Fourth Dimensional revelations by some Euclidean deity!
"The Metal Monster" by A. Merritt
In their non-Euclidean geometry the part is always greater than the whole.
"The Open Secret of Ireland" by T. M. Kettle
Accordingly the Euclidean property of space arises from the parabolic property of time.
"The Concept of Nature" by Alfred North Whitehead
She could find no hole in Elizabeth's arguments; it was founded as solidly as a Euclidean proposition.
"Miss Mapp" by Edward Frederic Benson
Non-Euclidean geometry, II, 83.
"A Budget of Paradoxes, Volume II (of II)" by Augustus de Morgan
The argument in a Euclidean demonstration would not be made clearer by being cast into formal Syllogisms.
"Logic, Inductive and Deductive" by William Minto
It was developed with rigorous mathematical logic and Euclidean conclusiveness.
"Education: How Old The New" by James J. Walsh
The geometry on such a surface is shown to be Euclidean, limit-lines replacing Euclidean straight lines.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
In elementary geometry, however, the Euclidean idea is still held.
"The Teaching of Geometry" by David Eugene Smith
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In Nat Friedman's "Pas de Deux," the perimeter straight-line Euclidean geometry plays the supporting role for the interesting inner fractal geometry.
Euclidean geometry, Fibonacci numbers, the digits of pi, the notion of algorithms, concepts of infinity, fractals , and other ideas furnished the mathematical underpinnings.
We consider the case where data is sampled from a low dimensional manifold which is embedded in high dimensional Euclidean space.
They determine the neighborhood graph using Euclidean distance so that they often fail to nicely deal with sparsely sampled or noise contaminated data.
CEREBELLUM'S FRONT AND BACK can be combined into single flat maps (shown here in Euclidean and hyperbolic views) to reveal details that are normally hidden in the brain's folds.
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The total number of such eigenvalues increases with the square root of the Euclidean four-volume.
Random Matrix Theory and Chiral Symmetry in QCD
The QCD partition function can be written as a Euclidean path integral that can be expressed as the expectation value of the fermion determinant, det(D + mf )E .
Random Matrix Theory and Chiral Symmetry in QCD
Here the average is over all gauge ﬁelds weighted by the Euclidean Yang-Mills action, the product is over quark ﬂavors of mass mf , and D is the Dirac operator, which we introduce in great detail in Sec. 2.
Random Matrix Theory and Chiral Symmetry in QCD
For the Euclidean QCD partition function, it is more natural to construct a random matrix model for the Dirac operator.
Random Matrix Theory and Chiral Symmetry in QCD
We now consider QCD in a ﬁnite Euclidean volume V4 = L4 .
Random Matrix Theory and Chiral Symmetry in QCD
Nf is the number of quark ﬂavors and SYM is the Euclidean Yang-Mills action.
Random Matrix Theory and Chiral Symmetry in QCD
The γµ are Euclidean gamma matrices with {γµ , γν } = 2δµν .
Random Matrix Theory and Chiral Symmetry in QCD
Similar results have been derived for a large-d strong coupling expansion of lattice QCD at ﬁnite density [197] (with d the Euclidean dimensionality).
Random Matrix Theory and Chiral Symmetry in QCD
More recently, Larralde et al. [3,4] and Havlin et al. studied the problem of evaluating SN (t) when N ≫ 1 noninteracting random walkers diffuse in Euclidean and fractal media, respectively.
I. Territory covered by N random walkers on deterministic fractals. The Sierpinski gasket
The mathematical techniques involved in the calculation are very similar to those corresponding to the Euclidean case and we will only outline the main steps.
I. Territory covered by N random walkers on deterministic fractals. The Sierpinski gasket
In these Euclidean media we found that the second-order asymptotic approximation gives rise to a signiﬁcant improvement in the estimate of SN (t) even for relatively small values of N .
I. Territory covered by N random walkers on deterministic fractals. The Sierpinski gasket
Xm=0 *n(m)Xi=1 n1 − [Γt (rm,i )]N o+ , where rm,i stands for the i-th site out of n(m) that are separated from the origin by a Euclidean distance between m∆r ≡ rm and (m + 1)∆r with ∆r small (say, of the order of the lattice spacing).
II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate
The chemical distance ℓ, the length of the shortest path between two sites along lattice bonds, is a more natural measure than the Euclidean distance in disordered systems.
II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate
It is known that both the propagator and the mortality function share the same asymptotic behavior for Euclidean lattices and for the Sierpinski lattice and we can expect that this coincidence also is the case for stochastic fractals (we will check this supposition in Sec.
II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate
On the other hand, a ﬁxed Euclidean distance r could correspond to many different chemical distances from one cluster to another.
II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate
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