From 07737d3368efb51fd78c71e3ade13ea2ed401939 Mon Sep 17 00:00:00 2001 From: Anthony Berg Date: Sat, 1 Mar 2025 14:40:30 +0100 Subject: [PATCH] Quartz sync: Mar 1, 2025, 2:40 PM --- public/404.html | 2 +- .../Cauchy-Schwarz-Inequality.html | 2 +- public/Definitions/Cauchy-Sequence.html | 2 +- .../Functions/Characteristic-Function.html | 2 +- .../Definitions/Functions/Direct-Product.html | 2 +- .../Functions/Inverse-Function.html | 2 +- public/Definitions/Functions/Metric.html | 2 +- public/Definitions/Functions/Power-Set.html | 2 +- public/Definitions/Functions/index.html | 2 +- public/Definitions/Hilbert-Spaces.html | 2 +- public/Definitions/Inner-Product.html | 2 +- .../Least-Upper-Bound-Property.html | 2 +- public/Definitions/Linear-Map.html | 2 +- .../Sigma-Algebra/Borel-Measurable.html | 2 +- .../Sigma-Algebra/Borel-Sets.html | 2 +- .../Sigma-Algebra/Measurable.html | 2 +- .../Measure-Theory/Sigma-Algebra/Measure.html | 2 +- .../Sigma-Algebra/Sigma-Algebra.html | 2 +- .../Measure-Theory/Sigma-Algebra/index.html | 2 +- public/Definitions/Measure-Theory/index.html | 2 +- public/Definitions/Metric-Spaces/Ball.html | 2 +- .../Metric-Spaces/Interior-Point.html | 2 +- .../Metric-Spaces/Metric-Space.html | 2 +- public/Definitions/Metric-Spaces/index.html | 2 +- public/Definitions/Nets.html | 2 +- public/Definitions/Norm.html | 2 +- public/Definitions/Number-Field.html | 2 +- public/Definitions/Period-of-a-Fraction.html | 2 +- .../Rational-Cauchy-Sequences.html | 2 +- public/Definitions/Sets/Complex-Numbers.html | 2 +- public/Definitions/Sets/Open-Cover.html | 2 +- public/Definitions/Sets/Open-Map.html | 2 +- public/Definitions/Sets/Open-Sets.html | 2 +- public/Definitions/Sets/index.html | 2 +- public/Definitions/Statements/And.html | 2 +- public/Definitions/Statements/Implies.html | 2 +- public/Definitions/Statements/Not.html | 2 +- public/Definitions/Statements/Or.html | 2 +- public/Definitions/Statements/index.html | 2 +- public/Definitions/Subcover.html | 2 +- public/Definitions/Subnet.html | 2 +- .../Terminology/Algebraically-Complete.html | 2 +- public/Definitions/Terminology/Bijective.html | 2 +- public/Definitions/Terminology/Bounded.html | 2 +- public/Definitions/Terminology/Countable.html | 2 +- public/Definitions/Terminology/Injective.html | 2 +- public/Definitions/Terminology/QED.html | 2 +- .../Definitions/Terminology/Surjective.html | 2 +- public/Definitions/Terminology/index.html | 2 +- .../Topological-Spaces/Continuous.html | 2 +- .../Topological-Spaces/Hausdorff.html | 2 +- .../Induced/Initial-Topology.html | 2 +- .../Induced/Product-Topology.html | 2 +- .../Induced/Separating-Points.html | 2 +- .../Induced/Weakest-Topology.html | 2 +- .../Topological-Spaces/Induced/index.html | 2 +- .../Terminologies/Compact.html | 2 +- .../Terminologies/Connected-Component.html | 2 +- .../Terminologies/Connected.html | 2 +- .../Terminologies/index.html | 2 +- .../Topological-Spaces/Topological-Space.html | 2 +- .../Topological-Spaces/Topology.html | 2 +- .../Topological-Spaces/Tychonoff-Theorem.html | 2 +- .../Definitions/Topological-Spaces/index.html | 2 +- .../Vector-Spaces/Complex-Vector-Space.html | 2 +- .../Vector-Spaces/Linear-Basis.html | 2 +- .../Vector-Spaces/Normed-Vector-Space.html | 2 +- .../Properties-of-a-Vector-Space.html | 2 +- public/Definitions/Vector-Spaces/index.html | 2 +- public/Definitions/index.html | 2 +- ...rawing-2025-02-13-10.57.31.excalidraw.html | 2 +- ...rawing-2025-02-13-11.02.27.excalidraw.html | 2 +- ...rawing-2025-02-13-11.26.08.excalidraw.html | 2 +- ...rawing-2025-02-13-11.47.33.excalidraw.html | 2 +- ...rawing-2025-02-13-12.01.57.excalidraw.html | 2 +- ...rawing-2025-02-13-12.07.00.excalidraw.html | 2 +- ...rawing-2025-02-13-14.01.44.excalidraw.html | 2 +- ...rawing-2025-02-13-14.06.25.excalidraw.html | 2 +- public/Excalidraw/Lecture-11/index.html | 2 +- ...rawing-2025-02-24-11.58.37.excalidraw.html | 2 +- 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quartz/.quartz-cache/transpiled-build.mjs.map | 2 +- 105 files changed, 220 insertions(+), 220 deletions(-) diff --git a/public/404.html b/public/404.html index e1b06140..2c434246 100644 --- a/public/404.html +++ b/public/404.html @@ -1,5 +1,5 @@ -Not Found

🪴 Quartz 4.0

Cauchy-Schwarz Inequality

1 min read

Definition

+Cauchy-Schwarz Inequality

ACIT4330 Lecture Notes

Cauchy-Schwarz Inequality

1 min read

Definition


diff --git a/public/Definitions/Cauchy-Sequence.html b/public/Definitions/Cauchy-Sequence.html index fa09bf1f..bcce6d5e 100644 --- a/public/Definitions/Cauchy-Sequence.html +++ b/public/Definitions/Cauchy-Sequence.html @@ -1,5 +1,5 @@ -Cauchy Sequence

🪴 Quartz 4.0

Cauchy Sequence

1 min read

Definition

+Cauchy Sequence

ACIT4330 Lecture Notes

Cauchy Sequence

1 min read

Definition

A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses.

Examples

Cauchy Sequence

diff --git a/public/Definitions/Functions/Characteristic-Function.html b/public/Definitions/Functions/Characteristic-Function.html index 84ed27ba..f3703a67 100644 --- a/public/Definitions/Functions/Characteristic-Function.html +++ b/public/Definitions/Functions/Characteristic-Function.html @@ -1,5 +1,5 @@ -Characteristic Function

🪴 Quartz 4.0

Characteristic Function

1 min read

, so defined as:

+Characteristic Function

ACIT4330 Lecture Notes

Characteristic Function

1 min read

, so defined as:

1, & \text{if}\ x \in Y\\ 0, & \text{if}\ x \notin Y \end{cases}$$

Graph View

Direct Product

1 min read


+Direct Product

ACIT4330 Lecture Notes

Direct Product

1 min read


where and , basically .

Graph View

Inverse Function

1 min read


+Inverse Function

ACIT4330 Lecture Notes

Inverse Function

1 min read


such that

Graph View

Metric

1 min read

Definition

+Metric

ACIT4330 Lecture Notes

Metric

1 min read

Definition

A function which measures distances between two points in a Metric Space.

Graph View

Power Set

1 min read

Definition

+Power Set

ACIT4330 Lecture Notes

Power Set

1 min read

Definition

of consists of all the subsets of .

Amount of Elements in a Power Set

Lets say we have :
diff --git a/public/Definitions/Functions/index.html b/public/Definitions/Functions/index.html index 734f5354..5f6c02ad 100644 --- a/public/Definitions/Functions/index.html +++ b/public/Definitions/Functions/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Functions

🪴 Quartz 4.0

Folder: Definitions/Functions

5 items under this folder.

ACIT4330 Lecture Notes

Folder: Definitions/Functions

5 items under this folder.

🪴 Quartz 4.0

Hilbert Spaces

1 min read

Definition

+Hilbert Spaces

ACIT4330 Lecture Notes

Hilbert Spaces

1 min read

Definition

These are Banach Spaces with norms given by an Inner Product.

diff --git a/public/Definitions/Inner-Product.html b/public/Definitions/Inner-Product.html index 57c63020..2f3d336b 100644 --- a/public/Definitions/Inner-Product.html +++ b/public/Definitions/Inner-Product.html @@ -1,5 +1,5 @@ -Inner Product

🪴 Quartz 4.0

Inner Product

1 min read

Definition

+Inner Product

ACIT4330 Lecture Notes

Inner Product

1 min read

Definition

diff --git a/public/Definitions/Least-Upper-Bound-Property.html b/public/Definitions/Least-Upper-Bound-Property.html index 8f051001..95991a38 100644 --- a/public/Definitions/Least-Upper-Bound-Property.html +++ b/public/Definitions/Least-Upper-Bound-Property.html @@ -1,5 +1,5 @@ -Least Upper Bound Property

🪴 Quartz 4.0

Least Upper Bound Property

1 min read


+Least Upper Bound Property

ACIT4330 Lecture Notes

Least Upper Bound Property

1 min read




diff --git a/public/Definitions/Linear-Map.html b/public/Definitions/Linear-Map.html index fe9d1c5e..a2504faa 100644 --- a/public/Definitions/Linear-Map.html +++ b/public/Definitions/Linear-Map.html @@ -1,5 +1,5 @@ -Linear Map

🪴 Quartz 4.0

Linear Map

1 min read

Definition

+Linear Map

ACIT4330 Lecture Notes

Linear Map

1 min read

Definition



Graph View

Borel Measurable

1 min read

Definition

+Borel Measurable

ACIT4330 Lecture Notes

Borel Measurable

1 min read

Definition

Say is continuous, is Borel measurable.

Graph View

Borel Sets

1 min read

Definition

+Borel Sets

ACIT4330 Lecture Notes

Borel Sets

1 min read

Graph View

Measurable

1 min read

Definition

+Measurable

ACIT4330 Lecture Notes

Measurable

1 min read

Definition

(Topological Space) is measurable if .

Graph View

Measure

1 min read

Definition

+Measure

ACIT4330 Lecture Notes

Measure

1 min read

Definition

A measure on is a function such that:

    2. diff --git a/public/Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra.html b/public/Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra.html index 804d8bd7..1de18b8d 100644 --- a/public/Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra.html +++ b/public/Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra.html @@ -1,5 +1,5 @@ -Sigma-Algebra

    🪴 Quartz 4.0

    Sigma-Algebra

    1 min read

    Definition

    +Sigma-Algebra

    ACIT4330 Lecture Notes

    Sigma-Algebra

    1 min read

    Definition

    A -algebra in a set is a collection of subsets, so called measurable sets, of such that (the requirements are):

      2. diff --git a/public/Definitions/Measure-Theory/Sigma-Algebra/index.html b/public/Definitions/Measure-Theory/Sigma-Algebra/index.html index 4e4d021b..93501714 100644 --- a/public/Definitions/Measure-Theory/Sigma-Algebra/index.html +++ b/public/Definitions/Measure-Theory/Sigma-Algebra/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Measure-Theory/Sigma-Algebra

      🪴 Quartz 4.0

      Folder: Definitions/Measure-Theory/Sigma-Algebra

      5 items under this folder.

      ACIT4330 Lecture Notes

      Folder: Definitions/Measure-Theory/Sigma-Algebra

      5 items under this folder.

      🪴 Quartz 4.0

      Folder: Definitions/Measure-Theory

      1 item under this folder.

      ACIT4330 Lecture Notes

      Folder: Definitions/Measure-Theory

      1 item under this folder.

      🪴 Quartz 4.0

      Ball

      1 min read

      Definition

      +Ball

      ACIT4330 Lecture Notes

      Ball

      1 min read

      Definition

      Let be a metric on a set .

      The (open) ball with centre and radius is .

      A sequence in converges to if it eventually belongs to any ball ; such that

      🪴 Quartz 4.0

      Interior Point

      1 min read

      Definition

      +Interior Point

      ACIT4330 Lecture Notes

      Interior Point

      1 min read

      Definition

      Points such that small enough balls centred around them are contained in .

      Graph View

      Metric Space

      1 min read

      Definition

      +Metric Space

      ACIT4330 Lecture Notes

      Metric Space

      1 min read

      Definition

      A Metric on a set is a function such that

        2. diff --git a/public/Definitions/Metric-Spaces/index.html b/public/Definitions/Metric-Spaces/index.html index c6d820ba..adc94efc 100644 --- a/public/Definitions/Metric-Spaces/index.html +++ b/public/Definitions/Metric-Spaces/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Metric-Spaces

        🪴 Quartz 4.0

        Folder: Definitions/Metric-Spaces

        3 items under this folder.

        ACIT4330 Lecture Notes

        Folder: Definitions/Metric-Spaces

        3 items under this folder.

        🪴 Quartz 4.0

        Nets

        1 min read

        ACIT4330 Lecture Notes

        Nets

        1 min read

        🪴 Quartz 4.0

        Norm

        1 min read

        Definition

        +Norm

        ACIT4330 Lecture Notes

        Norm

        1 min read

        Definition

        A norm on is a map such that

          2. diff --git a/public/Definitions/Number-Field.html b/public/Definitions/Number-Field.html index 178daf8f..e1867d18 100644 --- a/public/Definitions/Number-Field.html +++ b/public/Definitions/Number-Field.html @@ -1,5 +1,5 @@ -Number Field

          🪴 Quartz 4.0

          Number Field

          1 min read


          +Number Field

          ACIT4330 Lecture Notes

          Number Field

          1 min read




          Graph View

          Period of a Fraction

          1 min read

          Definition

          +Period of a Fraction

          ACIT4330 Lecture Notes

          Period of a Fraction

          1 min read

          Definition

          The period of a fraction is the digits that repeat themselves.

          The digits can be from , and the length of the period is determined by the denominator, n in .

          Examples

          diff --git a/public/Definitions/Rational-Cauchy-Sequences.html b/public/Definitions/Rational-Cauchy-Sequences.html index 5e7658cd..434362e4 100644 --- a/public/Definitions/Rational-Cauchy-Sequences.html +++ b/public/Definitions/Rational-Cauchy-Sequences.html @@ -1,5 +1,5 @@ -Rational Cauchy Sequences

          🪴 Quartz 4.0

          Rational Cauchy Sequences

          1 min read

          Definition

          +Rational Cauchy Sequences

          ACIT4330 Lecture Notes

          Rational Cauchy Sequences

          1 min read

          Definition

          They are sequences such that

          Such that
          diff --git a/public/Definitions/Sets/Complex-Numbers.html b/public/Definitions/Sets/Complex-Numbers.html index 990807e8..8d700de4 100644 --- a/public/Definitions/Sets/Complex-Numbers.html +++ b/public/Definitions/Sets/Complex-Numbers.html @@ -1,5 +1,5 @@ -Complex Numbers

          🪴 Quartz 4.0

          Complex Numbers

          1 min read


          +Complex Numbers

          ACIT4330 Lecture Notes

          Complex Numbers

          1 min read



          Operations

          diff --git a/public/Definitions/Sets/Open-Cover.html b/public/Definitions/Sets/Open-Cover.html index 16d0c655..0cc4bf48 100644 --- a/public/Definitions/Sets/Open-Cover.html +++ b/public/Definitions/Sets/Open-Cover.html @@ -1,5 +1,5 @@ -Open Cover

          🪴 Quartz 4.0

          Open Cover

          1 min read

          Definition

          +Open Cover

          ACIT4330 Lecture Notes

          Open Cover

          1 min read

          Graph View

          Open Map

          1 min read

          Definition

          +Open Map

          ACIT4330 Lecture Notes

          Open Map

          1 min read

          Graph View

          Open Sets

          1 min read

          Definition

          +Open Sets

          ACIT4330 Lecture Notes

          Open Sets

          1 min read

          Definition

          An open set in (means set with a metric) consists only of interior points.

          Then a sequence converges to it eventually belongs to any open set containing .

          Graph View

          Folder: Definitions/Sets

          4 items under this folder.

          ACIT4330 Lecture Notes

          Folder: Definitions/Sets

          4 items under this folder.

          🪴 Quartz 4.0

          And

          1 min read

          +And

          ACIT4330 Lecture Notes

          And

          1 min read

          Truth Table

          diff --git a/public/Definitions/Statements/Implies.html b/public/Definitions/Statements/Implies.html index 52be6c13..11f84b80 100644 --- a/public/Definitions/Statements/Implies.html +++ b/public/Definitions/Statements/Implies.html @@ -1,5 +1,5 @@ -Implies

          🪴 Quartz 4.0

          Implies

          1 min read

          +Implies

          ACIT4330 Lecture Notes

          Implies

          1 min read

          Truth Table

          diff --git a/public/Definitions/Statements/Not.html b/public/Definitions/Statements/Not.html index a136119e..a24317ab 100644 --- a/public/Definitions/Statements/Not.html +++ b/public/Definitions/Statements/Not.html @@ -1,5 +1,5 @@ -Not

          🪴 Quartz 4.0

          Not

          1 min read


          +Not

          ACIT4330 Lecture Notes

          Not

          1 min read


          The symbol () is called negation

          Truth Table

          diff --git a/public/Definitions/Statements/Or.html b/public/Definitions/Statements/Or.html index 674dc796..4d10c8f2 100644 --- a/public/Definitions/Statements/Or.html +++ b/public/Definitions/Statements/Or.html @@ -1,5 +1,5 @@ -Or

          🪴 Quartz 4.0

          Or

          1 min read

          +Or

          ACIT4330 Lecture Notes

          Or

          1 min read

          Truth Table

          diff --git a/public/Definitions/Statements/index.html b/public/Definitions/Statements/index.html index 4dc4e65a..01c839c6 100644 --- a/public/Definitions/Statements/index.html +++ b/public/Definitions/Statements/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Statements

          🪴 Quartz 4.0

          Folder: Definitions/Statements

          4 items under this folder.

          ACIT4330 Lecture Notes

          Folder: Definitions/Statements

          4 items under this folder.

          🪴 Quartz 4.0

          Subcover

          1 min read

          Definition

          +Subcover

          ACIT4330 Lecture Notes

          Subcover

          1 min read

          Definition

          Any subcollection with union

          Graph View

          Subnet

          1 min read

          Definition

          +Subnet

          ACIT4330 Lecture Notes

          Subnet

          1 min read

          Definition

          A subnet of a net is a net and a map such that and such that with .

          Example

          diff --git a/public/Definitions/Terminology/Algebraically-Complete.html b/public/Definitions/Terminology/Algebraically-Complete.html index e1f667c8..422eba05 100644 --- a/public/Definitions/Terminology/Algebraically-Complete.html +++ b/public/Definitions/Terminology/Algebraically-Complete.html @@ -1,5 +1,5 @@ -Algebraically Complete

          🪴 Quartz 4.0

          Algebraically Complete

          1 min read

          For Complex Numbers

          +Algebraically Complete

          ACIT4330 Lecture Notes

          Algebraically Complete

          1 min read

          For Complex Numbers

          Any equation

          with has roots, or solutions, counted with multiplicity.

          Graph View

          Bijective

          1 min read

          Definition

          +Bijective

          ACIT4330 Lecture Notes

          Bijective

          1 min read

          Definition

          It is both Injective and Surjective.

          Bijective Map

          An example could be
          diff --git a/public/Definitions/Terminology/Bounded.html b/public/Definitions/Terminology/Bounded.html index 0ba588b2..fa7860bc 100644 --- a/public/Definitions/Terminology/Bounded.html +++ b/public/Definitions/Terminology/Bounded.html @@ -1,5 +1,5 @@ -Bounded

          🪴 Quartz 4.0

          Bounded

          1 min read

          Definition

          +Bounded

          ACIT4330 Lecture Notes

          Bounded

          1 min read

          Definition

          such that .

          Graph View

          Countable

          1 min read

          What is countable?

          +Countable

          ACIT4330 Lecture Notes

          Countable

          1 min read

          What is countable?

          Natural Numbers

          Quite simple

          diff --git a/public/Definitions/Terminology/Injective.html b/public/Definitions/Terminology/Injective.html index d46090fc..913e3bb5 100644 --- a/public/Definitions/Terminology/Injective.html +++ b/public/Definitions/Terminology/Injective.html @@ -1,5 +1,5 @@ -Injective

          🪴 Quartz 4.0

          Injective

          1 min read


          +Injective

          ACIT4330 Lecture Notes

          Injective

          1 min read


          Usually a function would be mapped like:

          Definition

          diff --git a/public/Definitions/Terminology/QED.html b/public/Definitions/Terminology/QED.html index c7bdafd8..4fb46a8a 100644 --- a/public/Definitions/Terminology/QED.html +++ b/public/Definitions/Terminology/QED.html @@ -1,5 +1,5 @@ -QED

          🪴 Quartz 4.0

          QED

          1 min read

          Can be written as “Q.E.D.” or “QED”.

          +QED

          ACIT4330 Lecture Notes

          QED

          1 min read

          Can be written as “Q.E.D.” or “QED”.

          It is shortened in Latin from “quod erat demonstrandum” (that which was to be demonstrated).

          Definition

          A notation which is often placed at the end of a mathematical proof to indicate its completion

          Graph View

          Surjective

          1 min read


          +Surjective

          ACIT4330 Lecture Notes

          Surjective

          1 min read



          Graph View

          Folder: Definitions/Terminology

          7 items under this folder.

          ACIT4330 Lecture Notes

          Folder: Definitions/Terminology

          7 items under this folder.

          🪴 Quartz 4.0

          Continuous

          1 min read

          Definition

          +Continuous

          ACIT4330 Lecture Notes

          Continuous

          1 min read

          Definition

          We say is continuous (at every ) if is open for every open .

          We say is open if is open and open .

          If is a bijection that is both continuous and open, it is a homeomorphism, and and are homeomorphic, written ; they are the ‘same’ as topological spaces.

          diff --git a/public/Definitions/Topological-Spaces/Hausdorff.html b/public/Definitions/Topological-Spaces/Hausdorff.html index 6d19d2d5..028eac75 100644 --- a/public/Definitions/Topological-Spaces/Hausdorff.html +++ b/public/Definitions/Topological-Spaces/Hausdorff.html @@ -1,5 +1,5 @@ -Hausdorff

          🪴 Quartz 4.0

          Hausdorff

          1 min read

          Definition

          +Hausdorff

          ACIT4330 Lecture Notes

          Hausdorff

          1 min read

          Definition

          If there are two points and in a topological space that can be separated by neighbourhoods if there exists a neighbourhood of and a neighbourhood of , such that and are disjoint .

          is a Hausdorff space if any two distinct points in are separated by neighbourhoods.

          Graph View

          Initial Topology

          1 min read

          Definition

          +Initial Topology

          ACIT4330 Lecture Notes

          Initial Topology

          1 min read

          Definition

          The initial topology on induced by a family of functions into topological spaces is the weakest topologyon making all these functions continuous.

          Here: .

          Graph View

          Product Topology

          1 min read

          Definition

          +Product Topology

          ACIT4330 Lecture Notes

          Product Topology

          1 min read

          Definition

          The product topology on , topological spaces, is the initial topology induced by the family of projections

          diff --git a/public/Definitions/Topological-Spaces/Induced/Separating-Points.html b/public/Definitions/Topological-Spaces/Induced/Separating-Points.html index 8a90dd32..1f88ca5f 100644 --- a/public/Definitions/Topological-Spaces/Induced/Separating-Points.html +++ b/public/Definitions/Topological-Spaces/Induced/Separating-Points.html @@ -1,5 +1,5 @@ -Separating Points

          🪴 Quartz 4.0

          Separating Points

          1 min read

          Definition

          +Separating Points

          ACIT4330 Lecture Notes

          Separating Points

          1 min read

          Definition

          A family of functions on a set separating points in the set if for some

          Graph View

          Weakest Topology

          1 min read

          Definition

          +Weakest Topology

          ACIT4330 Lecture Notes

          Weakest Topology

          1 min read

          Definition

          Given . The weakest topology on that contains is the intersection of all the topologies that contains . This is a topology, and consists of , , and all unions of finite intersections of members from .

          diff --git a/public/Definitions/Topological-Spaces/Induced/index.html b/public/Definitions/Topological-Spaces/Induced/index.html index b0e715d8..c8e182f1 100644 --- a/public/Definitions/Topological-Spaces/Induced/index.html +++ b/public/Definitions/Topological-Spaces/Induced/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Topological-Spaces/Induced

          🪴 Quartz 4.0

          Folder: Definitions/Topological-Spaces/Induced

          4 items under this folder.

          ACIT4330 Lecture Notes

          Folder: Definitions/Topological-Spaces/Induced

          4 items under this folder.

          🪴 Quartz 4.0

          Compact

          1 min read

          Definition

          +Compact

          ACIT4330 Lecture Notes

          Compact

          1 min read

          Graph View

          Connected Component

          1 min read

          Definition

          +Connected Component

          ACIT4330 Lecture Notes

          Connected Component

          1 min read

          Graph View

          Connected

          1 min read

          Definition

          +Connected

          ACIT4330 Lecture Notes

          Connected

          1 min read

          Definition

          A topological space is connected if it is not a union of two non-empty open sets.

          i.e. if you draw the two non-empty open sets on the graph, if you have to lift your pen, it will not be connected.

          Examples

          diff --git a/public/Definitions/Topological-Spaces/Terminologies/index.html b/public/Definitions/Topological-Spaces/Terminologies/index.html index 1ddaa6f9..15f04289 100644 --- a/public/Definitions/Topological-Spaces/Terminologies/index.html +++ b/public/Definitions/Topological-Spaces/Terminologies/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Topological-Spaces/Terminologies

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          Topological Space

          1 min read

          Definition

          +Topological Space

          ACIT4330 Lecture Notes

          Topological Space

          1 min read

          Definition

          Let be a topological space

          1. A neighbourhood of is an open set with .
            2. diff --git a/public/Definitions/Topological-Spaces/Topology.html b/public/Definitions/Topological-Spaces/Topology.html index b1ee8ea2..35d08676 100644 --- a/public/Definitions/Topological-Spaces/Topology.html +++ b/public/Definitions/Topological-Spaces/Topology.html @@ -1,5 +1,5 @@ -Topology

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            Topology

            1 min read

            Definition

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            ACIT4330 Lecture Notes

            Topology

            1 min read

            Definition

            A collection of subsets of , called open sets, such that:

              2. diff --git a/public/Definitions/Topological-Spaces/Tychonoff-Theorem.html b/public/Definitions/Topological-Spaces/Tychonoff-Theorem.html index c2f7c0b0..1b0de688 100644 --- a/public/Definitions/Topological-Spaces/Tychonoff-Theorem.html +++ b/public/Definitions/Topological-Spaces/Tychonoff-Theorem.html @@ -1,5 +1,5 @@ -Tychonoff Theorem

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              Tychonoff Theorem

              1 min read

              Can be abbreviated as “THM”.

              +Tychonoff Theorem

              ACIT4330 Lecture Notes

              Tychonoff Theorem

              1 min read

              Can be abbreviated as “THM”.

              Definition

              The product of compact spaces is compact in the product topology

              Graph View

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              ACIT4330 Lecture Notes

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              Complex Vector Space

              1 min read

              Definition

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              Complex Vector Space

              1 min read

              Definition

              A complex vector space is a set with addition of vectors , , and scalar multiplication , satisfying the Properties of a Vector Space

              Graph View

              Linear Basis

              1 min read

              Definition

              +Linear Basis

              ACIT4330 Lecture Notes

              Linear Basis

              1 min read

              Definition

              A linear basis of is a subset such that every vector can be written uniquely as for finitely many non-zero .

              Graph View

              Normed Vector Space

              1 min read

              Definition

              +Normed Vector Space

              ACIT4330 Lecture Notes

              Normed Vector Space

              1 min read

              Definition

              Have a metric given by the Norm on a vector space as .

              Graph View

              Properties of a Vector Space

              1 min read

                +Properties of a Vector Space

                ACIT4330 Lecture Notes

                Properties of a Vector Space

                1 min read

                • ,
                • ,
                • ,
                  • diff --git a/public/Definitions/Vector-Spaces/index.html b/public/Definitions/Vector-Spaces/index.html index 79e7f05e..23ceaf39 100644 --- a/public/Definitions/Vector-Spaces/index.html +++ b/public/Definitions/Vector-Spaces/index.html @@ -1,5 +1,5 @@ -Folder: Definitions/Vector-Spaces

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                  Folder: Definitions

                  21 items under this folder.

                  ACIT4330 Lecture Notes

                  Folder: Definitions

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                  Drawing 2025-02-13 10.57.31.excalidraw

                  1 min read

                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

                  +Drawing 2025-02-13 10.57.31.excalidraw

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                  Drawing 2025-02-13 10.57.31.excalidraw

                  1 min read

                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

                  Excalidraw Data

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                  Drawing 2025-02-13 11.02.27.excalidraw

                  1 min read

                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  Drawing 2025-02-13 11.47.33.excalidraw

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  Drawing 2025-02-13 12.01.57.excalidraw

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  Drawing 2025-02-13 12.01.57.excalidraw

                  1 min read

                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ACIT4330 Lecture Notes

                  Drawing 2025-02-13 12.07.00.excalidraw

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ACIT4330 Lecture Notes

                  Drawing 2025-02-24 12.47.32.excalidraw

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

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                  diff --git a/public/Excalidraw/Lecture-13/index.html b/public/Excalidraw/Lecture-13/index.html index a05fb659..f21bed59 100644 --- a/public/Excalidraw/Lecture-13/index.html +++ b/public/Excalidraw/Lecture-13/index.html @@ -1,5 +1,5 @@ -Folder: Excalidraw/Lecture-13

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                  Lecture 1 - 1.1 Sets and Numbers

                  4 min read

                  Prime Numbers

                  +Lecture 1 - 1.1 Sets and Numbers

                  ACIT4330 Lecture Notes

                  Lecture 1 - 1.1 Sets and Numbers

                  4 min read

                  Prime Numbers

                  (natural numbers)

                  Prime Number - only divisible by and .

                  They are building blocks for multiplication; for instance
                  diff --git a/public/Lectures/Lecture-11.html b/public/Lectures/Lecture-11.html index 9fcf9829..18e4440f 100644 --- a/public/Lectures/Lecture-11.html +++ b/public/Lectures/Lecture-11.html @@ -1,5 +1,5 @@ -Lecture 11

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                  Lecture 11

                  5 min read

                  Last lecture talked about Nets

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                  ACIT4330 Lecture Notes

                  Lecture 11

                  5 min read

                  Last lecture talked about Nets

                  Proposition

                  A topological space is Hausdorff each net converges to at most one point.

                  Proof

                  diff --git a/public/Lectures/Lecture-12---Induced-Topologies.html b/public/Lectures/Lecture-12---Induced-Topologies.html index d7657744..41678b26 100644 --- a/public/Lectures/Lecture-12---Induced-Topologies.html +++ b/public/Lectures/Lecture-12---Induced-Topologies.html @@ -1,5 +1,5 @@ -Lecture 12 - Induced Topologies

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                  Lecture 12 - Induced Topologies

                  3 min read

                  Weakest Topology

                  +Lecture 12 - Induced Topologies

                  ACIT4330 Lecture Notes

                  Lecture 12 - Induced Topologies

                  3 min read

                  Weakest Topology

                  Definition of Weakest Topology defined in the lecture.

                  Initial Topology

                  Definition of Initial Topology, defined in the lecture.

                  diff --git a/public/Lectures/Lecture-13---Measure-Theory.html b/public/Lectures/Lecture-13---Measure-Theory.html index dfdbc0cb..8d5a9d6d 100644 --- a/public/Lectures/Lecture-13---Measure-Theory.html +++ b/public/Lectures/Lecture-13---Measure-Theory.html @@ -1,5 +1,5 @@ -Lecture 13 - Measure Theory

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                  Lecture 13 - Measure Theory

                  5 min read

                  Sigma Algebra

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                  ACIT4330 Lecture Notes

                  Lecture 13 - Measure Theory

                  5 min read

                  Sigma Algebra

                  Defined in the lecture: Sigma-Algebra
                  (See also Measurable)

                  Measure

                  diff --git a/public/Lectures/Lecture-2.html b/public/Lectures/Lecture-2.html index 3904cb08..629059ae 100644 --- a/public/Lectures/Lecture-2.html +++ b/public/Lectures/Lecture-2.html @@ -1,5 +1,5 @@ -Lecture 2

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                  Lecture 2

                  2 min read

                  Quotients

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                  ACIT4330 Lecture Notes

                  Lecture 2

                  2 min read

                  Quotients

                  Constructing Quotients

                  Have equivalence relation on given when .

                  Then is the set of rational numbers with addition and multiplication defined in an obvious way.
                  diff --git a/public/Lectures/Lecture-3.html b/public/Lectures/Lecture-3.html index 59b0866b..d8b7f1de 100644 --- a/public/Lectures/Lecture-3.html +++ b/public/Lectures/Lecture-3.html @@ -1,5 +1,5 @@ -Lecture 3

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                  Lecture 3

                  2 min read

                  Proposition 1.1.4

                  +Lecture 3

                  ACIT4330 Lecture Notes

                  Lecture 3

                  2 min read

                  Proposition 1.1.4

                  Each real number is a limit of a sequence of rational numbers.

                  As an ordered Number Field is complete, meaning that every Cauchy Sequence in converges to a real number.
                  diff --git a/public/Lectures/Lecture-4---1.2-Metric-Spaces.html b/public/Lectures/Lecture-4---1.2-Metric-Spaces.html index cdd5d871..fc20be4c 100644 --- a/public/Lectures/Lecture-4---1.2-Metric-Spaces.html +++ b/public/Lectures/Lecture-4---1.2-Metric-Spaces.html @@ -1,5 +1,5 @@ -Lecture 4 - 1.2 Metric Spaces

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                  Lecture 4 - 1.2 Metric Spaces

                  2 min read

                  The Inverse Image

                  +Lecture 4 - 1.2 Metric Spaces

                  ACIT4330 Lecture Notes

                  Lecture 4 - 1.2 Metric Spaces

                  2 min read

                  The Inverse Image

                  The Inverse Image uses a Inverse Function of written is .

                  Complex Numbers

                  In Complex Numbers, with usual addition of vectors

                  diff --git a/public/Lectures/Lecture-5.html b/public/Lectures/Lecture-5.html index 14c95b73..900fb62b 100644 --- a/public/Lectures/Lecture-5.html +++ b/public/Lectures/Lecture-5.html @@ -1,5 +1,5 @@ -Lecture 5

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                  Lecture 5

                  2 min read

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                  Lecture 5

                  2 min read

                  Info

                  diff --git a/public/Lectures/Lecture-6---2.1-Topology.html b/public/Lectures/Lecture-6---2.1-Topology.html index 1c36cec6..458ef641 100644 --- a/public/Lectures/Lecture-6---2.1-Topology.html +++ b/public/Lectures/Lecture-6---2.1-Topology.html @@ -1,5 +1,5 @@ -Lecture 6 - 2.1 Topology

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                  Lecture 6 - 2.1 Topology

                  1 min read

                  Open Sets

                  +Lecture 6 - 2.1 Topology

                  ACIT4330 Lecture Notes

                  Lecture 6 - 2.1 Topology

                  1 min read

                  Open Sets

                  The concept of balls is required to understand what an open sets are.

                  Definition

                  The definition can be found in Open Sets.
                  diff --git a/public/Lectures/Lecture-7.html b/public/Lectures/Lecture-7.html index e14685cf..ee332643 100644 --- a/public/Lectures/Lecture-7.html +++ b/public/Lectures/Lecture-7.html @@ -1,5 +1,5 @@ -Lecture 7

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                  Lecture 7

                  4 min read

                  ball topology
                  +Lecture 7

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                  Lecture 7

                  4 min read

                  ball topology
                  Say is a closed set in (in other words, open).

                  Say we have a sequence in and that .
                  Then .

                  diff --git a/public/Lectures/Lecture-8.html b/public/Lectures/Lecture-8.html index e3e4cc68..110fac95 100644 --- a/public/Lectures/Lecture-8.html +++ b/public/Lectures/Lecture-8.html @@ -1,5 +1,5 @@ -Lecture 8

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                  Lecture 8

                  2 min read

                  Defines in topological spaces:

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                  ACIT4330 Lecture Notes

                  Lecture 8

                  2 min read

                  Defines in topological spaces:

                  • Connected
                  • Connected Component
                    • diff --git a/public/Lectures/index.html b/public/Lectures/index.html index c596a429..08de196a 100644 --- a/public/Lectures/index.html +++ b/public/Lectures/index.html @@ -1,5 +1,5 @@ -Folder: Lectures

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                    Folder: Lectures

                    11 items under this folder.

                    ACIT4330 Lecture Notes

                    Folder: Lectures

                    11 items under this folder.

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                    index

                    1 min read

                    Chapter 1

                    +index

                    ACIT4330 Lecture Notes

                    index

                    1 min read

                    Chapter 1

                    1.1 Sets and Numbers

                    • Lecture 1 - 1.1 Sets and Numbers (complimentary written notes: ACIT4330-2025-01-06-Lecture 1.rnote)
                      • diff --git a/public/index.xml b/public/index.xml index ae04d530..2887af27 100644 --- a/public/index.xml +++ b/public/index.xml @@ -1,70 +1,70 @@ - 🪴 Quartz 4.0 + ACIT4330 Lecture Notes https://quartz.jzhao.xyz - Last 10 notes on 🪴 Quartz 4.0 + Last 10 notes on ACIT4330 Lecture Notes Quartz -- quartz.jzhao.xyz - Drawing 2025-02-13 10.57.31.excalidraw - https://quartz.jzhao.xyz/Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw - https://quartz.jzhao.xyz/Excalidraw/Lecture-11/Drawing-2025-02-13-10.57.31.excalidraw - ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. - Sat, 01 Mar 2025 13:27:35 GMT - Drawing 2025-02-13 11.26.08.excalidraw https://quartz.jzhao.xyz/Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw https://quartz.jzhao.xyz/Excalidraw/Lecture-11/Drawing-2025-02-13-11.26.08.excalidraw ⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. - 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2025-03-01T13:26:38.879Z + 2025-03-01T13:37:40.378Z \ No newline at end of file diff --git a/public/tags/excalidraw.html b/public/tags/excalidraw.html index 8bb6cb7f..8b56e018 100644 --- a/public/tags/excalidraw.html +++ b/public/tags/excalidraw.html @@ -1,5 +1,5 @@ -Tag: excalidraw

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                      ACIT4330 Lecture Notes

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