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Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c,...
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Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...
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A function's graph can be modified by changing its position or size without altering its overall shape. These transformations allow the graph to be moved across the coordinate plane while preserving its pattern and structure. One of the most common transformations is shifting, which repositions the graph without distorting it.When the output of a function is adjusted by adding or subtracting a constant, the graph shifts vertically. A positive value moves the graph upward, while a negative value...
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A function is continuous at a point a if three conditions are met: the function is defined at a, the limit of the function as x approaches a exists, and this limit equals the function’s value. Mathematically, this is written asThis definition ensures the graph of the function does not exhibit any breaks, holes, or jumps at that point. Discontinuities occur when any of these conditions fail. A removable discontinuity exists when the two-sided limit exists but the function is either...
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Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
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A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
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Fluctuation Relations Associated to an Arbitrary Bijection in Path Space.

Raphaël Chétrite1, Stefano Marcantoni2,3

  • 1Institut de Physique de Nice (INPHYNI), Université Côte d'Azur, CNRS, 17 rue Julien Lauprêtre, 06200 Nice, France.

Mathematical Physics, Analysis, and Geometry
|October 24, 2025
PubMed
Summary
This summary is machine-generated.

We present a framework to discover Fluctuation Relations for vector-valued observables in stochastic systems. This method identifies new relations by analyzing invertible trajectory transformations, generalizing existing theories.

Keywords:
Fluctuation RelationsLarge DeviationNon-Degenerate DiffusionsSemi-MarkovStochastic Processes

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Dynamical Systems

Background:

  • Stochastic dynamics govern many physical systems.
  • Fluctuation relations offer insights into non-equilibrium thermodynamics.
  • Existing relations often focus on specific symmetries.

Purpose of the Study:

  • To introduce a general framework for identifying Fluctuation Relations.
  • To extend Fluctuation Relations to vector-valued observables.
  • To uncover new types of Fluctuation Relations.

Main Methods:

  • Developing a framework based on entropic functionals.
  • Analyzing invertible, non-involutional transformations in trajectory space.
  • Applying the framework to canonical path probability and diffusion processes.

Main Results:

  • Identification of a general class of Fluctuation Relations.
  • Recovery of known isometric and spatial fluctuation relations as special cases.
  • Development of a method for discovering novel Fluctuation Relations.

Conclusions:

  • The proposed framework provides a unified approach to Fluctuation Relations.
  • New Fluctuation Relations can be systematically derived.
  • The findings are applicable to various stochastic processes over finite and asymptotic times.