These differ from other forms of force in being incommensurate with extension.
"The Religious Sentiment" by Daniel G. Brinton
Life introduces something indeterminate and incommensurable.
"Under the Maples" by John Burroughs
There is somewhat incommensurable in his works.
"Whitman" by John Burroughs
A non-terminating simple continued fraction must be incommensurable.
"Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2" by Various
Premeditations, Powers, fore-ordained Destinies, elaborate there together an incommensurable work.
"Toilers of the Sea" by Victor Hugo
Official data on manpower and employment in agriculture are incomplete and incommensurate.
"Area Handbook for Bulgaria" by Eugene K. Keefe, Violeta D. Baluyut, William Giloane, Anne K. Long, James M. Moore, and Neda A. Walpole
The incommensurable of Pythagoras and the paradoxes of Zeno present the "no thoroughfares" of ancient mathematical thought.
"Creative Intelligence" by John Dewey, Addison W. Moore, Harold Chapman Brown, George H. Mead, Boyd H. Bode, Henry Waldgrave, Stuart James, Hayden Tufts, Horace M. Kallen
But slavery and education of the people are incommensurable quantities.
"Speeches, Addresses, and Occasional Sermons, Volume 2 (of 3)" by Theodore Parker
The two are incommensurate and incommensurable.
"The Value of Money" by Benjamin M. Anderson, Jr.
Now he was enormously exercised in the search for a reconciliation between these, he felt, incommensurable factors.
"Marriage" by H. G. Wells
It is an incommensurable number which to five decimal places is equal to 2.71828.
"The New Gresham Encyclopedia" by Various
The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included.
"The Teaching of Geometry" by David Eugene Smith
But for the literary life of both poets the gain was incommensurable.
"Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 2" by Various
But milk is in units of liquid measure; gold in incommensurable units of Troy weight.
"Social Value" by B. M. Anderson
His sense of rhythm was incommensurate with hers.
"What Will People Say?" by Rupert Hughes
Spiritual and physical magnitudes are incommensurable.
"The Expositor's Bible: Ephesians" by G. G. Findlay
Man is incommensurable with the rest of the universe.
"The Expositor's Bible: The Book of Genesis" by Marcus Dods
Had he understood this he could have seen that his limitless ambition was incommensurate with his comparatively small capabilities.
"Modern Painting, Its Tendency and Meaning" by Willard Huntington Wright
They are in fact incommensurable magnitudes.
"The Place of Science in Modern Civilisation and Other Essays" by Thorstein Veblen
They were taken north and set to tasks that were again incommensurate with their force.
"Generals of the British Army" by Francis Dodd
***
We notice a minimum in the magnon dispersion relation occurring at an incommensurate wave-vector q∗ ≃ 3π/4, using the notation of Eq. (4).
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
The up arrows indicate an approximate location of the incommensurate wave-vector q∗ corresponding to the minimum of the magnon relation dispersion.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
Entering the RS phase by increasing further J⊥, the groundstate is now non-degenerate and the spectrum fully gapped with an incommensurate magnon branch as the lowest excitation.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
For J⊥ = J1 (Fig. 5(c)), this branch enters the continuum of many-magnons excitations for small and large kx, but for large enough transverse coupling (J⊥ = 3J1 in Fig. 5(d)), the incommensurate magnon branch is well separated from the continuum (as usually found for Heisenberg ladders).
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
This section is dedicated to a more detailed study of the magnon excitation branch and its main features : effect of the frustrating term, spin gap, correlations lengths and the onset of incommensurability.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
The onset of incommensurability across the phase diagram is obtained by choosing the de ﬁnition from real-space spin correlations, i.e. when q deviates from π .
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
In comparison to the standard ladder dispersion relation, frustration brings the harmonic 2k leading to incommensurability.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
The threshold and incommensurate wave-vector (11) are the ones that actually appear as the angle of the helical structure of the classical version of both the J1 -J2 chain and ladder models, hence in the spin-wave calculation.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
One may observe from Eq. (14) that the local maximum of the spin gap is actually realized for a value of J2 larger than the onset of the q∗ incommensurability wave-vector.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
III C that q∗ gets very different from q in the large-J⊥ limit so that the local maximum is actually not related to the true onset of incommensurability associated with q .
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
The onset of the incommensurability in spin correlations (q ), spin-structure factor ( ¯q ) and dispersion relation (q∗ ) are given when available. (a) the isotropic ladder situation J⊥ = J1, where we have ¯q ∼ q .
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
One may wonder whether the spin correlation length ξspin obtained from Eq. (9) has also a singular behavior as it has been observed for instance in the spin-1 chain38,39 and whether its location is related to the onset of incommensurability.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
From Fig. 7, we do observe that ξ−1 spin displays a local singular maximum close to the onset of incommensurability associated with q (and not q∗ or ¯q ).
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
These observations support that the physical onset of incommensurability is the one associated with q since it is associated with a singular behavior in ξspin, as for the MG point of a single chain.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
How the local minimum observed close to the transition line interpolates with the local maximum arising in the vicinity of the onset of incommensurability in the dispersion relation is a challenging question.
Melting of a frustration-induced dimer crystal and incommensurability in the J_1-J_2 two-leg ladder
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