Reference Book : Elements of Abstract and Linear Algebra

Author : E. H. Connell

Downloaded from : http://www.math.miami.edu/~ec/book/

 

I have a habbit of making notes when I read something that pleases me. I read first few pages of this book and here are the notes I have collected.

 

Symbols and their definitions in Mathematics

$             -           There exists

$!            -           There exists a unique

"            -            For each

Þ           -            Implies

È            -            Union

Ç            -            Intersection

Ì            -            Subset

Π           -            Belongs to

~             -            Is related to

N             -           Set of positive integers

Z              -           Ring of Integers

Q             -           Field of rational numbers {a/b: a, bÎ Z, b¹ 0}

R             -            Field of real numbers

C             -            Field of complex numbers

 

Properties of a Relation R:

Say, (a, b) ÎR then,

1.       " a ÎR  $ a ~ a, R is reflexive

2.       " a ~ b $ b ~ a, Þ R is a symmetric relation

3.       " a ~ b and b ~ c, $ a ~ c Þ R is a transitive relation

4.       " a ~ b there doesn’t exist b ~ a for a ¹ b, Þ R is a anti-symmetric relation

 

A relation X satisfying properties 1, 2 and 3 is called an equivalence relation.

 

 

Group theory:

Say, G is a non-void set and j is a function

                j: G ´ G ® G

Definition of a Multiplicative Group:

1.       " a, b, c ÎG,  a. (b. c) = (a. b).c

2.       " a  Î G, $ e Î G such that a. e = e. a = a

3.       " a  Î G, $ b Î G such that a. b = b. a = e

If the properties 1, 2 and 3 are true, then (G, j) constitute a Multiplicative Group.

4.       " a, b ÎG then a. b = b. a

If a multiplicative group also satisfies the property 4, then it is said to be multiplicative abelian or commutative group.

 

Definition of an Additive Group:

1.       " a, b, c ÎG,  a+ (b+ c) = (a+ b) +c

2.       " a  Î G, $ e Î G such that a+ e = e+ a = a

3.       " a  Î G, $ b Î G such that a+ b = b+ a = e

If the properties 1, 2 and 3 are true, then (G, j) constitute a additive Group.

4.       " a, b ÎG then a+ b = b+ a

If an additive group also satisfies the property 4, then it is said to be additive abelian or commutative group.

 

A lengthy day of unsettled thoughts, the fear of losing something very valuable and yet I wasn’t unable to understand for a long time the exact cause for which I was vexed.  The rooms appeared empty, the space appeared vacant and there was acute disturbance in the winds blowing around.  There was a degree of strangeness in everything that was visible to me. I kept asking myself, what has gone so wrong that it’s deeply affecting me, why this gloom over my face when everything is going so damn well? Just when I was about to conclude that I am worried over some anonymous trivial reason, a deep thought stuck me.  I was searching for peripheral reasons while the problem is deeply rooted within me.

 

Over the last one year, there has been enormous transformation in me. Transformation such as, a student to professional, an aggressive and arrogant individual to an unruffled and responsible guy, from being crazy about science to being crazy about beautiful ladies, I think I can go on and on about the things that have changed in my life. When I look at life, it appears to be totally changed.  No more long waits at bus-stops to reach college, no more working on technical papers, no more constructive discussions on Unified theory of physics and there are many such little activities that made my life which now have become extinct. Yes, some changes have been positive; however I am really afraid of losing the actual ME in this whole transformation process. Today, I speak to people basing on what they expect from me rather than what I want to speak. It’s a complicated sentence to understand and people around me will certainly find it difficult to grasp the meaning I want to communicate through that sentence.

 

What hurts me most is that there was a time when playing around with non-linear differential equations gave me pleasure, today they appear so damn dry and boring.  There was a time when I tried hard to avoid complicated scientific vocabulary in everyday talk, today I search for words and most often fumble with inappropriate grammar, there was a time I won’t have cared to think about a girl, today every other girl distracts me! Of course, over the time the MF (maal factor) in surrounding areas has increased, which explains the cause of distractions. Deep inside me, there is a fear building up of losing my actual characteristics and transforming into an insignificant person with no potential.  No doubt, I was very much insignificant earlier, however, I worked for my dreams to come true. Considering the situation I am today, I might have failed to convert those dreams into reality, nonetheless I did try. Today, the way I am wasting time is a clear indication that I have given up in life.

 

No, I am not born to live and die just like millions on this planet, I here to make a difference. I need to re-awaken myself and avoid getting distracted by temporary things around me. I know it’s going to hard to avoid the temptations, anyhow, I consider this situation as test for my will power. Rome was not built in a day, so is life. I hope to rebuild my life once again and I pray to God for the strength, will power and continuous luck to make this process successful.

In quantum mechanics, there is a concept of Heisenberg uncertainty due to which you cannot accurately calculate the values of two canonical observables simultaneously. One simple example of this is, in all realistic conditions, you cannot calculate accurately the exact location and velocity of a particle simultaneously. This might seem nothing but some theoretical nonsense, however there has been enough scientific evidence about its validity. The consequence of this effect is that our universe is no longer deterministic but rather probabilistic.

 

When this effect was highlighted, there were many people who were quite opposing to it. Some people believed God's playing Dice with us, some people believed God can't play dice with us and some felt we should not question what does! Before I further go ahead, let me briefly describe what actually deterministic nature of Universe is and what's its relation with God. Before the advent of quantum mechanics, it was believed that if you know the state of universe at one point of time then you could accurately describe the state of universe at all points by applying pure scientific laws. This was attributed as Deterministic state of Universe. The deterministic nature of the Universe downplayed the Role of God in functioning of Universe. Once I have the set of laws and known state of Universe, I can know everything… then what's the need of so called "God"? This was the question raised by lot of people including scientists. There were theories thrown up about self-organized structure of universe doubting the very existence of God.

 

It was quantum mechanics that strengthened the idea of existence of "God" indirectly by proposing some revolutionary ideas. Heisenberg Uncertainty principle abolished the idea of Determinism, i.e. even if you have set of scientific laws and a known state of Universe you cannot accurately calculate the state of universe at any unknown point. Irrespective of how it sounds, this effect has decreased the power of science! Now with science, I can only give the probability of what can happen but cannot accurately say what will happen. As bizarre as it sounds, quantum mechanics produces many such intriguing results that down plays the absolute power of science and indirectly strengthening the idea of God.

It is amazing to note how much a person can get disturbed when a simple thing doesn’t work the way he/she wanted to.  More often than not, I get perplexed with the way simple things work in my life. When I say simple things, it refers to day to day things like recharging my cell phone or keeping a comb in my pocket when leaving outside. I have absolutely no idea why I always forget to carry a comb to office! One blow of wind and hair goes unstructured. Considering that I am always very passionate about hair styles, forgetting to carry a comb along with me is so much disturbing. It makes me so uncomfortable; to put it in right words… it wouldn’t be so uncomfortable with my pant zip open than having an untidy hair.  Ever wondered the mathematics behind how your hair style can get chaotic when a blow of wind comes your way? Truly speaking, you can actually calculate this! It depends on average length of your hair, the hair style equations, direction of the wind, Wind speed, strength of the hair style (gel, oil, etc) and angle between your direction and wind direction. Let me know if anyone of you are able to get any accurate formula experimentally.

 

Now coming to recharging my cell phone, I really don’t understand why I always forget to recharge it. Either there is no charge in it or no talk time left over! This happens especially when someone sends a nice SMS and I have no talk time left even to reply that particular SMS.  Anyhow, I think that’s what my life is.

 

Extracted from my another blog : http://perfdude.spaces.live.com

 

Web Application Performance

My basic intension of starting this new blog space on Web Application Performance is to educate developers and software community about the importance of Performance in Web Applications.  Though the software community does realize the impact of performance in their web based business, it is seldom we see that an active Performance oriented start-to-end approach is adopted in Project life-cycles. I would be dealing in detail about the various performance related issues faced by a client while accessing a web application, how these issues can be detected before going in production and most importantly how these can be resolved.

My posts would be more intended towards the beginners and I would be adopting a very simple approach in explaining the various problems and solutions. Let me confess, I do not have high amount of technical knowledge in Web Performance and I am still a learner in Performance field. I would also be sharing some of my learning experiences, the issues I faced while doing performance analysis and more importantly the various things I have learnt from other experts in Performance field.

One of the free resources available on NET regarding Web Application Performance is http://www.codeplex.com/PerfTesting. If you intend to do performance testing on ASP.NET Web applications sometime in future then do read the free e-book called “Performance Testing Guidance for Web Application”, published by Patterns & Practices, Microsoft. You can download the PDF file of the book from http://www.codeplex.com/PerfTestingGuide.

In my next post I would be dealing with generic client problems faced while accessing a web application and how it impacts your business.

 

The Customer’s Prospective

Customer satisfaction is always a tough commitment given by a Service Provider to its clients but then it’s a must for a business to run. The behavior of a Customer is always unpredictable and it’s the responsibility of the Service Provider to be prepared for all possibilities. The situation is even more delicate when you are service provider for a Web based application. There is always a little enigma and distrust in the mind of the customer about Web applications. Currently we are in a transition phase from manual computing to web based computing and it’s will consume certain time to build trust and faith among customers in robustness of web applications. Even a little erratic behavior of your web application can drive away the customer causing a heavy damage to your business.

Money decides the direction of business and it appears that money is in “Web Applications”. The software firms are flocking up new innovative web applications to attract new customers however it’s all in the hands of customers to decide the fate of the Business. The big question before software firms is “what does the customer expect from us other the web application itself?” You might be having an exceptional web application in the market but then there are certain other norms to be satisfied by that application to catch the eye of the customer. The customer who is accessing your web application doesn’t understand the difficulty you faced in coding and making that particular application work, all that customer wants is whether it satisfies his/her needs. There are few things about the application that customer will be able to judge very quickly while looking at your application. These include look, ease of usage and Performance.  Look and Ease of Usage are purely based upon the design of the application. Now-a-days you find websites using different graphic techniques to make the website look attractive and elegant. The ease of usage is again very important for most of the application. Remember, not all the people in the world are enthusiastic about computers and as a service provider you cannot afford to lose even such people. It is of high importance to design a website which is easier to use.

Having gone through the look and ease of usage of the application, the next feature that customer would like to verify while accessing your application is its performance. The first feedback that customer gets about your application is its page load time. No customer would like to wait for a long duration for the page to load. Especially, in applications where you are selling or marketing a product, the page load time of the application is of high importance. The general statistical observation is that the customer spends up to 5 seconds for the page to load. If your application has pages that take more than 5 seconds to load then there is a potential chance that you might lose your customer. General tendency of the customer is to close the application and do other useful work rather than waiting for a page to load! And that’s what you and me will do while accessing any website on NET.

 

Page Load Time

It is unacceptable to have a page load time more than 5 seconds for your web application. Basically, you tend to lose away customers because they don’t want to sit idle till your sick slow page loads. There are multiple reasons why your web application can have a high page load time. In the next few paragraphs I shall discuss the various reasons that can cause High Page load time. There could be many more reasons which are not mentioned here, however, these are the common reasons that I have found.

 

High Data Transfer:

For typical .aspx pages, the recommended value for data transfer is 200KB. Any value higher than 200KB can affect the page load time due to network latency. Especially in cases where your pages has pictures, make sure that they are in compressed format. Recently I was surprised to find a website having header pictures in .bmp format! Make sure that pictures you are using in your .aspx pages are of JPEG or GIF format. One general observation is that resolution of actual image transferred and shown on the web page is not same. It means that you showing a lesser resolution image to viewer and yet transferring a image with higher resolution. Such cases should be avoided. In case, your .aspx page is going over 200KB then you should opt for IIS Compression. IIS Compression is a built in feature in IIS 6.0.  By default IIS compression is not enabled. I will post about IIS Compression in detail in my future posts.

 

Servers Unable to Handle Peak Load:

As the number of concurrent users accessing your website increases, there is an increase in the CPU utilization and Memory of the web server.  This is because each .aspx page is processed at Web server which results in certain amount CPU utilization. Say, when 100 users are concurrently accessing your website, it means about 100 connections are being made to Web server. For processing each connection certain amount of CPU and Memory is used depending on your application. There is always a threshold limit of concurrent users that your server can handle, if the concurrent users exceed that particular threshold then your Web server will take more time to process the request, hence causing delay in processing and increasing the page load time.  In any case, it should be seen that %CPU utilization at Web server/DB should not exceed 75%.

 

Deadlocks:

There is possibility that when more than one client is accessing your web applications, a deadlock might occur for usage of certain resources like database, etc. You need to make sure that no such deadlock occurs. Generally the deadlocks occur at SQL box. Say to clients are updating a same row, there is possibility that a deadlock might occur. Deadlocks are dangerous as they halt your application for certain duration of time.

 

WCF offers significant performance results compared to the other existing distributed technologies such as ASMX and WSE. In terms of throughput, i.e. number of transactions per second, WCF is 25-50% faster than traditional ASP.NET Web Services. A detailed study on performance comparisons of WCF with existing distributed communication technologies can be found at

 

http://msdn2.microsoft.com/en-us/library/bb310550.aspx

 

 

To achieve the best performance oriented results, one has to fine tune throttle parameters as per the requirements of the applications.  The default values of the throttle parameters are slightly on the lower side and you need to scale the values up as per the needs of your application and for better performance. Defining the optimum values of the throttle parameters can significantly increase the performance. The throttle parameters include MaxConcurrentCalls, MaxConcurrentSessions and MaxConcurrentInstances. MaxConcurrentCalls specifies the number of active messages being processed across the service host object and its default value is 16. MaxConcurrentSessions indicates maximum number of concurrent users/connections WCF can handle provided it has enough resources. If the number of client machines accessing the WCF service exceed the specified no. of MaxConcurrentSessions they get queued up until the free session are available. The default value of MaxConcurrentSessions is 10. MaxConcurrentInstances specifies the number of service objects each service can have.

 

High throughput can be achieved by keeping the values of throttle parameters as low as possible, however the low values can affect the number of users accessing the web service. For an application that needs to handle high number of concurrent users, setting the higher value for MaxConcurrentSessions can benefit at the cost of higher response time. To decide the optimum values for the throttle parameters for a particular application, one has to test it at different numbers.