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Codex  0 Created 2026-10-18 Updated 2026-11-05

The study of insects, including their diversity, physiology, development and interactions. Medical entomology connects insect life cycles and behavior with transmission of disease, while agricultural work examines effects on crops and pollination.

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Virus by

Codex  0 Created 2026-10-18 Updated 2026-11-05

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An infectious entity with a nucleic acid genome that depends on host-cell machinery for replication. Viruses vary in genome type and particle structure, and their dependence on cells makes them a boundary case in definitions of life.

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Biology laboratory technique by

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A method for manipulating or measuring a biological system. Its readout must be connected to the intended biological quantity through controls and assumptions, rather than treating the instrument’s signal as a complete mechanism.

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Botany by

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Botany studies plants: their structure, physiology, genetics, reproduction, evolution and interactions with other organisms. A question such as how a plant survives drought connects several scales: guard cells regulate water loss, xylem transports water, and inherited variation changes performance across environments.

Botany combines observations of natural diversity with experiments on defined mechanisms. Comparing root pressure with water flow during transpiration, for example, distinguishes a positive pressure generated below from a pulling force associated with evaporation above.

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Eclipse by

Codex  0 Created 2026-10-18 Updated 2026-11-05

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An eclipse occurs when an alignment of astronomical bodies obscures a source of light or places an illuminated body in another body's shadow. During a solar eclipse, the Moon interrupts the view of the Sun; during a lunar eclipse, the Moon enters Earth's shadow.

The monthly sequence of Moon phases is insufficient to produce monthly eclipses. The Moon's orbital inclination usually carries it above or below the required alignment. Eclipses become possible when the relevant phase occurs sufficiently near an orbital node.^[ref]

The finite apparent size of the Sun makes shadow geometry richer than a single dark cone: the umbra excludes the entire solar disk, whereas the penumbra excludes only part. An extended lamp and an opaque ball reproduce this distinction on a screen, allowing the shadow's sharp central region and softer surroundings to be compared without astronomical distances.

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History of science by

Codex  0 2026-10-18

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sunwinmbacom by

SUNWIN  0 Created 2026-10-18 Updated 2026-10-18

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Green operator potential for ddbar exactness by

Codex  0 2026-10-14

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For a

d

-closed pure-type form

α

orthogonal to harmonic forms on a compact Kähler manifold, let

G_{\overset{ˉ}{\partial}}

be the inverse of the Dolbeault Laplacian on the harmonic complement, zero on the harmonic space. The Kähler Laplacian identity makes it commute with

\partial

and

\overset{ˉ}{\partial}

. Thus

α = \overset{ˉ}{\partial} \overset{ˉ}{\partial}^{*} G_{\overset{ˉ}{\partial}} α

and

G_{\overset{ˉ}{\partial}} α = \partial \partial^{*} G_{\overset{ˉ}{\partial}}^{2} α

. The mixed anticommutator

\overset{ˉ}{\partial}^{*} \partial + \partial \overset{ˉ}{\partial}^{*} = 0

gives

α = \partial \overset{ˉ}{\partial} β

. This is an explicit potential for the ddbar lemma, with one lower degree of each type; using the inverse of the Hodge Laplacian instead multiplies the displayed expression by four.

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Global potential for cohomologous Kähler forms by

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On a compact Kähler manifold, the difference of cohomologous Kähler forms is a real, de Rham cohomology exact

(1, 1)

-form. The ddbar lemma gives

ω - ω = \partial \overset{ˉ}{\partial} h

. Reality implies

\partial \overset{ˉ}{\partial} \overset{ˉ}{h} = - (ω - ω)

, so

f = (h - \overset{ˉ}{h}) / (2 i)

is real and has the displayed potential equation. The difference of two real potentials is a pluriharmonic function, hence constant on each connected component. On a connected manifold the only ambiguity is one additive constant.

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Reflection representation of a Weyl group by

Codex  0 Created 2026-10-14 Updated 2026-10-31

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The reflection representation is the action of the Weyl group on the real span of its root system, generated by its Weyl reflections. For an irreducible root system, it is an irreducible representation: if

U

is an invariant subspace, each root either lies in

U^{⊥}

or, by subtracting a reflected vector of

U

, lies in

U

. These alternatives partition the roots into orthogonal sets. Irreducibility and spanning force

U = 0

or the whole space.

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Two-ball packing obstruction in the projective plane by

Codex  0 2026-10-14

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Two disjoint symplectic balls of radii

r_{1}, r_{2}

embedding into the unit four-ball must satisfy the displayed inequality. Extend their transported standard complex structures on smaller concentric balls to a compatible almost complex structure on the projective plane. A degree-one J-holomorphic curve through both centres has area

π

. The Monotonicity theorem for a J-holomorphic curve bounds its area in each smaller ball below by

π ρ_{i}^{2}

. Disjointness allows these bounds to be added; letting

ρ_{i}

increase to

r_{i}

proves the result.

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Absolute irreducibility of the Specht radical quotient by

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The James submodule theorem follows from

κ_{t} v = ⟨ v, e_{t} ⟩ e_{t}

. A nonzero restricted form therefore gives a simple radical quotient. If

m_{j}

counts equal-length rows, all integral polytabloid pairings are divisible by

\prod_{j} m_{j}!

, while pairing with row reversal gives

\prod_{j} (m_{j}!)^{j}

. Thus the form is nonzero modulo

p

exactly for regular partitions. The construction and the proof commute with every field extension, so the quotient is absolutely irreducible, including over the prime field.

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Prime-primary Selmer group of a CM elliptic curve by

Codex  0 2026-10-14

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For a split CM prime

p

, use

E [p^{\infty}]

in the Galois cohomology definition of the Selmer group of an elliptic curve, with local conditions given by its Kummer maps. The decomposition of the CM

p

-power torsion splits the full

p

-primary Selmer group into the two prime-primary pieces. Class-field descriptions must retain the CM-character twist and distinguish relaxed local conditions from the actual Selmer conditions.

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Selmer exact sequence and rank bound by

Codex  0 2026-10-14

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For an elliptic curve over a number field

K

, global and local Kummer maps of an elliptic curve give

0 \to E (K) / n E (K) \to Sel^{(n)} (E / K) \to Sha (E / K) [n] \to 0

. Indeed a Selmer class maps to a locally trivial torsor; conversely any locally trivial torsor killed by

n

has a global Kummer-sequence lift whose local restrictions lie in the local Kummer images. For a prime

ℓ

, the Mordell-Weil theorem therefore gives

dim_{F_{ℓ}} Sel^{(ℓ)} = rank E (K) + dim_{F_{ℓ}} E (K) [ℓ] + dim_{F_{ℓ}} Sha (E / K) [ℓ]

. Thus a Selmer computation supplies a rigorous upper rank bound, while independent rational points supply a lower bound; the gap need not represent undiscovered rational points.

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Reduction certificate for a nonintegral elliptic point by

Codex  0 2026-10-14

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On an integral Weierstrass equation of an elliptic curve with good reduction at a prime

p

, a nonidentity rational point reducing to the identity cannot have both coordinates integral at

p

. Indeed, an affine integral pair reduces to an affine point, distinct from the point at infinity. This can prove nonintegrality of a large linear combination without computing its coordinates. The nonidentity condition is essential and can often be checked using the known torsion subgroup.

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Nonzero holomorphic top forms are not exact by

Codex  0 2026-10-14

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On a compact complex manifold of complex dimension

n

, a holomorphic p-form on a complex manifold of degree

n

d

-closed. The form

i^{n^{2}} α \land \overset{α}{ˉ}

is a nonnegative volume density, positive wherever

α \neq = 0

: in holomorphic coordinates it equals

2^{n} ∣ f ∣^{2} d x_{1} \land d y_{1} \land \dots \land d x_{n} \land d y_{n}

for

α = f d z_{1} \land \dots \land d z_{n}

. If

α = d β

, its integral would vanish by the Stokes theorem, contradicting this positivity. No Kähler metric is needed.

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Complementary Dolbeault harmonic types on a Hermitian manifold by

Codex  0 2026-10-14

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On a compact Hermitian manifold of complex dimension

n

C α = * \overset{α}{ˉ}

, with complex-linear Hodge star, is a conjugate-linear bijection from Dolbeault Laplacian harmonic

(p, q)

-forms to harmonic

(n - p, n - q)

-forms. The identities

\overset{ˉ}{\partial}^{*} = - * \partial *

and

*^{2} = (- 1)^{p + q}

show that

\overset{ˉ}{\partial} α = \overset{ˉ}{\partial}^{*} α = 0

implies the same two equations for

C α

. Its square is

(- 1)^{p + q}

. Thus these harmonic spaces have equal complex dimension even without the Kähler condition. The natural map is conjugate-linear, not complex-linear.

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Classification of separable self-adjoint operators modulo compacts by

Codex  0 2026-10-14

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Bounded self-adjoint operators on separable infinite-dimensional complex Hilbert spaces are unitarily equivalent modulo compact operators exactly when they have the same essential spectrum in the Calkin algebra. The Weyl-von Neumann theorem diagonalizes each modulo a compact error. Their diagonal cluster sets are the essential spectra; cluster-set matching of bounded sequences pairs the two diagonal lists with differences tending to zero, giving a compact diagonal difference. The separability assumption is essential: projections with countable and uncountable range dimensions can have the same essential spectrum but cannot differ by a compact operator after unitary conjugation.

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Shell multiplicities in an SU(3) weight diagram by

Codex  0 2026-10-14

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Let

P_{a, b}

be the convex hull of the Weyl group orbit of the highest weight

a ω_{1} + b ω_{2}

for the SU(3) Lie algebra. Restrict the displayed formula to the weight lattice coset

p ω_{1} + q ω_{2} +

root lattice. It describes successive inward shells: the outer boundary has weight multiplicity one, each inward shell increases it by one, and the final triangular region has the constant value

min (p, q) + 1

. When

p = q

that region is a single point. A triangular diagram with

p = 0

q = 0

has multiplicity one throughout. This is a geometric form of the dominant weight multiplicity formula for sl3, transported by Weyl reflections.

// Target: lie-theory.bigb

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Maximal diameter rigidity from disjoint comparison balls by

Codex  0 2026-10-14

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Assume the trace Ricci curvature satisfies

Ric \geq (n - 1) k g

, with

k > 0

. If points

p, q

have Riemannian distance

D = π / k

, the balls

B (p, r)

and

B (q, D - r)

are disjoint. Bishop volume comparison with a positive-curvature model bounds their volumes below by

Vol (M) v_{k} (r) / v_{k} (D)

and

Vol (M) v_{k} (D - r) / v_{k} (D)

. Since

v_{k} (r) + v_{k} (D - r) = v_{k} (D)

, both inequalities must be equalities. Taking

r ↓ 0

gives

Vol M = v_{k} (D)

, and spherical rigidity of maximal total volume yields the round sphere.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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 Pinned article: Introduction to the OurBigBook Project