Skip to main content

हार्मोनल परिवर्तन के कारण कम उम्र में लड़कियों में हो रहा शारीरिक परिवर्तन

हार्मोनल परिवर्तन के कारण कम उम्र में लड़कियों में हो रहा शारीरिक परिवर्तन प्रदुषण और जंक फूड घटा रहा मुश्किल भरे दिन की उम्र लड़कियों में समय से पहले हार्मोनल चेंज के मामले बढ़ रहे हैं। शारीरिक बदलाव के लिए औसत उम्र 13 से 14 साल मानी जाती है, अब 8 से 11-12 साल में वजन बढ़ने जैसी समस्या आ रही है। कम उम्र में मुश्किल भरे दिनों का सामना करना पड़ रहा है। विशेषज्ञों की मानें तो समस्या प्रदुषण और जंक फूड के अत्यधिक सेवन से बढ़ रही है। शरीर में हार्मोनल परिवर्तन, किसी प्रकार की सिस्ट और ट्यूमर जैसे कारण सामने आ रहे हैं। इनके लिए मुख्य कारण निम्न हैं- लड़कियों में हार्मोनल परिवर्तन से पीरियड जल्दी आते हैं। आनुवंशिक समस्या इसके लिए जिम्मेदार हो सकती है। तनाव से भी हार्मोनल परिवर्तन होते हैं। आयरन तथा विटामिन-डी जैसे पोषण तत्वों की कमी भी हार्मोनल परिवर्तन के लिए जिम्मेदार हैं। प्रदुषण के सम्पर्क में रहने पर भी ये समस्या होती है। शरीर में कही भी सिस्ट या ट्यूमर होने पर भी ये सम्भव है। अगर 8 साल से 12 साल की उम्र के बीच किसी बालिका के शरीर में तेजी से परिवर्तन हो तो मुश्...
Post a Comment

Real Analysis | Mathematics | BSc

Real Analysis

Real Analysis and Theory of Convergence



Authors: Dr. Vimal Saraswat, Dr. Anil Kumar Menaria, Dr. Gajendrapal Singh Rathore

ISBN : 978-81-7906-338-5

Price: Rs. 395.00

Publisher:Himanshu Publications, Hiran Magri Udaipur; Himanshu Publications Prakash House, Ansari Road, New Delhi

E-mail : info@sacademy.co.in

Phone: +91 9664392614

To buy this book click on the link Real Analysis by Saraswat

This book includes the following topics 

Real Number System

  • Introduction
  • Field axiom
  • Uniqueness property
  • Cancellation law of addition and multiplication
  • Order axiom and ordered field
  • Positive class
  • Boundedness
    • Upper bound, Supremum, Lower bound, Infimum, Bounded set
  • Greatest and least element
  • Completeness axiom
  • Complete ordered field
  • Archimedean property of real numbers
    • Archimedean ordered field
  • Betweenness theorem
  • Dedekind's completeness axiom
  • Irrational numbers
  • Rational density theorem or denseness property
  • Absolute value of a real number or Modulus

Point Set Topology

  • Introduction
  • Neighbourhood (nbd) of a real number
  • Properties of neighbourhood
  • Interior and exterior point of a set
  • Interior of a set
  • Open set
  • Limit point of a set
  • Derived set and closed set
  • Closure
  • Open and closed interval
  • Nested interval property
  • Bolzano-Weierstrass theorem
  • Complement of set
  • Open cover, subcover and compact set
  • Heine Borel theorem
  • Connected and disconnected set

Countable Sets

  • Introduction
  • Equivalent sets
  • Finite and infinite set
  • Countable set
  • Uncountable set
  • Cantor ternary set
  • Binary representation
  • Ternary representation
  • Construction of Cantor ternary set
  • Properties of Cantor ternary set

Real Sequences

  • Introduction
  • Sequence
  • Range of a sequence
  • Bounded and unbounded sequence
  • Supremum and infimum of sequence
  • Monotonic sequence
  • Limit point of a sequence
  • Bolzano-Weierstrass theorem
  • Limit of a sequence
  • Convergent sequence
  • Divergent sequence
  • Oscillatory sequence
  • Theorems on convergence sequences
  • Theorems on convergence of monotonic sequences
  • Algebra of sequences
  • Sandwich theorem
  • Limit superior and limit inferior
  • Sub-sequence
  • Some theorems of sub-sequence
  • Cauchy's sequence or fundamental sequence
  • Some important theorems of Cauchy's sequence
  • Cauchy's general principle of convergence for sequence
  • Cauchy's first theorem on limits
  • Cauchy's second theorem on limits
  • Cesaro's theorem

Infinite Series

  • Introduction
  • Sequence of partial sums of series
  • Nature of an infinite series
  • Some important theorems
    • Cauchy's general principle of convergence Test of the convergence of geometric series
  • Comparative tests of the first type
  • Comparative tests of the second type
    • Ratio-comparison test; D' Alembert's ratio test; Raabe's test; de Morgan's and Bertrand's test; Logarithmic ratio test; Second logarithmic ratio test; Gauss's test
  • Some other useful tests
    • Cauchy's nth root test; Cauchy's condensation test
  • Alternating series
  • Absolute convergence
  • Conditionally convergence

Uniform Convergence

  • Introduction
  • Pointwise convergence of a sequence of functions
  • Uniform convergence
  • Series of functions
  • Cauchy's criterion for uniform convergence
  • Test for uniform convergence of a sequence and series of functions
  • Uniform convergence and continuity
  • Term by term integration
  • A sufficient condition for term by term integration of an infinite series
  • A sufficient condition for term by term differentiation of the series

Improper Integrals

  • Finite and infinite intervals
  • Bounded function
  • Improper integral
  • Types of improper integral
  • Convergence of improper integral of first kind
  • Convergence tests for the improper integral of first kind
  • Convergence of improper integral of second kind
  • Convergence tests for the improper integral of second kind
  • Convergence of improper integrals of third kind

Riemann Integration

  • Introduction
  • Partition of a closed interval
  • Norm of partition
  • Refinement of a partition
  • Supremum and infimum
  • Upper and lower Darboux sum
  • Theorems on Darboux sum
  • Upper and lower Riemann integral
  • Integral function
  • Riemann integral
  • Theorems of Riemann integral
    • Necessary and sufficient condition for a function to be R-integrable
  • Particular classes of Riemann integrable functions
  • Riemann integral as the limit of a sum
  • Properties of Riemann integral function
  • Integral function
  • Properties of integral function
  • Primitive
  • Mean value theorems of integral calculus
  • Fundamental theorem of integral
  • Techniques of integration

Fourier Series

  • Introduction
  • Perodic functions
  • Properties of definite integral
  • Some important definite integrals
  • Fourier series
  • Dirichlet's conditions for the expansion of a Fourier series
  • Even and odd functions
  • Fourier series for even and odd functions
  • Fourier's half range series
  • Other forms of Fourier series
Labels:

Comments

Post a Comment

Popular posts from this blog

Electric field due to circular loop of charge | Electromagnetics

Electric field due to circular loop of charge Electric field The space around a charged particle in which another charge experience a force is known as electric field. The source of electric field is either a charge or a time varying magnetic field. If the value of electric field does not change with time, then it will be uniform electric field, otherwise it will be non-uniform electric field. Electric field due to circular loop of charge If λ is linear charge density, then the charge on d l dq = λ d l      ⇒     dq = (q / 2πa) d l Electric field at P due to charge dq Special cases When P lies at the centre of the loop i. e., r = 0, then E = 0 When P lies very far from the centre of the loop i. e., r >> a, then E = kq / r 2 In this case circular loop behaves as a point charge. To know more about this topic please click on the link  https://youtu.be/54MIe0Ow43w   or...
Post a Comment
Read more

Thick lens and its cardinal points | Optics | General theory of image formation

Thick lens and its cardinal points Thick lens The combination of two spherical surfaces, is known as thick lens, if the distance between their poles can not be neglected in comparison to their radii. It can be treated as the combination of thin lenses. The thickness of these lenses are comparable to their focal lengths. Cardinal points of thick lens Let µ = refractive index of material of lens t = thickness of lens R 1 and R 2 = Radii of curvature of first and second curved surface of the given thick lens. F 1 and F 2 = First and second focal points respectively. H 1 and H 2 = First and second principal points respectively. f 1 and f 2 = First and second focal length of thick lens respectively. Cardinal points of thick lens Equivalent focal length If there is same medium on either side of thick lens H 1 F 1 = f 1 = - f , and H 2 F 2 = f 2 = f Refraction from surface PP 1 Refraction from surfa...
Post a Comment
Read more

Advantage and Disadvantage of Power Electronics

Advantage and Disadvantage of Power Electronics Advantage of Power electronics Power electronics is used in space shuttle power supplies Since there is very low loss in power electronic devices so its efficiency is very high. Power electronic converter systems are highly reliable. Since there is no moving parts in power electronic systems so it has long life and also its maintenance cost is very less. The power electronic systems has fast dynamic response in comparison to electromechanical converter systems. Since the power electronic system has small size and also it has less weight so they occupy less floor space and hence their installation cost is also less. Now these days power equipments are being mostly used, so power semiconductor devices are being produced on a large scale, resulting in lower cost of converter equipment. Power electronics are used in computer and office equipments. It is used in uninterruptible power supplies. Power...
Post a Comment
Read more

Adesterra