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6cadae1114e561f3577f562c3bfe4013f4f81673 | subsection | 36 | 59 | An energy inequality and a monotonicity formula | Then, for any C^1 vector field \zeta :\mathop {\rm \mathbb {R}}\nolimits ^n\times (0,T)\rightarrow \mathop {\rm \mathbb {R}}\nolimits ^m compactly supported in space,<\partial _tu,\nabla _{\zeta }u>_{L^2(\mathop {\rm \mathbb {R}}\nolimits ^n\times [t_1,t_2])}&=&<e_K(u),\mathop {\rm div}\nolimits \zeta >_{L^2(\mathop {\... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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5bd7645def62bc88b8ff9e8e3cf817518fc88493 | subsection | 37 | 59 | An energy inequality and a monotonicity formula | Then, on one hand:\int _{\mathop {\rm \mathbb {R}}\nolimits ^n\times [t_1,t_2]}<\partial _tu,\nabla _{\zeta }u>dxdt&=&\int _{\mathop {\rm \mathbb {R}}\nolimits ^n\times [t_1,t_2]}\left<\Delta u,\nabla _{\zeta }u\right>dxdt\\
&&-\int _{\mathop {\rm \mathbb {R}}\nolimits ^n\times [t_1,t_2]}\left<\frac{K}{t}\chi ^{\prime ... | {
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} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
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67e557390fc65679e7fbd026414a65fcdfc06d49 | subsection | 38 | 59 | An energy inequality and a monotonicity formula | Let (u(t))_{t>0} be a smooth solution to the Homogeneous Chen-Struwe flow coming out of u_0 such that (E_{K,x_0}(u(t))_{t>0} is continuous at t=0 for every x_0\in \mathop {\rm \mathbb {R}}\nolimits ^n. Then, for any z_0=(x_0,t_0)\in \mathop {\rm \mathbb {R}}\nolimits ^n\times \mathop {\rm \mathbb {R}}\nolimits _+ and 0... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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454255c411728e2051e81559b8f376aaa4a4aa16 | subsection | 39 | 59 | An energy inequality and a monotonicity formula | Then,\mathop {\rm div}\nolimits \zeta &=&\phi ^2_{x_0}\mathop {\rm div}\nolimits (G_{z_0}\cdot (x-x_0))+<\nabla \phi _{x_0}^2,G_{z_0}\cdot (x-x_0)>\\
&=&\phi ^2_{x_0}\left(-\frac{|x-x_0|^2}{2|t-t_0|}+n\right)G_{z_0}+<\nabla \phi _{x_0}^2,G_{z_0}\cdot (x-x_0)>,\\
\nabla _i(G_{z_0}(x-x_0))_j&=&\left(-\frac{(x-x_0)_i(x-x_... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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5d78277b0bdf9c73de0683ee8364bf7fae9505e9 | subsection | 40 | 59 | An energy inequality and a monotonicity formula | The monotonicity result for \Psi follows from the one for \Phi . | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
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434c296f4d5e76639713db3e7903bd9980ff68ec | subsection | 41 | 59 | An | Theorem 3.18
Let u_0:\mathop {\rm \mathbb {R}}\nolimits ^n\rightarrow (N,g)\subset \mathop {\rm \mathbb {R}}\nolimits ^m be a 0-homogeneous Lipschitz map.
Then there exists a radius R=R(\Vert \nabla u_0\Vert _{L^2_{loc}},n,m)>0 and a constant C=C(\Vert \nabla u_0\Vert _{L^2_{loc}},n,m)>0 such that if u is a smooth sol... | {
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... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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a660edf28e57f934433375d83a13184c946c7ede | subsection | 42 | 59 | An | Thanks to the monotonicity formula from Proposition REF , then one shows that for a given positive \varepsilon , there exists a positive \delta (\varepsilon ) such that\sigma ^{-n}\int _{P_{\sigma }(z_0)}e_K(u)dxdt\le c\Psi (u,z_1,R)+c( (R-\sigma )+\varepsilon )\Vert \nabla u_0\Vert ^2_{L^2(B(x_1,2))}.Now, by smoothnes... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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2fe2d8162d6e807830fd1615a23e1bb13dcd4e7d | subsection | 43 | 59 | Weighted | Proposition 3.19
There is a positive constant M uniform in \sigma \in [0,1] such that if V\in X is a fixed point of F_K^{\sigma } then \Vert fV\Vert _{C^0}\le M.Remark 3.20 Because of Proposition REF , it suffices to show this a priori bound outside a ball of radius independent of \sigma \in [0,1].Since V is fixed poi... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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043a195c17ba72b5f7349fbbeb3146263eca62d3 | subsection | 44 | 59 | Weighted | Outside a ball of radius sufficiently large (but independent of V and \sigma \in [0,1]), one has\Delta _f|V|^2&=&2|\nabla V|^2+2Kd_N(U_0^{\sigma }+V)\left<\nabla d_N(U_0^{\sigma }+V),V\right>,\\
|d_{U_0^{\sigma }+V}d_N(V)-d_{U_0^{\sigma }}d_N(V)|&\le & C(N,\Vert U_0^{\sigma }\Vert _{L^{\infty }})|V|^2,\\
|d_N(U_0^{\sig... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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f8a98270ca238b6e579510de286ffff1b0bf92f2 | subsection | 45 | 59 | Weighted | In particular, as V_{\varepsilon } is bounded independently of \varepsilon \in (0,1] and of \sigma \in [0,1] (and of K), one can choose a constant A sufficiently large such that on the boundary \partial B(0,R), \sup _{\partial B(0,R)}V_{\varepsilon }-Af^{-1}<0. By applying the maximum principle to the function V_{\vare... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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cca424f702aa8c46a76de1c826f573d0746dca08 | subsection | 46 | 59 | Weighted | Since U:=U_0^{\sigma }+V is an expander of the homogeneous Chen-Struwe flow and because of the previous remark, the gradient \nabla V satisfies the following equation outside a sufficiently large ball independent of \sigma\Delta _f\nabla V&=&-\frac{\nabla V}{2}+K\nabla \left(\nabla \left( \frac{d^2_{N}}{2}\right)(U)\ri... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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97fdb2ba27769e613fff5051eb1f4135e3c4ebf9 | subsection | 47 | 59 | Lipschitz initial conditions | In this section we give the proof of Theorem REF :[Proof of Theorem REF ]
Let (u_0^{\varepsilon })_{\varepsilon \in (0,1)} be a sequence of 0-homogeneous maps u_0^{\varepsilon }:\mathbb {R}^n\rightarrow N\subset \mathop {\rm \mathbb {R}}\nolimits ^m in C^3_{loc}(\mathop {\rm \mathbb {R}}\nolimits ^n\setminus \lbrace 0\... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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e76a06164e1d85ac4f792d9e57fe35d70686a0c9 | subsection | 48 | 59 | Lipschitz initial conditions | According to (REF ), there exist constants C and R uniform in \varepsilon such that the previous inequalities hold.Fix \varepsilon \in (0,1). As in , there exists a subsequence (still denoted by (u_{K}^{\varepsilon })_{K>0}) converging weakly to a map u^{\varepsilon }:\mathop {\rm \mathbb {R}}\nolimits ^n\times \mathop... | {
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... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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70faaa66b1483ba1086c234a528fad3b60917fe7 | subsection | 49 | 59 | Lipschitz initial conditions | To sum it up, we have obtained that&&|\mathop {\rm Lip}\nolimits (U^{\varepsilon })|(x)\le \frac{C}{|x|},\quad |x|\ge R,\\
&&\Vert \nabla u^{\varepsilon }(t)\Vert _{L^2(B(x_0,1))}\le C\left(n,m,\Vert \nabla u_0^{\varepsilon }\Vert _{L^2_{loc}(\mathop {\rm \mathbb {R}}\nolimits ^n)},t\right)\Vert \nabla u_0^{\varepsilon... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
3734c6435278e82db588909fb301ce1c5acf69b3 | subsection | 50 | 59 | Lipschitz initial conditions | Then, for K>0:\left|\int _{\mathop {\rm \mathbb {R}}\nolimits ^n}<u^{\varepsilon }_K(t)-u^{\varepsilon }_K(s),\psi _{x_0}>dx\right|&=&\left|\int _s^t\int _{\mathop {\rm \mathbb {R}}\nolimits ^n}<\partial _{\tau }u^{\varepsilon }_K,\psi _{x_0}>dxd\tau \right|\\
&\le &\int _s^t\int _{B(x_0,1)}|\partial _{\tau }u^{\vareps... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
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f2585924810615c22a1300ea2e1e74da3b569cfe | subsection | 51 | 59 | Lipschitz initial conditions | This ends the proof of the claim.The fact that U^{\varepsilon } is regular off a singular closed (hence compact by (REF )) set of finite (n-2) Hausdorff dimensional measure follows from [Sect. III, ] and . Finally, the fact that u^{\varepsilon } solves the harmonic map flow follows from [Sect. III, ] as well.The same s... | {
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... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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f10d6882488f54cbada87c92ab6f90357a8e16cf | subsection | 52 | 59 | Lipschitz initial conditions | The function F satifies the following differential inequality\Delta _fF&\ge & |\nabla ^2U|^4-c(1+|\nabla U|^2)F-8|\nabla ^2U|^2|\nabla U||\nabla ^3U|\\
&&+|\nabla ^3U|^2(|\nabla U|^2+a^2)-c|\nabla ^2U|^3(|\nabla U|^2+a^2)\\
&&-c|\nabla U|^4(|\nabla U|^2+a^2)(1+|\nabla U|^2)\\
&\ge &\frac{1}{2}|\nabla ^2U|^4-c(1+|\nabla... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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79881b706ee3d8ea8a46757df9258527d80e580a | subsection | 53 | 59 | Lipschitz initial conditions | By the previous differential inequality satisfied by F evaluated at x_1 together with the maximum principle0&=&\nabla G=F\nabla \phi +\phi \nabla F,\\
0&\ge & \phi \Delta _fG\ge \frac{G^2}{4a^4}-ca^2G-c(1+a^2)a^6-2G\frac{\vert \nabla \phi \vert ^2}{\phi }+G(\Delta \phi +\langle \nabla f,\nabla \phi \rangle ).Now,\nabla... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
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41f488b8adcc5689dcf872662f268285c1361a90 | subsection | 54 | 59 | Lipschitz initial conditions | By integrating along radial lines, U approaches u_0 at a linear rate:|U(x)-u_0(x/|x|)|=\textit {O}(|x|^{-1}),for every x far from the origin.Remark 3.23 It does not seem straightforward to improve the convergence rate in case u_0 is at least C^3_{loc}(\mathop {\rm \mathbb {R}}\nolimits ^n\setminus \lbrace 0\rbrace ). T... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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48961e404310754e12db5ef1165bca1dcb0480bd | subsection | 55 | 59 | Taylor expansions at infinity for expanders of the harmonic map flow | We gather necessary conditions at infinity on an expanding solution of the Homogeneous Ginzburg-Landau flow (REF ) or the harmonic map flow smoothly coming out of a 0-homogeneous map u_0:\mathbb {S}^{n-1}\rightarrow \mathbb {S}^{m-1}\subset \mathop {\rm \mathbb {R}}\nolimits ^m in C^{\infty }(\mathop {\rm \mathbb {R}}\... | {
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} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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1edd10442a0bbec28609c0c8a0af0359128c17fe | subsection | 56 | 59 | Taylor expansions at infinity for expanders of the harmonic map flow | Then, on one hand,\Delta _fU&=&\sum _{i=0}^k\left(|x|^{-2i-2}\Delta _{\mathbb {S}^{n-1}} u_i+\Delta _f(|x|^{-2i})u_i\right)+\textit {O}(|x|^{-2k-2})\\
&=&\sum _{i=0}^k\left(|x|^{-2i-2}\Delta _{\mathbb {S}^{n-1}} u_i+\left(2i(2(i+1)-n)|x|^{-2}-i\right)|x|^{-2i}u_i\right)+\textit {O}(|x|^{-2k-2})\\
&=&\sum _{i=1}^k|x|^{-... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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9aa07a7199836b51cc2f3bbaf7d4f2d5fcc75535 | subsection | 57 | 59 | Taylor expansions at infinity for expanders of the harmonic map flow | Then, since the maps (u_i)_{i\ge 0} are spherical,|\nabla U|^2&=&\left|\nabla \left(\sum _{i=0}^k|x|^{-2i}u_i\right)\right|^2+\textit {O}(|x|^{-2k-2})\\
&=&\left|\sum _{i=0}^k(-2i)|x|^{-2i-1}u_i\right|^2+\left|\sum _{i=0}^k|x|^{-2i-1}(\nabla ^{\mathbb {S}^{n-1}} u_i)\right|^2+\textit {O}(|x|^{-2k-2})\\
&=&\sum _{i=0}^{... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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d03c49da76c1852fb26da8b53713dea39ddebaef | subsection | 58 | 59 | Taylor expansions at infinity for expanders of the harmonic map flow | Indeed, as K goes to +\infty , one gets by () that:\lim _{K\rightarrow +\infty }u_1^K&=&\Delta _{\mathbb {S}^{n-1}} u_{0}-<\Delta _{\mathbb {S}^{n-1}} u_{0},u_0>u_0\\
&=&\Delta _{\mathbb {S}^{n-1}} u_{0}+|\nabla ^{\mathbb {S}^{n-1}} u_0|^2u_0,which is exactly formula (REF ) since 0=\Delta _{\mathbb {S}^{n-1}} |u_{0}|^2... | {
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"raw": "A. Deruelle. Asymptotic estimates and compactness of expanding gradient ricci solitons. Annali della Scuola Normale Superiore di Pisa, Vo... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
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8a8f149ae4d6cca0ca48e84114ad8f1ad0652d32 | abstract | 0 | 9 | Abstract | In this paper, we present a formalization of Matiyasevi\v{c}'s theorem, which
states that the power function is Diophantine, forming the last and hardest
piece of the MRDP theorem of the unsolvability of Hilbert's 10th problem. The
formalization is performed within the Lean theorem prover, and necessitated the
developm... | {
"cite_spans": []
} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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77f3786b5c75edab7b78b653d94afd2be762ce9a | subsection | 1 | 9 | Introduction | In 1900, David Hilbert presented a list of 23 unsolved problems to the International Congress of Mathematicians . Of these, the tenth was the following: To find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. This problem remained unsolved for... | {
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"raw": "Hilbert, D., Mathematichse Probleme, Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris 1900. Nachrichten Akad. Wiss. Gottingen, Math. -Phys. Kl. (1900) 253–297. English transl... | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f789ad1c4b7631d35e28a670adf92e6abccd7f61 | subsection | 2 | 9 | Introduction | The simple arithmetic functions and relations +,-,\times ,<,\le ,= are easily shown to be Diophantine, and conjunctions and disjunctions of Diophantine relations are Diophantine because of the equivalences:p(\bar{x})=0\wedge q(\bar{x})=0\iff p(\bar{x})^2+q(\bar{x})^2=0p(\bar{x})=0\vee q(\bar{x})=0\iff p(\bar{x})q(\bar{... | {
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} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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b71ec51e22382b74e9a23f67d775fb5628edd234 | subsection | 3 | 9 | The number theory | It was already recognized before Matiyasevič that the following “Julia Robinson hypothesis” would suffice to prove that the exponential function is Diophantine:There exists a Diophantine set D\subseteq \mathbb {N}^2 such that:(x,y)\in D\rightarrow y \le x^x
For every n, there exists (x,y)\in D such that y > x^n.Thus i... | {
"cite_spans": []
} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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1dccfb706723fba31979f0ce2c860636485c0681 | subsection | 4 | 9 | The ring | structure zsqrtd (d : ℕ) := (re : ℤ) (im : ℤ)
prefix `ℤ√`:100 := zsqrtdThis defines \mathbb {Z}[\sqrt{d}]\simeq \mathbb {Z}\times \mathbb {Z} with the projection functions called "re" and "im", by analogy to the real and imaginary part functions of \mathbb {Z}[i], although of course here both parts are real since d is ... | {
"cite_spans": []
} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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50645d51977a9ed0ab27701034dff3e9093a1275 | subsection | 5 | 9 | The ring | (Also, a definition of \mathbb {R} with a square root function did not exist at the time of writing of the formalization.) | {
"cite_spans": []
} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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ef9abe2d291d5d3e368655dfff25398adea4f820 | subsection | 6 | 9 | Pell's equation | Most of the concepts surrounding Pell's equation translate readily to statements about elements of \mathbb {Z}[\sqrt{d}]. There is a “conjugation” operation (a+b\sqrt{d})^*=a-b\sqrt{d} on \mathbb {Z}[\sqrt{d}], in terms of which (x,y) is a solution to Pell's equation iff N(x+y\sqrt{d})=1, where N(z)=zz^* is the norm on... | {
"cite_spans": []
} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.... | |
ec76e13a102dd9410ef9c2b8e56f239b2518e1c8 | subsection | 7 | 9 | Pell's equation | Thenx=x_k(a)\wedge y=y_k(a)\iff k\le y\wedge ((x=1\wedge y=0)\vee \varphi ),where\varphi :=\exists u,v,s,t,b\in \mathbb {N}\quad [&x^2-(a^2-1)y^2=1\ \wedge \\
&u^2-(a^2-1)v^2=1\ \wedge \\
&s^2-(b^2-1)t^2=1\ \wedge \\
&b>1\ \wedge \ b\equiv 1\pmod {4y}\ \wedge \ b\equiv a\pmod {u}\ \wedge \\
&v>0\ \wedge \ y^2\mid v\ \w... | {
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}
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} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a451cd6b8778e536fc6b53e2c25bbf37a739d5fc | subsection | 8 | 9 | Diophantine sets | The informal proof contains many theorems in the style of Theorem REF , where we have an equivalence between a target relation or function on one side, and a conjunction of disjunctions of quantified Diophantine predicates of Diophantine functions on the other, and will immediately conclude from such a theorem that the... | {
"cite_spans": []
} | 1802.01795 | A Lean formalization of Matiyasevi\v{c}'s Theorem | [
"Mario Carneiro"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
ab3fee6ab800f6d76bbbc373cbc6e2bf8f0048a7 | abstract | 0 | 40 | Abstract | We deform monomial space curves in order to construct examples of
set-theoretical complete intersection space curve singularities. As a
by-product we describe an inverse to Herzog's construction of minimal
generators of non-complete intersection numerical semigroups with three
generators. | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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8193f44ac8f851c95a983fcecdd9e1449e945f05 | subsection | 1 | 40 | Introduction | It is a classical problem in algebraic geometry to determine the minimal number of equations that define a variety.
The codimension is a lower bound for this number which is reached in case of set-theoretic complete intersections.
Let I be an ideal in a polynomial ring or a regular analytic algebra over a field {K}.
Th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1094,
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"raw": "R. C. Cowsik and M. V. Nori, Affine curves in characteristic p are set theoretic complete intersections, Invent. Math. 45 (1978), no. 2, 111–114. MR 0472835",
"source_ref_id": "ed45c7121cebbc... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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f2cd7ded2b4e252806a3010cc605889d12297faa | subsection | 2 | 40 | Introduction | If a,b\ge 3 and d_1+q-n\ge \gamma +\ell , then C defined by
p:=\gamma -1-\ell >m
is a non-monomial set-theoretic complete intersection.In the setup of Corollary REF Moh's third condition in (REF ) becomes ab<1 and is trivially false.
Corollary REF thus yields an infinite list of new examples of non-monomial set-theor... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 559,
"openalex_id": "",
"raw": "Charles Delorme, Sous-monoïdes d'intersection complète de N., Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 145–154. MR 0407038",
"source_ref_id": "5970223e5a3ddc67018bf7c1a2c585e85f1f1cb1",... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... |
13b30a8c5e7ddcbb37818c27afa229bd15f79787 | subsection | 3 | 40 | Ideals of monomial space curves | Let \ell ,m,n\in {N} generate a semigroup \Gamma ={\left\langle \ell ,m,n\right\rangle }\subset {N}.d=\gcd (\ell ,m).We assume that \Gamma is numerical, that is, \gcd (\ell ,m,n)=1.Let {K} be a field and consider the map\varphi \colon {K}[x,y,z]\rightarrow {K}[t],\quad (x,y,z)\mapsto (t^\ell ,t^m,t^n),whose image {K}[\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01273309",
"end": 595,
"openalex_id": "https://openalex.org/W2102265874",
"raw": "Jürgen Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175–193. MR 0269762",
"source_r... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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d4f4786c7621ef80ad2fdfb64dad87b203dbdfec | subsection | 4 | 40 | Ideals of monomial space curves | It is unique up to adding multiples of the first row to the second.
Overall there are 3 cases and an overlap case described equivalently by 3 matrices
\begin{pmatrix}
a & -b & 0 \\
a & 0 & c
\end{pmatrix},\quad \begin{pmatrix}
a & -b & 0 \\
0 & -b & c
\end{pmatrix},\quad \begin{pmatrix}
a & 0 & -c \\
0 & b & -c
\end{p... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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3ec03784ffa4fd4446781ab98e97a6aa32e506bb | subsection | 5 | 40 | Ideals of monomial space curves | Then g\in I if and only if g^{\prime }=0.
By (REF ), reductions by f_2 can be avoided in the calculation of g.
If r_2 and r_1 many reductions by f_1 and f_3 respectively are applied, theng^{\prime }=x^{\tilde{n}-a_1r_1-ar_2}y^{b_1r_2-r_1b_2}z^{r_1c+r_2c_2}-z^{\tilde{\ell }}and g^{\prime }=0 is equivalent to\tilde{\ell ... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... |
ea14c444ec2db648c0b5904f212f7b37ee61eef6 | subsection | 6 | 40 | Ideals of monomial space curves | Proposition 2.4 [(a)]
In case (REF ), a_1,a_2,b_1,b_2,c_1,c_2 arise through (REF ) from some numerical semigroup \Gamma ={\left\langle \ell ,m,n\right\rangle } if and only if e^{\prime }=1.
In this case, (\ell ,m,n)=(\ell ^{\prime },m^{\prime },n^{\prime }).
In case (REF ), a,b,c,a_1,b_2 arise through (REF ) from so... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... |
15f2665d1274fbfa5403e0ab8a30400432542fc2 | subsection | 7 | 40 | Ideals of monomial space curves | The difference of first rows of (REF ) and (REF ) is then a relation
\begin{pmatrix}a^{\prime }-a & b_1-b^{\prime } & c_2\end{pmatrix}
of (\ell ^{\prime },m^{\prime },n^{\prime }) with a^{\prime }-a<0, b_1-b^{\prime }<0 and c_2>0.
Then c_2\ge c^{\prime }\ge d^{\prime } by choice of c^{\prime }.
This contradicts (REF ... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... |
962a1d556775f0230820fec6295df9f0c8f6260f | subsection | 8 | 40 | Ideals of monomial space curves | The difference of first rows of (REF ) and (REF ) is then a relation
\begin{pmatrix}a^{\prime }-a & b_1-b^{\prime }_1 & c_2-c^{\prime }_2\end{pmatrix}
of (\ell ^{\prime },m^{\prime },n^{\prime }) with a^{\prime }-a\le 0, c_2-c^{\prime }_2<0 and hence b_1-b^{\prime }_1\ge b^{\prime } by choice of the latter.
This lead... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.01910245046019554,
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0.... |
2090cbc4e1200aff051c5565119caddae2db9761 | subsection | 9 | 40 | Ideals of monomial space curves | Then (\ell ^{\prime },m^{\prime },n^{\prime })=(3,4,5), but (a,-b,0) is not a minimal relation.
In fact the corresponding complete intersection {K}[\Gamma ] defined by the ideal {\left\langle x^3-y^4,z^2-x^2y\right\rangle } is the union of two branches x=t^3,y=t^4,z=\pm t^5.
Deformation with constant semigroup
Let \mat... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.02... |
2a01f284a1b70daeca84c9d2aa5c3388dc3ca33d | subsection | 10 | 40 | Ideals of monomial space curves | Then
{
{\overline{\mathcal {O}}}_{W,w}=({\overline{\mathcal {O}}},{\overline{\mathfrak {m}}})\cong ({K}{\left\lbrace t^{\prime }\right\rbrace },{\left\langle t^{\prime }\right\rangle })[r]^-\upsilon & {N}\cup {\left\lbrace \infty \right\rbrace }}
is a discrete valuation ring.
Denote by \mathfrak {m}_W and {\overline{... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... |
bb851bad8f2a6dcac291079c0a0a828a91df4cf6 | subsection | 11 | 40 | Ideals of monomial space curves | The graded sheaves \operatorname{gr}^\mathcal {F}\mathcal {O}_W\subset \operatorname{gr}^\mathcal {F}{\overline{\mathcal {O}}}_W are thus supported at w and the isomorphism
\operatorname{gr}^\mathcal {F}({\overline{\mathcal {O}}}_W)_w=\operatorname{gr}^F{\overline{\mathcal {O}}}\cong {K}[t^{\prime }]\cong {K}[{N}]
id... | {
"cite_spans": [
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"raw": "Séminaire Henri Cartan, 13ième année: 1960/61. Familles d'espaces complexes et fondements de la géométrie analytique. Fasc. 1 et 2: Exp. 1–21, 2ième édition, corrigée. École Normale Supérieure, Secr... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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c42c58082bc083e2cc7a0d142bc357a878bff6d7 | subsection | 12 | 40 | Ideals of monomial space curves | Remark 3.1 Teissier defines X as the analytic spectrum of \mathcal {A} over W\times L (see ).
This requires to interpret the \mathcal {O}_W-algebra \mathcal {A} as an \mathcal {O}_{W\times L}-algebra.
Remark 3.2 In order to describe (REF ) in explicit terms, embed
{
L\supset {\overline{W}}[r]^\nu & W\subset L^n
}
wi... | {
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{
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"doi": "10.1090/ulect/039",
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.026... |
16d678e397f3285312d26ee6350f8e67d85faf3b | subsection | 13 | 40 | Ideals of monomial space curves | This yields the finite extension of {K}-analytic domains
\mathcal {O}_S=\mathcal {O}_{X,\iota (0)}\subset \mathcal {O}_{Y,\iota (0)}.
We aim to describe \mathcal {O}_{Y,\iota (0)} and {K}-analytic algebra generators of \mathcal {O}_S.
In explicit terms \mathcal {O}_S is obtained from a presentation
I\rightarrow \mat... | {
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"raw": "Séminaire Henri Cartan, 13ième année: 1960/61. Familles d'espaces complexes et fondements de la géométrie analytique. Fasc. 1 et 2: Exp. 1–21, 2ième édition, corrigée. École Normale Supérieure, Sec... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.0... |
29b9341f59fa8831b2f9871c214f0a8cc1addfe7 | subsection | 14 | 40 | Ideals of monomial space curves | By the universal property of \operatorname{Spec}^\mathrm {an}, it follows that (see )
\operatorname{Spec}^\mathrm {an}_{{\overline{W}}}({\overline{\mathcal {B}}})
&=\operatorname{Spec}^\mathrm {an}_{{\overline{W}}}(\mathcal {O}_{{\overline{W}}}\otimes _{\nu ^*{\overline{\mathcal {O}}}_W}\nu ^*\mathcal {B})\\
&=\operat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.5802/aif.2207",
"end": 840,
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"raw": "Brian Conrad, Relative ampleness in rigid geometry, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1049–1126. MR 2266885",
"source_ref_id": "8c... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... |
7ad12c4086e7f248062023dc41c648a45b685757 | subsection | 15 | 40 | Ideals of monomial space curves | By choice of F_\bullet , there is a cartesian square
{
\llap {B=\,}{\overline{\mathcal {O}}}[t,s]@{^(->}[r] & {\overline{\mathcal {O}}}[s^{\pm 1}]\\
\llap {A=\,}\bigoplus _{i\in {Z}}(F_i\cap \mathcal {O})s^{-i}@{^(->}[u]@{^(->}[r] & \mathcal {O}[s^{\pm 1}]@{^(->}[u]
}
of finite type graded \mathcal {O}-algebras.
Thus... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-0.01532796304672956... |
b8c130ec70e36eb7917f1d089f31b9d5675af0ed | subsection | 16 | 40 | Ideals of monomial space curves | With \gamma ^{\prime } the conductor of \Gamma ^{\prime } and i=\gamma ^{\prime }+j, F_{\gamma ^{\prime }}\subset {\overline{\mathfrak {m}}}\cap \mathcal {O}=\mathfrak {m} and hence F_i=F_{\gamma ^{\prime }}F_j\subset \mathfrak {m}F_j.
Therefore these monomials generate A_j as \mathcal {O}-module by Nakayama's lemma.
I... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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d4d6556ce4bfae3d44d599e5f9c1f9c0f16f0385 | subsection | 17 | 40 | Ideals of monomial space curves | Set
\delta =\min {\left\lbrace \Delta {\underline{\ell }}\right\rbrace },\quad \Delta {\underline{\ell }}=\Delta \ell _1,\dots ,\Delta \ell _n.
With \deg (t)=1=-\deg (s) {\underline{\xi }} defines a map of graded {K}-algebras {K}[{\underline{x}},s]\rightarrow {K}[t,s] and a map of analytically graded {K}-analytic dom... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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0.0... |
f0f0b7899c4cba98c178e50c11a26ead9114342d | subsection | 18 | 40 | Ideals of monomial space curves | By flatness of \pi in Proposition REF , the relations {\underline{f}} of {\underline{\xi }}(t,0)=t^{{\underline{\ell }}} lift to relations {\underline{F}}\in {K}{\left\lbrace {\underline{x}},s\right\rbrace }^m of {\underline{\xi }}.
That is, {\underline{F}}({\underline{x}},0)={\underline{f}} and {\underline{F}}({\under... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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2cb7542dead8da1900e7da33bddbc6f87bf4cb61 | subsection | 19 | 40 | Ideals of monomial space curves | With (REF ) and homogeneity of {\underline{f}}^{\prime } it follows that \operatorname{ord}(h^{\prime })>k contradicting the maximality of k.
Remark 3.8 The proof of Proposition REF shows in fact that the condition \Gamma ^{\prime }=\Gamma is equivalent to the flatness of a homogeneous deformation of the parametrizatio... | {
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"source_ref_id": "699547f4f9b98682... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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9053e40a9dde0f6768dfe27d707f2e00993240e0 | subsection | 20 | 40 | Ideals of monomial space curves | Consider a matrix of indeterminates
M=
\begin{pmatrix}
Z_1 & X_1 & Y_1\\
Y_2 & Z_2 & X_2\\
\end{pmatrix}
and the system of equations defined by its maximal minors
F_1&=X_1X_2-Y_1Z_2,\\
F_2&=Y_1Y_2-X_2Z_1,\\
F_3&=X_1Y_2-Z_1Z_2.
By Schaps' theorem (see ), there is a solution with coefficients in {K}{\left\lbrace x,y,... | {
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"arxiv_id": "",
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"raw": "Mary Schaps, Deformations of Cohen-Macaulay schemes of codimension 2 and non-singular deformations of space curves, Amer. J. Math. 99 (1977), no. 4, 66... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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1eaf511e72549c0d783327aeee1646d75b267bb5 | subsection | 21 | 40 | Ideals of monomial space curves | The calculations are the same.
In the examples we favor powers of x in order to minimize the conductor \gamma +k\ell .
Series of examples
Redefining a,b suitably, we specialize to the case where the matrix in (REF ) is of the form
M_0=
\begin{pmatrix}
z & x & y\\
y^b & z & x^a\\
\end{pmatrix}.
By Proposition REF .(... | {
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"raw": "Ernst Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1970), 748–751. MR 0265353 (42 #263... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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daaf7b17887b698e6c91bbb8824c0f7251ceb34a | subsection | 22 | 40 | Ideals of monomial space curves | They are thus the largest elements of \Gamma \setminus \Gamma _1.
Their minimum attained at i=\left\lfloor \frac{m}{\ell }\right\rfloor then bounds
\gamma \le \gamma _1-1-\left\lfloor \frac{m}{\ell }\right\rfloor \ell .
Substituting \gamma _1+\ell -1=d_2 yields the first particular inequality.
The second one follows ... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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8a7aece3757c3a4bc3b63f7f06e90d6fcbf4f27a | subsection | 23 | 40 | Ideals of monomial space curves | Since O(t^7) lies in the conductor, it follows that C\cong C_0.
In all other cases, Corollary REF yields an infinite list of new examples.
a=3, b=2.
Consider the monomial curve C_0 defined by (x,y,z)=(t^4,t^7,t^9).
By Zariski's method from (REF ), we reduce to considering the deformation
(x,y,z)=(t^4,t^7,t^9+st^{10}... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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5f8db31267aa242631cf80e35158cc644171724d | subsection | 24 | 40 | Deformation with constant semigroup | Let \mathcal {O}=(\mathcal {O},\mathfrak {m}) be a local {K}-algebra with \mathcal {O}/\mathfrak {m}\cong {K}.
Let F_\bullet ={\left\lbrace F_i\mid i\in {Z}\right\rbrace } be a decreasing filtration by ideals such that F_i=\mathcal {O} for all i\le 0 and F_1\subset \mathfrak {m}.
Consider the Rees ringA=\bigoplus _{i\i... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... |
01ec23b36053f0f1902accdbb4c9e588086b7f3d | subsection | 25 | 40 | Deformation with constant semigroup | There are decreasing filtrations by ideal (sheaves)\mathcal {F}_\bullet ={\overline{\mathfrak {m}}}_W^\bullet \lhd {\overline{\mathcal {O}}}_W,\quad F_\bullet =\mathcal {F}_{\bullet ,w}={\overline{\mathfrak {m}}}^\bullet =\upsilon ^{-1}[\bullet ,\infty ]\lhd {\overline{\mathcal {O}}}.Setting t=t^{\prime }/s and identif... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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68f280003ade8e3cf6b708e9110bd3ab7892f246 | subsection | 26 | 40 | Deformation with constant semigroup | The graded sheaves \operatorname{gr}^\mathcal {F}\mathcal {O}_W\subset \operatorname{gr}^\mathcal {F}{\overline{\mathcal {O}}}_W are thus supported at w and the isomorphism\operatorname{gr}^\mathcal {F}({\overline{\mathcal {O}}}_W)_w=\operatorname{gr}^F{\overline{\mathcal {O}}}\cong {K}[t^{\prime }]\cong {K}[{N}]identi... | {
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"raw": "Séminaire Henri Cartan, 13ième année: 1960/61. Familles d'espaces complexes et fondements de la géométrie analytique. Fasc. 1 et 2: Exp. 1–21, 2ième édition, corrigée. École Normale Supérieure, Secr... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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f6bf714fbeef66f0c1a48589313591d099247f42 | subsection | 27 | 40 | Deformation with constant semigroup | Applying \operatorname{Spec}^\mathrm {an}_W to (REF ) yields a diagram of {K}-analytic spaces (see ){
X=\operatorname{Spec}^\mathrm {an}_W(\mathcal {A})[dr]^-\pi && \operatorname{Spec}^\mathrm {an}_W(\mathcal {B})=Y[ll]_-\rho \\
&L[ur]^-\iota }where \pi is flat with \pi \circ \rho \circ \iota =\operatorname{id} and\pi ... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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135268ddf13949ccf9b347ef7064f3f3b18b5084 | subsection | 28 | 40 | Deformation with constant semigroup | The map \rho in (REF ) becomes\rho (t,s)=(x_1(t^{\prime })/s^{\ell _1},\dots ,x_n(t^{\prime })/s^{\ell _n})for s\ne 0 and the fiber \pi ^{-1}(0) is the image of the map\rho (t,0)=((\xi _1(t),\dots ,\xi _n(t)),0),\quad \xi _k(t)=\lim _{s\rightarrow 0}x_k(st)/s^{\ell _k}=\sigma (x_k)(t).Taking germs in (REF ) this yields... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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a6ffe438411f8b7dc03d158e0f1bd4d3a0ba08ba | subsection | 29 | 40 | Deformation with constant semigroup | In explicit terms \mathcal {O}_S is obtained from a presentationI\rightarrow \mathcal {O}[{\underline{x}}]\rightarrow A\rightarrow 0mapping {\underline{x}}=x_1,\dots ,x_n to \iota (0)=A\cap \mathfrak {m}[s^{\pm 1}]+As as\mathcal {O}_S=\mathcal {O}{\left\lbrace {\underline{x}}\right\rbrace }/\mathcal {O}{\left\lbrace {\... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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851ff67b0dee8ee525525b800ff0fbd78ff51eb7 | subsection | 30 | 40 | Deformation with constant semigroup | If \Gamma ^{\prime }={\left\langle {\underline{\ell }}\right\rangle }, then \mathcal {O}={K}{\left\lbrace {\underline{\xi }}^{\prime }\right\rbrace } and \mathcal {O}_S={K}{\left\lbrace {\underline{\xi }},s\right\rbrace }.By choice of F_\bullet , there is a cartesian square{
\llap {B=\,}{\overline{\mathcal {O}}}[t,s]@{... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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e45fdd18afd7db8507dcabd8a51eadddc4c3a5e4 | subsection | 31 | 40 | Deformation with constant semigroup | For i\ge j,{
(A/As)_i=\operatorname{gr}^F_iA_i[r]^-{\cdot s^{i-j}}_-\cong & \operatorname{gr}^F_iA_j.
}Thus finitely many monomials in {\underline{\xi }},s generate any A_j/F_iA_j\cong F_j/F_i over {K}.
With \gamma ^{\prime } the conductor of \Gamma ^{\prime } and i=\gamma ^{\prime }+j, F_{\gamma ^{\prime }}\subset {\o... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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f8b8b32b013df23ae4caaaff4bfd706288a79c35 | subsection | 32 | 40 | Deformation with constant semigroup | Consider elements {\underline{\xi }}=\xi _1,\dots ,\xi _n defined by\xi _j=t^{\ell _j}+\sum _{i\ge \ell _j+\Delta \ell _j}\xi _{j,i}t^is^{i-\ell _j}\in {K}[t,s]\subset {\overline{\mathcal {O}}}[t,s]=Bwith \Delta \ell _j\in {N}\setminus {\left\lbrace 0\right\rbrace }\cup {\left\lbrace \infty \right\rbrace }.
Set\delta =... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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88a5d0cf50ba5ec3cd3c1ef74746e41140864049 | subsection | 33 | 40 | Deformation with constant semigroup | The flat deformation in Proposition REF is then defined by\mathcal {O}_S={K}{\left\lbrace {\underline{\xi }},s\right\rbrace }={K}{\left\lbrace {\underline{x}},s\right\rbrace }/{\left\langle {\underline{F}}\right\rangle },\quad {\underline{F}}={\underline{f}}-{\underline{f}}^{\prime }s.First let \Gamma ^{\prime }=\Gamma... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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7e83005ae6fe66f81452284ebda92c2e469a50b7 | subsection | 34 | 40 | Deformation with constant semigroup | Seth^{\prime }=h-\sum _{i=1}^mq_iF_i({\underline{x}},1)=h-\operatorname{inp}h+\sum _{i=1}^mq_if^{\prime }_i({\underline{x}},1).Then h^{\prime }({\underline{\xi }}^{\prime })=h({\underline{\xi }}^{\prime }) by (REF ) and hence \upsilon (h^{\prime }({\underline{\xi }}^{\prime }))=k^{\prime }.
With (REF ) and homogeneity ... | {
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"source_ref_id": "699547f4f9b98682... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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5c18a884aa1b463d0496a2b7ff7df2599976653a | subsection | 35 | 40 | Set-theoretic complete intersections | We return to the special case \Gamma ={\left\langle \ell ,m,n\right\rangle } of §.
Recall Bresinsky's method to show that \operatorname{Spec}({K}[\Gamma ]) is a set-theoretic complete intersection (see ).
Starting from the defining equations (REF ) in case (REF ) he computesf_1^c=(x^a-y^{b_1}z^{c_2})^c
&=x^ag_1\pm y^{b... | {
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"raw": "H. Bresinsky, Monomial Gorenstein curves in A^{4} as set-theoretic complete intersections, Manuscripta Math. 27 (1979), no. 4, 353–358. MR 534800",
"source_ref_id": "2ef075d24511a6cfbdc45cda77... | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
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507b0d9c66168b2afcdd77a4837f20b6636b6c9c | subsection | 36 | 40 | Set-theoretic complete intersections | Then the curve germ C defined by () is a set-theoretic complete intersection if\min (d_1,d_2,d_3)+\delta &\ge \gamma ,\\
\min (d_1,d_3)+\delta &\ge \gamma +k\ell ,or, equivalently,\min (d_1,d_2+k\ell ,d_3)+\delta &\ge \gamma +k\ell .By Lemma REF , the first inequality yields the assumption \Gamma ^{\prime }=\Gamma on (... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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e563a0873da8d17823980aeb1cec6a8cee44dcdf | subsection | 37 | 40 | Series of examples | Redefining a,b suitably, we specialize to the case where the matrix in (REF ) is of the formM_0=
\begin{pmatrix}
z & x & y\\
y^b & z & x^a\\
\end{pmatrix}.By Proposition REF .(REF ), these define \operatorname{Spec}({K}[{\left\langle \ell ,m,n\right\rangle }]) if and only if\ell =b+2,\quad m=2a+1,\quad n=ab+b+1(=(a+1)\... | {
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singularities | [
"Michel Granger",
"Mathias Schulze"
] | [
"math.AG"
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20060e4924e4db7ffe61fe3b94e222d6dcc557d8 | subsection | 38 | 40 | Series of examples | Their minimum attained at i=\left\lfloor \frac{m}{\ell }\right\rfloor then bounds\gamma \le \gamma _1-1-\left\lfloor \frac{m}{\ell }\right\rfloor \ell .Substituting \gamma _1+\ell -1=d_2 yields the first particular inequality.
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singularities | [
"Michel Granger",
"Mathias Schulze"
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5a15cead81b27d16ce163415f3189f1ef842609d | subsection | 39 | 40 | Series of examples | In all other cases, Corollary REF yields an infinite list of new examples.
a=3, b=2.
Consider the monomial curve C_0 defined by (x,y,z)=(t^4,t^7,t^9).
By Zariski's method from (REF ), we reduce to considering the deformation
(x,y,z)=(t^4,t^7,t^9+st^{10}).
While part REF of Corollary REF does not apply, C\lnot \cong... | {
"cite_spans": []
} | 10.5427/jsing.2018.17s | 1804.01316 | Deforming monomial space curves into set-theoretic complete intersection
singularities | [
"Michel Granger",
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"math.AG"
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37d94245d385b34c571b14d37afa84041394ad06 | abstract | 0 | 28 | Abstract | SiMRX is a MRX simulation toolbox written in MATLAB for simulation of
realistic 2D and 3D Magnetorelaxometry (MRX) setups, including coils, sensors
and activation patterns. MRX is a new modality that uses magnetic nanoparticles
(MNP) as contrast agent and shows promising results in medical applications,
e.g. cancer tre... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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63881dc029400fa10849937b28495125a3ad19e8 | subsection | 1 | 28 | Introduction | Many new and experimental treatment methods in medical applications use magnetic nanoparticles as a contrast agent.
These particles allow for multiple different approaches (, ), however for the named methods the exact knowledge about the particle distribution is crucial.
Here, Magnetorelaxometry (MRX) can be used to de... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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302ef0e8cab880f6aff3a5bd9536c8e515159ab7 | subsection | 2 | 28 | Model | This following section (including notation) is part of , which, for interested readers, provides an in depth look in the analysis of the following operator.
The magnetic field in w\in \Omega induced by a coil \alpha = (\varphi _\alpha , I_\alpha ) is given by\mathbf {B}_\alpha ^\textbf {coil}\colon \Omega \rightarrow \... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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654c077115f77e0e45f3aee18776fb0aa31ef605 | subsection | 3 | 28 | Discretization | First we consider the 3D case:
Here the conductor coil \varphi _\alpha is approximated by a set list of segments, with starting points a_k and ending points b_k for the k-th segment respectively.
Then the magnetic field in w\in \Omega is :\mathbf {B}_{\alpha ,k}^\textbf {coil}\colon \Omega &\rightarrow \mathbb {R}^3\\
... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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1232de61b17b9d4644a7e527182fed789f19eab3 | subsection | 4 | 28 | Structure | The SiMRX toolbox is a modular toolkit that provides tools for MRX experiment setup configuration and simulation as well as visualization of data.
SiMRX is capable of simulating synthetic or real setups and datasets.The processes required for simulation are handled in a sequence of modular functions, i.e. the creation ... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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30b5b295ac610559cc170ca7d9eb1fecb3e7759d | subsection | 5 | 28 | Features | The SiMRX toolbox is separated in the following submodules (each in its own subfolder):[Table: NO_CAPTION]We give a short overview of each module in the following sections.
For all provided functions, in detail information of syntax and features are available in each file header.
This documentation is also available us... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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] | [
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ef89ed7236d837792ce5763035c4551f17060b36 | subsection | 6 | 28 | Configuration | This module provides a set of functions to create setup and config files. | {
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"Lea Föcke"
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e1e79d097eef33a8a13b6768bb87b0a29ad7a8d8 | subsection | 7 | 28 | Conventions for 2D configurations: | SiMRX supports 2D and 3D systems.
However, at its core, the toolbox is designed for 3D simulations.
Therefore 2D setups are implemented as 3D setups with the following variances:all entities are on the same z-layer (z=0 recommended)
the region of interest in z direction is set accordingly (setup.roi.z = [0,0] recommen... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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bb38c6d2d2841e01f4edb2d0a337ef02837d4f1a | subsection | 8 | 28 | Setup file | In MATLAB a setup file is represented as struct.
It holds the following fields[Table: NO_CAPTION]Determined by the dimension of the system, setup.dim is either 2 or 3.
The region of interest setup.roi is a struct with fields x, y and z, that holds boundary information in an array with unit meter \left[{m}\right]. For i... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
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04f6c98f83d0c14f4c0660cd2d4e0e5c7f252153 | subsection | 9 | 28 | Config file | config files are, similar to setup files, structs in MATLAB. They hold the fields[Table: NO_CAPTION]The simulated voxel resolution of the region of interest is given by a three element array and stored in res.A setup may contain a lot of coils/sensors that are not used in a specific configuration.
To optimize the simul... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
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42b0aae717c10a6f405959cfc28c16c4d4a3662c | subsection | 10 | 28 | Setup and Config Validation | This toolbox also contain functions to check the created setup and config.
The functions isConfigValid.m and isSetupValid.m validate if all required fields are set and further include checks for data inconsistencies.
Additionally the function checkCompatibility.m checks if a given config is applicable for a given setup... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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8930a27118c1fd6d52e3133ceba875c820564a38 | subsection | 11 | 28 | Save and Load | The setup and config datasets can be stored using saveSetup.m and saveConfig.m.
These are stored using a custom .mrxsetup and .mrxcfg extension, that is based on the MATLAB internal .mat file format.
Corresponding load functions (loadSetup.m and loadConfig.m) are available.To store the created setup and config datasets... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
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b90f196ca8709a664e935850932e8ef2773b02c0 | subsection | 12 | 28 | Simulation | The simulation module is the core element of the toolbox.
All necessary files can be found in the folder Ay./simulation
.
The main script is provided by the file createSystemMatrix.m.
It processes a given MRX setup and configuration into a linear operator, namely a matrix A.
A valid setup file is required, that provi... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
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7ba660cfe68a83928068c6c2fefc24348ce8090d | subsection | 13 | 28 | Raw export/import | In general the simulation of the setup is the most time demanding step.
It is possible to dump the dataset ARaw into file, which allows for flexible experimentation with varying coil currents.
However this is not very suitable in case the set of active coils or sensors changes, since ARaw contains a fixed set of coils.... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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eafde05b863eafedd2dc4ce69ed15a8754bb6bd4 | subsection | 14 | 28 | Phantom | The function createPhantom.m provides multiple phantom options, including phantoms suited for reconstruction or resolution tests.
createPhantom.m uses the function phantom3.m, which is a 3D reimplementation of the MATLAB given phantom function and able to generate 3D phantoms that are composed of multiple ellipses.
For... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
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195854aff09f9ce47dda07df523243ac3484ec20 | subsection | 15 | 28 | Visualization | SiMRX provides tools to visualize the region of interest (drawROI.m), coils (drawCoils.m), sensors (drawSensors.m), magnetic fields (drawFields.m) and 3D phantoms as cuttable volume (drawVolume.m).The toolbox also contains visualization functions that combine these base functions to useful tools:The function visualizeS... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
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0722d183bfc4fdbd4cb9545fed544644b69fe58f | subsection | 16 | 28 | Setups | This toolbox contains three setups, namely 'default2D', 'default3D' and 'realistic3D'.
These can be used to test the functionalities of the SiMRX toolbox.
We will first outline the used folder structure in section REF and then give a short introduction to the included setups in section REF .Please note: Due to internal... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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a8b96b81226ef9daf47fc64edee1d5008596722e | subsection | 17 | 28 | Folder structure | The following folder structure is proposed for setup and its respective config datasets (see Listing REF as a reference and visual representation of the folder structure).For each unique setup a base folder Ay./setups/<my setup>
is created and all related configs, scripts and datasets are included in this folder.
At ... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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ed6e15c692e3ccd51147064ebb8f8bb30c4099fa | subsection | 18 | 28 | Example setups | The respective folder Ay./setups/<setup>/scripts/
contain scripts for the creation of the following datasets.
Furthermore the file README.m in Ay./setups/<setup>
can be executed to run all creation scripts for that setup at once. | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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163cab9d53ff6504134057990c12d1b8441c9869 | subsection | 19 | 28 | Fully synthetic dataset: | In SiMRX a 2D ('default2D') and a 3D ('default3D') example is available.
These can be created using the respective MATLAB scripts createSetup.m and createConfigs.m (the 2D scrips are shown in listing REF and REF as seen in section REF and REF ).
A visualization of both datasets is available in figure REF . | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
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46feb0ccdc945d835366db78971f16869ac18d9e | subsection | 20 | 28 | A 3D dataset from formatted text files: | With SiMRX it is possible to load datasets from text files, save in the .mrxsetup and .mrxcfg data structure, as well as pre-processing for simulation and reconstructions tasks.In the subfolder Ay./setups/realistic3D/scripts
the script createRawDataset.m is used to create the following files in folder Ay./setups/re... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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3875382c533fe898dd3f42a17b056563018aef50 | subsection | 21 | 28 | Sensor Information (Table | The table (see Table REF ) that is used in file sensors.dat) stores all necessary sensor information.
Each row defines a sensor unit with properties defined by the columns as follows. Columns 1-3 define the x, y and z position as a translation vector. Columns 4-6 define the orientation of the sensors measure direction.... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
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f2b9304c8f80796f4094dd89b278a3f2c232bc37 | subsection | 22 | 28 | Coil Information (Table | The file coilGrid.dat contains positioning information for every activation coil in the system (see Table REF ).
Each row defines a coil position with columns 1-3 defining the translation vector for direction x, y and z.
As described in Sections REF and REF , the coil is a composition of multiple conductor segments.
Th... | {
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} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
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7f3072dc9ace2e94e107cd4e644cd3fd3d885bb6 | subsection | 23 | 28 | Voxel Grid Information (Table | The file voxelGrid.dat contains a list of used voxels in the current setup (see Table REF ).
Again columns 1-3 define the positions of the voxel midpoints in x, y and z. | {
"cite_spans": []
} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
"cs.NA",
"math.NA"
] | 2,018 | en | Computer Science | [
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212d8b82b4ffecebd5e5dea94f3484dee3d3b2ee | subsection | 24 | 28 | Current Information (Table | The table REF is used as data scheme for file dataset.01.currents.dat and contains a list of currents in Ampere \left[{A}\right] that are applied to the coils.
Currently, for external data, only subsequent coil patterns are supported.
This means that the number of givencurrents has to fit the number of coils as of coil... | {
"cite_spans": []
} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
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4479e6c37189c345ba3192b93c7b83208bf2adfe | subsection | 25 | 28 | Measurements Information (Table | The data structure defined in Table REF ) is used in file dataset.01.relax.dat.
The first column contains the change in the magnetic response \Delta B \left[{fT}\right] after an coil activation (compare with equation (REF )).
Column 2-5 provide information about the sensorID, channelID, groupID and the coil/current pat... | {
"cite_spans": []
} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
"cs.NA",
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a7b7ed1ba6f182023c81d712bc7b58db898f0813 | subsection | 26 | 28 | Examples | The folder Ay./exmaples
contains scripts to illiustrate the workflow that is required to handle simulated and real experimenal data with the SiMRX toolbox.
It make use of the provided setups presented in section REF .ExampleA.m loads the 'default2D' setup and a eligible config file (see section REF ) and simulates t... | {
"cite_spans": []
} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
"cs.NA",
"math.NA"
] | 2,018 | en | Computer Science | [
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736c9a4bbf484e99026661e3d71ff22595f9ffe3 | subsection | 27 | 28 | Remarks | This simulation toolbox does not guarantee its correctness and closeness to reality.
SiMRX is not qualified for commercial usage. | {
"cite_spans": []
} | 1810.02286 | SiMRX -- A Simulation toolbox for MRX | [
"Lea Föcke"
] | [
"cs.CE",
"cs.NA",
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