text:
# Georeferencing

Georeferencing or georegistration is a type of coordinate transformation that binds a digital raster image or vector database that represents a geographic space (usually a scanned map or aerial photograph) to a spatial reference system, thus locating the digital data in the real world.[1][2] It is thus the geographic form of image registration. The term can refer to the mathematical formulas used to perform the transformation, the metadata stored alongside or within the image file to specify the transformation, or the process of manually or automatically aligning the image to the real world to create such metadata. The most common result is that the image can be visually and analytically integrated with other geographic data in geographic information systems and remote sensing software.

A number of mathematical methods are available, but the process typically involves identifying several sample ground control points with known locations on the image and the ground, then using curve fitting techniques to generate a parametric (or piecewise parametric) formula to transform the rest of the image.[3] Once the parameters of the formula are stored, the image may be transformed dynamically at drawing time, or resampled to generate a georeferenced raster GIS file or orthophoto.

The term georeferencing has also been used to refer to other types of transformation from general expressions of geographic location (geocodes) to coordinate measurements,[4] but most of these other methods are more commonly called geocoding. Because of this ambiguity, Georegistration is preferred by some to refer to the image transformation.[5]: 141–143  Occasionally, this process has been called rubbersheeting, but that term is more commonly applied to a very similar process applied to vector GIS data.[5]: 240

## Motivation

• Georeferencing is crucial to make aerial and satellite imagery, usually raster images, useful for mapping as it explains how other data, such as the above GPS points, relate to the imagery.
• Very essential information may be contained in data or images that were produced at a different point of time. It may be desired either to combine or compare this data with that currently available. The latter can be used to analyze the changes in the features under study over a period of time.
• Different maps may use different projection systems. Georeferencing tools contain methods to combine and overlay these maps with minimum distortion.

## Mathematics

Graphical view of the affine transformation.

The registration of an image to a geographic space is essentially the transformation from an input coordinate system (the inherent coordinates of pixels in the images based on row and column number) to an output coordinate system, a spatial reference system of the user's choice, such as the geographic coordinate system or a particular Universal Transverse Mercator zone. It is thus the extension of the typical task of curve fitting a relationship between two variables to four dimensions. The goal is to have a pair of functions of the form:

${\displaystyle x_{out}=F(x_{in},y_{in})}$
${\displaystyle y_{out}=G(x_{in},y_{in})}$

Such that for every pixel in the image (${\displaystyle x_{in},y_{in))$ being its column and row number, respectively), a corresponding real-world coordinate can be calculated.

Several types of functions are available in most GIS and remote sensing software for georeferencing.[6] As the simplest type of two-dimensional curve is a straight line, so the simplest form of coordinate transformation is a linear transformation, the most common type being the affine transformation:[7]: 171

${\displaystyle x_{out}=Ax_{in}+By_{in}+C}$
${\displaystyle y_{out}=Dx_{in}+Ey_{in}+F}$

Where A-F are constant coefficients set for the entire image. These formulas allow an image to be moved (the C and F coefficients specify the desired location of the top left corner of the image), scaled (without rotation, the A and E coefficients specify the size of each cell or spatial resolution), and rotated.[8]: 115  In the last case, if the cell size is r in both the x and y directions, and the image is to be rotated α degrees counter-clockwise, then ${\displaystyle A=E=r\cos(\alpha ),B=D=r\sin(\alpha )}$. The world file developed by Esri is a commonly used sidecar file that specifies these six coefficients for image georeferencing.

Higher order polynomial transformations are also commonly used. For example, a Second-order polynomial transformation would be:

${\displaystyle x_{out}=Ax_{in}+By_{in}+Cx_{in}^{2}+Dy_{in}^{2}+Ex_{in}y_{in}+F}$
${\displaystyle y_{out}=Gx_{in}+Hy_{in}+Ix_{in}^{2}+Jy_{in}^{2}+Kx_{in}y_{in}+L}$

The second-order terms (and third-order terms in a third-order polynomial) allow for the variable warping of the image, which is especially useful for removing the inherent distortion in aerial photographs.

In addition to global parametric formulas, piecewise formulas can also be used, which transform different parts of the image in different ways. A common example is a Thin plate spline transformation.[9]

## The GCP method

It is very rare that a user would specify the parameters for the transformation directly. Instead, most GIS and remote sensing software provides an interactive environment for visually aligning the image to the destination coordinate system. The most common method for doing this is to create a series of ground control points (GCP).[7]: 170  A ground control point is a location that can be identified on both the image and the ground, so that it has precise coordinates in both the image coordinate system (${\displaystyle x_{in))$ = pixel column, ${\displaystyle y_{in))$ = pixel row) and the ground coordinate system (${\displaystyle x_{out},y_{out))$). Easily visible locations that be precisely located are preferred as GCP's, such as a road intersection or the corner of a building. When very high accuracy registration is required, it is common to place or paint high-contrast markers on the ground at survey control monuments before the photography is taken, and use GNSS-measured coordinates for the output. In most software, these are entered by pointing at the location on the image, then pointing at the same location on a vector base map or orthophoto that is already in the desired coordinate system. This can then be moved and adjusted to improve accuracy.

With a minimal set of GCPs, the known coordinates can be entered into the mathematical equations for the desired type of transformation, which can then be solved using linear algebra to determine the coefficients and derive the formulas to use for the entire grid.[8]: 116  For example, the linear affine transformation above has six unknown coefficients, so six equations with known <${\displaystyle x_{in},y_{in},x_{out},y_{out))$> are needed to derive them, which will require three ground control points.[7]: 171  The second-order polynomial requires a minimum of six ground control points, and so on.

The entered GCPs are rarely perfectly located and are even more rarely perfectly representative of the distortion in the rest of the image, but the algebraic solution, which appears to be a perfect match, masks any error. To avoid this, it is common to create many more than the minimal required set (creating an overdetermined system), and use least squares regression to derive a set of function parameters that most closely matches the points.[8]: 116  This is almost never a perfect match, so the variance between each GCP location and the location predicted by the functions can be measured and summarized as a Root-mean-square error (RMSE). A lower RMSE thus means that the transformation formulas closely match the GCPs.

Once the function parameters are determined, the transformation functions can be used to transform every pixel of the image to its real-world location. Two options are usually available for making this transformation permanent. One option is to save the parameters themselves as a form of metadata, either in the header of the image file itself (e.g., GeoTIFF), or in a sidecar file stored alongside the image file (e.g., a world file). With this metadata, the software can perform the transformation dynamically as it displays the image, so that it appears to align with other data in the desired coordinate system. The alternative method is rectification, in which the image is resampled to create a new raster grid that is natively tied to the coordinate system. Rectification was traditionally the only option, until the computing power became available for the intense calculations of dynamic coordinate transformations; even now, drawing and analysis performance is better with a rectified image.

## References

1. ^ "What does "georeferenced" mean?". www.usgs.gov. U.S. Geological Survey. Retrieved 2022-01-04.
2. ^ Yao, Xiaobai A. (2020-01-01), "Georeferencing and Geocoding", in Kobayashi, Audrey (ed.), International Encyclopedia of Human Geography (Second Edition), Oxford: Elsevier, pp. 111–117, doi:10.1016/b978-0-08-102295-5.10548-7, ISBN 978-0-08-102296-2, S2CID 241797395, retrieved 2022-01-04
3. ^ Hackeloeer, A.; Klasing, K.; Krisp, J.M.; Meng, L. (2014). "Georeferencing: a review of methods and applications". Annals of GIS. 20 (1): 61–69. doi:10.1080/19475683.2013.868826. S2CID 38306705.
4. ^ Leidner, J.L. (2017). "Georeferencing: From Texts to Maps". International Encyclopedia of Geography: People, the Earth, Environment and Technology. vi: 2897–2907. doi:10.1002/9781118786352.wbieg0160. ISBN 9780470659632.
5. ^ a b Longley, Paul A.; Goodchild, Michael F.; Maguire, David J.; Rhind, David W. (2011). Geographic Information Systems & Science (3rd ed.). Wiley.
6. ^ a b "Overview of georeferencing". ArcGIS Pro Documentation. Esri. Retrieved 8 January 2023.
7. ^ a b c Bolstad, Paul (2019). GIS Fundamentals: A First Text on Geographic Information Systems. Ann Arbor, MI: XanEdu. ISBN 978-1-59399-552-2.
8. ^ a b c Chang, Kang-tsung (2014). Introduction to Geographic Information Systems (7th ed.). McGraw-Hill. pp. 50–57. ISBN 978-0-07-352290-6.
9. ^ a b "16.3 Georeferencer". QGIS 3.22 documentation. OSGEO. Retrieved 8 January 2023.

Georeferencing

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# A joker’s cap is in the form of a cone of radius 7 cm and height 24 cm. Find the area of the cardboard required to make the cap.

Last updated date: 20th Mar 2023
Total views: 307.5k
Views today: 4.86k
Given, radius of the cone-shaped joker’s cap $r = 7{\text{ cm}}$
Height of the cone-shaped joker’s cap $h = 24{\text{ cm}}$
As we know that the curved surface area of the cone with radius as $r$ and height as $h$ is given by $\pi rl$ where slant height is $l = \sqrt {{r^2} + {h^2}}$
$\Rightarrow$Area of the cardboard required to make the cap$=$Curved surface area of the cap$= \pi rl = \pi r\sqrt {{r^2} + {h^2}} = \dfrac{{22}}{7} \times 7 \times \sqrt {{7^2} + {{24}^2}} = 22 \times 25 = 550{\text{ c}}{{\text{m}}^2}$

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# Calculating increase in percentage

In this chapter, we will learn to calculate the increase in value using the percentage.

Here we will discuss the method to calculate the increase in percentage value along with examples.

At the end of the chapter, some problems are also given for your practice.

## Find increase in value using percentage

We will understand the concept with the helps of example.

Let the price of chips increased from 5$to 7$.

The percentage increase in price is calculated by using following steps;

(a) Calculate the increase in value.

⟹ $7 –$5

⟹ $2 Here, the price has been increased by 2$.

(b) Apply percentage increase formula;

\mathtt{Percentage\ increase\ =\frac{Increase\ in\ Value\ }{Original\ Value} \times 100}

Putting the values, we get;

\mathtt{Percentage\ increase\ }\\\ \\ \mathtt{\Longrightarrow \ \ \frac{2\$}{5\$} \times 100}\\\ \\ \mathtt{\Longrightarrow \ 40\%}

The price of chips has increased by 40%

Hence, using the above two steps, you can calculate the percentage increase in any given value.

Note:
The numerator and denominator in above formula should be in same format. For example, if numerator value is in dollars then denominator value should also be in dollars otherwise we will get faulty results.

Given below are some solved examples for further clarity.

### Percent increase – Solved examples

Example 01
Susan weight increased from 60 kg to 75 kg. Calculate the % increase in weight.

Solution
Increase in weight = 75 – 60 = 15

\mathtt{\%\ increase\ \ =\ \ \frac{15}{60} \times 100}\\\ \\ \mathtt{\%\ increase\ =\ 25\%}

Hence, Susan’s weight has been increased by 25%.

Example 02
The population of town in USA increased from 50,000 to 52,000. Calculate the percentage increase in population.

Solution
Original population = 50,000
New population = 52,000

Increase in population = 52,000 – 50,000 = 2000

\mathtt{\%\ increase\ =\frac{Increase\ in\ population\ }{Original\ population} \times 100}\\\ \\ \mathtt{\%\ increase\ \ =\ \ \frac{2000}{50000} \times 100}\\\ \\ \mathtt{\%\ increase\ =\ 4\%}

Hence, population of town is increased by 4%

Example 03
The price of Honda Accord is $23,000. After two years, the company increased the price to$25000. Calculate the percent increase in price.

Solution
Original Price = $23,000 Final Price =$ 25,000

Increase in Price = 25000- 23000 = $2000 The percentage increase is calculated by using following formula; \mathtt{\%\ increase\ =\frac{Increase\ in\ price\ }{Original\ price} \times 100}\\\ \\ \mathtt{\%\ increase\ \ =\ \ \frac{2000}{23000} \times 100}\\\ \\ \mathtt{\%\ increase\ =\ 8.69\%} Hence, the price of the car is increased by 8.69% Example 04 The height of a plant is 4 meters. After six months, the height is now increased to 7 meters. Calculate the % increase in height. Solution Original height = 4 meter New height = 7 meter Increase in height = 7 – 4 = 3 meter. To calculate the percentage increase, do the following calculation; \mathtt{\%\ increase\ =\frac{Increase\ in\ height\ }{Original\ height} \times 100}\\\ \\ \mathtt{\%\ increase\ \ =\ \ \frac{3}{4} \times 100}\\\ \\ \mathtt{\%\ increase\ =\ 75\%} Hence, height of plant is increased by 75%. Example 05 The price of candy is 10$. The company has decided to increase price by 40%. Calculate the final price.

Solution
Initial Price = 10$Price increased by 40%. Let the final price be x$

Use the following formula to calculate the final price.

\mathtt{\%\ increase\ =\frac{Final\ Price-\ Initial\ Price\ }{Initial\ Price} \times 100}\\\ \\ \mathtt{40\ =\ \ \frac{x-10}{10} \times 100}\\\ \\ \mathtt{\frac{40\times 10}{100} =\ x-10}\\\ \\ \mathtt{4\ =\ x-10}\\\ \\ \mathtt{x\ =\ 14}

Hence, the final price of candy is 14\$

I hope you understood the above examples. Given below are some problems for your practice.

## Percent increase – Solved problems

(01) Calculate the percent increase of below questions.

(a) Original number : 23
Final number : 50

(b) Original number : 100
Final number : 106

(c) Original number : 21
Final number : 25

(d) Original number : 80
Final number 100

(e) Original number 10
Final number 90

Solution

(a) Original number : 23
Final number : 50

\mathtt{\%\ increase\ =\ \ \frac{50-23}{23} \times 100}\\\ \\ \mathtt{\%\ increase=\ \frac{27}{23} \times 100}\\\ \\ \mathtt{\%\ increase=\ 117.4\%\ }

The number has been increased by 117.4%

(b) Original number : 100
Final number : 106

\mathtt{\%\ increase\ =\ \ \frac{106-100}{100} \times 100}\\\ \\ \mathtt{\%\ increase=\ \frac{6}{100} \times 100}\\\ \\ \mathtt{\%\ increase=\ 6\%\ }

Hence, the number has been increased by 6%

(c) Original number : 21
Final number : 25

\mathtt{\%\ increase\ =\ \ \frac{25-21}{21} \times 100}\\\ \\ \mathtt{\%\ increase=\ \frac{4}{21} \times 100}\\\ \\ \mathtt{\%\ increase=\ 19.04\%\ }

Hence, number has been increased by 19.04%

(d) Original number : 80
Final number 100

\mathtt{\%\ increase\ =\ \ \frac{100-80}{80} \times 100}\\\ \\ \mathtt{\%\ increase=\ \frac{20}{80} \times 100}\\\ \\ \mathtt{\%\ increase=\ 25\%\ }\

Hence, number has been increased by 25%.

(e) Original number 10
Final number 90

\mathtt{\%\ increase\ =\ \ \frac{90-10}{10} \times 100}\\\ \\ \mathtt{\%\ increase=\ \frac{80}{10} \times 100}\\\ \\ \mathtt{\%\ increase=\ 800\%\ }

Hence, the number has been increased by 800%

(02) The weight of a man increased from 80 kg to 85 kg. Find the % increase in weight.

Solution
Original weight = 80 Kg
Final weight = 85 Kg

Increased weight = 85 – 80 = 5 kg

The percentage increase is calculated by following formula;

\mathtt{\%\ increase\ =\frac{Final\ -Initial\ weight}{Initial\ weight} \times 100}\\\ \\ \mathtt{\%\ increase\ =\ \ \frac{5}{80} \times 100}\\\ \\ \mathtt{\%\ increase=\ 6.25\%}

The weight has been increased by 6.25%


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# Bond Convexity

Convexity of a bond is the phenomena that causes the increase in bond price due to a decrease in interest rates to be higher than the decrease in bond price owing to an increase in interest rates. It represents the change in duration that occurs due to change in bond yield. High convexity means higher sensitivity of bond price to interest rate changes.

A bond’ intrinsic value equals the present value of its future cash flows i.e. coupon payments and maturity value. The following equation mathematically expresses the relationship:

$$\text{P}=\frac{\text{c}}{\text{m}}\times \text{F}\times\frac{\text{1}-{(\text{1}+\frac{\text{r}}{\text{m}})}^{-\text{n}\times \text{m}}}{\frac{\text{r}}{\text{m}}}+\frac{\text{F}}{{(\text{1}+\frac{\text{r}}{\text{m}})}^{\text{n}\times \text{m}}}$$

Where P is the current price of the bond, c is the annual coupon rate, F is the face value of the bond, r is the annual yield on the bond, m is the number of coupon payments per year and n is the total years to maturity.

It follows from the above equation that the bond price P falls with increase in the market interest rate r and vice versa.

Let’s say you have a 10-year $1,000 face value bond paying 5% coupon rate twice a year. If the market interest rate is equal to the coupon rate i.e. 5%, the bond price exactly equals the face value as illustrated below: $$\text{P} _ \text{0}=\frac{\text{5%}}{\text{2}}\times\text{\1,000}\times\frac{\text{1}-{(\text{1}+\frac{\text{5%}}{\text{2}})}^{-\text{10}\times\text{2}}}{\frac{\text{5%}}{\text{2}}}+\frac{\text{\1,000}}{{(\text{1}+\frac{\text{5%}}{\text{2}})}^{\text{10}\times\text{2}}}=\text{\1,000}$$ If the market interest rates fall to 4.8%, the new bond price will be$1,015.74 representing a 1.57% change as shown below:

$$\text{P} _ \text{d}=\frac{\text{4.8%}}{\text{2}}\times\text{\1,000}\times\frac{\text{1}-{(\text{1}+\frac{\text{4.8%}}{\text{2}})}^{-\text{10}\times\text{2}}}{\frac{\text{4.8%}}{\text{2}}}+\frac{\text{\1,000}}{{(\text{1}+\frac{\text{4.8%}}{\text{2}})}^{\text{10}\times\text{2}}}=\text{\1,015.74}$$

However, if the interest rates increase by the same magnitude i.e. 0.2% to 5.2%, the bond price drops to $984.56 representing a decline of 1.54% from face value: $$\text{P} _ \text{i}=\frac{\text{5.2%}}{\text{2}}\times\text{\1,000}\times\frac{\text{1}-{(\text{1}+\frac{\text{5.2%}}{\text{2}})}^{-\text{10}\times\text{2}}}{\frac{\text{5.2%}}{\text{2}}}+\frac{\text{\1,000}}{{(\text{1}+\frac{\text{5.2%}}{\text{2}})}^{\text{10}\times\text{2}}}=\text{\984.56}$$ The following table shows the bond price at different market interest rates: Market Interest Rate Bond Price 0%$1,500
1% $1,380 2%$1,271
3% $1,172 4%$1,082
5% $1,000 6%$926
7% $858 8%$796
9% $740 10%$688
11% $641 12%$599
13% $559 14%$523
15% $490 16%$460
17% $432 18%$407
19% $383 20%$361

If we plot the bond price against the market interest rates, we get the following graph:

As you can see, the graph is curved which shows that the rate of change in price is different at different points on the graph. This is called convexity.

## Formula

The general formula for convexity is as follows:

$$\text{Convexity}=\frac{\text{1}}{\text{P}\times{(\text{1}+\text{y})}^\text{2}}\times\sum _ {\text{t}=\text{1}}^{\text{n}}\frac{{\rm \text{CF}} _ \text{n}\times \text{t}\times(\text{1}+\text{t})}{{(\text{1}+\text{y})}^\text{n}}$$

Where P is the bond price, y is the yield, CFn is the nth cash flow of the bond, t is the time difference between time 0 and the cash flow.

The effective convexity can be calculated using the following formula:

$$\text{Effective Convexity}=\frac{\text{P} _ \text{i}+\text{P} _ \text{d}-\text{2P} _ \text{0}}{\text{2}\times \text{P} _ \text{0}\times{\text{deltaY}}^\text{2}}$$

Where Pi is the bond price after increase in interest rate, Pd is the bond price after a decrease in interest rate, P0 is the bond price when the yield equals the coupon rate and deltaY is the change in yield.

Percentage change in a bond price can be estimated by a combination of modified duration and the bond convexity as follows:

$$\text{Change in Bond Price}=-\text{D}\times \text{deltaY}+\frac{{\text{deltaY}}^\text{2}}{\text{2}}\times \text{Convexity}$$

## Example

If duration of the bond discussed above is 7.8, the bond price increases to $1,172 after a 2% decrease in bond yield and decreases to$858 after a 2% increase in yield, calculate the change in bond price that will occur after a 1% increase in bond yield.

We first need to calculate the convexity of the bond using the following approximation formula:

$$\text{Effective Convexity}=\frac{\text{\858}+\text{\1,172}-\text{2}\times\text{\1,000}}{\text{2}\times\text{\1,000}\times{\text{0.2%}}^\text{2}}=\text{37.5}$$

We can work out the approximate change in bond price if the interest rates increase by 1% using the following formula:

$$\text{Change in Bond Price}=-\text{7.8}\times\text{1%}+\frac{{\text{1%}}^\text{2}}{\text{2}}\times\text{37.5}=-\text{7.61%}$$

Based on the above percentage, the expected bond price after 1 increase in yield (from 5% to 6%) is $924 (=$1,000 × (1 – 7.61%)). Looking at the table above of the bond prices at different rates, you can see that the bond price equivalent to 6% yield is \$926.


text:
# Singular stochastic Allen–Cahn equations with dynamic boundary conditions

@article{Orrieri2019SingularSA,
title={Singular stochastic Allen–Cahn equations with dynamic boundary conditions},
author={Carlo Orrieri and Luca Scarpa},
journal={Journal of Differential Equations},
year={2019}
}
• Published 12 March 2017
• Mathematics
• Journal of Differential Equations
16 Citations
A stochastic Allen-Cahn-Navier-Stokes system with singular potential
• Mathematics
• 2022
. We investigate a stochastic version of the Allen–Cahn–Navier–Stokes system in a smooth two-or three-dimensional domain with random initial data. The system consists of a Navier–Stokes equation
The Stochastic Viscous Cahn–Hilliard Equation: Well-Posedness, Regularity and Vanishing Viscosity Limit
• Luca Scarpa
• Mathematics
Applied Mathematics & Optimization
• 2020
Well-posedness is proved for the stochastic viscous Cahn–Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have
The Cahn-Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity
• Mathematics
SIAM J. Math. Anal.
• 2022
An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn–Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to
A variational approach to some classes of singular stochastic PDEs
This thesis contains an analysis of certain classes of parabolic stochastic partial differential equations with singular drift and multiplicative Wiener noise. Equations of this type have been
Random separation property for stochastic Allen-Cahn-type equations
• Mathematics
Electronic Journal of Probability
• 2022
We study a large class of stochastic p-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and
The stochastic Cahn–Hilliard equation with degenerate mobility and logarithmic potential
We prove existence of martingale solutions for the stochastic Cahn–Hilliard equation with degenerate mobility and multiplicative Wiener noise. The potential is allowed to be of logarithmic or
Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation
A Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for
Degenerate Kolmogorov equations and ergodicity for the stochastic Allen-Cahn equation with logarithmic potential
• Mathematics
• 2022
Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of

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text:
## Duke Mathematical Journal

### 4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings

#### Abstract

We construct examples of 4-dimensional manifolds $M$ supporting a locally CAT(0)-metric, whose universal covers $\tilde{M}$ satisfy Hruska’s isolated flats condition, and contain $2$-dimensional flats $F$ with the property that $\partial^{\infty}F\cong S^{1}\hookrightarrowS^{3}\cong\partial^{\infty}\tilde{M}$ are nontrivial knots. As a consequence, we obtain that the group $\pi_{1}(M)$ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonpositive sectional curvature. In particular, if $K$ is any compact locally CAT(0)-manifold, then $M\times K$ is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.

#### Article information

Source
Duke Math. J., Volume 161, Number 1 (2012), 1-28.

Dates
First available in Project Euclid: 30 December 2011

https://projecteuclid.org/euclid.dmj/1325264704

Digital Object Identifier
doi:10.1215/00127094-1507259

Mathematical Reviews number (MathSciNet)
MR2872552

Zentralblatt MATH identifier
1237.57015

#### Citation

Davis, M.; Januszkiewicz, T.; Lafont, J.-F. 4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings. Duke Math. J. 161 (2012), no. 1, 1--28. doi:10.1215/00127094-1507259. https://projecteuclid.org/euclid.dmj/1325264704

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# zbMATH — the first resource for mathematics

## Chen, Feixiang

Compute Distance To:
 Author ID: chen.feixiang Published as: Chen, F.; Chen, F. X.; Chen, Fei-xiang; Chen, Feixiang
 Documents Indexed: 54 Publications since 1986
all top 5

#### Co-Authors

 18 single-authored 4 Zuo, Zhanfei 3 Feng, Yuming 2 Leng, Gangsong 2 Liu, Xuefei 2 Wu, Shanhe 2 Xiang, Ruiyin 1 Chen, Xiaochun 1 Huang, Yi 1 Hui, Zhang 1 Li, Benxiu 1 Li, Guangzhou 1 Liu, Jinkui 1 Tu, Zhengwen 1 Wu, Zhongxiang 1 Yang, Wengui 1 Zhang, Hui 1 Zhang, Jianbing 1 Zhou, Zhi-wu
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#### Serials

 5 Journal of Applied Mathematics 4 Italian Journal of Pure and Applied Mathematics 4 Journal of Applied Mathematics & Informatics 3 International Journal of Mathematical Analysis (Ruse) 3 Journal of Mathematical Inequalities 3 Chinese Journal of Mathematics 2 International Mathematical Forum 2 International Journal of Contemporary Mathematical Sciences 2 Applied Mathematical Sciences (Ruse) 2 Journal of Nonlinear Science and Applications 1 Applied Mathematics and Computation 1 Geometriae Dedicata 1 Mathematics in Practice and Theory 1 Abstract and Applied Analysis 1 Journal of Computational Analysis and Applications 1 Journal of Computer Applications 1 General Mathematics Notes 1 Journal of Modern Methods in Numerical Mathematics 1 ISRN Applied Mathematics
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#### Fields

 18 Real functions (26-XX) 14 Operations research, mathematical programming (90-XX) 11 Numerical analysis (65-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Convex and discrete geometry (52-XX) 2 Operator theory (47-XX) 1 Ordinary differential equations (34-XX) 1 Approximations and expansions (41-XX) 1 General topology (54-XX) 1 Computer science (68-XX)

#### Citations contained in zbMATH

22 Publications have been cited 84 times in 74 Documents Cited by Year
Several complementary inequalities to inequalities of Hermite-Hadamard type for $$s$$-convex functions. Zbl 1329.26032
Chen, Feixiang; Wu, Shanhe
2016
A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals. Zbl 1332.26037
Chen, Feixiang
2014
Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals. Zbl 1339.26043
Chen, Feixiang
2016
An edge/face-based smoothed radial point interpolation method for static analysis of structures. Zbl 1403.74101
Feng, S. Z.; Cui, X. Y.; Chen, F.; Liu, S. Z.; Meng, D. Y.
2016
Extensions of the Hermite-Hadamard inequality for harmonically convex functions via fractional integrals. Zbl 1410.26034
Chen, Feixiang
2015
Orlicz-Brunn-Minkowski inequalities for Blaschke-Minkowski homomorphisms. Zbl 1368.52003
Chen, Feixiang; Leng, Gangsong
2017
On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity. Zbl 1298.26020
Chen, Feixiang
2014
Equivalent linearization method for the flutter system of an airfoil with multiple nonlinearities. Zbl 1426.74125
Chen, F. X.; Chen, Y. M.; Liu, J. K.
2012
New inequalities of Hermite-Hadamard type for functions whose first derivatives absolute values are $$s$$-convex. Zbl 1332.26038
Chen, Feixiang; Feng, Yuming
2014
A note on the Hermite-Hadamard inequality for convex functions on the co-ordinates. Zbl 1305.26043
Chen, Feixiang
2014
Refinements on the Hermite-Hadamard inequalities for $$r$$-convex functions. Zbl 1397.26007
Chen, Feixiang; Liu, Xuefei
2013
A note on Hermite-Hadamard inequalities for products of convex functions. Zbl 1397.26004
Chen, Feixiang
2013
Bifurcation analysis of aeroelastic systems with hysteresis by incremental harmonic balance method. Zbl 1308.74042
Liu, J. K.; Chen, F. X.; Chen, Y. M.
2012
Polynomial convergence of primal-dual algorithms for SDLCP based on the M-Z family of directions. Zbl 1242.90257
Chen, Feixiang
2011
Brunn-Minkowski type inequalities for $$L_p$$ Blaschke-Minkowski homomorphisms. Zbl 1388.52001
Chen, Feixiang; Leng, Gangsong
2016
Some new Chebyshev type quantum integral inequalities on finite intervals. Zbl 1339.26044
Chen, Feixiang; Yang, Wengui
2016
Fejér and Hermite-Hadamard type inequalities for harmonically convex functions. Zbl 1442.26024
Chen, Feixiang; Wu, Shanhe
2014
Indecomposable and noncrossed product division algebras over curves over complete discrete valuation rings. Zbl 1282.16021
Chen, F.
2013
Polynomial convergence of predictor-corrector for SDLCP based on the M-Z family of directions. Zbl 1226.90113
Chen, Feixiang; Zuo, Zhanfei
2011
A class of inequalities on matrix norms and applications. Zbl 1223.15027
Chen, Feixiang; Zuo, Zhanfei
2011
DIV-CURL vector quasi-interpolation on a finite domain. Zbl 1043.65508
Chen, F.; Suter, D.
1999
Fast evaluation of vector splines in three dimensions. Zbl 0926.65014
Chen, F.; Suter, D.
1998
Orlicz-Brunn-Minkowski inequalities for Blaschke-Minkowski homomorphisms. Zbl 1368.52003
Chen, Feixiang; Leng, Gangsong
2017
Several complementary inequalities to inequalities of Hermite-Hadamard type for $$s$$-convex functions. Zbl 1329.26032
Chen, Feixiang; Wu, Shanhe
2016
Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals. Zbl 1339.26043
Chen, Feixiang
2016
An edge/face-based smoothed radial point interpolation method for static analysis of structures. Zbl 1403.74101
Feng, S. Z.; Cui, X. Y.; Chen, F.; Liu, S. Z.; Meng, D. Y.
2016
Brunn-Minkowski type inequalities for $$L_p$$ Blaschke-Minkowski homomorphisms. Zbl 1388.52001
Chen, Feixiang; Leng, Gangsong
2016
Some new Chebyshev type quantum integral inequalities on finite intervals. Zbl 1339.26044
Chen, Feixiang; Yang, Wengui
2016
Extensions of the Hermite-Hadamard inequality for harmonically convex functions via fractional integrals. Zbl 1410.26034
Chen, Feixiang
2015
A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals. Zbl 1332.26037
Chen, Feixiang
2014
On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity. Zbl 1298.26020
Chen, Feixiang
2014
New inequalities of Hermite-Hadamard type for functions whose first derivatives absolute values are $$s$$-convex. Zbl 1332.26038
Chen, Feixiang; Feng, Yuming
2014
A note on the Hermite-Hadamard inequality for convex functions on the co-ordinates. Zbl 1305.26043
Chen, Feixiang
2014
Fejér and Hermite-Hadamard type inequalities for harmonically convex functions. Zbl 1442.26024
Chen, Feixiang; Wu, Shanhe
2014
Refinements on the Hermite-Hadamard inequalities for $$r$$-convex functions. Zbl 1397.26007
Chen, Feixiang; Liu, Xuefei
2013
A note on Hermite-Hadamard inequalities for products of convex functions. Zbl 1397.26004
Chen, Feixiang
2013
Indecomposable and noncrossed product division algebras over curves over complete discrete valuation rings. Zbl 1282.16021
Chen, F.
2013
Equivalent linearization method for the flutter system of an airfoil with multiple nonlinearities. Zbl 1426.74125
Chen, F. X.; Chen, Y. M.; Liu, J. K.
2012
Bifurcation analysis of aeroelastic systems with hysteresis by incremental harmonic balance method. Zbl 1308.74042
Liu, J. K.; Chen, F. X.; Chen, Y. M.
2012
Polynomial convergence of primal-dual algorithms for SDLCP based on the M-Z family of directions. Zbl 1242.90257
Chen, Feixiang
2011
Polynomial convergence of predictor-corrector for SDLCP based on the M-Z family of directions. Zbl 1226.90113
Chen, Feixiang; Zuo, Zhanfei
2011
A class of inequalities on matrix norms and applications. Zbl 1223.15027
Chen, Feixiang; Zuo, Zhanfei
2011
DIV-CURL vector quasi-interpolation on a finite domain. Zbl 1043.65508
Chen, F.; Suter, D.
1999
Fast evaluation of vector splines in three dimensions. Zbl 0926.65014
Chen, F.; Suter, D.
1998
all top 5


text:
# How to handle a feature vector that could be variable length?

I would like to train a machine learning model with several features as input as X[] and with one output as Y. For example Every sample has a Data frame like this: X[0], X[1], X[2], X[3], X[4], Y

Let's say One sample the followings Data is only one value: X[0], X[1], X[2], X[4], Y This is normal machine training problem.

But now, if I would like to set X[3] multiple values for example sample 1 Data is:

X[0] | X[1] | X[2] |         X[3]         | X[4] | Y
10  |  5   |   6  | [10, 20, 30, 40, 50] |  7   | 90

Data in sample 2 is:

X[0] | X[1] | X[2] |         X[3]         | X[4] | Y
11  |  7   |   5  | [20, 30, 40, 50, 60] |  3   | 80

Is this possible to follow the normal machine training process and got a model which could calculate a sample with other example with Data like:

X[0]   | X[1] |  X[2]  |         X[3]         | X[4] | Y
10.5  |  6   |   5.5  | [15, 25, 35, 45, 55] |  5   | ???

If the length for each X[3] is not long, it is possible to divide the X[3] into multiple new features, but if the length of X[3] is very long (len > 1000) with different distribution, making binary is also lead to too many new features. Is there any way to treat the X[3] directly without adding new features?

Really appreciate for your help.

• Basically, it looks like you want to encode feature X[3] in some way; i.e. generate some other features based on it. For example - the mean, min and max of values in it (reducing the >1000 to 3). The exact way to do it depends on the nature of X[3]... – Itamar Mushkin Jul 13 '20 at 10:38
• Thank you for your answer, just to confirm that your answer means it is not possible to directly use an "multiple value feature". Anyway have to transfer X[3] into multiple features. Am i understand right? – Crazy9 Jul 14 '20 at 2:52
• Yes, in my opinion. – Itamar Mushkin Jul 14 '20 at 6:00
• Does this answer your question? How to deal with categorical feature of very high cardinality? – 10xAI Jul 14 '20 at 7:36
• @10xAI that question deals with categorical features; OP's question is about a feature that is itself a vector. Can you elaborate on why are the two equivalent? – Itamar Mushkin Jul 15 '20 at 6:26


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# Question About Mass and Mass Flow Rate Relations

1. Nov 19, 2008

### womfalcs3

Mass is the product of density and volume.

Mass flow rate is the product density, velocity, and cross-sectional area. (It's the derivative of mass with respect to time.)

Bare with the syntax please...

Looking at a sphere within a larger sphere, the volume of the difference is V=(4/3)*pi*(R^3-r^3) where R is a constant inner radius of the larger sphere. r is the radius of the smaller sphere, and it's not constant.

Multiplying that by density gives us the mass of the region in between the spheres.

Taking the derivative of that mass, however, with respect time, how would the equation look in relation to the description I gave above in the second line?

This isn't homework as I'm not in school anymore. It's a personal question.

It's a situation where the smaller sphere is increasing in radius as time is increased.

Last edited: Nov 19, 2008
2. Nov 20, 2008

### aerospaceut10

Well, since the bigger sphere is kept constant, as least that's what I am assuming from the start....

You leave the constants in front, including the R.

So. dM/dt = -4*pi*r^2*density

3. Nov 20, 2008

### vector2

Out of curiosity, if the inner sphere is getting larger then is the mass exiting the annular space between the spheres or is the mass being compressed? Or does it matter...? dM/dt implies a rate of change of mass with respect to time. If mass is not exiting or entering the space between the spheres then dm/dt =0... Perhaps I've misunderstood the question.

4. Nov 20, 2008

### womfalcs3

To first reply, yes, the outer sphere is constant.

It matters. Adiabatic compression is taking place. Even though it matters, such effect will be taken into account once I have a rate equation in terms of mass.

The effect will reflect on density.

5. Nov 21, 2008

### mordechai9

Problem statement: A spherical shell of radius r1, where: { r1 = f(t) ; i.e. r1 is a function of time} is enclosed inside a spherical shell of constant radius R. A fluid (e.g. gas, liquid) fills the region in between, with density rho. Assume f'(t) > 0 and compute the density rho, where: {rho = g(r,t) ; i.e. rho is a function of time and the radial distance from the origin}.

Is this correct? I started to derive this but I want to clarify a few assumptions first. If we really want to do this, we should do it right.

0.) What are the initial conditions? (initial density distributions, pressures, etc.)
1.) What is the rate of motion of the inner sphere? How fast does it move, and does it have an acceleration?
2.) Shall we assume the density in the spherical "annulus" is uniform? (I.e. the density is constant throughout the annulus.)
3.) The fluid in between has constant transport properties; no thermodynamic change in state.

Which assumptions would you like to consider? I don't think I could solve the problem without assumption (2). Also, I would need assumption (3), or else we would have to go into some thermodynamic analysis of the gas.

Remark 1: Note the assumption of (2) implies we need not consider compression wave effects.
Remark 2: Without assumption (3), the compression of the gas in the annulus causes a pressure gradient to develop inside the sphere pointing towards the center of the sphere; i.e. a force develops which tends to resist the encroachment of the inner sphere.

This should give us a more clear understanding.

Last edited: Nov 21, 2008


text:
# Calculating gross profit

Is it possible to calculate gross profit with just the following information: year end amounts for ending inventory, net sales, accounts payable, total current assets and total current liabilities - just year end figures given - no other detail.?

Solution: From above information we are unable to find the gross profit because Gross profit can be find two ways1. Gross profit = Sales - Cost of Goods sold2. Gross profit = Sales - (opening...

## Related Questions in Accounting Concepts and Principles

• ### Accounting

(Solved) August 21, 2011

The following accounts are taken from the financial statements of Rosen, Inc., as of the end of the year 2010. The accounts are in alphabetical order. Accounts Payable 28000 Net Income 48000 Accounts Receivable 66000 Other Current Liabilities 17000 Cash 54000 Total Assets 250000 Gross profit

a. current ratio = current assets/current liabilities. current assets = 54000 66000 = 120,000 current liabilities = 28000 5000 17000 = 50,000 current ratio = 120000/50000 = 2.4 b
• ### Accounting

(Solved) November 11, 2012


text:
# Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x changes? Is there more to understanding this part of calculus, I feel as I'm missing the fundamentals behind differential calculus.

Also one thing that I didn't understand is when doing u-substitution integration if we let $u=2x +1$ for example, sometimes I see $\displaystyle du = 2dx \therefore dx = \frac{du}{2}$. What is this known as and why does it work? My teacher school finds $\displaystyle \frac{du}{dx}$ and rearranges this to make $dx$ the subject. Is this an incorrect practice? I have searched for this on here and cannot find a definite answer.

What is the correct notation to be used? When differentiating y = f(x) are we always operating on y as in $\displaystyle \frac{d}{dx} (y)$ = $\displaystyle \frac{dy}{dx}$. If you differentiate x^2 w.r.t x as in $\displaystyle \frac{d}{dx}x^2$, are you finding how much x^2 changes as x changes e.g if $x = 1, x^2 = 1, x = 2, x^2 = 4, x = 3, x^2 = 9$ so $x^2$ is 2 times the value of x? If we have something like $y^3$, what does it mean to differentiate $y^3$ with respect to x as in $\displaystyle \frac{d}{dx} y^3$ and how is it done?

thanks, I have been looking for the solutions to my problems for quite a while but cannot find an answer that leaves me satisfied. Sorry if questions likethese are not to be asked here.

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+1 these are mature questions for a calculus student to be asking – Jonathan Aug 25 '13 at 15:59

## 4 Answers

You are asking some excellent questions! First, you are correct in assessing the notation $\frac{dy}{dx}$ as the rate of change of $y$ with respect to $x$. It is important to keep in mind, however, that the derivative gives you an instantaneous rate of change. In basic algebra, we talk about lines and their slopes; the slope allows us to measure the rate of change of $y$ w.r.t $x$. If we now talk about more general functions -- not necessarily lines -- such as $y=f(x)=x^2$, we can formally define the derivative to be the limit $$\frac{df}{dx}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.$$ But this definition is essentially just the idea of slope! You take a $y$-value $y=f(x)$ and another $y$-value $y'=f(x+h)$ a little ways away, take the difference (this is your $\Delta y$) and do the same for the difference in $x$, $\Delta x=h$, and then divide. This gives you the slope of the secant line that passes through the curve $y=f(x)$ at $x$ and $x+h$. Now, the limit out front is just asking us to visualize the two points $y$ and $y'$ to be very close to each other -- the secant line thus becomes a tangent line, and instead of writing $\frac{\Delta y}{\Delta x}$, we now write $\frac{dy}{dx}$, for the slope of the line tangent to the curve at any $x$. Hence the derivative $\frac{dy}{dx}$ is a function, which when evaluated at $x$, yields the slope of the line tangent to the curve $y=f(x)$ at the point $x$.

In this sense, the derivative actually gives you information about the instantaneous rate of change of a function. Imagine the function $y=f(x)$ as the trajectory of a baseball. If we removed the limit out front from above (i.e. just computing some $\frac{\Delta y}{\Delta x}$) we'd have computed the slope of a secant line, i.e. the average velocity of the baseball during the time interval $(x,x+h)$. In taking the limit, however, we make this interval smaller and smaller until we're talking about the actual speed of the baseball at some time $x$. I hope this gives you some intuition how a derivative extends the concept of slope/rate-of-change.

As AlexR mentioned above, the $u$-substitutions we make when computing integrals are actually related to the chain rule, for which is there is a perfectly rigorous proof. You are justified in feeling uneasy about writing things like $du=2dx$, but rest assured that this is simply for notational convenience. You can think of this more rigorously as AlexR describes (or, in fact, if you learn differential geometry, the notation can be made rigorous in terms of differential forms).

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Is it appropriate to understand that when finding the derivative, as $dx \to 0$, $dy \to 0$. Since the gradient of a tangent at a point is equal to the rise/run (difference in y/difference in x) then this becomes $$\lim_{h\to 0}\frac{dy}{dx}$$ and since both dx and dy are approaching 0 they're said to become infinitesimally small so the gradient at that point is therefore a ratio of two infinitesimals? So the procedure my teacher used is correct assuming this understanding? – salman Aug 25 '13 at 16:52
@user90771: If $dx$ is an infinitesimal $x$-increment and $dy$ the corresponding $y$-increment (of the dependent variable $y$), then the ratio $\frac{dy}{dx}$ will be infinitely close to the derivative (or gradient, as you call it). What you are describing is the correct intuition, but to formalize the procedure one usually uses the standard part. The limit can then be defined as the standard part of the ratio of infinitesimals. – Mikhail Katz Aug 25 '13 at 17:01
Yes, that is the correct intuitive picture of what is happening. To summarize what you're saying in slightly tighter notation: you can compute rise/run ($\Delta y/\Delta x$), but if you take a limit $$\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}$$ you no longer are talking about rise/run. You're now (loosely speaking) thinking of infinitesimal changes, which is why we use the notation $dy/dx$ and not $\Delta y/\Delta x$. You can treat these as infinitesimals on their own (just $dy$ or just $dx$), and you will get the correct answer, but really what is happening is just the chain rule. – Nilay Kumar Aug 25 '13 at 17:03
@user72694 I havent studied hyperreals or much on that wikipedia page (and I will not be studying maths to university level). Do you think it's more appropriate for me to stop worrying about intuition and focus more of problem-solving or learning more topics in calculus? Sometimes I wish to learn of things from first principles (such as the origins of e) but if this requires knowledge which is out of scope it may not be a feasible option and maybe a waste of time. – salman Aug 25 '13 at 17:17
I think it's always worth spending time familiarizing yourself with the intuition behind mathematical concepts, even (especially) if you don't have the time to delve into the rigorous details. Learning things from first principles is always enlightening, but really what is important is understanding why certain mathematical statements are plausible -- why things should be the way they should be. Having a loose sketch of the math in your head is useful when it comes to problem solving. If there's a technical detail you don't quite remember from calculus, you can always just look it up again. – Nilay Kumar Aug 25 '13 at 17:23

While the other answers are precise and to the point, the OP may be looking for motivation for the calculational practices he observed in the classroom. To address specifically his question:

"one thing that I didn't understand is when doing u-substitution integration if we let $u=2x +1$ for example, sometimes I see $\displaystyle du = 2dx \therefore dx = \frac{du}{2}$. What is this known as and why does it work? My teacher school finds $\displaystyle \frac{du}{dx}$ and rearranges this to make $dx$ the subject. Is this an incorrect practice? I have searched for this on here and cannot find a definite answer."

I would point out that the proper justification of your teacher's procedure involves infinitesimals. In this approach, $dx$ and $du$ are infinitesimals and the expression $\frac{du}{dx}$ is a ratio. While there are some details to be ironed out, you should keep in mind that the mainstream approach adopted at a majority of universities avoids infinitesimals and formulates the calculus in terms of the real numbers and epsilon, delta procedures. In this approach it is difficult to account for $\frac{du}{dx}$ as a true ratio in a meaningful way. While you will have to learn the calculus at the university the way it is taught there, it is usually helpful to think of the procedures of the calculus in terms of infinitesimals.

A revealing study of this phenomenon from the educational viewpoint is the following article by R. Ely:

Ely, Robert: Nonstandard student conceptions about infinitesimal and infinite numbers. Journal for Research in Mathematics Education 41 (2010), no. 2, 117-146. See http://www.nctm.org/publications/article.aspx?id=26196 and http://u.cs.biu.ac.il/~katzmik/ely10.pdf

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$\frac{dy}{dx}$ is the rate of change of with respect to $x$. I will try to explain what this mean.

For example let us take $y=x^2$ And we are going to find out $\frac{d}{dx}(x^2)$.In other words we are going to find out rate of change of $x^2$ with respect to $x$. In other words we are going to find out how $x^2$ is changing when we introduce a change in $x$.

So, we introduce a change in $x$, say by an amount $h$, so as to make it $x+h$.Now instead of $x^2$ we have $(x+h)^2$. So how much change has been added to $x^2$?.

It is $$(x+h)^2-x^2=2xh+h^2$$ This is the amount change in $x^2$ when we introduced a change of $h$ in $x$.So what is the rate? It is $$\frac{\text{change in }x^2}{\text{change in }x}=\frac{2xh+h^2}{h}=2x+h$$ Now we note that the rate is dependent on how much change we are introducing.Actually we are not interested in large amount of change. What we want to find is an instantaneous rate of change, we want to to make the change or $h$ as small as possible,as close to zero as possible, but obviously not zero(then there would be no change!). So that $2x+h$ is is getting closer to $2x$ as $h$ gets closer to zero.

We formally write this as $$\underset {h\rightarrow 0}{lim}\text{ } 2x+h=2x$$ .

Now the formal definition of derivative, $$\frac{d}{dx}(f(x))=\underset {h\rightarrow 0}{lim}\frac{f(x+h)-f(x)}{h}$$or $$\frac{dy}{dx}=\underset{\Delta x\rightarrow 0}{lim}\frac{\Delta y}{\Delta x}$$ where $\Delta y$ is the change in $y$ as we change $x$ by $\Delta x$ .

So as $\Delta x$ gets closer to zero $\Delta y$ is also getting zero.Now we may view the derivative as ratio of two infinitesimals. This is the reason why, for instance when $\frac{dy}{dx}=2$ ,we write $dy=2dx$ . This is often useful for practical point of view. But it is not very mathematically rigorous, but one need not be worried about it most of the cases.

It must be clear now that derivative involves concept of limits, and I have explained it in a loose way.But that is okay to understand and to have a feel of what derivative is.

Now coming to your question of differentiating $y^3$ with respect to $x$, Since $y$ is a function of $x$ $y^3$ is also a function of $x$. So it should be possible to do the differentiation using the definition of derivative above.Now if you know $\frac{dy}{dx}$ there are rules known as chain rule to compute the derivative of some function of y with respect to x.But everything is basically derived from the base definition. So just for the sake of completeness $$\frac{dy^3}{dx}=\frac{d(y^3)}{dy}\frac{dy}{dx}=3y^2\frac{dy}{dx}$$

$\frac{dy}{dx}$ is the same as $\frac{d}{dx}(y)$. Also when you write $du=2xdx$ what it means is that the infinitesimal change in $u$, when you make an infinitesimal change in $x$ is 2 times $x$ times the infinitesimal change in $x$ .

I may add some more details.Say $y$ is some arbitrary function of function of $x$ ($y=f(x)$).When make change $x$ to $x+\Delta x$, f(x) will change in this way $$f(x+\Delta x)=f(x)+(\text{some expression in x})\Delta x +(\text{something}){\Delta x}^2+(\text{something else }){\Delta x}^3+......$$

You will see that by using the definition of derivative the coefficient of $\Delta x$ is the derivative of $f(x)$ with respect to $x$. Now $dy$ is ($(f(x+\Delta x)-f(x)$) as $\Delta x$ tends to zero . This can be seen to be equal to $(\text{derivative})\times \Delta x+ \text{other terms}$ as $\Delta x$ tends to zero. What we are doing is almost like ignoring the terms involving higher powers of $\Delta x$ .

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@user90771 I have added some more details to the answer to address your further concerns. hope it helps – jaseem Aug 25 '13 at 17:52
hey can I also ask, for $$\frac{d}{dx}y^3 = \frac{d}{dy}y^3 \frac{d}{dx}y$$ is it correct to think that the dy in the denominator and numerator cancel? – salman Aug 25 '13 at 19:07
@user90771 It is not correct think that way. But it's easy to think that way to solve most of the problems. – jaseem Aug 26 '13 at 9:18
@user90771 Letting $u=y^3$ what you are asking is whether it is correct to think of the identity $\frac{du}{dx}=\frac{du}{dy}\frac{dy}{dx}$ (chain rule) as "cancellation" of $dy$ in numerator and denominator on the right hand side. Again thinking in terms of infinitesimals, I would say that the intuition is correct though some technical details need to be worked out (see my remarks above). For second derivatives, the distinction between dependent variable and independent variable becomes more important and one can't cancel as easily. – Mikhail Katz Aug 26 '13 at 13:03
@user72694 yeah that is what I meant. Is this a "bad" way to look at this issue? Is there a definite answer about the infinitesimals cancelling? And what exactly do you mean by the second derivative? – salman Aug 26 '13 at 17:50

$$\frac{d}{dx} y(x)^3 = 3 (y(x))^2 \cdot y'(x)$$ by the chain rule of differentiation. The notation $du = 2dx$ is not mathematically clean. Actually you use in Substitution: $$\int_a^b y(x) dx = \int_{\phi^{-1}(a)}^{\phi^{-1}(b)} y(\phi(u)) \cdot \phi'(u) du$$ And you give $u = 2x + 1 \leadsto \phi(u) = \frac{u-1}{2}$ and $\phi'(u) = \frac{1}{2}$

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I, for one, find the notation $du=2dx$ mathematically perfectly clean (differentials of functions do belong to elementary calculus) – user8268 Aug 25 '13 at 16:14
@user8268 $u$ and $x$ are variables whereas $du, dx$ are associated Lebesgue-measures. Thus $du = 2 dx$ means $$\forall A \in \mathcal{B}: \lambda_u(A) = 2 \lambda_x(A)$$ which is not exactly "clean"... – AlexR Aug 25 '13 at 16:19
notation is overloaded (and thus perhaps not clean). I meant differentials of (differentiable) functions, which is far more elementary than measures. – user8268 Aug 25 '13 at 16:23
Then again $du$ and $dx$ may not be separated as both are no functions, but $\displaystyle \frac{du}{dx}$ is a differentiation of a function $u = u(x)$. – AlexR Aug 25 '13 at 16:27
The linear function on WHAT domain? See the Problem here? A linear function on the tangent space is not exactly what students are taught in calculus ;-) – AlexR Aug 25 '13 at 16:33


text:
# Active Filter Derivation

1. Apr 29, 2011

### dillonmhudson

Can I get some help with this? I don't even know where to begin.

2. Apr 29, 2011

### jegues

Bandpass filters usually employ both inductors and capictors.

The inductor cuts off the high frequencies, while the capacitor cuts off the low frequencies, ideal for a bandpass filter.

3. Apr 29, 2011

### jegues

Whoops, I just noticed that the the answer is given in the figure. I can see they did not use a capacitor!

I managed to derive the transfer function for this circuit and found the following,

$$T(jw) = \frac{-z}{(jwL + z)(jwL + R)}$$

Now we simply need to select our Z such that the requirement for resonant frequency is obatined.

If we select,

$$z = R_{z}$$

We rearrange the transfer function,

$$\frac{-\frac{1}{R}}{(1 + \frac{jwL}{R_{z}})(1 + \frac{jwL}{R})}$$

This becomes,

$$\frac{-\frac{1}{R}}{(1 + \frac{jw}{w_{L}})(1 + \frac{jw}{w_{H}})}$$

Now we know what wL and wH are we can solve for fL and fH since,

$$f = 2\pi w$$

Then our desired resonant frequency,

$$f_{r} = \sqrt{f_{L} \cdot f_{H}}$$

Simply solve for, $$R_{z}$$.

You should find that, $$R_{z} = \frac{100\pi^{2}L^{2}}{R}$$

Last edited: Apr 29, 2011


text:
Home » Posts tagged 'Real Analysis'

# Tag Archives: Real Analysis

## Proofs of Irrationality

“Irrational numbers are those real numbers which are not rational numbers!”

Def.1: Rational Number

A rational number is a real number which can be expressed in the form of $\frac{a}{b}$ where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero.
Following two statements are equivalent to the definition 1.
1. $x=\frac{a}{b}$ is rational if and only if $a$ and $b$ are integers relatively prime to each other and $b$ does not equal to zero.
2. $x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}$.

## The Area of a Disk

[This post is under review.]

If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius $R$ is $\pi R^2$.

The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular lamina. Radius for a disk is always same, irrespective of the location of point at circumference to which you are joining the center of disk. The area of disk is defined as the ‘measure of surface‘ surrounded by the round edge (circumference) of the disk.

The area of a disk can be derived by breaking it into a number of identical parts of disk as units — calculating their areas and summing them up till disk is reformed. There are many ways to imagine a unit of disk. We can imagine the disk to be made up of several concentric very thin rings increasing in radius from zero to the radius of disc. In this method we can take an arbitrary ring, calculate its area and then in similar manner, induce areas of other rings -sum them till whole disk is obtained. (more…)

## Triangle Inequality

Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If $a$, $b$ and $c$ be the three sides of a triangle, then neither $a$ can be greater than $b+c$, nor$b$ can be greater than $c+a$, nor $c$ can be than $a+b$.

Triangle

Consider the triangle in the image, side $a$ shall be equal to the sum of other two sides $b$ and $c$, only if the triangle behaves like a straight line. Thinking practically, one can say that one side is formed by joining the end points of two other sides.
In modulus form, $|x+y|$ represents the side $a$ if $|x|$ represents side $b$ and $|y|$ represents side $c$. A modulus is nothing, but the distance of a point on the number line from point zero.

Visual representation of Triangle inequality

For example, the distance of $5$ and $-5$ from $0$ on the initial line is $5$. So we may write that $|5|=|-5|=5$.

Triangle inequalities are not only valid for real numbers but also for complex numbers, vectors and in Euclidean spaces. In this article, I shall discuss them separately. (more…)

## A Trip to Mathematics: Part III Relations and Functions

‘Michelle is the wife of Barak Obama.’
‘John is the brother of Nick.’
‘Robert is the father of Marry.’
‘Ram is older than Laxman.’
‘Mac is the product of Apple Inc.’
After reading these statements, you will realize that first ‘Noun’ of each sentence is some how related to other. We say that each one noun is in a RELATIONSHIP to other. Mischell is related to Barak Obama, as wife. John is related to Nick, as brother. Robert is related to Marry, as father. Ram is related to Laxman in terms of age(seniority). Mac is related to Apple Inc. as a product.These relations are also used in Mathematics, but a little variations; like ‘alphabets’ or ‘numbers’ are used at place of some noun and mathematical relations are used between them. Some good examples of relations are:

is less than
is greater than
is equal to
is an element of
belongs to
divides
etc. etc.

Some examples of regular mathematical statements which we encounter daily are:

4<6 : 4 is less than 6.
5=5 : 5 is equal to 5.
6/3 : 3 divides 6.

For a general use, we can represent a statement as:
”some x is related to y”
Here ‘is related to’ phrase is nothing but a particular mathematical relation. For mathematical convenience, we write ”x is related to y” as $x \rho y$. x and y are two objects in a certain order and they can also be used as ordered pairs (x,y).
$(x,y) \in \rho$ and $x \rho y$ are the same and will be treated as the same term in further readings. If $\rho$ represents the relation motherhood, then $\mathrm {(Jane, \ John)} \in \rho$ means that Jane is mother of John.
All the relations we discussed above, were in between two objects (x,y), thus they are called Binary Relations. $(x,y) \in \rho \Rightarrow \rho$ is a binary relation between a and b. Similarly, $(x,y,z) \in \rho \Rightarrow \rho$ is a ternary (3-nary) relation on ordered pair (x,y,z). In general a relation working on an n-tuple $(x_1, x_2, \ldots x_n) \in \rho \Rightarrow \rho$ is an n-ary relation working on n-tuple.
We shall now discuss Binary Relations more rigorously, since they have solid importance in process of defining functions and also in higher studies. In a binary relation, $(x,y) \in \rho$, the first object of the ordered pair is called the the domain of relation ρ and is defined by
$D_{\rho} := \{x| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$ and also the second object is called the range of the relation ρ and is defined by $R_{\rho} := \{y| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$.
There is one more thing to discuss about relations and that is about equivalence relation.
A relation is equivalence if it satisfies three properties, Symmetric, Reflexive and Transitive.
I mean to say that if a relation is symmetric, reflexive and transitive then the relation is equivalence. You might be thinking that what these terms (symmetric, reflexive and transitive) mean here. Let me explain them separately:
A relation is symmetric: Consider three sentences “Jen is the mother of John.”; “John is brother of Nick.” and “Jen, John and Nick live in a room altogether.”
In first sentence Jen has a relationship of motherhood to John. But can John have the same relation to Jen? Can John be mother of Jen? The answer is obviously NO! This type of relations are not symmetric. Now consider second statement. John has a brotherhood relationship with Nick. But can Nick have the same relation to John? Can Nick be brother of John? The answer is simply YES! Thus, both the sentences “John is the brother of Nick.” and “Nick is the brother of John.” are the same. We may say that both are symmetric sentences. And here the relation of ‘brotherhood’ is symmetric in nature. Again LIVING WITH is also symmetric (it’s your take to understand how?).
Now let we try to write above short discussion in general and mathematical forms. Let X and Y be two objects (numbers or people or any living or non-living thing) and have a relation ρ between them. Then we write that X is related by a relation ρ , to Y. Or X ρ Y.
And if ρ is a symmetric relation, we might say that Y is (also) related by a relation ρ to X. Or Y ρ X.
So, in one line; $X \rho Y \iff Y \rho X$ is true.

A relation is reflexive if X is related to itself by a relation. i.e., $X \rho X$. Consider the statement “Jen, John and Nick live in a house altogether.” once again. Is the relation of living reflexive? How to check? Ask like, Jen lives with Jen, true? Yes! Jen lives there.
A relation is transitive, means that some objects X, Y, Z are such that if X is related to Y by the relation, Y is related to Z by the relation, then X is also related to Z by the same relation.
i.e., $X \rho Y \wedge Y \rho Z \Rightarrow X \rho Z$. For example, the relationship of brotherhood is transitive. (Why?) Now we are able to define the equivalence relation.
We say that a relation ρ is an equivalence relation if following properties are satisfied: (i) $X \rho Y \iff Y \rho X$
(ii) $X \rho X$
(iii) $X \rho Y \ Y \rho Z \Rightarrow X \rho Z$.

Functions: Let f be a relation (we are using f at the place of earlier used ρ ) on an ordered pair $(x,y) : x \in X \ y \in Y$. We can write xfy, a relation. This relation is called a function if and only if for every x, there is always a single value of y. I mean to say that if $xfy_1$ is true and $xfy_2$ is also true, then always $y_1=y_2$. This definition is standard but there are some drawbacks of this definition, which we shall discuss in the beginning of Real Analysis .
Many synonyms for the word ‘function’ are used at various stages of mathematics, e.g. Transformation, Map or Mapping, Operator, Correspondence. As already said, in ordered pair (x,y), x is called the element of domain of the function (and X the domain of the function) and y is called the element in range or co-domain of the function (and Y the range of the function).

Here I will stop myself. I don’t want a post to be long (specially when writing on basic mathematics) that reader feel it boring. The intermediate mathematics of functions is planned to be discussed in Calculus and advanced part in functional analysis. Please note that I am regularly revising older articles and trying to maintain the accuracy and completeness. If you feel that there is any fault or incompleteness in a post then please make a comment on respective post. If you are interested in writing a guest article on this blog, then kindly email me at mdnb[at]live[dot]in.

# Announcement

Hi all!
I know some friends, who don’t know what mathematics in real is, always blame me for the language of the blog. It is very complicated and detailed. I understand that it is. But MY DIGITAL NOTEBOOK is mainly prepared for my study and research on mathematical sciences. So, I don’t care about what people say (SAID) about the

A Torus

content and how many hits did my posts get. I feel happy in such a way that MY DIGITAL NOTEBOOK has satisfied me at its peak-est level. I would like to thank WordPress.com for their brilliant blogging tools and to my those friends, teachers and classmates who always encourage me about my passion. For me the most important thing is my study. More I learn, more I will go ahead. So, today (I mean tonight) I have decided to write some lecture-notes (say them study-notes, since I am not a lecturer) on MY DIGITAL NOTEBOOK. I have planned to write on Group Theory at first and then on Real Analysis. And this post is just to introduce you with some fundamental notations which will be used in those study-notes.

# Notations

Conditionals and Operators
$r /; c$ : Relation $r$ holds under the condition $c$.
$a=b$ : The expression $a$ is mathematically identical to $b$.
$a \ne b$ : The expression a is mathematically different from $b$.
$x > y$ : The quantity $x$ is greater than quantity $y$.
$x \ge y$ : The quantity $x$ is greater than or equal to the quantity $y$.
$x < y$ : The quantity $x$ is less than quantity $y$.
$x \le y$ : The quantity $x$ is less than or equal to quantity $y$.
$P := Q$ : Statement $P$ defines statement $Q$.
$a \wedge b$ : a and b.
$a \vee b$ : a or b.
$\forall a$ : for all $a$.
$\exists$ : [there] exists.
$\iff$: If and only if.
Sets & Domains
$\{ a_1, a_2, \ldots, a_n \}$ : A finite set with some elements $a_1, a_2, \ldots, a_n$.
$\{ a_1, a_2, \ldots, a_n \ldots \}$ : An infinite set with elements $a_1, a_2, \ldots$
$\mathrm{\{ listElement /; domainSpecification\}}$ : A sequence of elements listElement with some domainSpecifications in the set. For example, $\{ x : x=\frac{p}{q} /; p \in \mathbb{Z}, q \in \mathbb{N^+}\}$ $a \in A$ : $a$ is an element of the set A.
$a \notin A$: a is not an element of the set A.
$x \in (a,b)$: The number x lies within the specified interval $(a,b)$.
$x \notin (a,b)$: The number x does not belong to the specified interval $(a,b)$. Standard Set Notations
$\mathbb{N}$ : the set of natural numbers $\{0, 1, 2, \ldots \}$
$\mathbb{N}^+$: The set of positive natural numbers: $\{1, 2, 3, \ldots \}$
$\mathbb{Z}$ : The set of integers $\{ 0, \pm 1, \pm 2, \ldots\}$
$\mathbb{Q}$ : The set of rational numbers
$\mathbb{R}$: The set of real numbers
$\mathbb{C}$: The set of complex numbers
$\mathbb{P}$: The set of prime numbers.
$\{ \}$ : The empty set.
$\{ A \otimes B \}$ : The ordered set of sets $A$ and $B$.
$n!$ : Factorial of n: $n!=1\cdot 2 \cdot 3 \ldots (n-1) n /; n \in \mathbb{N}$

Other mathematical notations, constants and terms will be introduced as their need.

For Non-Mathematicians:
Don’t worry I have planned to post more fun. Let’s see how the time proceeds!

## Everywhere Continuous Non-derivable Function

Weierstrass had drawn attention to the fact that there exist functions which are continuous for every value of $x$ but do not possess a derivative for any value. We now consider the celebrated function given by Weierstrass to show this fact. It will be shown that if

$f(x)= \displaystyle{\sum_{n=0}^{\infty} } b^n \cos (a^n \pi x) \ \ldots (1) \\ = \cos \pi x +b \cos a \pi x + b^2 \cos a^2 \pi x+ \ldots$ where $a$ is an odd positive integer, $0 < b <1$ and $ab > 1+\frac{3}{2} \pi$, then the function $f$ is continuous $\forall x$ but not finitely derivable for any value of $x$.

G.H. Hardy improved this result to allow $ab \ge 1$.

We have $|b^n \cos (a^n \pi x)| \le b^n$ and $\sum b^n$ is convergent. Thus, by Wierstrass’s $M$-Test for uniform Convergence the series (1), is uniformly convergent in every interval. Hence $f$ is continuous $\forall x$.
Again, we have $\dfrac{f(x+h)-f(x)}{h} = \displaystyle{\sum_{n=0}^{\infty}} b^n \dfrac{\cos [a^n \pi (x+h)]-\cos a^n \pi x}{h} \ \ \ldots (2)$
Let, now, $m$ be any positive integer. Also let $S_m$ denote the sum of the $m$ terms and $R_m$, the remainder after $m$ terms, of the series (2), so that
$\displaystyle{\sum_{n=0}^{\infty}} b^n \dfrac{\cos [a^n \pi (x+h)]-\cos a^n \pi x}{h} = S_m+R_m$. By Lagrange’s mean value theorem, we have
$\dfrac{|\cos {[a^n \pi (x+h)]} -\cos {a^n \pi x|}}{|h|}=|a^n \pi h \sin {a^n \pi(x+\theta h)}| \le a^n \pi |h|$,
$|S_m| \le \displaystyle{\sum_{n=0}^{m-1}} b^n a^n \pi = \pi \dfrac {a^m b^m -1}{ab-1} < \pi \dfrac {a^m b^m}{ab-1}$. We shall now consider $R_m$.
So far we have taken $h$ as an arbitrary but we shall now choose it as follows:

We write $a^m x=\alpha_m+\xi_m$, where $\alpha_m$ is the integer nearest to $a^m x$ and $-1/2 \le \xi_m < 1/2$.
Therefore $a^m(x+h) = \alpha_m+\xi_m+ha^m$. We choose, $h$, so that $\xi_m+ha^m=1$
i.e., $h=\dfrac{1-\xi_m}{a^m}$ which $\to 0 \ \text{as} \ m \to \infty$ for $0< h \le \dfrac{3}{2a^m} \ \ldots (3)$
Now, $a^n \pi (x+h) = a^{n-m} a^m (x+h.) \\ \ =a^{n-m} \pi [(\alpha_m +\xi_m)+(1-\xi_m)] \\ \ =a^{n-m} \pi(\alpha_m+1)$

Thus $\cos[a^n \pi (x+h)] =cos [a^{n-m} (\alpha_m-1) \pi] =(-1)^{\alpha_{m+1}}$.
$\cos (a^n \pi x) = \cos [a^{n-m} (a^m \pi x)] \\ \ =\cos [a^{n-m} (\alpha_m+\xi_m) \pi] \\ \ =\cos a^{n-m} \alpha_m \pi \cos a^{n-m} \xi_m \pi - \sin a^{n-m} \alpha_m \pi \sin a^{n-m} \xi_m \pi \\ \ = (-1)^{\alpha_m} \cos a^{n-m} \xi_m \pi$ for $a$, is an odd integer and $\alpha_m$ is an integer.

Therefore, $R_m =\dfrac{(-1)^{\alpha_m}+1}{h} \displaystyle{\sum_{n=m}^{\infty}} b^n [2+\cos (a^{n-m} \xi_m \pi] \ \ldots (4)$
Now each term of series in (4) is greater than or equal to 0 and, in particular, the first term is positive, $|R_m| > \dfrac{b^m}{|h|} > \dfrac{2a^m b^m}{3} \ \ldots (3)$
Thus $\left| {\dfrac{f(x+h) -f(x)}{h}} \right| = |R_m +S_m| \\ \ \ge |R_m|-|S_m| > \left({\frac{2}{3} -\dfrac{\pi}{ab-1}} \right) a^mb^m$
As $ab > 1+\frac{3}{2}\pi$, therefore $\left({\frac{3}{2} -\dfrac{\pi}{ab-1}} \right)$ is positive.
Thus we see that when $m \to \infty$ so that $h \to 0$, the expression $\dfrac{f(x+h)-f(x)}{h}$ takes arbitrary large values. Hence, $f'(x)$ does not exist or is at least not finite.

### Reference

A course of mathematical analysis
SHANTI NARAYAN
PK MITTAL
S. Chand Co.

## Dedekind’s Theory of Real Numbers

Image via Wikipedia

# Intro

Let $\mathbf{Q}$ be the set of rational numbers. It is well known that $\mathbf{Q}$ is an ordered field and also the set $\mathbf{Q}$ is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists infinite number of elements of $\mathbf{Q}$. Thus the system of rational numbers seems to be dense and so apparently complete. But it is quite easy to show that there exist some numbers (?) (e.g., ${\sqrt{2}, \sqrt{3} \ldots}$ etc.) which are not rational. For example, let we have to prove that $\sqrt{2}$ is not a rational number or in other words, there exist no rational number whose square is 2. To do that if possible, purpose that $\sqrt{2}$ is a rational number. Then according to the definition of rational numbers $\sqrt{2}=\dfrac{p}{q}$, where p & q are relatively prime integers. Hence, ${\left(\sqrt{2}\right)}^2=p^2/q^2$ or $p^2=2q^2$. This implies that p is even. Let $p=2m$, then $(2m)^2=2q^2$ or $q^2=2m^2$. Thus $q$ is also even if 2 is rational. But since both are even, they are not relatively prime, which is a contradiction. Hence $\sqrt{2}$ is not a rational number and the proof is complete. Similarly we can prove that why other irrational numbers are not rational. From this proof, it is clear that the set $\mathbf{Q}$ is not complete and dense and that there are some gaps between the rational numbers in form of irrational numbers. This remark shows the necessity of forming a more comprehensive system of numbers other that the system of rational number. The elements of this extended set will be called a real number. The following three approaches have been made for defining a real number.

1. Dedekind’s Theory
2. Cantor’s Theory
3. Method of Decimal Representation

The method known as Dedekind’s Theory will be discussed in this not, which is due to R. Dedekind (1831-1916). To discuss this theory we need the following definitions:

Rational number A number which can be represented as $\dfrac{p}{q}$ where p is an integer and q is a non-zero integer i.e., $p \in \mathbf{Z}$ and $q \in \mathbf{Z} \setminus \{0\}$ and p and q are
relatively prime as their greatest common divisor is 1, i.e., $\left(p,q\right) =1$.

Ordered Field: Here, $\mathbf{Q}$ is, an algebraic structure on which the operations of addition, subtraction, multiplication & division by a non-zero number can be carried out.

Least or Smallest Element: Let $A \subseteq Q$ and $a \in Q$. Then $a$ is said to be a least element of $A$ if (i) $a \in A$ and (ii) $a \le x$ for every $x \in A$.

Greatest or Largest Element: Let $A \subseteq Q$ and $b \in Q$. Then $b$ is said to be a least element of $A$ if (i) $b \in A$ and (ii) $x \le b$ for every $x \in A$.

## Dedekind’s Section (Cut) of the Set of All the Rational Numbers

Since the set of rational numbers is an ordered field, we may consider the rational numbers to be arranged in order on straight line from left to right. Now if we cut this line by some point $P$, then the set of rational numbers is divided into two classes $L$ and $U$. The rational numbers on the left, i.e. the rational numbers less than the number corresponding to the point of cut $P$ are all in $L$ and the rational numbers on the right, i.e. The rational number greater than the point are all in $U$. If the point $P$ is not a rational number then every rational number either belongs to $L$ or $U$. But if $P$ is a rational number, then it may be considered as an element of $U$.

Def.

### Real Numbers:

Let $L \subset \mathbf{Q}$ satisfying the following conditions:

1. $L$ is non-empty proper subset of $\mathbf{Q}$.
2. $a, b \in \mathbf{Q}$ , $a < b$ and $b \in L$ then this implies that $a \in L$.
3. $L$ doesn’t have a greatest element.

Let $U=\mathbf{Q}-L$. Then the ordered pair $< L,U >$ is called a section or a cut of the set of rational numbers. This section of the set of rational numbers is called a real number.

Notation: The set of real numbers $\alpha, \beta, \gamma, \ldots$ is denote by $\mathbf{R}$.

Let $\alpha = \langle L,U \rangle$ then $L$ and $U$ are called Lower and Upper Class of $\alpha$ respectively. These classes will be denoted by $L(\alpha)$ and $U(\alpha)$ respectively.