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169fb3014f9dd0d20694773163aab461d317bdd4 | subsection | 76 | 80 | Proof of | \end{align*}
\end{}\end{align*}
\end{}\begin{}
We list some examples of E\left( H^{[n]}_m;t \right):
\begin{equation} \begin{array}{|c|c|c| c|}
\hline n & m=1 & m=2 & m=3 \\\hline 1 & t^2-1 & 1 & 0 \\ \cline {1-1}
2 & t^4+t^3-t^2-t & t^2+t & 0 \\\cline {1-1}
3 & t^{6}+t^{5}-2 t^3-t^2+t & t^4+2 t^3+t^2-t-1 & 1 \\\cline ... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.018762949854135513,
... | |
677e59dee14d3b2db70a3215fc4c669c3d72f9c6 | subsection | 77 | 80 | Proof of | \end{equation}
\begin{}[h]
\begin{array}{|c|cccc|}\hline n & m=2& m=3&m=4&m=5\\\hline 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
2 & 2 & 0 & 0 & 0 \\
3 & 2 & 1 & 0 & 0 \\
4 & 3 & 2 & 0 & 0 \\
5 & 2 & 5 & 0 & 0 \\
6 & 4 & 6 & 1 & 0 \\
7 & 2 & 11 & 2 & 0 \\\hline \end{array}
\caption {Examples of \chi \left( B^{[n]}_m\rig... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-0... | |
76954a7440ee0a9d72f0b4979bc2d24c46a843c7 | subsection | 78 | 80 | Proof of | Indeed, each strata B^{[1]}_2, B^{[3]}_3 and B^{[6]}_4 contains a single monomial ideal:
1{16pt}\begin{array}{ccc}
(:y,~:x) & (:y^2,~:<xy>,~~:<x^2>) & (:<y^3>,~:<y^2x>,~~:<x^2y>,~~~:<x^3>) \\
B^{[1]}_2=\lbrace \mathfrak {m}\rbrace & B^{[3]}_3=\lbrace \langle x^2,xy,y^2\rangle \rbrace & B^{[6]}_4=\lbrace \langle x^3,x^2... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
04056769494789adb9d7f6518d34e3d8ad94a871 | subsection | 79 | 80 | Proof of | Göttsche,
Hilbert schemes of points on surfaces, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 483–494, Higher Ed. Press, Beijing.L. Göttsche,
Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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dfa526767c836147b1e2145bf56444915b80215d | abstract | 0 | 30 | Abstract | We suggest an index-free formalism allowing to simplify many computations in
Riemann geometry. The main ingredients are forms with values in a Clifford
algebra and an action of the group $\mathfrak{sl}_2\times \mathfrak{sl}_2$ on
such forms. | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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46231c572ea06cef58bf88ae63a1fdd659984a4c | subsection | 1 | 30 | Introduction | Working with Levi-Civita connection, curvature, Weyl and Ricci tensors and in particular deriving Einstein equation out of the Hilbert action is a painful struggle with indices (see for example ). In the present note we suggest a version of Cartan-Palatini formalism allowing to make most of the computations without ind... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 196,
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"raw": "B.A.Dubrovin, A.T.Fomenko, and S.P.Novikov, Modern Geometry - Methods and Applications : Part I, 2nd Edition, Springer-Verlag, N.Y., 1992.",
"source_ref_id": "40f9ab95e282b33183b3684272d1500d4... | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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a77cbf7599094fb2c5d151c649a4c67a012edfea | subsection | 2 | 30 | Grassmann and Clifford algebras. | In this section we recall basic notions about Grassmann (exterior) and Clifford algebras and relations between them. Then we define the algebra of Clifford forms and the action of the algebra \mathfrak {sl}_2\times \mathfrak {sl}_2 thereon.A Clifford algebra \mathit {Cl}(\mathrm {V}) of a vector space \mathrm {V} provi... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
bc78591d77833e6ab7892bce7a0b6156e347043e | subsection | 3 | 30 | Grassmann and Clifford algebras. | The identification of \mathit {Cl}(\mathrm {V}) and \Lambda (\mathrm {V}) allows to consider * also as an automorphism of the Grassmann algebra (as a vector space). The actions v\wedge and v\vdash on the Clifford algebra transferred to the Grassmann algebra are just the usual external and internal multiplication by v.O... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1054,
"openalex_id": "",
"raw": "A.Losev, From Berezin integral to Batalin-Vilkovisky formalism. A mathematical physicist's point of view. in M.Shifman (ed.) Felix Berezin. Life and Death of the Mastermind of Supermathematics. Wor... | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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de58b454406e0948232ba4601a38d54013e10e95 | subsection | 4 | 30 | Action of the Lie algebra | Consider now the algebra \Omega ^{{\cdot }{\cdot }}=\mathit {Cl}(V)\otimes \Omega (M) of forms on an n-dimensional manifold M with values in the Clifford algebra. Let the vielbein \theta \in \Omega ^{11} defines an isomorphism between the tangent bundle to M and a trivial bundle with fiber \mathrm {V}. One can define t... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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2d34ab8dfcc6d977f1e52730672d6a9e60a0c666 | subsection | 5 | 30 | Action of the Lie algebra | At one point m\in M a Clifford form takes value in \mathit {Cl}(\mathrm {V})\otimes \Lambda (T_mM) which, using the isomorphism defined by \theta , can be identified with \mathit {Cl}(\mathrm {V})\otimes \Lambda (\mathrm {V}^*).To make explicit computation one can further identify this algebra with the algebra \Lambda ... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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6b25542a804a1ad6aeccb8e64d7cba09415d136c | subsection | 6 | 30 | Action of the Lie algebra | The property 3 follows from the computation similar to (REF ).2E^{\prime }x&=\theta x-(-1)^{p+q}x\theta =\xi _ia\otimes \xi ^i\alpha -(-1)^{p+q} a\xi _i\otimes \alpha \xi ^i=\\&=\xi _ia\otimes \xi ^i\alpha -(-1)^p a\xi _i\otimes \xi ^i\alpha =\xi _i\vdash a\otimes \xi ^i\alpha =\eta _{ij}\frac{\partial }{\partial \xi _... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0e16b28825dbccb5d81130a843c6e695d772a68d | subsection | 7 | 30 | Supertrace. | A supertrace of a Clifford algebra is a linear function on \mathit {Cl}(\mathrm {V}) satisfying the identity \operatorname{str}(ab)=(-1)^{\deg a\deg b}\operatorname{str}(ba) and normalized by the condition that \operatorname{str}\operatorname{{v}}=1. If n is even then the Clifford algebra has a unique representation an... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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836afab0abbb36a6bc4e2a8465374daf570bb59c | subsection | 8 | 30 | The action of the group | The invertible elements of g\in \mathit {Cl}(\mathrm {V}) such that g^{-1}\mathit {Cl}^1(\mathrm {V})g\subset \mathit {Cl}^1(\mathrm {V}) form a Lie group denoted by Pin(\mathrm {V}). The Lie algebra of this group is \mathit {Cl}^2(\mathrm {V}) with respect to the commutator. It is isomorphic to the Lie algebra \mathfr... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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912a87826abffd25dca46b54008b50a70725f3b3 | subsection | 9 | 30 | Use of the formalism | Let M be an n-dimensional manifold and \mathrm {V} an n-dimensional vector space with a nondegenerate quadratic form \eta . Let we are given two Clifford algebra valued 1-forms \theta \in \Omega ^{11} and \omega \in \Omega ^{21} such that the form \theta is nondegenerate in the sense that it induces an isomorphism at e... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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da77f5727bef3359a0c009d2d28cf3df62358a52 | subsection | 10 | 30 | Gauge group action. | A pair \theta ,\omega defines the same metric and connection as a pair \theta ^{\prime },\omega ^{\prime } if and only if
there exists a gauge transformation relating them, namely if there exists a function g on M with values in Pin(\mathrm {V})\subset \mathit {Cl}(\mathrm {V}) such that \theta ^{\prime }=g^{-1}\theta ... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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b7d1aff4a7cc8b8b98004b8e347310734518c367 | subsection | 11 | 30 | Curvature. | The connection \nabla can be extended in a standard way to \Omega ^{pq} by the formula \nabla e \alpha = (\nabla e) \alpha + ed\alpha for e\in \Omega ^{p0} and \alpha \in \Omega ^{0q}. It can be therefore written as \nabla x = dx + \omega x -(-1)^q x\omega for any x\in \Omega ^{pq}. This allows to compute \nabla ^2 x =... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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a8e31c5af20c88044da3007487d6b4dd6df44181 | subsection | 12 | 30 | Torsion. | The torsion can be defined as an element t\in \Omega ^{12} such that for any two vector fields X and Y on M we have i_Yi_Xt = i_X\nabla i_Y\theta -i_Y\nabla i_X\theta -i_{[X,Y]}\theta . Taking into account the identity i_Xi_Yd\alpha =i_Ydi_X\alpha -i_Xdi_Y\alpha +i_{[X,Y]}\alpha valid for any 1-form \alpha one can easi... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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ae36372a963a824b16fa18130fa960765d6a9694 | subsection | 13 | 30 | Bianchi identities. | Computing the covariant derivative of the torsion one gets\nabla t= dt+\omega t-t\omega =R\theta -\theta R=-2E^{\prime }R.In particular this identity implies that if the torsion vanishes we have E^{\prime }R=0. Since H^{\prime }R=0 we have F^{\prime }R=0 and therefore the curvature form is invariant with respect to the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.math/0204191",
"end": 619,
"openalex_id": "https://openalex.org/W1552534450",
"raw": "P.Ševera, A remark on the symmetries of the Riemann curvature tensor., arxiv:math/0204191.",
"source_ref_id": "192d05d58f4444b4de8b44... | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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4994e3f1be32652719b42137c626125bcf5cf21b | subsection | 14 | 30 | Conformal transformations and the Weyl tensor. | Let (\theta ,\omega ) be a pair of Clifford forms with vanishing torsion. Let \tilde{\theta }=e^\phi \theta be a conformal transformation of the form \theta . This transformation corresponds to a conformal transformation of the metric. Compute now the induced transformation of the form \omega and of the curvature R.Pro... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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fee1407b5867520cdd3c0935bb84ce1b56818b23 | subsection | 15 | 30 | Conformal transformations and the Weyl tensor. | The torsion and the curvature can be computed directly:&e^{-\phi }(d\tilde{\theta }+\tilde{\theta }\tilde{\omega }+\tilde{\omega }\tilde{\theta })=e^{-\phi }d(e^{\phi }\theta )+\theta \omega +\omega \theta +\\&+\theta (\theta \varepsilon -\varepsilon \theta )+(\theta \varepsilon -\varepsilon \theta )\theta =d\theta +\t... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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abbffc0c0511ada8b9f5090abf6edd444f6fee83 | subsection | 16 | 30 | Hilbert action. | On the space of pairs \theta ,\omega define the Hilbert functional:S(\theta , \omega )=\operatorname{str}\int _M \theta ^{n-2}(d\omega +\omega ^2)=\operatorname{str}\int _M E^{n-2}R.Proposition 3 The Hilbert functional is gauge invariant.
The variation of this functional is\frac{\delta S}{\delta \theta } =&(-1)^{(n^2+n... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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62b8bf50c5dcca020700997e5753e49f4bd10bf9 | subsection | 17 | 30 | Hilbert action. | Computing the variation of the Hilbert functional\delta S=\operatorname{str}\int _{\partial M} \theta ^{n-2}\delta \omega + \operatorname{str}\int _M \frac{\delta S}{\delta \theta }\delta \theta + \operatorname{str}\int _M \frac{\delta S}{\delta \omega }\delta \omegaone gets\frac{\delta S}{\delta \theta } =\Pi ^{n-1}\l... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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c264bb50e21e8018f22deab6872b1852908b2600 | subsection | 18 | 30 | Special case | The dimension 4 of the manifold M is special in particular since in this case HR=0 and the equation of motion (Einstein equation) amounts to ER=0. Together with the Bianchi identity E^{\prime }R=0 it implies that the curvature R is invariant under the action of both \mathfrak {sl}_2 algebras. In this dimension the acti... | {
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"doi": "10.1098/rspa.1978.0143",
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"raw": "M.F.Atiyah, N.J.Hitchin, I.M.Singer: Self-duality in four-dimensional Riemannian geometry, Proc.Roy.Soc.Lond.A 362 (1978), 425–461.",
"sou... | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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3d0f0d81550b18d6548f2f3bb13488a572c4f702 | subsection | 19 | 30 | Complex structure | In this section we generalize the construction for the case where the space \mathrm {V} is provided with a complex structure J compatible with the metric. In order to simplify notation we rename this space, vielbein, connection, torsion, curvature etc. into \mathrm {V}^{\mathbb {R}}, \theta ^{\mathbb {R}} t^{\mathbb {R... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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18b9e2201ae303bcdf72a15b2460ec5d18a02a8e | subsection | 20 | 30 | Complex structure | In this case one can define a decomposition of the differential d=\partial +\bar{\partial } with the property \partial :\Omega ^{p\bar{p}q\bar{q}}\rightarrow \Omega ^{p\bar{p}(q+1)\bar{q}} and \bar{\partial }:\Omega ^{p\bar{p}q\bar{q}}\rightarrow \Omega ^{p\bar{p}q(\bar{q}+1)}. | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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09a5abb8ea761b380cf2a827c4b058b846c29764 | subsection | 21 | 30 | Action of the algebra | Define the operators2E_0x&=\theta x+(-1)^{(p+\bar{p}+q+\bar{q})}x\theta ,&
2F_1x&=\theta x-(-1)^{(p+\bar{p}+q+\bar{q})}x\theta \\
2F_2x&=\bar{\theta } x+(-1)^{(p+\bar{p}+q+\bar{q})}x\bar{\theta }, &
2E_3&=\bar{\theta } x-(-1)^{(p+\bar{p}+q+\bar{q})}x\bar{\theta }Observe that E_3=*_1^{-1}F_2*_1 and F_1=*_1^{-1}E_0*_1. D... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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f2fd93ca93d27c2ad82e15d5f00f40492c176cc3 | subsection | 22 | 30 | Action of the algebra | These operators have degrees\deg {E_0}&=(1,0,1,0),& \deg {E_1}&=(0,1,-1,0),& \deg {E_2}&=(0,-1,0,-1),\\ \deg {E_3}&=(-1,0,0,1), &
\deg {F_0}&=(-1,0,-1,0),& \deg {F_1}&=(0,-1,1,0), \\ \deg {F_2}&=(0,1,0,1),&\deg {F_3}&=(1,0,0,-1), &\deg H_i&=(0,0,0,0).Proposition 4 The operators \lbrace E_i,F_i,H_i\rbrace generate the a... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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131e74d978e2b6df4faefa263dacf53e46a7b3ac | subsection | 23 | 30 | Action of the algebra | Taking into account that*^{-1}\frac{\partial }{\partial \xi _i}*=\eta ^{i\bar{j}}\bar{\xi }_j, && *^{-1}\frac{\partial }{\partial \bar{\xi }_i}*=\eta ^{\bar{i}j}\xi _j, && *^{-1}\xi _i* =\eta _{\bar{j}i} \frac{\partial }{\partial \bar{\xi }_j}, && *^{-1}\bar{\xi }_i* =\eta _{j\bar{i}} \frac{\partial }{\partial \xi _j},... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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b4009e5f53d8c2d05f316ca9a7fd65bd748df8bf | subsection | 24 | 30 | Action of the algebra | To prove () compute the commutators:[E_0,E_1]&=\eta ^{i\bar{j}}\,\xi _i\bar{\xi }_j\otimes 1,& [E_1,E_2]&=1\otimes \eta ^{i\bar{j}}\,\frac{\partial ^2}{\partial \xi ^i\partial \bar{\xi }^j},\\
[E_2,E_3]&=-\eta _{i\bar{j}}\,\frac{\partial ^2}{\partial \xi _i\partial \bar{\xi }_j}\otimes 1,&
[E_3,E_0]&=-1\otimes \eta _{i... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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fa0571e28a79afbb9a5e5dcaed1f70deda556520 | subsection | 25 | 30 | Kähler condition. | The Lie algebra of the unitary group (the group preserving both the complex structure J and the scalar product \eta ) is embedded into the Clifford algebra as \mathit {Cl}^{11}(\mathrm {V}^\mathbb {C}). The connection preserving unitary structure is therefore a form \omega ^{\mathbb {R}}=\omega +\bar{\omega } with \ome... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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445c4365ad665780fe15e1e44e4e232def5d5ded | subsection | 26 | 30 | Kähler condition. | The proof that \bar{\partial }w=0 is analogous and thus the form w is necessarily closed.Conversely, the condition dw=0, the relation t^{0111}=0 and the invertibility of E_3:\Omega ^{1110}\rightarrow \Omega ^{0111} imply that the condition t^{1020}=0 is also satisfied. Indeed the condition t^{0111}=0 implies \omega = -... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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7b006fc73591300686162c7c58affc5ca5c9aebd | subsection | 27 | 30 | Bianchi identity in the Kähler case. | In the Kahler case the first Bianchi identity implies that only the component R\in \Omega ^{1111} of the curvature R^\mathbb {R} is nonzero and that it is invariant under a subalgebra \widehat{\mathfrak {sl}}_2\times \widehat{\mathfrak {sl}}_2 of \widehat{\mathfrak {sl}}_4.Proposition 6 The curvature is given by R^{\ma... | {
"cite_spans": []
} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
"Pierre Goussard"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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3b6448cbebdef77c274e697aa49550efd31bc077 | subsection | 28 | 30 | Einstein equation in the Kähler case. | In the Kähler case the Einstein equation is equivalent to the equations F_0R=0 and E_2R=0. In four real dimension it is equivalent to the full \widehat{\mathfrak {sl}}_4 invariance of the curvature. Indeed, the Einstein equation reads as FR^\mathbb {R}=0. In the complex case it amounts to (F_0+E_2)R=0. Since the two te... | {
"cite_spans": []
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"Vladimir V. Fock",
"Pierre Goussard"
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12521fbb0629b615b89b52d07a05a45438deee4b | subsection | 29 | 30 | Conclusion. | First of all we hope that the developed formalism allows to simplify learning Riemann differential geometry for students as well as to better understand the logic of the theory. Emergence of the affine group \widehat{\mathfrak {sl}}_4 seems mysterious for us and requires better understanding of its consequences. In the... | {
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} | 1810.00239 | Riemann geometry without indices | [
"Vladimir V. Fock",
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623027b1c982101eb2e9716e7a277a4e0c039338 | abstract | 0 | 29 | Abstract | The way developers edit day-to-day code tends to be repetitive, often using
existing code elements. Many researchers have tried to automate repetitive code
changes by learning from specific change templates which are applied to limited
scope. The advancement of deep neural networks and the availability of vast
open-sou... | {
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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772c39b029699751d71596184ac5ca8d36126fa5 | subsection | 1 | 29 | 0pt | 8pt plus 4pt minus 2pt4pt plus 2pt minus 2ptplaintop
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
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0447d2363a0ebcc6e04ab32c8e47eacb839fa204 | subsection | 2 | 29 | Introduction | Developers edit source code to add new features, fix bugs, or maintain existing functionality (e.g., API updates, refactoring, etc.) all the time. Recent research has shown that these edits are often repetitive , , . Moreover, the code components (e.g., token, sub-trees, etc.) used to build the edits are often taken fr... | {
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"Saikat Chakraborty",
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8e8ead4a9d56c2eac65bee6cb93bb4785e17d244 | subsection | 3 | 29 | Introduction | In the second step, the model concretizes the previously generated code fragment by predicting the tokens conditioned on the AST that was generated in the first step: given the type of each leaf node in the syntax tree, our model suggests concrete tokens of the correct type while respecting scope information. We combin... | {
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f41a6938f7934ed514660e8fb3da56558275434f | subsection | 4 | 29 | Background | Modeling Code Changes.
Generating source code using machine learning models has been explored in the past , , , . These methods model a probability distribution p(c|\kappa ) where c is the generated code and \kappa is any contextual information upon which the generated code is conditioned. In this work, we generate cod... | {
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"Saikat Chakraborty",
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92fa67a54e898e9002d5a4a2e9b70822d853f6b2 | subsection | 5 | 29 | Motivating Example | fig:motiv illustrates an example of our approach.
Here, the original code fragment [rgb]1.00,0.00,0.00return super.equals(object) is edited to [rgb]0.0,0.6,0return object == this.
Codit takes these two code fragments along with their context, for training.
While suggesting changes, i.e., during test time, Codit takes a... | {
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"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
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14ea526a2b32edcfbfba607b55d87ae3fb46e1ba | subsection | 6 | 29 | Motivating Example | However, in contrast to traditional seq2seq where the generation of each token is only conditioned on the previously generated and source tokens, we additionally condition on the token type that has been predicted by the tree model and generate only tokens that are valid for that toke type. fig:motivtoken shows this st... | {
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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05ae9f53620e78e1b008361d04e3f2c5cf482bd7 | subsection | 7 | 29 | Tree-based Neural Translation Model | We decompose the task of predicting code changes in two stages:
First, we learn and predict the structure (syntax tree) of the edited code. Then, given the predicted tree structure, we concretize the code.
We factor the generation process as\vspace{-14.22636pt}
P(c_{n}|c_{p}) = P(c_n|t_n, c_p) P(t_n|t_p) P(t_p|c_p)and ... | {
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
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fba533e581ad4d8f3f724896940093f73cd2a377 | subsection | 8 | 29 | Tree Translation Model ( | The goal of {M}_{tree} is to model the probability distribution of a new tree (t_n) given a previous version of the tree (t_p).
For any meaningful code the generated tree is syntactically correct.
We represent the tree as a sequence of grammar rule generations following the CFG of the underlying programming language. T... | {
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e8a5d2a4f49c64366b6288e1affbb28439150dcc | subsection | 9 | 29 | Tree Translation Model ( | At a given decoding step k the decoder LSTM changes its internal state in the following way,{h}_k^n = f_{LSTM}({h}_{k-1}^n, {\psi }_k),where {\psi }_k is computed by the attention-based weighted sum of the inputs {h}^p_j as in , i.e.{\psi }_k = \sum \limits _{j=1}^{\tau }
softmax({{h}_{k-1}^n}^T {h}_j^p) {h}_j^pThen,
t... | {
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"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
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c10cca8a275efa8ffe78e0229f6b53aab97bb9cb | subsection | 10 | 29 | Token Generation Model ( | We now focus on generating a concrete code fragment c, i.e. a sequence of tokens (x_1, x_2, ...).
For the edit task, the probability of an edited token x^n_k depends not only on the tokens of the previous version (x_1^p, ..., x_m^p) but also on the previously generated tokens x_1^n, ..., x_{k-1}^n. The next token
x^n_k... | {
"cite_spans": []
} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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e59f47356f0a339bdcd21b1911d06e4bc1a70381 | subsection | 11 | 29 | Token Generation Model ( | Since the language grammar provides this information, we create a mask (mask(\theta _k^n)) that returns a -\infty value for masked entries and zero otherwise. Similarly, not all variable, method names, type names are valid at every position. We refine the mask based on the variables, method names and type names extract... | {
"cite_spans": []
} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
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83f87bb59a1ac4cd5d48123aafd84bc68790649b | subsection | 12 | 29 | Implementation | Our tree-based translation model is implemented as an edit recommendation tool, Codit. Codit learns source code changes from a dataset of patches.
Then, given a code fragment to edit, Codit predicts potential changes that are likely to take place in a similar context. We implement Codit extending OpenNMT based on PyTo... | {
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"s... | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
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6d10dd7a13a62ad685d2ae039c43059ff46459c6 | subsection | 13 | 29 | Implementation | Note that the losses of the two models are independent and thus we train each model separately.
In our preliminary experiment, we found that the quality of the generated code is not entirely correlated to the loss. To mitigate this, we used top-1 accuracy to validate our model.
We train the model for a fixed amount of ... | {
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"raw": "D. R. Reddy et al., “Speech understanding systems: A summary of results of the five-year research effort,” Department of Computer Science. Camegie-Mell University, Pittsburgh, PA, 1977.",
"sou... | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
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56049c67944ae2534cdf6795ec32ab12e48b49b6 | subsection | 14 | 29 | Experimental Design | We evaluate Codit for three different types of changes that often appear in practice:
(i) code change in the wild, (ii) pull request edits, and (iii) bug repair. For each task, we train and evaluate Codit on different datasets. tab:datasetsummary provides detailed statistics of the datasets we used.(i) Code Change Task... | {
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"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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520804c6a9d8d82871edcec0e931358cd7903f4b | subsection | 15 | 29 | Evaluation Metric | To evaluate Codit, we measure for a given code fragment, how accurately Codit generates patches. We consider Codit to correctly generate a patch if it exactly matches the developer produced patches. Codit produces the top K patches and we compute Codit's accuracy by counting how many patches are correctly generated in ... | {
"cite_spans": []
} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
] | [
"cs.SE"
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05de6c24faeb72cae75bab6d9214b8ee34bd2378 | subsection | 16 | 29 | Baseline | We consider several baselines to evaluate Codit's performance. Our first baseline in a vanilla LSTM based Seq2Seq model with attention mechanism . Results of this baseline indicate different drawbacks of considering raw code as a sequence of token.The second baseline, we consider, is proposed by Tufano et al. . For a ... | {
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"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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4073c9cd8c94edc0998388084a7d6797277401b0 | subsection | 17 | 29 | Results | We evaluate Codit's performances to generate concrete patches w.r.t. generic edits (RQ1) and bug fixes (RQ3).
In RQ2, we present an ablation study to evaluate our design choices.RQ1. How accurately can Codit suggest concrete edits?
[Table: Performance of Codit suggesting concrete patches. For Token Based models, pred... | {
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a003903f051354ef9ec6ecbfcc322aca383ff168 | subsection | 18 | 29 | Results | Instead, we present the context to the Codit through the token mask (see eqn:maskedprob). If we enable copy attention, Codit becomes highly biased by the tokens that are inside c_p.
[Figure: Patch size (Tree Edit-distance) histogram of correctly generated patches in different datasets.][Table: Examples of different typ... | {
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0553efb73b2bd9321dc59dccc0d2455d6896652b | subsection | 19 | 29 | Results | Other structural transformation that Codit include, but not limited to, include scoping (example 7 in examples), adding/deleting method parameters (example 3 in examples), changing method/variable access modifiers (example 9, 10 in examples), etc.Result : Codit suggests 15.94% correct patches for Code-Change-Data and 2... | {
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"Saikat Chakraborty",
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fdd349aee06f140d2f048f82c136cdc145d0a0c3 | subsection | 20 | 29 | Results | For example, 3050 test patches of Code-Change-Data, and 225 test patches of Pull-Request-Data do not have structural changes. When we use these patches to train {M}_{tree}, we essentially train the model to sometimes copy the input to the output and rewarding the loss function for predicting no transformation.
Thus, to... | {
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
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60e4e4551e1a055c677ecae1cc2fa2c23702db3d | subsection | 21 | 29 | Results | With this, Codit generates 820 (15.94%), and 177 (28.87%) correct patches from Code-Change-Data and Pull-Request-Data respectively (tab:replaceunksummary).Second, we test two configuration parameters related to the beam size, K_{tree} and K_{token} i.e. the number of trees generated by {M}_{tree} and number of concrete... | {
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
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e00fc238583edabfa535405e84de6cf004e025a0 | subsection | 22 | 29 | Results | The main reason behind such deterioration is diverse choice of token name across different projects. Developer tend to use project specific naming convention, api etc. This also indicates that the structural change pattern that developers follow are more ubiquitous across different projects than the token changes.Resul... | {
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3a732497f89ce21d37ac208e2a7887c09ac043d3 | subsection | 23 | 29 | Results | We see that, 48 out of 51 successful patches are generated within 20 minutes.We further manually compare the patches with the developer-provided patches: among 51 potential patches, 30 patches are identical and come from 25 different bug ids (See Table REF ). The bugs marked in [rgb]0.0,0.6,0green are completely fixed ... | {
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"Saikat Chakraborty",
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"Miltiadis Allamanis",
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4b8c26c25c977a19ac01a86a66d8d8a4d21b03d1 | subsection | 24 | 29 | Threats to validity | External Validity.
We built and trained Codit on real-world changes. Like all machine learning models, our hypothesis is that the dataset is representative of real code changes. To mitigate this threat, we collected patch data from different repositories and different types of edits collected from real world.Most NMT b... | {
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"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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8464291fa1842778545d89ec733599447c68f0dc | subsection | 25 | 29 | Related Work | Modeling source code.
Applying ML to source code has received increasing attention in recent years across many applications such as
code completion , , bug prediction , , , clone detection , code search , etc.
In these work, code was represented in many form, e.g., token sequences , , parse-trees , , graphs , , embedd... | {
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{
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"doi": "10.1145/3212695",
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"raw": "M. Allamanis, E. T. Barr, P. Devanbu, and C. Sutton, “A survey of machine learning for big code and naturalness,” ACM Computing Surveys, 2018.",
... | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
"Baishakhi Ray"
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5b4e0e8582b43e2248f35d59bab50fdebe51affe | subsection | 26 | 29 | Related Work | Automatic program repair is a well-researched field, and previous researchers proposed many generic techniques for general software bugs repair , , , , . There are two differnt directions in program repair research : generate and validate approach, and sysnthesis bases approach. In generate and validate approaches, can... | {
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{
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"doi": "10.1145/2568225.2568258",
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"raw": "S. Kaleeswaran, V. Tulsian, A. Kanade, and A. Orso, “Minthint: Automated synthesis of repair hints,” in Proceedings of the 36th International C... | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
"Yangruibo Ding",
"Miltiadis Allamanis",
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02b0db3ba2c666a6ec3fa02ad9f4f5e471e3e318 | subsection | 27 | 29 | Discussion and Future Work | Search Space for Code Generation.
Synthesizing patches (or code in general) is challenging . When we view code generation as a sequence of token generation problem, the space of the possible actions becomes too large. Existing statistical language modeling techniques endorse the action space with a probability distribu... | {
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"Saikat Chakraborty",
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bad47ff748055190b4aa75fd3339e500eba3c4a3 | subsection | 28 | 29 | Conclusion | In this paper, we proposed and evaluated Codit, a tree-based hierarchical model for suggesting eminent source code changes.
Codit's objective is to suggest changes that are similar to change patterns observed in the wild.
We evaluate our work against a large number of real-world patches. The results indicate that tree-... | {
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} | 10.1109/TSE.2020.3020502 | 1810.00314 | CODIT: Code Editing with Tree-Based Neural Models | [
"Saikat Chakraborty",
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4ccdcc29d6a0d4aed693d6db3e8a09c40c0cc3f7 | abstract | 0 | 98 | Abstract | The free closed semialgebraic set $D_f$ determined by a hermitian
noncommutative polynomial $f$ is the closure of the connected component of
$\{(X,X^*)\mid f(X,X^*)>0\}$ containing the origin. When $L$ is a hermitian
monic linear pencil, the free closed semialgebraic set $D_L$ is the feasible
set of the linear matrix i... | {
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} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0... |
6d4be751357da9061e50f94405c60c0bd2b85d4d | subsection | 1 | 98 | Introduction | Semidefinite programming (SDP) , is the main branch
of convex optimization to emerge in the last 25 years.
Feasibility sets of semidefinite programs are given by linear matrix inequalities (LMIs)
and are called spectrahedra.
We refer to the book for an overview of the substantial theory of LMIs and spectrahedra and the... | {
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} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
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0.0228... |
da6aaf6c8d2358385f595a10e08b86e0e8aca434 | subsection | 2 | 98 | Definitions | Let x=(x_1,\dots ,x_g) denote freely noncommuting variables
and x^*=(x_1^*,\dots ,x_g^*) their formal adjoints.
Let \!\mathop {<}\!x,x^*\!\mathop {>} denote the set of words in x and x^*
and \mathop {<}\!x,x^*\!\mathop {>} the free polynomials in (x,x^*)
equal the finite -linear combinations from <x,x*>.
For a positive... | {
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} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
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6e362d7864a8196e4d4d8bc087302ad75a2c9fd0 | subsection | 3 | 98 | Definitions | A monic (linear) pencil of size \delta is an
element L of \operatorname{M}_{{\delta }}(\mathop {<}\!x,x^*\!\mathop {>}) of the formL(x,x^*)
= I_\delta - A\operatorname{\raisebox {1pt}{{\bigodot }}}x - B\operatorname{\raisebox {1pt}{{\bigodot }}}x^*=
I_\delta - \sum _{j=1}^g A_j x_j - \sum _{j=1}^g B_j x_j^*.In the case... | {
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} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
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0.... |
cabe3771319256b1ffb80749c648d2b82faf9d87 | subsection | 4 | 98 | Main results | We are now ready to exposit our main
results.
Using the theory of realizations for
noncommutative rational functions , , , , , in Theorem REF we
explicitly and constructively describe the structure of noncommutative matrix polynomials f
whose invertibility set \mathcal {K}_f
is convex.
Each \delta \times \delta noncomm... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
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7208c652286f2911b942f9dd2fed0336d19a21fb | subsection | 5 | 98 | Main results | The converse is proved in Section REF .Theorem REF
implies that, for a monic linear pencil L, the invertibility set
\mathcal {K}_L is convex
if and only if
the semisimple part of a minimal size pencil L describing \mathcal {K}_L is
similar to a hermitian pencil.A non-invertible element f\in \operatorname{M}_{{\delta }... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
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df2545d1961b6256849fe7bc647d292371c48e18 | subsection | 6 | 98 | Main results | The proof of REF is based on
Theorem REF (see Subsection REF ), while the proof
of REF in Subsection REF uses
REF and new, of independent interest, (recursive) certificates
for invertibility of linear pencils on interiors of free spectrahedra.Theorem 1.4 (Nichtsingulärstellensatz)
Let L be a hermitian monic pencil,
an... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
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79d9d84a4b9c7dfe11fb8d2c7a856bf08f30931d | subsection | 7 | 98 | Main results | The matrix cube
problem of , is a notable example of this phenomena , . See also
, . Theorem REF provides another example as it gives
a computationally tractable relaxation for the problem of determining whether
a polynomial is of constant sign on the interior of a spectrahedron. | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
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0d73df0a057858f337ce1b8812fdb5c01c8aaeba | subsection | 8 | 98 | Reader's guide | Section contains background and some preliminary results on linear pencils, free
spectrahedra and realizations of noncommutative rational functions needed in the sequel.
The proof of Theorem REF is
given in Section , followed
by the proof of Theorem REF
and its corollary, Corollary REF ,
in Subsection REF . Corollary ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
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c0efec0d671f64f652924a9502f6b035a45dbda2 | subsection | 9 | 98 | Preliminaries | Let z=(z_1,\dots ,z_g,z_{g+1},\dots ,z_{2g})=(x_1,\dots ,x_g,y_1,\dots ,y_g)
denote 2g freely noncommuting variables. Replacing z_{g+j}=y_j with
x_j^* identifies \mathop {<}\!z \!\mathop {>} with \mathop {<}\!x,x^*\!\mathop {>}.
On the other hand, elements
f\in \mathop {<}\!z \!\mathop {>} are naturally evaluated at tu... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
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b339c60c0ef7f76c7e3b04e8b17d7a154c1811fe | subsection | 10 | 98 | Preliminaries | Therefore the results and definitions for matrix polynomials in z=(z_1,\dots ,z_{h}),
whose assumptions refer only to the structure, and not to evaluations, of polynomials,
directly apply to matrix polynomials in x_1,\dots ,x_g,x_1^*,\dots ,x_g^*.The free locus \mathcal {Z}_f of
f\in \mathop {<}\!z \!\mathop {>}^{\del... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
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"S. McCullough",
"J. Volčič"
] | [
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5473d2f3dfa344a54513d216a80576a642f799d4 | subsection | 11 | 98 | Preliminaries | If L and M are minimal and \mathcal {D}_L=\mathcal {D}_M, then L and M
are unitarily equivalent. (See Proposition REF .)
It is convenient to declare that the minimal pencil for the largest
free spectrahedron \mathcal {D}_I=\lbrace (X,X^*)\colon X\in \operatorname{M}_{n}(^n,n\in {\mathbb {N}}\rbrace is of size 0.
Every ... | {
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} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
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0.013020575977861881,
-0.022... |
6780459795417864b34c4337fde5b49dfbe9c459 | subsection | 12 | 98 | Free loci and spectrahedra | For h,n\in {\mathbb {N}}, let \Omega ^{(n)}=(\Omega ^{(n)}_1,\dots ,\Omega ^{(n)}_h) be an h-tuple of n\times n generic matrices, that is,\Omega ^{(n)}_j=(\omega _{j\imath \jmath })_{\imath \jmath },where \omega _{j\imath \jmath } for 1\le j\le h and 1\le \imath ,\jmath \le n are commuting indeterminates.Lemma 2.1
A l... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.07185816764831543,
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0.011462507769465446,
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0.025183938443660736,
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0.020177677273750305,
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0.0009491673554293811,
0.005906778387725353,
0.... |
63434f4fd596a2852f01ba05bff0aa143296588e | subsection | 13 | 98 | Free loci and spectrahedra | By an invariant subspace
for L, we mean an invariant subspace for \lbrace A_1,\dots ,A_g,A_1^*,\dots ,A_g^*\rbrace .
Since L is hermitian, any invariant subspace for L
is in fact reducing.
Hence L=\oplus _i L^i, where each L^i is a hermitian monic
pencil with no nontrivial invariant (equivalently reducing) subspaces.
T... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04889026656746864,
0.049714259803295135,
0.0013456649612635374,
-0.0212101973593235,
0.028183622285723686,
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0.03396683558821678,
0.003982634283602238,
-0.028168363496661186,
0.008758743293583393,
-0.009514070115983486,
0.01... |
bd58472b7358d518aa6527a881f245e1dc621259 | subsection | 14 | 98 | Free loci and spectrahedra | If L(X,X^*)\nsucc 0, then there exists t\in (0,1) such that \det L(\gamma (t))=0, contradicting \mathcal {Z}_f=\mathcal {Z}_L. Therefore L(X,X^*)\succ 0. A similar argument shows
L(X,X^*)\succ 0 implies (X,X^*)\in \mathcal {O}. Taking closures obtains \mathcal {K}_f=\mathcal {D}_L.Taking up items REF and REF , suppose ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.036162104457616806,
0.0355822928249836,
-0.02076650969684124,
-0.0430283285677433,
0.0006747017614543438,
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0.005328951869159937,
0.00232307193800807,
-0.0010814301203936338,
-0.014213080517947674,
-... |
5d56bf5f94a241aef75028e9322fc01083244c2f | subsection | 15 | 98 | Realization theory | Let \operatorname{M}_{{\delta }}(z \leavevmode \vtop {
{\hfil )\cr >#\hfil \cr )\crcr }} denote the \delta \times \delta noncommutative (nc) rational functions in z_1,\dots ,z_h , , .
Evaluations and the involution for polynomials
naturally extend to
\operatorname{M}_{{\delta }}(z \leavevmode \vtop {
{\hfil )\cr >#\hf... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.044670071452856064,
-0.003060769522562623,
-0.03899477422237396,
0.0007709134370088577,
-0.005038350820541382,
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0.03572995588183403,
0.06950712203979492,
0.022792721167206764,
0.01752934232354164,
-0.030405551195144653,
-0.017697159200906754,
0.0060986545868217945,
0... |
41ecbbedb45db0924f709e60e8cbfccc0e795b7a | subsection | 16 | 98 | Realization theory | Given a realization I+c^* (I-A\operatorname{\raisebox {1pt}{{\bigodot }}}z)^{-1}(b\operatorname{\raisebox {1pt}{{\bigodot }}}z)
there is a linear algebra algorithm – an extension of the Kalman decomposition – that
produces a minimal realization.
In the classical (commutative) one-variable
setting, if \mathbb {r}(\zet... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.020345428958535194,
0.0012878060806542635,
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0.012126363813877106,
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-0.0018038825364783406,
-0.024390093982219696,
0... |
1111b8b11a6eb66ee5ccd3615829ff188fad2b7d | subsection | 17 | 98 | Realization theory | By Remark REFREF ,
N_j:=A_j-b_jc^*, the coefficients of L+{\bf b}c^*, are the coefficients
in a minimal realization of the polynomial f. By Remark REFREF
, the N_j are jointly nilpotent.
Hence \det f(\Omega ^{(n)})=\det L(\Omega ^{(n)}) for all n.(2)
If 0\ne v\in \ker A, thenN_jv=-(c^*v)b_j,and c^*v\in \lbrace 0\rbrac... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.00402841717004776,
0.030716681852936745,
-0.006073144264519215,
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0.024780869483947754,
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0.02621523104608059,
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0.02703922614455223,
-0.01699870079755783,
0.0059434412978589535,
-0.04147438704967499,
0.030... |
838078328a01b2dce8f706543c74d324c8b771a8 | subsection | 18 | 98 | Proof of Theorem | We start the proof of Theorem REF
with a lemma.Lemma 3.1
Suppose \mathbb {r}\in x,x^* \leavevmode \vtop {
{\hfil )\cr >#\hfil \cr \setminus \crcr }} is defined at the origin and r(0)=1. Assume that r is hermitian and r=1+c* L-1b is a minimal FM realization, where b=j bjxj+j bjxj*. If L is irreducible and monic hermit... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.029198169708251953,
0.022409671917557716,
-0.014484671875834465,
-0.019404426217079163,
0.012669320218265057,
0.005285875406116247,
0.04417863115668297,
0.040792010724544525,
0.02016717940568924,
0.013912606984376907,
-0.028908323496580124,
-0.03194407746195793,
-0.02352328971028328,
0.... |
92eef1ab46b01762a186486874c0939475d90967 | subsection | 19 | 98 | Proof of Theorem | Hence by (REF ),(),() and the fact that w(A,A^*) span \operatorname{M}_{d}(, there exist \lambda _{jk}^1,\lambda _{jk}^2,\lambda _{jk}^3\in such that
\begin{alignat}{3}
{b}_j &=\lambda _{jk}^1 A_jc,&\qquad \widehat{b}_k&=\overline{\lambda _{jk}^1} A_k^*c \\
{b}_j &=\lambda _{jk}^2 A_jc,&\qquad {b}_k&=\overline{\lambda... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.0003222548693884164,
0.04060792550444603,
-0.024498997256159782,
-0.02323286049067974,
0.033133137971162796,
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0.03539083153009415,
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0.026832962408661842,
0.017542868852615356,
-0.03523828461766243,
-0.002707703737542033,
-0.0224243625998497,
0.0... |
56120c3d4201a22a40b1a4c3d2a8cfcb4b31f2c7 | subsection | 20 | 98 | Proof of Theorem | By Remark REFREF , f admits a minimal realizationf=1 - \varepsilon c^*\Big ( I-A(I-\varepsilon cc^*)\operatorname{\raisebox {1pt}{{\bigodot }}}x-A^*(I-\varepsilon cc^*)\operatorname{\raisebox {1pt}{{\bigodot }}}x^* \Big )^{-1}(A\operatorname{\raisebox {1pt}{{\bigodot }}}x+A^*\operatorname{\raisebox {1pt}{{\bigodot }}}x... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.0029938523657619953,
0.05092218890786171,
-0.025598391890525818,
-0.008504829369485378,
0.016140107065439224,
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0.09165383875370026,
0.017833445221185684,
0.030251258984208107,
0.008939605206251144,
-0.04475904628634453,
-0.01862671971321106,
-0.035422805696725845,
0.... |
bcb99c3fc53292e19234d3729d8a29bb3dae5b58 | subsection | 21 | 98 | Proof of Theorem | It follows that AP,A^*P are jointly nilpotent of order at most two and\Big (I-A(I-cc^*)\operatorname{\raisebox {1pt}{{\bigodot }}}x - A^*(I-cc^*)\operatorname{\raisebox {1pt}{{\bigodot }}}x^*\Big )^{-1}
=I+A(I-cc^*)\operatorname{\raisebox {1pt}{{\bigodot }}}x + A^*(I-cc^*)\operatorname{\raisebox {1pt}{{\bigodot }}}x^*.... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.010445975698530674,
0.008324733003973961,
-0.00274693313986063,
-0.047705069184303284,
0.020342566072940826,
-0.048620715737342834,
0.06140921637415886,
0.03284111246466637,
0.050146788358688354,
0.014261160977184772,
-0.02424931526184082,
0.0021651173010468483,
-0.0161000806838274,
-0.... |
a2d4d0c5d7c57121fea0b2141fa3bee9c7b2d571 | subsection | 22 | 98 | Proof of Theorem | If \mathcal {D}_f=\mathcal {D}_L, then L is irreducible and
there exists b_j,c\in d such that f^{-1}=I+ c^* L^{-1}{\bf b}
is a minimal FM realization.Write
L=I-A\operatorname{\raisebox {1pt}{{\bigodot }}}x-A^*\operatorname{\raisebox {1pt}{{\bigodot }}}x^*. By Proposition REFREF , \mathcal {Z}_f=\mathcal {Z}_L. After a ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.021681837737560272,
0.047452859580516815,
-0.0005311744753271341,
-0.02270413376390934,
0.046262722462415695,
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0.029326172545552254,
0.015021652914583683,
0.027754582464694977,
-0.03326277807354927,
-0.005378499161452055,
-0.020659541711211205,
0... |
869e2cdcd9e42d7472d4bfdcb40fe0057b731aea | subsection | 23 | 98 | Proof of Theorem | If \mathcal {D}_f is proper and convex, then f has degree two and is concave.Further, normalizing f(0)=1, if L is a minimal
hermitian monic pencil such that \mathcal {D}_f=\mathcal {D}_L,
then L is irreducible, f is a Schur complement of L and there
exist vectors c,b_1,\dots , b_{2g} such thatf^{-1} = 1+ c^* L^{-1}{\bf... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.07942263782024384,
0.03897874802350998,
0.015673913061618805,
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0.03047790192067623,
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0.017886880785226822,
0.03302662819623947,
-0.027318701148033142,
0.03314872458577156,
0.007920896634459496,
0.01997... |
70c9ac38f84361aacc9c18be36219e97142e3020 | subsection | 24 | 98 | Proof of Theorem | Therefore the set of irreducible components of \mathcal {Z}_{L}(n) contains the set of irreducible components of \mathcal {Z}_{\widetilde{L}}(n). Since\mathcal {Z}_L=\mathcal {Z}_{L^1}\cup \cdots \cup \mathcal {Z}_{L^\ell }and the \mathcal {Z}_{L^i}(n) are
irreducible hypersurfaces for all n large enough by Lemma REF ,... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04513552784919739,
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0.01509858202189207,
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0.024566667154431343,
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0.0018482219893485308,
-0.03105165623128414,
0... |
91dde4b889de7e58cb02365e6e58cf5f755f5abc | subsection | 25 | 98 | Proof of Theorem | If it is not similar to L^{i_k} for any k, then (REF ) implies\bigcap _k\mathcal {K}_{L^{i_k}}\subseteq \mathcal {K}_{L^m}.Hence \mathcal {K}_f=\mathcal {D}_{\widehat{L}} holds by (REF ).Remark 4.1
Given a factorization of f into atomic factors f=f_1\cdots f_t with f_j(0)=I, one can use the proof of Theorem REF to
ide... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.030024034902453423,
0.058919116854667664,
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0.019741414114832878,
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0.02889508381485939,
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0.00584309222176671,
-0.024638626724481583,
0... |
fb9ab18a17ff0f0a4a3add35ab4141606ed08343 | subsection | 26 | 98 | Proof of Theorem | By Remark REFREF ,
\mathcal {K}_f=\mathcal {K}_L. By assumption there exists a minimal hermitian monic pencil \widetilde{L} such that \mathcal {K}_L=\mathcal {D}_{\widetilde{L}}. By \partial \mathcal {K}_L(n) we denote the topological boundary of \mathcal {K}_L(n). Thus\mathcal {Z}_L(n)\supseteq \partial \mathcal {K}_L... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.038794293999671936,
0.0684928372502327,
0.006985872983932495,
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0.010881327092647552,
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0.0138267632573843,
0.02925596572458744,
-0.0028080842457711697,
0.014101467095315456,
-0.021930528804659843,
0.043006... |
7288fec9b6c632981f06224fcef3b35fcf5fc692 | subsection | 27 | 98 | Proof of Theorem | Then \mathcal {K}_{L^{i_k}}=\mathcal {D}_{\widetilde{L}^k} is convex for every k and therefore\mathcal {K}_L=\bigcap _k\mathcal {K}_{L^{i_k}}=\bigcap _k\mathcal {D}_{\widetilde{L}^k}=\mathcal {D}_{\widetilde{L}^1\oplus \cdots \oplus \widetilde{L}^s}.Moreover, L^{i_k} is similar to \widetilde{L}^k by .Recall that \wideh... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.025219494476914406,
0.056999415159225464,
-0.0076551008969545364,
-0.04857766255736351,
0.01934562623500824,
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0.0502864234149456,
-0.01497455220669508,
0.001595289446413517,
-0.0334429107606411,
0.05... |
fa1284674ad17f8d28e5514a93b32ebdbaf3bff0 | subsection | 28 | 98 | Proof of Theorem | As in Remark REF , for every i there
exists j_i such that \mathcal {Z}_{L^i}=\mathcal {Z}_{f_{j_i}}, whence \mathcal {K}_{L^i}=\mathcal {K}_{f_{j_i}}. If some L^i
is not similar to a hermitian monic pencil, then
{L}
is nontrivial and is invertible
on \operatorname{int}\mathcal {K}_{\widehat{L}} by
convexity of \mathcal... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.033106546849012375,
0.05696767196059227,
0.014775894582271576,
0.0026908605359494686,
0.002458199393004179,
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0.009329333901405334,
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0.008032533340156078,
-0.032831933349370956,
... |
95264ff7c2572df468c4bd9a82a456d4dd0379c5 | subsection | 29 | 98 | Finding an LMI representation for a convex | The main result of states that for
a hermitian matrix polynomial f\in \operatorname{M}_{{\delta }}(\mathop {<}\!x,x^*\!\mathop {>}) with f(0)\succ 0, the set
\mathcal {K}_f(n) is convex for all n if and only if \mathcal {K}_f is a free spectrahedron.
Actually, the version in does this for hermitian f with
bounded \math... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
0.011280768550932407,
0.010556175373494625,
-0.006582354661077261,
-0.019891982898116112,
0.026359928771853447,
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0.026070091873407364,
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0.03490249812602997,
0.029258299618959427,
-0.0224242452532053,
0.0028983717784285545,
-0.02066996693611145,
-0.... |
2f692c3df84bac1daea2d6b2b15e4c03f6442dbf | subsection | 30 | 98 | Algorithm | We next explain how the machinery developed in this paper produces an
explicit
minimal
LMI representation for a convex \mathcal {K}_f.
This efficient algorithm only involves linear
algebra and semidefinite programming (SDP) , .Compute the minimal realization
I+c^*L^{-1}{\bf b}
for f^{-1}.
To construct this realization... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.06003188341856003,
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0.03467016667127609,
0.015145308338105679,
0.042971476912498474,
0.04812927171587944,
-0.04474160820245743,
0.0016356551786884665,
0.015297906473279,
0.062107... |
6372960b514c7c438360595fe1a3bdf106dbfacf | subsection | 31 | 98 | Algorithm | Solve the following feasibility SDP:\begin{split}
\operatorname{tr}(\operatorname{Re}(D\widetilde{L})(0))&=1\\
\operatorname{Re}(D\widetilde{L})& = P_0 + \sum _k C_k^* L C_k
\quad \text{ for some } C_k,P_0, \text{ with }P_0\succeq 0.
\end{split}We note that (REF ) is a SDP.
Indeed, the first equation is simply a linear... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.02736601047217846,
0.04740592837333679,
-0.018620487302541733,
-0.005139712244272232,
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0.005731141660362482,
0.... |
5ca6b9fce1482c2d53e4f1a3a0a4a163fb3ddd33 | subsection | 32 | 98 | Algorithm | One first computes and mods out the radical of \mathcal {A} (corresponding to the \star entries) using the algorithm in ; then the
algorithm of is applied to find the irreducible blocks L^j. Alternatively, gives an algorithm for decomposing \mathcal {A} as a direct sum of minimal left ideals; after omitting the ideals ... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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-0.008086648769676685,
-0.004111984744668007,
0... |
a9e858fe131a0f6f4bd79598e8de92a0cf32961b | subsection | 33 | 98 | Algorithm | If \varepsilon ^{\prime }=\dim V>0, then let \widetilde{L}^{\prime } be the \delta \times \varepsilon ^{\prime } affine linear pencil whose coefficients are the restrictions of coefficients of \widetilde{L} to V. Then \widetilde{L} is of full rank on the interior of \mathcal {D}_L if and only if \widetilde{L}^{\prime }... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
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0.01232... |
c93293300a826348a2827a4ebf385e731a1e46f9 | subsection | 34 | 98 | Checking whether | As a side product of Theorem REF
and the Algorithm in Subsection REF
we obtain a procedure for checking whether \mathcal {K}_f is convex.Given f\in \operatorname{M}_{{\delta }}(\mathop {<}\!x,x^*\!\mathop {>}) with f(0)=I,
we construct the realization of f^{-1}
and identify its irreducible blocks L^i, choosing one fr... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.047156088054180145,
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... |
cafbd7f672e7b3dcf63a8a84a448876cb6f64c10 | subsection | 35 | 98 | Checking whether | Consequently \widetilde{L}(X,X^*) has full rank.Proposition 4.3
Let \delta \ge \varepsilon . If every solution of (REF ) satisfiesP_0=0,\quad C_k=0\quad \text{for all }k,then there exists X\in \operatorname{M}_{\max \lbrace d,\varepsilon \rbrace }(^g such that L(X,X^*)\succ 0 and \ker \widetilde{L}(X,X^*)\ne \lbrace 0... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.03118853271007538,
0.00004035178790218197,
-0.032287150621414185,
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0.004249514080584049,
0.00027322390815243125,
... |
5b6a5f0c4e633688da3b9b5ef2563bde91d9fc2f | subsection | 36 | 98 | Checking whether | Using the standard argument involving Caratheodory's theorem on convex hulls it is easy to show that \mathcal {C}+\mathcal {S} is closed in \mathcal {V}^{\rm h}_2; see e.g. .Lemma 4.4
Keep the notation from above. If every solution of (REF ) satisfiesP_0=0,\quad C_k=0\quad \text{for all }k,then \mathcal {U}\cap (\math... | {
"cite_spans": []
} | 10.1007/s10208-020-09465-w | 1808.06669 | Noncommutative polynomials describing convex sets | [
"J. W. Helton",
"I. Klep",
"S. McCullough",
"J. Volčič"
] | [
"math.FA",
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04986436292529106,
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0.030837170779705048,
-0.01942390762269497,
0.003122244495898485,
0.021727923303842545,
... |
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