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33eb6fff4beeb69658f6f438af46e2d0014f0672 | subsection | 13 | 40 | Comparison with the cyclic module of the algebra | Let \mathsf {H} be a cocommutative Hopf algebra.
In this section, we construct an injective cyclic module map
\tau :\mathrm {B}(\mathsf {H})\rightarrow C(\mathsf {H}), from the cyclic bar complex
\mathrm {B}(\mathsf {H}) of REF to the canonical cyclic k-module C(\mathsf {H}) of the algebra underlying \mathsf {H} (),
an... | {
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edf4c35422b7d678e32430fd642a1deb131e3594 | subsection | 14 | 40 | A natural section of the projection | Recall from Definition REF that \mathrm {M}^{\prime }=\mathrm {M}^{\prime }(\mathsf {H}) is a mixed
complex whose underlying chain complex is
HH(\mathrm {M}^{\prime })\!=\!(\mathrm {E}(\mathsf {H})_\mathrm {norm},\partial ^{\prime }), and write \pi ^{\prime } for the
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0c67c6f96fb44f96e82e6022d829d25e706c69d2 | subsection | 15 | 40 | A natural section of the projection | \quadThere is a sequence of \mathsf {H}-linear maps
\Upsilon ^{\prime n}:\mathrm {E}(\mathsf {H})\rightarrow \mathrm {E}(\mathsf {H})[2n], starting with \Upsilon ^{\prime 0}=1,
such that B^{\prime }(\Upsilon ^{\prime n}\,\partial ^{\prime }-\partial ^{\prime }\Upsilon ^{\prime n})=0.They induce maps on the normalized c... | {
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dad7dab2816488520ffd7b51ed091cd4289ab3ef | subsection | 16 | 40 | A natural section of the projection | By naturality of \kappa ^f in f, the
above construction also goes through with \mathrm {E}(\mathsf {H}) replaced by M^{\prime }(\mathsf {H}),
and the maps \Upsilon ^{\prime } on \mathrm {E}(\mathsf {H}) and HH(M^{\prime }) are compatible.We define maps \Upsilon ^{\prime }:HH(\mathrm {E}(\mathsf {H}))\rightarrow HN(\mat... | {
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a22584758a6f22bd9104c03869faa212173cc769 | subsection | 17 | 40 | A natural section of the projection | Thus \Upsilon ^{\prime }(1)=(0,\dots ,0,1) in HN(M^{\prime }).Recall from Definition REF that M=k\otimes _{\mathsf {H}}M^{\prime },
and that \mathrm {B}(A)=k\otimes _A\mathrm {E}(A).There are morphisms of chain k-modules,
\Upsilon \!:HH(\mathrm {B}(\mathsf {H}))\rightarrow HN(\mathrm {B}(\mathsf {H})) and
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810d25535f456f6ec11d28e8154227cf1212d91f | subsection | 18 | 40 | The lift | Recall that C(\mathsf {H}) denotes the canonical cyclic complex of the algebra
underlying \mathsf {H} (). We set \tau _0=\eta :k\rightarrow \mathsf {H}.Let \mathsf {H} be a cocommutative Hopf algebra. Then the k-linear map\tau :\mathrm {B}(\mathsf {H})\rightarrow C(\mathsf {H})\\
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6dd5c2b3576ef88dd05684f826334110659a91f6 | subsection | 19 | 40 | Passage to completion | If A is a filtered algebra, the induced filtration (REF )
on the canonical cyclic module C(A) makes it a cyclic filtered module.
Passing to completion we obtain a cyclic module C^{\rm top}(\hat{A}) with
C^{\rm top}_n(\hat{A})=\hat{A}^{\hat{\otimes }n+1}. In the spirit of subsection
REF , we write HH^{\rm top}(\hat{A}),... | {
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dafb5972567a11336432a3b6e16c9ae1258c4286 | subsection | 20 | 40 | The case of universal enveloping algebras of Lie algebras | Let \mathfrak {g} be a Lie algebra over a commutative ring k. Then the
enveloping algebra U\mathfrak {g} is a cocommutative Hopf algebra, so
the constructions of the previous sections apply to U\mathfrak {g}. In
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(REF... | {
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36e66b68f56dc73a598acead7ee3a7dea3550e7a | subsection | 21 | 40 | Chevalley-Eilenberg complex | The Chevalley-Eilenberg resolution of k as a U\mathfrak {g}-module has the
form (U\mathfrak {g}\otimes \wedge \mathfrak {g},d^{\prime }), and is given in .
Tensoring it over U\mathfrak {g} with k, we obtain a complex
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af134f86ba1dfccabde643dcb65c2f513cd5a98b | subsection | 22 | 40 | Chevalley-Eilenberg complex | Thus
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8eb5e4e91f849f320d0dd7305af9a510f48b5371 | subsection | 23 | 40 | Chevalley-Eilenberg complex | By definition (see REF ),\tau =(S\otimes 1^{\otimes n})\circ (\nabla ^{(n)}\otimes 1^{\otimes n})\circ \sigma \circ \Delta ^{\otimes n}in C(U\mathfrak {g}). Since the x_i are primitive,\Delta ^{\otimes n}(x_1\otimes \dots \otimes x_n)=(x_1\otimes 1+
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220fa0c1bba9ee7f3402238780dbbc2cd14c6da1 | subsection | 24 | 40 | The Loday-Quillen map | We can now show that \tau \,\psi factors through the Connes' complex C^\lambda (U\mathfrak {g})=\operatorname{coker}(1-t:U^{\otimes \ast }\rightarrow HH(U\mathfrak {g})).
We have a homomorphism \theta which lifts
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d9746e5bf21e9431db9ac39e30caaf964e7299a2 | subsection | 25 | 40 | The Loday-Quillen map | The following result is implicit in the proof of
for r=1, A=U\mathfrak {g}.\theta is a chain homomorphism \wedge \mathfrak {g}\rightarrow C^\lambda (U\mathfrak {g})[-1].To show that b\theta =-\theta d, we fix a
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d855c62131bbb29c42311ef90c0eac9cffdc24e6 | subsection | 26 | 40 | The Loday-Quillen map | Therefore
b\theta (x_0\wedge \dots \wedge x_n)=-\theta d(x_0\wedge \dots \wedge x_n).Warning:
the sign convention used for d in
differs by -1 from the usual convention, used here and in
and .It is well known and easy to see that, because Connes' operator B
vanishes on the image of 1-t, it induces a chain map
for ever... | {
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02afd77387840ad11386c9aae216160875b5bbeb | subsection | 27 | 40 | The Loday-Quillen map | We may interpret \theta (x_1\wedge \dots \wedge x_n) as the
shuffle product x_1\star B(x_2)\star \dots \star B(x_n)
(see ), and
the nonzero coordinate of (REF ) as
the shuffle product(1\otimes x_1)\star \dots \star (1\otimes x_n)
=B(x_1)\star \dots \star B(x_n) \qquad \qquad \\
~=~ B\bigl (x_1\star B(x_2)\star \dots \s... | {
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d4858b3ca8d93fb9304d10e8f6ba1b9812737f6e | subsection | 28 | 40 | Nilpotent Lie algebras and nilpotent groups | In this and the remaining sections we shall fix the ground ring k=\mathbb {Q}.
Let \mathfrak {g} be a nilpotent Lie algebra;
consider the completion \hat{U}\mathfrak {g} of its enveloping algebra with
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176abc85de04762da39ce31ced83b566dd96a794 | subsection | 29 | 40 | Nilpotent Lie algebras and nilpotent groups | Thus it suffices
to show that the composite \mathrm {E}(\mathbb {Q}[G])\rightarrow \mathrm {E}^{\rm top}(\hat{\mathsf {H}}) is
naturally homotopic to the map induced by the homomorphism
\mathbb {Q}[G]\rightarrow \hat{\mathsf {H}}. By definition, these maps agree on \mathrm {E}_0(\mathbb {Q}[G]);
thus their difference g... | {
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3414d639ecda8f1544b6f5ee2c97cece07048ef2 | subsection | 30 | 40 | Nilpotent Lie algebras and nilpotent groups | Consider the resulting mapj:HN(\mathbb {Q}[G],\mathcal {I}_G)\rightarrow HN(\hat{U}\mathfrak {g},\hat{\mathcal {I}})\rightarrow HN(A,J).Putting together Theorem REF with Proposition ,
we get a naturally homotopy commutative diagram (with G=(1+J)^\times ):{
\mathrm {B}(\mathbb {Q}[G],\mathcal {I}_G)[r]^-{c}[d]_{sw}& HN(... | {
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c4617fbb76d201b2cd2d190a0a1ecc495f63ca6c | subsection | 31 | 40 | The relative Chern character of a nilpotent ideal | In this section we establish Theorem REF , promised in
(REF ), that the two definitions (REF ) and
(REF ) of the Chern character K_*(A,I)\rightarrow HN_*(A,I)
agree for a nilpotent ideal I in a unital \mathbb {Q}-algebra A.
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to the construction of the ... | {
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6631b98ba4486b38c70351ce86857545f2d937aa | subsection | 32 | 40 | The absolute Chern character | Let A be a unital \mathbb {Q}-algebra,
and \mathrm {B}\mathrm {GL}(A) the classifying space of GL(A).
Now the plus construction \mathrm {B}\mathrm {GL}(A)\rightarrow \mathrm {B}\mathrm {GL}(A)^+ is a homology
isomorphism, and K_n(A)=\pi _n\mathrm {B}\mathrm {GL}(A)^+ for n\ge 1. In particular,
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bddd95ebafe68a238e792faeff985e00b7256350 | subsection | 33 | 40 | The absolute Chern character | Composing this with the maps HN(A)\rightarrow HC(A)[2n] yields the map
ch_{0,n}: K_0(A)\rightarrow HC_{2n}(A) of .
From Example REF above, with A=k, we see that
ch([k])=c(1), and ch_{0,n}([k])=(-1)^n(2n)!/n! in HC_{2n}(k)\cong k,
in accordance with . | {
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ad938db4039e52ee90c791386656f8a71ff12918 | subsection | 34 | 40 | Volodin models for the relative Chern character
of nilpotent ideals | In order to define the relative version ch_* of the absolute
Chern character, we need to recall a chunk of notation about Volodin models.
For expositional simplicity, we
shall assume that I is a nilpotent ideal in a unital \mathbb {Q}-algebra A.Let I a nilpotent ideal in a \mathbb {Q}-algebra A, and \sigma a
partial or... | {
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6f0c517a0ddd14e4db9d3ecf8bf68d37d1af6196 | subsection | 35 | 40 | Volodin models for the relative Chern character
of nilpotent ideals | On the other hand, by Example , we also have a mapC(\mathbb {Q}[G],\mathcal {I}_{G}) \ {j \over \longrightarrow }\ C(G), \quad \text{for~} G=\mathcal {T}_n^\sigma (A,I).From the definition of ch^-_A in (REF )
and the naturality of c,
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39d2b6521cf116ad07d335b9da6ef9a07d85b116 | subsection | 36 | 40 | The relative Chern character for rational nilpotent ideals | When I is a nilpotent ideal in an algebra A, we define
K(A,I) to be the homotopy fiber of
B\mathrm {GL}(A)^+\rightarrow B\mathrm {GL}(A/I)^+; K(A,I) is a connected space whose homotopy groups are the relative K-groups K_n(A,I) for all n.
We now cite Theorem 6.1 of for nilpotent I; the proof in
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68bb2a614b1920cb080660e07855adaf36424df4 | subsection | 37 | 40 | The rational homotopy theory character for nilpotent ideals | For a nilpotent ideal I, consider the chain subcomplex of the
Chevalley-Eilenberg complex \wedge \mathfrak {gl}(A),x(A,I)=\sum _{n,\sigma } \wedge \mathfrak {t}_n^\sigma (A,I).Because sw is natural in G, the family of maps
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... | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
"Charles Weibel"
] | [
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882a4fa99fea6cfe09767e6ead092baeb5d4f48f | subsection | 38 | 40 | Main theorem | Let I be a nilpotent ideal in a \mathbb {Q}-algebra A. The relative Chern
character ch of Definition REF induces the relative Chern
character ch_* of (REF ) on homotopy groups, and the
rational homotopy character ch^{\prime } of Definition REF induces
the character ch^{\prime }_* of (REF ) on homotopy groups.
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} | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
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71043cff195759957fc15d3a74c2c584820a36fe | subsection | 39 | 40 | Naturality | In order to formulate a naturality result for the homotopy between
ch and ch^{\prime }, it is necessary to give definitions for the maps ch and
ch^{\prime } which are natural in A and I.
Contemplation of Definitions REF and REF shows
that we need to find a natural inverse for the backwards quasi-isomorphism
of Theorem ... | {
"cite_spans": []
} | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
"Charles Weibel"
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974c2ea7c71e6aa34b9b36234ff4a5de10e1c0c6 | abstract | 0 | 9 | Abstract | Let $\Omega$ be a two-dimensional heat conduction body. We consider the
problem of determining the heat source $F(x,t)=\varphi(t)f(x,y)$ with $\varphi$
be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed.
By a specific form of Fourier transforms, we shall show that the heat source is
determine... | {
"cite_spans": []
} | 0807.1812 | Determine the source term of a two-dimensional heat equation | [
"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
] | [
"math.AP"
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e85f1edeacf81c12903356463e5487d25a724884 | subsection | 1 | 9 | Body | Determine the source term of a two-dimensional heat equation
DANG DUC TRONG ^1, TRUONG TRUNG
TUYEN ^2,PHAN THANH NAM ^1 and ALAIN PHAM NGOC DINH ^3^1 Mathematics Department, Natural Science HoChiMinh City University, Viet Nam^2 Mathematics Department, Indiana University, Rawles Hall , Bloomington, IN 47405^3 Mathemati... | {
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"Dang Duc Trong",
"Truong Trung Tuyen",
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ce3c0c1e9aa2772f09de1d7c2f34a13c2696675d | subsection | 2 | 9 | Body | If (u_i,f_i) (i=1,2) satisfy the system (1) with g_0,g_1\in L^2(\Omega ) and \varphi \in L^1 (0,1)\backslash \lbrace 0\rbrace thenWe also have a regularization result. Using the Tikhonov regularization and truncated integration, we can construct a regularized solution for all \varphi \lnot \equiv 0.Theorem 2 (Regulariz... | {
"cite_spans": []
} | 0807.1812 | Determine the source term of a two-dimensional heat equation | [
"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
] | [
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8674f1a5db40bec3254504ed6ac9dc49d6b2a739 | subsection | 3 | 9 | Body | Let g_{\varepsilon }=(g_{0\varepsilon },g_{1\varepsilon })\in (L^2(\Omega ))^2 and \varphi _{\varepsilon }\in L^1 (0,1) be measured data satisfyingF̉rom \lbrace g_{0\varepsilon },g_{1\varepsilon },\varphi _{\varepsilon }\rbrace , we can construct a regularized solution f_{2\varepsilon }\in L^2(R^2) such thatM̉oreover, ... | {
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e9a27c152169290e33c833e1082153daee643df3 | subsection | 4 | 9 | Body | Hence, for each n\in Z, the functionis also a nontrivial entire function.(i) For each n\in Z, since the zeros set of \phi _n is either finite or countable, D(\varphi )(\alpha ,n)=\phi _n(\alpha )\ne 0 for a.e \alpha \in R. Hence D(\varphi )(\alpha ,n)\ne 0 for all n\in Z and for a.e \alpha \in R.T̉o estimate the measur... | {
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"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
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c8e276cf393f57d09282176ecdc3dd5d66eb3455 | subsection | 5 | 9 | Body | Thusfor \varepsilon >0 small enough (depended on \varphi , q and \beta ).(ii) Note thatTherefore, there exists a constant R_1>0 (depended on \varphi , \lambda ) satisfying for either |\alpha |\ge R_1 or |n|\ge R_1 thatConsequently, for \varepsilon >0 small enough (depended on \varphi , q and q_1) and for all (\alpha ,n... | {
"cite_spans": []
} | 0807.1812 | Determine the source term of a two-dimensional heat equation | [
"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
] | [
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ea50504d8c60d84b64f288e20c613b716a90e1c7 | subsection | 6 | 9 | Body | Moreover, if w\in H^1(\Omega ) thenFor each w\in L^2(\Omega ), applying Lemma 4 we obtainIt implies that \mathop {\lim }\limits _{r \rightarrow + \infty } \mu (w,r) = 0.N̉ow, we consider w\in H^1(\Omega ). Sincewe getConsequently,Similarly, we haveTherefore,HenceNoting thatwe getandThusIn summary, we getT̉he proof is c... | {
"cite_spans": []
} | 0807.1812 | Determine the source term of a two-dimensional heat equation | [
"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
] | [
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92c298d71766a1bbf9bb80690f95c92bc59670a2 | subsection | 7 | 9 | Body | We haveWe getand similarly,Thus,We next considerIf \left| {D(\varphi _{ex} )} \right| \ge \varepsilon ^q thenIf \left| {D(\varphi _{ex} )} \right|<\varepsilon ^q thenwhere B(\varphi _{ex} ,R_\varepsilon ,\varepsilon ^q) is as in Definition 1.Ỉn summary, for all (\alpha ,n)\in R \times Z , we haveTo estimate \sum \limit... | {
"cite_spans": []
} | 0807.1812 | Determine the source term of a two-dimensional heat equation | [
"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
] | [
"math.AP"
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d28c3fb73edc92827c1fc113a784ec7aabfe6e8d | subsection | 8 | 9 | Body | A numerical experimentWe consider the exact dataThen the corresponding exact solution of the system (1) isFor all m=2,4,6,8,..., we consider the disturbed dataThen the corresponding disturbed solution of the system (1) isWe getIt means that, when m is large, a small error of data causes a large error of solution. Hence... | {
"cite_spans": []
} | 0807.1812 | Determine the source term of a two-dimensional heat equation | [
"Dang Duc Trong",
"Truong Trung Tuyen",
"Phan Thanh Nam",
"Alain Pham Ngoc Dinh"
] | [
"math.AP"
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c36678972b17b8773d98af69645e8986370c9a83 | abstract | 0 | 29 | Abstract | The aim of this paper is to provide a logic-based conceptual analysis of the
twin paradox (TwP) theorem within a first-order logic framework. A geometrical
characterization of TwP and its variants is given. It is shown that TwP is not
logically equivalent to the assumption of the slowing down of moving clocks,
and the ... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
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1b90d652b58a03daa6028fab44297aa49c8caa1b | subsection | 1 | 29 | Introduction | The twin paradox (TwP) theorem is one of the most famous predictions of
special relativity. According to TwP, if a twin makes a journey into
space, he will return to find that he has aged less than
his twin brother who stayed at home. However
surprising TwP is, it is not a contradiction. It is only a fact that
shows th... | {
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"Gergely Szekely"
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ae91875eb6317676bc3adac8bc35fd65a665b54b | subsection | 2 | 29 | Introduction | The
linearity of transformations between inertial observers (inertial reference frames)
can also be proven from some plausible assumptions, therefore it need
not be assumed as an axiom, see ,
.The usual approaches to special theory of relativity base the theory
on two postulates, namely, Einstein's principle of relativ... | {
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86ce4e678f4d9f41e2bc22eb6e98b3c33bd9be38 | subsection | 3 | 29 | Introduction | Hilbert's 6th problem . In our
perspective axiomatization is only a first step to logical and conceptual analysis
where the real fun begins.For good reasons, the foundation of mathematics was performed strictly
within FOL. A reason for this fact is that staying within
FOL helps to avoid tacit assumptions. Another reaso... | {
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9a7edfb157c620e1f3fad811a0d31d992588864c | subsection | 4 | 29 | A FOL axiom system of kinematics | Here we explain our basic concepts. We deal with kinematics, i.e.,
with the motion of bodies (anything which can move, e.g.,
test-particles, reference frames, electromagnetic waves or centers of
mass). We represent motion as the changing of spatial location in
time. Thus we use reference frames for coordinatizing event... | {
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4e6cd9d7e1a776114c6e30841fcc660b5a045b9f | subsection | 5 | 29 | A FOL axiom system of kinematics | \mathsf {IOb}(m), \mathsf {W}(m,b,x_1,\ldots ,
x_d), m=b, x_1=x_2 and x_1<x_2 are the so-called atomic
formulas of our FOL language, where m,b,x_1,\dots ,x_d can
be arbitrary terms of the required sorts. The formulas of our
FOL language are built up from these atomic formulas by using
the logical connectives not (\lnot... | {
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} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
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9b38b025b8eccbd7b5ebef52e5faf463f71d0db5 | subsection | 6 | 29 | A FOL axiom system of kinematics | If
p\in \mathsf {Q}^n, we assume that p=\langle p_1,\ldots ,p_n\rangle , i.e., p_i\in \mathsf {Q} denotes the i-th component of the n-tuple p.
Specially, we write \mathsf {W}(m,b,p) in place of \mathsf {W}(m,b,p_1,\dots ,p_d),
and we write \forall p in place of \forall p_1\dots \forall p_d,
etc. To abbreviate formulas,... | {
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"Gergely Szekely"
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442821af7803dde34fd4057bbe9c705d76321df3 | subsection | 7 | 29 | A FOL axiom system of kinematics | The set of positive elements of \mathsf {Q} (i.e.,
the set \lbrace x\in \mathsf {Q}:0<x\rbrace ) is denoted by \colorbox {defbgcolor}{\mathsf {Q}^+}.
[Figure: Illustration of the basic definitions]We need some definitions and notations to formulate our other axioms.
The set \mathsf {Q}^d is called the coordinate system... | {
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} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
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2d99d2e093722875777e8af430778f11fcfd2ac9 | subsection | 8 | 29 | A FOL axiom system of kinematics | Let Ev_m denote the set of nonempty events coordinatized by observer m, i.e.,\colorbox {defbgcolor}{Ev_m}\,\mbox{$:=$}\,\left\lbrace \,\mathsf {ev}_m(p) \::\: \mathsf {ev}_m(p)\ne \emptyset \,\right\rbrace ,and Ev denote the set of all observed events, i.e.,\colorbox {defbgcolor}{Ev}\,\mbox{$:=$}\,\left\lbrace \,e\in E... | {
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5cdb036db10ee87a10e0cebbace5339647d826b3 | subsection | 9 | 29 | A FOL axiom system of kinematics | Let us note that whenever we write \mathsf {time}_m, we
assume that the events in its argument have unique coordinates by
Convention REF .The coordinate-domain of observer m, in symbols Cd_m, is the set of coordinate points where m
observes something, i.e.,\colorbox {defbgcolor}{Cd_m}\,\mbox{$:=$}\,\lbrace \, p\in \mat... | {
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6a750f76a2144ab9e27d6888c7a33ab76b7ba6fa | subsection | 10 | 29 | A FOL axiom system of kinematics | The time-unit vector of k according to m is defined as\colorbox {defbgcolor}{1^k_m}\,\mbox{$:=$}\,w^k_m(1_t)-w^k_m(o).The world-line of body b according to observer m is defined as
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} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
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f9754db132f0d71d73ffa414fd06ab943a6189e5 | subsection | 11 | 29 | Geometrical Characterization of TwP | Since the axiom systems we use here deal only with inertial motions of
observers, we formulate the inertial version of TwP, which
is also called clock paradox in the literature.This inertial version
is the one that was formulated by Einstein in his famous 1905 paper,
see . Logical investigation of the
accelerated versi... | {
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2309238b2bd8062e4d5f03bdccce2ebf2a3a7465 | subsection | 12 | 29 | Geometrical Characterization of TwP | That does not mean
abandoning our FOL language. It is just simplifying the
formalization of our axioms. Instead of events we could speak about
observers and spacetime locations. For example, instead of \forall e\in Ev_m\phi we could write \forall p\in Cd_m\phi [e\!\leadsto \! \mathsf {ev}_m(p)], where none of p_1\ldots... | {
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c07df70e085873fb9327dcf795772e875252cf3f | subsection | 13 | 29 | Geometrical Characterization of TwP | Then&\mathsf {time}(\widehat{ac}<b)(e_a,e,e_c) &\quad &\Longleftrightarrow \ &\quad &\mathsf {Conv}({}^\ddag 1_m^a,{}^\ddag 1_m^b,{}^\ddag 1_m^c),\\ &\mathsf {time}(\widehat{ac}=b)(e_a,e,e_c)
& &\Longleftrightarrow \ &
&\mathsf {Bw}({}^\ddag 1_m^a,{}^\ddag 1_m^b,{}^\ddag 1_m^c),\\ &\mathsf {time}(\widehat{ac}>b)(e_a,e,... | {
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] | 2,008 | en | Physics | [
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63805e9a83ae849d9335ad87ca815f692b6e59d2 | subsection | 14 | 29 | Geometrical Characterization of TwP | From which, by axcolor\mathsf {AxLinTime}, it follows that\Big |\mathsf {time}_a(e_a,e)\Big |+\Big |\mathsf {time}_c(e,e_c)\Big | = \frac{|p-q|}{|a^\ddag |}+\frac{|q-r|}{|c^\ddag |}\\
=\frac{|p-s|+|s-r|}{|b^\ddag |}=\frac{|r-p|}{|b^\ddag |}= \Big |\mathsf {time}_c(e_a,e_c)\Big |.Hence \mathsf {time}(\widehat{ac}=b)(e_a... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
] | 2,008 | en | Physics | [
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445c7329707c64a40cf4a5e65b843cf6b7d30f08 | subsection | 15 | 29 | Geometrical Characterization of TwP | By the respective
definitions, it is easy to see that any nontrivial convex (flat,
concave) set intersects a halfline at most once.Let us define the Minkowski sphere here as
\colorbox {defbgcolor}{MS^\ddag _m}\,\mbox{$:=$}\,\big \lbrace \, {}^\ddag 1^k_m \::\: k\in \mathsf {IOb}\,\big \rbrace .Remark 3 Convexity as use... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
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"math.LO",
"math.MP"
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c85ed1a9dec90c68df1e65644a2f382fd39349e8 | subsection | 16 | 29 | Geometrical Characterization of TwP | Now we can reverse the
implications of Corollary REF .Theorem 1
Assume axcolor\mathsf {Kinem_0} and axcolor\mathsf {AxShift}. Then&{axcolor}{\mathsf {TwP}} &&\Longleftrightarrow \ && \forall m\in \mathsf {IOb}MS^\ddag _m \text{ is convex,}\\
&{axcolor}{\mathsf {noTwP}} & &\Longleftrightarrow \ & &\forall m\in \mathsf ... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
] | 2,008 | en | Physics | [
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9a364e342c88a793cb171f0a45b0d53f1252b44f | subsection | 17 | 29 | Consequences for Newtonian kinematics | Let us investigate the logical connection between No-TwP and the Newtonian assumption on the absoluteness of time.axcolor\mathsf {AbsTime}
Observers measure the same time elapsing between events:
\forall m,k\in \mathsf {IOb}\forall e_1,e_2\in Ev\quad \mathsf {time}_m(e_1,e_2)=\mathsf {time}_k(e_1,e_2).To strengthen ou... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
] | 2,008 | en | Physics | [
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f8bc7a269d766c36e960ebe7fbdc90d9ad3b945e | subsection | 18 | 29 | Consequences for Newtonian kinematics | A more experimental version of axiom
axcolor\mathsf {AxThExp^+} is the following:axcolor\mathsf {AxThExp^*}
Observers
can move in any direction at a speed which is arbitrarily close to any finite speed:
\forall m\in \mathsf {IOb}\forall p,q\in \mathsf {Q}^d\forall \varepsilon \in \mathsf {Q}^+ \quad p_\tau \ne q_\tau... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
] | 2,008 | en | Physics | [
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8a6cd26a433886dc9d48ce3237999bea1bd43b2f | subsection | 19 | 29 | Consequences for Newtonian kinematics | If the
hyperplane containing MS^\ddag _m were not horizontal, there would be
nonhorizontal lines parallel to it. Therefore MS^\ddag _m has to be a
subset of a horizontal hyperplane. If MS^\ddag _m were a porper
subset of this hyperplane, there would be nonhorizontal lines not
intersecting it. So MS^\ddag _m has to be a... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
] | 2,008 | en | Physics | [
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410113b1c68138e34ec321e925e5ebf0c2c56a66 | subsection | 20 | 29 | Consequences for Newtonian kinematics | Hence axcolor\mathsf {AbsTime} is
not true in this model, as we claimed. | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
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717646c094f9f443e4060645c0ad5a963ebe73ac | subsection | 21 | 29 | Consequences for special relativity theory | Now we are going to investigate the consequences of Theorem
REF for special relativity. To do so, let us extend our
language by a new unary relation \mathsf {Ph} on \mathsf {B} for photons
(light signals) and formulate an axiom on the constancy of the speed
of light. For convenience, this speed is chosen to be 1.axcolo... | {
"cite_spans": []
} | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
] | 2,008 | en | Physics | [
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bdbab3c4fd2a661368a2a4bceba10fa0b475eb35 | subsection | 22 | 29 | Consequences for special relativity theory | Every bijection from F^d to F^d
that transforms lines of slope 1 to lines of slope 1 is a Poincaré
transformation composed by a dilation and a field-automorphism-induced
map.For the proof of Theorem REF , see ,
. From this theorem we derive that the worldview
transformations between observers are Poincaré ones in the m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 212,
"openalex_id": "",
"raw": "P. G. Vroegindewey. An algebraic generalization of a theorem of E. C. Zeeman. Indag. Math., 36(1):77–81, 1974.",
"source_ref_id": "8a3ca74448a329316cc4bf6e6ec7ba5fc8491879",
"start": 174... | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
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f19392b8902fa8218420456e5da1083e177900d5 | subsection | 23 | 29 | Consequences for special relativity theory | Hence w^k_m is a
Poincaré transformation.Let us now formulate another famous prediction of relativity.axcolor\mathsf {SlowTime}
Relatively moving observers' clocks slow down:
\forall m,k\in \mathsf {IOb}\quad \mathsf {wl}_m(k)\ne \mathsf {wl}_m(m) \rightarrow \big |(1^k_m)_\tau \big |>1.To investigate the logical conn... | {
"cite_spans": [
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"end": 871,
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"raw": "H. Andréka, J. X. Madarász, and I. Németi. Logical axiomatizations of space-time. Samples from the literature. In A. Prékopa and E. Molnár, editors, Non-Euclidean geometries, pages 155–185. Springer... | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
"gr-qc",
"math.LO",
"math.MP"
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f93e1123d95f321fb2cb888555e999e4a7802dea | subsection | 24 | 29 | Consequences for special relativity theory | Hence \mathsf {time}_m(e)
is between \mathsf {time}_m(e_a) and \mathsf {time}_m(e_c) iff \mathsf {time}_b(e) is
between \mathsf {time}_b(e_a) and \mathsf {time}_b(e_c). This completes the proof
since the other parts of the definition of relation \mathsf {meetTwP} do not
depend on observers m and b.We cannot consistentl... | {
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"end": 550,
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"raw": "H. Andréka, J. X. Madarász, and I. Németi. Logical axiomatizations of space-time. Samples from the literature. In A. Prékopa and E. Molnár, editors, Non-Euclidean geometries, pages 155–185. Springer... | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
] | [
"math-ph",
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178ffca9d6cacfa8fecdfd8ea0cfd8d1c8197e36 | subsection | 25 | 29 | Consequences for special relativity theory | Let \mathsf {IOb}\,\mbox{$:=$}\,\lbrace \langle p,q\rangle \in \mathsf {B}\,:\,|p_\sigma -q_\sigma |<|p_\tau -q_\tau |\rbrace . It is easy to see that
there is a nontrivial convex subset M of \mathsf {Q}^d such that 1_t\in M
and |p_\tau |<1 for some p\in M. Let MS^\ddag _{\langle 1,0\rangle } be such a convex subset of... | {
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"doi": "",
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"source_ref_id": "6acc211e4f2450d43... | 10.1007/s11225-010-9253-7 | 0807.1813 | A Geometrical Characterization of the Twin Paradox and its Variants | [
"Gergely Szekely"
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1f1b31de23f57583892da4ce0b0ac8f58394bf9b | subsection | 26 | 29 | Consequences for special relativity theory | Then{axcolor}{\mathsf {SpecRel^-_\mathit {d}}}+{axcolor}{\mathsf {AxSymDist}}&\models {axcolor}{\mathsf {TwP}}, \text{ but}\\
{axcolor}{\mathsf {SpecRel^-_\mathit {d}}}+ {axcolor}{\mathsf {AxShift}}+{axcolor}{\mathsf {AxLinTime}} +{axcolor}{\mathsf {AxThExp}}+{axcolor}{\mathsf {TwP}}&\lnot \models {axcolor}{\mathsf {A... | {
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46ad8c5a3d20aa676119d56eee299be1908de87e | subsection | 27 | 29 | Consequences for special relativity theory | This
theorem is interesting because it shows that assuming only that all
moving clocks slow down to some degree implies the exact ratio of
the slowing down of moving clocks (since if \mathit {d}\ge 3,
{axcolor}{\mathsf {SpecRel^-_\mathit {d}}}+{axcolor}{\mathsf {AxSymDist}} implies that the worldview
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3978aa70082c0c6dfb9900bbee737a12647a4bbc | subsection | 28 | 29 | Concluding remarks | We have seen that (the inertial version of) TwP can be
characterized geometrically within a general axiom system of
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characterization; in particular, that TwP is logically weaker than axiom
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0bd8fdb6c5af3f42529205e0e3a4dec67c7c09dd | abstract | 0 | 9 | Abstract | A growing or shrinking disc will adopt a conical shape, its intrinsic
geometry characterized by a surplus angle $se$ at the apex. If growth is slow,
the cone will find its equilibrium. Whereas this is trivial if $se <= 0$, the
disc can fold into one of a discrete infinite number of states if $se$ is
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6218d16f4a09dfdf29bad04b13b88742a0a84c32 | subsection | 1 | 9 | Body | Conical defects in growing sheetsMartin Michael Müller
Martine Ben Amar
Laboratoire de Physique Statistique de l'Ecole Normale Supérieure
(UMR 8550), associé aux Universités Paris 6 et Paris 7 et au CNRS;
24, rue Lhomond, 75005 Paris, FranceJemal Guven
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de M... | {
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96ca155d6c669203ac94c200290edb6b47c9b2ac | subsection | 2 | 9 | Body | Whereas this state is an unremarkable circular cone in
the case of a deficit, when the deficit is turned to surplus, the folded shape–an
excess-cone (e-cone for short)–exhibits a surprisingly subtle behavior.
In this letter, we will describe the equilibrium states associated with these
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4683ee3a01b4e6856b76ca69bf300c4101af25e9 | subsection | 3 | 9 | Body | The constraint of unstretchability is implemented by adding a term to the energy
functional which fixes the metric via a set of local Lagrange multipliers T^{ab}
. These can be identified with a conserved tangential
stress.The shape equation and its solution.
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a75f8bc0d114b8ae0a0ec967ce861db3f652789a | subsection | 4 | 9 | Body | For small \varphi _{\text{e}}, we find k=a_{1}\varphi _{\text{e}}+{\mathcal {O}}(\varphi _{\text{e}}^{2}),
where a_{1} = -\frac{1}{2\pi }(1-\frac{1}{n^{2}}).
[Figure: Paper model (a) and calculated surface shapes for \varphi _{\text{e}}=2\pi with n=2 (b),n=3 (c), and n=4 (d).]Surface shapes.
The e-cone has to have two ... | {
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b25aa86ecf9c9fa60fe33d630a7bfc85b2346d6c | subsection | 5 | 9 | Body | REF the values of \varphi _{\text{e}}^{\text{kiss}} of various n-folds are
given.
[Table: Kissing points for different n-folds.]Interestingly, \varphi _{\text{e}}^{\text{kiss}} converges to \varphi _{\text{e}, \text{max}}^{\text{kiss}}\approx 35.23 from below if n is sent
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4342a137e1e682a2dbfc12312eade2bc7b4051d7 | subsection | 6 | 9 | Body | Investigated more carefully, however, a set
of adjacent intersection points is found which converge to \varphi _{\text{e}}\approx 7.47 for
n\rightarrow \infty . Above this region each curve reaches a maximum which diverges
quadratically with n. If n>5 and \varphi _{\text{e}} sufficiently large, C_{\Vert } is greater
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a3b05b095865334b1a143a0dbce5b75feb95d8e7 | subsection | 7 | 9 | Body | However, with increased crowding one begins to force up the average curvature:
above \varphi _{\text{e}}=8.27 the 3-fold possesses lower energy than the touching symmetrical
2-fold and the e-cone may flip from n=2 to n=3. Equivalent behavior is
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173c9079ec09d81155b705442f4a102141b4f887 | subsection | 8 | 9 | Body | These truncated cones can also be glued together to model surfaces which are not
flat: a surface of constant negative Gaussian curvature can be approximated by a
telescope formed by such annuli.Partial support from CONACyT grant 51111 as well as
DGAPA PAPIIT grant IN119206-3 is acknowledged.
The authors would like to t... | {
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fa94a615dc7fe00818a45b186dc819aa05ae4ce0 | abstract | 0 | 9 | Abstract | Recently, Matzkin claimed the construction of a hidden variable (HV) model
which is both local and equivalent with the quantum-mechanical predictions. In
this paper we will briefly present this HV model and argue, by identifying an
extra non-local "hidden" HV, why this model is not local | {
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e0245932ce723adbf2bbf3283fcfc41948d372b9 | subsection | 1 | 9 | Introduction | With the derivation of his well-known inequalities, Bell proved that any local model based on hidden variables (HV) can not reproduce the empirical predictions of quantum-mechanics. Recently however, Matzkin claimed that he has constructed a HV model which is both local and equivalent with the quantum-mechanical predic... | {
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ae4bccfbdcffd7a37f037d14202ea63d8cbcc6e3 | subsection | 2 | 9 | EPRB | In the EPRB-experiment, studied by both Bell and Matzkin, a pair of spin-\frac{1}{2} particles are formed in the singlet spin state \mathinner {|{\Psi _0}\rangle } :\mathinner {|{\Psi _0}\rangle } = \frac{1}{\sqrt{2}}( \mathinner {|{{z}\uparrow }\rangle } \otimes \mathinner {|{{z}\downarrow }\rangle } - \mathinner {|{{... | {
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55500d2e9e82875961bffb87b4463e69da96686c | subsection | 3 | 9 | Matzkins HV model | Matzkins HV model makes use of the following elements:HV. A single particle is specified by a hidden variable \vec{\lambda }: a normalized vector in \mathbb {R}^3.
For the two particles specified by (\vec{\lambda }_1, \vec{\lambda }_2) in the singlet spin state,
anti-correlation is described by the relation:
\vec{\lam... | {
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} | 0807.1815 | A local hidden-variable model violating Bell's inequalities: a reply to
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51107cf4355bc29ccd4d395000f659d18c198d0e | subsection | 4 | 9 | Matzkins HV model | For a system consisting of a single particle in a quantum state with a positive spin along the \vec{z}-axis (having an initial HV distribution R_{\pm \frac{1}{2}\vec{z}}) the probabilities of \mathcal {M}(R=R_{+\frac{1}{2}\vec{z}},\vec{u},\vec{\lambda }) = \pm \frac{1}{2} by measuring the spin component in direction \v... | {
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69fe03e1f37b87732d38fa4bc2c1a4bb0325e6e4 | subsection | 5 | 9 | A HV model for two particles | With these elements a HV-model for the EPRB experiment is constructed. The system consists of two particles, described with the HV (\vec{\lambda }_1, \vec{\lambda }_2) respecting correlation (REF ).
The initial distribution of HV for each particle in the singlet state is (as given by Matzkin) a uniform distribution R_{... | {
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d3f23432f799bb16ba9f167472fcd2566396e4cc | subsection | 6 | 9 | A HV model for two particles | Matzkin states that if we measure the spin component of particle 1 along axis \vec{a} (S_{\vec{a}}) and obtain, for example, result a = +\frac{1}{2}, we know (using (REF )) that \vec{\lambda }_1 \cdot \vec{a} \ge 0.
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} | 0807.1815 | A local hidden-variable model violating Bell's inequalities: a reply to
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8840cf44c9be13b7d3be804f711b32bfe8e9fc01 | subsection | 7 | 9 | A HV model for two particles | Both of these properties are in conflict with the definition of “locality” as given by Bell.Matzkin does not give us any local mechanism (based on information locally available) to explain the perturbation of the HV distribution of particle 2 after measuring particle 1.
So, we have to conclude that the initial HV distr... | {
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} | 0807.1815 | A local hidden-variable model violating Bell's inequalities: a reply to
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c945eed7d24b6b72d1864a27feb76e3713664a83 | subsection | 8 | 9 | Conclusion | We recognize that measurement outcomes in the HV model of Matzkin not only depend on the HV \vec{\lambda } and the the measurement direction \vec{u}, but also on a initial HV distribution R.
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885e1dba3d533f656fca05fe2c0aff55223669af | abstract | 0 | 8 | Abstract | The presence of exponential bulges and anti-truncated disks has been noticed
in many lenticular galaxies. In fact, it could be expected because the very
formation of S0 galaxies includes various processes of secular evolution. We
discuss how to distinguish between a pseudobulge and an anti-truncated disk,
and also what... | {
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} | 10.1017/S1743921308027567 | 0807.1817 | Exponential bulges and antitruncated disks in lenticular galaxies | [
"Olga K. Sil'chenko"
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7dea0486fe02e9fd3cb575f7f9c7ec6434c7abce | subsection | 1 | 8 | Introduction | Bulges have been traditionally thought to have de Vaucouleurs' brightness
profiles just as elliptical galaxies (e.g. ).
However John Kormendy (,
, )
has proved that there exists a type of bulges named `pseudobulges' which
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3e99245c08f222ded1bf6a036517c06b638441c7 | subsection | 2 | 8 | Introduction | But for
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8525fc0f8c78bae650f34fd3d5c219c2a5e15b3a | subsection | 3 | 8 | Observations | The photometric observations the results of which we discuss here have been
made with the focal reducer SCORPIO of the Russian 6m telescope
() in the direct-image mode.
The CCD detector EEV 42-40 with the size of 2048 \times 2048
has been used in binned mode of 2\times 2. The field of view
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67dea77b85de9debafd11ce67ef00fc8d249e513 | subsection | 4 | 8 | Photometric structure of the central group S0 galaxies | Two central group galaxies under consideration are typical
giant lenticular galaxies, with the blue absolute magnitudes of
about -21.6 - -21.7 (HYPERLEDA). Both are very red, (B-V)_e=1.07,
and are seen face-on, b/a>0.9.We have calculated azimuthally averaged surface brightness profiles for
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"do... | 10.1017/S1743921308027567 | 0807.1817 | Exponential bulges and antitruncated disks in lenticular galaxies | [
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81f6605677d1b00654f3549024d2e64e90904e37 | subsection | 5 | 8 | Photometric structure of the central group S0 galaxies | REF ).
The profiles of the colour and of the stellar velocity dispersion are
qualitatively similar! Certainly, we see a transition from the
(exponential) bulge to the inner disk at R\approx 10^{\prime \prime }.
Preliminary estimates of the scaleheight of the inner disk in NGC 524,
by treating the measured line-of-sight... | {
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a1adf10a9e8ed531d09c00b9ab71855becaacf56 | subsection | 6 | 8 | What can be the mechanisms to form anti-truncated disks in
lenticular galaxies? | It seems clear that anti-truncated disks are to be a result of matter
re-distribution along the radius of a disk galaxy, and the very event of
re-distribution must be rather fast and discrete. Several candidate
mechanisms can be proposed. Younger et al.
() simulates a minor
merger, and they obtain an anti-truncated ste... | {
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2932b390acc4aefae052f5481f73d6ead0759891 | subsection | 7 | 8 | Conclusions | If we assume that multi-tiers exponential profiles are formed by secular evolution
of galactic disks, the best place to search for them would be lenticular galaxies.
Lenticulars galaxies had to reform secularly their stellar disks during their
transformation from S to S0; hence S0s must be the hosts of both multi-tiers... | {
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2b2168fa9fdbe9e1a9da5bae37db136f430404b3 | abstract | 0 | 14 | Abstract | The statistical properties of the return intervals $\tau_q$ between
successive 1-min volatilities of 30 liquid Chinese stocks exceeding a certain
threshold $q$ are carefully studied. The Kolmogorov-Smirnov (KS) test shows
that 12 stocks exhibit scaling behaviors in the distributions of $\tau_q$ for
different thresholds... | {
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} | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
"Fei Ren",
"Liang Guo",
"Wei-Xing Zhou"
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48f6fd7f4cc0756840a49bf617c624e011885af0 | subsection | 1 | 14 | Introduction | In recent years many concepts and methods from statistical physics
have been applied to the study of financial markets
. The statistical analysis of the
waiting time between two successive events has drawn much attention.
Different definitions of event refer to a variety of
variables characterizing the properties of st... | {
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e8c4951afdb3aed71433a04bd7cde0772c58384a | subsection | 2 | 14 | Introduction | In addition, Lee et
al. investigated the return intervals of 1-min volatility data of
the Korean KOSPI index . They found
that the interval return distribution had a power-law tail and no
scaling was observed. However, it seems that they did not remove the
intraday pattern from the intraday volatility series, which wea... | {
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... | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
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1319316bcdf92179ea777721b29676998de720b9 | subsection | 3 | 14 | Preprosessing the data sets | Our analysis is based on the high-frequency intraday data of 30
most liquid stocks traded on the Shanghai Stock Exchange and the
Shenzhen Stock Exchange. These 30 stocks are most actively traded
stocks representative in a variety of industry sectors, and thus
have the largest sizes among all the stocks. The basic infor... | {
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1cbb9fbda2bf0b46215006238402c34336cba297 | subsection | 4 | 14 | Preprosessing the data sets | When the threshold q is very small, say
less than the minimum of r(t), then all \tau _q values equal to
1.
[Figure: Illustration of volatility return intervals for stock000625, where \tau _2, \tau _3 and \tau _4 correspond to returnintervals for q=2, 3 and 4.] | {
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} | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
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"Liang Guo",
"Wei-Xing Zhou"
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c3fd1d263dc9d5f3c61fb70e771e76cf93dd77d4 | subsection | 5 | 14 | Scaling and nonscaling behaviors of return interval distributions | Several empirical studies show that the probability distribution
function (PDF) of the return intervals obeys a scaling form:P_q(\tau )=\frac{1}{\bar{\tau }} f \left( \frac{\tau }{\bar{\tau }}
\right),where \bar{\tau } is the mean return interval which depends on the
threshold q. The scaling form could be approximated ... | {
"cite_spans": []
} | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
"Fei Ren",
"Liang Guo",
"Wei-Xing Zhou"
] | [
"q-fin.ST",
"physics.data-an",
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c22a272147fb3ad7835aba47aaaea3a7e9dbad1b | subsection | 6 | 14 | Complementary cumulative distributions | To make the observation of this possible scaling behavior
(REF ) more clear, we study the complementary cumulative
distribution function (CCDF) of the scaled return intervalsC_q(\tau /\bar{\tau })= \int _\tau ^\infty P_q(\tau )d\tau =\int _{\tau /\bar{\tau }}^\infty f(x)dx.If the PDFs for different q obey the scaling f... | {
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} | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
"Fei Ren",
"Liang Guo",
"Wei-Xing Zhou"
] | [
"q-fin.ST",
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95323d18a4dc8821dee290eae5067166b5c45129 | subsection | 7 | 14 | Kolmogorov-Smirnov test of scaling in return interval distributions | The eyeballing of complementary cumulative distributions offers a
qualitative way of distinguishing scaling and nonscaling behaviors.
Here we further adopt a quantitative approach, the
Kolmogorov-Smirnov (KS) test. We use the KS test to compare two
distributions for q=2 and q=5, which behave most differently
among all ... | {
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... | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
"Fei Ren",
"Liang Guo",
"Wei-Xing Zhou"
] | [
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8ea9c0aa91cc208c59ffc82af6ca0fd23e31ad5b | subsection | 8 | 14 | Kolmogorov-Smirnov test of the scaling function | We have demonstrated that the return distributions of 12 stocks
show scaling behaviors. To further study the particular form of the
scaling function, we perform the KS goodness-of-fit test
, .
Empirical studies have shown that the scaling form could be
approximated by a stretched exponential function as in
Eq. (REF ). ... | {
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"doi":... | 10.1016/j.physa.2008.12.005 | 0807.1818 | Statistical properties of volatility return intervals of Chinese stocks | [
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