Maths History-Notation

Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics
was written out in words, a painstaking process that limited mathematical
discovery.

In the 18th century, Euler was responsible for many of the notations in use today. Modern notation makes
mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict
syntax and encodes information that would be difficult to write in any other way.

Mathematical language can also be hard for beginners. Words such as or and only have more
precise meanings than in everyday speech. Additionally, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their
theorems to follow from axioms by means of systematic reasoning. This is to
avoid mistaken "theorems", based on fallible intuitions, of which many
instances have occurred in the history of the subject.The level of rigor
expected in mathematics has varied over time: the Greeks expected detailed
arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the
definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large
computations are hard to verify, such proofs may not be sufficiently rigorous. Axioms in traditional
thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of
an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a
firm axiomatic basis, but according to Gödel's incompleteness theorem every
(sufficiently powerful) axiomatic system has undecidable formulas; and so a
final axiomatization of mathematics is impossible.

Nonetheless mathematics is often imagined to be (as far as its formal content)
nothing but set theory in some axiomatization, in the sense that every mathematical
statement or proof could be cast into formulas within set theory.

Maths History - History

History

• The evolution of mathematics might be seen as an ever-increasing series of abstractions,
  or alternatively an expansion of subject matter.
• The first abstraction was probably that of numbers.
• The realization that two apples and two oranges have something in common was a
  breakthrough in human thought.
• In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Arithmetic
  (addition, subtraction,multiplication and division), naturally followed.
• Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.
• Numeralsystems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus. The Indus Valley civilization developed the modern decimal system, including the concept of zero.
• From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics into the studies of quantity, structure,space, and change.
• since been greatly extended, and there has been a fruitful interaction between
  mathematics and science, to the benefit of both.
• Mathematical discoveries have been made throughout history and continue to be made today.
• According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American
  Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year.
• The overwhelming majorityof works in this ocean contain new mathematical theorems and
  their proofs."

Maths History - Etymology

• The word "mathematics" (Greek: μαθηματικά or mathēmatiká)
  comes from the Greek μάθημα (máthēma),
• which means learning, study, science, and additionally came to have the narrower and
• more technical meaning "mathematical study", even in Classical times.
• Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical.
• In particular, μαθηματικὴτέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.
• The apparent plural form in English, like the French plural form les mathématiques (and
  the less commonly used singular derivative la mathématique), goes back
  to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα
  μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things
  mathematical".
• In English, however, the noun mathematics takes
  singular verb forms.
• It is often shortened to math in English-speaking
  North America and maths elsewhere.

Mathematics

• Mathematics is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them.
• Benjamin Peirce called it "the science that draws necessary conclusions".
• Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere.
• Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.
• Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life.
• Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds.
• Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.
  
• Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines.
• Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.

SET EXAM- Mathematics Paper III

Mathematics Paper III (Unit 16 to 24)

16. Linear Integral Equations :
Linear integral Equations of the first and second kind of Fredholm and Volterra type, solution by successive substitutions and successive approximations ; Solution of equations with separable kernels ; The Fredholm Alternative ; Holbert – Schmidt theory for symmetric kernels.
17. Numerical analysis :
Finite differences, interpolation ; Numerical solution of algebric equation ; Iteration ; Newton – Rephson method ; Solution on linear system ; Direct method ; Gauss elimination method ; Matrix – Inversion, elgenvalue problems ; Numerical differentiation and integration.Numerical solution of ordinary differential equation; iteration method, Picard’s method, Euler’s method and improved Euler’s method.
18. Integral Transform :
Laplace transform ; Transform of elementary functions, Transform of Derivatives, inverse Transform, Comrolution Theorem, Applications, Ordinary and Partial differential equations ; Fourier transforms ; sine and cosine transform, Inverse Fourier Transform, Application to ordinary and partial differential equations. 19. Mathematical Programming :
Revised simplex method, Dual simplex method, Sensitivity analysis and parametric linear programming. Kuhn – Tucker conditions of optimality. Quadratic programming ; methods due to Beale, Wofle and Vandepanne, Duality in quadranic programming, self duality, integer programming.
20. Measure Theory :
Measurable and measure spaces ; Extension of measures, signed measures, Jordan – Hahn decomposition theorems. Integration, monotone convergence theorem, Fatou’s lemma, dominated convergence theorem. Absolute continuity. Radon Nikodym theorem, Product measures, Fubini’s theorem.
21. Probability :
Sequences of events and random variables ; Zero – one laws of Borel and Kolmogorov. Almost sure convergence, convergence in mean square, Khintchine’s weak law of large numbers ; Kolmogorov’s inequality, strong law of large numbers.Convergence of series of random variables, three – series criterion. Central limit theorems of Liapounov and Lindeberg – Feller. Conditional expectation, martingales.
22. Distribution Theory :
Properties of distribution functions and characteristic functions ; continuity theorem, inversion formula, Representation of distribution function as a mixture of discrete and continuous distribution functions ; Convolutions, marginal and conditional distributions of bivariate discrete and continuous distributions.Relations between characteristic functions and moments ; Moment inequalities of Holder and Minkowski.
23. Statistical Inference and Decision Theory :
Statistical decision problem : non – randomized, mixed and randomized decision rules ; risk function, admissibility, Bayes’ rules, minimax rules, least favourable distributions, complete class, and minimal complete class. Decision problem for finite parameter class. Convex loss function. Role of sufficiency.Admissible, Bayes and minimax estimators ; Illustrations, Unbiasedness. UMVU estimators. Families of distributions with monotone likelihood property, exponential family of distributions. Test of a simple hypothesis against a simple alternative from decision – theoretic view point. Tests with Neymen structure. Uniformly most powerful unbiased tests. Locally most powerful tests. Inference on location and scale parameters estimation and tests. Equivariant estimators invariance in hypothesis testing.
24. Large sample statistical methods :
Various modes of convergence, Op and op, CLT Sheffe’s theorem, Polya’s theorem and Stutsky’s theorem. Transformation and variance stabilizing formula. Asymptotic distribution of function of sample moments. Sample quantities. Order statistics and their functions. Tests on correlations, coefficients of variation, skewness and kurtosis. Pearson Chi-square, contingency. Chi – square and likelihood ratio statistics. U – statistics. Consistency of Tests. Asymptotic relative efficiency.



Mathematics Paper III  (Unit 31 to 34)

31. Demography and Vital Statistics :
Measures of fertility and mortality, period and Cohort measures.Life tables and its applications ; Methods of construction of abridged life tables. Application of stable population theory to estimate vital rates. Population projections, Stochastic models of fertility and reproduction.
32. Industrial Statistics :
Control charts for variables and attributes ; Acceptance sampling by attributes ; single, double and sequential sampling plans ; OC and ASN functions, AOQL and ATI ; Acceptance sampling by varieties. Tolerance limits. Reliability analysis : Hazard function, distribution with DFR and IFR ; Series and parallel systems. Life testing experiments.
33. Inventory and Queueing theory :
Inventory ( S, S ) policy, periodic review models with stochastic demand. Dynamic inventory models. Probabilistic re-order point, lot size inventory system with and without lead time. Distribution free analysis. Solution of inventory problem with unknown density function. Warehousing problem. Queues ; Imbedded Markov Chain method to obtain steady state solution of M/G/1, G/M/1 AND M/D/C, Network models. Machine maintenance models. Design and control of queueing systems.
34. Dynamic Programming and Marketing :
Nature of dynamic programming, Deterministic processes, Non-sequential discrete optimization – allocation problems, assortment problems. Sequential discrete optimization long – term planning problems, multi stage production processes. Functional approximations. Marketing systems, application of dynamic programming to marketing problems. Introduction of new product, objective in setting market price and its policies, purchasing under fluctuating prices, Advertising and promotional decisions, Brands switching analysis, Distribution decisions.

Mathematics Paper III (Unit 1 to 6)

1. Real Analysis :
Riemann integrate functions ; improper integrate, their convergence and uniform convergence. Eulidean space R¯ , Boizano – Weleratrass theorem, compact. Subsets of R•, Heine – Borel theorem, Fourier series.Continuity of functions on R”, Differentiability of F : R• > Rm, Properties of differential, partial and directional derivatives, continuously differentiable functions. Taylor’s series. Inverse function theorem, implicit function theorem.Integral functions, line and surface integrals, Green’s theorem. Stoke’s theorem.
2. Complex Analysis :
Cauchy’s theorem for convex regions, Power series representation of Analytic functions. Liouville’s theorem, Fundamental theorem of algebra, Riemann’s theorem on removable singularities, maximum modulus principle. Schwarz lemma, Open Mapping theorem, Casoratti – Weierstrass – theorem, Weierstrass’s theorem on uniform convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions, Riemann Surfaces.
3. Algebra :
Symmetric groups, alternating groups, Simple groups, Rings, Maximal ideals, Prime ideals, integral domains, Euclidean domains, principal ideal domains, Unique Factorisation domains, quotient fields, Finite fields, Algebra of Linear Transformations, Reduction of matrices to Canonical Forms, Inner Product Spaces, Orthogonality, quadratic Forms, Reduction of quadratic forms.
4. Advanced Analysis :
Elements of Metric Spaces, Convergence, continuity, compactness, Connectedness, Weierstrass’s approximation Theorem. Completeness, Bare category theorem, Labesgue measure, Labesgue integral, Differentiation and integration.
5. Advanced Algebra :
Conjugate elements and class equations of finite groups, Sylow theorems, solvable groups, Jordan Holder Theorem, Direct Products, Structure Theorem for finite abelian groups, Chain conditions on Rings : Characteristic of Field, Field extensions, Elements of Galois theory, solvability by Radicals, Ruler and compass construction.
6. Functional Analysis :
Banach Spaces, Hahn – Banach Theorem, Open mapping and closed Graph Theorems. Principle of Uniform boundedness, Boundedness and continuity of Linear Transformations. Dual Space, Embedding in the second dual, Hilbert Spaces, Projections. Orthonormal Basis, Riesz – representation theorem. Bessel’s inequality, parsaval’s identity, self adjoined operators, Normal Operators.

Mathematics Paper III (Unit 25 - 30)

25. Multivariate Statistical Analysis :
Singular and non – singular multivariate distributions. Characteristics functions. Multivariate normal distribution ; marginal and conditional distribution, distribution of linear forms, and quadratic forms, Cochran’s theorem.
Inference on parameters of multivariate normal distributions : one – population and two – population cases. Wishart distribution. Hotellings T2, Mahalanobis D2, Discrimination analysis, Principal components, Canonical correlations, Cluster analysis.
26. Linear Models and Regression :
Standard Gauss – Markov models ; Estimability of parameters ; best linear unbiased estimates ( BLUE ); Method of least squares and Gauss – Markov theorem ; Variance – covariance matrix of BLUES.
Tests of linear hypothesis ; One – way and two – way classifications. Fixed, random and mixed effects models ( two – way classifications only ); variance components, Bivariate and multiple linear regression; Polynomial regression ; use of orthogonal polynomials. Analysis of covariance. Linear and nonlinear regression. Outliers.
27. Sample Surveys : Sampling with varying probability of selection, Hurwitz – Thompson estimator ; PPS sampling ; Double sampling, Cluster sampling. Non-sampling errors ; interpenetrating samples. Multiphase sampling. Ratio and regression methods of estimation.
28. Design of Experiments :
Factorial experiments, confounding and fractional replication. Split and strip plot designs ; Quesi – Latin square designs ; Youden square. Design for study of response surfaces ; first and second order designs. Incomplete block designs ; Balanced, connectedness and orthogonality, BIBD with recovery of inter-block information, PBIBD with 2 associate classes. Analysis of sense of experiments, estimation of residual effects. Construction of orthogonal – Latin squares, BIB designs, and confounded factorial designs. Optimality criteria for experimental designs.
29. Time – Series Analysis :
Discrete – parameter stochastic processes ; strong and weak stationarity ; autocovariance and autocorrelation, Moving average, autoregressive, autoregressive moving average and autoregressive integrated moving average processes. Box – Jenkins models. Estimation of the parameters in ARIMA models, forecasting. Perfodogram analysis.
30. Stochastic Processes :
Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution ; branching processes ; Random walk ; Gambler’s ruin. Markov processes in continuous time ; Poisson processes, birth and death processes, Wiener process.

SET EXAM Mathematics Paper II

1. Basic concepts of Real and Complex analysis :

Sequences and series, continuity, uniform continuity, Differentiability, Mean Value Theorem, sequences and series of functions, uniform convergence, Riemann integral – definition and simple properties. Algebra of Complex numbers, Analytic functions. Cauchy’s Theorem and integral formula, Power series, Taylor’s and Laurent’s series, Residues, Contour integration.


2.Basic Concepts of Linear Algebra :

Space of n-vectors, Linear dependence, Basis, Linear transformation, Algebra of matrices, Rank of a matrix, Determinants, Linear equations, Quadratic forms, Characteristic roots and vectors.

3.Basic concepts of probability :

Sample space, discrete probability, simple theorems on probability, independence of events, Bayes Theorem. Discrete and continuous random variables, Binomial, Paisson and Normal distributions ; Expectation and moments, independence of random variables, Chebyshev’s inequality.

4. Linear Programming Basic Concepts:

Convex sets. Linear Programming Problem ( LPP ). Examples of LPP, Hyperplane, open and closed half – spaces. Feasible, basic feasible and optimal solutions. Extreme point and graphical method.

5. Real Analysis :

Finite, countable and uncountable sets, Bounded and unbounded sets. Archimedean property, ordered field, completeness of R, Extended real number system, liens up and limits of a sequence, the epsilon – delta definition of continuity and convergence, the algebra of continuous functions, monotonic functions, types of discontinuities, infinite limits and limits at infinity, functions of bounded variation, elements of metric spaces.

6.Complex Analysis:

Riemann Sphere and Stereographic projection. Lines, Circles, crossratio. Mobius transformations, Analytic functions, Cauchy - Riemann equations, line integrals, Cauchy's theorem, Morera's theorem, Liouville's theorem, integral formula, zero-sets of analytic functions, exponential, sine and cosine functions, Power series representation, Classification of singularities, Conformal Mapping.


7. Algebra:

Group, subgroups, Normal subgroups, Quotient Groups, Homomorphisms, Cyclic Groups, permutation Groups, Cayley's Theorem, Rings, Ideals, Integral Domains, Fields, Polynomial Rings.


8.Linear Algebra:

Vector spaces, subspaces, quotient spaces, Linear independence, Bases, Dimension. The algebra of linear Transformations, kernel, range, isomorphism, Matrix Representation of a linear transformation, change of bases, Linear functionals, dual space, projection, determinant function, eigenvalues and eigen vectors, Cayley-Hamilton Theorem, Invariant Sub-spaces, Canonical Forms: diagonal form, Triangular form, Jordan Form, Inner product spaces.


9. Differential Equations:

First order ODE, singular solutions, initial value Problems of First Order ODE, General theory of homogeneous and non-homogeneous Linear ODE, Variation of Parameters. Lagrange's and Charpit's methods of solving first order Partial Differential Equations. PDE's of higher order with constant coefficients.


10.Data Analysis Basic Concepts:

Graphical representation, measures of central tendency and dispersion. Bivariate data, correlation and regression. Least squares - polynomial regression, Applications of normal distribution.


11.Probability: Axiomatic definition of probability. Random variables and distribution functions (univariate and multivariate); expectation and moments; independent events and independent random variables; Bayes' theorem; marginal and conditional distribution in the multivariate case, covariance matrix and correlation coefficients (product moment, partial and multiple), regression.
Moment generating functions, characteristic functions; probability inequalities (Tchebyshef, Markov, Jensen). Convergence in probability and in distribution; weak law of large numbers and central limit theorem for independent identically distributed random variables with finite variance.
12.Probability Distribution: Bernoulli, Binomial, Multinomial, Hypergeomatric, Poisson, Geometric and Negative binomial distributions, Uniform, exponential, Cauchy, Beta, Gamma, and normal (univariate and multivariate) distributions Transformations of random variables; sampling distributions. t, F and chi-square distributions as sampling distributions, Standard errors and large sample distributions. Distribution of order statistics and range.
13. Theory of Statistics: Methods of estimation: maximum likelihood method, method of moments, minimum chi-square method, least-squares method. Unbiasedness, efficiency, consistency. Cramer-Rao inequality. Sufficient Statistics. Rao-Blackwell Theorem. Uniformly minimum variance unbiased estimators. Estimation by confidence intervals. Tests of hypotheses: Simple and composite hypotheses, two types of errors, critical region, randomized test, power function, most powerful and uniformly most powerful tests. Likelihood-ratio tests. Wald's sequential probability ratio test.
14. Statistical methods and Data Analysis: Tests for mean and variance in the normal distribution: one-population and two- population cases; related confidence intervals. Tests for product moment, partial and multiple correlation coefficients; comparison of k linear regressions. Fitting polynomial regression; related test. Analysis of discrete data: chi-square test of goodness of fit, contingency tables. Analysis of variance: one-way and two-way classification (equal number of observations per cell). Large-sample tests through normal approximation. Nonparametric tests: sign test, median test, Mann-Whitney test, Wilcoxon test for one and two-samples, rank correlation and test of independence.
15. Operational Research Modelling: Definition and scope of Operational Research. Different types of models. Replacement models and sequencing theory, Inventory problems and their analytical structure. Simple deterministic and stochastic models of inventory control. Basic characteristics of queueing system, different performance measures. Steady state solution of Markovian queueing models: M/M/1, M/M/1 with limited waiting space M/M/C, M/M/C with limited waiting space.
16.Linear Programming: Linear Programming, Simplex method, Duality in linear programming. Transformation and assignment problems. Two person-zero sum games. Equivalence of rectangular game and linear programming.
17. Finite Population: Sampling Techniques and Estimation: Simple random sampling with and without replacement. Stratified sampling; allocation problem; systematic sampling. Two stage sampling. Related estimation problems in the above cases.
18. Design of Experiments: Basic principles of experimental design. Randomisation structure and analysis of completely randomised, randomised blocks and Latin-square designs. Factorial experiments. Analysis of 2n factorial experiments in randomised blocks.

SET EXAM - Mathematics Paper III (Unit 7-15)

7. Topology:
Elements of Topological Spaces, Continuity, Convergence, Homeomorphism, Compactness, Connectedness, Separation Axioms, First and Second Countability, Separability, Subspaces, Product Spaces, quotient spaces. Tychonoff's Theorem, Urysohn's Metrization theorem, Homotopy and Fundamental Group.
8.Discrete Mathematics:
Partially ordered sets, Lattices, Complete Lattices, Distributive lattices, Complements, Boolean Algebra, Boolean Expressions, Application to switching circuits, Elements of Graph Theory, Eulerian and Hamiltonian graphs, planar Graphs, Directed Graphs, Trees, Permutations and Combinations, Pigeonhole principle, principle of Inclusion and Exclusion, Derangements.
9.Ordinary and Partial Differential Equations:
Existence and Uniqueness of solution dy/dx =f(x,y) Green's function, sturm Liouville Boundary Value Problems, Cauchy Problems and Characteristics, Classification of Second Order PDE, Separation of Variables for heat equation, wave equation and Laplace equation, Special functions.
10.Number Theory:
Divisibility; Linear diophantine equations. Congruences. Quadratic residues; Sums of two squares, Arithmetic functions Mu, Tau, Phi and Sigma ( and ).
11.Mechanics:
Generalised coordinates; Lagranges equation; Hamilton's cononical equations; Variational principles - Hamilton's principles and principles of least action; Two dimensional motion of rigid bodies; Euler's dynamical equations for the motion of rigid body; Motion of a rigid body about an axis; Motion about revolving axes.
12.Elasticity:
Analysis of strain and stress, strain and stress tensors; Geometrical representation; Compatibility conditions; Strain energy function; Constitutive relations; Elastic solids Hookes law; Saint-Venant's principle, Equations of equilibrium; Plane problems - Airy's stress function, vibrations of elastic, cylindrical and spherical media.
13.Fluid Mechanics:
Equation of continuity in fluid motion; Euler's equations of motion for perfect fluids; Two dimensional motion complex potential; Motion of sphere in perfect liquid and motion of liquid past a sphere; vorticity; Navier-Stokes's equations for viscous flows-some exact solutions.
14.Differetial Geometry:
Space curves - their curvature and torsion; Serret Frehat Formula; Fundamental theorem of space curves; Curves on surfaces; First and second fundamental form; Gaussian curvatures; Principal directions and principal curvatures; Goedesics, Fundamental equations of surface theory.
15.Calculus of Variations:
Linear functionals, minimal functional theorem, general variation of a functional, Euler- Lagrange equation; Variational methods of boundary value problems in ordinary and partial differential equations.