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From Wikipedia, the free encyclopedia Part of a series on Jainism

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Some Sthanakvasi monks from Gujarat.

Sthānakavāsī is a sect of Śvetāmbara Jainism which was created in the medieval era as a result of a misinterpretation of the Śvetāmbara canon. The Sthanakvasi, whose name refers to the sect’s preference for performing religious duties at a secular place such as a monks’ meeting house (sthanak) rather than at a temple, is different from the Murtipujaka sect in that it rejects idolatry. It believes that idol worship is not essential in the path of soul purification and attainment of Nirvana/Moksha. Sthānakavāsī accept thirty-two of the Jain Agamas, the Śvetāmbara canon, contending that the scriptures make no mention of idol worship and temples.[1] However, since the original texts of the ancient Ardhamagadhi canon contain numerous references to idolatry in the past, they have, over time, also modified the 32 texts they accept, to establish their view.

In the 15th century, Loṅkā Śāh, a rebellious scribe in the Gujarat region, accessed manuscripts of the Śvetāmbara canon illegitimately, and started the Sthanakavasi tradition.[2] Armed with access to numerous Jain scriptures and manuscripts, Loṅkā misinterpreted them as lacking references to temple construction or image worship, despite these practices being extremely well-detailed even in the untouched ancient anga sutras and prevalent at the time. He asserted that such practices were spiritually hazardous, violating the principle of ahiṃsā (non-injury) central to Jain philosophy.[2] Loṅkā argued that building temples led to the destruction of microscopic organisms, and ritualistic pūjā (worship) involved subtle forms of harm through material offerings like flowers or incense.[3] However, such minute levels of violence for spiritual upliftment was totally permissible for householders. The Murtipujakas assert that since ascetics are required to completely abstain from violence, they do not perform worshipping of idols using materialistic objects (Dravya Puja). Besides, the mention of an idol of a Tirthankara in the Hathigumpha inscription proves the existence of idolatry in Jainism in the 2nd century BCE.[4]

The Sthanakvasi sect was founded in the 17th century by Lava of Surat, a follower of Loṅkā. Today, both the Sthānakavāsī and Terāpanthī sects align with Loṅkā, asserting that mental worship (bhāva-pūjā) is the most appropriate form of religious practice. They argue that reliance on images and temples signifies an attachment to material objects that is spiritually counterproductive.[5]

In contrast, Mūrtipūjaka Jains respond to these criticisms by highlighting the scriptural prevalence of image worship and emphasizing the necessity of images for the spiritual practices of laypeople. A notable figure in this discourse is the Jaina scholar, Ātmārām (1837 – 1896), initially a Sthānakavāsī monk who later became the mendicant leader Ācārya Vijayānandasūri. Upon studying early Jain texts in Prakrit and their Sanskrit commentaries, Ātmārām discovered abundant references to image worship.[4] This revelation led him to challenge the non-Mūrtipūjaka position, asserting that it contradicted Jain scriptures.[4] Notes

Jains in the World, Religious Values and Ideology in India, John E. Cort, p. 46 Dundas, Paul (2002). The Jains. London, UK: Routledge. p. 246. ISBN 978-0415266062. Cort, John (2010). Framing the Jina: Narratives of Icons and Idols in Jain History. Oxford, UK: Oxford University Press. pp. 5. ISBN 978-0195385021. Cort, John (2010). Framing the Jina: Narratives of Icons and Idols in Jain History. Oxford, UK: Oxford University Press. pp. 6. ISBN 978-0195385021.

Long, Jeffrey (2009). Jainism: An Introduction. London, UK: I.B. Tauris & Co. Ltd. p. 20. ISBN 978-1845116262.

References

Dundas, Paul (2002), The Jains, Routledge, ISBN 978-0-415-26605-5

Cort, John (2010). Framing the Jina: Narratives of Icons and Idols in Jain History. Oxford, UK: Oxford University Press. pp. 5. ISBN 978-0195385021.
Flügel, Peter (2008). "The Unknown Loṅkā: Tradition and the Cultural Unconscious". In Nalini Balbir; Colette Caillat (eds.). Jaina Studies. 12th World Sanskrit Conference, Helsinki, 13–18 July 2003. Delhi: Motilal Banarsidas. pp. 181–279. ISBN 978-8120832473.
Long, Jeffrey (2009). Jainism: An Introduction. London, UK: I.B. Tauris & Co. Ltd. ISBN 978-1845116262.
Wiley, Kristi L. (2004). The A to Z of Jainism. Lanham, MD: The Scarecrow Press, Inc. ISBN 978-0810868212.

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Jain monks and nuns

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Fabius function

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From Wikipedia, the free encyclopedia This is the current revision of this page, as edited by ~2026-16141-58 (talk) at 22:15, 13 March 2026 (​                                                                ▢                                                                ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦                                                                ▢                                                                ​). The present address (URL) is a permanent link to this version. (diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff) Graph of the Fabius function on the interval [0,1].

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).

This function satisfies the initial condition f ( 0 ) = 0 {\displaystyle f(0)=0}, the symmetry condition f ( 1 − x ) = 1 − f ( x ) {\displaystyle f(1-x)=1-f(x)} for ⁠ 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1}⁠, and the functional differential equation

f ′ ( x ) = 2 f ( 2 x ) {\displaystyle f'(x)=2f(2x)}

for ⁠ 0 ≤ x ≤ 1 / 2 {\displaystyle 0\leq x\leq 1/2}⁠. It follows that f ( x ) {\displaystyle f(x)} is monotone increasing for ⁠ 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1}⁠, with f ( 1 / 2 ) = 1 / 2 {\displaystyle f(1/2)=1/2} and f ( 1 ) = 1 {\displaystyle f(1)=1} and f ′ ( 1 − x ) = f ′ ( x ) {\displaystyle f'(1-x)=f'(x)} and ⁠ f ′ ( x ) + f ′ ( 1 2 − x ) = 2 {\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2}⁠.

It was also written down as the Fourier transform of

f ^ ( z ) = ∏ m = 1 ∞ ( cos ⁡ π z 2 m ) m {\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^{m}}}\right)^{m}}

by Børge Jessen and Aurel Wintner (1935).

The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of

∑ n = 1 ∞ 2 − n ξ n , {\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}

where the ξn are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of 1 2 {\displaystyle {\tfrac {1}{2}}} and a variance of ⁠ 1 36 {\displaystyle {\tfrac {1}{36}}}⁠. Extension of the function to the nonnegative real numbers.

There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f(x) = 0 for x ≤ 0, f(x + 1) = 1 − f(x) for 0 ≤ x ≤ 1, and f(x + 2r) = −f(x) for 0 ≤ x ≤ 2r with r a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.

The Rvachëv up function[1] is closely related to the Fabius function f: u ( t ) = { f ( t + 1 ) , | t | < 1 0 , | t | ≥ 1 . {\displaystyle u(t)={\begin{cases}f(t+1),\quad |t|<1\0,\quad |t|\geq 1\end{cases}}.} It fulfills the delay differential equation[2] d d t u ( t ) = 2 u ( 2 t + 1 ) − 2 u ( 2 t − 1 ) . {\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).} (See Delay differential equation for another example.) ▢ Values

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:[3][4]

f ( 1 ) = 1 {\displaystyle f(1)=1}
f ( 1 2 ) = 1 2 {\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}
f ( 1 4 ) = 5 72 {\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}
f ( 1 8 ) = 1 288 {\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}
f ( 1 16 ) = 143 2073600 {\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}
f ( 1 32 ) = 19 33177600 {\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}
f ( 1 64 ) = 1153 561842749440 {\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}
f ( 1 128 ) = 583 179789679820800 {\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}

with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757. Asymptotic

log ⁡ f ( x ) = − log 2 ⁡ x 2 log ⁡ 2 + log ⁡ x ⋅ log ⁡ ( − log ⁡ x ) log ⁡ 2 − ( 1 2 + 1 + log ⁡ log ⁡ 2 log ⁡ 2 ) log ⁡ x − log 2 ⁡ ( − log ⁡ x ) 2 log ⁡ 2 + log ⁡ log ⁡ 2 ⋅ log ⁡ ( − log ⁡ x ) log ⁡ 2 + ( 6 γ 2 + 12 γ 1 − π 2 − 6 log 2 ⁡ log ⁡ 2 12 log ⁡ 2 − 7 log ⁡ 2 12 − log ⁡ π 2 ) + log 2 ⁡ ( − log ⁡ x ) 2 log ⁡ 2 ⋅ log ⁡ x − log ⁡ log ⁡ 2 ⋅ log ⁡ ( − log ⁡ x ) log ⁡ 2 ⋅ log ⁡ x + O ( 1 log ⁡ x ) {\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O!\left({\frac {1}{\log x}}\right)\end{aligned}}}

for ⁠ x → 0 + {\displaystyle x\to 0^{+}}⁠, where γ {\displaystyle \gamma } is Euler's constant, and γ 1 {\displaystyle \gamma _{1}} is the Stieltjes constant. Equivalently,

log ⁡ f ( 2 − n ) = − n 2 log ⁡ 2 2 − n log ⁡ n + ( 1 + log ⁡ 2 2 ) n − log 2 ⁡ n 2 log ⁡ 2 + ( 6 γ 2 + 12 γ 1 − π 2 12 log ⁡ 2 − 7 log ⁡ 2 12 − log ⁡ π 2 ) − log 2 ⁡ n 2 n log 2 ⁡ 2 + O ( 1 n ) {\displaystyle \log f!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)-{\frac {\log ^{2}n}{2n\log ^{2}2}}+O!\left({\frac {1}{n}}\right)}

for ⁠ n → ∞ {\displaystyle n\to \infty }⁠. References

"A288163 – Oeis". Juan Arias de Reyna (2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT]. Sloane, N. J. A. (ed.). "Sequence A272755 (Numerators of the Fabius function F(1/2^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Sloane, N. J. A. (ed.). "Sequence A272757 (Denominators of the Fabius function F(1/2^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Fabius, J. (1966), "A probabilistic example of a nowhere analytic C∞-function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174, doi:10.1007/bf00536652, MR 0197656, S2CID 122126180
Jessen, Børge; Wintner, Aurel (1935), "Distribution functions and the Riemann zeta function", Trans. Amer. Math. Soc., 38: 48–88, doi:10.1090/S0002-9947-1935-1501802-5, MR 1501802
Dimitrov, Youri (2006). Polynomially-divided solutions of bipartite self-differential functional equations (Thesis).
Arias de Reyna, Juan (2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT].
Arias de Reyna, Juan (2017). "An infinitely differentiable function with compact support: Definition and properties". arXiv:1702.05442 [math.CA]. (an English translation of the author's paper published in Spanish in 1982)
Alkauskas, Giedrius (2001), Dirichlet series associated with Thue–Morse sequence, preprint.
Rvachev, V. L.; Rvachev, V. A. (1979), Non-classical methods of the approximation theory in boundary value problems (in Russian), Kiev: Naukova Dumka

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White hole

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From Wikipedia, the free encyclopedia For other uses, see White hole (disambiguation). Artist's impression of a white hole. General relativity Spacetime curvature schematic G μ ν + Λ g μ ν = κ T μ ν {\displaystyle G{\mu \nu }+\Lambda g{\mu \nu }={\kappa }T_{\mu \nu }}

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In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy, matter, light and information can escape from it. In this sense, it is the reverse of a black hole, from which energy, matter, light and information cannot escape. White holes appear in the theory of eternal black holes. In addition to a black hole region in the future, such a solution of the Einstein field equations has a white hole region in its past.[1] This region does not exist for black holes that have formed through gravitational collapse, however, nor are there any observed physical processes through which a white hole could be formed.

Supermassive black holes (SMBHs) are theoretically predicted to be at the center of every galaxy and may be essential for their formation. Stephen Hawking[2] and others have proposed that these supermassive black holes could spawn supermassive white holes.[3] Overview

Like black holes, white holes have properties such as mass, charge, and angular momentum. They attract matter like any other mass, but objects falling towards a white hole would never actually reach the white hole's event horizon (though in the case of the maximally extended Schwarzschild solution, discussed below, the white hole event horizon in the past becomes a black hole event horizon in the future, so any object falling towards it will eventually reach the black hole horizon). Imagine a gravitational field, without a surface. Acceleration due to gravity is the greatest on the surface of any body. But since black holes lack a surface, acceleration due to gravity increases exponentially, but never reaches a final value as there is no considered surface in a singularity.

In quantum mechanics, the black hole emits Hawking radiation and so it can come to thermal equilibrium with a gas of radiation (not compulsory). Because a thermal-equilibrium state is time-reversal-invariant, Stephen Hawking argued that the time reversal of a black hole in thermal equilibrium results in a white hole in thermal equilibrium (each absorbing and emitting energy to equivalent degrees).[4][further explanation needed] Consequently, this may imply that black holes and white holes are reciprocal in structure, wherein the Hawking radiation from an ordinary black hole is identified with a white hole's emission of energy and matter. Hawking's semi-classical argument is reproduced in a quantum mechanical AdS/CFT treatment,[5] where a black hole in anti-de Sitter space is described by a thermal gas in a gauge theory, whose time reversal is the same as itself. History A diagram of the structure of the maximally extended black hole spacetime. The horizontal direction is space and the vertical direction is time.

In the 1930s, physicists Robert Oppenheimer and Hartland Snyder introduced the idea of white holes as a solution to Einstein's equations of general relativity. These equations, the foundation of modern physics, describe the curvature of spacetime due to massive objects. Whereas black holes are born from the collapse of stars, white holes represent the theoretical birth of space, time, and potentially even universes. At the center, space and time do not end into a singularity, but continue across a short transition region where the Einstein equations are violated by quantum effects. From this region, space and time emerge with the structure of a white hole interior, a possibility already suggested by John Lighton Synge.[6]

The possibility of the existence of white holes was put forward by cosmologist Igor Novikov in 1964,[7] developed by Nikolai Kardashev.[8] White holes are predicted as part of a solution to the Einstein field equations known as the maximally extended version of the Schwarzschild metric[clarification needed] describing an eternal black hole with no charge and no rotation. Here, "maximally extended" implies that spacetime should not have any "edges". For any possible trajectory of a free-falling particle (following a geodesic) in spacetime, it should be possible to continue this path arbitrarily far into the particle's future, unless the trajectory hits a gravitational singularity like the one at the center of the black hole's interior. In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region, which allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. For an observer outside using Schwarzschild coordinates, infalling particles take an infinite time to reach the black hole horizon infinitely far in the future, while outgoing particles that pass the observer have been traveling outward for an infinite time since crossing the white hole horizon infinitely far in the past (however, the particles or other objects experience only a finite proper time between crossing the horizon and passing the outside observer). The black hole/white hole appears "eternal" from the perspective of an outside observer, in the sense that particles traveling outward from the white hole interior region can pass the observer at any time, and particles traveling inward, which will eventually reach the black hole interior region can also pass the observer at any time.

Just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black-hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see light that fell in from the other one), and likewise particles from the interior white-hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal–Szekeres coordinates (see figure).[9]

In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') and draw an "embedding diagram" depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein-Rosen bridge" or Schwarzschild wormhole.[9] Depending on where the space-like hypersurface is chosen, the Einstein-Rosen bridge can either connect two black hole event horizons in each universe (with points in the interior of the bridge being part of the black hole region of the spacetime), or two white hole event horizons in each universe (with points in the interior of the bridge being part of the white hole region). It is impossible to use the bridge to cross from one universe to the other, however, because it is impossible to enter a white hole event horizon from the outside, and anyone entering a black hole horizon from either universe will inevitably hit the black hole singularity.

Note that the maximally extended Schwarzschild metric describes an idealized black hole/white hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from a collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's history, it removes the part of the diagram corresponding to the white hole interior region.[10] But because the equations of general relativity are time-reversible – they exhibit Time reversal symmetry – general relativity must also allow the time-reverse of this type of "realistic" black hole that forms from collapsing matter. The time-reversed case would be a white hole that has existed since the beginning of the universe, and that emits matter until it finally "explodes" and disappears.[11] Despite the fact that such objects are permitted theoretically, they are not taken as seriously as black holes by physicists, since there would be no processes that would naturally lead to their formation; they could exist only if they were built into the initial conditions of the Big Bang.[11] Additionally, it is predicted that such a white hole would be highly "unstable" in the sense that if any small amount of matter fell towards the horizon from the outside, this would prevent the white hole's explosion as seen by distant observers, with the matter emitted from the singularity never able to escape the white hole's gravitational radius.[12] Properties

Depending on the type of black hole solution considered, there are several types of white holes. In the case of the Schwarzschild black hole mentioned above, a geodesic coming out of a white hole comes from the "gravitational singularity" it contains. In the case of a black hole possessing an electric charge (Reissner-Nordström black hole) or an angular momentum (Kerr black hole), then the white hole happens to be the "exit door" of a black hole existing in another universe.