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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...
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Some solids can transition directly into the gaseous state, bypassing the liquid state, via a process known as sublimation. At room temperature and standard pressure, a piece of dry ice (solid CO2) sublimes, appearing to gradually disappear without ever forming any liquid. Snow and ice sublimate at temperatures below the melting point of water, a slow process that may be accelerated by winds and the reduced atmospheric pressures at high altitudes. When solid iodine is warmed, the solid sublimes...
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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Phase Transitions: Melting and Freezing02:39

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The physical form of a substance changes on changing its temperature. For example, raising the temperature of a liquid causes the liquid to vaporize (convert into vapor). The process is called vaporization—a surface phenomenon. Vaporization occurs when the thermal motion of the molecules overcome the intermolecular forces, and the molecules (at the surface) escape into the gaseous state. When a liquid vaporizes in a closed container, gas molecules cannot escape. As these gas phase molecules...
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Rigorous location of phase transitions in hard optimization problems.

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Researchers developed a rigorous method to locate phase transitions in computationally hard optimization problems. This confirms statistical physics predictions and shows solutions remain beyond current algorithms.

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Area of Science:

  • Theoretical Computer Science
  • Statistical Physics
  • Computational Complexity

Background:

  • Many optimization problems are computationally intractable, with exhaustive search being the most efficient known algorithm.
  • Understanding the extreme 'hardness' of these problems has been a focus of computer science, mathematics, and physics for two decades.
  • A key area of investigation is the link between computational complexity and phase transitions in random constraint satisfaction problems.

Purpose of the Study:

  • To develop a mathematically rigorous method for precisely locating phase transitions in constraint satisfaction problems.
  • To validate heuristic predictions from statistical physics regarding the behavior of these hard problems.
  • To establish the limits of algorithmic solvability for random instances of constraint satisfaction problems.

Main Methods:

  • Analysis of the distribution of distances between solution pairs as constraints are incrementally added.
  • Identification of critical behavior in the evolution of this solution distance distribution.
  • Application of the method to pinpoint phase transition thresholds for problems like random k-SAT and random graph coloring.

Main Results:

  • A rigorous method for locating phase transitions in constraint satisfaction problems was successfully developed and applied.
  • The critical threshold locations for key problems, including random k-SAT and random graph coloring, were precisely identified.
  • The results provide mathematical proof supporting the accuracy of statistical physics heuristic predictions in this domain.

Conclusions:

  • The developed method rigorously confirms the heuristic predictions of statistical physics concerning phase transitions in random constraint satisfaction problems.
  • Random instances of constraint satisfaction problems possess solutions that are demonstrably beyond the reach of any currently analyzed algorithms.
  • This research deepens the understanding of computational hardness and the fundamental limits of algorithmic problem-solving.