Saturday 29 July 2017

Zero (शून्य smoke)

शून्य (0) एक अंक है जो संख्याओं के निरूपण के लिये प्रयुक्त आजकी सभी स्थानीय मान पद्धतियों का अपरिहार्य प्रतीक है। इसके अलावा यह एक संख्या भी है। दोनों रूपों में गणित में इसकी अत्यन्त महत्वपूर्ण भूमिका है। पूर्णांकों तथा वास्तविक संख्याओं के लिये यह योग का तत्समक अवयव(additive identity) है।


  • किसी भी वास्तविक संख्या को शून्य से गुणा करने से शून्य प्राप्त होता है। (x * 0 = 0)
  • किसी भी वास्तविक संख्या को शून्य से जोड़ने या घटाने पर वापस वही संख्या प्राप्त होती है। (x + 0 = x ; x - 0 = x)

शून्य का आविष्कार किसने और कब किया यह आज तक अंधकार के गर्त में छुपा हुआ है, परंतु सम्पूर्ण विश्व में यह तथ्य स्थापित हो चुका है कि शून्य का आविष्कार भारत में ही हुआ। ऐसी भी कथाएँ प्रचलित हैं कि पहली बार शून्य का आविष्कार बाबिल में हुआ और दूसरी बार माया सभ्यता के लोगों ने इसका आविष्कार किया पर दोनो ही बार के आविष्कार संख्या प्रणाली को प्रभावित करने में असमर्थ रहे तथा विश्व के लोगों ने इन्हें भुला दिया। फिर भारत में हिंदुओंने तीसरी बार शून्य का आविष्कार किया। हिंदुओं ने शून्य के विषय में कैसे जाना यह आज भी अनुत्तरित प्रश्न है। अधिकतम विद्वानों का मत है कि पांचवीं शताब्दी के मध्य में शून्य का आविष्कार किया गया। सर्वनन्दि नामक दिगम्बर जैन मुनि द्वारा मूल रूप से प्रकृत में रचित लोकविभागनामक ग्रंथ में शून्य का उल्लेख सबसे पहले मिलता है। इस ग्रंथ में दशमलव संख्या पद्धति का भी उल्लेख है


अर्थात् "एक, दश, शत, सहस्र, अयुत, नियुत, प्रयुत, कोटि, अर्बुद तथा बृन्द में प्रत्येक पिछले स्थान वाले से अगले स्थान वाला दस गुना है।"[1] और शायद यही संख्या के दशमलव सिद्धान्त का उद्गम रहा होगा। आर्यभट्ट द्वारा रचित गणितीय खगोलशास्त्र ग्रंथ 'आर्यभट्टीय' के संख्या प्रणाली में शून्य तथा उसके लिये विशिष्ट संकेत सम्मिलित था (इसी कारण से उन्हें संख्याओं को शब्दों में प्रदर्शित करने के अवसर मिला)। प्रचीन बक्षाली पाण्डुलिपि में, जिसका कि सही काल अब तक निश्चित नहीं हो पाया है परन्तु निश्चित रूप से उसका काल आर्यभट्ट के काल से प्राचीन है, शून्य का प्रयोग किया गया है और उसके लिये उसमें संकेत भी निश्चित है। उपरोक्त उद्धरणों से स्पष्ट है कि भारत में शून्य का प्रयोग ब्रह्मगुप्त के काल से भी पूर्व के काल में होता था।
शून्य तथा संख्या के दशमलव के सिद्धान्त का सर्वप्रथम अस्पष्ट प्रयोग ब्रह्मगुप्त रचित ग्रंथ ब्राह्मस्फुटसिद्धान्त में पाया गया है। इस ग्रंथ में ऋणात्मक संख्याओं (negative numbers) और बीजगणितीय सिद्धान्तों का भी प्रयोग हुआ है। सातवीं शताब्दी, जो कि ब्रह्मगुप्त का काल था, शून्य से सम्बंधित विचार कम्बोडिया तक पहुँच चुके थे और दस्तावेजों से ज्ञात होता है कि बाद में ये कम्बोडिया से चीनतथा अन्य मुस्लिम संसार में फैल गये।
इस बार भारत में हिंदुओं के द्वारा आविष्कृत शून्य ने समस्त विश्व की संख्या प्रणाली को प्रभावित किया और संपूर्ण विश्व को जानकारी मिली। मध्य-पूर्व में स्थित अरब देशों ने भी शून्य को भारतीय विद्वानों से प्राप्त किया। अंततः बारहवीं शताब्दी में भारत का यह शून्य पश्चिम में यूरोप तक पहुँचा।
भारत का 'शून्य' अरब जगत में 'सिफर' (अर्थ - खाली) नाम से प्रचलित हुआ। फिर लैटिनइटैलियनफ्रेंच आदि से होते हुए इसे अंग्रेजी में 'जीरो' (zero) कहते हैं
From Wikipedia...

PROFIT AND LOSS FORMULLA (लाभ और हानि )

FORMULLA OF PROFIT AND LOSS-

In case of profit,
profit = selling price – cost priceselling price = cost price + profitcost price = selling price - prof

In case of loss,
loss = cost price - selling price or c-s

selling price = cost price - loss or c-loss

cost price = selling price + loss or s+loss


Profit percentage and loss percentage

profit percentage=profit×100cost price or p/c *100

selling price=cost price+cost price×profit percentage100=cost price(100+profit percentage)100

cost price=100×selling price100+profit percentage



एक बिजली के आयरन को 15% के लाभ पर बेचा गया यदि उसे 600 रूपये में बेचा जाय तो 20% का लाभ होगा ताे उ‍सका बिक्रय मूल्य क्या था
हल -
frofit and loss question solve with pdf click here

Thursday 27 July 2017

Ratio (अनुपात )





Maths related problem in Hindi.. click here
a:b=1:2  b:c=1:3 find a:b:c=?

a:b=1:2
b:c=1:3 multiple with 2
So b:c =2:6

Therefore a:b:c= 1:2:6


a:b=2:5  b:c=2:3 find a:b:c=?

a:b=2:5 multiple with 2
b:c=2:3 multiple with 5
a:b=4:10
So b:c =10:15

Therefore a:b:c= 4:10:15..

State whether the given statements are true or false:
(a) 12 : 18 = 28 : 56
(b) 25 persons : 130 persons = 15kg : 78kg

Solution: (a) False, Because
12:18 = 2:3 (divided from 6)

and

28:56 = 1:2 (divide it from 28)

These are not equal..

Question-The length and breadth of a steel tape are 10m and
2.4cm, respectively. The ratio of the length to the
breadth is.
 solution- since 1 miter = 100 cm
therefore 10 m = 1000 cm
so rario of lenth and breadth is l:b = 1000: 2.4 =10000:24 =2500:6 (divided it from 4)
                                                      = 1250:3 (divided it from 2)
question :-For a proportion of a: b:: c :d ,
d is the fourth proportional of a, b, c
c is called third proportional to a, b
Mean proportional between a and b is root ab
Invertendo of a/b=c/d. is b/a=d/c
Alternendo of a/b=c/d is a/c=b/d...

A quantity M divided in the ratio of a : b, then each part of the quantity is
I. The first part is M*a/a+b
2  The second part is M*b/a+b
If a : b = x : y and b : c = p : q, then a : b : c = px : py : yq














Mathematics modelling

Mathematical Modeling





1  Why mathematical modeling?

Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application.


Mathematical modeling
  • is indispensable in many applications
  • is successful in many further applications
  • gives precision and direction for problem solution
  • enables a thorough understanding of the system modeled
  • prepares the way for better design or control of a system
  • allows the efficient use of modern computing capabilities
Learning about mathematical modeling is an important step from a theoretical mathematical training to an application-oriented mathematical expertise, and makes the student fit for mastering the challenges of our modern technological culture.



2  A list of applications

In the following, I give a list of applications whose modeling I understand, at least in some detail. All areas mentioned have numerous mathematical challenges.
This list is based on my own experience; therefore it is very incomplete as a list of applications of mathematics in general. There are an almost endless number of other areas with interesting mathematical problems.
Indeed, mathematics is simply the language for posing problems precisely and unambiguously (so that even a stupid, pedantic computer can understand it).
Anthropology
  • Modeling, classifying and reconstructing skulls
Archeology
  • Reconstruction of objects from preserved fragments
  • Classifying ancient artifices
Architecture
  • Virtual reality
Artificial intelligence
  • Computer vision
  • Image interpretation
  • Robotics
  • Speech recognition
  • Optical character recognition
  • Reasoning under uncertainty
Arts
  • Computer animation (Jurassic Park)
Astronomy
  • Detection of planetary systems
  • Correcting the Hubble telescope
  • Origin of the universe
  • Evolution of stars
Biology
  • Protein folding
  • Humane genome project
  • Population dynamics
  • Morphogenesis
  • Evolutionary pedigrees
  • Spreading of infectuous diseases (AIDS)
  • Animal and plant breeding (genetic variability)
Chemical engineering
  • Chemical equilibrium
  • Planning of production units
Chemistry
  • Chemical reaction dynamics
  • Molecular modeling
  • Electronic structure calculations
Computer science
  • Image processing
  • Realistic computer graphics (ray tracing)
Criminalistic science
  • Finger print recognition
  • Face recognition
Economics
  • Labor data analysis
Electrical engineering
  • Stability of electric curcuits
  • Microchip analysis
  • Power supply network optimization
Finance
  • Risk analysis
  • Value estimation of options
Fluid mechanics
  • Wind channel
  • Turbulence
Geosciences
  • Prediction of oil or ore deposits
  • Map production
  • Earth quake prediction
Internet
  • Web search
  • Optimal routing
Linguistics
  • Automatic translation
Materials Science
  • Microchip production
  • Microstructures
  • Semiconductor modeling
Mechanical engineering
  • Stability of structures (high rise buildings, bridges, air planes)
  • Structural optimization
  • Crash simulation
Medicine
  • Radiation therapy planning
  • Computer-aided tomography
  • Blood circulation models
Meteorology
  • Weather prediction
  • Climate prediction (global warming, what caused the ozone hole?)
Music
  • Analysis and synthesis of sounds
Neuroscience
  • Neural networks
  • Signal transmission in nerves
Pharmacology
  • Docking of molecules to proteins
  • Screening of new compounds
Physics
  • Elementary particle tracking
  • Quantum field theory predictions (baryon spectrum)
  • Laser dynamics
Political Sciences
  • Analysis of elections
Psychology
  • Formalizing diaries of therapy sessions
Space Sciences
  • Trajectory planning
  • Flight simulation
  • Shuttle reentry
Transport Science
  • Air traffic scheduling
  • Taxi for handicapped people
  • Automatic pilot for cars and airplanes




3 Basic numerical tasks

The following is a list of categories containing the basic algorithmic toolkit needed for extracting numerical information from mathematical models.
Due to the breadth of the subject, this cannot be covered in a single course. For a thorough education one needs to attend courses (or read books) at least on numerical analysis (which usually covers some numerical linear algebra, too), optimization, and numerical methods for partial differential equations.
Unfortunately, there appear to be few good courses and books on (higher-dimensional) numerical data analysis.
Numerical linear algebra
  • Linear systems of equations
  • Eigenvalue problems
  • Linear programming (linear optimization)
  • Techniques for large, sparse problems
Numerical analysis
  • Function evaluation
  • Automatic and numerical differentiation
  • Interpolation
  • Approximation (Padé, least squares, radial basis functions)
  • Integration (univariate, multivariate, Fourier transform)
  • Special functions
  • Nonlinear systems of equations
  • Optimization = nonlinear programming
  • Techniques for large, sparse problems
Numerical data analysis (= numerical statistics)
  • Visualization (2D and 3D computational geometry)
  • Parameter estimation (least squares, maximum likelihood)
  • Prediction
  • Classification
  • Time series analysis (signal processing, filtering, time correlations, spectral analysis)
  • Categorical time series (hidden Markov models)
  • Random numbers and Monte Carlo methods
  • Techniques for large, sparse problems
Numerical functional analysis
  • Ordinary differential equations (initial value problems, boundary value problems, eigenvalue problems, stability)
  • Techniques for large problems
  • Partial differential equations (finite differences, finite elements, boundary elements, mesh generation, adaptive meshes)
  • Stochastic differential equations
  • Integral equations (and regularization)
Non-numerical algorithms
  • Symbolic methods (computer algebra)
  • Sorting
  • Compression
  • Cryptography
  • Error correcting codes



4 The modeling diagram

The nodes of the following diagram represent information to be collected, sorted, evaluated, and organized.

Information flow diagram


The edges of the diagram represent activities of two-way communication (flow of relevant information) between the nodes and the corresponding sources of information.
S. Problem Statement
  • Interests of customer/boss
  • Often ambiguous/incomplete
  • Wishes are sometimes incompatible
M. Mathematical Model
  • Concepts/Variables
  • Relations
  • Restrictions
  • Goals
  • Priorities/Quality assignments
T. Theory
  • of Application
  • of Mathematics
  • Literature search
N. Numerical Methods
  • Software libraries
  • Free software from WWW
  • Background information
P. Programs
  • Flow diagrams
  • Implementation
  • User interface
  • Documentation
R. Report
  • Description
  • Analysis
  • Results
  • Validation
  • Visualization
  • Limitations
  • Recommendations
Using the modeling diagram
  • The modeling diagram breaks the modeling task into 16=6+10 different processes.
  • Each of the 6 nodes and each of the 10 edges deserve repeated attention, usually at every stage of the modeling process.
  • The modeling is complete only when the 'traffic' along all edges becomes insignificant.
  • Generally, working on an edge enriches both participating nodes.
  • If stuck along one edge, move to another one! Use the general rules below as a check list!
  • Frequently, the problem changes during modeling, in the light of the understanding gained by the modeling process. At the end, even a vague or contradictory initial problem description should have developed into a reasonably well-defined description, with an associated precisely defined (though perhaps inaccurate) mathematical model.



5  General rules


  • Look at how others model similar situations; adapt their models to the present situation.
  • Collect/ask for background information needed to understand the problem.
  • Start with simple models; add details as they become known and useful or necessary.
  • Find all relevant quantities and make them precise.
  • Find all relevant relationships between quantities ([differential] equations, inequalities, case distinctions).
  • Locate/collect/select the data needed to specify these relationships.
  • Find all restrictions that the quantities must obey (sign, limits, forbidden overlaps, etc.). Which restrictions are hard, which soft? How soft?
  • Try to incorporate qualitative constraints that rule out otherwise feasible results (usually from inadequate previous versions).
  • Find all goals (including conflicting ones)
  • Play the devil's advocate to find out and formulate the weak spots of your model.
  • Sort available information by the degree of impact expected/hoped for.
  • Create a hierarchy of models: from coarse, highly simplifying models to models with all known details. Are there useful toy models with simpler data? Are there limiting cases where the model simplifies? Are there interesting extreme cases that help discover difficulties?
  • First solve the coarser models (cheap but inaccurate) to get good starting points for the finer models (expensive to solve but realistic)
  • Try to have a simple working model (with report) after 1/3 of the total time planned for the task. Use the remaining time for improving or expanding the model based on your experience, for making the programs more versatile and speeding them up, for polishing documentation, etc.
  • Good communication is essential for good applied work.
  • The responsibility for understanding, for asking the questions that lead to it, for recognizing misunderstanding (mismatch between answers expected and answers received), and for overcoming them lies with the mathematician. You cannot usually assume your customer to understand your scientific jargon.
  • Be not discouraged. Failures inform you about important missing details in your understanding of the problem (or the customer/boss) - utilize this information!
  • There are rarely perfect solutions. Modeling is the art of finding a satisfying compromise. Start with the highest standards, and lower them as the deadline approaches. If you have results early, raise your standards again.
  • Finish your work in time.
Lao Tse: ''People often fail on the verge of success; take care at the end as at the beginning, so that you may avoid failure.''



6  Conflicts

Most modeling situations involve a number of tensions between conflicting requirements that cannot be reconciled easily.

  • fast - slow
  • cheap - expensive
  • short term - long term
  • simplicity - complexity
  • low quality - high quality
  • approximate - accurate
  • superficial - in depth
  • sketchy - comprehensive
  • concise - detailed
  • short description - long description
Einstein: ''A good theory'' (or model) ''should be as simple as possible, but not simpler.''

  • perfecting a program - need for quick results
  • collecting the theory - producing a solution
  • doing research - writing up
  • quality standards - deadlines
  • dreams - actual results
The conflicts described are creative and constructive, if one does not give in too easily. As a good material can handle more physical stress, so a good scientist can handle more stress created by conflict.
''We shall overcome'' - a successful motto of the black liberation movement, created by a strong trust in God. This generalizes to other situations where one has to face difficulties, too.
Among other qualities it has, university education is not least a long term stress test - if you got your degree, this is a proof that you could overcome significant barriers. The job market pays for the ability to persist.



7 Attitudes


  • Do whatever you do with love. Love (even in difficult circumstances) can be learnt; it noticeably improves the quality of your work and the satisfaction you derive from it.
  • Do whatever you do as a service to others. This will improve your attention, the feedback you'll get, and the impact you'll have.
  • Take responsibility; ask if in doubt; read to confirm your understanding. This will remove many impasses that otherwise would delay your work.
Jesus: ''Ask, and you will receive. Search, and you will find. Knock, and the door will be opened for you.''



8  References

For more information about mathematics, software, and applications, see, e.g., my home page, athttp://www.mat.univie.ac.at/~neum/