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fca7925c311e0cd63f08ec578f361da42be603fe | subsection | 4 | 20 | Recursions for | Let A_n = (1,0,1,0,\dots ) be the alternating word of length n starting with 1.
(By symmetry, probabilities for alternating words starting with 0 are the same as for A_n.)
In this section, we first compute the v_n given in (REF ), and
then we prove Theorem 1(a). The computation of v_n is done recursively; the
recursion... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.004851870238780975,
0.015310855582356453,
0.002429749583825469,
-0.013205326162278652,
0.01067258883267641,
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0.02160455472767353,
0.03509214520454407,
0.0006594081060029566,
-0.051051441580057144,
-0.007510481867939234,
-0.019148103892803192,
... | |
7c84dae8037c7d9c91302ed8171d606e68bf1979 | subsection | 5 | 20 | Recursions for | For example, if M=5, the larger root of
f is .9978\dots .A Bernoulli sequence Y contributing to v_{n,k} must satisfy Y_k=1,
and the preceding sequence (Y_1,\dots ,Y_{k-1}) cannot contain 1 and 0 in that order, since
if it did, there would be an admissible embedding starting before position k.
Therefore, (Y_1,\dots ,Y_{... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.011068807914853096,
0.022351210936903954,
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-0.003718844847753644,
-0.04097976163029671,
0.006114162504673004,
0.004001095425337553,
... | |
62bdd829d4beecbd6411a6503f1ef7acb89c111e | subsection | 6 | 20 | Recursions for | This givesw_{m,2} \le \tfrac{1}{4} w_{m-1,2} + \tfrac{1}{4} w_{m-1}.Note that equality need not hold in (REF ).
Suppose that both Y and W_m begin with the letters aa, and that (Y_3,Y_4,\dots )
contains an admissible embedding of both W_{m-2} and W_{m-1}. In this
case, Y does not contribute to the left side of (REF ) bu... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.02662915736436844,
0.05365508422255516,
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-0.011544388718903065,
0.012986483052372932,
-0.... | |
b2534f3606e64a9f7dbe81e10c74951b641a7b67 | subsection | 7 | 20 | Relations to the spacing random variables | In this section, we prove several results relating M-seen finite words to
inequalities satisfied by the spacing variables \tau _k and their partial sums.
When the word is either constant or alternating, these are equivalences, and
were stated as Theorem REF in the Introduction. For general words, we only have
one direc... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.0038542612455785275,
0.025677276775240898,
0.0036845288705080748,
-0.031886812299489975,
0.014097328297793865,
0.012838638387620449,
0.007006707601249218,
0.024258388206362724,
0.025433165952563286,
0.03524332121014595,
-0.005587820895016193,
-0.0342668816447258,
-0.006365919951349497,
... | |
1b2fb97b9c9987bdb4976360d98f2dc233832993 | subsection | 8 | 20 | Relations to the spacing random variables | As T_n \le m_n by Proposition REF , this
implies \tau _i\le M for all i\le n.Proposition 8 Let W=(w_1,\dots , w_n).
Suppose that W is seen at (m_1,\dots ,m_n) and
that 0<m_{i+1}-m_i\le M for each i. If w_k\ne w_{k+1}=\cdots =w_l for some k+1 \le l
and if \tau _l>M, then T_l\le m_{k+1}.Let a=w_{l}. In between locations
... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
0.015397991053760052,
0.042516157031059265,
0.0015594472642987967,
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0.00986600760370493,
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0.020846040919423103,
0.038334742188453674,
0.0030464064329862595,
-0.039525073021650314,
-0.0637284591794014,
0.04... | |
3dab8373892c61de91592c574429fb670287d77e | subsection | 9 | 20 | Relations to the spacing random variables | As (Y_i: T_j\le i<T_k) consists of k-j constant blocks,
the interval [T_j,T_k) contains at most k-j consecutive
elements from m_1,\dots , m_n, i.e.
m_l < T_j \le m_{l+1} \le m_{l+r} < T_k \le m_{l+r+1}
for some l and r \le k-j.
So T_k - T_j < m_{l+r+1} - m_1 = \sum _{i=l}^{l+r} (m_{i+1} - m_i)
\le (r+1)M.Finally, assum... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.014392927289009094,
0.040324617177248,
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0.026649046689271927,
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0.00658976286649704,
-0.05088655278086662,
0.04789... | |
4deba721c245bef087a2072fe75f2c4b51dbfeac | subsection | 10 | 20 | Two-block words | We prove Theorem 1(b) in this section. For p,q,j\ge 0,
define&\sigma _{p,j}=P(\tau _1\le M,\dots ,\tau _p\le M,\, T_{p+j}>pM), \\
&\sigma _{p,j}^{\prime }=P(\tau _1\le M,\dots ,\tau _p\le M,\, T_{p+j} \le pM)andu_{p,q}= \alpha ^{p+q}+\beta \sum _{j=1}^{q} \alpha ^{q-j} \sigma _{p,j}^{\prime }
= \alpha ^p-\beta \sum _{j... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.014467021450400352,
0.028949303552508354,
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0.03210823982954025,
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0.00041179696563631296,
-0.024371132254600525,
... | |
8220d391456e855912d5ce2cc5d26d20abe749d3 | subsection | 11 | 20 | Two-block words | Considering the largest j \ge 1 (if any) for which \tau _{p+j}>M, we see thatP(W_{p,q}\text{ is seen})
\le P(\tau _1\le M,\dots ,\tau _{p+q}\le M)
+\sum _{j=1}^{q} \tilde{\sigma }_{p,j},where\tilde{\sigma }_{p,j} = P(\tau _1\le M,\dots ,\tau _p\le M,\tau _{p+j}>M,\tau _{p+j+1}\le M,\dots ,\tau _{p+q}\le M, \,T_{p+j}\le... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.033424798399209976,
0.017765508964657784,
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0.03177645057439804,
0.0004430884146131575,
0.005807367619127035,
-0.02635827660560608,
0.0053... | |
c2cef8b5b0c8f86733bfdddf07875c55ac513e8b | subsection | 12 | 20 | Two-block words | We will do so by computing the generating function of this expression as a function of q.By Lemma REF ,\sigma _{p,j}=\sum _{i=0}^p\binom{p}{i}(-\beta )^{p-i}P(T_{p+j}>iM).Since T_{m} is a sum of m independent geometric random variables
and can thus be interpreted as the waiting time for the m-th success
in a sequence o... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.0405561588704586,
0.038999829441308975,
-0.01956094615161419,
-0.01124525535851717,
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0.02490129880607128,
0.024428296834230423,
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-0.005035188980400562,
-0.04730026051402092,
0.03536839038133621,
-0.02879212610423565,
0.... | |
43f9823b0b875b4b2b58f9d56b028b2c277156f4 | subsection | 13 | 20 | Two-block words | \end{aligned}Therefore,(1-x)\sum _{q=1}^{\infty }w_{p,q}x^{q-1}=\frac{1-x}{1-\alpha x}\sum _{j=1}^{\infty }\sigma _{p,j}x^{j-1}
=\frac{1}{1-\alpha x}\beta ^px^{-p}[(1+x)^M-1]^p.Using this expression, we can write(1-x)\sum _{q=0}^{\infty }\Delta w_{p,q}x^{q-1}=\frac{\beta ^{p+1}x^{-p}}{x^2(1-\alpha x)}
[(1+x)^M-1]^pP(x)... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.03423048555850983,
0.01699320785701275,
-0.047562677413225174,
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0.020349137485027313,
-0.0234915092587471,
-0... | |
17f506015e91e50119ec4c2d76da055289f8c51c | subsection | 14 | 20 | Two-block words | Therefore, we need to check that the following expression is nonnegative for l\ge 2:&\binom{M}{2}-2(\alpha -M\beta )+\sum _{k=3}^l\bigg [\binom{M}{k}-(\alpha -\beta )\binom{M}{k-1}\bigg ]\\
&\hspace{56.9055pt}=\binom{M}{l}+2\beta \sum _{k=2}^{l-1}\binom{M}{k}-2(\alpha -M\beta )=\binom{M}{l}-2\beta \sum _{k=l}^M\binom{M... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.04491136595606804,
0.006639834027737379,
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0.0085429223254323,
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... | |
a3ed9652513b0c28c0ce0d872da4928ffd43ce30 | subsection | 15 | 20 | Two-block words | Therefore, since 0\le M-2\alpha \le M(\alpha -\beta )
(which is equivalent to M\beta \le \alpha ), (REF ) can be written as\delta _{p+1,q+1}-M\beta \delta _{p,q+1}\ge \frac{M-2\alpha }{M}[\delta _{p+1,q}-M\beta \delta _{p,q}]\ge 0,where the final inequality follows from the induction hypothesis. This proves that
\delta... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.051354363560676575,
0.015806036069989204,
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... | |
12ed10c7f5a69776270193b6b9788e7d674ea5ba | subsection | 16 | 20 | Independence of parameter choice | Let X = (X_n: n \ge 1) and Y = (Y_n : n \ge 1)
be two independent Bernoulli sequences with parameters
p_X and p_Y respectively, and let W \in \lbrace 0,1\rbrace ^\mathbb {N}
be an arbitrary infinite word.Theorem 13
(a) The validity of
the assertion “\ \forall M \ge 2: P(X \text{ is $M$-seen\ in } Y) = 0"
does not depe... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.018539495766162872,
0.036956921219825745,
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-0.009689366444945335,
... | |
4387e3d034bd285448f618d8d742d6def7ebfefa | subsection | 17 | 20 | Independence of parameter choice | Thus by (a) there exist Bernoulli sequences
X^{(0)}, \ldots , X^{(k)} such that X^{(i)} has parameter p_i
and X^{(i+1)} can be 3-seen in X^{(i)} for all i.
Thus X^{\prime } := X^{(k)} can be 3^k-seen in X := X^{(0)}.[Proof of Theorem REF ]
(a) Let p_X,p_Y,p_X^{\prime },p_Y^{\prime } \in (0,1). By Lemma
REF there are Be... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.018855297937989235,
0.02800835482776165,
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0.011845582164824009,
-0... | |
dabc58bb2c5f9568b446a6315fea4a5f4a253e3a | subsection | 18 | 20 | Variance of the number of embeddings | We prove Theorem REF .
Let W = (w_1,w_2,\dots ,w_n) be a word of length n, and let
j=(j_0,j_1,\dots ,j_n) and k=(k_0,k_1,\dots ,k_n) be strictly increasing sequences
of integers with j_0=k_0=0 and gaps not exceeding M. ThenE(N_n^2) &= \sum _{j,k}P(\text{$j$ and $k$ are $M$-admissible embeddings})\\
&= \sum _{j,k} (\tfr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1646,
"openalex_id": "",
"raw": "Feller, W., An Introduction to Probability Theory and its Applications, Volume II, Wiley (1966).",
"source_ref_id": "c42c439a07b584f6d1e10786a143ea47056bd329",
"start": 1567
}
]
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0.039144594222307205,
0.003432398196309805,
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0.037252962589263916,
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-0.0050570666790008545,
-0.015544950030744076,... | |
f6a13b6ea5a1a9ae6d570e5a7d500fba911a133c | subsection | 19 | 20 | Variance of the number of embeddings | When M=2, Ex^{\tau }=1-\sqrt{1-x} (see, for example,
Section XIV.4 of ), so c_2=\frac{4}{3}.In order to get full benefit from the Tauberian theorem, we need to know that the sequence E(2^{Z_n})c^{-n} is
monotone; otherwise we would only have convergence in the Cesàro sense.
We check this next. Let u_n=P(J_n-K_n=0), v_0... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 92,
"openalex_id": "",
"raw": "Feller, W., An Introduction to Probability Theory and its Applications, Volume I, 3rd edition, Wiley (1968).",
"source_ref_id": "8971b27a038b88879e9ccc25a44ee915f873b78a",
"start": 65
... | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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aa5c9168968a7c7ab5929265429e21f2fda67d53 | abstract | 0 | 33 | Abstract | We are interested in knowing what type of manifolds are obtained by doing
Dehn surgery on closed pure 3-braids in the 3-sphere. In particular, we want to
determine when we get the 3-sphere by surgery on such a link. We consider links
which are small closed pure 3-braids; these are the closure of 3-braids of the
form $(... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
-0.025487929582595825,
0.028921931982040405,
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0.035377856343984604,
-0.004628272261470556,... | |
3e2ff7b4bd6c984e2a029e8ced0f9dbbeee89ef1 | subsection | 1 | 33 | Introduction | We are interested in knowing what type of
manifolds are obtained by doing Dehn
surgery on closed pure 3-braids in S^3. In
particular, when is possible to obtain the 3-sphere by
Dehn surgery on a closed pure 3-braid.By the fundamental theorem of surgery proved by
Lickorish and Wallace , , ,
we know that any closed, conn... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 416,
"openalex_id": "",
"raw": "W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Annals of Mathematics 76, (1962), 531-538.",
"source_ref_id": "df96af6e72f68ac868f9399b757673c11bbd0521",
"s... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
0.012741819955408573,
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0.04690821096301079,
-0.007851097732782364,
0.0... | |
e6109caebc78b52843ce78b04c8216bd5724f07d | subsection | 2 | 33 | Introduction | We indicate surgeries on this link
by \mathcal {L}(1/e_1,1/f_1,1/e,m,n,p), as indicated in Figure 2, which implicitly is
giving an order to the components of the link.
[Figure: NO_CAPTION]Note that the link \mathcal {L} is strongly invertible, an involution axis is shown in
Figure 2. The quotient of the exterior of \ma... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 715,
"openalex_id": "",
"raw": "C. McA. Gordon and J. Luecke, Knots with unknotting number 1 and essential Conway spheres, Algebraic and Geometric Topology 6 (2006), 2051-2116.",
"source_ref_id": "c34aa55f0dfce371919325d1b4f... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.02629... | |
55d8668de3746d076a6d9f493d5137804885b40d | subsection | 3 | 33 | Introduction | The capital letters A,B,C,D,E,F denote boundary components in the
hexatangle, and \alpha ,\beta ,\gamma ,\delta ,\epsilon ,\eta denote fillings of the hexatangle
with rational tangles, so for example, \mathcal {H}(\alpha ,\beta ,*,*,*,\eta ) denote the
tangle obtained by filing the components A, B and F with the ration... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1071,
"openalex_id": "",
"raw": "J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems is Abstract A, J.Leech, ed., Pergamon Press, Oxford and New York, 1969, pp.329-358.... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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... | |
ff5be63bbc0b91496633c12517f75c1e5c57ffb0 | subsection | 4 | 33 | Introduction | In fact, the 3-chain link is the closure the pure 3-braid
{\sigma _1^{4}\sigma _2^{4}({\sigma _2\sigma _1\sigma _2})^{-2}}. In
all the exceptional fillings of the 3-chain link are determined; these results can be verified
by looking at the corresponding fillings of the hexatangle. It would be also interesting
to deter... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 282,
"openalex_id": "",
"raw": "B. Martelli and C. Petronio, Dehn filling of the “magic\" 3-manifold, Comm. Anal. Geom. 14 (2006), 969-1026.",
"source_ref_id": "93d148c567c2e09c99bc20515c2b39d3dae9a580",
"start": 124
... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0... | |
2f8993cf64a0735da21d92c0e8dcf6544ea40f72 | subsection | 5 | 33 | Introduction | Also, given two pairs of adjacent
(opposite) boxes there is a symmetry that takes one pair to the other.
[Figure: NO_CAPTION]Our results are the following:Theorem 1.1
Suppose an integral filling of the hexatangle produces the trivial knot,
then the parameters are exactly as shown in Tables 1, 2 and 3, up to symmetries... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.0... | |
4969aff9bc0c10732134f76a89eb239e36df03ae | subsection | 6 | 33 | When a parameter is 0 | In this and next section we do some rational tangles computations and rely on
known facts about rational tangles and knots. We follow the conventions of
.
We denote by R(p/q) the rational tangle determined by p/q, and by K(p/q)
the rational knot or 2-bridge knot, which is the numerator of the rational tangle
R(p/q). As... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 154,
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"raw": "M. Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries, Topology Appl. 121 (2002), 119–141.",
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"start": 124
... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.0018438356928527355,
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... | |
174ae42a5b5a11b546a5143c414e7e470df8a7ae | subsection | 7 | 33 | When a parameter is 0 | Then
\mathcal {H} (\alpha ,\beta ,\gamma ,\delta , \epsilon ,0) is the trivial
knot if and only if the parameters are as in Tables 1 and 2, up to symmetries.Proof The proof is a rational tangles calculation. Note that if
4 or more of the parameters are 0, then the link obtained has more than one
component. So suppose a... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.015974795445799828,
0.022978072986006737,
0.0150... | |
fd03f60a09211e13c795e28c2104d89d30961fa8 | subsection | 8 | 33 | When a parameter is 0 | \beta =0, \eta = 0In this case the knot is a sum of 2-string tangles. It is made of the Montesinos tangles
M(-1/\alpha ,-1/\delta ) and M(-1/\epsilon ,-1/\gamma ), and well, it is also a Montesinos
knot. For this to be a trivial knot, one of the two tangle must be a trivial tangle,
and we can assume, because of the sym... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.009323270060122013,
0.0027046639006584883,
... | |
0ba92e3c1d112fe153d9149440c14a488d8458b8 | subsection | 9 | 33 | When a parameter is 0 | To get the trivial knot, one of the tangles has to be
trivial, and because of the symmetries, we can assume that the tangle D(-1/\alpha ,-1/\delta )
is trivial. Then \alpha =\pm 1, or \delta =\pm 1.
[Figure: NO_CAPTION]Case E. \alpha =1The knot looks like the Montesinos knot
M((\delta +1)/(-\beta \delta -\beta -\delta ... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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-0.006849923171103001,
0.01... | |
bf96b5c4c8a28607edc08d0a119b8d158dc21bf2 | subsection | 10 | 33 | When a parameter is 0 | We get the solutions
\epsilon =1, \gamma =2; \epsilon =2, \gamma =1, which correspond to lines
17-18 in Table 2.TABLE 2
[Table: NO_CAPTION][Table: NO_CAPTION]Case E.2. The tangle R(-1/\epsilon ) is integralCase E.2.1. \epsilon =1The knot is the 2-bridge knot K[\delta ,1,\beta ,1,\gamma ]=K((\gamma \beta \delta +\gamma ... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.01... | |
edabdbd66aa561ed0f13fe62c4b9f10db72981a0 | subsection | 11 | 33 | When a parameter is | Theorem 1.4 Suppose that \alpha , \beta , \gamma ,
\epsilon and \eta are not 0, \delta =-1 and \eta =1.
\mathcal {H} (\alpha ,\beta ,\gamma ,-1, \epsilon ,1) is the trivial
knot if and only the parameters are as in Table 3, up to symmetries.Proof
In this case the knot looks like a Montesinos knot, see Figure 7.
In fact... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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-0.... | |
9abfb81cc19a449b7ff4f57a9273e5e2998189d1 | subsection | 12 | 33 | When a parameter is | These correspond to lines 9-11 of Table 3.Case D. \alpha =-2, \epsilon =-1In this case the knot is the 2-bridge knot
K[\gamma ,2,\beta -2]=K((2\gamma \beta + \beta -3\gamma -2)/(2\gamma +1)).
For this to be trivial
we need that 2\gamma \beta + \beta -3\gamma -2=\pm 1, and this is possible only
in the following cases: \... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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... | |
cb6b7852cde5d897fab3d16aab5d2ca1f755698f | subsection | 13 | 33 | The reduction Lemmas | In this section we prove the following theorem.Theorem 1.3 Suppose that all of
\alpha , \beta \gamma , \delta ,
\epsilon and \eta are different from 0. If
\mathcal {H} (\alpha ,\beta ,\gamma ,\delta , \epsilon ,\eta ) is the trivial
knot, then there is a pair of adjacent boxes,
say \delta and \eta ,
so that \delta =-1 ... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0... | |
40cc3e883b10176f0f58f1c3ff89f7f4de5d0785 | subsection | 14 | 33 | The first reduction | Let V_1, V_2, V_3, V_4 be solid tori. Let M_1= V_1 \cup _A V_2,
where V_1 and V_2 are glued along an annulus
A\subset \partial V_i, i=1,2, and suppose that A goes at least
twice longitudinally on each solid tori. Let M_2= V_3 \cup _B V_4,
where V_3 and V_4 are glued along an annulus
B\subset \partial V_i, i=3,4, and su... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.0... | |
9d3ba028b4432fdc23f127ae7f8b315c276ae8ff | subsection | 15 | 33 | The first reduction | As \hat{T} and R
are incompressible, they can be isotoped so that their intersection consists
of simple closed curves which are essential in both surfaces.
These divide R into a collection of annuli A_1,\dots ,A_n.
Suppose A_1 lies on M_1-int\, \eta (k_1). Look now at the intersections
between A_1 and E_1=A-int\, \eta ... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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0.02284282... | |
bc6c609514f2e68a401815f6744f50e3b31c2456 | subsection | 16 | 33 | The first reduction | There is a pair of opposite boxes, say \eta and \beta , so that all the other
boxes are different from \pm 1. Furthermore there are the following cases, up to
symmetries and mirror images:
a) \beta =1, \eta =2
b) \beta =-1, \eta =2
c) \beta =1, \eta =1
d) \beta =-1, \eta =12. There is a pair of adjacent boxes, ... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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... | |
b09d64e4b1a92490afc259510d99dcd148dfe549 | subsection | 17 | 33 | The first reduction | It follows that \tilde{\mathcal {H}} (\alpha , *,\gamma ,\delta , \epsilon ,\eta )=
\mathcal {L}(1/\eta ,*,1/\epsilon ,-\gamma ,-\alpha ,-\delta ) is a manifold
as M in Lemma REF . The knot to be
removed from \mathcal {L}(1/\eta ,1/0,1/\epsilon ,-\gamma ,-\alpha ,-\delta )
to get M is shown with dotted lines in Figure ... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 655,
"openalex_id": "",
"raw": "C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom. 3 1995, 597-644.",
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"start... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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d9c91769f445f7d90a56e902b38f1dcdeca36627 | subsection | 18 | 33 | The first reduction | By Lemma 4.1.1, \tilde{\mathcal {H}}(\alpha ,1,\gamma ,\delta ,\epsilon ,*) is a hyperbolic manifold,
in fact the exterior of a hyperbolic knot in S^3 by hypothesis. It also has a toroidal filling,
corresponding to \eta =0, which is at distance 2 from the filling \eta =2 that produces the
3-sphere. So, by the main resu... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 468,
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"raw": "C. McA. Gordon and J. Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004), 417-485.",
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"start": 300
... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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9466cf4ad2d7f1f69abf9192080b8660f0ab1895 | subsection | 19 | 33 | The first reduction | It follows from , 5.4, or , 3.1 that the EM-knot
K(\ell ,m,n,p) is the same as S(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime },\delta ^{\prime }) , where \alpha ^{\prime }, \beta ^{\prime }, \gamma ^{\prime },
\delta ^{\prime } are as follows:
p=0 : \ \ \alpha ^{\prime }= -\frac{1}{\ell },\ \beta ^{\prime }=\fra... | {
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... | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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2a2b2f6c175041db9ded0986af8ab75bc9b8f014 | subsection | 20 | 33 | The first reduction | \gamma =-2 and -1/\epsilon =(3n-2)/(4n-3)
In this case we have 3n-2=\pm 1, which is possible only if n=1, but then \epsilon =-1, which is not possible by hypothesis.
Case A.2. m=-2
Looking again at the other pair of tangles in the decomposition, we get that M(-1/\epsilon ,(-\gamma -1)/\gamma ) is equivalent to M((3-5n)... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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-0.012126103043556213,
-0.05412103235721588,
0.01030222699046135,
-0.0161935742944479,
-0.01... | |
961debf13ebbfeb341ad887661b54eeb75b759d0 | subsection | 21 | 33 | The first reduction | In this section we prove the following:
Proposition 4.4 If all of \vert \alpha \vert , \vert \gamma \vert ,
\vert \delta \vert ,
\vert \epsilon \vert are
\ge 2, then \mathcal {Q}_2 (\alpha ,\gamma ,\delta ,\epsilon ) cannot be the trivial knot.
Proof The proof follows the same lines as Proposition REF , just
note that... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
] | 2,008 | en | Mathematics | [
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... | |
0a278c2378703f789880061d2eea4cf21802709f | subsection | 22 | 33 | The first reduction | Case B. n=0
In this case we have M((m-1)/(2m-1),-1/2) = M(-1/\epsilon ,(\gamma -1)/\gamma ) =
M((\epsilon -1)/\epsilon ,-1/\gamma ) = M((2\epsilon -1)/\epsilon , (-\gamma -1)/\gamma ).
Case B.1. \gamma =-2, and then (m-1)/(2m-1)=(2\epsilon -1)/\epsilon
A simple calculation shows that this case is not possible.
Case B.... | {
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} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
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00ec0044be1f68306740d5991a8b0cd7e28e6ad5 | subsection | 23 | 33 | The first reduction | Proof Consider the tangle \mathcal {Q}_3(\alpha ,\gamma ,\delta ,*), then
\tilde{\mathcal {Q}}_3(\alpha ,\gamma ,\delta ,*) is the exterior of a knot in S^3.
Note that \mathcal {Q}_3(\alpha ,\gamma ,\delta ,-1) looks like a composite knot.
In fact, it is the connected sum of two-bridge knots
K(\alpha +1)\# K((\delta \g... | {
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"raw": "C. McA. Gordon and J. Luecke, Only integral Dehn surgeries can yield reducible manifolds, Proc. Cambridge Philos. Soc. 102 (1987), 94-101.",
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1c9ec7f8af0bf71c6ccbb83e8ddfe19fddd7d7de | subsection | 24 | 33 | The first reduction | Then \tilde{\mathcal {Q}}_4(\alpha ,\gamma ,\delta ,*) is the exterior of a knot in S^3.
Claim 4.8 The tangle \mathcal {Q}_4(\alpha ,\gamma ,\delta ,*)
is trivial.
Proof Note that
\mathcal {Q}_4(\alpha ,\gamma ,\delta ,0) is a 2-bridge knot.
Then the knot \tilde{\mathcal {Q}}_4(\alpha ,\gamma ,\delta ,*) has a Dehn su... | {
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} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
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fe477dcd009e6cf001793b7edc73c2ab4d6d1b0d | subsection | 25 | 33 | The first reduction | Note that \mathcal {Q}_4(-2,\gamma ,\delta ,1)
looks like the Montesinos knot M(\delta /(1-\delta \gamma ),-1/2,1/3), and for this to be a 2-bridge
knot, we need that \delta \gamma =2, so one of \delta or \gamma must be \pm 1,
which is not possible.
The knots \mathcal {Q}_5 (\alpha ,\beta ,\gamma ,\epsilon )
Let \mat... | {
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60502228c2dbd2262f592d689bcef17c53b4e42e | subsection | 26 | 33 | The first reduction | To do that we first find the Schreier unique
representative of the conjugacy class of the braid (see , §7). We have the following cases:
Case A. \alpha , \beta and \gamma are positive
In this case the braid \sigma _1^{-\alpha }\sigma _2\sigma _1^{-\beta }\sigma _2\sigma _1^{-\gamma }\sigma _2
is already the Schreier un... | {
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0a139773da1ed2ca53845671fdb736d41b0a99f4 | subsection | 27 | 33 | The first reduction | A composite link which is a closed 3-braid has as Schreier representative of the conjugacy class
of a 3-braid representing it, one of the following: \sigma _1^{-u}\sigma _2^v, where u\ge v\ge 2, or
C^{-1}\sigma _1^{-u}\sigma _2\sigma _1^{-v}\sigma _2, where u\ge v\ge 0 (see ,§7). None of these
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34178b13e7e5d53c25f79107fdab0a7e8310b68d | subsection | 28 | 33 | The first reduction | Any surgery on \tilde{\mathcal {Q}}_5 (\alpha ,\beta , \gamma ,*) at distance one
from 1/0 must produce a lens space.
So \tilde{\mathcal {Q}}_5 (\pm 3 ,\beta , -\beta ,-1) must be a lens space,
and then \mathcal {Q}_5 (\pm 3, \beta , -\beta ,-1) is a 2-bridge link.
But note that it looks like the Montesinos link M(3/4,... | {
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} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
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] | [
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5c53fc8d4ab340aa04e693d851c0f687c746465d | subsection | 29 | 33 | The first reduction | Note that it
looks like the Montesinos knot M(\gamma /(1-\gamma ),\beta /(1-\beta ),\alpha /(1+\alpha )),
so to be composite one of the tangles that
form it must be 1/0, so we have that either \beta =1 or \gamma =1 (or \alpha =-1,
but in this case we finish). Note also that both cases are symmetric,
so we can assume th... | {
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} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
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b75a7a93a728d95153a3329dd3dc7980dcccdaa1 | subsection | 30 | 33 | The first reduction | \gamma =-3
\mathcal {Q}_5 (\alpha ,2, -3,-1/2) looks like the sum of
two Montesinos tangles, T(-1/2,1/3) and T((\alpha +1)/(\alpha +2),-1/2),
so to be a 2-bridge knot we need that \alpha =- 1 or
\alpha =-3. But \mathcal {Q}_5 (-3,2, -3,-1/2) is the Montesinos knot
M(-2/3,-1/2,1/3), which is not a 2-bridge knot.
Case F.... | {
"cite_spans": []
} | 0807.1677 | The Hexatangle | [
"Lorena Armas-Sanabria",
"Mario Eudave-Munoz"
] | [
"math.GT"
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7ea389e9abf47f0a1666cc1c70f024acdd95ba72 | subsection | 31 | 33 | The first reduction | As before, we indicate surgeries on this link
by \mathcal {L}(1/e_1,1/f_1,1/e,m,n,p), as indicated in Figure 2, which implicitly is
giving an order to the components of the link. Note that a surgery \mathcal {L}(1/0,*,*,*,*,*),
or a surgery \mathcal {L}(1,*,-1,*,*,*) produces a non-hyperbolic manifold. So, any entry in... | {
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2771da77e98f2b1285ec85302985d90058cc336a | subsection | 32 | 33 | The pure 3-braids with trivial surgeries | In this section we return to the original problem of determining which small closed pure 3-braids
produce S^3 by surgery. All information is contained in Tables 1, 2, 3. Any entry in the tables produces many braids with a trivial surgery, because of the symmetries. We will not reproduce all such braids here. Now we wil... | {
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18bb5cdbbe3cbedf6c2bbea9d8d934e5dd8f1392 | abstract | 0 | 22 | Abstract | The logarithmic Sobolev inequality for the Hamming cube {0,1}^n states that
for any real-valued function f on the cube holds E(f,f) \ge 2 Ent(f^2), where
E(f,f) is the appropriate Dirichlet form (also known as "sum of influences").
We show that the constant C = 2 at the right hand side of this inequality can
be replac... | {
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} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
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4badbf6b3b2b90b703687116206014c6cec093f2 | subsection | 1 | 22 | Isoperimetric problems on the Hamming cube | This paper deals with discrete isoperimetric inequalities on graphs. Let a graph G = (V,E) be given, and let A \subseteq V be a set of vertices in G. An isoperimetric inequality addresses the question of how small the boundary \partial A of A can be, given the cardinality of A, by lower bounding the size of \partial A ... | {
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... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
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"Alex Samorodnitsky"
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"math.CO"
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78e4c9c6e808f470063679a3f9f85070a69379ff | subsection | 2 | 22 | Isoperimetric problems on the Hamming cube | For a non-zero
function f:~\lbrace 0,1\rbrace ^n\rightarrow \mathbb {R} holds\mathbb {E}_x \sum _{y \sim x} (f(x) - f(y))^2 \ge 2 \cdot \mathbb {E}f^2 \log \frac{\mathbb {E}f^2}{\mathbb {E}^2 |f|}Choosing f in (REF ) to be the
characteristic function of a subset A of the cube, we recover the
edge-isoperimetric inequali... | {
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5dd4a4f65a3a29aa567f844c432c564f7ca17003 | subsection | 3 | 22 | Bounds on binary error correcting codes | A binary error-correcting code of length n and minimal distance d
is a subset C of the boolean cube \lbrace 0,1\rbrace ^n such that the distance between
any two distinct points in C is at least d. In other words, the
points in C can be taken as centers in a disjoint packing of Hamming
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"Alex Samorodnitsky"
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1af3bc245358de9bfd1b732306f966f3e0cf36c2 | subsection | 4 | 22 | Bounds on binary error correcting codes | It is shown in that|\partial ^* B | \le 4\left(\frac{n}{2} - \sqrt{r(n-r)} +o(n)\right) \cdot \frac{|B|}{2^n}Combined with Theorem REF , this shows that a
binary error-correcting code with minimal distance d is at most as large, up to
negligible multiplicative factors, as a Hamming ball of radius r =
n/2 - \sqrt{d(n-d)... | {
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"Alex Samorodnitsky"
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efb813da4702b3eb1a74f1c728f6495bb3515e3e | subsection | 5 | 22 | An isoperimetric problem for the Hamming cube | In order to obtain the best possible bounds on codes via Theorem REF , we need to find subsets of the Hamming cube with the smallest possible fractional edge-boundary. In particular, an existence of subsets whose fractional boundary is noticeably smaller than that of Hamming balls of the same cardinality, would imply a... | {
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... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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4aa826b319b8ab50dd3ebca6ca64f502137cae76 | subsection | 6 | 22 | Main results | Our main technical result is a modified version of the logarithmic Sobolev inequality (REF ). Let H(x) = -x\log x - (1-x) \log (1-x) be the "natural" (i.e., using natural logarithms) entropy function.Theorem 1.2 :Let f:~\lbrace 0,1\rbrace ^n\rightarrow \mathbb {R} be a non-zero function, and let \rho = \frac{1}{n} \fra... | {
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"start": 1292
... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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9ca91318e44d72b31c23ad53caaeb1a0e77021bc | subsection | 7 | 22 | Main results | It implies, in particular, that as the ratio \frac{\mathbb {E}f^2}{\mathbb {E}^2 f} grows (the function f becomes less "flat") its edge-isoperimetric constant approaches the isoperimetric constant C = \frac{2}{\log 2} in the edge-isoperimetric inequality (REF ) for 0-1 functions. One possible partial explanation for th... | {
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"raw": "R. J. McEliece, E. R. Rodemich, H. Rumsey, Jr., and L. R. Welch, New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities, IEEE Trans. Inform. Theory, vol. IT-23, 1977, 157-... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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54dfbfc4f98995e54127cbf828a03fa5fb1b657a | subsection | 8 | 22 | Main results | Then for any subset A of \lbrace 0,1\rbrace ^n holds
|\partial ^* A| \ge 4n \left(\frac{1}{2} - \sqrt{H^{-1}\left(\frac{\log |A|}{n}\right)\left(1 - H^{-1}\left(\frac{\log |A|}{n}\right)\right)}\right) \cdot \frac{|A|}{2^n}
On the other hand, let B be a Hamming ball. Then
|\partial ^* B| \le 4n \left(\frac{1}{2} - ... | {
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"start": 271
... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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81e239f77e029810b6c8adea509dc2f650d81e68 | subsection | 9 | 22 | Main results | Let 0 \le t \le 1 and let T = T_t be
a linear operator taking a function f = \sum _S \hat{f}(S) w_S to Tf
= \sum _S t^{|S|} \hat{f}(S) w_S. Then ()\Vert Tf\Vert _2 \le \Vert f\Vert _{1+t^2}Substituting (REF ) instead of (REF ) in the proof in leads to a modified version of (REF ). It turns out that the exponent 2 on th... | {
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applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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58887cc5a1deab53216c3d91b0945d8cacc215f9 | subsection | 10 | 22 | The proof of Theorem | Let us start with a brief overview.
The main goal of this section is to prove the logarithmic Sobolev inequality (REF ).
Our proof follows the outline of the proof of (REF ) in . We will prove an inequality (REF ), which will imply (REF ) as a corollary, first for the base case n=1, and then for general n, using subadd... | {
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applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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70849c14129fba8e3427283eefaabb49c27999cc | subsection | 11 | 22 | The proof of Theorem | The function c(t) has an explicit representation
c(t) = \frac{4}{t} \cdot \left(\frac{1}{2} - \sqrt{H^{-1}(\log 2 - t)\Big (1 -
H^{-1}(\log 2 - t)\Big )}\right).
In other words c = C, where C is the function in (REF ).Note that the second claim of Theorem REF follows from the third and the fourth claims of this lemma... | {
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"ar... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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8f7a2ea1a9890662e9fe6ce7b582d8fdbeea2eb6 | subsection | 12 | 22 | The proof of Theorem | Let f^{(x)}_i be the restriction of f to this one-dimensional cube.Recall () that entropy is subadditive, namely\sum _{i=1}^n \mathbb {E}_{x} Ent\left(f^{(x)}_i\right) \ge Ent(f),while D^2(f) is additive, that is\sum _{i=1}^n \mathbb {E}_{x} D^2\left(f^{(x)}_i\right) = D^2(f).K^2(f) is also
additive, since \sum _{i=1}^... | {
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"raw": "M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Seminaire de Probabilites, XXXIII, Lecture Notes in Math., 1709, Springer 1990, pp. 120-216.",
"source_ref_id": "8daf8... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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cad654bec4ef136142abaa42528c8019db972674 | subsection | 13 | 22 | The proof of Theorem | Then (REF ) remains valid (as equality) in the one-dimensional case. It is easy to see that this formal definition does not affect the computation below.The general case. Let n \ge 1. Then, by the base case, by
subadditivity of the entropy, and by convexity and monotonicity of \phi :D^2(f) = \sum _{i=1}^n \mathbb {E}_x... | {
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"raw": "D. Falik, A. Samorodnitsky, A combinatorial proof of a theorem of Kahn, Kalai, and Linial, CPC, to appear.",
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"start": 1000
... | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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7a7b22ee9ad92f27887b10535961526e63f93e3d | subsection | 14 | 22 | The proof of Theorem | ThenD^2(f) = m Ent\left(f^2\right) \ge m \rho n.This means K^2(f) = n \mathbb {E}f^2 - 1/4 D^2(f) \le \left( 1 - (m\rho )/4\right) n, and therefore \frac{Ent\left(f^2\right)}{K^2(f)} \ge \frac{4\rho }{4 - m\rho }.Recall \phi is an increasing convex function on [0,2 \log 2] with
\phi (0) = 0. Therefore the function \tau... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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77841a9499257d7cfb54f51e7248dad8c57684f7 | subsection | 15 | 22 | Proof of Lemma | Leth(t) = \frac{1}{2} (1-t)^2 \log (1-t)^2 + \frac{1}{2} (1+t)^2 \log (1+t)^2 - \left(1+t^2\right)\log \left(1+t^2\right).In other words, h(t) = \psi \left(t^2\right). We will start with some useful properties of the function h.Lemma 3.1 :h^{\prime } \ge h \ge 0
\left(1-t^2\right) h^{\prime } \ge t h^{\prime \prime... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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bcf2bd175729dcab050391139cba76a0a3deea4e | subsection | 16 | 22 | Proof of Lemma | Substituting these series in the inequality above, we need to showt^3 \cdot \sum _{k=0}^{\infty } \frac{t^{4k+2}}{2k+1} + \left(1-t^4\right) \cdot \sum _{k=0}^{\infty } \frac{t^{2k+1}}{2k+1} \ge t,and this is easily verified by observing that all the higher coefficients of the power series on the left hand side are non... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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a0a0c45306f767a23a223d9bffb05ab0ddbb3087 | subsection | 17 | 22 | Proof of Lemma | Therefore, it suffices to prove\psi \ge (1+t) \psi ^{\prime } + 2 \psi ^{\prime \prime }Again, since \psi is concave with \psi (0) = 0, we have \psi \ge t \psi ^{\prime }. Therefore, we only need to prove-2 \psi ^{\prime \prime } \ge \psi ^{\prime }Writing this in terms of h, this is equivalent to \left(1-s^2\right) h^... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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ae72bd46b601f51153fb635e799dba9c77e1e5b7 | subsection | 18 | 22 | Proof of Lemma | Consequently, \alpha ^{\prime }(\xi (t)) =
\frac{1}{\xi ^{\prime }(t)}, and \alpha ^{\prime \prime }(\xi (t)) =
-\frac{\xi ^{\prime \prime }(t)}{\left(\xi ^{\prime }(t)\right)^3}. Therefore, \alpha ^{\prime \prime } >
2\left(\alpha ^{\prime }\right)^2 is equivalent to -\xi ^{\prime \prime } > 2\xi ^{\prime }, which is ... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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f0ab9ae330bdfc12c4359ce86c3fa120ec77954e | subsection | 19 | 22 | Proof of Lemma | As in the proof of Claim 1, expressing \psi and its derivatives in terms of h, leads to the following equivalent inequality:2x(h^{\prime })^2 > \left(3-x^2\right)hh^{\prime } + x\left(1+x^2\right) h h^{\prime \prime }From now on we concentrate on the proof of (REF ). It will be convenient to write h and its derivatives... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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4229ad90a8f7922647701eefa135ee79dbc8f05f | subsection | 20 | 22 | Proof of Lemma | For all odd k starting from k = 3:
\ell _{2k+1} > r_{2k+1}~~~and~~~\ell _{2k+1} + \ell _{2k+3} > r_{2k+1} + r_{2k+3}This will implyF(x) - G(x) = 4\cdot \sum _{k=3}^{\infty } \left(\ell _{2k+1} - r_{2k+1}\right) x^{2k+1} = 4\cdot \sum _{odd~k \ge 3} \left(\left(\ell _{2k+1} - r_{2k+2}\right) - \left(\ell _{2k+3} - r_{2... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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02ab10f8e89dfaf697352b6ea3554c69e4c57863 | subsection | 21 | 22 | Proof of Lemma | For completeness sake, we do list explicit expressions for the coefficients belowOur apologies to the reader..For an odd k \ge 3:
\ell _{2k+1} = \left(\frac{8k - 20}{(2k-3)(2k+1)}\right) \cdot \sum _{m=1}^{(k-1)/2} \frac{1}{4m-3} ~~~+~~~ \frac{4}{2k-1} \cdot \sum _{m = 1}^{k-2} \frac{1}{2m+1} ~~~+~~~
\left(\frac{3}{... | {
"cite_spans": []
} | 0807.1679 | A modified logarithmic Sobolev inequality for the Hamming cube and some
applications | [
"Alex Samorodnitsky"
] | [
"math.CO"
] | 2,008 | en | Mathematics | [
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b2e43483feb3f6f0a76b1d291b28c3cfd49de522 | abstract | 0 | 7 | Abstract | The production of prompt photons at high energies provides a direct probe of
the dynamics of the strong interactions. In particular, one expect that it
could be used to constrain the behavior of the nuclear gluon distribution in
$pA$ and $AA$ collisions. In this letter we investigate the influence of
nuclear effects in... | {
"cite_spans": []
} | 10.1103/PhysRevC.78.037901 | 0807.1680 | Nuclear shadowing and prompt photons at relativistic hadron colliders | [
"C. Brenner Mariotto",
"V. P. Goncalves"
] | [
"hep-ph"
] | 2,008 | en | Physics | [
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193af2427b39dc59adabd92cd6c645fba05cec1e | subsection | 1 | 7 | Body | Nuclear shadowing and prompt photons at relativistic hadron colliders
13.85.Qk; 12.38.-t
C. Brenner Mariotto ^{a} and V.P. Gonçalves
^{b}^a Departamento de Física, Universidade Federal do Rio GrandeCaixa Postal 474, CEP 96201-900, Rio Grande, RS, Brazil^b Instituto de Física e Matemática, Universidade Federal de
Pelota... | {
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{
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"raw": "See, e.g., U. Heinz, Nuc. Phys. A 685, 414 (2000).",
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... | 10.1103/PhysRevC.78.037901 | 0807.1680 | Nuclear shadowing and prompt photons at relativistic hadron colliders | [
"C. Brenner Mariotto",
"V. P. Goncalves"
] | [
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] | 2,008 | en | Physics | [
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97de9d8540db8e4b3405a9af2bfa3fc3da3052e1 | subsection | 2 | 7 | Body | It is demonstrated by the analysis of the Fig. REF , where we present the results for the ratio R_g \equiv xg_A/A.xg_N predicted by the EKS , DS , HKN , and EPS parameterizations at Q^2 = 2.5 GeV^2 and A = 208. These four groups realize a global analysis of the nuclear experimental data using the DGLAP evolution equati... | {
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d517f43324f36bc13b4e8208f045d85da071e6eb | subsection | 3 | 7 | Body | In this case, the prompt
photon production cross section is given by where x_T=2p_T/\sqrt{s}, y and p_T are the rapidity and
transverse momentum of the produced photon, f_i (x,Q^2) are the
parton densities, x_1 and x_2 are the momentum fractions of the
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x_2... | {
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a210b438cd2df6b5a6d43777ce6db0fcba2c3351 | subsection | 4 | 7 | Body | The same happens for the HKN one in this limit, but this growth starts only at x>10^{-2}, with R_g being flat for 10^{-5}<x< 10^{-2}. At x\approx 10^{-1}, we have that behaviors predicted by the EKS and EPS parameterizations are similar, with R_g exceeding 1.2. The main distinction between these parameterizations is th... | {
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} | 10.1103/PhysRevC.78.037901 | 0807.1680 | Nuclear shadowing and prompt photons at relativistic hadron colliders | [
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07ab2d77ff59dd3048a8e52876106d06220c65e4 | subsection | 5 | 7 | Body | From these results we can conclude that the study of the ratio R_{pA} at RHIC can be useful to determine the presence or not of the antishadowing and constrain its magnitude.
On the other hand, at LHC energies all parameterizations predict a ratio smaller than one, with the suppression directly associated to the magnit... | {
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... | 10.1103/PhysRevC.78.037901 | 0807.1680 | Nuclear shadowing and prompt photons at relativistic hadron colliders | [
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8dc823c51fe4951aaacd00ed0699359c311492cf | subsection | 6 | 7 | Body | As shown in Ref. , where the prompt photon production in pp collisions was studied considering the solution of GLR-MQ evolution equation for the proton gluon distribution obtained in , new effects are expected when nonlinear PDFs are used in the calculations. A similar expectation is valid for the nuclear case.In summa... | {
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... | 10.1103/PhysRevC.78.037901 | 0807.1680 | Nuclear shadowing and prompt photons at relativistic hadron colliders | [
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8e05005163bb7351acfba12a355b4fb063432340 | abstract | 0 | 6 | Abstract | The Eyring-Kramers law describes the mean transition time of an overdamped
Brownian particle between local minima in a potential landscape. In the
weak-noise limit, the transition time is to leading order exponential in the
potential difference to overcome. This exponential is corrected by a prefactor
which depends on ... | {
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} | 0807.1681 | The Eyring-Kramers law for potentials with nonquadratic saddles | [
"Nils Berglund",
"Barbara Gentz"
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21c9160ef31a158d8657f64428bbdb68025eb50a | subsection | 1 | 6 | Introduction | Consider the stochastic differential equation\operatorname{d}\!x_t = -\nabla V(x_t)\hspace{0.0pt}\operatorname{d}\!t + \sqrt{2\varepsilon } \hspace{0.0pt}\operatorname{d}\!W_t\;,where V:\mathbb {R}\hspace{0.0pt}^d\rightarrow \mathbb {R}\hspace{0.0pt} is a confining potential. The Eyring–Kramers law
(, ) describes the e... | {
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} | 0807.1681 | The Eyring-Kramers law for potentials with nonquadratic saddles | [
"Nils Berglund",
"Barbara Gentz"
] | [
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94691ba71a789b96b687480b21893f8943e2019d | subsection | 2 | 6 | Introduction | Unfortunately, it turns out that when translated into terms
of semiclassical analysis, the problem of proving the Eyring–Kramers formula
becomes a particularly intricate one, known as “tunnelling through
non-resonant wells”. The first mathematically rigorous proof
of (REF ) in arbitrary dimension (and its generalisatio... | {
"cite_spans": []
} | 0807.1681 | The Eyring-Kramers law for potentials with nonquadratic saddles | [
"Nils Berglund",
"Barbara Gentz"
] | [
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511309f94625bad1d7f8df6a12bedb37aaf0c4b1 | subsection | 3 | 6 | Introduction | Financial support by the French Ministry of Research, by way of the
Action Concertée Incitative (ACI) Jeunes Chercheurs,
Modélisation stochastique de systèmes hors équilibre, and the
German Research Council (DFG), by way of the CRC 701 Spectral Structures and Topological Methods in Mathematics, is
gratefully acknowledg... | {
"cite_spans": []
} | 0807.1681 | The Eyring-Kramers law for potentials with nonquadratic saddles | [
"Nils Berglund",
"Barbara Gentz"
] | [
"math.PR"
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2baeb887748126f4e561542331d046f92c64d562 | subsection | 4 | 6 | Classification of nonquadratic saddles | We consider a continuous, confining potential V:\mathbb {R}\hspace{0.0pt}^d\rightarrow \mathbb {R}\hspace{0.0pt}, bounded below by
some a_{0}\in \mathbb {R}\hspace{0.0pt} and having exponentially tight level sets, that is,\int _{\lbrace x\in \mathbb {R}\hspace{0.0pt}^d\colon V(x)\mathrel a\rbrace } \operatorname{e}^{-V... | {
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} | 0807.1681 | The Eyring-Kramers law for potentials with nonquadratic saddles | [
"Nils Berglund",
"Barbara Gentz"
] | [
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9ee57b2629973a67eb5fc4d05191c70ebd63d6c7 | subsection | 5 | 6 | Topological definition of saddles | We start by introducing the notion of a gate between two sets A and B.
Roughly speaking, a gate is a set that cannot be avoided by those
paths
going from A to B which stay as low as possible in the potential
landscape. Saddles will then be defined as particular points in
gates.It is useful to introduce some terminology... | {
"cite_spans": []
} | 0807.1681 | The Eyring-Kramers law for potentials with nonquadratic saddles | [
"Nils Berglund",
"Barbara Gentz"
] | [
"math.PR"
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6821aebee765d902be4395dd89c50813c2f987f6 | abstract | 0 | 6 | Abstract | The near threshold production of K+K- pairs in proton-proton collisions has
been investigated at the cooler synchrotron COSY below and above the threshold
for the $\phi$ meson using the COSY-11 and ANKE facilities, respectively. The
excitation function determined for the pp-->ppK+K- reaction revealed a
statistically si... | {
"cite_spans": []
} | 10.1016/j.nuclphysbps.2008.09.035 | 0807.1682 | Study of the hadronic production of kaon pairs below the threshold for
the phi meson | [
"P. Moskal",
"M. Silarski",
"A. Budzanowski",
"E. Czerwinski",
"R. Czyzykiewicz",
"D. Gil",
"D. Grzonka",
"M. Janusz",
"L. Jarczyk",
"B. Kamys",
"A. Khoukaz",
"P. Klaja",
"W. Oelert",
"C. Piskor-Ignatowicz",
"J. Przerwa",
"B. Rejdych",
"J. Ritman",
"T. Sefzick",
"M. Siemaszko",
... | [
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1221c254eb437a36b5c825c1d9520b8a97f50726 | subsection | 1 | 6 | Introduction | A primary motivation for measuring cross sections for the pp\rightarrow ppK^{+}K^{-} reaction near the kinematical threshold
was the study of the hadronic interaction between K^{+} and K^{-} mesons in order to understand the structure of the
scalar resonances f_{0}(980) and a_{0}(980) .
Such measurements have been
made... | {
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55dd1ef2f1d0273a24e31630dd88c3f864cad64b | subsection | 2 | 6 | Measurements of the | The measurements of the pp\rightarrow ppK^{+}K^{-} reaction close to threshold
have been conducted using the cooler synchrotron COSY and the
COSY-11 detection system shown schematically in Fig. REF .
The target, being a beam of H_2 molecules grouped inside clusters of up to 10^5 atoms ,
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9efe9852594e8fb6bba7ffb7f5f5d0f863673ee4 | subsection | 3 | 6 | Measurements of the | The background, however, can be completely reduced by demanding a signal
in the silicon pad detectors (mounted inside the dipole) at the position where the K^{-} meson originating from the
pp\rightarrow ppK^{+}K^{-} reaction is expected (lower panel of Figure 2).
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a92ffc6b3c114d63cc5ca4f032685aff9dddc1b5 | subsection | 4 | 6 | Goldhaber plot analysis: generalization of the Dalitz plot for four
particle final state | Usage of the Dalitz plot for extracting information
about the interaction among particles
in the case of three body final states
is well known.
It was introduced by Dalitz in a nonrelativistic application and then
extended to the relativistic case by Fabri .
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0.019788792356848717,
0.01... |
db7b8dd8038b9248c10719df2ee7860c117701df | subsection | 5 | 6 | Goldhaber plot analysis: generalization of the Dalitz plot for four
particle final state | It is important to note that the event density in the Goldhaber plot
is not homogeneous and goes to zero on the entire boundary
of the plot given by the following equations: M_{12}+M_{34}=\sqrt{s},~M_{12}=m_{1}+m_{2},~M_{34}=m_{3}+m_{4}~.Figure REF (upper panel) presents the simulated distribution
for the ppK^{+}K^{-}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 238,
"openalex_id": "",
"raw": "P. Nyborg et al., Phys. Rev. 140 (1965) 914.",
"source_ref_id": "b1ce6daeb3a9d82d748e6b7311dd5d02174df11d",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end": 89... | 10.1016/j.nuclphysbps.2008.09.035 | 0807.1682 | Study of the hadronic production of kaon pairs below the threshold for
the phi meson | [
"P. Moskal",
"M. Silarski",
"A. Budzanowski",
"E. Czerwinski",
"R. Czyzykiewicz",
"D. Gil",
"D. Grzonka",
"M. Janusz",
"L. Jarczyk",
"B. Kamys",
"A. Khoukaz",
"P. Klaja",
"W. Oelert",
"C. Piskor-Ignatowicz",
"J. Przerwa",
"B. Rejdych",
"J. Ritman",
"T. Sefzick",
"M. Siemaszko",
... | [
"nucl-ex"
] | 2,008 | en | Physics | [
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-0.028890201821923256,
-0.04224409908056259,
-0.009576651267707348,
-0.04880658537149429,
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0.028401831164956093,
-0.022846609354019165,
... |
9896226fb1bd378734c3db76f75c4b998b5fe718 | abstract | 0 | 16 | Abstract | We investigate the Hubbard model on the triangular lattice at half-filling
using the dynamical cluster approximation (DCA) and dual fermion (DF) methods
in combination with continuous-time quantum Monte carlo (CT QMC) and
semiclassical approximation (SCA) methods. We study the one-particle properties
and nearest-neighb... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
-0.06507883965969086,
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0.00021379273675847799,
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0.002409550128504634,
... |
85ec23d7622c1fc55d20d7e02e6fdd2058e2f72e | subsection | 1 | 16 | Introduction | The physics of systems which exhibit strong electronic correlations and geometric
frustration at the same time is still unclear and therefore interesting.
Recent experiments, such as discovery of the pyrochlore compound {\rm LiV_{2}O_{4}} which show heavy fermion behavior
and organic materials \kappa -(BEDT-TTF)_{2}X w... | {
"cite_spans": [
{
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"end": 423,
"openalex_id": "",
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},
{
"arxiv_id": "",
"doi": "",
"end"... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
-0.04475841298699379,
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0... |
62f60ee94a3f2fd92ad3d821bd7f0027ef46cc9a | subsection | 2 | 16 | Model | We consider the two-dimensional Hubbard model on the triangular lattice.H=-t\sum _{\langle i,j \rangle \sigma }c^{\dag }_{i\sigma }c_{j\sigma }
+ U \sum _{i}n_{i\uparrow }n_{i\downarrow }where c_{i\sigma }(c^{\dag }_{i\sigma }) is the annihilation (creation) operator
of an electron with spin \sigma at the i-th site, t ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1192,
"openalex_id": "",
"raw": "H. M. et al, J. Phys. Soc. Jpn 71, 2109 (2002).",
"source_ref_id": "d3cdf07f4b3ec0e83110c058d961fbe678c35a6c",
"start": 973
},
{
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"doi": "",
"en... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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0.011144866235554218,
0.02650052309036255,
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0.007796066347509623,
0.03197760507464409,
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0.026790397241711617,
-0.008642802014946938,
0.03792763501405716,
0.014897963032126427,
0.01662... |
b34c2df6a9f01e515001deffb5a3543a22b17384 | subsection | 3 | 16 | DCA method | The DCA method , assumes that the self-energy in the
first BZ is constant and the coarse-grained Green's
function (DCA equation) is given by Eq. (2).\overline{G}_{\sigma }({\bf K},z)=\frac{1}{N}\sum _{\tilde{\bf K}}
\frac{1}{z-\epsilon _{{\bf K+\tilde{K}}}-\Sigma _{\sigma }({\bf K},z)},where N is the number of lattice ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 413,
"openalex_id": "",
"raw": "M. Hettler, A. Tahrildar-zadeh, M. Jarrell, T. Pruschken, and H. Krishnamurthy, Phys. Rev. B 58, R7475 (1998).",
"source_ref_id": "e46acd72349a2bcd456f591e596db72e0852af7a",
"start": 0
... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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0.07055521756410599,
0.007541504688560963,
-0.0024737813510000706,
0.022399451583623886,
... |
e9ac69f118b70b75b2ae91135366cb006ce2150d | subsection | 4 | 16 | DF method | The DF method , , is a relatively new method which can describe non-local correlations based
on the single-site DMFT method. The basic idea of the DF method is to convert
the hopping of different fermions into an effective coupling to an
auxiliary field. Each lattice site can be viewed as an impurity which is
easily de... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 125,
"openalex_id": "",
"raw": "A. Rubtsov, M. Katsnelson, and A. Lichtenstein, Phys. Rev. B 77, 033101 (2008).",
"source_ref_id": "7fc7012d5fa3e03cef01df294425afc52dd49f05",
"start": 0
},
{
"arxiv_id": "... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
-0.044820625334978104,
-0.01881001703441143,
-0.05717756971716881,
-0.021647537127137184,
0.0024980863090604544,
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0.044485002756118774,
-0.03569784387946129,
0.025812286883592606,
-0.02329513058066368,
... |
6ef5061e7dc97156ef6f1317c26b8ed05c882766 | subsection | 5 | 16 | DF method | We obtain the
lattice Green's function via Eq. (5).
Even though the DF method is an approximate method, it considers not only the short range but also the long
range correlations. Moreover, the calculation of the
two-particle properties does not introduce serious computational burden and
fermionic sign problem. | {
"cite_spans": []
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
-0.03106597252190113,
0.01876775361597538,
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0.036406390368938446,
-0.02851783111691475,
0.... |
e637780683d32f8b07c3f5c7c45f53c5f7b2028a | subsection | 6 | 16 | SCA method | At high temperature the Monte carlo integration over the auxillary classical
field \phi (\tau ) can be approximated by assuming \phi (\tau )\approx
const. This approximation is useful because it allows to check the QMC
results at temperature quickly. In this part we introduce the SCA method , as
impruity solver for DC... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 330,
"openalex_id": "",
"raw": "S. Okamoto, A. Fuhrmann, A. Comanac, and A. Millis, Phys. Rev. B 71, 235113 (2005).",
"source_ref_id": "133aadc093be463a0324aa3f1abe9a9e3d88f464",
"start": 252
},
{
"arxiv_... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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0.0354895144701004,
-0.0012492279056459665,
0.044... |
26f12ca18643aa22520455266fc433d460804960 | subsection | 7 | 16 | SCA method | By a Grassmann integration we can rewrite the partition function which is
represented as a four-dimensional integration in terms of \phi _j and the fermionic
Matsubara frequency.Z=\int d{\vec{\phi _{j} }}e^{-S_{eff}[{\bf a}(i\omega ),\phi _{j}]},where the effective action S_{eff}=\beta V is defined byV(\vec{\phi })=\fr... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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0.034295063465833664,
0.027719810605049133,
0... |
fd1bfe81a8f65a0738f117cef1730a5da7bfbe0b | subsection | 8 | 16 | CT QMC method | Here we describe the CT QMC method . The starting
point is action can be split into an unperturbed action S_0 and an interaction part W. By Taylor expansion of
partition function in powers of the interaction U, we can reexpress the partition
functionZ=\sum _{k,\sigma }Z_0 \frac{(-U)^k}{k!}\int dr_{1\sigma } \dots dr_{k... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 36,
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"raw": "A. Rubtsov, V. Savkin, and A. Lichtenstein, Phys. Rev. B 72, 035122 (2005).",
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"start": 0
}
]
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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... |
46a5451571a4405db099da3d6a0bab6b1d9285ed | subsection | 9 | 16 | CT QMC method | The Green's function is defined byG(r,r^{\prime })=\frac{\langle Tc^{\dagger }_{r^{\prime }}c_{r}c^{\dagger }_{r^{{\prime }_{1\sigma }}}c_{r_{1\sigma }}
\dots c^{\dagger }_{r^{{\prime }_{k\sigma }}}c_{r_{k\sigma }} \rangle }{\langle c^{\dagger }_{r^{{\prime }_{1\sigma }}}c_{r_{1\sigma }}
\dots c^{\dagger }_{r^{{\prime ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 495,
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"raw": "A. Rubtsov, V. Savkin, and A. Lichtenstein, Phys. Rev. B 72, 035122 (2005).",
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"start": 0
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{
"arxiv_id": "",
... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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0.03... |
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