, optionally also plotting the uncertainty of the forecast, as well as the past values (both true and predicted) of the time series.
○ time_series (Optional[TimeSeries]) – the time series over whose timestamps we wish to make a forecast. Exactly one of time_series or time_stamps should be provided.
○ time_stamps (Optional[List[int]]) – a list of timestamps we wish to forecast for. Exactly one of time_series or time_stamps should be provided.
○ time_series_prev (Optional[TimeSeries]) – a TimeSeries immediately preceding time_stamps. If given, we use it to initialize the time series model. Otherwise, we assume that
time_stamps immediately follows the training data.
○ plot_forecast_uncertainty – whether to plot uncertainty estimates (the inter-quartile range) for forecast values. Not supported for all models.
○ plot_time_series_prev – whether to plot time_series_prev (and the model’s fit for it). Only used if time_series_prev is given.
○ figsize – figure size in pixels
merlion.models.anomaly.forecast_based.arima module
Classic ARIMA (AutoRegressive Integrated Moving Average) forecasting model, adapted for anomaly detection.
class merlion.models.anomaly.forecast_based.arima.ArimaDetectorConfig(order=(4, 1, 2), seasonal_order=(0, 0, 0, 0), max_forecast_steps: int = None, target_seq_index: int = None, transform:
TransformBase = None, max_score: float = 1000, threshold=None, enable_calibrator=True, enable_threshold=True, **kwargs)
Bases: ArimaConfig, DetectorConfig
Configuration class for Arima. Just a Sarima model with seasonal order (0, 0, 0, 0).
Base class of the object used to configure an anomaly detection model.
☆ order – Order is (p, d, q) for an ARIMA(p, d, q) process. d must be an integer indicating the integration order of the process, while p and q must be integers indicating the AR and MA
orders (so that all lags up to those orders are included).
☆ seasonal_order – (0, 0, 0, 0) because ARIMA has no seasonal order.
☆ max_forecast_steps – Max # of steps we would like to forecast for. Required for some models like MSES and LGBMForecaster.
☆ target_seq_index – The index of the univariate (amongst all univariates in a general multivariate time series) whose value we would like to forecast.
☆ transform – Transformation to pre-process input time series.
☆ max_score – maximum possible uncalibrated anomaly score
☆ threshold – the rule to use for thresholding anomaly scores
☆ enable_calibrator – whether to enable a calibrator which automatically transforms all raw anomaly scores to be z-scores (i.e. distributed as N(0, 1)).
☆ enable_threshold – whether to enable the thresholding rule when post-processing anomaly scores
class merlion.models.anomaly.forecast_based.arima.ArimaDetector(config)
Bases: ForecastingDetectorBase, Arima
alias of ArimaDetectorConfig
merlion.models.anomaly.forecast_based.sarima module
Seasonal ARIMA (SARIMA) forecasting model, adapted for anomaly detection.
class merlion.models.anomaly.forecast_based.sarima.SarimaDetectorConfig(order=(4, 1, 2), seasonal_order=(2, 0, 1, 24), max_forecast_steps: int = None, target_seq_index: int = None, transform:
TransformBase = None, max_score: float = 1000, threshold=None, enable_calibrator=True, enable_threshold=True, **kwargs)
Bases: SarimaConfig, DetectorConfig
Config class for Sarima (Seasonal AutoRegressive Integrated Moving Average).
Base class of the object used to configure an anomaly detection model.
☆ order – Order is (p, d, q) for an ARIMA(p, d, q) process. d must be an integer indicating the integration order of the process, while p and q must be integers indicating the AR and MA
orders (so that all lags up to those orders are included).
☆ seasonal_order – Seasonal order is (P, D, Q, S) for seasonal ARIMA process, where s is the length of the seasonality cycle (e.g. s=24 for 24 hours on hourly granularity). P, D, Q are as
for ARIMA.
☆ max_forecast_steps – Max # of steps we would like to forecast for. Required for some models like MSES and LGBMForecaster.
☆ target_seq_index – The index of the univariate (amongst all univariates in a general multivariate time series) whose value we would like to forecast.
☆ transform – Transformation to pre-process input time series.
☆ max_score – maximum possible uncalibrated anomaly score
☆ threshold – the rule to use for thresholding anomaly scores
☆ enable_calibrator – whether to enable a calibrator which automatically transforms all raw anomaly scores to be z-scores (i.e. distributed as N(0, 1)).
☆ enable_threshold – whether to enable the thresholding rule when post-processing anomaly scores
class merlion.models.anomaly.forecast_based.sarima.SarimaDetector(config)
Bases: ForecastingDetectorBase, Sarima
alias of SarimaDetectorConfig
merlion.models.anomaly.forecast_based.ets module
ETS (error, trend, seasonal) forecasting model, adapted for anomaly detection.
class merlion.models.anomaly.forecast_based.ets.ETSDetectorConfig(max_forecast_steps=None, target_seq_index=None, error='add', trend='add', damped_trend=True, seasonal='add', seasonal_periods=None,
transform: TransformBase = None, enable_calibrator=False, max_score: float = 1000, threshold=None, enable_threshold=True, **kwargs)
Bases: ETSConfig, NoCalibrationDetectorConfig
Configuration class for ETS model. ETS model is an underlying state space model consisting of an error term (E), a trend component (T), a seasonal component (S), and a level component. Each
component is flexible with different traits with additive (‘add’) or multiplicative (‘mul’) formulation. Refer to https://otexts.com/fpp2/taxonomy.html for more information about ETS model.
Base class of the object used to configure an anomaly detection model.
☆ max_forecast_steps – Number of steps we would like to forecast for.
☆ target_seq_index – The index of the univariate (amongst all univariates in a general multivariate time series) whose value we would like to forecast.
☆ error – The error term. “add” or “mul”.
☆ trend – The trend component. “add”, “mul” or None.
☆ damped_trend – Whether or not an included trend component is damped.
☆ seasonal – The seasonal component. “add”, “mul” or None.
☆ seasonal_periods – The length of the seasonality cycle. None by default.
☆ transform – Transformation to pre-process input time series.
☆ enable_calibrator – False because this config assumes calibrated outputs from the model.
☆ max_score – maximum possible uncalibrated anomaly score
☆ threshold – the rule to use for thresholding anomaly scores
☆ enable_threshold – whether to enable the thresholding rule when post-processing anomaly scores
class merlion.models.anomaly.forecast_based.ets.ETSDetector(config)
Bases: ForecastingDetectorBase, ETS
alias of ETSDetectorConfig
merlion.models.anomaly.forecast_based.prophet module
Adaptation of Facebook’s Prophet forecasting model to anomaly detection.
class merlion.models.anomaly.forecast_based.prophet.ProphetDetectorConfig(max_forecast_steps=None, target_seq_index=None, yearly_seasonality='auto', weekly_seasonality='auto', daily_seasonality=
'auto', seasonality_mode='additive', holidays=None, uncertainty_samples=100, transform=None, max_score=1000, threshold=None, enable_calibrator=True, enable_threshold=True, **kwargs)
Bases: ProphetConfig, DetectorConfig
Configuration class for Facebook’s Prophet model, as described by Taylor & Letham, 2017.
Base class of the object used to configure an anomaly detection model.
☆ max_forecast_steps (Optional[int]) – Max # of steps we would like to forecast for.
☆ target_seq_index (Optional[int]) – The index of the univariate (amongst all univariates in a general multivariate time series) whose value we would like to forecast.
☆ yearly_seasonality (Union[bool, int]) – If bool, whether to enable yearly seasonality. By default, it is activated if there are >= 2 years of history, but deactivated otherwise. If int,
this is the number of Fourier series components used to model the seasonality (default = 10).
☆ weekly_seasonality (Union[bool, int]) – If bool, whether to enable weekly seasonality. By default, it is activated if there are >= 2 weeks of history, but deactivated otherwise. If int,
this is the number of Fourier series components used to model the seasonality (default = 3).
☆ daily_seasonality (Union[bool, int]) – If bool, whether to enable daily seasonality. By default, it is activated if there are >= 2 days of history, but deactivated otherwise. If int, this
is the number of Fourier series components used to model the seasonality (default = 4).
☆ seasonality_mode – ‘additive’ (default) or ‘multiplicative’.
☆ holidays – pd.DataFrame with columns holiday (string) and ds (date type) and optionally columns lower_window and upper_window which specify a range of days around the date to be included
as holidays. lower_window=-2 will include 2 days prior to the date as holidays. Also optionally can have a column prior_scale specifying the prior scale for that holiday. Can also be a
dict corresponding to the desired pd.DataFrame.
☆ uncertainty_samples (int) – The number of posterior samples to draw in order to calibrate the anomaly scores.
☆ transform – Transformation to pre-process input time series.
☆ max_score – maximum possible uncalibrated anomaly score
☆ threshold – the rule to use for thresholding anomaly scores
☆ enable_calibrator – whether to enable a calibrator which automatically transforms all raw anomaly scores to be z-scores (i.e. distributed as N(0, 1)).
☆ enable_threshold – whether to enable the thresholding rule when post-processing anomaly scores
class merlion.models.anomaly.forecast_based.prophet.ProphetDetector(config)
Bases: ForecastingDetectorBase, Prophet
alias of ProphetDetectorConfig
merlion.models.anomaly.forecast_based.lstm module
Adaptation of a LSTM neural net forecaster, to the task of anomaly detection.
class merlion.models.anomaly.forecast_based.lstm.LSTMDetectorConfig(max_forecast_steps, nhid=1024, model_strides=(1,), target_seq_index=None, transform=None, max_score=1000, threshold=None,
enable_calibrator=True, enable_threshold=True, **kwargs)
Bases: LSTMConfig, DetectorConfig
Configuration class for LSTM.
Base class of the object used to configure an anomaly detection model.
☆ max_forecast_steps (int) – Max # of steps we would like to forecast for. Required for some models like MSES and LGBMForecaster.
☆ nhid – hidden dimension of LSTM
☆ model_strides – tuple indicating the stride(s) at which we would like to subsample the input data before giving it to the model.
☆ target_seq_index – The index of the univariate (amongst all univariates in a general multivariate time series) whose value we would like to forecast.
☆ transform – Transformation to pre-process input time series.
☆ max_score – maximum possible uncalibrated anomaly score
☆ threshold – the rule to use for thresholding anomaly scores
☆ enable_calibrator – whether to enable a calibrator which automatically transforms all raw anomaly scores to be z-scores (i.e. distributed as N(0, 1)).
☆ enable_threshold – whether to enable the thresholding rule when post-processing anomaly scores
class merlion.models.anomaly.forecast_based.lstm.LSTMDetector(config)
Bases: ForecastingDetectorBase, LSTM
config (LSTMConfig) – model configuration
alias of LSTMDetectorConfig
merlion.models.anomaly.forecast_based.mses module
MSES (Multi-Scale Exponential Smoother) forecasting model adapted for anomaly detection.
class merlion.models.anomaly.forecast_based.mses.MSESDetectorConfig(max_forecast_steps, online_updates=True, max_backstep=None, recency_weight=0.5, accel_weight=1.0, optimize_acc=True, eta=0.0, rho=
0.0, phi=2.0, inflation=1.0, target_seq_index=None, transform=None, max_score=1000, threshold=None, enable_calibrator=True, enable_threshold=True, **kwargs)
Bases: MSESConfig, DetectorConfig
Configuration class for an MSES forecasting model adapted for anomaly detection.
Letting w be the recency weight, B the maximum backstep, x_t the last seen data point, and l_s,t the series of losses for scale s.
\[\begin{split}\begin{align*} \hat{x}_{t+h} & = \sum_{b=0}^B p_{b} \cdot (x_{t-b} + v_{b+h,t} + a_{b+h,t}) \\ \space \\ \text{where} \space\space & v_{b+h,t} = \text{EMA}_w(\Delta_{b+h} x_t) \\ &
a_{b+h,t} = \text{EMA}_w(\Delta_{b+h}^2 x_t) \\ \text{and} \space\space & p_b = \sigma(z)_b \space\space \\ \text{if} & \space\space z_b = (b+h)^\phi \cdot \text{EMA}_w(l_{b+h,t}) \cdot \text
{RWSE}_w(l_{b+h,t})\\ \end{align*}\end{split}\]
☆ max_forecast_steps (int) – Max # of steps we would like to forecast for. Required for some models like MSES and LGBMForecaster.
☆ max_backstep – Max backstep to use in forecasting. If we train with x(0),…,x(t), Then, the b-th model MSES uses will forecast x(t+h) by anchoring at x(t-b) and predicting xhat(t+h) = x
(t-b) + delta_hat(b+h).
☆ recency_weight – The recency weight parameter to use when estimating delta_hat.
☆ accel_weight – The weight to scale the acceleration by when computing delta_hat. Specifically, delta_hat(b+h) = velocity(b+h) + accel_weight * acceleration(b+h).
☆ optimize_acc – If True, the acceleration correction will only be used at scales ranging from 1,…(max_backstep+max_forecast_steps)/2.
☆ eta – The parameter used to control the rate at which recency_weight gets tuned when online updates are made to the model and losses can be computed.
☆ rho – The parameter that determines what fraction of the overall error is due to velcity error, while the rest is due to the complement. The error at any scale will be determined as rho *
velocity_error + (1-rho) * loss_error.
☆ phi – The parameter used to exponentially inflate the magnitude of loss error at different scales. Loss error for scale s will be increased by a factor of phi ** s.
☆ inflation – The inflation exponent to use when computing the distribution p(b|h) over the models when forecasting at horizon h according to standard errors of the estimated velocities
over the models; inflation=1 is equivalent to using the softmax function.
☆ target_seq_index – The index of the univariate (amongst all univariates in a general multivariate time series) whose value we would like to forecast.
☆ transform – Transformation to pre-process input time series.
☆ max_score – maximum possible uncalibrated anomaly score
☆ threshold – the rule to use for thresholding anomaly scores
☆ enable_calibrator – whether to enable a calibrator which automatically transforms all raw anomaly scores to be z-scores (i.e. distributed as N(0, 1)).
☆ enable_threshold – whether to enable the thresholding rule when post-processing anomaly scores
class merlion.models.anomaly.forecast_based.mses.MSESDetector(config)
Bases: ForecastingDetectorBase, MSES
config (MSESConfig) – model configuration
alias of MSESDetectorConfig
property online_updates
train(train_data, anomaly_labels=None, train_config=None, post_rule_train_config=None)
Trains the forecaster on the input time series.
○ train_data (TimeSeries) – a TimeSeries of metric values to train the model.
○ train_config – Additional training configs, if needed. Only required for some models.
Return type
the model’s prediction on train_data, in the same format as if you called ForecasterBase.forecast on the time stamps of train_data
get_anomaly_score(time_series, time_series_prev=None)
Returns the model’s predicted sequence of anomaly scores.
○ time_series (TimeSeries) – the TimeSeries we wish to predict anomaly scores for.
○ time_series_prev (Optional[TimeSeries]) – a TimeSeries immediately preceding time_series. If given, we use it to initialize the time series anomaly detection model. Otherwise, we
assume that time_series immediately follows the training data.
Return type
a univariate TimeSeries of anomaly scores | {"url":"https://opensource.salesforce.com/Merlion/v1.1.3/merlion.models.anomaly.forecast_based.html","timestamp":"2024-11-11T21:08:39Z","content_type":"text/html","content_length":"122228","record_id":"<urn:uuid:0b32af55-9a66-4cf1-b203-8330781cdd07>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00282.warc.gz"} |
Mathematics and Statistics
[ 514MAST21 ] Subject Mathematics and Statistics
Workload Mode of examination Education level Study areas Responsible person Coordinating university
6 ECTS Accumulative subject examination B1 - Bachelor's programme 1. year Mathematics Elisabeth Gaar Johannes Kepler University Linz
Detailed information
Original Bachelor's programme International Business Administration 2021W
study plan
The students have knowledge of the basic mathematical and statistical concepts that are used in business administration. They are able to describe and solve basic business administration
issues in a precise way. They have extensive knowledge of linear and exponential functions. The students know how to differentiate a function and why derivatives play a role in business
Objectives administration. They can deal with interest calculus and with systems of linear equations. They have profound knowledge of matrix and vector algebra and linear programming. They are
familiar with the basic concepts of graph theory. They know the basics of probability theory, density functions and cumulative distribution functions. They have extensive knowledge of
scale and location parameters. They can perform a regression analysis and interpret its results.
The students have knowledge of first-order logic, linear functions, exponential functions, differential calculus, interest calculus, systems of linear equations, matrix and vector
Subject algebra, linear programming, basic graph theory, probability theory, probability density functions, cumulative distribution functions, scale and location parameters and regression | {"url":"https://studienhandbuch.jku.at/162073","timestamp":"2024-11-04T23:59:13Z","content_type":"application/xhtml+xml","content_length":"14894","record_id":"<urn:uuid:6f36c910-fcc5-4b2d-a2d8-e909bffc798a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00120.warc.gz"} |
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Upgrading Custom Hydraulic Blocks to Use the Isothermal Liquid Domain
If your model contains custom blocks with hydraulic ports, you can rewrite the underlying component source to adapt them to using the isothermal liquid domain.
This change may lead to numerical changes in the block behavior. Using mass flow rate, instead of volumetric flow rate, as the Through variable reduces the potential for small errors in mass
conservation to accumulate over time due to the conversion between mass and volumetric quantities, which results in increased accuracy.
To rewrite the component source, follow these steps:
1. Replace the nodes of type foundation.hydraulic.hydraulic with foundation.isothermal_liquid.isothermal_liquid.
2. In the variables section, replace the Through variable q with mdot. q represents volumetric flow rate and has units of volume over time, such as m^3/s. mdot represents mass flow rate and has
units of mass over time, such as kg/s.
3. Add an intermediate, rho, to represent fluid density. Use the provided library function to calculate density based on pressure at the port, for blocks with a single fluid port, or based on
average port pressure, for blocks with two or more fluid ports. To view the source file of this function, at the MATLAB^® command prompt, type:
open([matlabroot '/toolbox/physmod/simscape/library/m/+foundation/+isothermal_liquid/mixture_density.ssc'])
4. Rewrite the equations by replacing q with mdot/rho.
For example, consider this custom component, which models a hydraulic linear resistance.
component custom_linear_resistance
% Custom Linear Hydraulic Resistance
% This block represents a hydraulic resistance where pressure loss
% is directly proportional to flow rate.
% Connections A and B are conserving hydraulic ports associated
% with the block inlet and outlet, respectively. The block positive
% direction is from port A to port B. This means that the flow rate is
% positive if fluid flows from A to B, and the pressure loss is determined
% as p = p_A - p_B.
% Copyright 2005-2023 The MathWorks, Inc.
A = foundation.hydraulic.hydraulic; % A:left
B = foundation.hydraulic.hydraulic; % B:right
variables (Access = protected)
q = { 1e-3 , 'm^3/s' }; % Flow rate
p = { 0 , 'Pa' }; % Pressure differential
q : A.q -> B.q;
resistance = { 1, 'GPa/(m^3/s)' }; % Resistance
% Assertion
assert(resistance >= 0)
p == A.p - B.p;
p == resistance * q;
To adapt this component to use the isothermal liquid domain:
1. Declare nodes A and B as foundation.isothermal_liquid.isothermal_liquid.
2. Under variables, replace q with mdot.
3. Add the rho_avg intermediate, which calculates density based on average port pressure. The density calculation uses the Foundation library function foundation.isothermal_liquid.mixture_density.
4. Rewrite the equation p == resistance * q; by replacing q with mdot/rho_avg.
The new component, custom_linear_resistance_il, now models an isothermal liquid linear resistance.
component custom_linear_resistance_il
% Custom Linear Resistance (IL) :
% This block represents a hydraulic resistance where pressure loss
% is directly proportional to flow rate.
% Connections A and B are conserving isothermal liquid ports associated
% with the block inlet and outlet, respectively. The block positive
% direction is from port A to port B. This means that the flow rate is
% positive if fluid flows from A to B, and the pressure loss is determined
% as p = p_A - p_B.
% Copyright 2005-2023 The MathWorks, Inc.
A = foundation.isothermal_liquid.isothermal_liquid; % A:left
B = foundation.isothermal_liquid.isothermal_liquid; % B:right
variables (Access = protected)
mdot = { 0.1 , 'kg/s' }; % Mass flow rate
p = { 0 , 'Pa' }; % Pressure differential
mdot : A.mdot -> B.mdot;
resistance = { 1, 'GPa/(m^3/s)' }; % Resistance
% For logging
intermediates (Access = private)
rho_avg = foundation.isothermal_liquid.mixture_density((A.p + B.p)/2, ...
A.bulk_modulus_model, A.air_dissolution_model, A.rho_L_atm, A.beta_L_atm, ...
A.beta_gain, A.air_fraction, A.rho_g_atm, A.polytropic_index, A.p_atm, ...
A.p_crit, A.p_min); % Average liquid density
% Assertion
assert(resistance >= 0)
p == A.p - B.p;
p == resistance * mdot/rho_avg;
See Also
hydraulicToIsothermalLiquid | hydraulicToIsothermalLiquidPostProcess | Interface (H-IL) | Simulation Data Inspector
Related Topics | {"url":"https://uk.mathworks.com/help/simscape/ug/upgrading-custom-hydraulic-blocks-to-use-isothermal-liquid-domain.html","timestamp":"2024-11-07T10:09:49Z","content_type":"text/html","content_length":"75106","record_id":"<urn:uuid:75dfbd3f-0450-4a55-b641-5a08d0956116>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00585.warc.gz"} |
Multiplication a number with 99 Shortcut Tricks - Math Shortcut Tricks
Multiplication a number with 99 Shortcut Tricks
Shortcut Tricks are very important things in competitive exam. Time takes a huge part in competitive exams. If you know time management then everything will be easier for you. Most of us miss this
thing. Here in this page we give few examples on Multiplication a number with 99 shortcut tricks. These shortcut tricks cover all sorts of tricks on Multiplication a number with 99. We request all
visitors to read all examples carefully. These examples here will help you to better understand shortcut tricks on multiplication a number with 99. | {"url":"https://www.math-shortcut-tricks.com/multiplication-a-number-with-99/","timestamp":"2024-11-06T17:57:22Z","content_type":"text/html","content_length":"206613","record_id":"<urn:uuid:7e3262ab-6a6c-4035-985e-20c2edcc900e>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00460.warc.gz"} |
5.3.3 Torus with 1 hole
In our last example, we consider a pentagon with two pairs of edges identified. As we saw in Section 2.3, identification of the edges produces a torus with a hole. In this case there are five
vertex-neighbourhoods to fit together, as shown in Figure 109. Identification of edges a and b shows that neighbourhoods 2, 3 and 4 fit together as shown. Identification of the edges b shows that
neighbourhood 5 fits alongside neighbourhood 2, and identification of edges a shows that neighbourhood 1 fits alongside neighbourhood 4. However, neighbourhoods 1 and 5 do not fit together, and so we
do not obtain a disc-like neighbourhood – rather we obtain a half-disc-like neighbourhood and the resulting vertex lies on the boundary of the hole.
Finally consider a point x that lies on the edge c of the pentagon (see Figure 110). This has a half-disc-like neighbourhood. After identification of edges, this point lies on the boundary of the
hole, and maintains its half-disc-like neighbourhood.
From these examples we can deduce that:
• if a point lies on an edge that is identified with another edge, but not at a vertex of that edge, then after identification that point has a disc-like neighbourhood;
• if a point lies on a part of the boundary that is not identified with another part, then after identification that point has a half-disc-like neighbourhood.
However, if a point is vertex of an edge that is identified with another edge, we have seen that after identification that point can have a disc-like or half-disc-like neighbourhood. We need to be
sure that these are the only possibilities.
Consider the vertices x[1], x[2], …, x[k] of a polygon that are identified to the same point x by the edge identifications. In the polygon, each vertex x[i] has a half-disc-like neighbourhood that
can be represented as in Figure 111. We shall call such a piece a wedge.
Some of the edges that come into each vertex x[i] are to be identified. Let us see what can happen. First, notice that the edge identifications that involve the vertices x[i] cannot have wedges that
fit together to form two or more separate pieces: this is because, in such a case, the edge identifications would not identify all the vertices x[i] to a single point x. Also, because whole edges and
not just vertices are identified, we cannot produce a situation such as that illustrated in Figure 112. Thus, the edge identifications that involve the vertices x[i] are such that all the wedges fit
together to form a single neighbourhood. Either the point x is in the interior of the object so formed, as for the torus and Klein bottle, giving rise to a disc-like neighbourhood formed from the
wedges, or the point x lies on the boundary, as for the torus with 1 hole. Thus:
• if a point lies at a vertex of an edge that is identified with another edge, then after identification that point has a disc-like or half-disc-like neighbourhood.
We have therefore demonstrated that every point of polygon with edge identifications has, after identification, a disc-like or half-disc-like neighbourhood. Furthermore, given any two points of the
object obtained after identification, by choosing disc-like or half-disc-like neighbourhoods of sufficiently small radius, we can find neighbourhoods that do not intersect – so the object is
Hausdorff. Combining these results with those of Section 5.2 tells us that an object formed by identifying edges of a polygon is a surface, as defined in Section 2.5. Furthermore, we know that such a
surface is homeomorphic to the identification space of the polygon with edge identifications. | {"url":"https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/surfaces/content-section-5.3.3","timestamp":"2024-11-05T08:43:46Z","content_type":"text/html","content_length":"107962","record_id":"<urn:uuid:497e2875-0eba-4b50-b36b-a9a6b411fa67>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00258.warc.gz"} |
Everyday maths 2
1.3 Circumference of a circle
You may have noticed that a new term has slipped in to the title of this section. The term circumference refers to the distance around the outside of a circle – its perimeter. The perimeter and the
circumference of a circle mean exactly the same, it’s just that when referring to circles you would normally use the term circumference.
Download this video clip.Video player: bltl_4_1_3_circumference.mp4
The circumference of a circle is the distance around its edge. That is, its perimeter. Before learning how to work out the circumference of a circle, look at these two key terms: diameter and radius.
Radius, or r, is the distance from the centre of the circle to the edge. Diameter, or d, is the distance from one edge of the circle to the other, passing through the centre of the circle.
You can see the diameter is always double the radius. If you know either the radius or the diameter, you can always work out the other. And you can also work out the circumference. Here is the basic
formula you need to work out the circumference of a circle: Circumference = pi x diameter. This can also be written as C = pi x d, or pi d.
Pi, or this symbol (which is the Greek letter used to represent it), is a constant number that is around the value of 3.142. It's a number that goes on for ever, so you tend to shorten it to the more
manageable number of 3.14 or 3.142. In technical terms, pi is the number you get if you divide a circle's circumference by its diameter, and it's the same for every circle.
Let's look at an example of calculating circumference. This circle has a diameter of 5 centimetres. You can put this into the formula, so you get C = pi x 5. Look for the pi key on your calculator,
or you can use the shortened version, 3.142. So the circumference, C = 3.142 x 5, which equals 15.71 centimetres.
Here's another example, but this time the radius is labelled. How would you calculate the circumference? Remember that the formula for circumference uses diameter, so you'll need to work this out
first. Since the radius is 12 centimetres, the diameter will be double this. 12 x 2 = 24 centimetres, so d equals 24 centimetres. Now using the formula, circumference = 3.142 x 24, which equals
Now, try the examples in the next activity.
End transcript
Interactive feature not available in single page view (
see it in standard view
Activity 3: Finding the circumference
1. You have made a cake and want to decorate it with a ribbon.
The diameter of the cake is 15 cm. You have a length of ribbon that is 0.5 m long. Will you have enough ribbon to go around the outside of the cake?
2. You have recently put a pond in your garden and are thinking about putting a fence around it for safety. The radius of the pond is 7.4 m.
What length of fencing would you require to fit around the full length of the pond? Round your answer up to the next full metre.
1. d = 15 cm
Using the formula C = πd
□ C = 3.142 × 15
□ C = 47.13 cm
Since you need 47.13 cm and have ribbon that is 0.5 m (50 cm) long, yes, you have enough ribbon to go around the cake.
2. C = πd
C = 3.142 × 14.8
C = 46.5016 m which is 47 m to the next full metre.
You should now be feeling confident with finding the perimeter of all types of shapes, including circles. By completing Activity 3, you have also re-capped on using formulas and rounding.
The next part of this section looks at finding the area (space inside) a shape or space. As mentioned previously, this is incredibly useful in everyday situations such as working out how much carpet
or turf to buy, how many rolls of wallpaper you need or how many tins of paint you need to give the wall two coats.
In this section you have learned:
• that perimeter is the distance around the outside of a space or shape
• how to find the perimeter of simple and more complex shapes
• how to use the formula for finding the circumference of a circle. | {"url":"https://www.open.edu/openlearn/mod/oucontent/view.php?id=83664§ion=1.3","timestamp":"2024-11-04T03:05:45Z","content_type":"text/html","content_length":"130763","record_id":"<urn:uuid:1d394658-f55f-40bd-b642-be2cf0a2fff3>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00535.warc.gz"} |
Free Cash Flow Approach
10.4 Free Cash Flow Approach
PLEASE NOTE: This book is currently in draft form; material is not final.
Learning Objectives
1. Calculate the valuation of a company based on FCFs and WACC.
2. Calculate the value of a share of stock using the FCF method.
The free cash flow (FCF) approach for valuing a company is very much related to the dividend discount model explained in section 2. The key difference is that we look at all of the cash flows
available for distribution to the investors and use them to arrive at a value for the entire company. Since we are using the cash flows for all investors, we need to discount them not using just our
expected return on equity, but on the weighted average cost of capital (WACC)An average of the returns required by equity holders and debt holders weighted by the company’s relative usage of each..
As the name implies, this is an average of the returns required by equity holders and debt holders weighted by the company’s relative usage of each. Arriving at the WACC will be the topic of a later
Equation 10.7 Value of Company Using Discounted FCF
$VC= PV of Future FCFs+PV of Terminal Value of CompanyVC=FCF1(1+WACC)1+FCF2(1+WACC)2+…+FCFn(1+WACC)n+VTerminal(1+WACC)n$
Finding the terminal value for a company has some of the same headaches as finding the future expected stock price. A common method is to assume a long-term growth rate for FCF, and use a variation
of the perpetuity with growth formula:
Equation 10.8 Terminal Value of Company Using Discounted FCF
$VTerminal= FCFn+1WACC−g= FCFn(1+g)WACC−g {for g<WACC}$
This method can be extremely sensitive to the assumption used for the long-term growth rate. Once the value of the entire company is determined, we need to subtract the market values of our debt and
preferred stock to arrive at the value of the residual due to common shareholders:
Equation 10.9 Value of Stock
Value of Company = Value of Debt + Value of Equity
= Value of Debt + (Value of Pref. Stock + Value of Common Stock) V[C] = V[D] + V[E] = V[D] + (V[ps] + V[s]) therefore V[C] − V[D] − V[ps] = V[s]
Once the value of the common stock is obtained, dividing by the number of shares outstanding should lead to an appropriate price per share.
Of course, a company might have a negative FCF currently but still be a good investment, if FCF is expected to turn positive in the future. This can happen particularly with corporations that are
experiencing rapid growth, necessitating a large investment in capital to support future revenues. Since FCF for such companies tends to turn positive well before dividends are paid, this approach
typically provides a superior estimate for stock value over the DDM.
Key Takeaways
• Calculating the value of a company using the FCF method tends to be more accurate, so it is used in practice much more than the DDM.
• The FCF method can be very sensitive to assumed long-term growth rates.
1. Our company projects the following FCFs for the next 3 years: $5 million, $5.5 million, $6 million. Future growth is expected to slow to 5% beyond year 3. What is the terminal value of the
company in year 3 if the WACC is 8%? What is the value of the company today? What is the company worth if the projected growth rate is only 3% beyond year 3?
2. If a company’s value is $250 million, and the company has $100 million market value of debt outstanding and no preferred stock, what is the value of its common stock? If there are ten million
shares of stock outstanding, what is should be the price of one share of stock? | {"url":"https://2012books.lardbucket.org/books/finance-for-managers/s10-04-free-cash-flow-approach.html","timestamp":"2024-11-01T20:47:50Z","content_type":"text/html","content_length":"15380","record_id":"<urn:uuid:e0bb245d-7371-4aae-bd74-3ffeae909be2>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00665.warc.gz"} |
5 Best Ways to Find Python Indices of Numbers Greater Than k
π ‘ Problem Formulation: You have a list of integers and a number k. Your task is to find the indices of all numbers in the list that are greater than k. For instance, given the list [1, 5, 7, 3, 8]
and k as 4, the desired output would be the indices [1, 2, 4].
Method 1: Using a for-loop and list append
This approach involves iterating over the list with a for-loop and appending the index of each element that is greater than k to a new list. It’s straightforward and easy to understand for those new
to Python.
Here’s an example:
def find_indices(lst, k):
indices = []
for i, num in enumerate(lst):
if num > k:
return indices
# Example usage
indices = find_indices([9, 2, 6, 3, 5], 3)
[0, 2, 4]
This code snippet defines a function find_indices(lst, k) that returns a new list of indices. It uses enumerate() to get both the index and the value for elements in the original list, making it easy
to tell if a number is greater than k and then capture its index.
Method 2: Using list comprehensions
List comprehensions provide a concise way to create lists. They are typically more compact and faster than normal for-loops because they are optimized for Python’s interpreter.
Here’s an example:
lst = [10, 12, 3, 5, 17]
k = 8
indices = [i for i, n in enumerate(lst) if n > k]
[1, 4]
The list comprehension iterates over lst using enumerate() to retrieve both the index and number. The if condition filters out those numbers not greater than k, resulting in a list of appropriate
Method 3: Using the filter and lambda functions
This method combines filter() with a lambda function to isolate the indices of elements greater than k. While not as readable as list comprehensions, it can be useful for larger datasets.
Here’s an example:
lst = [4, 18, 10, 5, 6]
k = 7
indices = list(filter(lambda i: lst[i] > k, range(len(lst))))
[1, 2]
The lambda function takes an index and returns true if the corresponding list element is greater than k. filter() applies this lambda to every index and filters in those that meet the condition.
Method 4: Using NumPy arrays
For numerically intensive computing, NumPy provides efficient storage and manipulation of arrays. Using NumPy, you can filter indices quickly, especially when working with large datasets.
Here’s an example:
import numpy as np
arr = np.array([2, 3, 15, 7, 9])
k = 6
indices = np.where(arr > k)[0]
[2, 4]
The code example uses NumPy’s np.where() function to find indices where the condition is true. This function is highly optimized and can be significantly faster than Python loops for large arrays.
Bonus One-Liner Method 5: Using itertools.compress
The itertools.compress function offers another way to filter list indices based on a condition. It can lead to very compact code but sacrificing some readability for those unfamiliar with itertools.
Here’s an example:
from itertools import compress
lst = [6, 7, 10, 2, 5]
k = 5
indices = list(compress(range(len(lst)), (n > k for n in lst)))
[1, 2]
itertools.compress accepts two iterators: the first one is the list of indices, and the second is a generator expression that evaluates the condition for each element. It returns only those indices
for which the condition is True.
• Method 1: For-loop and list append. Easy for beginners to understand. Can be slow for large lists.
• Method 2: List comprehensions. Concise and Pythonic. Faster than for-loops but may consume more memory.
• Method 3: Filter with lambda. Good for large lists. Less readable than list comprehensions.
• Method 4: Using NumPy arrays. Best for numerical computations and large datasets. Requires an external library.
• Method 5: Using itertools.compress. Compact code for one-liners. Not as intuitive for those new to Python’s itertools. | {"url":"https://blog.finxter.com/5-best-ways-to-find-python-indices-of-numbers-greater-than-k/","timestamp":"2024-11-03T06:54:02Z","content_type":"text/html","content_length":"71462","record_id":"<urn:uuid:72aec0c0-417c-4100-89ac-5cfd251b0c9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00257.warc.gz"} |
How to Define Input And Output Tensors In Pytorch?
In PyTorch, input and output tensors are defined by specifying the shape and datatype of the tensors. The shape of a tensor refers to the dimensions of the tensor, while the datatype refers to the
type of data stored in the tensor (e.g. float, integer, etc.).
To define input tensors in PyTorch, you need to create a tensor using the torch.Tensor() function and specify the shape and datatype of the tensor. For example, to create a 2D tensor with dimensions
3x3 and datatype float, you can use the following code:
1 import torch
3 input_tensor = torch.Tensor(3, 3).float()
Similarly, to define output tensors in PyTorch, you can create a tensor with the desired shape and datatype using the torch.Tensor() function. For example, to create a 1D tensor with dimensions 5 and
datatype integer, you can use the following code:
1 output_tensor = torch.Tensor(5).int()
Once you have defined the input and output tensors, you can use them to perform operations in your PyTorch model, such as feeding the input tensor into the model and using the output tensor to store
the results of the model's predictions. Defining input and output tensors in PyTorch is an important step in building and training neural networks using the PyTorch framework.
How to check the dimensions of input and output tensors in PyTorch?
In PyTorch, you can check the dimensions of input and output tensors using the size() method. Here is an example of how to do it:
1 import torch
3 # Create an input tensor with size (batch size, channels, height, width)
4 input_tensor = torch.randn(1, 3, 224, 224)
6 # Define a neural network model
7 model = torch.nn.Conv2d(3, 10, kernel_size=3, padding=1)
9 # Pass the input tensor through the model
10 output_tensor = model(input_tensor)
12 # Check the dimensions of the input and output tensors
13 print("Input tensor size:", input_tensor.size())
14 print("Output tensor size:", output_tensor.size())
This code snippet shows how to create an input tensor with a specific size and pass it through a convolutional neural network model. You can then use the size() method to check the dimensions of both
the input and output tensors.
How to manipulate input and output tensors in PyTorch for data preprocessing?
In PyTorch, you can manipulate input and output tensors for data preprocessing using various built-in functions and operations. Here are some common techniques for data preprocessing in PyTorch:
1. Normalizing data: You can normalize input tensors to have zero mean and unit variance by using the torch.nn.functional.normalize function or by directly subtracting the mean and dividing by the
standard deviation.
2. Reshaping tensors: You can reshape input tensors using the torch.Tensor.view or torch.Tensor.reshape functions to match the required input shape for your model.
3. Padding tensors: If you need to pad input tensors to match a certain size or shape, you can use the torch.nn.functional.pad function to add zeros or other padding values.
4. Augmenting data: You can augment input tensors by applying transformations such as rotation, flipping, or cropping using the torchvision.transforms module or custom functions.
5. Converting data types: You can convert input and output tensors to different data types such as float, integer, or double using the torch.Tensor.to function.
6. Handling missing values: If your input data contains missing values, you can mask those values and replace them with a certain value using the torch.Tensor.masked_fill function.
By using these techniques, you can effectively preprocess input and output tensors in PyTorch to improve the performance of your neural network model.
What is the relationship between data types and input/output tensors in PyTorch?
In PyTorch, data types, also known as data types or dtype, define the type of data stored in a tensor. This can be integer, float, complex, boolean, etc. Data types in PyTorch are important as they
determine the precision of the computations performed on the tensors, as well as the amount of memory consumed by the tensors.
The input and output tensors in PyTorch also have data types associated with them. When performing operations or calculations on tensors in PyTorch, it is important that the input and output tensors
have compatible data types. If the data types of the input tensors are not compatible with the desired data type of the output tensor, PyTorch will throw an error.
Therefore, when working with PyTorch, it is important to pay attention to the data types of input and output tensors to ensure that operations can be performed smoothly and efficiently.
How to handle imbalanced data in input and output tensors in PyTorch?
There are several techniques that can be used to handle imbalanced data in input and output tensors in PyTorch:
1. Weighted Loss Function: One common approach is to use a weighted loss function, where the loss for each class is weighted based on its frequency in the dataset. This way, the model pays more
attention to the minority classes and helps prevent bias towards the majority classes.
2. Over/Under Sampling: Another approach is to oversample the minority class or undersample the majority class to balance the dataset. This can be done using techniques such as SMOTE (Synthetic
Minority Over-sampling Technique) for oversampling or Random Under Sampling for undersampling.
3. Class Weights: PyTorch provides a "weight" parameter in the loss function that allows you to assign different weights to each class. These weights can be set inversely proportional to the class
frequencies to account for the class imbalance.
4. Focal Loss: Focal loss is a modification of the cross-entropy loss function that focuses on hard-to-classify examples and downweights easy-to-classify examples. This can be particularly useful
for imbalanced datasets.
5. Data Augmentation: Data augmentation techniques such as flipping, rotating, or scaling can help increase the diversity of the dataset and improve the model's ability to learn from the minority
class samples.
By using one or a combination of these techniques, you can help address the issues caused by imbalanced data in your PyTorch models. | {"url":"https://freelanceshack.com/blog/how-to-define-input-and-output-tensors-in-pytorch","timestamp":"2024-11-14T00:12:10Z","content_type":"text/html","content_length":"396589","record_id":"<urn:uuid:d35715cd-49d2-4660-9183-b6b5724e1c2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00002.warc.gz"} |
What is the formula of principal or present value?
What is the formula of principal or present value?
NPV Formula. It’s important to understand exactly how the NPV formula works in Excel and the math behind it. NPV = F / [ (1 + r)^n ] where, PV = Present Value, F = Future payment (cash flow), r =
Discount rate, n = the number of periods in the future.
What is the present value of the principal?
Present Value of a Bond’s Maturity Amount. The second component of a bond’s present value is the present value of the principal payment occurring on the bond’s maturity date. The principal payment is
also referred to as the bond’s maturity value or face value.
Is the principle the present value?
A comparison of present value with future value (FV) best illustrates the principle of the time value of money and the need for charging or paying additional risk-based interest rates. Simply put,
the money today is worth more than the same money tomorrow because of the passage of time.
Is principal PV or FV?
Pv (required argument) – The present value or total amount that a series of future payments is worth now. It is also termed as the principal of a loan. Fv (optional argument) – This is the future
value or a cash balance we want to attain after the last payment is made.
What is the formula of present value in simple interest?
For both simple and compound interest, the PV is FV divided by 1+i.
Is present value the same as principal amount?
Compound Interest Formula Compound Interest = total amount of principal and interest in future (or future value) less the principal amount at present, called present value (PV). PV is the current
worth of a future sum of money or stream of cash flows given a specified rate of return.
What is a principal loan?
Principal is the money that you originally agreed to pay back. Interest is the cost of borrowing the principal. If you plan to pay more than your monthly payment amount, you can request that the
lender or servicer apply the additional amount immediately to the loan principal.
Which is the correct formula for present value?
Present value refers to today’s value of a future amount. Present Value Formula: S. P = ————. (1+rt) Instead of beginning with the principal which is invested, you could start from what you want to
accumulate in the future, and then work backward to see the amount that you must invest to reach the required amount.
What do you mean by present value of money?
Present value refers to today’s value of a future amount. Instead of beginning with the principal which is invested, you could start from what you want to accumulate in the future, and then work
backward to see the amount that you must invest to reach the required amount.
What is the present value of a sum?
Present Value (PV) is the current value given a specified rate of return of a future sum of money or cash flow. The Present Value takes the Future value and applies a rate of discount or interest
that could be earned if it is invested.
How is present value different from future value?
The Present Value takes the Future value and applies a rate of discount or interest that could be earned if it is invested. Future Value tells you what an investment will be worth in the future,
while Present Value tells you how much you would need to earn a specific amount in the future in today’s dollars. | {"url":"https://eyebulb.com/what-is-the-formula-of-principal-or-present-value/","timestamp":"2024-11-08T17:26:40Z","content_type":"text/html","content_length":"111187","record_id":"<urn:uuid:a642d04a-a610-4d55-9fc2-3cfc5add7569>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00336.warc.gz"} |
In this chapter, we will learn how to compute the Union of overlapping rectangles in 2-D plane with all their base (lower horizontal boundary line) at the same y-coordinate first.
We will discuss the concept here by solving Skyline problem.
We will see how we can effectively leverage Sweep Line technique to solve this kind of problems.
A city's skyline is the outer contour of the silhouette formed by all the buildings in that city when viewed from a distance
Now suppose you are given the x-coordinates and heights of all the buildings as shown on the left side of the image below. Our objective is to think through and come up with an algorithm to output
the skyline formed by these buildings collectively, as shown on the right side of the image below.
The geometric information of each building is represented by a triplet of integers [Li, Ri, Hi], where Li and Ri are the x coordinates of the left and right edge of the ith building, respectively,
and Hi is its height. It is guaranteed that 0 ≤ Li, Ri ≤ INT_MAX, 0 < Hi ≤ INT_MAX, and Ri - Li > 0. You may assume all buildings are perfect rectangles
grounded on an absolutely flat surface at height 0
For instance, the dimensions of all buildings in the image below could be represented as: [[2,9,10], [3,6,15], [5,12,12], [13,16,10], [13,16,10], [15,17,5]] .
The output is a list of "key points" (red dots in Figure B) in the format of [ [x1,y1], [x2, y2], [x3, y3], ... ] that uniquely defines a skyline. A key point is the left endpoint of a horizontal
line segment. Note that the last key point, where the rightmost building ends, is merely used to mark the termination of the skyline, and always has zero height. Also, the ground in between any two
adjacent buildings should be considered part of the skyline contour. For instance, the output skyline for the image below should be represented as: [[2,10], [3,15], [6,12], [12,0], [13,10], [16,5],
In-depth Algorithm Discussion and Code Implementation:
Login to Access Content
Time Complexity:
If there are n number of buildings, then there are 2n vertical lines (start line and end line) we need to process.
Sorting is O(2nlog2n) = O(nlogn).
Insertion and deletion from a heap are O(logn) operations. For 2n elements we have O(nlogn).
Overall time complexity = O(nlogn) + O(nlogn) = O(nlogn).
Related Must-Read Topics: | {"url":"https://thealgorists.com/Algo/TwoDIntervalsMerging","timestamp":"2024-11-10T07:21:48Z","content_type":"text/html","content_length":"28921","record_id":"<urn:uuid:4a457113-7256-40ef-a65b-5bb8604e4c75>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00622.warc.gz"} |
is a Channel4 game show with standard number and word puzzles. In this post we'll look at the
Numbers game
. The rules are simple, given 6 numbers (between 1 and 999 inclusive), calculate the target number (between 100 and 999 inclusive). You can use +, -, / and * to get the numbers and you have a 30
second time limit to do so.
To solve this in
we'll start with a
brute force search
of all the possibilities and see if that's good enough to solve it.
One approach I tried initially was just to build up the Clojure expression tree for all possible examples, using code like this:
(def *operators* ['+ '- '/ '*])
(defn expr
"A list of expressions for a and b"
[a b]
(map (fn [x] (list x a b)) *operators*))
The idea would be that I could then just
(map eval (expr 4 5))
and get all the possible results. This turned out to be
slow. Generally you want to avoid calling
at run-time if you can help it.
To solve this I defined a simple structure to keep track of running expressions and their value. As the expressions are built up, the values are calculated in sync.
(def *operators* {'+ + '- - '/ / '* *})
(defn is-valid [op a b]
(= + op) true
(= - op) (> a b)
(= * op) true
(= / op) (= (mod a b) 0)))
(defstruct node :expression :value)
(defn value
(if (map? x)
(x :value)
(defn expression
(if (map? x)
(x :expression)
(defn expr
"A list of expressions for a and b"
[a b]
(let [nodea (map? a) nodeb (map? b)]
(filter (fn [x] (not (nil? x)))
(map (fn [x] (when (is-valid (second x) (value a) (value b))
(struct node
(list (first x) (expression a) (expression b))
((second x) (value a) (value b)))))
Why is
a map? That's simply because "+" doesn't print very nicely e.g.
countdown> +
#<core$_PLUS___3180 clojure.core$_PLUS___3180@61dd1c39>
I also added a check to prune entries out that results in floating point or negative numbers, that just helps keep the number of combinations down a little.
Armed with a function that calculates all the possible expressions for a pair of expressions, how do we now use that to generate all the possible expressions?
(defn make-expressions-helper
"Given a lst, build up all valid Countdown expressions"
(< (count x) 2) (list (struct node (first x) (first x)))
(= 2 (count x)) (apply expr x)
(let [exps (apply expr (take 2 x))
remd (drop 2 x)]
(mapcat make-expressions-helper (map (fn [x] (cons x remd)) exps)))))
This is a recursive definition with the following logic:
• A singleton list (1) can only evaluate to itself so the only possibility is [expr=1 value=1]
• A list of size two just uses the expr function to generate all the expressions
• Any other list builds all the possible expressions for the first two elements, and then for all of these (mapcat) calls make-expressions-helper on the rest.
Note that this just builds up the possible expressions with the numbers in this particular order. For example.
countdown>(make-expressions-helper '(1 2 3))
({:expression (+ (+ 1 2) 3), :value 6} {:expression (/ (+ 1 2) 3), :value 1}
{:expression (* (+ 1 2) 3), :value 9} {:expression (+ (* 1 2) 3), :value 5}
{:expression (* (* 1 2) 3), :value 6})
countdown> (count (make-expressions-helper '(1 2 3 4 5 6)))
So now we need to apply the helper function to all possible combinations. Thankfully,
Clojure Contrib
already has a few
returns a lazy list of all possible permutations of the supplied list.
(defn make-expressions [lst]
(if (nil? lst)
(mapcat make-expressions-helper (permutations lst))
(mapcat make-expressions (drop-one lst)))))
So this algorithm applies the helper function to all permutations of the input, and then applies itself to all combinations of the remainder of the list.
is a helper function which gives a list of all combinations of a list without one element.
So how many valid Countdown expressions are there?
countdown> (count (make-expressions '(1 2 3 4 5 6)))
countdown> (time (count (make-expressions '(1 7 8 25 50 75))))
"Elapsed time: 2653.618442 msecs"
Note that the number is different because we rule out cases which result in floating point or negative numbers. The elapsed time is just under three seconds which is pretty fast! Remember that this
time includes all the calculation of the results too, not just generating the expressions. So finally, all we need is a solver function.
(defn solve
"Solve the countdown problem"
[numbers target]
(filter (fn [x] (= (x :value) target)) (make-expressions numbers)))
This will return all the combinations that lead to the right results. Let's try it out with a toy examples:
countdown> (time (solve '(4 5 6) 15))
"Elapsed time: 0.281907 msecs"
({:expression (+ (+ 4 5) 6), :value 15} {:expression (+ (+ 4 6) 5), :value 15}
{:expression (+ (+ 5 4) 6), :value 15} {:expression (+ (+ 5 6) 4), :value 15}
{:expression (+ (+ 6 4) 5), :value 15} {:expression (+ (+ 6 5) 4), :value 15})
Notice that we've returned all possible + expressions that make 15. We've not taken any notice of the
properties of addition. Taking advantage of these properties is explored in
"The Countdown Problem"
[PDF] by Graham Hutton.
How does it fair on bigger solutions?
countdown> (time (solve '(7 5 9 25 40 10) 753))
"Elapsed time: 222.632493 msecs"
{:expression (- (* (- (- (* 5 25) 9) 40) 10) 7), :value 753}
With the code as it stands we could add additional operators (exponent for example) without any code changes, but more operators would probably require something more sophisticated than brute force.
As usual, any suggestions for making the code clearer (or finding any bugs!) are greatly appreciated. Full code is on
my Git repository | {"url":"http://www.fatvat.co.uk/2009/02/countdown.html","timestamp":"2024-11-07T03:11:29Z","content_type":"text/html","content_length":"73458","record_id":"<urn:uuid:d1e6d7f7-7662-496f-a3ec-160337d83405>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00442.warc.gz"} |
Cal Teach Math - Key Issues in K-12 Math
Math 73XP - Key Issues in K-12 Mathematics
Math 73XP (formerly Math 73SL) aims to introduce students to K-12 mathematics activity in the United States through exploration of the sequences of mathematical content and habits of mind taught in
these grades as well as the cognitive aspects of learning this mathematics.
Coursework will include:
• analysis of the sequences of topics in the current California State Standards in Mathematics (CCSS-M) and the mathematical structures that underlie these sequences
• experience with the habits of mind of a professional mathematician outlined in the California Standards for Mathematical Practice including proof and mathematical modeling
• experience with effective strategies for teaching mathematics to diverse student groups with fieldwork in local mathematics classrooms
Class meetings will include presentations by guest speakers who, as a whole, represent the diversity of career paths available to students in the K-12 mathematics education profession. One class
meeting will consist of students engaging with current teachers as they participate in a professional development activity.
Fieldwork in either a local elementary, middle or high school school classroom will include observation as well as presenting an original mathematical task written by the student.
Course Meetings: 1 hour 50 minutes per week, with 20 hours of observation in local school.
3 units - P/NP
Students should be making regular progress toward a STEM major (Science, Technology, Engineering and Math). Other majors are welcome to apply and your eligibility is based on the amount of math
classes you have completed.
How to Apply?
You need to fill out a short online application in order to receive permission to enroll in the seminar though MyUCLA. Once accepted into the seminar, we will send you a PTE number that will enable
you to enroll in the course.
In our main menu, go to "Apply" and choose "Online Applications" from the drop down menu. In "Course Applications" page, select link for the quarter and course of interest.
Note that you may need to provide proof of negative TB status, proof of covid vaccination, and weekly negative covid tests prior to visiting a school (read more about TB requirements).
You must be in good academic standing to enroll in a course as pass/ no pass - see UCLA Academic Regulations.
Please contact the Cal Teach staff at cateach@chem.ucla.edu with any questions. | {"url":"http://cateach.ucla.edu/?q=content/cal-teach-math-key-issues-k-12-math","timestamp":"2024-11-11T22:35:25Z","content_type":"text/html","content_length":"28654","record_id":"<urn:uuid:388d5cca-89e0-43ca-ab44-7ce1a37b7061>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00510.warc.gz"} |
Five Cards
How To Play
□ Choose the appropriate card pack.
○ The first example uses a 1-10 pack and the second a 1-6 pack.
□ Shuffle the cards.
○ Younger children can spread the cards out on the table face down, swirl them around and push them back into a pile.
□ Place the shuffled pack between the two players.
□ Players take turns to select cards and place them face up in front of themselves until they have five cards.
□ Player A tries to use any combination of +, -, x or ÷ to make their cards into a pile worth five. A calculator may be used to help.
○ A pile, called a 'trick', can be any number of cards, including just one card.
○ If a player has cards left after making a trick they try to make more tricks.
□ If a player can't make any tricks they can ask the pack for help and pick up the top card.
○ Asking for help can only be done once each turn.
○ A 'help' card can be used straight away or the player can choose to stop and wait for their next turn.
○ If a player decides not to use the 'help' card and wait for their next turn, their turn is over.
□ If Player A uses all their cards to make tricks, they go 'out' and don't play again until Player B has also gone 'out'.
□ If Player A doesn't use all their cards to make tricks they take the top card from the deck and wait for their next turn.
□ Now it's Player B's turn.
□ Play keeps going until both Players are out.
□ The game is over when both players are 'out'. When it's over:
○ Count the number of tricks that were made.
○ Record the number of 'tricks' in each game.
○ You can record the total number of 'tricks' between you or just your own number.
○ Choose one of your tricks and quickly sketch its cards.
○ Write the equation you created from these cards.
Shuffle the cards and begin another game.
Co-operation or Competition?
□ Co-operation
Work together so that both players go out in the smallest number of tricks. Play three games and record the total number of tricks. Do this at least three times in a week and do it for at least
three weeks. In your journal keep a table of results and draw a graph from the table.
□ Competition
Here are two ways to make the game competitive.
1. First person out wins the game. Agree before starting that the winner will be the person who wins most out either three or five games.
2. The player who goes out first scores one point for each trick and five (5) extra points for finishing first. The player who doesn't go out first is allowed two more turns before the game is
over. Play three games and the player with the higher total score wins.
Care With The Calculator
This activity offers opportunity to use calculators thoughtfully - in fact, to teach how to use the calculator thoughtfully.
Suppose a player has the cards 2, 6, 4 & 1. They might see that 2 & 6 could be an addition to make 8 and 4 & 1 could be a subtraction to make 3. Then 8 - 3 gives 5. The player would probably put
their cards down and say: 2 plus 6 equals 8 take away 4 minus 1 equals 5.
However, checking this on any calculator by typing in the order said, ie: 2 + 6 - 4 - 1, gives the answer 3. What?
The player is likely to be quite sure the answer is 5 and should be encouraged to ask why their calculator doesn't get the 'right' answer. This is the opportunity to discuss the use of brackets,
which on most simple calculators is handled by using the memory buttons. The 'brackets' are likely to be indicated by the way the player has actually laid the cards down, placing the 2 and 6
together and the 4 and 1 together but separated from the others.
What Happens If...?
□ What happens if you play the same rules but Jack, Queen and King are included in the pack with values of 11, 12 and 13?
□ What happens if the rules are the same but the tricks have to equal seven (7) ... or 8 or 11 or any number up to 20 that you choose?
□ What happens if you are allowed to take one rule out of the game (or add one new rule to the game)?
Just Before You Finish
□ Draw an oval in your journal.
Change it into a face that shows how you feel about Five Cards.
Add a speech bubble if you wish.
□ What do you know now that you didn't know when you started Five Cards?
1 - 10 Pack
Ready to start. Left player will play first.
Left makes a one card pile with 5.
Unable to make five with the remaining 9, 9, 8 & 3,
perhaps because they didn't see that 8 - 3 = 5,
Left draws the top card from the deck (it was another 5)
and decides to wait for their next turn.
NB: If playing a co-operative game the other
player would tell Left player about the 8 - 3 before
the deck card was drawn. Why? Because they are working
together to end the game in the lowest total number of tricks.
Right's turn, but unable to make five from 2, 2, 6, 8, 10,
although some players might see (10 + 8 + 2) ÷ (6 - 2),
Right asks for help from the deck and receives 1 (Ace).
Right has an 'aha' moment and creates five with 10 - 2 - 2 - 1.
(Left realises their first trick should have been turned over by now.)
Right has finished their turn without using all cards,
so takes the top card from the deck (6 H) and waits for their next turn.
It's Left's turn.
Left might play the obvious single pile 5 card,
and draw the top card because they can't think of another equation;
or might play the 5 and also create 9 x 8 ÷ 9 - 3,
using all their cards by making two more tricks and going out. | {"url":"https://mathematicscentre.com/mathsathome/challenges/5cards.htm","timestamp":"2024-11-08T09:03:10Z","content_type":"text/html","content_length":"12028","record_id":"<urn:uuid:f36b24a3-f142-48f1-ab99-42e8024d7f36>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00195.warc.gz"} |