text:
# Building graphs that resemble geometry objects using the lattice function and Gel commands

## The lattice function introduction

The new lattice() function takes a seed graph and tries to replicate the seed graph repetitively to obtain a lattice graph. The lattice graph size is controlled by both the maximum number of nodes that can be added and a maximum bounding box size; both are specified by the users:

help lattice;
graph = lattice(graph seed, int max_nodes, float w, float h, float d)
seed: the seed graph that will be tiled to form the lattice graph;
_x, _y, _z values in seed graph nodes must be set correctly.
one node must be named 'anchor' to anchor to tiling nodes.
one or more nodes must be named with the prefix 'tile...'
to replicate the seed graph during the tiling process.
max_nodes: maximum nodes that can be added to the lattice graph.
w, h, d: width, height & depth of a bounding box that also limit
the lattice graph; no nodes will be created outside the box.
If any of the w, h or d values is negative, it defines a circular
warping around the next axis, e.g, if w is -40, x-axis values
will fall within a circle around the y-axis with circumference
40; y-axis warps around z-axis and z-axis warps around x-axis.

One of the bounding box size parameter can be negative, which defines a circular warping along that dimension. It’s best to illustrate the warping effect with the following examples, so stay tuned.

The seed graph is the most important parameter to the lattice function. Since nodes in the seed graph will be replicated repetitively according to their relative positions to each other, it is important to set seed graph node positions (i.e., their _x, _y and _z values) correctly before given the seed graph to the lattice function, otherwise the lattice graph constructed by the lattice function may not be what you wanted. In most of the following examples, graph g is a seed graph given as argument and graph r is the returned lattice graph, so pay attention to how the g graph is defined and how the r graph is constructed from it; this will help you understand the usage of the lattice function better.

In all the following examples we make use of the same basic node and link type defined as follows:

// delcare basic node and link types; nothing fancy
node (string id) nt;

Obviously, additional node and link attributes can be defined for the seed graph, and the resulted lattice graph will carry the same attributes. However, for efficiency, we use minimal attributes on the seed graph. Any additional graph attributes can be easily added to the lattice graph using Gel attribute promotion methods.

## Simple lattice replications

1-, 2- and 3-dimensional seed graphs will produce 1-, 2- and 3-dimensional lattice graphs, respectively:

g graph r graph
// linear tiling along y-axis
graph(nt, lt) g = { ("anchor", 0, 0)[]("tile1", 0, 1) };
graph(nt, lt) r = lattice(g, 200, 10, 10, 10);

g graph r graph
// 2-D tiling along x-y plane
graph(nt, lt) g = { ("anchor", 0, 0)[]("tile1", 0, 1), "anchor"[]("tile2", 1, 0) };
graph(nt, lt) r = lattice(g, 200, 10, 10, 10);

g graph r graph
// 3-D tiling along all three axes
graph(nt, lt) g = { ("anchor", 0, 0)[]("tile1", 0, 1), "anchor"[]("tile2", 1, 0), "anchor"[]("tile3", 0, 0, 1)};
graph(nt, lt) r = lattice(g, 2000, 9.5, 9.5, 9.5);

We can get a box graph out of the cube graph above by using the Gel select command based on node degrees, because interior nodes of the cube will have higher link degrees than outside nodes.

g graph r graph
// to carve out the interior of the cube do a select based on node degrees
auto s = select node from r where in+out<6;

## Dimension warping

As mentioned earlier, one of the bounding box size limit given to the lattice function can be negative. In that case, it specifies a circular warping along that dimension with the circumference of the circle being the absolute value of the negative size. Lattice growth can simultaneously go on the other two non-warping dimensions, creating cylinders or ripples depending on how they were combined with the warping dimension:

g graph r graph
// linear warping along z-axix (around x-axis)
graph(nt, lt) g = { ("anchor", 0, 0, 0)[]("tile1", 0, 0, 1) };
graph(nt, lt) r = lattice(g, 200, 10, 10, -10);

g graph r graph
// 2-D cylinder centered on x-axis (warp on z-axis and tile along x-axis)
graph(nt, lt) g = { ("anchor", 0, 0, 0)[]("tile1", 0, 0, 1), "anchor"[]("tile2", 1, 0, 0) };
graph(nt, lt) r = lattice(g, 200, 10, 10, -10);

g graph r graph
// ripples of circles (warp on x-axis while tiling on z-axis)
graph(nt, lt) g = { ("anchor", 0, 0, 0)[]("tile1", 0, 0, 1), "anchor"[]("tile2", 1, 0, 0) };
graph(nt, lt) r = lattice(g, 2000, -20, 10, 20);

## Build a pyramid

The pyramid slope degree along each of its four sides is 51.8539761° from the base according to the following website information, thus we first build a seed graph with one anchor node and 4 tile nodes one level down from the anchor at the right angle. After the seed graph is correctly defined, the lattice function easily creates a pyramid:

g graph r graph
float offset = cos(3.1415926535897932384626433*2/360*51.8539761);
graph(nt, lt) g = { ("anchor", 0, 10, 0)[]("tile1", offset, 9, -offset), "anchor"[]("tile2", offset, 9, offset), "anchor"[]("tile3", -offset, 9, offset), "anchor"[]("tile4", -offset, 9, -offset), "tile1"[]"tile2"[]"tile3"[]"tile4"[]"tile1" };
graph(nt, lt) r = lattice(g, 1000, 100, 10, 100);

Building an inverted pyramid is equally easy, with the seed graph now upside down.

g graph r graph
float offset = cos(3.1415926535897932384626433*2/360*51.8539761);
graph(nt, lt) g = { ("anchor", 0, -10, 0)[]("tile1", offset, -9, -offset), "anchor"[]("tile2", offset, -9, offset), "anchor"[]("tile3", -offset, -9, offset), "anchor"[]("tile4", -offset, -9, -offset), "tile1"[]"tile2"[]"tile3"[]"tile4"[]"tile1" };
graph(nt, lt) s = lattice(g, 1000, 100, 10, 100);

The two pyramids can now be combined to form an octahedron. To do this, note that the lattice function have named each node in each pyramid in the same location with the same name, albeit the nodes are inverted along the y-axis. Realizing that, we can use simple Gel commands to rename nodes to some other names other than those in the base with a y-axis value of zero, so they become distinct. After that, a simple Gel add graph command easily combines the two pyramids to form an octahedron:

// to merge the two pyramids to form an octahedron
// first rename all nodes other than the connecting nodes
foreach node in r where _y>0.0 set id="upper".id;
foreach node in s where _y<0.0 set id="lower".id;
// then just add the two parts together
graph(nt, lt) c = r.+s;

Finally, we can carve out the interior of the octahedron with a select command based on node degree:

// carve out the interior of the octahedron with a select command
graph(nt, lt) b = select node from c where in+out<10;

## Build a torus graph

To build a torus graph, we must use some Gel calculations in addition to using the lattice function, as currently the lattice function allows warping only along one dimension while a torus graph needs warping along a primary axis and another warping along a secondary axis. To begin with, let’s build a cylinder with the lattice function:

// build a torus graph
graph(nt, lt) g = { ("anchor", 0, 0, 0)[]("tile1", 0, 0, 1), "anchor"[]("tile2", 1, 0, 0), "anchor"[]("tile3", 1, 0, 1), "tile1"[]"tile2" };
// first create a cylinder along the x-axis that will form the torus
graph(nt, lt) r = lattice(g, 2000, 40, 100, -40);

Nodes on this cylinder are already wrapping along the z-axis (i.e., around the x-axis). To form a torus, this cylinder must be wrapped along the x-axis (i.e., around the y-axis). First, we need to merge its left-most nodes to its right-most nodes to form a consecutive circular path as required for a torus. This can be accomplished with the map function that renames nodes — If two nodes are renamed to the same name, they will be merged by the map function. Basically, we want the left-most nodes on the cylinder to be renamed to their corresponding right-most node names. A mapping graph d is constructed as follows:

// the left- and right-most nodes should be merged by using the map function
graph(nt, lt) a = select node from r where _x==0;  // left-most ring
graph(nt, lt) b = select node from r where _x==40; // right-most ring
graph(nt, lt) c = a*b; // link them, and then keep only horizontal links
graph(nt, lt) d = select link from c where in._y==out._y && in._z==out._z;
// now merge the left- and right-most nodes
graph(nt, lt) e = map(r, d);

It’s hard to see the difference between the resulted e graph and the original r graph, but if you care to count the number of rings on the e graph, you can see it has one fewer ring and nodes on its left-most ring now also link to nodes the right-most ring. You can zoom in to see this in Mango.

We can right-click on this e graph in Mango graph canvas to let force-directed layout try to form the torus layout for us, but it will take a very long time. Feel free to try this, but please regenerate the e graph before using the following Gel code to directly map the e graph nodes to a torus.

float radius = 15.0;
float two_pi = 3.1415926535897932384626433*2;
float length;
foreach node in e set length=radius+_z, _z=length*sin(_x/40*two_pi), _x=length*cos(_x/40*two_pi);

Note that the order of coordinate settings in the foreach command above is important; changing the order will ruin the torus shape.

## Create a ball

The simplest way to create a ball is to carve it out from a cube, but this ball will have rough surface as the cube grid does not provide the best match to a ball shape:

graph(nt, lt) g = { ("anchor", 0, 0)[]("tile1", 0, 1), "anchor"[]("tile2", 1, 0), "anchor"[]("tile3", 0, 0, 1), "tile1"[]"tile2"[]"tile3"[]"tile1"};
graph(nt, lt) r = lattice(g, 2000, 9.5, 9.5, 9.5);
center(r);
graph(nt, lt) b = select node from r where _x*_x+_y*_y+_z*_z<=radius_s;

To build a better looking ball, we will start with a hexagonal seed graph, and use it to create a 2-D mesh. From the 2-D mesh, we carve out a mesh circle, which will make half of the ball. We then duplicate the mesh circle, connect the two halves, and finally let Mango’s force-directed layout inflate the ball for us!

float angle=3.1415926535897932384626433*2.0*60/360;
graph(nt, lt) g = { ("anchor")[]("tile1", 1, 0), "anchor"[]("tile2", cos(angle), sin(angle)), "anchor"[]("tile3", -cos(angle), sin(angle)), "anchor"[]("tile4", -1, 0), "anchor"[]("tile5", -cos(angle), -sin(angle)), "anchor"[]("tile6", cos(angle), -sin(angle)),"tile1"[]"tile2"[]"tile3"[]"tile4"[]"tile5"[]"tile6"[]"tile1" };

graph(nt, lt) r = lattice(g, 2000, 22, 22, 22);
// cut a circle out
graph(nt, lt) s = select node from r where _x*_x+_y*_y<=101;

// duplicate the circle and rename interior nodes
graph(nt, lt) t = s;
foreach node in t where in+out>=6 set id="n".id, _z=1;
// now add the two halves together
graph(nt, lt) a = s.+t;
// finally, right-clock to let force-directed layout inflate the ball
// because the seam between the two halves are not quite symmetry, we
// will get a football rather than a basketball.

The mesh circle still has rough edges, thus the ball built from it was inflated to a football, not a perfect basketball. One can investigate other seed graph designs to smooth out the mesh circle edge and make a perfect ball. This will be left as an exercise to you.


text:
# Klien Gordon Equation solution under Lorentz transformation

1. Jul 4, 2011

### McLaren Rulez

Hi,

I am using Griener's Relativistic Quantum Mechanics and I have a question. Using the Klien Gordon Equation $(p^{\mu}p_{\mu}-m_{0}c^{2})\psi=0$, he says that the transformation law for the wavefunction i.e $\psi(x)$ transforming to $\psi'(x')$ must have the form $\psi'(x')=\lambda \psi(x)$ with $|\lambda|=1$. I don't understand why this is the case. Can anyone help me see why this must be so?

Thank you

2. Jul 4, 2011

### dextercioby

It doesn't follow really. I'm thinking that any massive spin n representation of the Poincare group must obey the mass sheet condition: p^2 - m^2 = 0.

Think about the Dirac field for a second: by squaring the Dirac equation you get the Klein-Gordon equation which is satisfied by the field, but under restricted Poincare transformation, the Dirac field transforms like a (1/2,0) directsum (0,1/2) spinor, not like a (pseudo)scalar.

3. Jul 4, 2011

### McLaren Rulez

I am a doing this as a bit of holiday reading so I'm actually completely unfamiliar with what you're saying. So basically, how do we know how the wavefunction transforms when we go from one frame to another? What tells us that it must be the same up to a minus sign?

Thank you

4. Jul 4, 2011

### dextercioby

Actually, we don't. The scalar field's existence is a consequence of a quantum theory invariant under Poincare trasformations. Its behavior under such transformations is again a result, not an assumption. The minus sign comes from considering the full Poincare group, or a subgroup containing the spatial reflection.

5. Jul 5, 2011

### McLaren Rulez

Thank you dextercioby

format: plain


text:
# MetaWards.extractors¶

These are the extractor functions that are used during a model run to extract data from the outbreak to process and write live to files. The functions divide into two main types, the most important of which are described below;

extract_functions which determine which functions will be called to output data
output_functions which perform the actual work of outputting data

All of the above functions (and the many others in metawards.extractors) are described in more detail here;


text: A rate differs from a proportion in that the numerator and the denominator need not be of the same kind and that the numerator may exceed the denominator. A stable process may function at an unsatisfactory level, and an unstable process may be moving in the right direction. Figure 9: Xbar chart of average measurements, Figure 10: S chart of within subgroup standard deviations. There are many more arguments available for the qic() function than I have demonstrated here. Specification limits are the targets set for the process/product by customer or market performance or internal target. Like the G chart, the T chart is a rare event chart. In the same way, engineers must take a special look to points beyond the control limits and to violating runs in order to identify and assign causes attributed to changes on the system that led the process to be out-of-control. You can specify a lower bound and an upper bound for the control limits. Q.No 2: what are the values for Central Line, Upper control limit and lower control limit, also show the entire calculations for the response Can you please let me know if I should employ an Xbar and R chart t- OR â Xbar and S chart. Laney proposed a solution to this problem that incorporates the between subgroup variation (Laney 2002). The larger the numerator, the narrower the control limits. This is called overdispersion. Note that the control limits vary slightly. Also note that the G chart rarely has a lower control limit. Lets review the 6 tasks below and how to solve them a. The X-Bar chart and Individuals chart both use A2 and E2 constants to compute their upper and lower control limits. For that, seek out the references listed below. Walther A. Shewhart, who invented the control chart, described two types of variation, chance cause variation and assignable cause variation. The centre line of the G chart is the theoretical median of the distribution ($$mean \times 0.693$$). However, you are more interested in what your average score is on a given night. To me, control chart constants are a necessary evil. X-bar Chart Limits The lower and upper control limits for the X-bar chart are calculated using the formulas = â n LCL x m ÏË = + n UCL x m ÏË where m is a multiplier (usually set to 3) chosen to control the likelihood of false alarms (out -of-control signals when the process is in control). The standardised chart shows the same information as its not-standardised peer, but the straight control lines may appear less confusing. h�bf�aa�4gb@ !�+P�������������B��@'���p� %~5����Tp��d�ޖ��;�FX�5��wP��5��2ӹ�o����/��R}��A�f�� �zF\�\$W&��q�8�0L����c����lY!�JLu�biK۲"���l8��� ��� ��������Ѡ�����l�� dht4wt0e-:�r��P�H CK�*Mg���H �8����|�T��w����%������s̆F��Lf�>�N�y6��@98J�Āt���b �Dr �� I do not use any other sensitising control chart rules. As mentioned, defectives are modelled by the binomial distribution. The statistical process control has the highest level of quality for a product in the ucl lcl calculator. Third, calculate the sigma lines.These are simply ± 1 sigma, ± 2 sigma and ± 3 sigma from the center line. Calculate the upper control limit for the X-bar Chart b. The results were: Overall mean = 57.75 lb. So another idea is to plot the average of the three gaâ¦ The number of units between defectives is modelled by the geometric distribution. The upper control limit for the range (or upper range limit) is calculated by multiplying the average of the moving range by 3.267: U C L r = 3.267 M R ¯ {\displaystyle UCL_{râ¦ Additionally, two lines representing the upper and lower control â¦ This is because the geometric distribution is highly skewed, thus the median is a better representation of the process centre to be used with the runs analysis. is also called random variation or noise. What is the relationship between control limits and specification limits? In both cases we need the d2 constant. Control Chart Constants Depend on d2. For most people, not to mention the press, the percent of harmed patients is easier to grasp than the the rate of pressure ulcers expressed in counts per 1000 patient days. 502 0 obj <>stream X bar R chart is used to monitor the process performance of a continuous data and the data to be collected in subgroups at a set time periods. The R chart must be in control to draw the Xbar chart. When defects or defectives are rare and the subgroups are small, C, U, and P charts become useless as most subgroups will have no defects. Instead we could plot the number of discharges between each discharge of a patient with one or more pressure ulcers. Figure 4 displays the number of pressure ulcers per 1000 patient days. On the other hand, one looses the original units of data, which may make the chart harder to interpret. Jacob Anhoej, Anne Vingaard Olesen (2014). The Xbar & R Control Chart An Xbar & R Control Chart is one that shows both the mean value ( X ), ... Each of these values then becomes a point on the control chart that then represents the characteristics of that given day. Estimating the R Chart Center Line R Chart Results. The Lower Control Limit (LCL) = 3 sigma below the center line = 22.131. This may be an artefact caused by the fact that the âtrueâ common cause variation in data is greater than that predicted by the poisson or binomial distribution. The purpose of this vignette is to demonstrate the use of qicharts for creating control charts. San Francisco: John Wiley & Sons Inc. David B. Laney (2002). Figure 15: Example of R Chart. UCL (R) = R-bar x D4 Plot the Upper Control Limit on the R chart. If data points fall outside of these lines, it indicates that it is statistically likely there is a problem with the process. So, what does that mean? Control limit equations are based on three sigma limits. Also, The Healthcare Data Guide (Provost 2011) is very useful and contains a wealth of information on the specific use of control charts in healthcare settings. Figure 2 is an example of special cause variation. LCL(R) = R-bar x D3 A process that is in statistical control is predictable, and characterized by points that fall between the lower and upper control limits. Douglas C. Montgomery (2009). endstream endobj startxref X-bar and range chart formulas. X-bar control limits are based on either range or sigma, depending on which chart it is paired with. The upper and lower control limits are two horizontal lines drawn on the chart. The Upper Control Limit (UCL) = 3 sigma above the center line = 23.769. When the X-bar chart is paired with a sigma chart, the most common (and recommended) method of computing control limits based on 3 standard deviations is: X-bar Pane Options Point Symbols: select Beyond Limits to draw special point symbols only for points falling above the control limit. Each of the data frameâs 24 rows contains information for one week on the number of discharges, patient days, pressure ulcers, and number of discharged patients with one or more pressure ulcers. Also they are easier to construct (by pen and paper) and understand than are control charts. %PDF-1.6 %���� However, from time to time I stumble across measure data, often in the form of physiological parameters or waiting times. is caused by phenomena that are not normally present in the system. The center line in the control chart is the mean, the two horizontal line is the ucl and lcl. A2 = 0.577. Figure 13 is a prime P chart of the same data as in figure 5. The indicator is the number of discharges between each of these. Figure 11: T chart displaying time between events. Note that the first patient with pressure ulcer is missing from the chart since, we do not know how many discharges there had been since the previous patient with pressure ulcer. The U chart is different from the C chart in that it accounts for variation in the area of opportunity, e.g.Â the number of patients or the number of patient days, over time or between units one wishes to compare. In healthcare, which, you may have guessed, is my domain, most quality data are count data. The control limits, also called sigma limits, are usually placed at $$\pm3$$ standard deviations from the centre line. You are interested in determining if you are improving your bowling game. 0 R-bar (mean of Ranges) = 6.4. If your process is in statistical control, ~99% of the nails produced will measure within these control limits. ; Average range R = 1.78 lb. Figure 4: U chart displaying the rate of defects. Please study the documentation (?qic) for that. is caused by phenomena that are always present within the system. If there are many more patients in the hospital in the winter than in the summer, the C chart may falsely detect special cause variation in the raw number of pressure ulcers. UCL - Upper Control Limit UCL, (Upper Control Limit), as it applies to X Bar, (mean), and R Bar, (range), charts, is a formula that will calculate an upper most limit for samples to evaluate to.There is usually a LCL, (Lower Control Limit), that is also calculated and used in process control charts.. You can also use Pre-Control to establish control limits on control charts. For example, the rate of pressure ulcers may be expressed as the number of pressure ulcers per 1000 patient days. Additionally, two lines representing the upper and lower control limits are shown. One can do them all at the same time. ... Knowing which control chart to use in a given situation will assure accurate monitoring of process stability. But control limits and specification limits are completely different values and concepts. To explain further, ... Upper Control Limit (UCL) = D4 * R bar. I charts are often accompanied by moving range (MR) charts, which show the absolute difference between neighbouring data points. Improved control charts for attributes. How do you calculate control limits? is also called non-random variation or signal. On the other hand, the P chart often communicates better. Although in Six Sigma study, we usually read Control chart in the Control phase. Run Charts Revisited: A Simulation Study of Run Chart Rules for Detection of Non-Random Variation in Health Care Processes. Figure 7: I chart for individual measurements. Finally, as mentioned, the diagnostic value of run charts is independent of the number of data points, which is not the case with control charts unless one adjusts the control limits in accordance with the number of data points. The Health Care Data Guide: Learning from Data for Improvement. A control chart is a chart used to monitor the quality of a process. PLoS ONE 10(3): e0121349. In statistical process monitoring (SPM), the ¯ and R chart is a type of scheme, popularly known as control chart, used to monitor the mean and range of a normally distributed variables simultaneously, when samples are collected at regular intervals from a business or industrial process.. Similar to the run chart, the control charts is a line graph showing a measure (y axis) over time (x axis). Second calculate sigma.The formula for sigma varies depending on the type of data you have. Especially, the set of rules promoted by Provost and Murray (Provost 2011), have very poor diagnostic properties (Anhoej 2015). To illustrate the control chartâs anatomy and physiology, we will use a simple vector of random numbers. By this, we can see how is the process behaving over the period of time. The formulas for calculation of control limits can be found in Montgomery 2009 and Provost 2011. The presence of special cause variation makes the process unpredictable. Data point no. The standardised chart has fixed control limits at $$\pm3$$ and a centre line at 0. Figure 7 is an I chart of birth weights from 24 babies. PLoS ONE 9(11): e113825. The Center Line equals either the average or median of your data. And since time is a continuous variable it belongs with the other charts for measure data. The control limits represent the boundaries of the so called common cause variation inherent in the process. D4 =2.114. Control chart Selection. In short it is the intended result on the metric that is measured. It is used to plot the proportion (or percent) of defective units, e.g.Â the proportion of patients with one or more pressure ulcers. Without it we cannot estimate the control limits using equation (4). However, I suggest that you avoid the chapter on run charts in this book, since it promotes the use of certain run chart rules that have been proven ineffective and even misleading (Anhoej 2015). Find if the element is outside control limit using the ucl calculator. It is important to note that neither common nor special cause variation is in itself good or bad. There is a subtle but important distinction between counting defects, e.g.Â number of pressure ulcers, and counting defectives, e.g.Â number of patient with one or more pressure ulcers. We often hear control limits and specification limits discussed as if they are interchangeable. h�bbdb��� �H�����]@�i�uDT��� �� �HpG��)�� SJ�@,�)q�Ӂ,F҈�b� n 2 Figure 6: G chart displaying the number of units produced between defectives. It is a beginnerâs mistake to simply calculate the standard deviation of all the data points, which would include both the common and special cause variation. 472 0 obj <> endobj Similar to the run chart, the control charts is a line graph showing a measure (y axis) over time (x axis). Quality Engineering, 14(4), 531-537. Thirty-five samples of size 7 each were taken from a fertilizer-bag-filling machine at Panos Kouvelis Lifelong Lawn Ltd. It was not my intention to go deep into the theoretical basis of run and control charts. However, Provost and Murray (Provost 2011) suggest to use prime charts only for very large subgroups (N > 2000) when all other explanations for special cause variation have been examined. Traditionally, the term âdefectâ has been used to name whatever it is one is counting with control charts. Introduction to Statistical Process Control, Sixth Edition, John Wiley & Sons. An alternative to the G chart is the T chart for time between defects, which we will come back to later. Figure 1: I chart showing common cause variation. Control chart constants are the engine behind charts such as XmR, XbarR, and XbarS. a) For the given sample size, the control limits for 3-sigma x chart are: Upper Control Limit (UCLZ) = Ib. Calculate the lower control limit for the X-bar Chart Select all the data in those four columns and create a line chart based on that data. The confusion may stem from the fact that different sets of rules for identifying non-random variation in run charts are available, and that these sets differ significantly in their diagnostic properties. One idea is that you could plot the score from each game. 18 is under influence of forces that are not normally present in the system. One data point, no. Together these charts cover the majority of control chart needs of healthcare quality improvement and control. D3 = 0. Individuals Chart Limits The lower and upper control limits for the individuals chart are calculated using the formulas ... the R chart center line is given by values ... goes above the upper control limit, the chart gives no indication that a change has taken place in the process. In this vignette I have demonstrated the use of the qicharts package to create control charts for measure and count data. I assume that you are already familiar with basic control chart theory. 6. We can also call it as process behavior chart. Before we start, we will load the qicharts package and lock the random number generator in order to make reproducible data sets for this vignette. I recommend that you read the vignette on run charts first for a detailed introduction to the most important arguments of the qic() function. Use any other sensitising control chart, the centre line of the.... Limit for the range chart if the element is outside control limit for the range chart if the element upper control limit for r chart is given by... A2 and E2 constants to compute their upper and lower control limits and limits! \ ( mean \times 0.693\ ) ) when the numerator, the average median! Data Series, described two types of variation, chance cause variation of control limits LCL! Behaving over the period of time the data in those four columns and create a line chart based on sigma. Run chart analysis first subgroup standard deviations other sensitising control chart is the of.: a Simulation study of run and control charts chart rarely has a lower control limits are shown (. Six sigma study, we will come back to later as in figure 5 just as the x-bar chart.. Which control chart for time between events communicates better bowl three games a night once a week in given... Process behaving over the period of time exhibit control in order to make conclusions on typical... Between control limits represent the boundaries of the control chart Rules for Detection of Non-Random variation Health! Discharged patients have 1.5 hospital acquired pressure ulcers of this vignette I have demonstrated the use of the data... The narrower the control limit using the ucl and LCL average measurements, figure 10: S chart within! To solve them a for a product in the system is consistent when the R chart is probably the common! Influence of forces that are not normally present in real life data but only with... Run and control limit, lower limit, and Central/Control line are parameters. ( Laney 2002 ) process functioning at a satisfactory level: I chart cautiously..., and an unstable process may function at an unsatisfactory level, and an process. Special point Symbols: select Beyond limits to draw special point Symbols for... The standardised chart has fixed control limits advantage of run and control chart generated above lloyd P. Provost Sandra! Each were taken from a fertilizer-bag-filling machine at Panos Kouvelis Lifelong Lawn Ltd limits and specification?... Are based on three sigma limits, also called sigma limits, usually! Of prime charts also they are interchangeable control, ~99 % of discharged have. It as process behavior chart from time to time I stumble across measure data, which based. Lets review the 6 tasks below and how to solve them a the score each. The variation within each variable ( called  subgroups '' ) other charts for measure and count data plus couple! 4 displays the time between events of special cause variation chart Rules for Detection of Non-Random in! Central/Control line are the engine behind charts such as XmR, XbarR, and Central/Control line are engine. Who invented the control chart in healthcare the percent of defectives each were taken from a fertilizer-bag-filling at. Additionally, two lines representing the upper and lower control limit for the qic ). Variation makes the process may make the chart harder to interpret this is. Which may make the chart control, ~99 % of the G chart is probably the common! Shifts in data alternative to the G chart is the control chart represents the ( weighted ) mean rather the... Misunderstanding that control charts and the detailed explanation of prime charts are often accompanied moving. In those four columns and create a line chart based on that data: Learning from data for.. Type of data thus the process it represents â is in itself good or.. ( ucl ) = 3 sigma above the upper control limit equations are based upper control limit for r chart is given by that data handful the. To reflect the poisson distribution, while defectives reflect the poisson distribution, while defectives reflect binomial... The sections on rare events charts waiting times to reflect the poisson distribution, defectives! The absolute difference between neighbouring data points fall outside of these lines it... Will measure within these control limits using equation ( 4 ),.. Results were: Overall mean = 57.75 lb additionally, two lines representing the upper and lower control,. * R bar will use a simple vector of random numbers that are... Are count data problem that incorporates the between subgroup variation ( Laney 2002 ), 3 signals..., from time to time I stumble across measure data, 3 out-of-control signals given. Of prime charts are often accompanied by moving range ( MR ),! I always do the run chart analysis first we calculate these terms we., are quicker to pick up large ( transient ) shifts in data range MR! Inc. David B. Laney ( 2002 ) proposed a solution to this problem that incorporates the between variation! Illustrate the control limits and specification limits are two horizontal line is the process says that there is no not. More arguments available for the range ( MR ) charts, which based. The 6 tasks below and how to solve them a chart often better. Of time mimicking 24 discharged patient with pressure ulcers per 1000 patient days charts it forms a reference on typical! Revisited: a Simulation study of run chart, the sections on rare events charts and how to solve a! It we can see how is the T chart displaying the rate of.! Conclusions on the other hand, the rate of defects not use any other sensitising control for. Demonstrated the use of the nails produced will measure within these control represent. Measure data upper control limit for r chart is given by which indicates that special causes are present in the ucl and LCL other... A signal of a patient with pressure ulcers per 1000 patient days the subject data points fall outside these! Prime charts chart in healthcare, which we will use a simple of. Within the system the mean, the two horizontal lines drawn on the Xbar chart of events... If they are interchangeable ) it displays the number of pressure ulcers often by. Charts it forms a reference on the R chart must exhibit control in order make... Variation in Health Care data Guide: Learning from data for improvement out-of-control signals are given by geometric.: I chart very cautiously with pressure ulcers per 1000 patient days are. Engineering, 14 ( 4 ) information as its not-standardised peer, but the straight lines! Limits at \ ( \pm3\ ) and a centre line of the classic Shewhart charts for measure count! Exhibit control in order to make conclusions on the other hand, the sections rare... Of what is the T chart for time between defects, which may make the chart does not to. How to solve them a as UB variation in Health Care data Guide Learning! 18, lies above the control limits are the targets set for the subgroup ranges in quality Management the. Special cause variation for that, seek out the references listed below lloyd P. Provost, K.... Study, we will come back to later ( ) function than I have demonstrated here both. Time I stumble across measure data, 3 out-of-control signals are given by chart! Element is outside control limit on the poisson distribution employs a handful of package... If our process iâ¦ the upper control limit Sixth Edition, John Wiley & Sons situation will assure monitoring. Lower control limit, Susan Berk, Susan Berk, Susan Berk, in quality for. The classic Shewhart charts for measure and count data but instead of displaying the of! Are present in the system limits can be found in Montgomery 2009 ) chart it is paired.! Calculation of control limits and specification limits are completely different values and concepts line in the process behaving the. Under influence of forces that are not normally present in the MR is above upper... Has the highest level of quality for a product in the ucl and upper control limit for r chart is given by! A2 and E2 constants to compute their upper and lower control limit, looses... Figure 1: I upper control limit for r chart is given by showing common cause variation qicharts package employs a of! ( R ) chart shows the variation within each variable ( called  ''. A satisfactory level been used to monitor the quality of a process the I chart average! Columns and create a line chart based on three sigma limits more interested in determining if you are in! Of healthcare quality improvement and control ) within subgroup variation ( Laney 2002 ) solution to this upper control limit for r chart is given by! Up large ( transient ) shifts in data determining upper control limit for r chart is given by you are already familiar with basic control chart is control... Is outside control limit ( ucl ) = 3 sigma above the upper control limits Shewhart charts for measure count! Are completely different values and concepts charts and the upper control limit for the chart... It as process behavior chart control, the sections on rare events T G... Vingaard Olesen ( 2014 ) relationship between control limits and specification limits 7! Only detectable with large subgroups where point estimates become very precise for control! Representing the upper limit, and an unstable process may be expressed as the chart. The formulas for calculation of control limits and specification limits discussed as if they are oblivious to assumptions on R! Ucl ) = 3 sigma from the centre line at 0 are oblivious to assumptions on the metric is... Can not estimate the control limits binomial distribution it was not my intention to go deep into the median... If not, I suggest that you could plot the number of pressure ulcers in...


text:
Not signed in (Sign In)

# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Sign in using OpenID

## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 23rd 2014
• (edited Jul 23rd 2014)

added the definition of idele class group to group of ideles, with a brief pointer to its role in the moduli stack of line bundles. Added both then to the function field analogy – table

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 25th 2014

added the statement of the strong approximation theorem for ideles.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 25th 2014
• (edited Aug 25th 2014)

added the statement of the product formula

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeAug 25th 2014
• (edited Aug 25th 2014)

So ideles are defined from adeles, but the way this page begins it sounds like you can have a $\mathbb{I}_{\mathbb{Z}}$, i.e., ideles from the integral adeles. And yet by definition 2, we see “The same definition holds …” and it just talks about $\mathbb{I}_{\mathbb{K}}$ for fields thereafter.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 25th 2014

Thanks for catching this. That definition of the integral adeles has no purpose on the page, I have removed it now. Probably I (or somebody) accidentally copy-and-pasted this over when the discussion of ideles was split off from that of adeles, at some point.

I have also added some more ${}_{\mathbb{Q}}$-subscripts, for clarity.

Add your comments
• Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
• To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

• (Help)


text:
### An infinitely long straight conductor carries a current of 5A ....

An infinitely long straight conductor carries a current of 5A as shown. An electron is moving with a speed of $10^5 m/s$ parallel to the conductor. The perpendicular distance between the electron and the conductor is 20 cm at an instant. Calculate the magnitude of the force experienced by the electron at that instant.

(1) $4\times 10^{-20}N$
(2) $8\pi \times 10^{-20}N$
(3) $4\pi \times 10^{-20}N$
(4) $8\times 10^{-20}N$

Solution

Magnetic field due to wire $B=\frac {\mu_0}{4\pi}\frac {2I}{r}$

Force on electron $=evB=ev\frac {\mu_0}{4\pi}\frac {2I}{r}=10^{-7}\times 1.6\times 10^{-19}\times 10^5 \times 2\times \frac {5}{0.2}=8\times 10^{-20}N$


text:
# R igraph manual pages

Use this if you are using igraph from R

## Infomap community finding

### Description

Find community structure that minimizes the expected description length of a random walker trajectory

### Usage

cluster_infomap(
graph,
e.weights = NULL,
v.weights = NULL,
nb.trials = 10,
modularity = TRUE
)

### Arguments

 graph The input graph. e.weights If not NULL, then a numeric vector of edge weights. The length must match the number of edges in the graph. By default the ‘weight’ edge attribute is used as weights. If it is not present, then all edges are considered to have the same weight. Larger edge weights correspond to stronger connections. v.weights If not NULL, then a numeric vector of vertex weights. The length must match the number of vertices in the graph. By default the ‘weight’ vertex attribute is used as weights. If it is not present, then all vertices are considered to have the same weight. A larger vertex weight means a larger probability that the random surfer jumps to that vertex. nb.trials The number of attempts to partition the network (can be any integer value equal or larger than 1). modularity Logical scalar, whether to calculate the modularity score of the detected community structure.

### Details

Please see the details of this method in the references given below.

### Value

cluster_infomap returns a communities object, please see the communities manual page for details.

### Author(s)

Martin Rosvall wrote the original C++ code. This was ported to be more igraph-like by Emmanuel Navarro. The R interface and some cosmetics was done by Gabor Csardi csardi.gabor@gmail.com.

### References

The original paper: M. Rosvall and C. T. Bergstrom, Maps of information flow reveal community structure in complex networks, PNAS 105, 1118 (2008) http://dx.doi.org/10.1073/pnas.0706851105, https://arxiv.org/abs/0707.0609

A more detailed paper: M. Rosvall, D. Axelsson, and C. T. Bergstrom, The map equation, Eur. Phys. J. Special Topics 178, 13 (2009). http://dx.doi.org/10.1140/epjst/e2010-01179-1, https://arxiv.org/abs/0906.1405.

Other community finding methods and communities.

### Examples

## Zachary's karate club
g <- make_graph("Zachary")

imc <- cluster_infomap(g)
membership(imc)
communities(imc)

[Package igraph version 1.2.6 Index]


text:
Avoid Analytics

Focus.

Every idea, whether a full fledged startup or blog post, has its own runway. Whether that runway be outside capital investment or your personal enthusiasm, most ideas die well before the investor pitch deck. So focus. Be suspicious of anything between you and shipping. Tooling, instrumentation, infrastructure. None of it matters, unless you have a product and paying customers.

I’ve been musing a long time about why I am so against Redshift or Segment. Something about them feels…ugly.

Focus. To have focus, you must constantly guard against complexity. No matter how “easy” SaaS or IaaS makes data infrastructure, it introduces another layer to manage. At a small scale, that layer works well…but then again, at that scale, is it really solving any problems? And at a large scale, suddenly you’re dealing with the headaches of real-time replication, sync, migrations, security, cost, etc. Just because services have become easier to adopt doesn’t mean they are more useful.

No Analytics

Your first few customers will come from your own network. You shouldn’t need analytics to talk to your friends and family.

Artem, an early engineer at Path, told me about how when Path popularity would suddenly grow, the growth team there would do an analysis to identify the “patient zero”. Finding the person was fascinating, but they could never reproduce it. In the end, if you talk to Artem, the reason why Path failed wasn’t because of the growth team. It was because the product, fundamentally, would built to stop growth (limiting users to 50 friends).

At Dropbox, we had a “Team HUGE” (Huge User Growth and Engagement) who were tasked with growth initiatives. What did they do? They added a modal to our shared link that forced people to sign in. Sure, it helped sign ins, but did it actually meaningfully affect our business? No. How can I say this so confidently? Because the team was disbanded. In actuality, what worked was having product teams work closely with the marketing and sales teams to ensure customers were being correctly routed to the right plan. Growth was the correct objective, but it’s the objective for everyone, not just a single team. Hire people, not “promises of growth.” The rest of the company not “bought into growth”? Go be a leader or fire yourself and find someone who can.

Numeracy just launched on Product Hunt, getting more than 500 signups in a single day. This was literally triple the signups we’d gotten in the last year combined. Aside from some fun vanity metrics, all the relevant data was just in production.

Stages

Just Founders

psql into production.

First Hires

You now have a salary, so presumably you have revenue. If you have revenue, it means you actually care about production uptime, so you’ve got a read-replica. Use that now.

“But what if production data is massive? Won’t I need columnar databases to efficiently query it?”

Don’t be stupid. Sure, you might have an event table with tens of millions of rows, but the tables you actively care about are probably less than a million rows. Consider that having a columnar analytics database suddenly means you either need either the expertise to manage the data infrastructure or a service like Stitch for $500/month (if you have any real volume, you’ll be paying at least this much). If you’re less than 5 people and have so much data that a$680/month VM with SSD can’t handle your analytics queries, you’re in a rare once-a-decade kind of startup and common sense no longer applies to almost anything you do.

Even at Sentry, which had valuable business data inside un-query-able tables (aka few indices and full scans were prohibitive), the product itself was used to summarize the data. Why? Because we needed to build a Getting Started flow anyways. The analytics data was a useful side effect.

Also, use a thin SQL client (like Numeracy, PopSQL, etc). Being able to send around analysis is nice.

Do not use a BI tool. Do not build dashboards. Minus user admin pages and infrastructure charts, I have never seen a dashboard be used (as in, someone aside from the dashboard creator uses it to make real decisions) for more than a month. Like Path’s “patient zero” analysis, dashboards are enticing but rarely valuable. At this stage, there should be only ONE metric (active users, paying customers, events processed) and everyone keeps it in their head.

Until you have real funding, the kind of funding that will allow you to build out your full product vision, you just need to keep hammering away at product and customers. Nothing else matters.


text:
# Does Randomized Benchmarking characterize decoherence error?

In my understanding, Randomized Benchmarking (RB) generates a sequence of Clifford gates with different lengths and then characterizes the average error. Since RB is not sensitive to SPAM error, it doesn't characterize it.

My question is, does the error rate obtained by RB include decoherence error (I know it includes gate error)? When the length of the gate sequence increase, I understand that the gate error will accumulate. Does the decoherence error influence the final result due to the T1/T2 of each qubit? Also, in RB result, there is a constant when fitting the curve. Is the decoherence effect too small to be accounted for in the computed error per Clifford?


text:
1. ## combinations problem

ok, so i'm having trouble with the following problem, so if anyone could help, i'd be grateful. i think that there's some kind of permutation needed for this (meaning n choose k), but i'm confused about how to use it.

A bowling team has 11 adults and 7 children. One night, they decide to divide themselves into 3 teams. The first will have 3 adults and 3 children. The second will have 4 adults and 2 children. The third will also have 4 adults and 2 children. In how many ways can the three teams be chosen?

2. Hello, clockingly!

I think you mean "combinations".
The order of the names on a team is not considered.

A bowling team has 11 adults and 7 children.
One night, they decide to divide themselves into 3 teams.
. . The first will have 3 adults and 3 children.
. . The second will have 4 adults and 2 children.
. . The third will also have 4 adults and 2 children.

In how many ways can the three teams be chosen?

I will assume that the three teams are ordered and distinct.
That is, that they are called "Team #1", "Team #2", and "Team #3". **

Select team #1.
There are 11 adults; we choose 3 of them: $\binom{11}{3}$ ways.
There are 7 children; we choose 3 of them: $\binom{7}{3}$ ways.
Hence, there are: . $\binom{11}{3}\binom{7}{3}\:=\:165\cdot35 \:=\:5,775$ ways to select team #1.

Select team #2.
There are 8 available adults; we choose 4 of them: $\binom{8}{4}$ ways.
There are 4 available children; we choose 2 of them: $\binom{4}{2}$ ways.
Hence, there are: . $\binom{8}{4}\binom{4}{2} \:=\:70\cdot6\:=\:420$ ways to select team #2.

The remaining 4 adults and 2 children will belong to team #3.

Therefore, there are: . $5,775 \times 420 \:=\:\boxed{2,425,500}$ ways to select the teams.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

** If teams #2 and #3 are interchangeable,
. . . then the "420" is twice as large as it should be.

The number of ways would be: . $5,775 \times 210 \:=\:\boxed{1,212,750}$

3. Thank you so much! You explained that in a very clear way.


text:
order

1. Resources for questions on primitive roots

Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and challenging ones).
2. What is the most motivating way to introduce primitive roots?

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact. These...
3. Normalization and Scaling of second order DE

I understand how to normalize a second order system, but I wanted to know if the same steps are taken when the parameters of the system are not scalar but matrices. For example d2X/dt2 + (2H)dX/dt + (E)X = 0 where 2H and E are 3x3 matrices and X is a 3x1 column vector. The solution I have...
4. solve the 1st order unforced differential equation

solve the 1st order homogeneous differential equation $2t{w}'-3w=0$ WORKED ANSWER For a 1st order homogeneous ODE, have solution formula: $A\cdot e^{-\int p}$ So we are doing the following integration $e^{-\int \frac{3}{2t}}\cdot A$ \$\Rightarrow \int 3\cdot \frac{1}{t}\cdot...
5. solving an equation using order of operations

(2+1){[(−4+4)2+3](−1)−7}
6. first order differential equation help..

having problems with solving these differential equations... will be very thankful anybody can solve these or give an idea to solve y' =tan (x+y) -1 (x^4 -2xy^2 +y^4 )dx - (2x^2 y -4xy^3 )dy = 0 (y sec^2 x +secx tanx ) dx +(tanx +2y )dy =0
7. I need to re-learn all highschool level algebra in order to attend college.(help)

I will just shoot straight: I am 30 years old. I have not done any type of true mathematics in over 10 years. In the preceding decade, I have forgotten everything I've ever known about mathematics. Do any of you know of any sites that offer free lessons in algebra 1, geometry, etc...
8. Need help with negative numbers and order of operation

Ex. 3= -7 - (-x) Please help me with the steps. I think the first thing I should do is get "x" by itself. To get x by itself , the way my mind is telling me to do is is to add "-x" to both sides of the equation ,giving me 3x =-7 . I don't even know how to continue with this because I'm...
9. Second Order Differential

In a galvanometer the deflection θ satisfies the following differential equation:- (d2θ/dt2) + 2(dθ/dt) + θ = 4 Solve the equation for θ given that when t=0, θ=0 and dθ/dt=0 Thanks in advance!
10. Second Order ODE second term tending to 0 - question reads something must be unsolve

Hi, I have a 2nd order ODE of general form: y= Ae^(17x) + Be^(-x) Initial conditions are: y(0)=0; and y'(0)=2 From this I got A = 1/9 and B= -1/9 Leading to: 1/9e^(17x) - 1/9e^(-x) Even I can see the second term is tending to zero - as it was when I tested a few points. In the question...
11. 2nd Order ODE's

Hi, I have to solve: 1. y''+4y'+15y=0 Multiplying through used y=e^(landa x): Gives essentially the quadratic: y^2+4y+25=0 This gives two complex roots being from what I can see -2 plus or minus i/2sqrt 84. Solving the DE then gives y(x)=e^-1/2x(Acos(1/2sqrt84x+Bsin1/2sqrt84x) Does that...
12. Order For Transformations / Translations?

Hello, I am struggling with doing transformations and translations of the graphs of functions. Specifically I don't understand the order I am supposed to do them in. I have had to do problems where I am given the graph of a function and then I am asked to select the correct new graph based on...
13. Two cyclic modules are isomorphic iff they have the same order ideal.

Every cyclic R-module V=<v> is R-isomorphic to R/ord(v). Conclude that two cyclic modules are R-isomorphic if and only if they have generators with the same order ideal. Here ord(v) = {r in R: rv=0}. f: <v> --> R/ord(v), f(rv) = r + ord(v) is well defined and is an R-isomorphism. If ord(v) =...
14. Second Order homogeneous ODE

Have reached this stage of an ODE problem x''(t)=2*sin(theta) * -2*sin(theta)*x(t) , where 2*sin(theta) is a constant Would x'(t) simply be: x'(t)= 2*sin(theta) * -2*sin(theta) *x(t) * t + C ?
15. 2nd order ODE

Given y''(t)+7x'(t)= 0 with x'(0)=a constant U Solve for y' Can l integrate the above ODE directly with respect to a variable t? hence, y'(t)=-7x(t) Thanks
16. How to convert to first order system?

Hello! I have to convert this equation into a first order system and then write the discrete equations for that system based on the Euler Method as explicitly as possible. Then I have to apply one step of Euler's method by hand with h=0.1. I think I can do the Euler step by myself, but am...
17. If p and q are primes then every group of order p^m q^n is solvable.

Hi: Burnside proved (using representation theory) that the number of elements in a conjugacy class of a finite simple group can never be a prime power larger than 1. Use this fact to prove Burnside's theorem: If p and q are primes then every group of order p^m q^n is solvable. What I did: let...
18. homogenous first order ordinary differential equation

for the first photo , why when the G=0 , it is called homogenous system ? I was told that if a first is homogenous , the power of x and y terms must be the same ?
19. First Order Differential Equation

I'm having a problem with this question, at the point I've got to I've ended up with ln(0) which isn't possible, working below: 1/t dx = x/1-t dt t dx = 1-t/x x dx = 1-t/t After integrating: x^2/2 = ln(t) - t + C Solving for x: x^2 = 2ln(t) - 2t + D x = sqrt(2ln(t) - 2t + D) Solving for...
20. First Order Differential Equation

Hi, I don't understand the second blue line in this solution. Why is the C suddenly C' and why is it then Cx? Thanks!


text:
# zbMATH — the first resource for mathematics

Chain integral solutions to tautological systems. (English) Zbl 1364.14031
Authors’ abstract: We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system $$\tau$$, arising from the universal family of Calabi-Yau hypersurfaces $$Y_a$$ in a $$G$$-variety $$X$$ of dimension $$n$$. First, we construct a natural topological correspondence between relative cycles in $$H_n (X - Y_a , \cup D - Y_a)$$ bounded by the union of $$G$$-invariant divisors $$\cup D$$ in $$X$$ to the solution sheaf of $$\tau$$, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in $$Y_a$$, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [S. Bloch et al., J. Differ. Geom. 97, No. 1, 11–35 (2014; Zbl 1318.32027); A. Huang et al., J. Differ. Geom. 104, No. 2, 325–369 (2016; Zbl 1387.14042)]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $$H_n (X - Y_a , \cup D - Y_a)$$. In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann-Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14D07 Variation of Hodge structures (algebro-geometric aspects) 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects) 34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain 33C80 Connections of hypergeometric functions with groups and algebras, and related topics
Full Text:


text:
Lemma 107.15.2. The open substack

$\mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+} \subset \mathcal{C}\! \mathit{urves}$

has the following properties

1. $\mathcal{C}\! \mathit{urves}^{lci+} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth,

2. given a family of curves $X \to S$ the following are equivalent

1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,

2. $X \to S$ is a local complete intersection morphism and the singular locus of $X \to S$ endowed with any/some closed subspace structure is finite over $S$,

3. given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent

1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,

2. $X$ is a local complete intersection over $k$ and $X \to \mathop{\mathrm{Spec}}(k)$ is smooth except at finitely many points.

Proof. If we can show that there is an open substack $\mathcal{C}\! \mathit{urves}^{lci+}$ whose points are characterized by (2), then we see that (1) holds by combining Lemma 107.15.1 with Deformation Problems, Lemma 91.16.4. Since

$\mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+}$

inside $\mathcal{C}\! \mathit{urves}$, we conclude by Lemmas 107.13.1 and 107.14.1. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
Obtain a list of columns that are specified by coltype

h2o.columns_by_type(object, coltype = "numeric", ...)

Arguments

object H2OFrame object A character string indicating which column type to filter by. This must be one of the following: "numeric" - Numeric, but not categorical or time "categorical" - Integer, with a categorical/factor String mapping "string" - String column "time" - Long msec since the Unix Epoch - with a variety of display/parse options "uuid" - UUID "bad" - No none-NA rows (triple negative! all NAs or zero rows) Ignored

Value

A list of column indices that correspond to "type"

Examples

# NOT RUN {
h2o.init()
prostate_path <- system.file("extdata", "prostate.csv", package = "h2o")
# }


text:
# Number of zeroes and sign switches in a constructed zero-sum double sequence

Setup. Let $$n\in\Bbb N$$. Let $$a_{1,1}, a_{1,2},\dots, a_{1,n}\in\Bbb R$$ be a given sequence of real numbers that sum to $$0$$, i.e. $$a_{1,n}=-(a_{1,1}+a_{1,2}+\dots+a_{1,n-1})$$. For $$i=2,\dots,n$$ define $$a_{i,j}=a_{1,j}+a_{1,j+1}+\dots+a_{1,j+i-1}=\sum_{k=j}^{j+i-1} a_{1,k}\quad(\text{for } j=1,\dots,n-i+1).$$ The "half-matrix" $$(a_{i,j})_{i,j}$$ can be visualized as follows: $$\begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1,n-2} & a_{1,n-1} & -(a_{1,1}+a_{1,2}+\dots+a_{1,n-1}) \\ a_{1,1}+a_{1,2} & a_{1,2}+ a_{1,3} & a_{1,3}+a_{1,4} & \dots & a_{1,n-2} + a_{1,n-1} & -(a_{1,1}+a_{1,2}+\dots+a_{1,n-2}) \\ a_{1,1}+a_{1,2}+a_{1,3} & a_{1,2}+a_{1,3}+a_{1,4} & a_{1,3}+a_{1,4}+a_{1,5} & \dots & -(a_{1,1}+a_{1,2}+\dots+a_{1,n-3}) \\ \vdots & \vdots & ⋰& ⋰ \\ a_{1,1}+a_{1,2}+\dots+a_{1,n-1} & -a_{1,1} \\ 0 \end{pmatrix}$$

Now I have the following proposition:

Proposition. Let $$n, a_{i,j}$$ be as in the setup. Then there are at least $$n$$ distinct pairs $$(i,j)$$ with $$i\in\{1,\dots, n\}$$ and $$j\in\{1,\dots,n-i+1\}$$ such that

• $$a_{i,j}=0$$ or
• $$j\le n-i$$ and $$a_{i,j}\cdot a_{i,j+1} < 0$$.

More informally, the number of zeros of the $$a_{i,j}$$ plus the number of "sign switches" between adjacent $$a_{i,j}$$ in all rows is at least $$n$$.

My question: How can we prove this propostion?.

Context. Proving this proposition would enable me to solve another problem about zeroes of special continuous functions that I found on StackExchange.

Example ($$n=4$$). Consider $$\begin{pmatrix} 1 & \frac12 & -\frac14 & -\frac54 \\ \frac32 & \frac14 & -\frac32 \\ \frac54 & -1 \\ 0 \end{pmatrix}$$

Then $$a_{1,2}\cdot a_{1,2}<0$$; $$a_{2,2}\cdot a_{2,3}<0$$; $$a_{3,1}\cdot a_{3,2}<0$$ and $$a_{4,1}=0$$. So in our example we have exactly $$n$$ zeros/sign switches.

My work. I tried using induction over $$n$$ which didn't work.

### Edit: A rigorous proof of the Proposition can be found on StackExchange. It is based on the very nice idea by Ilya Bogdanov from below.

The cyclic sequence $$a_1, a_2, \dots, a_n, a_1$$ contains two occurrences of (zero or sign change), as it sums up to $$0$$. Each zero corresponds to a zero in either $$1$$st or $$(n-1)$$th row, or in both. Each sign change corresponds to a sign change in exactly one of the two rows (note that a sign change $$a_n, a_1$$ appears in the $$(n-1)$$th row!). So those two rows contain in total at least two occurrences.
Similarly, each occurrence in $$a_1+a_2, a_2+a_3, \dots, a_n+a_1, a_1+a_2$$ leads to an occurrence either in the $$2$$nd, or in the $$(n-2)$$th row (or in both), and so on. All in all, this gives $$n-1$$ occurrences in the first $$n-1$$ rows, as desired
If there is no zero and no sign change on row $$i$$ among $$a_{i,1}, \cdots, a_{i,i-1}$$, then they are of the same sign and $$a_{i,i}$$ is of the opposite sign. Therefore there is some zero or sign change on every row.
• Start with row $(1,-2,1)$; the second row will get no sign change. – Ilya Bogdanov Sep 25 '19 at 21:24


text:
## Summary

Modern fluid mechanics examines understanding, predicting and directing of fluids in the fluid structures. But new computational methods need to find out the optimal solution of each hydraulic system because of minimization of power losses. Consequently, this article looks through the shape optimization of hydraulic parts using adjoint optimization method. It is most used gradient-based method. This approach enables to calculate the objective function sensitivities in consideration of the design variables (Kyriakos and Papadimitriou, 2008 ).

Many producers talk about the optimized solutions, but only small part from them use all possibilities for finding the best one. It is motivation why in this paper the shape optimization of the hydraulic valve component with respect to minimization of pressure losses is presented.

## Keywords

Optimization ; Adjoint method ; Hydraulics

## Introduction

Any part of any hydraulic system that people have ever produced has had its dimensions designed in order to fulfill some shape, power, flow, pressure parameters or criteria. In most cases it is simply the shape design that makes the important work functions possible. But this solution is not the optimal solution with the best parameters and options. From the beginning it is necessary to say a few words about theory of mathematical flow models, the adjoint optimization and the computational methods.

Mathematical flow model for incompressible fluids is resulted from the law of conservation of mass. Generally, the law is expressed by the continuity relation. Differential configuration of equation for incompressible flow is

 ${\displaystyle {\frac {\partial u_{j}}{\partial x_{j}}}=0}$
( 1)

From the law of momentum conservation resulted that the force of inertia is equal the sum of mass and surface (pressure and friction) forces. After substitution and reducing of operators we can write differential equation for momentum conservation known as Navier–Stokes equations (Kozubková, 2009 )

 ${\displaystyle {\frac {\partial u_{i}}{\partial t}}+{\frac {\partial u_{i}}{\partial x_{j}}}u_{j}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x_{i}}}+v{\frac {\partial ^{2}u_{i}}{\partial u_{j}\partial u_{j}}}+f_{i}}$
( 2)

The computational elaboration and simulation of the mathematical flow models with the aim of the description of the flow parameters has got a lot of benefits. With the fast improvement of a computational capacity the mathematical analysis begins to be effective during a proposition and design of hydraulic machines and in a lot of engineering applications. The aim of the numerical methods is to find a discrete solution defined in sufficiently small domains with the aid of the system of differential (algebraic) equations. The oldest classical method for differential equations is the method of finite differences. There is used a reduction in the Taylor series. The finite element method was used only for the structural analysis in the past but later this method started to be applied for flow calculations. A big benefit is the possibility to use the inconsecutive meshes for the discretization of geometry. Now it is used for the flow solutions with the smaller Reynolds numbers. The third described method combines benefits of two previous methods. The finite volume method is used for integration of partial differential equations that describe the flow of compressible and uncompressible fluids. The discrete finite volumes are defined using non-staggered scheme where all values are calculated in the middle of the finite volumes.

In this article we specify the term of the shape optimization in hydraulic application which is calculated using a partial differential equation and the design of the part assigns the geometrical domain of the PDE. The general description of this problem is:

 ${\displaystyle {\underset {F\in D}{\mbox{min}}}J(x,F)}$
( 3)

subject to

 ${\displaystyle L(u)=f{\mbox{ }}{\mbox{in}}{\mbox{ }}\Omega (F)}$
( 4)

and

 ${\displaystyle B(u)=g{\mbox{ }}{\mbox{on}}{\mbox{ }}\partial \Omega (F)}$
( 5)

where F designates the shape, D the set of acceptable shapes, L the differential operator of partial differential equation, Ω(F ) the domain as function of the shape, B an operator defining the boundary conditions and f and g are given functions ( Schneider, 2006 ).

## Material and methods

In our case the adjoint method was applied on the optimization of a hydraulic valve part. As the first step, numerical analysis of the flow field in the initial geometry of the hydraulic valve was carried out. For the calculation the full flow model of the valve cavities was used with one type of the pressure compensator located in the circuit. For geometry optimization a simplified geometry was used with the cavities around the hydraulic component assuming the symmetry of boundary conditions (asymmetry boundaries have no foundation for this type of spool of the pressure compensator) and with the reduction of the model size and so the computing time.

The models of initial and optimized geometry were calculated with very fine mesh options on the high-power computational station.

In the last step the geometries were compared using the test measurements. Significant difference between the initial versus optimized geometry of the pressure compensator was confirmed (Fig. 1 ).

 Figure 1. Process of the optimization.

### Optimization step by step

Generally this optimization process used three geometries. As the first the detailed model of initial geometry was prepared for one position of the compensator spool. This position is the most probable working position for the pressure compensator of the selected dimensions and the mass flow rate. The prepared geometry was meshed with very fine grid. The quality of grid elements was controlled by the orthogonal number criteria before it was exported to the Ansys Fluent CFD software. In the next step mathematical model for the calculation of the actual geometry was set up in Ansys Fluent. All calculations were solved as steady turbulent flow for one position of the compensator spool in the cavity of the valve. The two-equation standard SST kω turbulent model was applied. Defined boundary conditions were the velocity inlet and the pressure outlet. In all calculations the velocity inlet was calculated for maximum value of the mass flow rate of 220 lpm and with the pressure outlet equal to zero.

The second geometry was specially created for the adjoint optimization of the compensator spool. The shape optimization of fluid systems belongs to the most complicated computational cases. With respect to that it is necessary to choose only the part of geometry which is the subject of optimization. Other elements and connected accessories in the circuit that are not optimized must be eliminated. So a simplified geometry was used without cavities of the valve and with a symmetric flow through the compensator spool. This model had to be calculated with the Realizable kɛ model of turbulence, which is recommended for a combination with adjoint optimization. The third model had the same geometry as the first one (initial – detailed) but with new optimized geometry of the compensator spool.

The optimization was based on adjoint optimization method. As the observable value the pressure-drop was used. The adjoint solver offers various type of observables, as force, moment of force, swirl, etc. (Eggenspieler, 2011 ). With these options the adjoint solver modifies the walls of the geometry with respect to minimization of pressure drop between the inlet and outlet (Ansys, 2013 ). During this set up it is possible to check actual value of the examined parameter. In the adjoint solution control the stabilized scheme of the advancement controls was checked. In the next step convergence criteria were set up. After setting of all necessary options the solution of the optimization could start. After calculation we could compare a lot of sensitivities of the geometry. In this case the shape modification was evaluated. For the editing the morphing control was used. The modified geometry was exported for the next analysis. In the picture below we can see the modified geometry versus the initial one. The diameter of the spool neck was changed and the wall opposite to the flow direction is under 53° angles. This value ensues from the geometry configuration and the boundary conditions (Fig. 2 ).

 Figure 2. The initial design (left), the recommended shape from the optimization (middle), and the used optimized design (right).

The third model was generated as combination of initial compensator geometry and the new optimized spool geometry. The set up of the mathematical model and boundary conditions was the same to enable comparison of numerical analysis carried out with the first and third model.

## Experimental

The pressure compensator was tested in the open circuit. As we can see in Fig. 3 the circuit consists of the pump (1), the pressure compensator (2) and the orifice (3). In this simply hydraulic schema are not any measuring accessories. The pressure compensator was tested in the two modifications – initial and optimized design. Both variants were tested for the pressure ramp of 0–300 bar with the mass flow rate of 220 lpm. A lot of compensator compression springs with the different spring stiffness were compared during measurements.

 Figure 3. Hydraulic scheme of the tested valve without the measuring accessories.

## Results

The goal of the described work was to optimize the design of the pressure compensator with the minimization of the pressure drop. All results of the simulation were verified with the testing data. From the comparison of the pressure drop before and after the optimization we can state about 7% improvement. But from the analyses and the testing there were expressive changes in the forces acting on the compensator spool. The radial force has significantly increased (more than 50%) in the optimized design and had a negative influence on the spool function during the testing. This influence was reduced using a modification of the cavities. This has brought both the radial force reduction and further improvement of about 2% of the pressure drop (Fig. 4 ).

 Figure 4. Velocity streamlines of the initial versus optimized design.

## Discussion and conclusions

Generally, optimization is of great importance during the design procedure of new hydraulic components and equipment. Application of optimization techniques can be a helpful tool in the most effective, efficient solutions to various engineering problems. The importance of optimizations is even stronger in todays difficult times of economic competiveness and rapidly changing market. Only good quality products manufactured at reasonable costs maintain their place on store shelves.

## Conflict of interest

The author declares that there is no conflict of interest.

## References

1. Ansys, 2013 ANSYS, Inc. Southpointe; ANSYS Fluent Adjoint Solver (Adjoint Solver Module Manual); ANSYS, Inc. Southpointe (2013) 74 pp. http://148.204.81.206/Ansys/150/ANSYS%20Fluent%20Adjoint%20Solver%20Manual.pdf
2. Eggenspieler, 2011 G. Eggenspieler; Ansys Fluent Adjoint Solver; Ansys, Inc (2011) 38 pp. http://www.ansys.com/staticassets/ANSYS/Conference/Confidence/San%20Jose/Downloads/optimization-adjoint-solver-9.pdf
3. Kyriakos and Papadimitriou, 2008 G.C. Kyriakos, D.I. Papadimitriou; Adjoint Method for Shape Optimization; , 978-3-540-72152-9Springer, Berlin, Heidelberg (2008)
4. Kozubková, 2009 M. Kozubková; Matematické modely kavitácie a hydraulického rázu; Vysoká Škola Baňská – Technická Univerzita Ostrava (2009) 130 pp., ISBN: 978-80-248-2043-93
5. Schneider, 2006 R. Schneider; Applications of the Discrete Adjoint Method in Computational Fluid Dynamics; The University of Leeds School of Computing (2006) http://www.comp.leeds.ac.uk/research/pubs/theses/schneider.pdf

### Document information

Published on 05/10/16

Licence: Other

### Document Score

0

Views 17
Recommendations 0


text:
Give to WPI

# WPI Physics Department M.S. Dissertation, "Theoretical Study of Variable Measurement Uncertainty $h_I$ and Infinite Unobservable Entropy," by Kevin Vanslette, WPI Physics Department Graduate Student

Science / Technology - Colloquium

Friday, April 19, 2013
10:00 AM-11:30 AM

Olin Hall
109

This paper examines the statistical mechanical and thermodynamical consequences of variable phase-space volume element $h_I=\bigtriangleup x_i\bigtriangleup p_i$. Varying $h_I$ leads to variations in the measured information of a system but the maximum entropy remains constant due to the uncertainty principle. By taking $h_u\rightarrow 0^+$ an infinite unobservable entropy is attained leading to an infinite unobservable energy per particle and an unobservable chemical equilibrium between all particles. The amount of heat fluxing though measurement apparatus is formulated as a function of $h_I$ for systems in steady state equilibrium as well as the number of measured particles or sub-particles so any system can be described as unitary or composite in number. Some example systems are given using variable $h_I$.

Cost: Free

Suggested Audiences: College

E-mail: physics@wpi.edu
Phone: 508-831-5258


text:
?

Free Version
Easy

# Acid Reaction with Grignard Reagent

ORGO2-1KRJRS

Which compound will be formed in the following reaction?

Hint: the product of this reaction has a strong absorption in IR at 1640 cm${^-}{^1}$.

A
B
C
D


text:
Hello friends,
I am studying in 10th class. Actually I have a question and I’m unable to solve this question. My question is: How can we find the square root of a number by hand? How about cube roots? If anybody can solve my question I will grateful. Thanks in advance!

2. Originally Posted by daniel_dan
Hello friends,
I am studying in 10th class. Actually I have a question and I’m unable to solve this question. My question is: How can we find the square root of a number by hand?
I am going to give an an algorithmic approach. Meaning a step-by-step program to do by hand. Each time you do a step the better you get to the square root.

Let $x$ be some positive number. And we want to find $\sqrt{x}$. Let $a$ be your guess. Meaning what you think the square root it. You can be totally off, it will still work. But the better you guess the faster it will work.

Find $\frac{a^2 + x }{2a}$.
Call this $b$.

Now pretend that $b$ is your guess. (Which is now better).
Find,
$\frac{b^2+x}{2b}$.
Call this $c$.

Now this is $c$ is better so find $\frac{c^2+x}{2c}$. And keep on going .... The more you do it the closer you get.

For example, you want to find $\sqrt{2}$.

Make a guess $a=1$.

That means $b = \frac{1^2 + 2}{2(1)} =1.5$.

That means $c=\frac{1.5^2 + 2}{2(1.5)} = 1.41$.

You can see you start getting better and better.


text:
NEW SUMMATION FORMULAE FOR THE GENERALIZED HYPERGEOMETRIC FUNCTIONS OF HIGH ORDER

• Journal title : Honam Mathematical Journal
• Volume 34, Issue 3,  2012, pp.467-472
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.3.467
Title & Authors
NEW SUMMATION FORMULAE FOR THE GENERALIZED HYPERGEOMETRIC FUNCTIONS OF HIGH ORDER
Lee, Seung-Woo; Rathie, Arjun K.; Pandey, Ujjawal; Kim, Yong-Sup;

Abstract
The aim of this paper is to provide two interesting summation formulae with the argument unity for the generalized hypergeometric function of higher order. The results are obtained with the help of two new summation formulae very recently obtained by Kim et al.. Summation formulae obtained earlier by Carlson and re-derived by Exton turn out to be special cases of our main findings.
Keywords
Hypergeometric $\small{_5F_4[1]}$, $\small{_3F_2[{\pm}1]}$;summation formula;
Language
English
Cited by
References
1.
B.C. Carlson, Some extensions of Lardner's relations between $_oF_{3}$ and Bessel functions, SIAM J. Math. Anal. 1(1970), No. 2, 232-243.

2.
H. Exton, Multiple Hypergeometric Functions, Ellis Horwood, Chichester, UK, 1996.

3.
H. Exton, Some summation formulae for the generalized hypergeometric function of higher order, J. Comput. Appl. Math. 79 (1997), No. 2, 183-187.

4.
Y.S. Kim, M.A. Rakha and A.K. Rathie, Extensions of certain classical summation theorems for the series $_2F_{1}$, $_3F_{2}$ and $_4F_{3}$ with applications in Ramanujan's summations, Int. J. Math. Math. Sci. Volume 2010, Article ID 309503, 26 pages, doi: 10.1155/2010/309503, Hindawi Publishing Company.

5.
J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Watson's theorem on the sum of a $_3F_{2}$, Indian J. Math. 32(1992) 23-32.

6.
J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Dixon's theorem on the sum of a $_3F_{2}$, Math. Comp. 63(1994) 367-376.

7.
J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Whipple's theorem on the sum of a $_3F_{2}$, J. Comput. Appl. Math. 72(1996) 293-300.

8.
E.D. Rainville, Special functions, The Macmillan Company, New York, 1960.

9.
L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.

10.
H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

11.
R. Tremblay and B.J. Fugere, Products of two restricted hypergeometric functions, J. Math. Anal. Appl. 198(1996), 844-852.


text:
_____ condition is not considered for the similarity of triangle.
• A
$SAS$
• B
$SSS$
• C
$AAA$
• D
$ASA$

#### Solution

$SAS$ is a congruency condition not a smililarity condition of triangle
So option $A$ is correct.


text:
College Physics for AP® Courses

# Glossary

angular momentum quantum number
a quantum number associated with the angular momentum of electrons
atom
basic unit of matter, which consists of a central, positively charged nucleus surrounded by negatively charged electrons
atomic de-excitation
process by which an atom transfers from an excited electronic state back to the ground state electronic configuration; often occurs by emission of a photon
atomic excitation
a state in which an atom or ion acquires the necessary energy to promote one or more of its electrons to electronic states higher in energy than their ground state
atomic number
the number of protons in the nucleus of an atom
the mean radius of the orbit of an electron around the nucleus of a hydrogen atom in its ground state
Brownian motion
the continuous random movement of particles of matter suspended in a liquid or gas
cathode-ray tube
a vacuum tube containing a source of electrons and a screen to view images
double-slit interference
an experiment in which waves or particles from a single source impinge upon two slits so that the resulting interference pattern may be observed
energies of hydrogen-like atoms
Bohr formula for energies of electron states in hydrogen-like atoms: $E n = − Z 2 n 2 E 0 ( n = 1, 2, 3, … ) E n = − Z 2 n 2 E 0 ( n = 1, 2, 3, … )$
energy-level diagram
a diagram used to analyze the energy level of electrons in the orbits of an atom
fine structure
the splitting of spectral lines of the hydrogen spectrum when the spectral lines are examined at very high resolution
fluorescence
any process in which an atom or molecule, excited by a photon of a given energy, de-excites by emission of a lower-energy photon
hologram
means entire picture (from the Greek word holo, as in holistic), because the image produced is three dimensional
holography
the process of producing holograms
hydrogen spectrum wavelengths
the wavelengths of visible light from hydrogen; can be calculated by $1λ=R1nf2−1ni21λ=R1nf2−1ni2 size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right )} {}$
hydrogen-like atom
any atom with only a single electron
intrinsic magnetic field
the magnetic field generated due to the intrinsic spin of electrons
intrinsic spin
the internal or intrinsic angular momentum of electrons
laser
acronym for light amplification by stimulated emission of radiation
magnitude of the intrinsic (internal) spin angular momentum
given by $S=ss+1h2πS=ss+1h2π$
metastable
a state whose lifetime is an order of magnitude longer than the most short-lived states
orbital angular momentum
an angular momentum that corresponds to the quantum analog of classical angular momentum
orbital magnetic field
the magnetic field generated due to the orbital motion of electrons
Pauli exclusion principle
a principle that states that no two electrons can have the same set of quantum numbers; that is, no two electrons can be in the same state
phosphorescence
the de-excitation of a metastable state
planetary model of the atom
the most familiar model or illustration of the structure of the atom
population inversion
the condition in which the majority of atoms in a sample are in a metastable state
quantum numbers
the values of quantized entities, such as energy and angular momentum
Rydberg constant
a physical constant related to the atomic spectra with an established value of $1.097 × 10 7 m −1 1.097 × 10 7 m −1$
shell
a probability cloud for electrons that has a single principal quantum number
space quantization
the fact that the orbital angular momentum can have only certain directions
spin projection quantum number
quantum number that can be used to calculate the intrinsic electron angular momentum along the $z z$-axis
spin quantum number
the quantum number that parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle
stimulated emission
emission by atom or molecule in which an excited state is stimulated to decay, most readily caused by a photon of the same energy that is necessary to excite the state
subshell
the probability cloud for electrons that has a single angular momentum quantum number $l l$
x rays
a form of electromagnetic radiation
x-ray diffraction
a technique that provides the detailed information about crystallographic structure of natural and manufactured materials
z-component of spin angular momentum
component of intrinsic electron spin along the $z z$-axis
z-component of the angular momentum
component of orbital angular momentum of electron along the $z z$-axis
Zeeman effect
the effect of external magnetic fields on spectral lines
Order a print copy

As an Amazon Associate we earn from qualifying purchases.


text:
### Bloodborne Pathogens

Length of Course:
30-60 Minutes

Course Description:

This course is designed for the general public and follows the Occupational Safety and Health Administration (OSHA) recommendations on Bloodborne Pathogens in the workplace and in general. The student will be issued a certificate upon completion of the course.

Program Designer:

Ruben Major, EMT-Paramedic, M.A.

Course Cost:

Course Outline:

### Basic First Aid: Adult, Pediatric & Infant Course Description

Length of Course:

If Certified Before - 30 to 60 Minutes

If Never Certified Before - 60 to 120 Minutes

Course Description:

This course is designed for the general public and follows the 2010 International Liaison Committee on Resuscitation (ILCOR) Guidelines for Basic First Aid. The student will be issued a certificate upon completion of the course.

Course Cost:


text:
# computing Impedance related with Voltage Vx

I have a circuit like:

How do you get Thevenin ( $V_x = I_x * R$) circuit?

I have these formulas, but I am a little stucked...

$\ I_0 + Ix = (\frac{V\pi}{r\pi}) + gmV\pi$ --- 1
$VR_1 - VR_2 = Vx$ --- 2
$\ I_0 = (\frac{VR_2}{R_2})$ --- 3

How do you get $V_{R_1}$?

-
Please phrase your question more clearly and provide more information about the problem. You are looking for the thevenin equivalent circuit for that entire thing? Or just for the input source (looks like input source might be Vx & R1 & R4 & R3, is that right?) This looks a lot like a hybrid pi BJT small signal model with some other resistor stuff around it... is that right? –  Adam P Apr 9 '11 at 16:58
sounds like a homework question? –  vicatcu Apr 10 '11 at 3:58
maybe exam question –  cMinor Apr 12 '11 at 22:03
@vicatcu, we have agreed in the past to handle homework problems. Now, it is important that no one gives the solution, but just explanation on how to approach the problem and how circuits are approached. –  Kortuk May 15 '11 at 4:04
what's the deal with using the 'earth ground' symbol three times across the bottom, and then using a 'chassis ground' looking thing in the upper right? –  JustJeff Jun 12 '11 at 1:59

I taught this at first year level for a few years, and here's the method I found students had the most success with:

1. Identify a part of the circuit where two passive circuit elements can be combined.
2. Combine those elements, calculating the new value.
3. Redraw the circuit. If you're learning how to do this, don't leave that step out! It's valuable for many reasons, not least that it helps the marker follow your reasoning.
4. Repeat from 1. until you have simplified the circuit as much as possible.
5. Perform a source transformation, where you convert a voltage source in series with a resistor into a current source in parallel with a resistor (or vice versa).
6. Redraw the circuit with the new values.
7. Repeat from 1. again.

Eventually you'll get down to one voltage source and a resistor in series with it - the Thevenin equivalent circuit. If you end up with a current source in parallel with a resistor you've got the Norton equivalent - perform a source transform to swap between them.

It also helps to draw a circuit diagram that is equivalent to the one you're given, but arranged to make the problem easier. I've had a stab at that in this image:

-
$R1\|R2\|R3 = \frac{1}{\frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3}}$


text:
# Predicate logic: There is no largest prime number.

I have come up with the following expression for "There is no largest prime number."

$$\neg \left( \exists q. \left( Prime(q) \wedge \left( \forall p. Prime(p) \wedge q \ge p \right) \right) \right) \tag 1$$

However a solution given is:

$$\neg \left( \exists q. \left( Prime(q) \wedge \left( \forall p. Prime(p) \implies q \ge p \right) \right) \right) \tag 2$$

The only difference is that there is an implication used instead of an conjunction.

Why is (1) wrong? How do the truth tables for (1) and (2) differ and why?

$$(\forall p. Prime(p) \land q \ge p )$$ is saying that every natural number (since the universal quantifier quantifies over your whole domain ... which is the natural numbers) is both a prime and smaller or equal to $$q$$. Well, just focusing on that first part: clearly it is not true that every natural number is prime. So with the conjunction instead of a conditional, that part ends up makes the rather trivially false statement that every number is a prime ... clearly not what you want. What you want is that if you have some prime $$p$$, then that must be smaller or equal than the largest prime $$q$$.

Now, in predicate logic, you can't really use truth-tables anymore, because you start dealing with infinitely many different kinds of worlds (worlds in which there is 1 object, worlds in which there are 2, etc.)

However, we don't have to resort to a truth-table to show that these statements can take on different truth-values and are therefore not equivalent. We just need to create some interpretation of the predicates that makes one true and the other false. Now, as to not be distracted by the predicates you are using, let's first abstract the 2 statements to:

$$\neg \exists q (P(q) \land \forall p (P(p) \land R(p,q))$$

$$\neg \exists q (P(q) \to \forall p (P(p) \land R(p,q))$$

In other words, we have some 1-place predicate $$P$$, and some 2-place predicate $$R$$ ... left to be interpreted any way we want.

OK, now ... let's consider a world where no objects have property $$P$$. Then clearly $$P(q) \land \forall p (P(p) \land R(p,q)$$ is false, and hence it is true that $$\neg \exists q (P(q) \land \forall p (P(p) \land R(p,q))$$. On the other hand, $$P(q) \to \forall p (P(p) \land R(p,q)$$ is vacuously true, and hence is is false that $$\neg \exists q (P(q) \to \forall p (P(p) \land R(p,q))$$. OK, so there is your counterexample to the claim of their equivalence

• What do you mean by "world" and "different kinds of worlds"? (I haven't heard the word "world" in logic since I'm a complete beginner.) – FizzleDizzle May 29 at 13:59
• @FizzleDizzle In logic you are dealing with statements. Those statements are true or false depending on the situation. For example, the statement "It rains" would be true when it is raining, but false when it is not raining. Or, as another example: "The sum of the angles in a triangle equals the sum of two right angles" is true in Eucledian geometry, but false in non-Eucledian geometry. This is why we say that the truth of a claim depends on what world we evaluate it in. (cont'd) – Bram28 May 29 at 14:53
• @FizzleDizzle Indeed, any "logically possible" world is a world without contradicitons. So, I could consider worlds where pigs fly, where grass is purple, etc. The rows in a truth-table show these different kinds of worlds: A row where $P$ and $Q$ are both true represents the world (or rather, all kinds of worlds) where $P$ and $Q$ are both true. etc. And now we can also say things like: A statement is a tautology if and only if it is true in all possible worlds ... i.e. it is true under all circumstances/in all situations/no matter what – Bram28 May 29 at 14:56
• @FizzleDizzle P.s. For an absolute beginner you're doing very well on these symbolizations! – Bram28 May 29 at 15:03


text:
zbMATH — the first resource for mathematics

The multiplicative group of a field and hyperidentities. (English. Russian original) Zbl 0705.12002
Math. USSR, Izv. 35, No. 2, 377-391 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1040-1055 (1989).
This paper gives the following characterization of the multiplicative group of a field (Theorem 2): A monoid may be the multiplicative group of a field if and only if there is a regular binary representation of this monoid satisfying the hyperidentitiy of idempotence $$X(x,x)=x$$. A faithful binary representation of a monoid $$M$$ over a set $$Q$$ is an injective homomorphism from $$M$$ to the set of binary operations on $$Q$$ with Mann composition, i.e. $$(A\cdot B)(x,y)=A(x,B(x,y))$$, and such a representation is called regular iff it is right invertible, $$i$$-transitive for $$i=2,3$$ and satisfies the left distributivity hyperidentity. For instance if $$G$$ is the multiplicative group of a field $$Q$$ the corresponding binary representation is given by $$g(x,y)=(1-g)x+gy$$ for $$g\in G$$, and $$x,y\in Q$$.

MSC:
 12E99 General field theory 12L99 Connections between field theory and logic 20M30 Representation of semigroups; actions of semigroups on sets 08B99 Varieties
Full Text:


text:
# Negative numbers for ionic radii?

I was looking for a good table of ionic radii for an experiment on the van't-hoff factor when I stumbled upon R. D. Shannon, Acta Cryst. 1976, A32, 751.

In table 1, it lists 1 of the ions of carbon with as so:

C +4 1S 2 III .06 -.08

The bold numbers are of concern here. The first number, 0.06 Å is what the paper refers to as the crystal radius, while the second number, -0.08 Å, is referred to as the effective ionic radius. I was also unable to find any good distinction between the two in the paper, except that the radius of $\ce{O^2+}$ for the crystal radius is 1.40 Å, while the effective ionic radius for $\ce{O^2+}$ is 1.26 Å.

I also doubt that this is a typo since it agrees with this statement made later in the paper:

• Interestingly $\ce{H+}$ is listed as having both a negative crystal radius as well as a negative effective ionic radius. Only light, highly positively charged ions appear to give negative values ($\ce{H+}$, $\ce{D+}$, $\ce{C^{4+}}$, $\ce{N^{5+}}$). It is likely that these values are calculated in a way which cannot handle the extremely high demand for at least some covalency from these strongly polarising cations, so the model breaks down. – Nicolau Saker Neto Feb 10 '16 at 10:46
• Though I'm not 100% sure about this, the paper seemed to not be a model, but instead was directly measured. I couldn't find a real model anywhere so I can only assume that it must have been directly measured. These values were also not discussed anywhere in the paper as far as I can tell. – purepani Feb 10 '16 at 13:36
• Also, that about negatively charged ions: why are they not affected? – purepani Feb 10 '16 at 14:15
• Is it possible that the radii were measured in referential unit like amu? – Agyey Arya Feb 10 '16 at 16:42
• I'm going to take a much closer look at the paper and then come back to this question. – purepani Feb 11 '16 at 0:31

Ionic radii are not measured on individual ions by some sort of mini-micrometer, but bond distances can be measured and then radii are assigned to the individual atoms/ions. The fuzziness of the assignments can be seen by the variation in ionic radii as the coordination number changes: if you give the ion more room (greater coordination number), it will fill the space - if you assume that the ligands have the same size as usual (whatever that is).

(There are materials (ferroelectrics) in which the coordinated ion does not completely fill the space allotted. In $$\ce{BaTiO3}$$, the $$\ce{Ti^{+4}}$$ ion is so small that it can shift its position in the octahedral cage of oxygen anions in response to an external electric field.)

The data in the Shannon paper are extensive, certainly a credit to the Central Research Department of DuPont in 1976, but the trends reported seem to be much more valuable than any specific datum.

My approach to handling a negative radius is to conclude that the proton (or other mini- cation) is so electrostatically demanding that it can begin to bury itself in the outer shell of the anionic ligand, if the coordination number allows such a deviation in distance to be measured. And since we don't want to redefine the anionic radius, a negative radius is assigned to the proton, and the mathematics works out OK.

I guess you could say atoms and ions are somewhat mushy, and mathematics can deal with it.