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00ccd0b6cf984fffb259b4c7da6b2a97148aa69c | subsection | 6 | 29 | Paraquaternionic submersions | Following the analogue definition given in the quaternionic context (cf. and ), we introduce the following.Definition 3.1
Let (M,\sigma ) and (M^{\prime },\sigma ^{\prime }) be
almost paraquaternionic manifolds. A smooth map \pi :M\rightarrow M^{\prime } is said to be a (\sigma ,\sigma ^{\prime })–paraholomorphic map ... | {
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7125d7ba8946f87c6e6214ea35c986b9bc1a7b37 | subsection | 7 | 29 | Paraquaternionic submersions | We shall call \pi a paraquaternionic submersion if it is both a semi–Riemannian submersion and
a (\sigma ,\sigma ^{\prime })–paraholomorphic map.Proposition 3.5 Let (M,\sigma ,g) and (M^{\prime },\sigma ^{\prime },g^{\prime }) be almost paraquaternionic Hermitian manifolds, and \pi :M\rightarrow M^{\prime } a paraquate... | {
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92a5dc582da8e6b926e8aad332e7b287978c657a | subsection | 8 | 29 | Paraquaternionic submersions | Since (M,\sigma ,g) is a paraquaternionic Kähler manifold, by Remark REF ,
there exist 1-forms (\omega _a)_{a=1,2,3} on U, such that for each cyclic permutation (a,b,c) of (1,2,3):\nabla J_{a}=-\tau _{c}\omega _{c}\otimes J_{b}+\omega _{b}\otimes J_{c},where \tau _{1}=\tau _{2}=-1=-\tau _{3}. The map \pi :U\rightarrow ... | {
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4041698289a3cfac014896284d5f3eace37b7356 | subsection | 9 | 29 | Paraquaternionic submersions | Clearly, by (REF ) one has, for each X^{\prime },Y^{\prime }\in \Gamma (TU^{\prime }):(\nabla _{X^{\prime }}^{\prime }J_{a}^{\prime })(Y^{\prime })=-\tau _{c}\omega _{c}^{\prime }(X^{\prime })J^{\prime }_{b}Y^{\prime }+\omega _{b}^{\prime }(X^{\prime })J^{\prime }_{c}Y^{\prime }that is, by Proposition REF , the manifol... | {
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2432ad7765f160bd1725fc68eadf0a87590f2fbb | subsection | 10 | 29 | Paraquaternionic submersions | Then using v\circ J_{a}=J_{a}\circ v, by the definition of the O'Neill tensor field A, for any E,F\in \Gamma (\mathcal {H}), and any a\in \left\lbrace
1,2,3\right\rbrace one gets (A_{E}J_{a})(F))=v((\nabla _{E}J_{a})(F))=0, hence A_{E}(J_{a}F)=J_{a}(A_{E}F).
Using the anticommutativity of A on the horizontal distribut... | {
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f8a6c335f1b10fe6b56084d076359acbf6b50a33 | subsection | 11 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Definition 4.1 Let (M,\sigma ) be an almost paraquaternionic manifold, and \mathcal {D} a distribution on M. We say that the distribution \mathcal {D} is \sigma –invariant if, for any x\in M and any J\in \Sigma _{x}, one has J(\mathcal {D}_{x})\subset \mathcal {D}_{x}.It is easy to see that a distribution \mathcal {D} ... | {
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b48c83c675e404feb78f2fb8493f38f9b51ed982 | subsection | 12 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Given two open sets U_{1}\subset M_{1} and U_{2}\subset M_{2}, on which local bases (J^{(1)}_{a})_{a=1,2,3} and (J^{(2)}_{a})_{a=1,2,3} for \sigma _{1} and \sigma _{2}, respectively, are defined, then on U:=U_{1}\times U_{2}\subset M we define, for any a\in \lbrace 1,2,3\rbrace , a (1,1)–type tensor field J_{a} by putt... | {
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175e449d60402c702e8018bcf70960c554c9e961 | subsection | 13 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Furthermore, defining 1-forms (\omega _{a})_{a=1,2,3} on U by putting \omega _{a}(X):=\omega ^{(1)}_{a}(P_{1}X)+\omega ^{(2)}_{a}(P_{2}X), then the following identities hold, for any X,Y\in \Gamma (TU) and any cyclic permutation (a,b,c) of (1,2,3) (cf. ):2(\nabla _{X}J_{a})Y=-\tau _{c}\omega _{c}(X)J_{b}Y+\omega _{b}(X... | {
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a4de091bc05dc182d8cc0a5b66e4c40c3b65235a | subsection | 14 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Being U^{\prime } an open set of M, it is clear that (J^{\prime }_{a})_{a=1,2,3} is not only a local basis for \sigma ^{\prime }, but also a local basis for \sigma . Therefore, we may say that the structure \sigma ^{\prime } is spanned by the family of all the local bases (J^{\prime }_{a})_{a=1,2,3} for \sigma , define... | {
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85ba43e0dca9d2410a6dddc9acc659627b068f3c | subsection | 15 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Furthermore, being U_1\subset L^{(1)}_x and U_2\subset L^{(2)}_x, we have U^{\prime }_1=U^{\prime }\cap L^{(1)}_x and U^{\prime }_2=U^{\prime }\cap L^{(2)}_x, Thus, if we set J^{\prime }_{a}{}^{(1)}:=(J^{\prime }_a)|_{U^{\prime }_1} and J^{\prime }_{a}{}^{(2)}:=(J^{\prime }_a)|_{U^{\prime }_2}, for any a\in \lbrace 1,2... | {
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90c5254235b3ec29a57bf8cc7ee9bed8e615bdfc | subsection | 16 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | We find, in this way, that the local basis (J^{\prime }_{a})_{a=1,2,3}, chosen at the beginning, is of product type.Being (M,\sigma ,g) a paraquaternionic Kähler manifold, and (J^{\prime }_{a})_{a=1,2,3} a local basis for \sigma , there exist 1–forms (\omega _{a})_{a=1,2,3} on U^{\prime }, such that(\nabla _{X}J^{\prim... | {
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82640d58114a9d710b8db7d6c6593e7237e319b9 | subsection | 17 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Since (J^{\prime }_{a})_{a=1,2,3} is a structure of product type, if we define on U^{\prime } 1–forms (\bar{\omega }_{a})_{a=1,2,3} by putting, for any a\in \lbrace 1,2,3\rbrace and any X\in \Gamma (TU^{\prime }), \bar{\omega }_{a}(X):=\omega _{a}^{(1)}(P_{1}X)+\omega _{a}^{(2)}(P_{2}X), then, by (REF ), we have2(\nabl... | {
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... |
ce9c2401b31fa818d345a87e4dd34a8850744117 | subsection | 18 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Comparing (REF ) and (REF ), and decomposing X and Y along U^{\prime }_1 and U^{\prime }_2, we get\tau _{c}\omega _{c}(P_{1}X)J^{\prime }_{b}(P_{2}Y)+\tau _{c}\omega _{c}(P_{2}X)J^{\prime }_{b}(P_{1}Y)-\omega _{b}(P_{1}X)J^{\prime }_{c}(P_{2}Y)-\omega _{b}(P_{2}X)J^{\prime }_{c}(P_{1}Y)=0for any X,Y\in \Gamma (TU^{\pri... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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e50c4a4e9275ecc980dc38fadc2b5c0939a9e562 | subsection | 19 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Using the definition of J^{\prime }_{a}{}^{(1)} and J^{\prime }_{a}{}^{(2)}, we have\left\lbrace \begin{array}{l}
\tau _{c}\omega _{c}(P_{1}X)J^{\prime }_{b}{}^{(2)}-\omega _{b}(P_{1}X)J^{\prime }_{c}{}^{(2)}=0 \\
\tau _{c}\omega _{c}(P_{2}X)J^{\prime }_{b}{}^{(1)}-\omega _{b}(P_{2}X)J^{\prime }_{c}{}^{(1)}=0
\end{arra... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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b4973b0a40907b49716b6241bb08b01b57e96325 | subsection | 20 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | The integrability of both distributions is equivalent to the vanishing of the Nijenhuis tensor field N_F related to the structure F (see ), and in this case the tensor field F is called a product structure, or a locally product structure.An indefinite Riemannian almost product structure (cf. ) on M is a pair (g,F), whe... | {
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"doi": "",
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"raw": "F. Etayo and R. Santamaría, (J^2=\\pm 1)-metric manifolds, Publ. Math. Debrecen 57 (2000), no. 3-4, 435–444.",
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... | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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9159b8a32d897f863f47dc08405ed08e8c77da50 | subsection | 21 | 29 | Paraquaternionic Kähler structures and semi-Rieman-nian products | Since \nabla _X(F\circ J_a)(Y)=(\nabla _XF)(J_aY)+F(\nabla _XJ_a)(Y) and \nabla _X(J_a\circ F)(Y)=(\nabla _XJ_a)(FY)+J_a(\nabla _XF)(Y), using again the \sigma -invariance of F and (REF ), one has (\nabla _XF)(J_aY)=J_a(\nabla _XF)(Y), for any a\in \lbrace 1,2,3\rbrace , and any X,Y\in \Gamma (TU). If we suppose that (... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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e6af057124358d41da722e1e3c79891ad57b4ce2 | subsection | 22 | 29 | An example of paraquaternionic submersion | Let M be an m-dimensional manifold and (TM,\pi ,M) its tangent bundle. We recall the following basic properties of the vertical and horizontal lifts of vector fields, following , , and . If \nabla is a linear
connection on M, then we have, for any X,Y\in \Gamma (TM), and any \xi \in TM: \left[ X^{v},Y^{v}\right] _{\xi ... | {
"cite_spans": [
{
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"end": 187,
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... | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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ae56154f9a4b5ab8e9e1d7380d5f04d0096e1ac2 | subsection | 23 | 29 | An example of paraquaternionic submersion | Then the
canonical projection \pi :(TM,G)\rightarrow (M,g) is a semi–Riemannian
submersion with totally geodesic fibers; moreover, the horizontal
distribution \mathcal {H}TM is integrable if and only if the metric g is
flat, and in this case its integral manifolds are totally geodesic submanifolds of TM.Let (M,\sigma ,... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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61c513e71f0b1b961ca65c8d8bc6a1e2b25f8020 | subsection | 24 | 29 | An example of paraquaternionic submersion | If X\in \Gamma (T(U_{i}\cap U_{j})), using (REF ) and the above equality, we have \tilde{J}^{i}_{a}(X^{v})=((s_{ij})^{b}_{a}\circ \pi )\tilde{J}^{j}_{b}(X^{v}), for any a\in \lbrace 1,2,3\rbrace . Analogously, for X^{h}. Putting \tilde{s}_{ij}:=s_{ij}\circ \pi , we get a smooth map \tilde{s}_{ij}: \pi ^{-1}(U_{i})\cap ... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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d0c35abd2422758c20bdfdf4baa1a226981cf1ab | subsection | 25 | 29 | An example of paraquaternionic submersion | To this end, let us consider a point \xi \in TM. By the definition of the structure \tilde{\sigma } induced on (TM,G) from \sigma , there exists an open neighbourhood U of \pi (\xi ), on which a local basis (J_{a})_{a=1,2,3} for \sigma is defined, such that on \pi ^{-1}(U) the local basis (\tilde{J}_{a})_{a=1,2,3} for ... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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477ac3872c16dde6e083274ecc5e8efac07ad230 | subsection | 26 | 29 | An example of paraquaternionic submersion | Let us consider an open neighbourhood U\subset M of x, on which a local basis (J_{a})_{a=1,2,3} for \sigma is defined, and then let us take the local basis (\tilde{J}_{a})_{a=1,2,3} for \tilde{\sigma } on \pi ^{-1}(U), induced from (J_{a})_{a=1,2,3}. Being M a paraquaternionic Kähler manifold, there exist 1–forms (\ome... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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b86947eae867d401cbe74c11430a5fca61c4dd38 | subsection | 27 | 29 | An example of paraquaternionic submersion | Since g is flat, using (REF ) and (REF ), from (REF ) it follows:\begin{array}{l}
(\tilde{\nabla }_{X^{v}}\tilde{J}_{a })(Y^{v})_{\xi }=0, \qquad (\tilde{\nabla }_{X^{v}}\tilde{J}_{a })(Y^{h})_{\xi }=0,\\\\
(\tilde{\nabla }_{X^{h}}\tilde{J}_{a })(Y^{h})_{\xi }=-\tau _{c}\omega _{c}(X)_{\pi (\xi )}(\tilde{J}_{b})(Y^{h})... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1253,
"openalex_id": "",
"raw": "O. Kowalski, Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math. 250 (1971), 124–129.",
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manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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aacb4dd6d0aee058819056160820c50748166de6 | subsection | 28 | 29 | An example of paraquaternionic submersion | Conversely, suppose that (M,\sigma ,g) is locally hyper paraKähler, with g flat. Take \xi _{0}\in TM, and put \pi (\xi _{0})=x. Let us consider an open neighbourhood U\subset M of x, on which a local basis (J_{a})_{a=1,2,3} for \sigma is defined, and then let us take the local basis (\tilde{J}_{a})_{a=1,2,3} for \tilde... | {
"cite_spans": []
} | 10.1007/s10440-009-9549-7 | 0807.1824 | On paraquaternionic submersions between paraquaternionic K\"ahler
manifolds | [
"Angelo V. Caldarella"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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e7b5b67ce10c93f34b3681ec163071c06d149776 | abstract | 0 | 18 | Abstract | We prove an optimal Hardy inequality for the fractional Laplacian on the
half-space. | {
"cite_spans": []
} | 0807.1825 | The best constant in a fractional Hardy inequality | [
"Krzysztof Bogdan",
"Bartłomiej Dyda"
] | [
"math.AP",
"math.PR"
] | 2,008 | en | Mathematics | [
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eae1427c33d956c686882480ed3aaf953c8c50f6 | subsection | 1 | 18 | Main result and discussion | Let 0<\alpha <2 and d=1,2,\ldots .
The purpose of this note is to prove the following Hardy-type inequality in
the half-space D=\lbrace x=(x_1,\ldots ,x_d)\in {\mathbb {R}^d}:\,x_d>0\rbrace .Theorem 1
For every u\in C_c(D),\frac{1}{2}
\int _D \! \int _D
\frac{(u(x)-u(y))^2}{|x-y|^{d+\alpha }} \,dx\,dy
\ge {\kappa _{d,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1228,
"openalex_id": "",
"raw": "B. Dyda. A fractional order Hardy inequality. Ill. J. Math., 48(2):575–588, 2004.",
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"start": 680
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"arxiv_i... | 0807.1825 | The best constant in a fractional Hardy inequality | [
"Krzysztof Bogdan",
"Bartłomiej Dyda"
] | [
"math.AP",
"math.PR"
] | 2,008 | en | Mathematics | [
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ca51a10ed0cd873fc0c5be659dfca14dc43bd281 | subsection | 2 | 18 | Main result and discussion | For instance, in the proof of
Theorem REF we will use w(x)=x_d^{(\alpha -1)/2}.
Full details of (REF ), and a converse result are given in .
Recall that\mathcal {E}(u,v)=-(Lu,v)\,,\quad \mbox{ if } u\in Dom(\mathcal {L})\,,\;v\in Dom(\mathcal {E})\,,(, ). Therefore, equality holds
in (REF ) if u=w\in Dom(\mathcal {L}),... | {
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d4b73f5144c26f9266fed37eb6f709189db4da84 | subsection | 3 | 18 | Main result and discussion | If w\notin Dom(\mathcal {L}), or \mathcal {L}w does not belong to the
underlying L^2 space, then, as we shall see, the optimality of \nu =-\mathcal {L}w/w
critically depends on the choice of w.According to , the Dirichlet form of the
censored \alpha -stable process in D is\mathcal {C}(u,v)=\frac{1}{2}\mathcal {A}_{d,-\... | {
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a81042f59b894d5a62b2515d6b7eb34db564084b | subsection | 4 | 18 | Main result and discussion | Decomposing {\mathbb {R}^d}=D\cup D^c, one obtains\mathcal {K}(u,u)=\mathcal {C}(u,u)+\int _Du^2(x)\kappa _D(x)dx\,,\quad u\in C^\infty _c(D)\,,where (the density of the killing measure for D is)\kappa _D(x)=\int _{D^c}\mathcal {A}_{d,-\alpha }
|x-y|^{-d-\alpha } \,dy=
\frac{1}{\alpha }\mathcal {A}_{d,-\alpha }
\frac{\... | {
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9307cf30d3ef1323e79cb371f860430ea1decc1c | subsection | 5 | 18 | Main result and discussion | Theorem REF and the
results obtained to date for Laplacian and fractional Laplacian suggest
possible strengthenings to weights with additional
terms of lower-order boundary asymptotics (, , ),
and
extensions to other
specific or more general
domains (, ).
To discuss the latter problem, we consider open \Omega \subset D... | {
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044ef09b611acc879250a8ee5ce9088d60c2f855 | subsection | 6 | 18 | Main result and discussion | However, \mathcal {A}_{d,-\alpha }\kappa _{d,\alpha }\rightarrow 1/4 and
{\Gamma ^2(\frac{1+\alpha }{2})/\pi }\rightarrow 1/4 as
\alpha \rightarrow 2,
an agreement with the classical Hardy inequality for
Laplacian ()
related to the fact that for u\in C^\infty _c(D), \Delta _D^{\alpha /2}u\rightarrow \Delta u
and \mathc... | {
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d36eed744f9cc71bc45d19053f6306dd59a259d7 | subsection | 7 | 18 | Main result and discussion | For d\ge 2 we occasionally write x=(x^{\prime },x_d), where
x^{\prime }=(x_1,\ldots ,x_{d-1}), and we let \Vert x^{\prime }\Vert =\max _{k=1,\ldots ,d-1}
|x_k|, the supremum norm on \mathbb {R}^{d-1}.
[Proof of Theorem REF ]
For u, v\in C^\infty _c(D) we define (Dirichlet form)\mathcal {E}(u,v)=\frac{1}{2}
\int _D \!\i... | {
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602a475134d90c9e45e4386e3256f9ca1609c277 | subsection | 8 | 18 | Main result and discussion | By Lemma REF below, (REF ), and
(REF ), we obtain (REF ) for u\in C^\infty _c(D)\subset Dom(\mathcal {C}), with \kappa _{d,\alpha } given by (REF ).
The case of general u\in C_c(D) is obtained by an approximation.Since the setups of and are rather
complex, we like to give the following elementary proof of (REF ).
Let w... | {
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5a33275c5671574e885d37a80cf9f15fad6c5a46 | subsection | 9 | 18 | Main result and discussion | If \alpha \ge 1 then we consider functions v_n such that(i)
v_n=1 on [-n^2,n^2]^{d-1}\times [\frac{1}{n}, 1],
(ii)
\operatorname{supp}v_n \subset [-n^2-1,n^2+1]^{d-1}\times [\frac{1}{2n}, 2],
(iii)
0\le v_n \le 1,
|\nabla v_n(x)|\le cx_d^{-1} and
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"Krzysztof Bogdan",
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0f971c05d4b1e546c516b46a2a3f9c04b744b9f0 | subsection | 10 | 18 | Appendix | Lemma 2
For 0<\alpha <2,\gamma (\alpha ,\frac{\alpha -1}{2}) =
-\frac{1}{\alpha }\left[
B(\frac{1+\alpha }{2},\frac{2-\alpha }{2})2^{-\alpha }-1\right]\,.Since\gamma (\alpha ,p) = \int _0^1
\frac{t^p-t^{\alpha -1}-1+t^{\alpha -p-1}}{(1-t)^{1+\alpha }}\, dt\,,we are led to consideringB_\kappa (a,b)=\int _0^\kappa t^{a-... | {
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514fe405fbe17c5f6818211431a4c872e7596382 | subsection | 11 | 18 | Appendix | For \alpha \ne 1 we have,&&
B_\kappa (p+1,-\alpha )-B_\kappa (\alpha ,-\alpha )-B_\kappa (1,-\alpha )+B_\kappa (\alpha -p,-\alpha )=
\frac{1}{\alpha (\alpha -1)}\times \\
&&
\left\lbrace
(p+1-\alpha )(p+1-\alpha +1)B_\kappa (p+1,2-\alpha )
-(\alpha -\alpha )(\alpha -\alpha +1)B_\kappa (\alpha ,2-\alpha )\right.\\
&&\l... | {
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"Krzysztof Bogdan",
"Bartłomiej Dyda"
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b70f3b15c5e020b91f714beed15fc1ad5a97e584 | subsection | 12 | 18 | Appendix | For \alpha \ne 1 we get\gamma (\alpha ,p)&=&\frac{1}{\alpha (\alpha -1)}\left\lbrace
(p+1-\alpha )(p+2-\alpha )B(p+1,2-\alpha )\right.\\
&&\left.-(1-\alpha )(2-\alpha )B(1,2-\alpha )
+p(p-1)B(\alpha -p,2-\alpha )\right\rbrace \,.By the duplication formula
\Gamma (2z)=(2\pi )^{-1/2}\, 2^{2z-1/2}\, \Gamma (z)\, \Gamma (... | {
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} | 0807.1825 | The best constant in a fractional Hardy inequality | [
"Krzysztof Bogdan",
"Bartłomiej Dyda"
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3d8e1cc1cd3846f1ff022549d0cfc97e6df99ee0 | subsection | 13 | 18 | Appendix | If a\ge x_d/2 then\int _{D\setminus B(x,a)}
\frac{y_d^r}{|x-y|^{d+\alpha }}\,dy &\le &
c \sum _{k=0}^\infty \int _{D\cap B(x,2^ka, 2^{k+1}a)}
\frac{y_d^r}{(2^k a)^{d+\alpha }}\,dy \\&\le &
c^{\prime } \sum _{k=0}^\infty (2^ka)^{r-\alpha } = c^{\prime \prime }a^{r-\alpha }.If a<x_d/2 then\int _{D\cap B(x,a,x_d)}
\frac{y... | {
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"Krzysztof Bogdan",
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] | [
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6c877542c5be9cb9e4f6a88b96f39d8919835657 | subsection | 14 | 18 | Appendix | We thus haveI_1&=& \int _D \int _{B(x,\frac{1}{4n})}
\frac{(v(x)-v(y))^2}{|x-y|^{d+\alpha }}\,w(x)w(y)\,dy\,dx\\
&\le &
2\int _{K_n} \int _{B(x,\frac{1}{4n})}
\frac{(v(x)-v(y))^2}{|x-y|^{d+\alpha }}\,w(x)w(y)\,dy\,dx\\
&\le &
c \int _{K_n} \int _{B(x,\frac{1}{4n})}
\frac{x_d^{2{\bf p}-2}}{|x-y|^{d+\alpha -2}} \,dy\,dx ... | {
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"Krzysztof Bogdan",
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26fad08c5227c80003aecb475f972ce43ee13837 | subsection | 15 | 18 | Appendix | We obtainI_4&=&
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02dc1ba42d6244ae7f4c1d81a12043b858f2d201 | subsection | 16 | 18 | Appendix | We haveI_6 &=&
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\frac{(v(x)-v(y))^2}{|x-y|^{d+\alpha }}\,w(x)w(y)\,dy\,dx\\
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eeb822f8dde800307b5e07e85e2ec37071d59793 | subsection | 17 | 18 | Appendix | We haveI&=&\int \limits _D\int \limits _D \frac{(v(x)-v(y))^2}{|x-y|^{d+\alpha }}\,w(x)w(y)\,dx\,dy\\
&\le &
\int _D \int _{B(x,\frac{1}{4})}
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d3726b94a0d45b10864124f0d8e1b701cd172972 | abstract | 0 | 37 | Abstract | We introduce the notion of quantum duplicates of an (associative, unital)
algebra, motivated by the problem of constructing toy-models for quantizations
of certain configuration spaces in quantum mechanics. The proposed (algebraic)
model relies on the classification of factorization structures with a
two-dimensional fa... | {
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4a87d5a435fc49552be0794383e4acf211c333b0 | subsection | 1 | 37 | Introduction | Consider a manifold M representing some physical system. From a dual point of
view, this manifold can also be represented by some algebra of functions A
(that could be taken, for instance, to be A=C^\infty (M), the algebra of smooth
functions on M) over some base field k (usually k=\mathbb {R} or
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9c7f9699d7740925e8c0534c25b5e5d44c387cd6 | subsection | 2 | 37 | Introduction | In the particular case of algebras, a well known result (independently proven many times) establishes a one-to-one correspondence between the set of factorization structures admitting two given algebras A and B as factors and the set of so-called twisting maps, which are linear maps \tau :B\otimes A\rightarrow A\otimes... | {
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4905e0e90b8be90c7503480cc437e848678dcc24 | subsection | 3 | 37 | Introduction | Finally, the remaining case of quantum duplicates obtained using a quadratic field extension have similar properties to complexifications of real algebras, hinting the possibility of thinking about them as noncommutative scalar extensions.In Section we introduce the definition of quantum duplicates of an algebra A as t... | {
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20870dd6f83264dd5be4be891560a9b989ec11f1 | subsection | 4 | 37 | Introduction | An algebra X is a factorization structure of the algebras A and B if there exist two injective algebra maps i_{A} : A \hookrightarrow X and i_{B} : B \hookrightarrow X and the map \varphi : A \otimes B \rightarrow X defined by \varphi (a \otimes b) = i_{A}(a) \cdot i_{B}(b) is a linear isomorphism.A k-linear map \tau :... | {
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52456d74e985ae1fd7929c65bd284639ef85205b | subsection | 5 | 37 | Generalities about quantum duplicates | Let A and B be two
(unitary) k-algebras, with \mathrm {dim}_k B=2, so that we may consider it given as a quotient B=k[x]/(p(x)), where p(x) is a
polynomial of degree two.
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9386ce7b628bf7b4dfe60bb1269e0f559447ada5 | subsection | 6 | 37 | Basic definitions and properties | Our purpose is to describe the twisting maps between A and B,
that is, the k-linear maps\tau : k[x]/(p(x))\otimes A\longrightarrow A\otimes k[x]/(p(x))verifying the twisting conditions (REF ) and (REF ).
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87fe95ccb5d101f563bb9980a596e99b24213390 | subsection | 7 | 37 | Basic definitions and properties | Taking the linear transformation
\phi (x)=(\alpha /2) x+1, we obtain that p(\phi (x))=x^2+\gamma
and k[x]/(p(x))\cong k[x]/(x^2+\gamma ). Thus equations (REF ), () are rewritten as follows:\delta ^2=\gamma (f^2-id_A) \\
f\delta +\delta f=0 | {
"cite_spans": []
} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
] | [
"math.QA",
"math.RA",
"math.RT"
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fc6fda994039acc4e132edc325aae41e371d6aee | subsection | 8 | 37 | Characterization of certain quantum duplicates | When we have a real vector space V, we can construct the complexification of V, called V^\mathbb {C}, as the tensor product V\otimes _{\mathbb {R}} \mathbb {C} . The original vector space V remains a real vector subspace of V^\mathbb {C}, and can be recovered if we take advantage of the canonical conjugation map \chi :... | {
"cite_spans": []
} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
] | [
"math.QA",
"math.RA",
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d4bafcf2f35e0d587aaf77413578287929b58012 | subsection | 9 | 37 | Characterization of certain quantum duplicates | Thus b=\varphi (a_1\otimes 1+a_2\otimes \eta ) and \varphi is surjective.Let \alpha \in B\otimes C, we may write \alpha =a\otimes +b\otimes \eta with a,b\in A, and then \varphi (\alpha )=\varphi (a\otimes 1)+\varphi (b\otimes \eta )=a+b\cdot i. Assume \varphi (\alpha )=0, that is, a+b\cdot \eta =0. Applying \sigma , a-... | {
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{
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"raw": "A. C̆ap, H. Schichl, and J. Vanz̆ura, On twisted tensor products of algebras, Comm. Algebra 23 (1995), 4701–4735.",
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"start": ... | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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8cc7c9c6bf0ec26babe5b258466352ba5d6aab53 | subsection | 10 | 37 | Lifting of endomorphisms and involutions | Let A be an algebra, \varphi :A\rightarrow A an algebra map, and A\otimes _\tau B a quantum duplicate of A, with B=k[x]/(p(x)), induced by the couple (f,\delta ). The map \varphi admits a natural lifting \tilde{\varphi }:A\otimes _\tau B\rightarrow A\otimes _\tau B defined by \tilde{\varphi }(a\otimes b):=\varphi (a)\o... | {
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} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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"math.QA",
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8c5f8703b5ca313d947cbdf42cee1bd8198e3539 | subsection | 11 | 37 | Lifting of endomorphisms and involutions | An easy computation shows that if f is an algebra map, then so is \overline{f}, and if \delta is a left f-derivation, then the conjugate \overline{\delta } is a right \overline{f}-derivation.Any involution j defined on a twisted tensor product A\otimes _{\tau } B which is compatible with the ones existing in A and B mu... | {
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"start": 588
... | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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3525dfcdc11738fa65f8c17e2aa4a7a8e09038d9 | subsection | 12 | 37 | Quantum duplicates of | In this section we
describe and classify all quantum duplicates of k^{n} for
some natural number n \ge 2. Denote by \lbrace e_{1}, \ldots , e_{n}\rbrace
the canonical basis of k^{n}. Following Cibils' procedure , the set
of algebra morphisms f:k^{n} \rightarrow k^{n} is in one-to-one
correspondence with the set of set... | {
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} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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ec8e127439a7c7d98a69aab48f9c3ad2613c7352 | subsection | 13 | 37 | Quantum duplicates of | For instance, in the figure below we have a 3-cycle.(0,0)*+{\circ }="a1",(6,-11)*+{\circ }="a3",(20,-10)*+{\circ }="a2",
(-3,0)*+{i_1},(6,-14)*+{i_3},(23,-10)*+{i_2},
(-3,10)*+{\circ }="b",
(3,10)*+{\circ }="c",
(-6,20)*+{\circ }="d",
(-3,20)*+{\circ }="e",
(0,20)*+{\circ }="f",
(20,0)*+{\circ }="h",
(18,10)*+{\circ }=... | {
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} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
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9c31376a009acaa38818bed366ca9a215b19007d | subsection | 14 | 37 | Quantum duplicates of | Observe that, according to this nomenclature, an strict 1-cycle component is nothing more than a single vertex with a loop whilst
an strict 2-cycle is the round-trip quiver.Therefore we have the following result, see or :Lemma 2.2 For any algebra map f:k^n\rightarrow k^n, each connected component of the quiver Q_f is a... | {
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a672c16a4ecf42748ab578f8b0e5a5321a16e597 | subsection | 15 | 37 | Quantum duplicates of | It is easy to see
that the study of the possible colorations of the
quiver Q_f can be reduced to the study of each
connected component separately. Let us start by showing up the case of an strict cycle.Proposition 2.3 Let p(x)=x^2-\alpha x+\beta \in k[x] be a polynomial of degree two.
The set of twisting maps (f,\delta... | {
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} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
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"math.QA",
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f244398a6a97e00e24b3ccce8f10a36b31ae112d | subsection | 16 | 37 | Quantum duplicates of | Then (REF ) and () applied to the vertex e_i reduces to(a_{i}a_{j} - \beta )e_{i} - a_{j}(a_{j}+a_{i}+\alpha ) e_{j} + (a_{i}^{2} + \alpha a_{i} + \beta )e_{i} = 0 \\
(a_i+a_j+\alpha ) e_i- (a_i+a_j+\alpha ) e_j=0Clearly, these equations hold if and only if a_i+a_j=-\alpha .Let us now consider an strict s-cycle with s... | {
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} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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"math.QA",
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2e656030c96434c0590191aca22aaa833a1804ff | subsection | 17 | 37 | Quantum duplicates of | If Q_i is a strict 2-cycle, any valid coloration must satisfy
{*+[o][F-]
{
\text{\scriptsize $a$}} @/^4pt/@<0.5 ex>[r] & *+[o][F-]{\text{\scriptsize $b$}} @/^4pt/@<0.5 ex>[l]
} \hspace{14.22636pt} \text{where} \hspace{14.22636pt} a+b=-\alpha .
Connected components of this kind give rise to a one-parameter family of t... | {
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} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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"math.QA",
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fceb21c9ed66605ecd4bd7cf9ea3cb8117df2bdf | subsection | 18 | 37 | Quantum duplicates of | Now, let us consider an
arrow {*+[o][F-]{\text{\scriptsize $e_i$}} [r] & *+[o][F-]{\text{\scriptsize $e_j$}}} inside one of
the trees engaged to the loop vertex. Then () in the
component i provides us the equation a_i+a_j+\alpha =0. That is,
for one of these trees, if one level is colored by a root r_1, the
next level ... | {
"cite_spans": []
} | 10.1112/jlms/jdp055 | 0807.1826 | Factorization structures with a 2-dimensional factor | [
"Óscar Cortadellas",
"Javier López Peña",
"Gabriel Navarro"
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f05410a69a4c3fecf16f596c34420cfa238c8346 | subsection | 19 | 37 | Quantum duplicates of | A round-trip connected component is colored by
{*+[o][F-]{\text{\scriptsize $a$}}
@/^4pt/@<0.5 ex>[r]&
*+[o][F-]{\text{\scriptsize $b$}} @/^4pt/@<0.5 ex>[l]},
where a+b=-2s.Now we are going to describe the isomorphism classes of the algebras
that we have obtained. There is no loss of generality on assuming
that the qu... | {
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95c4dc1886ac8868a4e6fe6a857e767f668d2d5a | subsection | 20 | 37 | Quantum duplicates of | Let us denote the
elements of Q as follows:@C=40pt{
*+[o][F-]{\text{\scriptsize $u$}} @/^4pt/@<0.5ex>[r]^-{R} & *+[o][F-]{\text{\scriptsize $v$}} @/^4pt/@<0.5 ex>[l]^-{S}
}Then we consider the isomorphism of k-algebras \Phi : kQ_{<2}
\rightarrow k^{2} \otimes _{(f,\delta )}k[x]/(p(x)) given by:u\mapsto e_1 \qquad v\map... | {
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cb6aedace888c2a3173fcd4e5945d7d1469f20aa | subsection | 21 | 37 | Quantum duplicates of | For any vertex e_i in Q^{\mathrm {op}}, the twisting map given by (f, \delta ), say \tau , verifies that\tau (x\otimes e_i)=\delta (e_i)\otimes 1+f(e_i)\otimes x=\displaystyle \sum _{e_j}\epsilon _j(e_j\otimes 1)-\epsilon _i(e_i\otimes 1)+\displaystyle \sum _{e_j}e_j\otimes x,where the e_j's are the target of the arrow... | {
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1738bea8194ea5f563b855261e55d32280390b78 | subsection | 22 | 37 | Quantum duplicates of | \widehat{\Phi }({*+[o][F-]{\text{\scriptsize $1$}}})=e_0\otimes \left(\frac{x-r_1}{r_2-r_1}\right) and \widehat{\Phi }({*+[o][F-]{\text{\scriptsize $2$}}})=e_0\otimes \left(\frac{x-r_2}{r_1-r_2}\right), where
e_0 is the loop vertex in Q.
For any arrow \alpha _i that does not start neither at {*+[o][F-]{\text{\scriptsi... | {
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de4150d867a86e91e377e378a6930b1dcfa4dada | subsection | 23 | 37 | Quantum duplicates of | If \tau is given by a pair (f,\delta ),
the only possibilities for the quiver Q_f are:\begin{array}{clclcl}
(Q_1) &
{
*+[o][F-]{} @(ur,ul)[]&
*+[o][F-]{} @(ur,ul)[]&
*+[o][F-]{} @(ur,ul)[]} \qquad & (Q_2) &
{
*+[o][F-]{} [r] &
*+[o][F-]{} @(ur,ul)[]&
*+[o][F-]{} @(ur,ul)[]} \qquad & (Q_3) &
{
*+[o][F-]{} [r] &
*+[o][F-... | {
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3b15f0e1f3ad063f86f250a606577d47584c152e | subsection | 24 | 37 | Quantum duplicates of | Then
R\cong (k[x]/(p(x)))^3 or R\cong \mathcal {M}_2(k) \times k[x]/(p(x)). Otherwise, if Q_f\ne T, R is isomorphic to one of the (truncated) path
algebras of the opposite quivers of Q_1, Q_2, Q_3, Q_4, Q_5 ,Q_6, \widehat{Q_1}, \widehat{Q_2}, \widehat{Q_3}, \widehat{Q_5} or \widetilde{Q_5}. In case of Q_f=T, R depends ... | {
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4b2088fdf068c246032c978c2a1d3f2878b6a09c | subsection | 25 | 37 | Factorization structures of dimension 4 | The simplest nontrivial algebras that can be factorized as twisted tensor product ought to have factors of dimension at least 2, and thus the dimension of the product has to be greater or equal than 4. In the present Section, our purpose is to classify, up to isomorphism, all the algebras of dimension 4 that can be fac... | {
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db12e2e62ab6d555802357c0456c419f34f47c60 | subsection | 26 | 37 | Twisted tensor products of the form | Every twisted tensor product of the form k^2\otimes _\tau k^2 is isomorphic to one of the following algebras:The commutative algebra k^4.
The algebra of matrices \mathcal {M}_2(k).
The quotient kQ_{< 2} of the path algebra kQ of
the round-trip quiver
Q=
{ \circ @/^4pt/@<0.5ex>[r]& \circ @/^4pt/@<0.5 ex>[l]
}
The p... | {
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18863201ec13160177074478d6da5cd6f263f27f | subsection | 27 | 37 | Twisted tensor products of the form | If we consider the two copies of k[\xi ] respectively generated by x and y with x^2=y^2=0, and identify x and y with their images in the twisted tensor product k[\xi ]\otimes _\tau k[\xi ], the twisting map is given by yx = a + bx + cy + dxy, and imposing the twisting conditions we obtain a system of equations that can... | {
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0a42069c8e42e4b05edf1abd6efb7622b29c45f7 | subsection | 28 | 37 | Twisted tensor products of the form | The matrix algebra \mathcal {M}_2(k).For l a quadratic field extension of k, by Lemma REF , twisted tensor products of the form k[\xi ]\otimes _\tau l are given by couples (f,\delta ), being f an algebra endomorphism of l, and \delta and f–derivation such that\delta ^2 & = & 0 \\
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3761134ca50a7d8393c5460c4e0e5bb54e125384 | subsection | 29 | 37 | Twisted tensor products of the form | Thus, the only \sigma -derivations providing valid twisting maps are of the form \delta _q(\eta ) = q for some q\in k, and we get a 1-parameter family of twisting maps given by the couples (\sigma ,\delta _q), leading to the family of algebrasB_q := k\langle x,y|\ x^2=0, y^2 = \gamma , xy+yx = q\ranglewhere in order to... | {
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dd53a087267db59e9f7c69e26cde11f29d3ce293 | subsection | 30 | 37 | Twisted tensor products of the form | Then from the above equations we obtain b=0, yielding \delta (\eta )=q \in k, which gives us exactly the same family of algebrasB_q := k\langle x,y|\ x^2=0, y^2 = \gamma , xy+yx = q\ranglepreviously mentioned. Same proof as in Lemma REF tells us that B_q\cong \mathcal {M}_2(k) whenever q\ne 0, but in this case for q=0 ... | {
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9728c361a7bdecc7d69ea20feec13b25421337fe | subsection | 31 | 37 | Twisted tensor products of the form | As it happened in the last case, for fields of characteristic 2 weird phenomena may show up, so we will study them separately.So, take k such that \operatorname{char}k\ne 2, and assume (without loss of generality) that the field extensions l and l^{\prime } are given as splitting fields of the polynomials x^2 - \alpha ... | {
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e6f96fe9ed04b5e17b843eb5b83be8a00579e418 | subsection | 32 | 37 | Twisted tensor products of the form | In particular, the algebras C_q form a family of linked quaternion algebras.Take the isomorphism C_q\rightarrow {^{\alpha }}k^{t} given byx \mapsto i,\quad y \mapsto \frac{q}{2\alpha }i + ij,where i and j are the generators of ^{\alpha }k^{t}=k\langle i,j|\ i^2 = \alpha , j^2 = t, ij + ji = 0\rangle .
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15f7b5a02ff02ee4d7d5ef88b04fc72200fea4c2 | subsection | 33 | 37 | Twisted tensor products of the form | Applied to our concrete situation, and taking into account that for a field extension l=k(\sqrt{\alpha }) the norm map is given by N_{l/k}(x+y\sqrt{\alpha })=x^2 - \alpha y^2, we obtain the following result:Theorem 3.9 Let q,h\in k such that 4\alpha \beta -q\ne 0, 4\alpha \beta - h\ne 0.The algebras C_q and C_{h} are i... | {
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fd3c1b785dba4e4b3eb73bc684ea25f9d5915499 | subsection | 34 | 37 | Twisted tensor products of the form | In this case, we use the isomorphism C_{2\alpha }\otimes l \rightarrow lQ_{< 2} given byx\longmapsto \sqrt{\alpha }u - \sqrt{\alpha }v + R + S, \quad y\longmapsto \sqrt{\alpha }u - \sqrt{\alpha }v.For the family of central simple algebras (1), as previously mentioned, the number of isomorphism classes (or orbits of the... | {
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44d71f2da826d394188723dad1ff0aeecc9a1c68 | subsection | 35 | 37 | Twisted tensor products of the form | However, doing some computations (left to the reader) similar to the ones at the end of section REF , we obtain the following result:Theorem 3.11 Let k be a field with \operatorname{char}k=2, and let l and l^{\prime } be quadratic field extensions of k generated by polynomials p(x)=x^2+\alpha x + \beta and p^{\prime }(... | {
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aa6f44f70af59eb74950573a46d036c2ef18e425 | subsection | 36 | 37 | Twisted tensor products of the form | We reproduce the scheme given there, highlighting the “decomposable” algebras putting them into a box.@C=7pt@R=10pt{ & & & & & & & *+[F-]{\text{\scriptsize $k^4$}} [d] & \\
& & & & & & & k^2\times k[\xi ] [rd][ld]& \\
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b4f6546e7c1b8e3bed1027a5b12184f22ef9eb8f | abstract | 0 | 26 | Abstract | We show that for each \eta>0 every digraph G of sufficiently large order n is
Hamiltonian if its out- and indegree sequences d^+_1\le ... \le d^+_n and d^-
_1 \le ... \le d^-_n satisfy
(i) d^+_i \geq i+ \eta n or d^-_{n-i- \eta n} \geq n-i and
(ii) d^-_i \geq i+ \eta n or d^+_{n-i- \eta n} \geq n-i for all i < n/2.
... | {
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2d21cf34c10cc43ae4c5f7f89f1306d15ddf3650 | subsection | 1 | 26 | Introduction | Since it is unlikely that there is a characterization of all those graphs which
contain a Hamilton cycle it is natural to ask for sufficient conditions
which ensure Hamiltonicity. One of the most general of these is Chvátal's
theorem that characterizes all those degree sequences which
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33f3b5432f011d809237117c3e92e17bae75e79d | subsection | 2 | 26 | Introduction | Indeed, consider the digraph obtained from the complete digraph K
on n-2\ge 4 vertices by adding two new vertices v and w which both send an edge to
every vertex in K and receive an edge from one fixed vertex u\in K.The following example shows that the
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1e6b05f541b509e83d0330412f9976f5f4502550 | subsection | 3 | 26 | Introduction | The following approximate version of Conjecture REF
is an immediate consequence of Theorem REF .Corollary 4
For every \eta >0 there exists an integer n_0 =n_0 (\eta ) such that every digraph G
on n \ge n_0 vertices with d^+ _i, d^-_i \ge i+ \eta n for all i < n/2
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c0d2cd5e6f444148ce682ac9a1109befd100e42e | subsection | 4 | 26 | Introduction | Note that Conjecture REF is a weakening of the following conjecture of Kelly (see e.g. , , ).Conjecture 7 (Kelly)
Every regular tournament on n vertices can be decomposed
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6864ee7360d5f749d8fa800b83f7d000414405b4 | subsection | 5 | 26 | Extremal examples for Conjecture | The example given in the introduction does not quite imply that Conjecture REF
would be best possible, as for some k it violates both (i) and (ii) for i=k.
Here is a slightly more complicated example which only violates one of the conditions
for i=k (unless n is odd and k=\lfloor n/2 \rfloor ).Suppose
n\ge 5 and 1\le ... | {
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5938a26db1e45770fc6b21affbf51519b9f97dd5 | subsection | 6 | 26 | Extremal examples for Conjecture | However, it turns out that it makes sense to replace the strong connectivity assumption with an additional
degree condition (condition (iii) below). If true, the following conjecture would provide
the desired characterization.Conjecture 8
Suppose that G is a digraph on n \ge 3 vertices
such that for all i < n/2(i)
d^+... | {
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"Daniela Kühn",
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f009b258a0c23cf36077a2fce30a8280b6d24031 | subsection | 7 | 26 | Notation and the proof of Corollary | We begin this section with some notation.
Given two vertices x and y of a digraph G, we write xy for the edge directed from x to y.
The order |G| of G is the number of its vertices.
We denote by N^+ _G (x) and N^- _G (x) the out- and the inneighbourhood of x
and by d^+_G(x) and d^-_G(x) its out- and indegree.
We will w... | {
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} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
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cff9945016820c0d8f290e4e1f82d4d7e5a410bb | subsection | 8 | 26 | Notation and the proof of Corollary | Thus at least n-i=n^{\prime }+1-i vertices in G^{\prime } have outdegree at least s-1 and so
d^+ _{i,G^{\prime }} \ge s-1. Thus for all i < n/2 the degree sequences of G^{\prime }
satisfyd^+ _{i,G^{\prime }} \ge i+ \eta n -1 or d^- _{n-i- \eta n, G^{\prime }} \ge n-i-1 ,
d^- _{i,G^{\prime }} \ge i+ \eta n -1 or d^+ _... | {
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"Daniela Kühn",
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5577ee8102ac80e95878b7897b1b96d65a77ee07 | subsection | 9 | 26 | Notation and the proof of Corollary | Add a new vertex x which sends an edge to all vertices in V_1 and receives
an edge from all vertices in K. Add all possible edges from V_i to V_{i+1}
(but no edges from V_{i+1} to V_i) for each i \le k-3.
Finally, add all possible edges going from vertices in K to other vertices
and add all edges from V_{k-2} to K.
The... | {
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"Daniela Kühn",
"Deryk Osthus",
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12b89c151b20e03b81b8a7fbcadf44b4e3b2a899 | subsection | 10 | 26 | Degree sequences for Hamilton cycles in oriented graphs | In Section we mentioned Ghouila-Houri's theorem which gives a bound on the
minimum semi-degree of a digraph G guaranteeing a Hamilton cycle. A natural question raised
by Thomassen is that of determining the minimum
semi-degree which ensures a Hamilton cycle
in an oriented graph. Häggkvist conjectured that every orie... | {
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9b8ee12be254e8dbb5a6f5e2b011a20625258f89 | subsection | 11 | 26 | Degree sequences for Hamilton cycles in oriented graphs | Furthermore, currently, d^+ _G (a)=n/4-1, d^- _G (a) =n/2 +1, d^+ _G (d)=n/2 and
d^- _G (d) =n/4-1 for all a \in A and all d \in D.Partition A into A^{\prime } and A^{\prime \prime } where |A^{\prime \prime }|=c and thus |A^{\prime }|=n/4-c. Write
A^{\prime }=:\lbrace x_1,x_2, \dots , x_{n/8-c/2},y_1,y_2, \dots , y_{n/... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
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c5dc3e4631323a3c1386319f88e6be9faf13f1a0 | subsection | 12 | 26 | Degree sequences for Hamilton cycles in oriented graphs | Thus, d^+ _G (a^{\prime }) \ge (n/4-1)+(n/8-c/2-1)+c/2+2=3n/8 for all a^{\prime } \in A^{\prime } andd^+ _G (a^{\prime \prime }) \ge (n/4-1)+(n/8-c/2-s)=3n/8 -c/2-n/(2c)+1 \ge \alpha nfor all a^{\prime \prime } \in A^{\prime \prime }.Partitioning D into D^{\prime } and D^{\prime \prime } (where |D^{\prime \prime }|=c) ... | {
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} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
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6b4020646c26a3bc1cf1fa15f2dfae9b6d9248fd | subsection | 13 | 26 | The Diregularity lemma and other tools | In the proof of Theorem REF we will use the directed version of
Szemerédi's Regularity lemma. Before we can state it we need some more definitions.
The density of an undirected bipartite graph G=(A,B) with vertex classes A and B is defined to bed_G (A,B):=\frac{e_G(A,B)}{|A||B|}.We will write d(A,B) if this is unambigu... | {
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... | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
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"math.CO"
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5af3447b049b15f8e23bc5d57cac0ea668743e23 | subsection | 14 | 26 | The Diregularity lemma and other tools | The reduced digraph R of G with parameters \varepsilon , d and M^{\prime } is the digraph whose
vertices are V_1, \dots , V_k and in which V_i V_j is an edge precisely when (V_i,V_j)_{G^{\prime }}
is \varepsilon -regular and has density at least d.Given 0<\nu \le \tau <1, we call a digraph G a (\nu ,\tau )-outexpander ... | {
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90ccae9bf1e990cdbc7198963e024d944fc38be5 | subsection | 15 | 26 | The Diregularity lemma and other tools | Let G be a digraph on n\ge n_0 vertices with(i)
d^{+} _i\ge i + \eta n or d^{-} _{n-i-\eta n} \ge n-i,
(ii)
d^{-} _i\ge i + \eta n or d^{+} _{n-i-\eta n} \ge n-ifor all i <n/2. Then \delta ^0(G)\ge \eta n and G is a robust (\tau ^2,\tau )-outexpander.Proof. Clearly, if d^+_1\ge 1+\eta n then \delta ^+(G)\ge \eta n.
If ... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
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253a8a8aeb3fb016fcc91c56294d113403f28a49 | subsection | 16 | 26 | The Diregularity lemma and other tools | If |RN^+_{\tau ^2,G}(S)|< |S|+2\tau ^2 n
then V(G)\setminus RN^+_{\tau ^2,G}(S) contains such a vertex x.
But then x has at least \tau ^2 n neighbours in S, i.e. x\in RN^+_{\tau ^2,G}(S), a contradiction.If |S|= n/2+\lfloor \tau n\rfloor then considering the outneighbourhood of a subset of S of
size |S|-1 shows that |R... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
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27ce3a173b473de1200b9a7ad1bdc469373cbfaf | subsection | 17 | 26 | The Diregularity lemma and other tools | Since |N^-_{G^{\prime }}(x)\cap S^{\prime }|\ge |N^-_{G}(x)\cap S^{\prime }|-(d+\varepsilon )n
\ge \nu n/2 for every x\in RN^+_{\nu ,G}(S^{\prime }) this implies that|RN^+_{\nu /2,G^{\prime }}(S^{\prime })|\ge |RN^+_{\nu ,G}(S^{\prime })|\ge |S^{\prime }|+\nu n\ge |S|m+\nu mk.However, in G^{\prime } every vertex x\in R... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
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8a0fd9a6cf7b4dde45e058a1c212f9ebf4ab1838 | subsection | 18 | 26 | The Diregularity lemma and other tools | Given a vertex x of R, we write N^{\pm }_R(x) for the set of all those vertices of R
which are both out- and inneighbours of x and define N^{\pm }_H(x) similarly.
Let H^*:=H\cap R^*. Clearly, d^+_{H^*}(x), d^-_{H^*}(x)\ge \eta n/4
if |N^{\pm }_H(x)|\le 3\eta n/4. So suppose that |N^{\pm }_H(x)|\ge 3\eta n/4. Let
X:=|N^... | {
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"Daniela Kühn",
"Deryk Osthus",
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4b80aca70b0ca7a5989abce5fe55b5bf0a5f3f96 | subsection | 19 | 26 | The Diregularity lemma and other tools | So as before, a Chernoff estimate gives\mathbb {P}(x \text{ fails})\le \mathbb {P}(|N^-_{R^*}(x)\cap N^{\pm }_R(x)\cap S|<\nu n/12)\le 2{\rm e}^{-c\nu n/8}=:p.Let Y be the number of all those vertices x\in ERN^{\pm }_R(S) which fail.
Then \mathbb {E}Y \le p|ERN^{\pm }_R(S)| \le pn.
Note that the failure of distinct ver... | {
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{
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"openalex_id": "",
"raw": "N. Alon and J. Spencer, The Probabilistic Method (2nd edition), Wiley-Interscience 2000.",
"source_ref_id": "e66471fa0be4ad38e10edbff69f2f8c139814497",
"start": 432
}
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} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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df1c832c8ab683402fc59a56134ec4b56602a68d | subsection | 20 | 26 | The Diregularity lemma and other tools | But since S_2 is good this implies that all but at most \nu n/6 vertices in S_1\cap N
are contained in ERN^+_{\nu /12,R^*}(S_2)\subseteq RN^+_{\nu /12,R^*}(S).
Similarly, since S_1 is good, all but at most \nu n/6 vertices in S_2\cap N
are contained in ERN^+_{\nu /12,R^*}(S_1)\subseteq RN^+_{\nu /12,R^*}(S).
Altogether... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
] | [
"math.CO"
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e5939c1bc282a535dc6944b0562044505d3c60d6 | subsection | 21 | 26 | Proof of Theorem | As indicated in Section , instead of proving Theorem REF directly,
we will prove the following stronger result. It immediately implies Theorem REF
since by Lemma REF any digraph G as in Theorem REF is a robust outexpander
and satisfies \delta ^0(G)\ge \eta n.Theorem 16
Let n_0 be a positive integer and \nu ,\tau ,\et... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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