60 Matching Annotations
  1. Last 7 days
    1. the M-actions on finitesets correspond to right admissible equivalences on M of finite index.

      It's yoneda lemma!

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  2. Jan 2025
    1. Mathematics

      Functional Analysis

      Optimization by Vector Space Methods<br /> Lectures and exercises on functional Analysis<br /> Convex Functional Analysis<br /> Functional Analysis for Probability and Stochastic Processes<br /> Abstract Calculus A Categorical Approach <br /> Banach-Hilbert Spaces, Vector Measures and Group Representations

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      Clifford algebra, geometric algebra, and applications<br /> Mathematical Structures From Linear Algebra over Rings to Geometry with Sheaves<br /> Diffeology<br /> Global Calculus<br /> Manifolds, Tensor Analysis, and Applications<br /> Manifolds, Sheaves, and Cohomology

      Algebra

      Basic algebra groups, rings and fields<br /> Further Algebra and Application<br /> Abstract Algebra - Paul Garrett.

      Linear Algebra

      Algebra An Approach via Module Theory<br /> Linear Algebra via Exterior Products<br /> Rings, Modules, and Linear Algebra<br /> Modules and Homological Algebra<br /> Module theory an approach to linear algebra<br /> 代数学方法I

      Measure, Probability and Statistics

      A Basic Course in Probability Theory<br /> Probability Theory An Analytic View<br /> Probability and Stochastics<br /> Measure-Theoretic Probability<br /> Measure-Theoretic Calculus in Abstract Spaces<br /> Plane Answers to Complex Questions<br /> Mathematical Statistics<br /> Time Series Theory and Methods<br /> The Coordinate-Free Approach to Linear Models<br /> Multivariate Statistics A Vector Space Approach<br /> hypo

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      THE OPEN LOGIC TEXT<br /> Categorical Logic and Type Theory

      Algebraic Topology

      Fundamentals of Algebraic Topology<br /> Algebraic Topology - A Structural Introduction<br /> Homology, Cohomology, and Sheaf<br /> Combinatorial Algebraic Topology<br /> Algebraic Foundations for Applied Topology and Data Analysis<br /> Topology A Categorical Approach<br /> hypo

      Combinatorics and Discrete

      Introduction to the Theory of Species of Structures<br /> Discrete Calculus Methods for Counting<br /> Combinatorics the rota way

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      Cisco CCNA Simplified

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      低压电工上岗速成

    1. In the previous stats module, we could haveassigned the mean function to module.exports.mean instead ofexports.mean.

      note that xxx.yyy is required to assign property to exported value

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    1. Accessing super.x behaves like Reflect.get(Object.getPrototypeOf(objectLiteral), "x", this), which means the property is always sought on the object literal/class declaration's prototype, and unbinding and re-binding a method won't change the reference of super

      super will remember the object literal it was defined on, so rebinding object won't change the output, but rebinding prototype of object itself or class' prototype can work. Note that when rebinding prototype, it's best to also rebind the class(constructor) function so that static methods are also rebound.

    1. Q1. Previous Brainbow tools work quite well in various applications. Why did the author bother to improve it further?

      Q2. What issues do Brainbow 3.x address? And how did the authors achieve that? (Hint: 6 sentences won’t be able to cover all the improvements that 3.x made. Please describe only two aspects).

      Q3. What is the purpose of making Flpbow? Can you give an example of using it in combination with Brainbow?

    1. It's worth mentioning that javascript a.p = (b.p = c) will set a.p = c, i.e. the return value in b.p setter is meaningless

    1. This sample complexity result admits the following equivalent statement as a gen-eralization bound: for any , δ > 0, with probability at least 1 − δ,R(hS ) ≤ 1m(log |H| + log 1δ).

      \(\epsilon\) here is not necessary. The statement is more precise

    2. false positives

      Positive/Negative stands for whether support the alternative hypothesis, in which case always exist.

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    1. githack is a proxy tool for changing content-type header and is useful with via.hypothes.is for pdf

    1. Positions the dropdownmenu below the togglebutton

      Two alternative css position:absolute bottom: 0; transform: translateY(100%); css /* remove top, bottom simple position:absolute would suffice*/

    2. On the absolutely positioned dropdown-menu, you used left: 0 to align its left sidewith the left side of the dropdown. Then you used top: 2.1em to place its top edgebeneath the label (with the padding and border, the label is about 2.1 em tall). A

      Both are not necessary: absolute without specified value will cause element to appear at its normal flow positon without affecting other.

    3. remove the word close from your markup and replaceit with an ×, but this would introduce an accessibility problem: a

      best practice: use aria-label on the button

    4. inset-inline: 20%

      Percentage for fixed element refer the viewport

    5. Its initial value is auto,

      flex-basis initial value is auto, note short-hand empty set it 0%

  3. Dec 2024
    1. 16-4.7.

      Use the limit structure of \(\mathbb{Q}, \mathbb{R}\) , wecan see that additive property leads to positive linar form

    2. Thereforeyu: F” — K is a well-defined extension

      By showing \( \mu \) is not affected by the selection of \(p, q\)

    3. Clearly everypositive linear form is real

      Note \(C - C = F^r \), apply linearity to see the form is indeed real.

    4. 16-3.10.

      Column/Row Decomposition for two family of vector in lattice

    5. 16-3.8. Theorem

      Vector in vector lattice can be boundly decomposed if it's bound above by sum of two other

    6. (b)

      $$ (x + y)^+ = x^+ + y^+ $$ Link

    7. 16-3.4.

      Meet distribution

    8. (d)

      join and meet operation are compatible with lattice order

    9. following calculation:

      The de morgan law apply to minus, join, meet as well

    10. imaginary part

      Obviously imaginary part is indeed real (its conguation equal to itself)

    11. 16-2.7. Lemma

      If any one of the following exists, the mid one exist. The equality follows from the fact that the two net are equivalent in cauchy sense.

    1. converges uniformly with limit f : E → X if and only if

      For functions whose range is uniform space, the function then thus be defined as uniform space: $$ (f_1 \times f_2) (X) \subseteq U $$ Where \(U\) is entourage. We can apply Moore-Osgood Theorem to such function \( f(x, y) \) if \(f_x \rightrightarrows f\)

    1. inline style

      External <link><br /> Internal head > style<br /> inline <tag style="...">

    2. a selector has more IDs,

      It's actually possible. see Link

    3. hree different types, or origins, of stylesheets

      User-agent, User, author

    1. .Ii + Ij = R

      In term of congruence, it means \(I_i \vee I_j = R\), which make decomposition map for algebra with permutable congruence surjective

    2. Mathematical Structures From Linear Algebra over Rings to Geometry with Sheaves