Number theory Full Course [A to Z]

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). In this number theroy course following topics hav been explained in a very comprehensive way.

⭐️ Table of Content ⭐️
⌨️ (00:00) Introduction to number theory
⌨️ (02:07) The principle of mathematical induction
⌨️ (06:04) Basic representation theorem
⌨️ (10:14) The division algorithm
⌨️ (13:49) The divisibility
⌨️ (17:31) The euclidean algorithm
⌨️ (22:48) Linear Diophantine Equations
⌨️ (26:03) The fundamental theorem of arithemetic
⌨️ (30:38) Permutations and combinations
⌨️ (36:10) Fermat’s Little theorem
⌨️ (38:51) Wilson’s Theorem
⌨️ (42:42) Computer Programming
⌨️ (49:18) Basic properties of congruences
⌨️ (52:51) Residue Systems
⌨️ (58:12) Linear Congruences
⌨️ (1:01:54) Fermat’s little theorem and wilson’s theorem
⌨️ (1:06:31) The Chinese remainder theorem
⌨️ (1:10:50) The Eular Phi Function Part 1
⌨️ (1:14:34) The Eular Phi Function Part 2
⌨️ (1:19:12) Multiplicative function
⌨️ (1:23:51) The mobious inversion formula
⌨️ (1:28:08) Order of Elements
⌨️ (1:33:49) Primitive roots modolo
⌨️ (1:37:03) The prime counting function
⌨️ (1:41:39) The Eular’s criterion
⌨️ (1:45:44) The Legendre symbol
⌨️ (1:48:42) Quadratic Reciprocity part 1
⌨️ (1:53:17) Quadratic Reciprocity part 2
⌨️ (1:59:33) Application of quadratic reciprocity
⌨️ (2:02:13) Consicutive Residues
⌨️ (2:06:54) Consicutive triples of Residues part 1
⌨️ (2:09:03) Consicutive triples of Residues part 2
⌨️ (2:13:40) Sums of two squares
⌨️ (2:16:02) Sums of four squares
⌨️ (2:22:16) Gauss circle problem
⌨️ (2:25:38) Dirichlet’s devisor problem
⌨️ (2:29:55) Infinity Conclusion

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Number theory Full Course [A to Z]
Dylan  Iqbal

Dylan Iqbal

1630996646

A Look at an ES2022 Feature: Class Static Initialization Blocks

ECMAScript class static initialization blocks

Class static blocks provide a mechanism to perform additional static initialization during class definition evaluation.

This is not intended as a replacement for public fields, as they provide useful information for static analysis tools and are a valid target for decorators. Rather, this is intended to augment existing use cases and enable new use cases not currently handled by that proposal.

Status

Stage: 4
Champion: Ron Buckton (@rbuckton)

For detailed status of this proposal see TODO, below.

Authors

  • Ron Buckton (@rbuckton)

Motivations

The current proposals for static fields and static private fields provide a mechanism to perform per-field initialization of the static-side of a class during ClassDefinitionEvaluation, however there are some cases that cannot be covered easily. For example, if you need to evaluate statements during initialization (such as try..catch), or set two fields from a single value, you have to perform that logic outside of the class definition.

// without static blocks:
class C {
  static x = ...;
  static y;
  static z;
}

try {
  const obj = doSomethingWith(C.x);
  C.y = obj.y
  C.z = obj.z;
}
catch {
  C.y = ...;
  C.z = ...;
}

// with static blocks:
class C {
  static x = ...;
  static y;
  static z;
  static {
    try {
      const obj = doSomethingWith(this.x);
      this.y = obj.y;
      this.z = obj.z;
    }
    catch {
      this.y = ...;
      this.z = ...;
    }
  }
}

In addition, there are cases where information sharing needs to occur between a class with an instance private field and another class or function declared in the same scope.

Static blocks provide an opportunity to evaluate statements in the context of the current class declaration, with privileged access to private state (be they instance-private or static-private):

let getX;

export class C {
  #x
  constructor(x) {
    this.#x = { data: x };
  }

  static {
    // getX has privileged access to #x
    getX = (obj) => obj.#x;
  }
}

export function readXData(obj) {
  return getX(obj).data;
}

Relation to "Private Declarations"

The Private Declarations proposal also intends to address the issue of privileged access between two classes, by lifting the private name out of the class declaration and into the enclosing scope. While there is some overlap in that respect, private declarations do not solve the issue of multi-step static initialization without potentially exposing a private name to the outer scope purely for initialization purposes:

// with private declarations
private #z; // exposed purely for post-declaration initialization
class C {
  static y;
  static outer #z;
}
const obj = ...;
C.y = obj.y;
C.#z = obj.z;

// with static block
class C {
  static y;
  static #z; // not exposed outside of class
  static {
    const obj = ...;
    this.y = obj.y;
    this.#z = obj.z;
  }
}

In addition, Private Declarations expose a private name that potentially allows both read and write access to shared private state when read-only access might be desireable. To work around this with private declarations requires additional complexity (though there is a similar cost for static{} as well):

// with private declarations
private #zRead;
class C {
  #z = ...; // only writable inside of the class
  get #zRead() { return this.#z; } // wrapper needed to ensure read-only access
}

// with static
let zRead;
class C {
  #z = ...; // only writable inside of the class
  static { zRead = obj => obj.#z; } // callback needed to ensure read-only access
}

In the long run, however, there is nothing that prevents these two proposals from working side-by-side:

private #shared;
class C {
  static outer #shared;
  static #local;
  static {
    const obj = ...;
    this.#shared = obj.shared;
    this.#local = obj.local;
  }
}
class D {
  method() {
    C.#shared; // ok
    C.#local; // no access
  }
}

Prior Art

Syntax

class C {
  static {
    // statements
  }
}

Semantics

  • A static {} initialization block creates a new lexical scope (e.g. var, function, and block-scoped declarations are local to the static {} initialization block. This lexical scope is nested within the lexical scope of the class body (granting privileged access to instance private state for the class).
  • A class may have any number of static {} initialization blocks in its class body.
  • static {} initialization blocks are evaluated in document order interleaved with static field initializers.
  • A static {} initialization block may not have decorators (instead you would decorate the class itself).
  • When evaluated, a static {} initialization block's this receiver is the constructor object of the class (as with static field initializers).
  • It is a Syntax Error to reference arguments from within a static {} initialization block.
  • It is a Syntax Error to include a SuperCall (i.e., super()) from within a static {} initialization block.
  • A static {} initialization block may contain SuperProperty references as a means to access or invoke static members on a base class that may have been overridden by the derived class containing the static {} initialization block.
  • A static {} initialization block should be represented as an independent stack frame in debuggers and exception traces.

Examples

// "friend" access (same module)
let A, B;
{
  let friendA;

  A = class A {
    #x;

    static {
        friendA = {
          getX(obj) { return obj.#x },
          setX(obj, value) { obj.#x = value }
        };
    }
  };

  B = class B {
    constructor(a) {
      const x = friendA.getX(a); // ok
      friendA.setX(a, x); // ok
    }
  };
}

References

TODO

The following is a high-level list of tasks to progress through each stage of the TC39 proposal process:

Stage 1 Entrance Criteria

  • Identified a "champion" who will advance the addition.
  • Prose outlining the problem or need and the general shape of a solution.
  • Illustrative examples of usage.
  • High-level API.

Stage 2 Entrance Criteria

Stage 3 Entrance Criteria

Stage 4 Entrance Criteria

For up-to-date information on Stage 4 criteria, check: #48

  • Test262 acceptance tests have been written for mainline usage scenarios and merged.
  • Two compatible implementations which pass the acceptance tests:
  • A pull request has been sent to tc39/ecma262 with the integrated spec text.
  • The ECMAScript editor has signed off on the pull request.

Download Details:
Author: tc39
The Demo/Documentation: View The Demo/Documentation
Download Link: Download The Source Code
Official Website: https://github.com/tc39/proposal-class-static-block 
License: BSD-3
#javascript #es2022 #ecmascript 

navin prakash

1609588950

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