Vanishing fraction

Definitions

  • Webster's Revised Unabridged Dictionary
    • Vanishing fraction (Math) a fraction which reduces to the form 0/0 for a particular value of the variable which enters it, usually in consequence of the existence of a common factor in both terms of the fraction, which factor becomes 0 for this particular value of the variable.
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Usage

In literature:

He caught sight of her, hesitated the fraction of a moment, and as he came forward again the cloud in his eyes vanished.
"A Modern Chronicle, Complete" by Winston Churchill
I lowered my arm an almost imperceptible fraction of a second before he sprang up and vanished.
"Romance" by Joseph Conrad and F.M. Hueffer
They existed for the barest fraction of a second, then vanished.
"The Status Civilization" by Robert Sheckley
In a fraction of a second every beaver had vanished beneath the shining surface.
"The Haunters of the Silences" by Charles G. D. Roberts
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In science:

The key insight in terms of estimation, is that in cases where networks are sparse, then the fraction of incidentally generated subgraphs vanishes.
Tractable and Consistent Random Graph Models
The proof of the theorem involves showing that under the nicely-ordered and sparsity conditions the fraction of incidentally generated subnetworks vanishes for each ‘, and so the observed counts of subnetworks converge to the true ones.
Tractable and Consistent Random Graph Models
This is particularly interesting in dimensions 3 and higher, where the random walk is transient, and visits only a vanishing fraction of the space.
Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth
Sinai’s regime, in which a vanishing fraction of the sites are replaced by ‘fast points’ with drift δ . A different, but related, perturbation of Sinai’s regime was introduced in [18, 19], in which the perturbation occurs at every site, but is asymptotically small in magnitude.
Random walk in mixed random environment without uniform ellipticity
This means that the Fractional Fourier Transform, i.e., the Path Integral, has real and imaginary parts, and its imaginary part only vanishes for special values of ω.
Complex Path Integrals and the Space of Theories
This is reminiscent of percolation: at high densities of edges the giant component is larger in disassortative networks, but in assortative ones a non-vanishing fraction of nodes remain interconnected even at densities below the usual percolation threshold (Newman, 2002, 2003a).
Interplay between Network Topology and Dynamics in Neural Systems
However, under the random-access algorithm, the network might spend a non-vanishing fraction of time in the schedules M\M∗ , which can cause instability as ρ approaches 1.
Queue-Based Random-Access Algorithms: Fluid Limits and Stability Issues
The fact that fV is vanishing almost as fast as ξ−2 suggests that these small loops represent only a small fraction of the full vortex density over the field range studied here.
Topological Defects in the Random-Field XY Model and the Pinned Vortex Lattice to Vortex Glass Transition in Type-II Superconductors
The calculated shape of the support of ρ(ω ) matches the data well; the fact that a finite fraction of the eigenvalues have vanishing imaginary part is a finite-size effect, analyzed in detail in Ref .
Diffusion in a Random Velocity Field: Spectral Properties of a Non-Hermitian Fokker-Planck Operator
Extrapolating to large fractions x, the magnon energy scale ˜J = J (1 − 2x) hence vanishes only close to the percolation threshold xperc ≈ 0.4.
Disorder and Impurities in Hubbard-Antiferromagnets
This corresponds to shifting the support of the density of states by unity along the negative real axis, so that all (except a vanishing fraction) of the eigenvalues have negative real parts.
Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
The average is dominated by such realizations (the so called rare events), which have a very large contribution, but their fraction is vanishing in the thermodynamic limit.
Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions
However, since the channel loss on the way from the source to the Faraday mirror inevitably increases the empty pulses fraction, the main advantage of photon pairs vanishes in such a configuration.
Quantum Cryptography
For instance, in a random landscape where the fitness of each molecule is a random number chosen from a uniform distribution in the interval [Wmin , Wmax ], a non-vanishing fraction of sequences has a fitness within a relative distance of the order 1 − e−µN of Wmax .
Biological Evolution and Statistical Physics
The diffusion equation could be mapped on the mean-field theory of Bose condensation, leading through an analytical calculation to a stationary state where a non-vanishing fraction of the population has an identical genome .
Biological Evolution and Statistical Physics
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