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Euler's Formula for Pin-Ended Columns01:21

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In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
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The perpendicular-axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular concurrent axes lying in the plane of the body.
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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
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Euler's formula is very important in the field of structural engineering, providing a foundation for understanding the critical loading conditions of pin-ended columns. This formula links the modulus of elasticity, the moment of inertia of the cross-section, and the column's length, offering a precise calculation of the critical load at which a column is prone to buckling.
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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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Updated: May 15, 2025

Observation and Analysis of Blinking Surface-enhanced Raman Scattering
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Backbone Exponent and Annulus Crossing Probability for Planar Percolation.

Pierre Nolin1, Wei Qian1, Xin Sun2

  • 1City University of Hong Kong, China.

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Summary
This summary is machine-generated.

We derived the backbone exponent for 2D percolation, revealing it as a transcendental number. This finding advances understanding of critical phenomena and connects to conformal field theory via Liouville quantum gravity.

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Area of Science:

  • Statistical Physics
  • Probability Theory
  • Conformal Field Theory

Background:

  • Percolation theory studies systems near a phase transition.
  • Exactly solved percolation exponents are typically rational numbers.
  • The asymptotic behavior of percolation systems is crucial for understanding critical phenomena.

Purpose of the Study:

  • To derive the backbone exponent for 2D percolation.
  • To investigate the mathematical nature of this exponent.
  • To explore connections between percolation theory and conformal field theory.

Main Methods:

  • Utilizing Schramm-Loewner evolution (SLE) curves.
  • Coupling SLE with Liouville quantum gravity (LQG).
  • Leveraging the integrability of Liouville conformal field theory (CFT).

Main Results:

  • The backbone exponent for 2D percolation is a transcendental number.
  • This exponent is a root of an elementary equation.
  • An exact formula for the probability of two disjoint paths crossing an annulus was derived.

Conclusions:

  • The backbone exponent governs the leading asymptotic behavior.
  • Other roots of the equation capture remaining asymptotic terms.
  • This suggests the backbone exponent is part of a CFT with a bulk spectrum related to these roots.