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Parametrization of ϵ-rational curves: error analysis
[article]

2010
*
arXiv
*
pre-print

The authors deeply thank

arXiv:1004.2148v1
fatcat:yojqlv6mwnecdjxrou42uixknq
*Sonia*Pérez-Díaz and J. Rafael Sendra for many useful discussions on the topics treated in this paper. ... Let*L*1 and*L*2 be the real asymptotes of C parallel to*L*1 and*L*2 respectively. We the value of η = max{H(*L*1 ,*L*1 ), H(*L*2 ,*L*2 ))} for all the non compact curves of F in the next table. ... The same process is repeated for y = j, to obtain the negative and positive integers τ 3 , τ 4 , respectively such that min{|ρ R 2 (a, j) − H(*L*1 ,*L*1 )|, |ρ R 2 (a, j) − H(*L*2 ,*L*2 )| / (a, j) ∈ Ω j ...##
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Finite dimensional representations of invariant differential operators
[article]

2003
*
arXiv
*
pre-print

χ and

arXiv:math/0305279v1
fatcat:rhy7nu56qfgmjctrsajyak5jxq
*L*µ =*L*ν. ... Then O(Y ) G = span{Q α ∈ O(Y )|*L*α = 0} ∼ = kΣ*L*the semigroup algebra on Σ*L*. If Γ, ∆ are as in the lemma and*L*′ = Γ*L*∆, there is an isomorphism Σ*L*−→ Σ*L*′ (17) given by x → ∆ −1 x. ...##
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Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves
[article]

2009
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arXiv
*
pre-print

We say that an algebraic plane curve has proper degree

arXiv:0901.0320v1
fatcat:xtissooypzfo5furlcd2yn6puy
*ℓ*if its defining polynomial has proper degree*ℓ*. The notion of ǫ-point is as follows. Definition 1.2. ... Let*L*be the algebraic closure of C(t), and C 1 , C 2 two plane projective curves over*L*with defining polynomials G 1 (x, y, z), G 2 (x, y, z) ∈ C[t][x, y, z], respectively. ...##
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On the computation of graded components of Laurent polynomial rings
[article]

2006
*
arXiv
*
pre-print

Consider the n ×

arXiv:math/0605567v1
fatcat:fjtc6xmulbf27kzmdvx6gk4kwq
*l*matrix E = 0 0 0 D I*l*′ 0 (8) with D as in section 2, the integer*l*′ =*l*− t and I*l*′ the*l*′ ×*l*′ identity matrix. ... If p > s then permute the first r columns of*L*to obtain*L*c so that rank(subm(*L*c , 1 . . . n,*l*+ 1 . . . r)) = p − s. 3.*L*:=*L*c . ...##
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A perturbed differential resultant based implicitization algorithm for linear DPPEs
[article]

2010
*
arXiv
*
pre-print

Then M φ (

arXiv:1003.4375v1
fatcat:a3cmwxduo5dgtcj27eg4h6q45m
*L*h ) is an*L*h ×*L*h matrix with elements in K[p] and there exists an*L*h ×*L*h matrix M φ with elements in K such that M φ (*L*h ) = M(*L*h ) − p M φ . ... We have A D =*L*1 (x 1 ) +*L*2 (x 2 ) +*L*3 (x 3 ) +*L*4 (x 4 ) with*L*1 = −864∂ 3 + 972∂ 2 − 972∂ 4 = −108∂ 2 (8∂ − 9 + 9∂ 2 ),*L*2 = −216∂ 2 ,*L*3 = −972∂ 3 + 648∂ 2 = −324∂ 2 (3∂ − 2),*L*4 = 540∂ 2 ... For this purpose let S and M*L*−1 be the leading and principal matrices of P(X, U) respectively. Let P φ (X, U) be a perturbed system of P(X, U) of degree D φ ≥ 0. ...##
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Factorization of KdV Schrödinger operators using differential subresultants
[article]

2019
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arXiv
*
pre-print

*Rueda*has been partially supported by the "Ministerio de Economía y Competitividad"under the project MTM2014-54141-P. M.A. Zurro is partially supported by Grupo UCM 910444. ... Computing the commutator with

*L*s we obtain 0 = [A 2s+1 − µ 0 ,

*L*s ] = [QL s ,

*L*s ] − [λ 0 ,

*L*s ] = [Q,

*L*s ]

*L*s . ... Since P 2n+3 = O + P 2n−1

*L*where O = v n+1 ∂ − 1 2 ∂(v n+1 ), we have [

*L*, P 2n+3 ] = [

*L*, O] + [

*L*, P 2n+1 ]

*L*= [

*L*, O] − 2∂(v n+1 )

*L*, with [

*L*, O] = −2∂(v n+1 )∂ 2 + (1/2)∂ 3 (v n+1 ) − v n+1 u ′ , 2∂(v ...

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ACTIONS OF TORI AND FINITE FANS

2006
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Algebras, Rings and Their Representations
*

Let I

doi:10.1142/9789812774552_0019
fatcat:rhqumed7d5bntlo4ap56csxfrq
*l*be the identity*l*×*l*matrix and E the n ×*l*matrix with I*l*in the last*l*rows and zeroes in the first m rows. Then*L*E = 0. We define E := P E . ... With respect to B the vectors v*l*+1 , . . . , v n have coordinates v j = (− 1 d b j−*l*,1 , . . . , − 1 d b j−*l*,*l*), j =*l*+ 1, . . . , n. (19) Let m = r −*l*. ...##
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Linear sparse differential resultant formulas
[article]

2012
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arXiv
*
pre-print

Given

arXiv:1112.3921v3
fatcat:qfqhqzexprbtda6v6adkgrz5a4
*l*∈ T , if*l*∈ T m , by (36), η*l*= η m • ρ(m,*l*) and {η*l*(t) | t ∈ T*l*} ⊆ {η m (t) | t ∈ T }, else there exist*l*1 , . . . ,*l*p ∈ {1, . . . , m − 1} such that*l*∈ T lp ,*l*k ∈ T*l*k−1 , k = 2 ... , . . . , p and*l*1 ∈ T m , by (36) {η*l*(t) | t ∈ T*l*} ⊆ {η lp (t) | t ∈ T \{*l*p }} = {η*l*p−1 (t) | t ∈ T \{*l*p−1 }} = · · · = {η*l*1 (t) | t ∈ T \{*l*1 }} ⊆ {η m (t) | t ∈ T }. ...##
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Linear sparse differential resultant formulas

2013
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Linear Algebra and its Applications
*

Given

doi:10.1016/j.laa.2013.01.016
fatcat:t4jgs3a5l5gnvchdp6kekmy2tu
*l*∈ T , if*l*∈ T m , by (36), η*l*= η m • ρ(m,*l*) and {η*l*(t) | t ∈ T*l*} ⊆ {η m (t) | t ∈ T }, else there exist*l*1 , . . . ,*l*p ∈ {1, . . . , m − 1} such that*l*∈ T lp ,*l*k ∈ T*l*k−1 , k = 2 ... by S(*L*) = {k ∈ N 0 | α k ̸ = 0}, and define ldeg(*L*) := min S(*L*), deg(*L*) := max S(*L*). ...##
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Approximate parametrization of plane algebraic curves by linear systems of curves

2010
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Computer Aided Geometric Design
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Moreover, we introduce the polynomials X>i(t,s) = /(£i(t,s))€

doi:10.1016/j.cagd.2009.12.002
fatcat:q6iwmfnanvfitpldhcpafd7ody
*l*(t)[s], P 2 (fl,*l*),s) = /(£ 2 ((i,*l*),s))eC(C)[s], where R(t) denotes the algebraic closure of R(t) and C(C) the field of rational functions ... z] such that*L*=UiGi(x,y,0) + U 2 G 2 (x,y,0)+zU 3 . ...##
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Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves
[article]

2013
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arXiv
*
pre-print

Λ), where

arXiv:1308.4466v1
fatcat:6yi74zk66ja2zb5c7ejnjdfgv4
*L*is a finite algebraic extension of K of degree k. ... [General Parametrization Theorem] There exists a rational proper parametrization of Curve(H(n, D)) with coefficients in*L*(Λ), where*L*is a finite algebraic extension of K of degree at most n. ...##
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Commuting Ordinary Differential Operators and the Dixmier Test
[article]

2019
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arXiv
*
pre-print

Dixmier, to give necessary conditions for an operator M to be in the centralizer of

arXiv:1902.01361v1
fatcat:de3m2tvd2nbk7miarnuaibml2e
*L*. Whenever the centralizer equals the algebra generated by*L*and M, we call*L*, M a Burchall-Chaundy (BC) pair. ... Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator*L*in normal form, following the approach of K.R. ... C D (*L*) = {p 0 (*L*) + p 1 (*L*)B | p 0 , p 1 ∈ C[*L*]} = C[*L*, B]. ...##
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An algorithm to parametrize approximately space curves

2013
*
Journal of symbolic computation
*

We introduce the following polynomial A(t) = { c 1 (t) ∏

doi:10.1016/j.jsc.2013.04.002
fatcat:5ahvnkfgdje5dhbhg3sfnj5zgi
*ℓ*i=2 u i,1 (t)q i (t) + · · · + c*ℓ*(t) ∏*ℓ*−1 i=1 u i,*ℓ*(t)q i (t) if*ℓ*> 1 c 1 (t) if*ℓ*= 1. ... Let*ℓ*= deg xn (G 1 ). Since*ℓ*= tdeg(G 1 ), Lemma 2.2. ... If*ℓ*= 1 the result is trivial. Let*ℓ*> 1 and let R(t), Q(t) be the remainder and quotient of the division of A(t) by q(t), respectively. Clearly deg(R) < deg(q). ...##
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Thermal Noise in Electro-Optic Devices at Cryogenic Temperatures
[article]

2020
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arXiv
*
pre-print

The quantum bits (qubits) on which superconducting quantum computers are based have energy scales corresponding to photons with GHz frequencies. The energy of photons in the gigahertz domain is too low to allow transmission through the noisy room-temperature environment, where the signal would be lost in thermal noise. Optical photons, on the other hand, have much higher energies, and signals can be detected using highly efficient single-photon detectors. Transduction from microwave to optical

arXiv:2008.08764v1
fatcat:7a5rz2vmjzehjpwyvvzdo3t7oa
## more »

... requencies is therefore a potential enabling technology for quantum devices. However, in such a device the optical pump can be a source of thermal noise and thus degrade the fidelity; the similarity of input microwave state to the output optical state. In order to investigate the magnitude of this effect we model the sub-Kelvin thermal behavior of an electro-optic transducer based on a lithium niobate whispering gallery mode resonator. We find that there is an optimum power level for a continuous pump, whilst pulsed operation of the pump increases the fidelity of the conversion.##
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Differential elimination by differential specialization of Sylvester style matrices

2015
*
ACM Communications in Computer Algebra
*

To start, through derivations, a system ps(P) of

doi:10.1145/2768577.2768635
fatcat:skxdncfggzasdewztx74mhzlqm
*L*polynomials in*L*− 1 algebraic variables is obtained, which is non sparse in the order of derivation. ... Furthermore*L*∑*l*=1 τ*l*∑ h=1 r*l*,h y σ*l*,h P*l*= y α*L*∑*l*=1 τ*l*∑ h=1 r*l*,h C*l*α−σ*l*,h = y α D. Since D ∈ Q[C], we have D = y −α*L*∑*l*=1 τ*l*∑ h=1 r*l*,h y σ*l*,h P*l*∈ (ags(P)) ∩ Q[C]. ... This proves g 1,1*L*∑*l*=1 τ*l*∑ h=1 r*l*,h (y σ*l*,h P*l*− C*l*α−σ*l*,h y α ) = γ*L*∑*l*=1 τ*l*∑ h=1 g*l*,h (y σ*l*,h P*l*− C*l*α−σ*l*,h y α ) = 0, and g 1,1 ̸ = 0 in the domain Q[C][Y ± ] implies*L*∑*l*=1 τ*l*...
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