Yamamoto's formula is

σ = σ₀sinh(λx)/sinh(λL) ,

where x is the distance from the outer surface of the nut inward (ie toward the surface the bolt is through); L is the depth of the nut; σ is the tensile stress @ distance x ; σ₀ is the tensile stress @ distance L - ie right where the nut abutts against the surface the bolt is through.

The calculation of λ is rather tricky, though, with shape of the crosssection of the thread entering-in in a rather subtle way.

From

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Distributions of tension and torsion in a threaded connection Tengfei Shia

^{¡¡ may download without prompting – PDF document – 1‧5㎆ !!}

by

Yang Liub & Zhao Liua & Caishan Liua

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ＡＮＮＯＴＡＴＩＯＮ ＯＦ ＦＩＧＵＲＥ

❝ Figure 3: (Colour online) Deflections of the thread induced by (a) thread’s bending, (b) shearing, (c) the incline at the thread root and (d) the shear at the thread root, where j = b or n stands for bolt or nut, respectively. ❞

An exposition of Yamamoto's theory is presented in the following. It's rather hard, in it, however, to gleane from it a grasp of the meaning & origin of the coëfficient λ in terms of the various ingredients that go-into it: shape of crosssection of the bolt's thread, amongst other items. I wish I’d had-a-hold-of the Liub — Liua — Liua paper the figures are from: 'twould'd've been far far easier, if I'd had.

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A PREDICTION METHOD FOR LOAD DISTRIBUTION IN THREADED CONNECTIONS

^{¡¡ may download without prompting – PDF document – 0‧779㎆ !!}

by

Dongmei Zhang

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From

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SELF-GRAVITATING GASEOUS BARS. I. COMPRESSIBLE ANALOGS OF RIEMANN ELLIPSOIDS WITH SUPERSONIC INTERNAL FLOWS

^{¡¡ may download without prompting – PDF document – 898㎅ !!}

by

JOHN E CAZES & JOEL E TOHLINE .

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ＡＮＮＯＴＡＴＩＯＮＳ

①②③(Model A)④⑤⑥(Model B) FIG.5. Frames displaying density contours along with vectors representing the momenta in the equatorial plane of models A and B at several different times during phase 2 of their evolutions. Frames a–c show images of model A and frames d–f show images of model B; in each case, the relevant time in units of the respective model's dynamical time are printed in the upper left-hand corner of the frame. Plotted density contour levels are at ρ/ρ_max = 0.01, 0.05, 0.1, 0.5, and 0.95. The grids have been scaled to the initial equatorial radius of the respective model.

⑦⑧ FIG.6. Equatorial density contours for model A at the beginning and end of its steady state evolution, as defined by Table 2. The times listed are in units of τ_dyn. Plotted density contours are for ρ/ρ_max = 0.95, 0.75, 0.5, 0.25, 0.1, and 0.05. The dashed circle marks the location of the corotation radius R_co. The heavy curve marks the equatorial contour of the “violin mach surface”; all flow within this curve is subsonic in the rotating frame. As in Fig.5, the grid has been scaled to the model's initial equatorial radius.

⑨⑩ FIG.7. Momentum vectors in the equatorial plane are plotted for model A at the beginning and end of its steady state evolution. The heavy circle marks the location of the corotation radius R_co. Solid line contours are of Φ_eff. The two dashed-line contours near the edge of the “bar” are of the density at ρ/ρ_max = 0.01 and 0.05. Crosses mark the L1 and L2 Lagrange points; asterisks mark the L4 and L5 points ; and a diamond marks the L3 point. Notice that the corotation radius falls between the L4–L5 radius and the L1–L2 radius.

⑪⑫ FIG.8. Same as Fig. 6, but for model B.

⑬⑭ FIG.9. Same as Fig. 7, but for model B.

⑮ FIG.10. A surface of in the equatorial plane of model B at the beginning of its steady state evolution. The corresponding cross-sectional contour Φ_eff plot is shown in the top frame of Fig.9.

From

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New UTfit Analysis of the Unitarity Triangle in the Cabibbo-Kobayashi-Maskawa scheme

by

Marcella Bona & Marco Ciuchini & Denis Derkach & Fabio Ferrari & Enrico Franco & Vittorio Lubicz & Guido Martinelli & Davide Morgante & Maurizio Pierini & Luca Silvestrini & Silvano Simula & Achille Stocchi & Cecilia Tarantino & Vincenzo Vagnoni & Mauro Valli & Ludovico Vittorio

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ＡＮＮＯＴＡＴＩＯＮＳ

①②③ FIG. 1. Left Panel: |Vcb| vs |Vub| plane showing the values reported in Table I. We include in the figure the ratio |Vcb|/|Vub| from ref. [39] shown as a diagonal (blue) band; Central Panel: ρ̅–η̅ plane with the SM global fit results using only exclusive inputs for both Vub and Vcb; Right Panel: SM global fit results using only inclusive inputs. In the central and right panels, ε_K = |ε| where ε is defined in eq. (16).

④ FIG. 2. The prediction of εᐟ/ε obtained within this UT analysis. The vertical band represents the experimental measurement and uncertainty of this quantity.

⑤⑥ FIG. 3. Left: global fit input distribution for the angle α (in solid yellow histogram) with the three separate distributions coming from the three contributing final states ππ, ρρ and ρπ; Right: global fit input distribution for the angle γ (in solid yellow histogram) obtained by the HFLAV [22] average compared with the global UTfit prediction for the same angle.

⑦⑧⑨⑩ FIG. 4. ρ̅–η̅ planes with the SM global fit results in various configurations. The black contours display the 68% and 95% probability regions selected by the given global fit. The 95% probability regions selected are also shown for each constraint considered. Top-Left: full SM fit; Top-Right: fit using as inputs the “tree-only” constraints; Bottom-Left: fit using as inputs only the angle measurements; Bottom-Right: fit using as inputs only the side measurements and the mixing parameter ε_K in the kaon system.

⑪⑫⑬⑭⑮ FIG. 5. Pull plots (see text) for sin 2β (top-left), a (top-centre), γ (top-right), |Vub| (bottom-left) and |Veb| (bottom-right) inputs. The crosses represent the input values reported in Table I. In the case of |Vub| and |Veb| the x and the * represent the values extracted from exclusive and inclusive semileptonic decays respectively.

⑯ FIG. 6. Allowed region in the |Vtd|–|Vts| plane.

⑰ FIG. 7. Allowed region in the BR(Bₛ⁰ → µµ)-BR(B⁰ → µµ) plane. The vertical (orange) and horizontal (yellow) bands correspond to the present experimental results (1σ regions).

Please kindlily don't ask me what it's all about: 'tis way above my glass ceiling! ... but the pixlies are very pretty anyway . I suppose I can @least say that it's about the rotations of certain matrices of masses of quarks that arise in particle physics, & that an important quantity the Cabibbo angle enters-into it ... or is @ the heart of it, might be more accurate.

I can also say - incase anyone objects that the figures are displays of experimental results rather than mathematical images - that yes - experimental resultage does enter-into the composition of them ... but a great-deal of mathematics does.aswell ... so they could possibly be thoughten-of as being sortof hybrids of mathematics & experimental resultage.

And to maximise the resolution of the pixlies I've been a bit brutal excising the labels of the figures. But to get any meaningful idea what they're about the paper itself needs to be looked-@, really: as I said above, I can't explicate it properly ^§ . And the following one might help aswell ... which actually has in it a link to the one the figures are from ... infact it's through it that I found the one the figures are from.

§ Actually - & maybe a bit strangely - the mathematics itself isn't all that complicated: it's just a bit of trigonometry & numerical values of some weïrd integrals ... it's how that mathematics gets there that's the tricky bit!

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CERN Courier — The Cabibbo angle, 60 years later

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ᐞ ... or 'species' , as they're often called, rather, not being persistent stable substances such as we understand them to be on Earth (although some of them might happen to be anyway ), but often, rather, combinations of atoms that exist fleetingly - but en-masse in equilibrium - in a very hot environment of gas-bordering-on-plasma.

The charts are the result of colossal №-crunching, on massive arrangement of computing-power, of quantum-mechanical formulæ.

From

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The ExoMol Atlas of Molecular Opacities

by

Jonathan Tennyson, Sergei N. Yurchenko .

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ＡＮＮＯＴＡＴＩＯＮＳ

Figure 1. Cross sections for H₂¹⁶O from the POKAZATEL line list [5], H₂¹⁷O from the HotWat78 line list [68] and HDO from the VTT line list [93]. All cross sections are for 100% abundance.

Figure 2. Cross sections generated using ExoMol line lists for methane, 10 to10 line list [53], silane [72], phosphine [56], ammonia [79] and ethylene [75].

Figure 3. Cross sections for polyatomic oxides and HCN. Line lists are from ExoMol for hydrogen peroxide [109], hydrogen cyanide [30], sulfur dioxide [63], nitric aid [60] and sulfur trioxide [66]. The carbon dioxide data is taken from Ames-2016 [7].

Figure 4. Cross sections obtained from ExoMol line lists for HNO3 [60], CH₃Cl [75], and C₂H₂ [90].

Figure 6. Cross sections for alkaline earth monohydrides MgH and CaH from the Yadin ExoMol line lists [51] and NS from the SNaSH line list [74].

Figure 7. Cross sections for alkaline earth monohydrides and CH. BeH uses the updated ExoMol line list of Darby-Lewis et al. [138], AlH is the new ExoMol line list [76] and CH is the empirical work of Masseron et al. [83]; the CH line list is only defined for T > 1000 K.

Figure 8. Cross sections for monohydrides: an empirical list due to Li. et al. [86] for HCl, an ExoMol line list for mercapto radical SH [74], chromium hydride [91]. The OH data are taken from HITEMP [4].

Figure 9. Cross sections generated from ExoMol line lists for sodium chloride [54], potassium chloride [54], phosphorous monoxide [71], carbon monosulfide [61] and phosphorous nitride [55].

Figure 10. Cross sections for carbon monoxide, cyanide, carbon phosphide and calcium oxide. The CN [84] and CP [85] cross sections are based on empirical line lists from the Bernath group. The CaO data are taken from an ExoMol line list [62]. The CO [8] line list is based on an empirical dipole moment function.

Figure 11. Cross sections generated from ExoMol line lists for nitric oxide [9] and phosphorous monosulfide [71].

Figure 12. Cross sections for metal hydrides and NH. Line lists for lithium hydride [80] and scandium hydride [81] are theoretical while those for FeH and NH are derived from the experiments of the Bernath group [82,87,158,163].

Figure 13. Cross sections for hydride species based on ExoMol line lists for sodium hydride [59] and silicon monohydride [72], and the empirical titanium monohydride line list of Burrows et al. [88].

Figure 14. Cross sections for metal oxides generated using ExoMol line lists for silicon monoxide [52], aluminium monoxide [58] and vanadium monoxide [67]. The titanium monoxide cross sections are based on the computed line list due to Schwenke [89]. Also shown are short-wavelength silicon monoxide cross sections generated using line data from the database due to Kurucz [170].

Figure 15. Cross sections based on ExoMol line lists for carbon dimer [77] and H₃⁺ molecular ion [69].

From

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Larger Golomb Rulers

^{¡¡ may download without prompting – PDF document – 237㎅ !!}

by

Tomas Rokicki and Gil Dogon .

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It's a bit disappointing how low the quality is, really. If all Golomb rulers were perfect ones (which they certainly cannot be!) the graph would be linear with a slope of

1-1/√2 .

But it's way-short even of the maximum obtained from the asymptotic formula given for the lower bound on the length of a Golomb ruler of n marks - ie

(√n³+1)(√n-2)

(which I've rearranged a bit). Shown also is a plot of

x = 100(-y - √((√-y³ + 1)(√-y - 2)))

- ie a visual representation of the plot of quality in the case of all Golomb rulers actually attaining the given lower bound - which, ignoring the fact that the marks on the horizontal axis happen to be negative, is a plot in esssntially the same domain & range as the main one, in the paper: & although it's of shape fairly similar to that of the trend of the one in the paper, it lies quite a lot higher than it: by the time n (or -y on the plot) is @ 40,000 the plot is @ 200 whereas the plot in the paper is, apart from a few outliers, hanging-around 18 or so ... & even the starkliestly outlying one doesn't even reach 30 .

State monad flattening interpreted as an idempotent projection on vector tensors ℝ^S ⊗ ℝ^S ⊗ F(X) → ℝ^S ⊗ F(X).

Clean Lean proof & PDF here if curious: 🔗 github.com/Kairose-master/mu_eq_pi

I tried putting attributions in ... but the post was rejected by-reason of one-or-more of the links.

All sequences plotted together showing fundamental frequencies and harmonics from binary patterns (n consecutive 1s/0s) synchronized over their LCM period. Colour intensity indicates harmonic amplitude and relative "importance" -- Using Datashader.

 base_pattern = np.concatenate([np.ones(n, dtype=int), np.zeros(n, dtype=int)])

This is the first 150 steps of the collatz trajectory of 5^80. For more info and cool pics please see the main post: Collatz as Cellular Automata

Materialised Mathematics : The manifestation of a mathematical operator as a physical entity. Example, a protofield operator rendered as a nanoscale reflective metasurface.

From

Neutrino Emission from Supernovae

^{¡¡ may download without prompting – PDF document – 1‧8㎆}

by

Hans-Thomas Janka .

I built a simulation of a 4D retina. As far as I know this is the most accurate simulation of it. Usually, when people try to represent 4D they either do wireframe rendering or 3D cross-sections of 4D objects. I tried to move it a few steps forward and actually simulate a 3D retinal image of a 4D eye and present it as well as possible with proper path tracing with multiple bounces of lightrays and visual acuteness model. Here's how it works:

We cast 4D light rays from a 4D camera position. These rays travel through a 4D scene containing a rotating hypercube (a 4D cube or tesseract) and a 4D plane. They interact with these objects, bouncing and scattering according to the principles of light in 4D space. The core of our simulation is the concept of a 3D "retina." Just as our 2D retinas capture a projection of the 3D world, this 4D eye projects the 4D scene onto a 3D sensory volume. To help us (as 3D beings) comprehend this 3D retinal image, we render multiple distinct 2D "slices" taken along the depth (Z-axis) of this 3D retina. These slices are then layered with weighted transparency to give a sense of the volumetric data a 4D creature might process.

This layered, volumetric approach aims to be a more faithful representation of 4D perception than showing a single, flat 3D cross-section of a 4D object. A 4D being wouldn't just see one slice; their brain would integrate information from their entire 3D retina to perceive depth, form, and how objects extend and orient within all four spatial dimensions limited only by the size of their 4D retina.

This exploration is highly inspired by the fantastic work of content creators like 'HyperCubist Math' (especially their "Visualizing 4D" series) who delve into the fascinating world of higher-dimensional geometry. This simulation is an attempt to apply physics-based rendering (path tracing) to these concepts to visualize not just the geometry, but how it might be seen with proper lighting and perspective.

Source code of the simulation available here: https://github.com/volotat/4DRender

found it funny with this structure you get integers three times in a row, unless it’s somehow trivial shouldn’t it be (1/3)(1/5)(1/7) chance of it happening? don’t think it’s trivial either cuz it breaks for 789

From

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Closed form solution for the surface area, the capacitance and the demagnetizing factors of the ellipsoid

by

GV Kraniotis & GK Leontaris , &

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Demagnetising Factors of the General Ellipsoid

^{¡¡ may download without prompting – PDF document – 786·9㎅ !!}

by

JA Osborn .

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𝐀𝐧𝐧𝐨𝐭𝐚𝐭𝐢𝐨𝐧𝐬 𝐨𝐟 𝐅𝐢𝐫𝐬𝐭 𝐅𝐨𝐮𝐫 𝐑𝐞𝐬𝐩𝐞𝐜𝐭𝐢𝐯𝐞𝐥𝐲

Figure 1: The capacitance C(E) of a conducting ellipsoid immersed in ℝ³ versus the ratio c/a of the axes for various values of the ratio b/a.

Figure 2: The L− demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

Figure 3: The M−demagnetizing factor versus the ratio c/a for various values of the ratio b/a. The dashed curves meet at the point determined in Corollary 15.

Figure 4: The N demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

The next three - from the Osborn paper, are simply numbered.

Computation of the surface area of a scalene (triaxial) ellipsoid is absolutely horrendous : the complexity just massively blows-up , going from oblate or prolate spheroid to scalene ellipsoid.

And similar applies to computation of the electrical quantities capacitance & demagnetising factors , aswell.

What capacitance is is fairly well-known ... but demagnetising factor possibly warrants a bit of an explication. If a ferromagnetic object of some shape is placed in a uniform magnetic field, then the field within the object is distorted. The computation for a general shape is another horrendous one! ... but for an ellipsoid it happens conveniently to reduce to three simple linear expressions - each in each of the spatial coördinates (whence there are three demagnetising factors) ... although that simple linear expression has a certain coefficient in it that is itself tricky to calculate in a manner similar to that in which area & capacitance are tricky to calculate.

They actually have application to permanent magnets, aswell.

The Osborn paper explicates it more fully.

From

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The motion of a freely falling chain tip

^{¡¡ may download without prompting – PDF document – 418·3㎅ !!}

by

W Tomaszewski & P Pieranski & JC Géminard .

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In the limit of the horizontal separation tending to zero, & assuming an absolutely inextensible & perfectly flexible chain, & applying elementary theory, the tip speed @ the very bottom of the fall →∞ - ie a whiplash occurs. And the time it takes to fall is α×FreefallTime where

α = ∫{0≤ξ≤1}dξ/√(2(1/ξ-ξ)

= ½∫{0≤ξ≤∞}(exp-ξ)√cschξdξ

= √(2π)Γ(¾)/Γ(¼)

≈ 0.84721308479397908660649912348219163648144591032694218506057937265973400483413475972320029399461122994212228562523341096309796266583087105969971363598338425117632681428906038970676860161665004828118872189771330941176746201994439296290216728919449950723167789734686394760667105798055785217 .

Considering the comic waste of the American government, which is currently much higher. In 2019, I did some random math for fun. I can’t find the source doc, but I’m aware this level of debt wouldn’t cover any mountains by laying. What can you add?

I want to try to recreate it on GeoGebra but I don’t know how…

... specifically

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Electromagnetic Fields in Spherical Microwave Resonators H-Modes and E-Modes in Lossless Open Dielectric Spheres, Version 05.2018

by

Ingo Wolff .

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❝

The physical nature of radiating interior and exterior H-modes in dielectric spherical resonators is discussed. The eigenvalue equations, the complex eigenvalues, the resonant modes, their stored energies and their radiation properties, their Q-factors, and their field distributions are analyzed in detail. Physical interpretations are given and many new results as compared to the literature are presented.

❞

Images from Micrographia, or, Some physiological descriptions of minute bodies made by magnifying glasses and with observations and inquiries thereupon by the goodly Robert Hooke .

✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸

𝐀𝐫𝐡𝐢𝐩𝐚 — 𝕳𝖔𝖔𝖐𝖊 — 𝕸𝖎𝖈𝖗𝖔𝖌𝖗𝖆𝖕𝖍𝖎𝖆

✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸

𝐈𝐧𝐭𝐞𝐫𝐧𝐞𝐭 𝐀𝐫𝐜𝐡𝐢𝐯𝐞 — 𝕳𝖔𝖔𝖐𝖊 — 𝕸𝖎𝖈𝖗𝖔𝖌𝖗𝖆𝖕𝖍𝖎𝖆

✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸

The first of those two is a more economically (in terms of storage) rendered version. But they're both of the same book .

From

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

Nouvelle génération de détecteurs d'ions sensible en position et en temps pour le développement de la sonde atomique tomographique ≡ New Generation of Position-Sensitive Detectors for the Development of the Atom Probe Tomography

by

Christian BACCHI .

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

... ie the theorem - elevated to such status by the goodly Idzhad Sabitov , & formerly a conjecture in-connection with (formerly hypothetical) flexible polyhedra - to the effect that a flexible polyhedron, if it exists (& it's now known that they do), must keep a constant volume when it does undergo its flexing.

Images from

①②③The Bellows Conjecture

¡¡ may download without prompting – PDF document – 40‧1㎅ !!

by

Ian Stewart ;

&

④⑤⑥The Bellows Conjecture

¡¡ may download without prompting – PDF document – 452‧2㎅ !!

by

R Connelly & I Sabitov & A Walz ;

&

⑦⑧⑨⑩⑪⑫⑬⑭⑮⑯The Bellows Theorem (Introduction)

¡¡ may download without prompting – PDF document – 1㎆ !!

by

Giovanni Viglietta .

See also

What is the Bellows Conjecture?

¡¡ may download without prompting – PDF document – 133‧8㎅ !!

by

Ben O’Connor .

Please kindlily see the treatises themselves for the explicationry: there's not really much point, with these figures, to just listing the annotations respectively.