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+{"text":"\\section{Introduction}\n\nThe study of uniform embeddings of metric spaces into Hilbert space was\ninitiated by Gromov.\n\n\\begin{defn}\\label{def1}\nLet $(\\Gamma,d)$ be a metric space. Let $\\mathcal{H}$ be a separable\nHilbert space. A map $f\\colon\\Gamma\\to\\mathcal{H}$ is said to be a\n{\\em uniform embedding\\\/}~{\\rm\\cite{Gr}} if there exist non-decreasing\nfunctions $\\rho_1$, $\\rho_2$ from $[0,\\infty)$ to itself such that\n\\begin{itemize}\n\\item[(1)] $\\rho_1(d(x,y))\\le \\|f(x)-f(y)\\|_{\\mathcal{H}}\\le \\rho_2(d(x,y))$\nfor all $x,y\\in\\Gamma$;\n\\item[(2)] $\\lim_{r\\to+\\infty}\\rho_i(r)=+\\infty$ for $i=1,2$.\n\\end{itemize}\n\\end{defn}\n\nGromov suggested that a finitely generated group $G$ (viewed as a metric space with a\nword length metric) uniformly embeddable into Hilbert space should satisfy the Novikov\nConjecture~\\cite[page~67]{Gr1}. That was proved in \\cite{Yu}, see also~\\cite{skandalis,Hig}.\n\n\n\nGuoliang Yu introduced a F\\o lner-type condition on finitely generated groups $G$,\ncalled property A, which is a weak form of amenability and which guaranties the\nexistence of a uniform embedding of the metric space into Hilbert space~\\cite{Yu}.\nThat property is interesting in itself because it turned out to be equivalent to\nthe exactness of $G$, and to the existence of an amenable action of $G$ on a compact\nspace (see \\cite{Tu} for a detailed discussion of property A).\n\nAmong ``classical\"  groups for which property A has not been proved so far, is the\nR.\\,Thompson group $F$, which is the group of all piecewise linear orientation\npreserving self-homeomorphisms of the unit interval with finitely many dyadic\nsingularities and all slopes integer powers of $2$.\n\nIt is not known whether $F$ is amenable or not, so the question about a weak\namenability is interesting.\n\nIt is worth noticing that, by a result of Niblo and Reeves, a group acting properly\nand cellularly by isometries on a CAT(0) cubical complex has the Haagerup\nproperty~\\cite{Niblo}. In particular, by a result of Farley \\cite{Farley}, this holds\nfor diagram groups over finite semigroup presentations. Moreover, this holds for all\ncountable diagram groups because they can be embedded into a certain ``universal\"\ndiagram group~\\cite{GS3} . It follows immediately from the definition of the Haagerup\nproperty that such a group can be embedded uniformly into Hilbert space. However,\nthe interaction between the Haagerup property and Guoliang Yu's property A is rather\ncomplicated~\\cite[Ch.~1.3]{cherix},~\\cite{GuKa,skandalis}. Notice that amenable groups\nsatisfy both the Haagerup property and property A.\n\n\\small\n$$\n\\begin{array}{ccccc}\nG\\ \\hbox{has property A}\\hspace{-1cm} & &\\Longrightarrow & &G\\ \\hbox{{\\small is\nuniformly }}\\\\\n                         &  &      &        & \\strut \\ \\hbox{{\\small  embedded\ninto }}\\\\\n\\Big\\Uparrow\\vcenter{\\rlap{{\\bf ?}}} &  &      &        &\\strut  \\ \\hbox{{\\small\nHilbert space}}\\\\\n& & &    & \\Big\\Uparrow\\\\\n& & &    & \\strut \\\\\nG\\ \\hbox{is a countable} & \\Longrightarrow& G\\ \\hbox{{\\small acts properly and}} &\n\\Longrightarrow & G \\ \\hbox{has Haagerup}\\\\\n\\ \\hbox{diagram group}& &\\strut  \\ \\hbox{cellularly by isometries }& &\\strut\\\n\\hbox{property}\\\\\n & &\\strut \\ \\hbox{on a CAT(0) }  & &\\\\\n & &\\strut \\ \\hbox{cubical complex}  & &\n\\end{array}\n$$\n\\normalsize\n\\bigskip\n\nGuentner and Kaminker \\cite{GuKa} introduced a natural quasi-isometry invariant of a\ngroup that shows how close to a quasi-isometry can a uniform embedding of a group\ninto a Hilbert space be.\n\n\\begin{defn}{\\rm (cf.\\cite[Def.~2.2]{GuKa})}\nThe Hilbert space compression of a finitely generated discrete group $G$ is the number\n$R(G)$, which is the supremum of all $\\alpha\\ge0$ for which there exists a uniform\nembedding of $G$ into a Hilbert space with $\\rho_1(n)=Cn^\\alpha$ with a constant $C>0$\nand linear $\\rho_2$ (see Definition \\ref{def1}).\n\\end{defn}\n\nBy ~\\cite{GuKa}, Hilbert space compression strictly greater than 1\/2 implies property A.\nAny group that is not uniformly embeddable into a Hilbert space (such groups exist\nby \\cite{random}) has Hilbert space compression 0. It is proved in \\cite{GuKa} that\nfree groups have Hilbert space compression 1 (although there are no quasi-isometric\nembeddings of a free group of rank $>1$ into a Hilbert\nspace~\\cite{bourg,linial}\\footnote{We thank D.~Sonkin for bringing these papers to our\nattention.}). More generally, by Campbell and Niblo \\cite{nibloS}, any discrete group\nacting properly, co-compactly on a finite dimensional CAT(0) cubical complex has Hilbert\nspace compression 1.\n\nNote that till now there were no examples of groups with compression\nstrictly between 0 and 1.\n\nOne of our main results is the following.\n\n\\begin{thm}\\label{Fhalf}\nThe Hilbert space compression of R.\\,Thompson's group $F$ equals $1\/2$.\n\\end{thm}\n\nIn particular, this shows that the  result by Guentner and Kaminker\non groups with compression strictly greater than $1\/2$ cannot be\napplied to $F$. However, one can extract a stronger fact\nfrom~\\cite{GuKa}. Suppose that a finitely generated group $G$ has a\nuniform embedding into Hilbert space with linear $\\rho_2(n)$ and\n$\\rho_1\\gg\\sqrt{n}$ (i.e.\n$\\lim_{n\\to\\infty}\\rho_1(n)\/\\sqrt{n}=\\infty$). The proof\nof~\\cite[Theorem~3.2]{GuKa} shows that property A holds for $G$ in\nthis case. For R.\\,Thompson's group $F$ we actually show that\n$\\rho_1\\le C\\sqrt{n} \\log n$ for some constant $C$.\n\n\n\\begin{quest}\n\\label{sqd}\nIs it true that there exists a uniform embedding of $F$ into Hilbert space\nwith linear $\\rho_2$ and $\\rho_1\\gg\\sqrt{n}$?\n\\end{quest}\n\nIf the answer is positive, then $F$ has property A.\n\\vspace{1ex}\n\n\nIt is known \\cite{GS1,GS5,Farley} that geometric properties of $F$\nare better understood when $F$ is considered as a diagram group.\nRecall that a diagram group is the fundamental group of the space of\npositive paths on a directed 2-complex~\\cite{GS3} (an equivalent\ndefinition in terms of semigroup presentations from~\\cite{GS1} is\ngiven below). For example, $F$ is the diagram group of the dunce hat\nconsidered as a directed $2$-complex with one vertex, one edge $x$ and\none directed cell $x=x^2$.\n\nOur interest in uniform embeddings for diagram groups was stimulated\nin part by a connection with certain metric properties of diagram\ngroups which have been studied independently before. Recall that\nelements of a diagram group $G$ can be represented by diagrams, which are\nplane cellular complexes subdivided into a number of regions (cells).\nThis allows one to introduce a canonical {\\em diagram metric\\\/} ${\\mathrm{dist}}_d$ on\n$G$ such that the ${\\mathrm{dist}}_d(g_1,g_2)$ is the number of cells in the diagram\nwith minimal number of cells representing the element $g_1^{-1}g_2$. This\nmetric is left invariant. It is proper provided the corresponding directed\n2-complex (semigroup presentation) is finite or when the diagram group is\nfinitely generated. It is known (this is proved in Burillo~\\cite{burillo} using\na different terminology) that for the R.\\,Thompson group $F$ considered as a\ndiagram group of the dunce hat, the diagram metric is bi-Lipschitz equivalent\nto the word metric. We say that a finitely generated diagram group satisfies\n{\\em property B\\\/} if its diagram metric is bi-Lipschitz equivalent to the word\nmetric.\n\n\\begin{thm}\\label{dghalf}]\nThe Hilbert space compression of a finitely generated diagram group with property B\nis at least $1\/2$.\n\\end{thm}\n\n\\begin{quest} \\label{prb} Do all finitely generated diagram groups satisfy property B?\n\\end{quest}\n\nWe think that is a very interesting question.\n\\vspace{2ex}\n\nIn \\cite{GS1,GS3}, it is proved that there are (finitely presented) diagram groups\ncontaining all countable diagram groups as subgroups. Such diagram groups are called\n{\\em universal\\\/}. One of these diagram groups (in this paper it is denoted by $U$)\ncorresponds to the semigroup presentation $\\langle\\, x,a\\mid  x^2=x^3,ax=a\\,\\rangle$.\n\n\n\\begin{thm}\\label{BurU}\nThe universal diagram group $U$ satisfies property B.\n\\end{thm}\n\n\n\nNotice that Theorem \\ref{BurU} does not imply (at least directly) a\npositive answer to Question \\ref{prb}. Indeed, $U$ contains the direct\nproduct $G={\\mathbb F}_2\\times{\\mathbb F}_2$ of two free groups of rank 2\n(which is a countable diagram group). By a result of Mikhailova \\cite{Mikh},\nthe group $G$ contains a non-recursively distorted subgroup $H$. Hence the word\nmetric on a diagram group may not be equivalent to the metric induced by the\nembedding of $H$ into $U$. Certainly Theorem \\ref{BurU} implies that all\nundistorted subgroups of $U$ satisfy property B. Clearly the diagram metric\non a finitely generated diagram group does not exceed a constant times the word\nmetric. An easy argument shows that for any finitely generated diagram group,\nthere exists a recursive function $f(n)$ such that the word metric does not\nexceed $f({\\mathrm{dist}}_d)$.\n\nBesides the ~R.\\,Thompson group $F$, the restricted wreath product\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ is another typical representative of the class of diagram groups.\nIt corresponds to the presentation\n$\\langle\\, a,b,b_1,b_2,c\\mid ab=a,bc=c,b=b_1,b_1=b_2, b_2=b\\,\\rangle$ \\cite{GS5}.\nIt also satisfies property B. Hence Theorem \\ref{dghalf} can be applied. The\ngroup ${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ is amenable, so it satisfies property A, but the\nnext theorem shows that the compression of that group is not $1$.\n\n\\begin{thm} \\label{wr}\nThe Hilbert space compression of the restricted wreath product\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ belongs to $[1\/2,3\/4]$.\n\\end{thm}\n\n\\begin{quest} What is the Hilbert space compression of ${\\mathbb{Z}}\\wr{\\mathbb{Z}}$?\n\\end{quest}\n\nFinally we prove the following general result.\n\n\\begin{thm} \\label{wr1} Let $H$ be a finitely generated group with\na su\\-per-po\\-ly\\-no\\-mial growth function. Then the Hilbert space compression of\n${\\mathbb{Z}}\\wr H$ is at most $1\/2$.\n\\end{thm}\n\nThis theorem immediately implies\n\n\\begin{cy} The Hilbert space compression of the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$\nis at most $1\/2$.\n\\end{cy}\n\nNote that the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$ is amenable and so it satisfies property A.\nThus property A does not imply that Hilbert space compression is bigger than $1\/2$.\nIt is not known whether ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$ is a diagram group (the most probable answer\nseems to be negative). Thus Theorem \\ref{dghalf} does not apply and we do not know if the\nHilbert space compression of this group is exactly $1\/2$.\n\n\\begin{quest} What is the Hilbert space compression of the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$?\nAre there amenable groups with Hilbert space compression 0 or arbitrary close to 0?\n\\end{quest}\n\n\nFinally, we would like to mention that our proof of Theorem~\\ref{dghalf} yields actually a\nstronger result. Namely, a lower bound on the the {\\it equivariant\\\/} Hilbert space\ncompression $R_G(G)$ which is the Hilbert space compression defined by restricting\nto $G$-equivariant uniform embeddings of $G$ into Hilbert spaces equiped with actions\nof $G$ by {\\it affine isometries\\\/}~\\cite[Section 5]{GuKa}.\n\n\\begin{thm}\\label{B-equiv}\nThe equivariant Hilbert space compression of a finitely generated diagram group\nwith property B is at least $1\/2$.\n\\end{thm}\n\nThe estimate $1\/2$ cannot be improved since $R_G(G)>1\/2$ implies amenability~\\cite{GuKa}\nand there are non-amenable diagram groups with property B~\\footnote{We thank Y. de Cornulier\nfor this remark.}. Moreover, one can see from the proof that the diagram metric of a\nfinitely generated diagram group is a conditionally negative definite function on the group.\nRecall that a function $\\psi : G\\to \\mathbb{R}$, satisfying $\\psi(g)=\\psi(g^{-1})$\nis said to be conditionally negative definite if $\\sum\\psi(g_ig_j^{-1})c_ic_j\\le 0$\nfor all $n\\in\\mathbb{N}$ and $g_1,\\dots ,g_n\\in G, c_1, \\dots , c_n\\in \\mathbb{R}$ such that\n$\\sum c_i=0$. The existence of a continious conditionally negative definite function $\\psi : G\\to [0,\\infty)$ with\n$\\lim_{g\\to\\infty}\\psi(g)=\\infty$ is equivalent to the a-T-menability ~\\cite{ch-b-v}, hence\nto the Haagerup approximation property~\\cite{cherix}. For more applications see~\\cite{GuK} and references therein.\n\n\n\n\\section{Preliminary information about diagram groups}\n\n\\subsection{Diagram groups}\\label{diagram}\n\nLet us briefly recall the concept of a diagram group and some\nterminology from this area. Details can be found in~\\cite{GS1}. For\nthe reasons of the present paper, it is enough to use the definition\nin terms of semigroup presentations given in \\cite{GS1} rather than\nan equivalent definition from \\cite{GS3}.\n\nLet $X$ be an alphabet. We denote by $X^*$  the set of all words over $X$\nand by $X^+$ the set of all nonempty words. The length of a word $w$\nis denoted by $|w|$.\n\nLet ${\\cal P}=\\langle\\, X\\mid{\\cal R}\\,\\rangle$ be a semigroup presentation. Here ${\\cal R}$ consists of\nordered pairs of nonempty words over $X$. We always assume that if $(u,v)\\in{\\cal R}$,\nthen $(v,u)\\notin{\\cal R}$.\n\nA triple of the form $(p,u=v,q)$ is called an {\\em atomic $2$-path\\\/}, where\n$p,q\\in X^*$, $(u,v)\\in{\\cal R}\\cup{\\cal R}^{-1}$. The following labelled plane graph is\ncalled an {\\em atomic diagram\\\/} (over ${\\cal P}$):\n\n\\begin{center}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(110.66,26.33)\n\\put(0.00,13.33){\\circle*{1.33}}\n\\put(40.00,13.33){\\circle*{1.33}}\n\\put(70.00,13.33){\\circle*{1.33}}\n\\put(110.00,13.33){\\circle*{1.33}}\n\\bezier{200}(40.00,13.33)(55.00,33.33)(70.00,13.33)\n\\bezier{200}(40.00,13.33)(55.00,-6.67)(70.00,13.33)\n\\put(20.00,17.33){\\makebox(0,0)[cc]{$p$}}\n\\put(55.00,25.33){\\makebox(0,0)[cc]{$u$}}\n\\put(90.00,17.33){\\makebox(0,0)[cc]{$q$}}\n\\put(55.00,0.00){\\makebox(0,0)[cc]{$v$}}\n\\put(0.33,13.33){\\line(1,0){39.67}}\n\\put(70.00,13.33){\\line(1,0){40.00}}\n\\end{picture}\n\\end{center}\n\n\\noindent Here each segment labelled by a word $w$ is subdivided into $|w|$ edges\nlabelled by the letters of $w$. (Segments have a left-to-right orientation.)\n\nNotice that each atomic diagram has the {\\em top path\\\/} (in our case it is\nlabelled by $puq$) and the {\\em bottom path\\\/} (labelled by $pvq$). This atomic\ndiagram is defined uniquely by a $2$-path $(p,u=v,q)$ so we will often use\nthis notation for the atomic diagram itself.\n\nSuppose that we have a sequence $\\Delta_1$, \\dots, $\\Delta_k$ of atomic\ndiagrams. Let us consider the case when the bottom label of each $\\Delta_i$\n($1\\le i<k$) coincides with the top label of $\\Delta_{i+1}$. We can thus\nidentify the bottom path of $\\Delta_i$ with the top path of $\\Delta_{i+1}$\n($1\\le i<k$) to obtain the new labelled plane graph denoted by\n$\\Delta:=\\Delta_1\\circ\\cdots\\circ\\Delta_k$ (the {\\em concatenation of\\\/}\nthe above {\\em atomic diagrams\\\/}). One can naturally define the top path\n(which will be the top path of $\\Delta_1$) and the bottom path (the bottom\npath of $\\Delta_k$) of $\\Delta$.\n\nFor every word $w\\in X^+$, we define the {\\em trivial diagram\\\/} denoted by\n$\\varepsilon(w)$, which is the interval labelled by $w$. Its top and bottom paths\ncoincide with itself.\n\n\\begin{defn}\nA {\\em diagram $\\Delta$ over a semigroup presentation ${\\cal P}$\\\/} is either a trivial\ndiagram, or a concatenation of atomic diagrams over ${\\cal P}$. The diagrams are\nconsidered identical whenever they are isotopic as labelled plane graphs.\n\\end{defn}\n\nFor a diagram $\\Delta$, its top and bottom paths will be denoted by\n$\\topp{\\Delta}$ and $\\bott{\\Delta}$, respectively. They start at the\nsame vertex denoted by $\\iota(\\Delta)$ and end at the same vertex\ndenoted by $\\tau(\\Delta)$. The diagram itself is situated ``between\"\nits top and bottom paths. If $w'$ is the label of the top path of\n$\\Delta$ and $w''$ is the label of its bottom path, then we say that\n$\\Delta$ is a $(w',w'')$-diagram.\n\nGiven two diagrams $\\Delta'$ and $\\Delta''$, one can define their\n{\\em concatenation\\\/} denoted by $\\Delta'\\circ\\Delta''$, provided the\nlabels of $\\bott{\\Delta'}$ and $\\topp{\\Delta''}$ coincide.\n\nFor every word $w\\in X^+$, the set of all $(w,w)$-diagrams over ${\\cal P}$\nforms a monoid with $\\varepsilon(w)$ as a unit.\n\nFor any diagram $\\Delta$, by $\\Delta^{-1}$ we denote the mirror image of\n$\\Delta$ with respect to the axis connecting $\\iota(\\Delta)$ and $\\tau(\\Delta)$).\nClearly, the top (bottom) path of $\\Delta$ becomes the bottom (top) path of\n$\\Delta^{-1}$.\n\\vspace{1ex}\n\nHere is an example of two diagrams over $\\langle\\, x\\mid x^3=x^2\\,\\rangle$:\n\n\\begin{center}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(126.33,25.67)\n\\put(80.33,12.00){\\circle*{1.33}}\n\\put(95.33,12.00){\\circle*{1.33}}\n\\put(110.33,12.00){\\circle*{1.33}}\n\\put(125.33,12.00){\\circle*{1.33}}\n\\bezier{200}(80.33,12.00)(95.33,32.00)(110.33,12.00)\n\\bezier{212}(95.33,12.00)(110.33,-10.00)(125.33,12.00)\n\\put(88.33,20.00){\\circle*{1.33}}\n\\put(102.33,20.00){\\circle*{1.33}}\n\\put(102.33,4.00){\\circle*{1.33}}\n\\put(118.33,4.00){\\circle*{1.33}}\n\\put(4.33,1.67){\\circle*{1.33}}\n\\put(19.33,1.67){\\circle*{1.33}}\n\\put(34.33,1.67){\\circle*{1.33}}\n\\put(49.33,1.67){\\circle*{1.33}}\n\\put(64.33,1.67){\\circle*{1.33}}\n\\bezier{180}(19.33,1.67)(34.33,18.67)(49.33,1.67)\n\\put(27.33,8.00){\\circle*{1.33}}\n\\put(41.33,8.67){\\circle*{1.33}}\n\\bezier{152}(27.33,8.00)(10.33,18.67)(4.33,1.67)\n\\bezier{152}(41.33,8.67)(60.33,17.67)(64.33,1.67)\n\\put(7.33,7.67){\\circle*{1.33}}\n\\put(62.33,6.67){\\circle*{1.33}}\n\\put(18.33,11.67){\\circle*{1.33}}\n\\put(51.33,11.67){\\circle*{1.33}}\n\\bezier{228}(7.33,7.67)(16.33,34.67)(27.33,8.67)\n\\bezier{240}(41.33,8.67)(54.33,35.67)(62.33,6.67)\n\\put(11.50,17.00){\\circle*{1.33}}\n\\put(22.33,17.67){\\circle*{1.33}}\n\\put(46.33,17.67){\\circle*{1.33}}\n\\put(58.33,17.67){\\circle*{1.33}}\n\\put(34.33,17.67){\\circle*{1.33}}\n\\put(4.33,1.67){\\line(1,0){60.33}}\n\\put(22.33,17.67){\\line(1,0){24.00}}\n\\put(80.33,12.00){\\line(1,0){45.00}}\n\\end{picture}\n\\end{center}\n\nThe label of each edge of these diagrams is $x$.\n\nNotice that each cell in a diagram can be considered as a diagram\nitself. If the cell corresponds to a relation $u=v$, we shall denote\nthis diagram by $(u=v)$.\n\nLet $\\Delta$ be a diagram over ${\\cal P}$. Suppose that it contains a pair\nof cells $\\pi'$, $\\pi''$ such that the bottom path of $\\pi'$ coincides\nwith the top path of $\\pi''$ and the top label of $\\pi'$ equals the\nbottom label of $\\pi''$. In this case we say that the cells $\\pi'$\nand $\\pi''$ form a {\\em dipole\\\/} in $\\Delta$. One can define the\noperation of removing the dipole. Namely, given a dipole formed by\n$\\pi'$, $\\pi''$, we first remove the common boundary of these cells\nand then glue $\\topp{\\pi'}$ with $\\bott{\\pi''}$ (the paths we glue\nhave the same label). We get a new diagram over ${\\cal P}$ that has fewer\ncells. Proceeding in such a way, we get to a diagram that has no dipoles,\ncalled a {\\em reduced\\\/} diagram.\n\nKilibarda \\cite{KilDiss,Kil} proved that the process of cancelling\ndipoles in diagrams is confluent, that is, the result does not\ndepend on the order in which we remove the dipoles. Two diagrams are\ncalled {\\em equivalent} if the corresponding reduced diagrams are\nthe same.\n\nFor every word $w\\in X^+$, the set of all reduced $(w,w)$-diagrams\nwith operation ``concatenation followed by removing all dipoles\" is\na group ~\\cite[Lemma~5.2]{GS1}, denoted by ${\\cal D}({\\cal P},w)$.\n\n\\begin{defn}\nThe group ${\\cal D}({\\cal P},w)$ is called a {\\em diagram group\\\/} over a semigroup\npresentation ${\\cal P}$ with {\\em base\\\/} $w$.\n\\end{defn}\n\n\\begin{example}{\\rm \\cite[Ex.~6.4]{GS1}}\nThe diagram group ${\\cal D}({\\cal P},x)$ over the semigroup presentation\n${\\cal P}=\\langle\\, x\\mid x^2=x\\,\\rangle$ with base $x$ is R.\\,Thompson's group $F$\nthat has the following group presentation:\n$$\n\\langle\\, x_0,x_1,x_2,\\ldots\\mid x_jx_i=x_ix_{j+1}\\ (j>i)\\,\\rangle.\n$$\n\\end{example}\n\nThere is one more important binary operation on diagrams. Given two diagrams\n$\\Delta_1$, $\\Delta_2$ over ${\\cal P}$, one can identify $\\tau(\\Delta_1)$ and\n$\\iota(\\Delta_2)$. This gives a new diagram over ${\\cal P}$ denoted by\n$\\Delta_1+\\Delta_2$. This operation is associative but not commutative.\n\n\n\\subsection{Universal diagram groups}\\label{universal}\n\n\nIn \\cite{GS3}, Guba and Sapir found fi\\-ni\\-tely presented diagram\ngroups each of which contains every countable diagram group as a\nsubgroup. Such diagram groups are called {\\em universal\\\/}. For\ninstance, the diagram group $U={\\cal D}({\\cal P},a)$, where ${\\cal P}=\\langle\\, x,a\\mid\nx^3=x^2,ax=a\\,\\rangle$ is universal. It has the following Thompson-like\ngroup presentation:\n$$\n\\langle\\, x_0,x_1,x_2,\\ldots\\mid x_jx_i=x_ix_{j+1}\\ (j-i>1)\\,\\rangle.\n$$\nIt is easy to see that $U$ can be generated by $x_0$, $x_1$, $x_2$.\nIndeed, $x_n=x_2^{x_0^{n-2}}$ for all $n\\ge3$ ($a^b=b^{-1}ab$ by\ndefinition). (In fact this group is finitely presented \\cite{GS3}.)\n\nWe are going to describe explicitly the procedure of expressing an\nelement of $U$ as a word in generators $x_0$, $x_1$, $x_2$, \\dots\\,.\nLet $g\\in U$ be represented by an $(a,a)$-diagram $\\Delta$ over ${\\cal P}$.\nOne can decompose $\\Delta$ into a product of atomic diagrams. It is easy\nto see that the atomic diagrams that can occur have the form\n$(1,a=ax,x^m)^{\\epsilon}$, where $m\\ge0$, $\\epsilon=\\pm1$ or the form\n$(ax^k,x^2=x^3,x^m)^{\\epsilon}$, where $k,m\\ge0$, $\\epsilon=\\pm1$. In the first case\nwe assign the identity to the atomic diagram and in the second case we assign\nto it the element $x_m^{\\epsilon}$. Multiplying these elements, we get an expression\nfor the element $g\\in U$, see~\\cite[Section~6]{GS3}.\n\nGiven a diagram $\\Xi$ over $\\langle\\, x\\mid x^3=x^2\\,\\rangle$, we can canonically\nassign a diagram $\\Delta$ over $\\langle\\, x,a\\mid x^3=x^2,ax=a\\,\\rangle$ as follows.\nFirst we take the sum $\\varepsilon(a)+\\Xi$ and then add a number of cells of\nthe form $a=ax$ on the top to make the top label equal $a$. Then we add\na number of cells of the form $ax=a$ on the bottom in order to make the\nbottom label also equal $a$. The result is an $(a,a)$-diagram that\nrepresents an element of the group $U$.\n\nOn the above picture, where two diagrams over $\\langle\\, x\\mid x^3=x^2\\,\\rangle$ are\ndrawn, the described operation leads to the following elements of $U$,\nrespectively: $x_3x_1^{-1}x_0^{-1}x_3^{-2}x_1^{-1}$ and $x_1^{-1}x_0$.\nNotice that the way to decompose a diagram into the product of atomic\nfactors is not unique in general. For the first case, we always chose the\nrightmost cell that can be included into the next atomic factor. This\nprocedure will be described later in details.\n\n\n\\section{Main Results}\n\n\\begin{defn}\nLet $M$ be a set and $p,q\\colon M\\to[0,\\infty)$ be arbitrary functions. Then\n$p\\preceq q$ if there exists a constant $c>0$ such that $p(m)\\le cq(m)$ for every\n$m\\in M$. These functions are equivalent, $p\\sim q$, if $p\\preceq q$ and $q\\preceq p$.\n\\end{defn}\n\nLet $G$ be a group generated by a set $\\Sigma$. For any $g\\in G$, we denote by\n$\\ell_\\Sigma(g)$ the length of the shortest word over $\\Sigma^{\\pm1}$ that\nrepresents $g$. Notice that if $G$ is finitely generated, then all functions of the\nform $\\ell_\\Sigma$ are equivalent for all finite sets of generators. Therefore, the\nfunction $g\\mapsto\\ell(g)$ is unique up to equivalence. The function $\\ell$ on $G$\ndefines the {\\em word length metric\\\/} on $G$ as follows: the (word) distance\nbetween $g_1,g_2\\in G$ is $\\ell(g_1^{-1}g_2)$.\n\nWe will also omit the subscript $\\Sigma$ on $\\ell$ if the generating set of $G$ is clear.\n\nLet $G={\\cal D}({\\cal P},w)$ be a diagram group over a semigroup presentation ${\\cal P}$ with base $w$.\nLet $\\psi\\colon H\\hookrightarrow G$ be an embedding of a finitely generated group $H$\ninto $G$.\n\n\\begin{defn}\nWe say that $\\psi\\colon H\\hookrightarrow G$ is a $B$-{\\em embedding\\\/}, whenever\n$\\#(h)\\sim\\ell(h)$ for all $h\\in H$. Here $\\#(h)$ is the number of cells in the\nreduced $(w,w)$-diagram representing $\\psi(h)$.\n\\end{defn}\n\nAn obvious argument implies that $\\#(g)\\preceq\\ell(g)$ in all cases. Indeed, let $C$\nbe the maximum number of cells that represent generators of $H$. Then any element\nthat has length $n$ in these generators can be represented by a diagram over ${\\cal P}$\nwith at most $Cn$ cells.\n\nHowever, not every embedding of the above form is a $B$-embedding.\n\n\\begin{example}\nLet us consider the direct product $G={\\mathbb F}_2\\times{\\mathbb F}_2$ of\ntwo free groups of rank $2$. It is known that this is a diagram group, see,\nfor example, \\cite[Section~8]{GS1}. By a well-known result of Mikhailova \\cite{Mikh},\nthere is a finitely generated subgroup $H$ in $G$ with undecidable membership problem.\nThis implies that no recursive function $f(n)$ can have the property\n$\\ell(h)\\le f(\\#(h))$ for all $h\\in H$.\n\\end{example}\n\nThe most important case is when $G={\\cal D}({\\cal P},w)$ is a finitely generated diagram\ngroup and $H=G$ with the identical embedding $\\psi$.\n\n\\begin{defn}\nA finitely generated diagram group $G={\\cal D}({\\cal P},w)$ over a semigroup\npresentation ${\\cal P}$ with base $w$ {\\em has property\\\/} B whenever the\nidentical embedding $G\\hookrightarrow G$ is a $B$-embedding.\n\\end{defn}\n\nIn other words, for all $g\\in G$, one has $\\#(g)\\sim\\ell(g)$, where $\\#(g)$\nis the number of cells in the reduced diagram representing $g$.\n\nNotice that we cannot simply say that $G$ has property B itself.\nThis concept depends on the presentation ${\\cal P}$ and base $w$.\n\nR.\\,Thompson's group $F$ has property B as a diagram group over $\\langle\\, x\\mid\nx^2=x\\,\\rangle$ with base $x$. This immediately follows from a result of\nBurillo \\cite{burillo}.\n\n\\begin{proof}[Proof of Theorem \\ref{dghalf}] Let $G$ be a finitely generated\ngroup that can be B-embedded into a diagram group. We need to show\nthat the Hilbert space compression of $G$ is at least $1\/2$.\n\nGiven a diagram group ${\\cal D}({\\cal P},w)$, let us build the following geometric object\nassociated with this group.\nLet us take all reduced diagrams over ${\\cal P}$ that have $w$ as a top label. We\nidentify all top paths of these diagrams. This gives a 1-path $p$\nlabelled by $w$.\n\nSuppose that there are two reduced $(w,\\cdot)$-diagrams $\\Delta_1$,\n$\\Delta_2$ with decompositions of the form\n$\\Delta_i\\equiv\\Delta_i'\\circ\\Delta_i''$, where $i=1,2$ and\n$\\Delta_1'$, $\\Delta_2'$ are isotopic. Then we identify $\\Delta_1'$\nwith $\\Delta_2'$ via this isotopy. We do that for all pairs of\ndiagrams and all decompositions of them with the above property. The\nobject we get as a result is a directed 2-complex $\\cal T$, which turns\nout to be a rooted 2-tree in the sense of \\cite{GS3}. This directed\n$2$-complex can be viewed as a semigroup presentation if we assign\ndifferent labels to different edges and consider pairs of words\nwritten on the boundaries of the cells as relations. It is proved in\n\\cite{GS3} that for any path $q$ in $\\cal T$ with the same endpoints as\n$p$ there exists a unique reduced $(p,q)$-diagram over $\\cal T$.\n\nLet $\\mathbf F$ be the set of all geometric 2-cells of $\\cal T$. By\n${\\mathbb R}^{\\mathbf F}$ we denote the vector space over ${\\mathbb R}$\nwith ${\\mathbf F}$ as a basis. Clearly, this vector space is a\nsubset in Hilbert space ${\\cal H}=\\ell_2(\\mathbf F)$.\n\nEvery element $g$ of the diagram group ${\\cal D}({\\cal P},w)$ can be uniquely\nrepresented by a reduced $(w,w)$-diagram $\\Delta$. This diagram can\nbe naturally embedded into $\\cal T$. (The top of the diagram is identified\nwith the path $p$ under this embedding.) Let us assign to $g$ a function\n$\\phi_g$ from $\\mathbf F$ to $\\mathbb R$, where $\\phi_g(f)=1$ if\n$f\\in\\mathbf F$ is contained in the image of $\\Delta$ under the\nabove embedding and $\\phi_g(f)=0$ otherwise.\n\nSo we have a mapping $\\phi\\colon{\\cal D}({\\cal P},w)\\to{\\cal H}$ from the diagram group\n${\\cal D}({\\cal P},w)$ to ${\\cal H}$ defined by the rule $g\\mapsto\\phi_g$.\n\nAs we have already mentioned in the introduction, one can define a\ncanonical {\\em diagram metric\\\/} on ${\\cal D}({\\cal P},w)$ as follows: given\ntwo elements $g_1,g_2\\in{\\cal D}({\\cal P},w)$, one can define the {\\em diagram\ndistance\\\/}, denoted by ${\\mathrm{dist}}_d(g_1,g_2)$ between these elements as\nthe number of cells in the reduced diagram over ${\\cal P}$ representing\n$g_1^{-1}g_2$.\n\nSuppose that the diagram distance between two diagrams $\\Delta_1$,\n$\\Delta_2$ from ${\\cal D}({\\cal P},w)$ equals $n$. Let us consider the images\nof the diagrams $\\Delta_i$ ($i=1,2$) in $\\cal T$. They can be\ndecomposed as $\\Delta_i\\equiv\\Psi\\circ\\bar\\Delta_i$ ($i=1,2$), where\n$\\Psi$ is the ``greatest common divisor\" of $\\Delta_1$ and\n$\\Delta_2$. In this case we see that $\\bar\\Delta_1$, $\\bar\\Delta_2$\ndo not have common cells in $\\cal T$. The total number of cells in\n$\\bar\\Delta_1$ and $\\bar\\Delta_2$ equals $n$. Since\n$\\phi_{g_1}-\\phi_{g_2}$, as an element of ${\\mathbb R}^{\\mathbf F}$\nis a vector whose coordinates are $0, 1$ or $-1$, and the number of\nnon-zero coordinates is $n$, we conclude that the norm of\n$\\phi_{g_1}-\\phi_{g_2}$ is $\\sqrt{n}$.\n\nNow given a finitely generated group $G$, which is B-embedded into a\ndiagram group ${\\cal D}({\\cal P},w)$, we see that the word length metric in\n$G$ is equivalent to the diagram metric induced on $G$ as a subset\nin ${\\cal D}({\\cal P},w)$. Therefore, for some positive constants $C_1$, $C_2$\none has inequalities\n$$C_1\\sqrt{{\\mathrm{dist}}(g_1,g_2)}\\le\\|\\phi_{g_1}-\\phi_{g_2}\\|\\le\nC_2\\sqrt{{\\mathrm{dist}}(g_1,g_2)}\\le C_2{\\mathrm{dist}}(g_1,g_2).$$ Hence the Hilbert\nspace compression of $G$ is at least $\\frac12$. \n\\end{proof}\n\n We will use\nthe following result which is a generalization of the well-known\nparallelogram theorem to higher dimensions \\cite{DeMa}. Namely, let\n$E^n\\subset {\\mathbb R}^n$ be an $n$-dimensional hypercube. Suppose\nthat we have a mapping of the set of vertices of $E^n$ into a metric\nspace $M$. In this case we will say that we have a {\\em skew cube\\ }\nin $M$. For every edge of $E^n$ (there are exactly $2^{n-1}n$ of\nthem), by an edge of the skew cube we will mean the distance in $M$\nbetween the images of the endpoints of the edge. Similarly, for each\n(long) diagonal of $E^n$ (which connects opposite vertices of $E^n$)\nwe consider the corresponding {\\em diagonal\\\/} of the skew cube.\n\n\\begin{lm}{\\rm (Skew Cube Inequality \\cite{DeMa})}\\label{cube}\nFor every skew cube in a Hilbert space, the sum of squares of its\ndiagonals does not exceed the sum of squares of its edges.\n\\end{lm}\n\n\\begin{proof}[Proof of Theorem~\\ref{Fhalf}] We need to prove that the\nHilbert space compression of R.\\,Thompson's group $F$ equals $1\/2$.\n\nThe fact that the compression is at least $1\/2$, follows from\nTheorem \\ref{dghalf}.\n\nFor any $n\\ge0$, let us define $2^n$ elements of $F$ that commute\npairwise. All these elements will be reduced $(x,x)$-diagrams over\n${\\cal P}=\\langle\\, x\\mid x^2=x\\,\\rangle$. For $n=0$, let $\\Delta$ be the diagram\nthat corresponds to the generator $x_0$. Namely, if $\\pi$ is\nthe diagram that consists of one cell of the form $x=x^2$, then\n$\\Delta$ is $\\pi\\circ(\\varepsilon(x)+\\pi)\\circ(\\pi+\\varepsilon(x))^{-1}\\circ\\pi^{-1}$ by\ndefinition. It has 4 cells.\n\nSuppose that $n\\ge1$ and we have already constructed diagrams $\\Delta_i$\n($1\\le i\\le2^{n-1}$) that commute pairwise. For every $i$ we consider two\n$(x^2,x^2)$-diagrams: $\\varepsilon(x)+\\Delta_i$ and $\\Delta_i+\\varepsilon(x)$. We get $2^n$\nspherical diagrams with base $x^2$ that obviously commute pairwise. It remains\nto conjugate them to obtain $2^n$ spherical diagrams with base $x$ having the\nsame property. Namely, we take $\\pi\\circ(\\varepsilon(x)+\\Delta _i)\\circ {\\pi}^{-1}$ and\n$\\pi\\circ(\\Delta_i+\\varepsilon(x))\\circ\\pi ^{-1}$.\n\nLet us denote the elements of $F$ obtained in this way by $g_i$ ($1\\le i\\le2^n$).\nThese elements define a $2^n$-dimensional skew cube in $F$. It follows easily from\nthe construction that each $g_i$ has exactly $2n+4$ cells as a diagram. So the word\nlength of each $g_i$ is $O(n)$. Now for any $\\epsilon_i=\\pm1$ ($1\\le i\\le2^n$) we consider\nthe product of the form $g=g_1^{\\epsilon_1}\\cdots g_{2^n}^{\\epsilon_{2^n}}$. It is easy to see\nfrom definitions that the diagram that represents $g$ has the form\n$\\Gamma _n\\circ(\\Delta^{\\epsilon _1}+\\cdots+\\Delta^{\\epsilon_{2^n}})\\circ\\Gamma _n^{-1}$,\nwhere $\\Gamma_n$ is defined by induction in the following way: $\\Gamma _0=\\pi$,\n$\\Gamma_{k+1}=\\pi\\circ(\\Gamma_k+\\Gamma_k)$ ($k\\ge0$). In particular,\nthe number of cells in $\\Gamma_n$ equals $2^n-1$ and so $g$ is represented by a\ndiagram with exactly $2(2^n-1)+4\\cdot2^n=3\\cdot2^{n+1}-2$ cells. Since $F$ satisfies\nproperty $B$, the word length of an element $g=g_1^{\\epsilon_1}\\cdots g_{2^n}^{\\epsilon_{2^n}}$\nwill be at least $C2^n$, where $C>0$ is a constant that does not depend on the\n$\\epsilon_i$'s.\n\nNow consider a uniform embedding of $F$ into a Hilbert space ${\\cal\nH}$ with linear $\\rho_2$. In the image of our skew cube in $F$\nformed by $g_1$, \\dots, $g_{2^n}$, each edge will be equal to\n$O(n)$. The Skew Cube Inequality implies that there exists a\ndiagonal of the corresponding skew cube in ${\\cal H}$ that does not\nexceed $O(n)\\cdot\\sqrt{2^n}=O(n2^{\\frac{n}{2}})$. This means that\nsome points in $F$ that were at distance $d\\ge C2^n$ from each other\nwill be mapped to points in ${\\cal H}$ at distance\n$O(n2^{\\frac{n}{2}})=O(\\sqrt{d}\\log_2 d)$. Therefore, the\ncompression cannot exceed $1\/2$. \\end{proof}\n\n\n\nIn order to prove Theorem \\ref{BurU}, we need the  following simple\nconstruction, called the {\\em rightmost decomposition\\\/} of a\ndiagram. The idea of such a decomposition applied to the\npresentation $\\langle\\, x\\mid x^2=x\\,\\rangle$ was used in \\cite{GS2} to get a\nnew normal form for the elements of R.\\,Thompson's group $F$.\n\n\\begin{defn}{\\rm (Rightmost decomposition of a diagram)}\nLet $\\Delta$ be a diagram over a semigroup presentation ${\\cal P}$. A\n{\\em rightmost decomposition\\\/} of the diagram $\\Delta$ into a\nproduct of atomic factors is defined as follows. We proceed by\ninduction on the number of cells in $\\Delta$. If $\\Delta$ has no\ncells, then the product has no factors. Let $\\Delta$ have cells.\nThen it has at least one top cell (a cell whose top path is a\nsubpath in the top path of $\\Delta$). Let us consider the rightmost\ntop cell $\\pi$ of $\\Delta$. Let $\\topp{\\Delta}=p\\topp{\\pi}q$. Then\n$\\Delta$ is a concatenation of an atomic diagram\n$\\Pi=\\varepsilon(p)+\\pi+\\varepsilon(q)$ and some diagram $\\Delta'$ that has fewer\ncells. (Here $\\Delta'$ is obtained from $\\Delta$ by deleting the\npath $\\topp{\\pi}$.) Taking the product of $\\Pi$ and the rightmost\ndecomposition of $\\Delta'$, we obtain the rightmost decomposition of\n$\\Delta$.\n\\end{defn}\n\n\\begin{proof}[Proof of Theorem~\\ref{BurU}] Let ${\\cal P}=\\langle\\, x,a\\mid\nx^3=x^2,ax=a\\,\\rangle$. We need to prove that the universal diagram group\n$U={\\cal D}({\\cal P},a)$ has property B.\n\nLet $\\Delta$ be an $(a,a)$-diagram over ${\\cal P}$ with $N$ cells. It\nsuffices to prove that the element $g\\in U$ represented by $\\Delta$\nhas length at most $KN$ in generators $x_0$, $x_1$, $x_2$, where\n$K>0$ is a constant independent on $g$.\n\nLet $e_0$ be the top edge of $\\Delta$. Suppose that there is an\n$(a,ax)$-cell whose top edge coincides with $e_0$. Then we denote by\n$e_1$ the edge labelled by $a$ on the bottom path of this cell. If\n$e_1$ is the top path of an $(a,ax)$-cell, then $e_2$ denotes the\nedge on the bottom path of this cell labelled by $a$. Proceeding in\nsuch a way, we finally obtain a sequence of edges $e_0$, \\dots,\n$e_k$ ($k\\ge0$).\n\nAnalogously, changing in the previous paragraph top by bottom, we define the sequence\nof edges $f_0$, \\dots, $f_m$  ($m\\ge0$), where $f_0$ is the bottom edge of $\\Delta$.\nA very easy geometric observation is that $e_k$ must coincide with $f_m$. It is also\neasy to see that $\\Delta$ is a concatenation of the form\n$\\Delta=\\Delta_1\\circ(\\varepsilon(a)+\\Delta')\\circ\\Delta_2$, where $\\Delta_1$ consists of\n$(a,ax)$-cells only, $\\Delta_2$ consists of $(ax,a)$-cells only and $\\Delta'$ is an\n$(x^k,x^m)$-diagram over ${\\cal P}'=\\langle\\, x\\mid x^3=x^2\\,\\rangle$.\n\nLet us apply the rightmost decomposition to $\\varepsilon(a)+\\Delta'$. Each factor is an\natomic diagram of the form $(ax^s,x^2=x^3,x^t)^\\epsilon$, where $s,t\\ge0$, $\\epsilon=\\pm1$.\nAccording to the description of $U$ given in subsection~\\ref{universal}, this atomic\ndiagram corresponds to $x_t^\\epsilon$. (Notice also that no generators correspond to atomic\ndiagram with $(a,ax)$-cells and\/or their inverses.)\n\nTherefore, the rightmost decomposition of $\\Delta'$ allows us to decompose $g$ as a\nproduct (in $U$) of the form\n\\be{epss}\ng=x_{j_1}^{\\epsilon_1}\\cdots x_{j_r}^{\\epsilon_r}\\hbox{ with }\\epsilon_i=\\pm 1,\n\\end{equation}\nwhere $r=N-k-m$ is the number of cells in $\\Delta'$. We need to establish some easy\nproperties of the subscripts and the superscripts in (\\ref{epss}).\n\nSuppose that (\\ref{epss}) contains a subword of the form $x_i^{\\epsilon}x_j$. Then we claim\nthat $i\\le j+1$. Indeed, otherwise the cell that corresponds to $x_j$ is located to\nthe right of the cell corresponding to $x_i^{\\epsilon}$. (One can also see that $i>j+1$\nwould imply that $x_i^{\\epsilon}x_j$ equals $x_jx_{i+1}^{\\epsilon}$ in $U$.)\n\nNow suppose that (\\ref{epss}) contains a subword of the form\n$x_i^{\\epsilon}x_j^{-1}$. In this case we claim that $i\\le j+2$. (Now\n$i>j+2$ would also imply that the cell that corresponds to $x_j^{-1}$ is located to\nthe right of the one corresponding to $x_i^{\\epsilon}$. The element $x_i^{\\epsilon}x_j^{-1}$\nwould be also equal to $x_j^{-1}x_{i-1}^{\\epsilon}$ in $U$.)\n\nFor every $j\\ge0$ we replace each letter $x_j$ in (\\ref{epss}) by\nthe word $u_j^{-1}v_ju_j$, where $u_j=1$, $v_j=x_j$ if $j=0,1$ and\n$u_j=x_0^{j-2}$, $v_j=x_2$ if $j\\ge2$. We get the following word\n$W$ in generators $x_0$, $x_1$, $x_2$:\n$$\nW=u_{j_1}^{-1}v_{j_1}^{\\epsilon_1}u_{j_1}u_{j_2}^{-1}v_{j_2}^{\\epsilon_2}\\cdots\nv_{j_r}^{\\epsilon_r}u_{j_r}.\n$$\n\nSince $x_j=u_j^{-1}v_ju_j$ in $U$, the word $W$ represent the same\nelement $g\\in U$. Now let $W_i$ denote the freely irreducible form\nof the word $u_{j_i}u_{j_{i+1}}^{-1}$ for all $1\\le i<r$. It is\nalso convenient to define the words $W_0=u_{j_1}^{-1}$ and\n$W_r=u_{j_r}$. We are going to estimate the length of the word\n$$\n\\overline{W}=W_0v_{j_1}^{\\epsilon_1}W_1\\cdots v_{j_r}^{\\epsilon_r}W_r.\n$$\n\nWe know that $|W_0|=j_1\\le k-2$ (the first cell chosen in\n$\\Delta'$ is at a distance $j_1$ from the rightmost point of\n$\\Delta$ and the length of its top path is at least $2$).\nAnalogously, $|W_r|=j_r\\le m-2$. Also $|v_{j_1}|+\\cdots+|v_{j_r}|=r$. So\nwe estimate some part of the length of $\\overline{W}$ as follows:\n\\be{wv}\n|W_0|+|W_r|+|v_{j_1}|+\\cdots+|v_{j_r}|\\le k+m+r-4<N.\n\\end{equation}\n\nIt remains to estimate the sum $|W_1|+\\cdots+|W_{r-1}|$. It is\neasy to see from the definitions that $|W_i|\\le|j_{i+1}-j_i|$ for\nall $1\\le i<r$. Let $I$ be the set of all $1\\le i<r$ such that\n$j_i>j_{i+1}$. Then $|W_1|+\\cdots+|W_{r-1}|\\le\\sum_{1\\le i<r}|j_{i+1}-j_i|=S_1+S_2$,\nwhere $S_1=\\sum_{i\\in I}(j_i-j_{i+1})$ and $S_2=\\sum_{i\\notin I}(j_{i+1}-j_i)$.\n\nFor every $i\\in I$, we know that $j_i-j_{i+1}\\le2$. Therefore,\n$S_1\\le2|I|\\le2(r-1)<2r$. On the other hand,\n$$\nS_2-S_1=\\sum_{1\\le i<r}(j_{i+1}-j_i)=j_r-j_1\\le j_r\\le m-2<k+m=N-r.\n$$\nThis gives $S_2=S_1+(S_2-S_1)<N+r$ and so $S_1+S_2<N+3r\\le4N$.\nUsing (\\ref{wv}), we finally have\n$$\n|\\overline{W}|=(|W_0|+|W_r|+|v_{j_1}|+\\cdots+|v_{j_r}|)+(|W_1|+\\cdots+|W_{r-1}|)<5N.\n$$\nSo our statement is true for $K=5$.\n\nNotice that the constant here is not optimal. \\end{proof}\n\nLet us recall the definition of the (restricted) wreath product of groups. Let $G$, $H$\nbe two groups. If $\\phi$ is a function from $H$ to $G$, then the {\\em support\\\/} of $\\phi$\nis the set $\\mathop{\\mathbf{supp}}(\\phi)=\\{\\,b\\in H\\mid \\phi(b)\\ne1\\,\\}$. The restricted wreath product\n$G\\wr H$ of the above groups will consist of all pairs of the form $(b,\\phi)$, where $b\\in H$\nand $\\phi$ is a function from $H$ to $G$ with finite support. The group operation is defined\nin the standard way, that is, $(b_1,\\phi_1)(b_2,\\phi_2)=(b_1b_2,\\phi_1^{b_2}\\phi_2)$, where\n$\\phi^\\beta(b)=\\phi(b\\beta)$ by definition ($\\beta\\in H)$. For each of the groups $G$, $H$ we\nhave a canonical embedding into $G\\wr H$. Namely, $a\\mapsto(1,\\phi_a)$, where $\\phi_a(1)=a$,\n$\\phi_a(b)=1$ whenever $b\\ne1$. Also $b\\mapsto(b,\\mathbf{1})$, where $\\mathbf{1}$ is the\nfunction from $H$ to $G$ that takes all elements to 1.\n\nWe will deal with restricted wreath products of finitely generated groups. If we fix some\ngenerating sets for $G$ and for $H$, then the union of these sets (via canonical embeddings)\nwill generate $G\\wr H$. Hence we have three length functions: on $G$, on $H$, and on $G\\wr H$.\nFor simplicity, we will always freely speak about lengths in each of these three groups\nprovided the generating sets for $G$ and for $H$ are clear. The same will concern Cayley\ngraphs of these groups. The same symbol $\\ell$ will be used to denote the length function\nfor each of these groups. This will not lead to a confusion.\n\nWorking with lengths, we need the following fact proved in \\cite{Parry} (see Theorem 1.2).\n\n\\begin{lm}\n\\label{Parr}\nLet $G$, $H$ be finitely generated groups with fixed generating sets. Then the length\nof an element $(b,\\phi)\\in G\\wr H$ is equal to the sum $\\sum\\ell(\\phi(\\beta))+M$, where\n$\\ell(a)$ is the length of $a\\in G$ and $M$ is the length of the shortest path in the\nCayley graph of $H$ starting at the identity, ending at $b$ and visiting each element\nof $\\mathop{\\mathbf{supp}}(\\phi)$ at least once. The sum is taken over all $\\beta\\in\\mathop{\\mathbf{supp}}(\\phi)$.\n\\end{lm}\n\nLet $G={\\cal D}({\\cal P},w)$ be a diagram group. We say that the base $w$ is {\\em protected\\\/}\nif $w$ is a letter such that $w=uwv$ modulo ${\\cal P}$ implies that both words $u$, $v$ are\nempty (for any $u$, $v$). For any diagram group $G={\\cal D}({\\cal P},w)$, one can construct a new\nsemigroup presentation ${\\cal P}'$ adding 3 new letters $a$, $b$, $c$ to the alphabet and one\nnew defining relation $a=bwc$. It is not hard to see that $G$ will be isomorphic to the\ndiagram group ${\\cal D}({\\cal P}',a)$, where the base $a$ is now protected.\n\nIt is shown in \\cite[Theorem 11]{GS5} that, given a diagram group $G={\\cal D}({\\cal P},y)$ with\nprotected base $y$, one can construct a new semigroup presentation $\\bar{\\cal P}$ adding two\nnew letters $x$, $z$ to the alphabet and two new defining relations $x=xy$, $yz=z$\n(see also Example 10 before the theorem we refer to). In this situation, the diagram\ngroup ${\\cal D}(\\bar{\\cal P},xz)$ will be isomorphic to the restricted wreath product $G\\wr{\\mathbb{Z}}$.\n(The concept of a protected base we gave slightly differs from the original one although\nthe proof goes without any essential changes.) Now we are going to prove that if $G$ has\nproperty B (as a diagram group over ${\\cal P}$) then $G\\wr{\\mathbb{Z}}$ also has property B (as a diagram\ngroup over $\\bar{\\cal P}$).\n\n\\begin{lm}\n\\label{GwrZB}\nIn the above notation, if $G={\\cal D}({\\cal P},y)$ has property B, then $G\\wr{\\mathbb{Z}}={\\cal D}(\\bar{\\cal P},xz)$\nalso has property B.\n\\end{lm}\n\n\\proof\nIt suffices to prove that any diagram over $\\bar{\\cal P}$ with $N$ cells has length $O(N)$ in\nthe group $G\\wr{\\mathbb{Z}}$. Any reduced $(xz,xz)$-diagram over $\\bar{\\cal P}$ can be obtained as\nfollows. Let us take finitely many $(y,y)$-diagrams over ${\\cal P}$ and let $\\Xi$ be their\nsum. Now let us add a few cells of the form $x=xy$ and a few cells of the form $z=yz$ to\nthe diagram $\\varepsilon(x)+\\Xi+\\varepsilon(z)$ on the top in order to make the top label equal $xz$. Then\nwe add a few cells of the form $xy=x$ and $yz=z$ on the bottom to make the bottom label also equal $xz$.\nThe result is an $(xz,xz)$-diagram $\\Delta$. Let us denote by\n$o$ the vertex on the bottom path of $\\Delta$ that splits this path into the product of two\nedges. Clearly, the vertex $o$ is contained in $\\Xi$ and splits it into a sum. Let\n$\\Xi=\\Xi_k+\\cdots+\\Xi_0+\\Xi_{-1}+\\cdots+\\Xi_{-m}$, where each\n$\\Xi_j$ ($-m\\le j\\le k$) is a $(y,y)$-diagram and the vertex $o$ subdivides $\\Xi$\ninto the sum of the two summands. All the $\\Xi_j$ that form the first summand satisfy\n$j\\ge0$, the ones that form the second summand satisfy $j<0$. (Notice that each of the\nsummands can be empty.)\n\nLet $g_j\\in G$ be the element of ${\\cal D}({\\cal P},y)$ represented by $\\Xi_j$ ($-m\\le j\\le k$).\nWe also define $g_j=1$ for any integer $j$ not in $[-m,k]$. Then, according to\n\\cite[Example 10]{GS5}, the diagram $\\Delta$ represents the element $h$ of  the group\n$G\\wr\\langle\\, t\\,\\rangle$ of the following form: $(t^d,\\phi)$, where $d$, $\\phi$ are defined as\nfollows. The exponent $d$ on $t$  is the difference between the number of $(z,yz)$-cells\nand the number of $(yz,z)$-cells in $\\Delta$. The function $\\phi$ from ${\\mathbb{Z}}=\\langle\\, t\\,\\rangle$ is\ndefined by the rule $\\phi(t^j)=g_j$ for any integer $j$.\n\nIt remains to estimate the length of $h$. We will use Lemma \\ref{Parr}. We see that\nthe length of $h$ equals\n\\be{sumss}\n\\sum\\limits_{j=-m}^k\\ell(g_j)+M,\n\\end{equation}\nwhere $M$ is the length of the shortest path in the Cayley graph of ${\\mathbb{Z}}=\\langle\\, t\\,\\rangle$ that\nstarts at the identity, ends at $t^d$ and visits all those vertices $t^j$ for which\n$g_j\\ne1$. It is clear that $M\\le2(k+m+1)+|d|$. Obviously, $|d|$ does not exceed the\nnumber of cells in $\\Delta$. It is also clear that $k+m+1$ is the number of $(x,xy)$-cells\nplus the number of $(z,yz)$-cells in $\\Delta$. This implies $M=O(N)$.\n\nSince $G={\\cal D}({\\cal P},y)$ has property B, there exists a constant $C>0$ such that the\nlength of each $g\\in G$ does not exceed $C\\cdot \\#(g)$, where $\\#(g)$ denotes the\nnumber of cells in the reduced diagram over ${\\cal P}$ representing $g$. Therefore, the\nfirst summand in~(\\ref{sumss}) does not exceed\n$C(\\#(g_{-m})+\\cdots+\\#(g_0)+\\cdots+\\#(g_k))\\le CN$. Thus we finally have that the\nlength of $h$ in $G\\wr{\\mathbb{Z}}$ is $O(N)$.\n\\endproof\n\n\\begin{cy}\n\\label{zwrzb}\nLet ${\\cal P}=\\langle\\, a,b,b_1,b_2,c\\mid ab=a,bc=c,b=b_1,b_1=b_2,b_2=b\\,\\rangle$. Then\nthe restricted wreath product ${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ has property B as a diagram group\n${\\cal D}({\\cal P},ac)$.\n\\end{cy}\n\nIndeed, ${\\mathbb{Z}}$ is the diagram group over $\\langle\\, b,b_1,b_2\\mid b=b_1,b_1=b_2,b_2=b\\,\\rangle$\nwith protected base $b$.\n\\vspace{2ex}\n\nLet $H$ be a finitely generated group with fixed generating set. By its\n{\\em growth function\\\/} $\\gamma(n)$ (with respect to this set of generators)\nwe mean the number of elements in the ball of radius $n$ around the identity\nin the Cayley graph of $H$.\n\nTheorems \\ref{wr} and \\ref{wr1} will follow from the next result.\n\n\\begin{thm}\\label{wr2}\nLet $H$ be a finitely generated group with fixed generating set. Suppose\nthat its growth function satisfies the condition $\\gamma(n)\\succeq n^k$ for\nsome $k>0$. Then the Hilbert space compression $\\alpha$ of the group ${\\mathbb{Z}}\\wr H$\nsatisfies the following inequality:\n$$\n\\alpha\\le\\frac{1+k\/2}{1+k}.\n$$\n\\end{thm}\n\n\\proof Let $x$ be a generator of ${\\mathbb{Z}}$ in the wreath product ${\\mathbb{Z}}\\wr H$. Every\nelement in ${\\mathbb{Z}}\\wr H$ is a pair $(b,\\phi)$ where $b\\in H$, $\\phi$ is a function\n$g\\mapsto x^{m_g}$ from $H$ to ${\\mathbb{Z}}$ with finite support.\n\nLet $B_n$ be the ball of radius $n$ around $1$ in the Cayley graph of $H$.\nConsider the set\n$$\nX_n=\\{\\,w_b=b^{-1}x^{2n+1-|b|}b\\mid b\\in B_n\\,\\},\n$$\nwhere $\\ell(b)$ is the length of $b$. Obviously, $w_b$ has the form $(1,\\phi_b)$ in\n${\\mathbb{Z}}\\wr H$, where the support of $\\phi_b$ is $\\{\\,b\\,\\}$ and $\\phi_b(b)=x^{2n+1-\\ell(b)}$.\nNotice that $\\ell(x^r)=|r|$ in $\\langle\\, x\\,\\rangle$ for all $r\\in{\\mathbb{Z}}$. Hence by Lemma~\\ref{Parr},\nthe length of $w_b$ in ${\\mathbb{Z}}\\wr H$ equals $2n+1-\\ell(b)+M$, where $M=2\\ell(b)$.\nTherefore, $\\ell(w_b)=2n+1+\\ell(g)$ is always between $2n+1$ and $3n+1$. Clearly, all\nelements of $X_n$ pairwise commute. The number of them is exactly $\\gamma(n)$.\n\nIt is easy to see that for every choice of $\\epsilon_b\\in\\{\\,1,-1\\,\\},b\\in B_n$, the\nlength of the element $w=\\prod_{b\\in B_n}w_b^{\\epsilon_b}$ is greater than $n\\gamma(n)$.\nIndeed, $w$ has the form $(1,\\phi)$ in ${\\mathbb{Z}}\\wr H$, where $\\mathop{\\mathbf{supp}}(\\phi)=B_n$ and\n$\\phi(b)=x^{2n+1-\\ell(b)}$ for all $b\\in B_n$. Hence the length of $w$ is at least\n$$\n\\sum_{b\\in B_n}\\ell(\\phi(b))=\\sum_{b\\in B_n}(2n+1-\\ell(b))>n\\gamma(n)\n$$\nby Lemma~\\ref{Parr}.\n\nConsider the skew cube spanned by $X_n$ in ${\\mathbb{Z}}\\wr H$ and its image\nin ${\\cal H}$ under the uniform embedding $f$. The sides of the skew\ncube in $\\cal H$ do not exceed $Cn$ for some constant $n$ since our\ngroup is finitely generated. So the sum of squares of sides does not\nexceed $C^22^{\\gamma(n)-1}\\gamma(n)n^2$. On the other hand, the number of\ndiagonals is $2^{\\gamma(n)-1}$. Hence by the Skew Cube Inequality,\nthere exists a diagonal of the skew cube in $\\cal H$ which does not\nexceed $Cn\\sqrt{\\gamma(n)}$. Thus there exist two points in the\nCayley graph of $G$ at distance $d\\ge n\\gamma(n)$ whose images under\n$f$ are at distance at most $Cn\\sqrt{\\gamma(n)}$. If $\\rho_1(d)=Kd^\\alpha$\nfor some constant $K>0$ in the definition of the uniform embedding, then\n$Cn\\sqrt{\\gamma(n)}\\ge K(n\\gamma(n))^\\alpha$. Hence\n$$\nn^{1-\\alpha}\\succeq(\\gamma(n))^{\\alpha-1\/2}\\succeq n^{k(\\alpha-1\/2)}.\n$$\nThis implies $1-\\alpha\\ge k(\\alpha-1\/2)$ and so $\\alpha\\le(1+k\/2)\/(1+k)$.\n\\endproof\n\n\\begin{thm}\n\\label{wrprod} The Hilbert space compression of the restricted wreath product\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ belongs to $[1\/2;3\/4]$.\n\\end{thm}\n\n\\proof The lower bound follows from Theorem \\ref{dghalf} and the fact that\n${\\mathbb{Z}}\\wr{\\mathbb{Z}}$ satisfies property B as a diagram group ${\\cal D}({\\cal P},ac)$ over\n${\\cal P}=\\langle\\, a,b,b_1,b_2,c\\mid ab=a,bc=c,b=b_1,b_1=b_2,b_2=b\\,\\rangle$\n(Corollary~\\ref{zwrzb}).\n\nThe upper bound follows from Theorem \\ref{wr2} with $G={\\mathbb{Z}}$. Indeed, the growth\nfunction of ${\\mathbb{Z}}$ is linear, that is, $\\gamma(n)\\succeq n^k$ for $k=1$. Hence\n$\\alpha\\le(1+k\/2)\/(1+k)=3\/4$.\n\\endproof\n\nRecall that a finitely generated group has a {\\em su\\-per-po\\-ly\\-no\\-mial\\\/}\ngrowth whenever its growth function $\\gamma(n)$ exceeds $n^k$ for all $k\\ge1$.\n\n\\begin{thm}\nLet $H$ be a finitely generated group with su\\-per-po\\-ly\\-no\\-mial growth. Then the\nHilbert  space compression of ${\\mathbb{Z}}\\wr H$ is at most $1\/2$.\n\\end{thm}\n\n\\proof Indeed, $\\gamma(n)\\succeq n^k$ implies $\\alpha\\le(1+k\/2)\/(1+k)$. The right-hand\nside of this inequality approaches $1\/2$ as $k\\to\\infty$.\n\\endproof\n\n\\begin{cy} The Hilbert space compression of the group ${\\mathbb{Z}}\\wr({\\mathbb{Z}}\\wr{\\mathbb{Z}})$\nis at most $1\/2$.\n\\end{cy}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:Intro} Introduction}\n\nIt is common to determine the oscillation frequencies of atoms in a magnetic trap by exciting their motion. A sudden displacement of the trap excites a dipole (centre-of-mass) oscillation of the cloud at the trap frequency, while a sudden compression or de-compression induces quadrupole (width) oscillation of the cloud. In many experiments, the trapped cloud is long and thin, with transverse frequencies of 1\\,kHz or above. The oscillation amplitude is then typically below the resolution of the imaging system and therefore difficult to observe directly. In that case, the oscillation may be driven, and the excitation of the cloud may be detected through the increase in its length, which results from heating. This exhibits a resonant behaviour, with a maximum temperature at a specific driving frequency, related to the natural transverse oscillation frequency.\n\nThis paper presents temperature resonances of magnetically trapped atoms, induced by dipole oscillation, with six different trap frequencies. These traps are formed on an atom chip using the field of a permanently magnetised videotape. A full characterisation of the videotape traps is carried out with particular emphasis on the effects of anharmonicity. Our experimental results are compared with a numerical model that provides valuable insight into the dynamics of the cold atoms moving in the real confining potential of the trap under the important influences of collisions and atom loss.\n\nSeveral references already consider the dipolar excitation of trapped particles by shaking the trap centre \\cite{st1,st2,st10,ions1,st13}. The first three are theory papers \\cite{st1,st2,st10} that describe heating due to laser noise in harmonic optical traps far from resonance.  Ref. \\cite{st2} calculates the evolving energy distribution, accounting for atom loss, in a truncated harmonic trap. Ref. \\cite{ions1} describes measurements of the resonant frequencies and laser cooling rates for ions in a shaken Penning trap. In Ref.\\,\\cite{st13} M. Kumakura et al.  investigate the excitation of neutral atoms in a shaken cloverleaf magnetic trap. They measure resonances in atom loss and temperature and discuss how different ratios of atomic temperature to effective trap depth result in either heating or cooling after shaking. They use a classical 1D equation of motion, without collisions, to illustrate some features of their experiment.\n\nThere have also been discussions of parametric excitation, i.e. modulation of the trap frequency, in various contexts. These include heating and cooling of neutral atoms in optical dipole traps \\cite{st1,st2,st10,st3,st6,st5}, one-dimensional optical lattices \\cite{st4,st6,st10} or magnetic traps \\cite{st17}, and measurement of oscillation frequencies for electrons in a Penning trap \\cite{st11,st12}, ions in a quadrupole trap \\cite{st16} or neutral atoms in a MOT \\cite{st14,st15}. We do not study parametric resonance here because there is no straightforward way to modulate the frequency of our videotape trap without also shaking its position. The same is true of any permanent-magnet atom trap.\n\n\nThe experiments described here determine both the position and the shape of the dipolar temperature resonances. We also investigate these resonances theoretically with the aid of a three-dimensional numerical model that accounts simultaneously for anharmonicity of the trap, atom loss due to finite trap depth, interatomic collisions and evolution of the collision rate during the excitation. All these aspects of our model go beyond what has been done before. Our model yields good quantitative agreement between experiment and theory, even though we use a rather simplified collision model. We anticipate that our simplifying assumption could be exported to a number of ensemble dynamic problems in order to achieve faster simulations.\n\n\\section{\\label{sec:experim} The experiment}\n\n\\begin{figure*}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot1x.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot1y.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot1z.eps}\n\\end{minipage}\n\\\\\n\\vspace{0.3cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot2x.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot2y.eps}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{plot2z.eps}\n\\end{minipage}\n\\caption{Total field strength of magnetic trap versus position. The transverse bias field is $B_b=5.9\\,\\mathrm{G}$ along $\\hat{x}$. Upper row: overview along each Cartesian direction. Lower row: zoom into the centre shows the extrema of the modulation induced by the shaking field $b=105\\,$mG.}\n\\label{fig:howPotChanges}\n\\end{figure*}\n\nOur experiment uses the field gradient of a magnetized videotape to trap the atoms. The tape, lying in the $xz$ plane, has magnetisation $M_1\\cos(kx)\\hat{x}$, where $k=(2\\pi\/110)\\,\\mu\\mbox{m}^{-1}$. Above the surface, this magnetisation  produces a transverse (i.e. in the $xy$ plane) magnetic field of $\\{B_{x},B_{y}\\}=B_{1}e^{-k y}\\{-\\cos(kx),\\sin(ky)\\}$ \\cite{HindsHughesReview} with $B_1\\simeq 100\\,$G.  The addition of a transverse bias field, $B_{b}\\{\\cos(\\theta),\\sin(\\theta)\\}$, produces an array of zeros in the total transverse field located at a height $y=\\ln(B_{1}\/B_{b})\/k$. On Taylor expanding the field around any of these zeros one finds a quadrupole field in the $xy$ plane whose orientation depends on the angle $\\theta$ of the bias field. The magnitude of the field grows linearly with the cylindrical radius $r$ according to $|B|=k B_{b} r$, providing a cylindrically symmetrical, linear confining potential $\\mu_{B}g_{F}m_{F}|B|$ for weak-field seeking atoms with magnetic quantum number $m_{F}$ and $g$-factor $g_{F}$ ($\\mu_B$ is the Bohr magneton).  With the addition of a further bias field $B_{z}\\hat{z}$, the trapping potential for transverse displacements becomes approximately harmonic in the region close to the axis, where $k B_{b} r\\ll B_{z}$. In that region, atoms oscillate in the trap with a transverse frequency of\n\\begin{equation}\\label{eqn:radialVideofreq}\nf_{r}\\approx \\frac{k B_b}{2\\pi}\\sqrt{\\frac{\\mu_B g_F m_F}{m\nB_{z}}}\\, ,\n\\end{equation}\nwhere $m$ is the mass of the atom. We use $^{87}$Rb atoms in the $(F=2, m_{F}=2)$ ground state, for which $g_{F}m_{F}=1$. The bias field $B_{b}$ is tuneable over the range $6-36$\\,G, which varies the distance from the trap to the videotape between $50$ and $18\\,\\mu$m. Over this range of positions the axial bias $B_{z}$ drops from $2.58$ to $2.51\\,$G. These parameters give transverse harmonic oscillation frequencies in the range $3-20\\, \\mathrm{kHz}$. The axial bias field has a minimum value near the centre of the videotape and increases roughly quadratically with $z$ to form a weak axial trap with a frequency of $15\\,$Hz. In summary, this combination of videotape fields and bias fields produces an array of cigar-shaped 3D traps with strong transverse confinement and weaker trapping along $\\hat{z}$. For more details of videotape traps, see \\cite{HindsHughesReview,myVideoPaper}.\n\nThe upper graphs in Figure\\,\\ref{fig:howPotChanges} show how the total magnetic field strength $|B|$ varies along the three Cartesian axes through the centre of the trap. These are calculated using a full numerical model of the apparatus, with $B_b=5.9\\,$G, directed along $\\hat{x}$. The trap along $x$ repeats every $110\\,\\mu$m because of the periodic magnetisation of the videotape, but only one of these is shown as only one trap is used in the experiments. Along $y$, we see an asymmetric trap, with a strong repulsive wall as the videotape is approached and an asymptote far from the tape that is equal to the total bias field strength. Along $z$, there is the weak trapping due to the inhomogeneous axial bias field.\n\n\n\n\nWe load one such trap using an experimental sequence similar to that described in \\cite{myVideoPaper}. The temperature of the atom cloud depends on the bias field and ranges from $13\\,\\mu$K at $B_b=5.9\\,$G up to $130\\,\\mu$K at $B_b=35.6\\,$G. We then add a modulation field $b \\cos(2\\pi f t)$ along $\\hat{x}+\\hat{y}$, to displace the trap by a small distance $-\\tfrac{b}{k B_b}\\cos(\\omega t)$ along $\\hat{x}+\\hat{y}$. We choose $b=105\\,$mG, which gives a shaking amplitude of $50-300\\,$nm over the range of transverse bias fields used. Zooming into the centre of the trap, the three lower graphs in Fig.\\,\\ref{fig:howPotChanges} show the extrema of this modulation. The first two show the (equal) movements along $x$ and $y$ resulting from our particular arrangement of fields. The third shows the absence of movement along $z$ and also illustrates the modulation of the minimum field, due to the variation of $B_z$ with transverse displacement of the trap. Since $B_z$ affects the transverse trap frequency, this could cause parametric heating, but the effect is considered in \\cite{MyThesis} and found to be negligible.\n\n\\begin{figure}[b]\n\\includegraphics[width=0.4\\textwidth]{resonances.eps}\n\\caption{Temperature of cloud after shaking for $5\\,$s, plotted as a function of shaking frequency $f$. (a) Experimental results. The points are measurements taken at six transverse bias fields, $B_b$. Lines are Lorentzian fits to the data.  (b) Simulation. Points are calculated and lines are Lorentzian fits.}\n\\label{fig:resonances}\n\\end{figure}\n\n\nWith $B_b$ initially set at $5.9\\,$G, the atoms are shaken for $5\\,\\mathrm{s}$ and held for $1\\,$s, after which the temperature of the trapped cloud is determined by measuring its density distribution along the $z$-direction. This is done using a CCD camera to record the absorption of resonant laser light \\cite{myVideoPaper}, taking into account the inhomogeneous Zeeman shift of the trapped atoms. The experiment is repeated for a range of shaking frequencies to map out a resonance curve, shown in the leftmost peak of Fig. \\ref{fig:resonances}(a). The next two curves are obtained in the same way with $B_b=8.9\\,$G and $B_b=14.8\\,$G. At still higher transverse bias fields ($B_b=21.9, 29.6$ and $35.6\\,$G), the atoms are too close to the videotape to yield clean absorption images. In these three cases the trap is moved away from the surface by lowering the bias to $3\\,$G over $3\\,$s before taking the image. This weakens the trap, thereby cooling the cloud, and it is the lower temperature in this final trap that we plot in Fig. \\ref{fig:resonances}(a).\n\nThese temperature resonances show that the atoms absorb energy most efficiently near a particular frequency. In the case of a one-dimensional harmonic trap, the physics would be that of a driven, weakly damped harmonic oscillator, whose Lorentzian resonance would be centred on the oscillator frequency with a width given by the collision rate. The actual widths are much greater than the collision rate, but nevertheless, motivated by this thought, we fit a Lorentzian to each resonance curve and plot the centre frequencies (the red squares) as a function of $B_b$ in Fig.\\,\\ref{fig:FreqVsBias}. For comparison, the line in Fig.\\,\\ref{fig:FreqVsBias} shows the frequency of small transverse oscillations, given by Eq.\\,\\ref{eqn:radialVideofreq}, with $B_z$ evaluated at the centre of the trap for each value of transverse bias.  The data points lie below this line because (i) the most energetic atoms move out of the region of small ${x,y}$ where the harmonic approximation of Eq.\\,(\\ref{eqn:radialVideofreq}) is valid and (ii) the atoms are also displaced from the centre along $z$, where the increased value of $B_z$ reduces the radial frequency. These effects also produce inhomogeneous broadening of the temperature resonances, making them wider for the hotter clouds at higher bias fields, as seen in Fig.\\,\\ref{fig:resonances}(a).\n\n\\begin{figure}[b]\n  \\includegraphics[width=0.9\\columnwidth]{FreqVsBias.eps}\n  \\caption{Frequency of temperature resonance versus bias magnetic field. Solid line: calculated harmonic frequency of transverse oscillations at trap centre. Red squares: measured centre frequencies of the temperature resonances in Fig.\\,\\ref{fig:resonances}(a). Blue circles: centre frequencies of the simulated temperature resonances plotted in Fig.\\,\\ref{fig:resonances}(b).}\n  \\label{fig:FreqVsBias}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{minipage}{0.9\\linewidth}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{FreqVsAmpliBias2.eps}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.9\\linewidth}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{FreqvsAmpliBias7.eps}\n\\end{minipage}\n\\caption{Frequency of oscillations along $\\hat{x}$ through centre of trap as a function of amplitude. Upper: $B_b=5.9\\,\\mathrm{G}$. Lower: $B_b=35.6\\,\\mathrm{G}$. Solid red line: frequency calculated from numerical solution to full equation of motion. Dotted green line: Harmonic approximation. Dashed black line: linear potential approximation. Circles: rms cloud radius at initial temperature.  Squares: rms cloud radius after heating at resonance.}\n\\label{fig:FreqsVsAmpliBias2and7}\n\\end{figure}\n\nNumerically integrating the equations of motion in the full trap potential, we have calculated the period of oscillation along the $x$ direction through the centre of the trap. The inverse of this is plotted as a function of amplitude $A_x$  by the solid lines in Fig.\\,\\ref{fig:FreqsVsAmpliBias2and7}, the upper(lower) panel being for the case of $B_b=5.9(35.6)\\,$G. At small amplitude, the frequency coincides with the harmonic approximation indicated by the dotted line. At large amplitude, the potential approaches that of  linear trap, and the frequency tends correspondingly to $\\tfrac{1}{4}\\sqrt{\\mu_B k B_b\/(2 m A_x)}$, indicated by the dashed line. The filled circles(squares) mark the rms radius of the cloud in this direction before(after) resonant heating. These show that the atoms explore the anharmonic region of the transverse trap even before the cloud is heated, and move further into this region after heating. These frequency shifts are larger when the bias field is larger. Figure\\,\\ref{fig:RadialFreqVsZ} shows how \\emph{axial} displacement from the centre of the trap lowers the harmonic frequency for small transverse oscillations. The essence of this effect is already captured in Eq.\\,(\\ref{eqn:radialVideofreq}) through the dependence of $f_r$ on $B_z$, but here we show the result of the full numerical model of our experiment. Again, the circles(squares) represent the rms size, this time along $z$ before(after) heating. Both of these mechanisms contribute appreciably to the inhomogeneous broadening seen in Fig.\\,\\ref{fig:resonances}(a) and the lowering of the trap frequency seen in Fig.\\,\\ref{fig:FreqVsBias}.\n\nFor a more quantitative understanding of the resonances it is necessary to build a dynamical model that allows the atoms to collide. In the next section we develop such a model and use it to simulate the data presented in Fig.\\,\\ref{fig:resonances}(a).\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.36\\textwidth]{freqradialVsZ.eps}\n  \\caption{Frequency of small (harmonic) transverse oscillations as a function of displacement $z$ from the trap centre. \\textbf{Circles}: rms cloud radius (half-length) at initial temperature.  \\textbf{Squares}: rms cloud radius (half-length) after heating at resonance.}\n  \\label{fig:RadialFreqVsZ}\n\\end{figure}\n\n\n\\section{\\label{sec:simul} Simulation}\n\\subsection{\\label{sec:simul:a} The role of collisions}\nOur goal here is to find a numerical model that is simplified as far as possible, while still reproducing the data of Fig.\\,\\ref{fig:resonances}(a). We use the known currents and videotape magnetisation to determine an accurate 3D  potential,  $U(x,y,z,t)$, for the shaken magnetic trap. We note that the inclusion of gravity has no significant effect because these traps are so strong vertically. The cloud of approximately $10^5$ atoms is represented by an ensemble of 500-5000 point particles, this being sufficient to represent the average properties of the ensemble. The three initial velocity components for each particle are chosen at random from the Maxwell-Boltzmann distribution corresponding to initial temperature $T_i$, which is the baseline temperature in Fig.\\,\\ref{fig:resonances}(a) for that particular trap. Similarly, the initial positions are distributed with a probability density proportional to $\\exp [-U(x,y,z,0)\/(k_{B}T_{i})]$, where $k_{B}$ is the Boltzmann constant. The classical equations of motion are then integrated numerically to follow the movement of the particles.\n\nThe two curves in Fig.\\,\\ref{fig:peak5}(a) show how the energy of the cloud increases with time when the trap having $B_b = 21.9\\,$G  is shaken at 7.2kHz. These parameters correspond to the peak of the pale blue resonance curve at $7\\,$kHz in Fig.\\,\\ref{fig:resonances}(a). A simple model without collisions produces the green (lower) curve in Fig.\\,\\ref{fig:peak5}(a). Particles are placed in the trap at time $t=0$, which remains static for the first $0.1\\,$s. Then the shaking is switched on and the energy rises rapidly, increasing by $20\\%$ over the next $0.1\\,$s. This is  due to the excitation of particles whose transverse oscillation frequency is close to the drive frequency.  Once they are sufficiently excited, the anharmonicity moves these particles out of resonance. Because the resonant group has been depleted, there is almost no subsequent energy increase even though the shaking continues until $t=5.1\\,$s. The shaking is then switched off leaving the cloud to evolve freely over the last second of the simulation. This behaviour disagrees with the experiment. In reality, the temperature doubles (though not in Fig.\\,\\ref{fig:resonances}(a) because there the trap was relaxed before recording the temperature). The discrepancy is removed when we allow the simulation to redistribute the momentum through collisions.\n\n\\begin{figure}[t]\n\\begin{minipage}{0.65\\linewidth}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{Peak5-1.eps}\n\\end{minipage}\\\\\n\\begin{minipage}{0.65\\linewidth}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{Peak5-2.eps}\n\\end{minipage}\\\\\n\\caption{Simulation of $10^5$ atoms in a trap having $B_b=21.9\\,\\mathrm{G}$.  The trap is shaken by a modulation field of  $105\\,$mG amplitude and  $7.2\\,\\mathrm{kHz}$ frequency over the time interval  $t=0.1-5.1\\,$s. The initial temperature is $71\\,\\mu \\mathrm{K}$. These parameters correspond to the resonant peak of the pale blue curve in Fig.\\,\\ref{fig:resonances}(a). This simulation uses 500 particles. Green curves: no collisions. Blue curves: momentum is redistributed through collisions. (a) Increase of total energy with time. (b) Number of atoms remaining within the volume of the trap, with the initial atom number being $N_i=10^5$.}\n\\label{fig:peak5}\n\\end{figure}\n\n\nThe following very simple model of momentum redistribution is sufficient for our purpose. The thermally averaged atom-atom scattering cross-section is $\\sigma=8 \\pi a^{2}\/(1+2\\pi a^2 m k_{B} T \/\\hbar^2)$, where $a=5.53\\times10^{-9}\\,\\mathrm{m}$ is the s-wave scattering length \\cite{scatteringLength} and $T$ is the temperature of the cloud. The average relative velocity between particles is $v = \\sqrt{16 k_{B} T\/(\\pi m)}$ \\cite{PhysChem}. Knowing the mean density of trapped atoms, $ n $, we obtain a mean collision rate per atom,  $ n \\sigma v$. At appropriate time intervals (short compared with the inverse collision rate), the numerical integration is paused and a random number is generated for each particle to determine whether it has had a collision. Ideally, this should account for the local variations of density \\cite{Bird94}, which we ignore here in order to keep the model simple. If there is no collision, the atom continues unperturbed. Otherwise, the momentum of the particle is redirected by the collision, according to some angular distribution. For the data presented here, we used that of elastic scattering from an infinitely heavy sphere. However, we find that the results are quite insensitive to the chosen distribution and there is nothing special about this particular one. Given our experimental conditions, the average time between collisions is $30-100\\, \\mathrm{ms}$ at the start of the excitation, and this becomes longer as the cloud heats up.\n\nWith the momentum redistribution thus incorporated,  we obtain the dark blue (higher) curve in Fig.\\,\\ref{fig:peak5}(a).  Now, the ensemble is able to continue absorbing energy after the initial absorption because the depleted velocity group is steadily replenished through collisions. The refilling rate slows down as the atoms become more energetic, and this is responsible for the saturation of heating, seen in Fig.\\,\\ref{fig:peak5}(a). In this case, the energy doubles over the $5\\,$s of shaking and our model approximates well  the measured heating of the atom cloud.\n\nFigure \\,\\ref{fig:peak5}(b) plots the number of atoms in the trap as a function of time, with collisions (blue, lower) and without (green, higher). In the collision-free case, few atoms leave the trap because the heating is weak and because the phase space is not efficiently sampled in the absence of collisions.  By contrast, when collisions are included, the cloud heats much more strongly and the energetic atoms are more easily able to find an exit route from the trap. There is competition between the heating rate due to resonant excitation and the cooling rate due to evaporation from the trap. In this example, the heating is dominant because the trap is deep in comparison with the mean energy absorbed by each atom.\n\n\\begin{figure}\n \\includegraphics[width=0.4\\textwidth]{energySpectra.eps}\n \\caption{Simulated energy spectra of $10^5$ atoms, initially at $71\\,\\mu \\mathrm{K}$ in a trap having $B_b=21.9\\,\\mathrm{G}$. They are shaken over the time interval  $t=0.1-5.1\\,$s by a $7.2\\,\\mathrm{kHz}$  field of  amplitude $b=105\\,$mG. These are the same parameters used in Fig.\\,\\ref{fig:peak5} and at the resonant peak of the pale blue curve in Fig.\\,\\ref{fig:resonances}(a). This simulation uses 5000 particles. (a) Without collisions. (b) With collisions. Red curves: Initial thermal distribution just before shaking. Green curves: After $0.1\\,$s of shaking. Blue curves: After $5\\,$s of shaking.}\n \\label{fig:energySpectra}\n\\end{figure}\n\n\nFigure \\ref{fig:energySpectra}(a) shows calculated energy spectra in the absence of collisions. The ordinate is the probability density for a given energy, normalised to unity, while the abscissa shows that energy,  normalised to the trap depth $U_0=1.2\\,$mK. The red curve ($t=0.1\\,$s), showing the initial distribution just before the trap starts to shake, has a single peak just below $E=0.2 U_0$. After only $100\\,$ms of shaking (green curve, $t=0.2\\,$s) a deep notch appears in the distribution close to the energy of the initial peak. This shows that atoms close to that energy, having an oscillation period close to the period of the drive, are the ones excited by the shaking. Their excitation causes a second peak in the distribution at approximately $E\/U_{0}=0.3$. There is no further significant change in the distribution, even after $5\\,$s of shaking, as shown by the blue curve. This behaviour is to be compared with Figure \\ref{fig:energySpectra}(b), which shows energy distributions for the same simulated experiment when collisions are included. The initial distribution is the same, as is the notch appearing at $0.2\\,$s, but in this case continued shaking does produce additional heating because the collisions refill the velocity group that absorbs energy from the drive. This is clearly seen in the growing probability on the high-energy end of the spectrum.\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.4\\textwidth]{simulatedResonAtomLoss.eps}\\\\\n  \\caption{Simulated ratio of final atom number $N$ to initial number $N_i$ as a function of the excitation frequency. The colour code corresponds to that of Fig. \\ref{fig:resonances}, with coloured arrows marking the temperature resonance frequencies. For the lowest three bias fields, the lowest barrier to escape is along \\emph{y} ($360\\, \\mu \\mathrm{K}$, $520\\, \\mu \\mathrm{K}$ and $880\\, \\mu \\mathrm{K}$ in order of increasing bias field), while for the highest three, it is along \\emph{z} ($\\sim 1200\\, \\mu \\mathrm{K}$). For comparison, the measured initial cloud temperatures before the excitation take values of $13.4\\, \\mu \\mathrm{K}$, $13.3\\, \\mu \\mathrm{K}$, $16.7\\, \\mu \\mathrm{K}$, $71\\, \\mu \\mathrm{K}$, $98\\, \\mu \\mathrm{K}$ and $132\\, \\mu \\mathrm{K}$, in order of increasing bias field.}\n  \\label{fig:simulatedResonAtomLoss}\n\\end{figure}\n\n\\subsection{\\label{sec:simul:b} Simulated Resonances}\n\nFigure \\ref{fig:resonances}(b) shows our simulated temperature resonances at each of the six different values of the bias field. After the shaking stops in the experiment, we allow the atoms to thermalise for one second before measuring the temperature. In the simulation, we simply take the final energy as a measure of the temperature that would be reached in equilibrium. The central frequencies of these simulated resonances agree very closely with the experiment, as indicated by the blue circles in Fig.\\,\\ref{fig:FreqVsBias}.\n\nThe peak temperature rises are also remarkably well reproduced by the simulations, given the simplicity of the model. In particular, these model collisions redistribute momentum, but do not permit re-thermalisation of the energy distribution because the energy of a given atom is conserved in the collision. Despite that, the energy increase in the model reproduces all the measured temperature rises to well within a factor of two and that remains the case for a variety of model angular distributions.\n\nThe widths of the simulated resonant peaks are a little too large - by a factor of $1.5 - 2$. In searching for an explanation we simulated the yellow ($8.9\\,$G) resonance with the initial temperature reduced from $13.3\\,\\mu$K to $7\\,\\mu$K. This only reduced the width by $10\\%$, so we do not think a temperature calibration error can explain the discrepancy between the measured and calculated widths. It could well be that our very simple collision model causes the resonances to be too broad, although we do not see a clear reason why that should be so.\n\nFigure \\ref{fig:simulatedResonAtomLoss} shows the simulated resonances in atom loss for the same six bias fields used in Fig.\\,\\ref{fig:resonances}. We see that the dips in atom number are on the low-frequency side of the temperature resonances indicated by arrows. This is because the energetic atoms most likely to be driven out of the trap are also those most able to explore the anharmonic regions and hence to oscillate at  lower frequencies. The same behaviour has been reported by several groups for both parametric shaking \\cite{st4,st5,st9,st17,st19}, and for shaking of the trap centre \\cite{st13}, in agreement with the results of our simulations.\\\\\n\n\\section{\\label{sec:Conclusions} Summary and Conclusions}\nWe have measured the temperature rise in a cigar-shaped cloud of cold atoms after shaking it sinusoidally in the transverse direction. Unlike most previous measurements, we have modulated the position of the trap, not its curvature. We have recorded temperature resonances as a function of frequency for several values of the bias field that controls the curvature of the transverse trapping potential.  Essential to the interpretation of our measurements is the understanding that,  at the temperatures involved, atoms oscillate with a wide range of transverse frequencies in the trap and hence that the central frequencies of the observed resonances lie below the calculated harmonic frequencies for small oscillations.\n\nWe have developed a simple numerical model that has provided a clear quantitative understanding of the driven dynamics. When compared with other models and simulations in the literature, this provides one of the most detailed and complete attempts to reproduce the observed resonances. No other models include both the collisions between particles and the anharmonicity of the trapping potential, yet both of these are shown to be essential for a full understanding of the behaviour.\n\n\\begin{acknowledgments}\nThe authors thank Robert Nyman and Michael Trupke for useful discussions. We are indebted to the FastNet and AtomChips European networks and to the UK EPSRC and Royal Society for their funding support.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplemental Material: Controlling spin relaxation in hexagonal BN-encapsulated graphene with a transverse electric field}\n\n\n                            \n                             \n\\maketitle\n\n\\subsection{Fabrication of the stacks}\nThe hBN-graphene-hBN stacks were fabricated using a pick-up method \\cite{DeanScience2013} described elsewhere \\cite{WeesAppliedPhysicsLetters2014} in more details.\nThe graphene and hBN flakes were obtained by mechanical cleavage of HOPG (SPI Supplies) and hBN powder (HQ Graphene).\nThe preparation of the stacks starts with the exfoliation and optical microscopy selection of hBN on a glass mask covered by a thin layer of polycarbonate (HQ Graphene).\nThe single layer graphene and bottom hBN flakes are exfoliated on Si\/SiO$_{2}$ substrates and selected by optical microscopy.\nThe mask containing the top hBN flake is then aligned and pressed against the substrate containing the graphene flake and heated to $\\approx$75 $^{\\circ}$C.\nWhen the mask is retracted from the substrate the graphene flake adhere strongly to the hBN flake and releases from the substrate sticking to the mask.\nThe mask containing the hBN-graphene stack is then aligned to the bottom hBN flake, brought in contact to the substrate and heated to temperatures up to $\\approx$150 $^{\\circ}$C.\nAt these temperatures the polycarbonate film melts on the substrate and releases from the mask together with the stack.\nThe sample is then left in chloroform for at least 15 hours in order to completely dissolve the polycarbonate film and obtain a clean graphene surface ensuring good graphene-contact interfaces.\n\n\\subsection{Contact deposition}\nThe contacts and top gate electrodes in this work were all fabricated at the same step to avoid further contamination of the non-encapsulated graphene regions.\nThe electrodes were patterned using standard e-beam lithography techniques using a PMMA 950K (270 nm thick) resist.\nThe markers for the e-beam lithography were patterned prior to the electrodes in the same PMMA resist and developed leaving openings which were used for the alignment of the electrodes.\nWe then used an e-beam evaporator with a base pressure below 10$^{-6}$ Torr to evaporate a 0.4 nm thick layer of Ti which was then naturally oxidized by inserting pure oxygen gas in the vacuum chamber to achieve pressures above 1 Torr.\nAfter 15 minutes the chamber was pumped down once again to the initial base pressure and the procedure was repeated to deposit and oxidize another 0.4 nm thick Ti layer.\nThis procedure is then followed by the deposition of a 67 nm thick Co layer with a capping layer of Al (5 nm) to protect the Co layer from oxidizing.\nThe large thickness of the Co layer is necessary to both overcome the height of the bottom hBN flake and also to facilitate the alignment of the electrodes by the perpendicular magnetic field.\nAfter the resist lift-off and wire bonding, the samples were loaded in a continuous He flow cryostat which stands in a room temperature electro-magnet.\nThe contact resistances were in the range 2 - 100 k$\\Omega$.\n\n\\subsection{Transport measurements}\nThe charge transport measurements were performed by standard low-frequency lock-in techniques with currents up to 100 nA.\nAll spin transport measurements presented here were done using the non-local geometry in which the charge contribution to the measured signal is minimized by separating the charge current path from the voltage detection circuit.\nWe used standard low-frequency lock-in techniques with applied currents of 1 $\\mu$A in order to preserve the high resistive contact interface barriers and avoid device heating. \n\n\\subsection{Magnetoresistance effects on the nonlocal signal}\nTo be sure that the magneto-resistance contribution is minimal in our nonlocal signals we characterize the graphene sheet resistance as a function of the applied magnetic field.\nIn Fig. \\ref{fig:SI-magres} we show a typical curve of the $R_{sq}$ versus $V_{tg}$ for B = 0 and 1 T measured in a 4-probe geometry.\nIt can be seen that the change in $R_{sq}$ with $B$ is maximum around the charge neutrality point and small at high values of $n$.\nThis observation also reflects on our nonlocal measurements in which we observe the monotonic increase in the nonlocal resistance as a function of $B$ characteristic to magneto-resistance effects only for low $n$.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{SI-magres.pdf}\n\t\\caption{Graphene square resistance as a function of $V_{tg}$ for $V_{bg}=-16.65$ V for sample 1 at 4.2 K and B = 0 (dashed blue line) and 1 T (red solid line). \\textit{Inset:} Ratio between the two curves in the main graph showing the magneto-resistance in the sample.}\n\t\\label{fig:SI-magres}\n\\end{figure}\n\n\\subsection{Simulation of Hanle precession data for non-homogeneous systems}\n\nIn order to describe the different regions in our system (encapsulated and non-encapsulated) we use a model for spin transport in an inhomogeneous system described elsewhere \\cite{GuimaraesNanoLetters2012}.\nWe solve the Bloch equations for spin diffusion in one dimension for a system with three distinct regions connected to one another.\nEach one of these regions can have independent values for the relevant parameters.\nWe assume the outer regions identical, with square resistance $R_{o}$, spin diffusion coefficient $D_{o}$ and spin relaxation time $\\tau_{o}$.\nThe inner region has distinct parameters $R_{i}$, $D_{i}$ and $\\tau_{i}$.\nWe assume that the spins are injected at the left boundary of the inner region and detected on the right, which results in a nonlocal spin signal $R_{sim}$.\nFor this work we get $R_{o}$, $D_{o}$, $\\tau_{o}$, $R_{i}$ and $D_{i}$ by charge and spin transport experiments as described in the main text.\nWe then fit the simulated curves for $R_{sim}$ using the solution for the Bloch equation for homogeneous systems (exactly in the same way we analyze our experimental results).\nFrom this fit we extract an effective spin diffusion coefficient $D_{fit}$ and relaxation time $\\tau_{fit}$.\nAn example of such simulated curve and fitting is shown in Fig. \\ref{fig:SI-simhanle}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.39\\textwidth]{SI-simhanle.pdf}\n\t\\caption{Hanle precession curve calculated using the model for a non-homonegeous system (black dots) with a fit using the expression for the solution of the Bloch equation in a homogeneous system.}\n\t\\label{fig:SI-simhanle}\n\\end{figure}\n\nFor a fixed $V_{bg}$, the effect of $V_{tg}$ is to change the spin resistance ratio between the inner and outer regions, $\\frac{R_{i}\\lambda_{i}}{R_{o}\\lambda_{o}}$, where $\\lambda_{i(o)}$ is the spin relaxation length for the inner (outer) regions.\nThe change of this ratio affects strongly the extracted spin relaxation times $\\tau_{fit}$ but leaves $D_{fit}$ mostly unaffected \\cite{GuimaraesNanoLetters2012}.\nIn order to get a good estimation of $\\tau_{i}$, we take the experimental values for $V_{bg}$=-52.5 V and simulate a few sets of points with only changing $R_{i}$.\nAs it can be seen in Fig. 3b in the main text, we find a close match between our simulated values of $\\tau_{fit}$ to our experimentally obtained values of $\\tau_{s}$ for $\\tau_{i}$ ranging from 3 to 5 ns.\nMeaning that the spin relaxation time for the encapsulated region is higher than, but still within the same order of magnitude of, the experimentally obtained values using the solution for the Bloch equations in a homogeneous system.\n\nIt is known that $R_{c}$ can affect the measurement of $\\tau_{s}$, especially in the case of very long values of $\\lambda_{s}$.\nHowever, even though we have a wide range of contact resistances $R_{c}$ 2 - 100 k$\\Omega$ with one sample having all contact resistances above 50 k$\\Omega$, we did not see any clear correlation between $R_{c}$ and the measured $\\tau_{s}$.\nWe would expect that, in the case of contact induced spin relaxation or poor graphene quality in the regions underneath the contacts, $\\tau_{s}$ would be mostly affected in the non-encapsulated regions.\nTherefore, the measured spin relaxation time for the non-encapsulated will be an effective spin relaxation time that includes all these effects, and consequently is taken into account in our model by setting $\\tau_{o}$ as the measured spin relaxation time for these regions.\n\n\\subsection{Experimental Hanle precession data}\n\nFor most of our devices we observe a small asymmetry in our Hanle precession curves between negative and positive perpendicular magnetic fields \\ref{fig:SI-hanle} which might arise due to some anisotropy in the magnetization of the electrodes.\nAlthough we cannot be certain about the reason for this asymmetry, the parameters extracted by fitting just one or the other side of the curve do not change significantly.\nComparing the parameters extracted from each side, $\\tau_{s}$ deviates by a factor of $\\approx$ 1.06 and $D_{s}$ by a maximum of a factor 2.\nSince the values for $D_{s}$ extracted by fitting the whole Hanle curve match the values for $D_{c}$ extracted by charge transport measurements, we believe that our procedure is justified.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[width=0.39\\textwidth]{SI-hanle-alt.pdf}\n\t\\caption{Top: Experimental data for $R_{s}=(R_{nl}^{P}-R_{nl}^{AP})\/2$ at $V_{bg}$ = -52.5V and $V_{tg}$ = +3.744 V, where $R_{nl}^{P(AP)}$ is the non-local resistance obtained for the electrodes in a (anti)parallel magnetization (dots) as a function of an applied perpendicular magnetic field. The fit (red line) used to extract $D_{s}$ and $\\tau_{s}$ is also shown. In the inset it is shown the data for $R_{nl}^{P(AP)}$ in black (red). Bottom: Hanle precession curve for $V_{bg}$ = $V_{tg}$ = 0 V with the respective non-local spin-valve shown in the inset.}\n\t\\label{fig:SI-hanle}\n\\end{figure}\n\nThe measured spin polarization of our electrodes were in the range $P$ = 1 - 6 $\\%$.\nThese values for $P$ are in the lower range of the values reported in literature \\cite{MihaiPhys.Rev.B2009,KawakamiPhys.Rev.Lett.2011}, however we believe that this value can be improved by increasing the quality of the high resistive barriers \\cite{KawakamiPhys.Rev.Lett.2010}.\n\n\\subsection{Results for spin and charge transport at 4.2 K and room temperature}\n\nOur results presented in the main text were all performed in sample 1 at 4.2 K.\nFor comparison we also include here the square resistance as a function of $V_{tg}$ and $V_{bg}$ and the extracted spin transport parameters for the same sample at room temperature (Fig. \\ref{fig:samples}a) and another sample (sample 2) at room (Fig. \\ref{fig:samples}b) and liquid Helium (Fig. \\ref{fig:samples}c) temperatures.\nThe samples dimensions and their electronic mobilities at room temperature and 4.2 K are given in table \\ref{fig:samples}.\nSample 3 (not shown here) showed similar results.\n\n\\begin{figure*}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.8\\textwidth]{samples-diftemp.pdf}\n\t\\caption{Charge and spin transport measurements for (a) sample 1 at room temperature and sample 2 at (b) room temperature and (c) 4.2 K.}\n\t\\label{fig:samples}\n\\end{figure*}\n\n\\begin{table*}[h]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n & $\\ell_{tg}$ & $\\ell_{enc}$ & $L$ & $W$ & $\\mu_{RT}$ & $\\mu_{4K}$ \\\\\n\\hline\nSample 1 & 10.35 $\\mu$m & 12 $\\mu$m & 13.8 $\\mu$m & 1.3 $\\mu$m & 1.5 m$^{2}$\/Vs & 2.3 m$^{2}$\/Vs \\\\\n\\hline\nSample 2 & 4.5 $\\mu$m & 7.6 $\\mu$m & 8.72 $\\mu$m & 2.8 $\\mu$m & 2.4 m$^{2}$\/Vs & 3.3 m$^{2}$\/Vs \\\\\n\\hline\n\\end{tabular}\n\\label{tab:samples}\n\\caption{Length of the top gated region ($\\ell_{tg}$), encapsulated region ($\\ell_{enc}$), inner electrodes spacing ($L$), graphene flake width ($W$) and electronic mobilities at room temperature ($\\mu_{RT}$) and 4.2 K ($\\mu_{4K}$) for samples 1 and 2.}\n\\end{table*}\n\n\\subsection{Estimation of $\\Delta_{R}$}\n\nIn order to give a rough estimation for the spin orbit coupling strength $\\Delta_{R}$ we assume that the spin relaxation is composed of the sum of an electric field independent and electric field dependent terms: $1 \/ \\tau_{\\bot} = 1 \/ \\tau_{ind} + 1 \/ \\tau_{E}$.\nThe spin relaxation time $\\tau_{||}$ is taken equal to the spin relaxation experimentally ($\\tau_{s}$) and $\\tau_{ind}$ is $\\tau_{s}$ multiplied by the ratio $r = \\tau_{\\bot} \/ \\tau_{s}$ at \\linebreak $\\bar{E}$ = 0 V\/nm: $\\tau_{ind} = 0.75 \\tau_{s}$.\nIf we assume a D'Yakonov-Perel spin relaxation mechanism for $1 \/ \\tau_{E} = \\frac{4 \\Delta_{R}^2}{\\hbar^{2}} \\tau_{p}$, we have: $\\Delta_{R}=\\zeta \\bar{E}=(\\hbar \/ 2) (\\tau_{E} \\tau_{p})^{-1\/2}$ \\cite{FabianPhys.Rev.B2009a}, where $\\tau_{p}=2 D_{c} \/ v_{F}$ is the momentum relaxation time, $D_{c}$ the charge diffusion constant and $v_{F}\\approx 10^{6}$ m\/s the Fermi velocity.\nWe further take $D_{c} \\approx D_{s}$.\nSolving it for $\\tau_{E}$ we have:\n\n\\begin{equation}\n\\tau_{E} = \\tau_{s} \\left( \\frac{1}{r} - \\frac{1}{0.75} \\right)^{-1},\n\\end{equation}\n\nand\n\n\\begin{equation}\n\\zeta = \\frac{\\Delta_{R}}{\\bar{E}} = \\frac{\\hbar}{2 \\bar{E}} \\frac{1}{\\sqrt{\\tau_{s} \\tau_{p}}} \\left( \\frac{1}{r} - \\frac{1}{0.75} \\right)^{-1\/2}\n\\end{equation}\n\n\\section{D'Yakonov-Perel \\textit{versus} Elliott-Yafet mechanisms for spin relaxation}\nTwo mechanisms are often considered in order to try to explain spin relaxation in graphene: they are the D'Yakonov-Perel (DP) and Elliott-Yafet (EY) mechanisms for spin relaxation.\nThese two mechanisms show different behavior of the spin relaxation time as a function of the momentum relaxation time ($\\tau_{p}$): $\\tau_{EY} \\propto \\tau_{p}$ and $\\tau_{DP} \\propto 1 \/ \\tau_{p}$, with $\\tau_{EY(DP)}$ been the spin relaxation time due to the EY (DP) mechanism.\n\nIt was theoretically demonstrated that $\\tau_{EY}$ for the specific case of graphene for many different type of scatterers is given by \\cite{GuineaPhys.Rev.Lett.2012}: \n\n\\begin{equation}\n\\tau_{EY} \\approx \\frac{E_{F}^{2}}{\\Delta_{SO}^2} \\tau_{p},\n\\end{equation}\n\n\\noindent where $E_{F}$ is the Fermi energy and $\\Delta_{SO}$ the spin orbit coupling.\nThe DP mechanism obeys:\n\n\\begin{equation}\n\\tau_{DP} \\approx \\frac{\\hbar^{2}}{\\Delta_{SO}^2} \\frac{1}{\\tau_{p}}.\n\\end{equation}\n\nFollowing the reasoning of Huertas-Hernando et al. \\cite{BrataasPhys.Rev.Lett.2009}, to obtain the relative importance of these mechanisms we take the ratio:\n\n\\begin{equation}\n\\frac{\\tau_{EY}}{\\tau_{DP}} \\approx \\left( \\frac{E_{F} \\tau_{p}}{\\hbar} \\right)^{2}.\n\\end{equation}\n\nIf $\\tau_{EY} \/ \\tau_{DP} >$ 1, the spin relaxation time due to the DP mechanism is smaller and it dominates over the EY mechanism.\nUsing typical values for our samples of $E_{F} \\approx$ 30 meV (equivalent to a charge carrier density $n \\approx$ 10$^{12}$ cm$^{-2}$) and $\\tau_{p} \\approx$ 10$^{-13}$ s, we obtain $\\tau_{EY} \/ \\tau_{DP} \\approx$ 20, indicating a higher importance of the DP mechanism for the spin relaxation in our samples.\n\nAlthough this is not a definite proof that the DP is the dominant mechanism for spin relaxation in graphene, we believe that it is the most appropriate way to obtain the spin orbit coupling strength due to the electric field in our devices.\n\n\n\n\\section{Introduction}\nThe generation, manipulation and detection of spin information has been the target of several studies due to the implications for novel spintronic devices \\cite{WolfScience2001,ZuticActaPhysicaSlovaca2007}.\nIn the recent years graphene has attracted a lot of attention in spintronics due to its theoretically large intrinsic spin relaxation time and length of the order of $\\tau_{s}\\approx$ 100 ns and $\\lambda_{s}\\approx$ 100 $\\mu$m respectively \\cite{BrataasPhys.Rev.Lett.2009,BarnafmmodePhys.Rev.B2011}.\nAlthough experimental results still fall short of these expectations \\cite{MihaiPhys.Rev.B2009,KawakamiPhys.Rev.Lett.2011,OezyilmazPhys.Rev.Lett.2011,GuimaraesNanoLetters2012}, graphene has already achieved the longest measured nonlocal spin relaxation length \\cite{KawakamiPhys.Rev.Lett.2011,WojtaszekPhys.Rev.B2013} and furthest transport of spin information at room temperature \\cite{ZomerPhys.Rev.B2012}.\nHowever, the mechanisms for spin relaxation in graphene are still under heavy debate with various theoretical models proposed \\cite{BrataasPhys.Rev.Lett.2009,BarnafmmodePhys.Rev.B2011,FabianPhys.Rev.Lett.2014,WuNewJournalofPhysics2012,RocheJPD2014,GuineaPhys.Rev.Lett.2012}.\n\nTo take advantage of the long spin relaxation times in graphene, e.g. for spin logic devices, one requires easy control of the spin information, for example by an applied electric field.\nSingle layer graphene is an ideal system for this purpose, not only because of its high mobilities and low intrinsic spin-orbit fields (SOF), but also due to the simple relation between the carriers' wavevector, the applied perpendicular electric field and the induced Rashba SOF \\cite{BarnafmmodePhys.Rev.B2011,FabianPhys.Rev.B2009,BrataasPhys.Rev.Lett.2009,BrataasPhys.Rev.B2006,MelePhys.Rev.Lett.2005,MacDonaldPhys.Rev.B2006,RashbaPhys.Rev.B2009,FabianPhys.Rev.B2009a}.\nIn bilayer graphene a more complicated behavior is expected when spin-orbit coupling is considered \\cite{FabianPhys.Rev.B2012}.\n\nHere we report nonlocal spin transport measurements on single layer graphene in which we address both topics specified above.\nOur devices consist of a single layer graphene flake on hexagonal Boron Nitride (hBN) of which a central region is encapsulated with another hBN flake and hence protected from the environment.\nThe presence of a top and bottom gate give rise to two independent electric fields that are experienced by the graphene: $E_{tg}= - \\epsilon_{tg}(V_{tg}-V_{tg}^{0})\/d_{tg}$ and $E_{bg}=\\epsilon_{bg}(V_{bg}-V_{bg}^{0})\/d_{bg}$, respectively \\cite{WangNature2009}, where $\\epsilon_{tg(bg)}\\approx$ 3.9 is the dielectric constant, $d_{tg(bg)}$ is the dielectric thickness and $V_{tg(bg)}^{0}$ the position of the charge neutrality point for the top (bottom) gate.\nTheir difference controls the carrier density in the graphene ($n = (E_{bg} - E_{tg}) \\epsilon_{0} \/ e$) and their average gives the effective electric field experienced by the graphene ($\\bar{E}=(E_{tg}+E_{bg})\/2$), which breaks the inversion symmetry in the encapsulated region, where $\\epsilon_{0}$ is the electric constant and $e$ the electric charge.\nOur devices show enhanced spin relaxation times of at least 2 ns and also, due to the higher electronic mobility, spin relaxation lengths above 12 $\\mu$m at room temperature (RT) and 4.2 K.\nBy a simple model we show that the measured spin relaxation times are a lower bound due to the influence of the non-encapsulated regions.\n\nBy comparing the spin relaxation time for spins out-of-plane ($\\tau_{\\bot}$) to spins in-plane ($\\tau_{||}$) as a function of the electric field we get insight on the nature of the SOF that cause spin relaxation in graphene.\nFor SOF pointing preferentially in the graphene plane, e.g. for adatoms and impurities, we expect: $\\tau_{\\bot}\\approx 0.5\\tau_{||}$ \\cite{ZuticActaPhysicaSlovaca2007,BrataasPhys.Rev.Lett.2009,TombrosPhys.Rev.Lett.2008,FabianPhys.Rev.B2009a}.\nIf the SOF point out-of-plane, as for ripples\\cite{BrataasPhys.Rev.Lett.2009}, we have: $\\tau_{\\bot} \\gg \\tau_{||}$.\nHowever, if the main relaxation mechanism is through random magnetic impurities or defects, no preferential direction for the spins is expected: $\\tau_{||} \\approx \\tau_{\\bot}$.\nHere we obtain $\\tau_{\\bot} \/ \\tau_{||}\\approx$ 0.75 at $\\bar{E}$ = 0 V\/nm$^{-1}$.\nThis ratio decreases with increasing $\\bar{E}$, in agreement with an electric field induced Rashba SOF pointing in the graphene plane.\n\nDevice number 1 is illustrated in Fig. \\ref{fig:figure-1}a and b.\nThe hBN-graphene-hBN stack sits on a 300 nm thick SiO$_{2}$ layer on a heavily doped Si substrate which is used as a back-gate.\nThe sample preparation is described in detail in the supplementary information and follows Ref. \\cite{DeanScience2013,WeesAppliedPhysicsLetters2014}.\nWe use Co electrodes with a thin TiO$_{2}$ interface barrier to perform spin transport measurements.\nThree devices were studied, all showing similar results.\nHere we show the results for the device with the longest encapsulated region ($\\approx$ 12 $\\mu$m) and spacing between the inner contacts (13.8 $\\mu$m).\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-1.pdf}\n\t\\caption{(a) Side-view schematics and (b) Top-view optical microscope image of a hBN encapsulated graphene spin-valve. The numbers show the contact electrodes and the top (bottom) gates electrodes are indicated as TG (BG). The graphene is outlined by the dashed line. (c) Square resistance (R$_{sq}$) as a function of $V_{tg}$ and $V_{bg}$.}\n\t\\label{fig:figure-1}\n\\end{figure}\n\nThe charge transport properties of the encapsulated region are measured by applying a current between electrodes 1 and 5, Fig. \\ref{fig:figure-1}a and b, and scanning the top and bottom gate voltages ($V_{tg}$ and $V_{bg}$, respectively) while recording the voltage between electrodes 2 and 3.\nFig. \\ref{fig:figure-1}c shows the square resistance ($R_{sq}$) as a function of $V_{tg}$ and $V_{bg}$.\nThe charge neutrality point depends on both $V_{tg}$ and $V_{bg}$ in a linear fashion.\nThe slope of the line gives the ratio between the bottom and top gate capacitances: $\\alpha_{bg}\/\\alpha_{tg}\\approx$ 0.036.\nA small top gate independent resistance peak around $V_{bg}$ = -16.6 V (not visible in Fig. \\ref{fig:figure-1}c) arises from the non-top gated regions between the two inner contacts.\nThe electronic mobility for this device is $\\mu\\approx$ 1.5 m$^{2}$\/Vs at RT and $\\mu\\approx$ 2.3 m$^{2}$\/Vs at 4.2 K.\nAlthough the mobilities of our devices are above the best devices based on SiO$_{2}$ they are still one order of magnitude lower than the best devices on hBN \\cite{DeanScience2013} which can be attributed to small bubbles or contamination visible on the graphene\/hBN stack.\n\nSpin dependent measurements are performed using a standard nonlocal geometry in which the current path is separated from the voltage detection circuit \\cite{MihaiPhys.Rev.B2009}.\nThe current is driven between electrodes 1 and 2 and the voltage measured between electrodes 3 and 5, which are on the other side of the encapsulated region (Fig. \\ref{fig:figure-1}a and b).\nTo obtain the spin relaxation time ($\\tau_{s}$) and the spin diffusion coefficient ($D_{s}$) we perform Hanle precession measurements where the nonlocal signal is measured as a function of a perpendicular magnetic field $B$.\nWe then fit the data with the solution to the Bloch equations \\cite{MihaiPhys.Rev.B2009}.\n\nThe results for $D_{s}$, $\\tau_{s}$ and the spin relaxation length ($\\lambda_{s}=\\sqrt{D_{s}\\tau_{s}}$) as a function of the $V_{tg}$ for three values of $V_{bg}$ at 4.2 K are shown in Fig. \\ref{fig:figure-2}.\nA similar set of measurements was performed at RT and for other samples where only a small difference was observed (see supplementary information \\cite{supinfo}).\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-2.pdf}\n\t\\caption{(Spin) diffusion coefficient ($D_{s}$), spin relaxation time $\\tau_{s}$ and spin relaxation length $\\lambda_{s}$ as a function of $V_{tg}$ for three different values of $V_{bg}$. The error bars are smaller than the dot size. The dashed red (blue) lines show the charge diffusion coefficient $D_{c}$ for $V_{bg}$ = +52.5 V (-52.5 V).}\n\t\\label{fig:figure-2}\n\\end{figure}\n\nDue to our device mobility, $D_{s}$ is higher than for regular graphene devices on SiO$_{2}$ ($D_{s}$ $\\approx$ 0.02 m$^{2}$\/s) \\cite{KawakamiPhys.Rev.Lett.2011,MihaiPhys.Rev.B2009} and comparable to suspended \\cite{GuimaraesNanoLetters2012} and non-encapsulated hBN supported devices \\cite{ZomerPhys.Rev.B2012} ($D_{s}$ $\\approx$ 0.05 m$^{2}$\/s).\nAs an extra confirmation, we check that $D_{s}$ agrees with the charge diffusion coefficient $D_{c}=[R_{sq}e^{2}\\nu(E_{F})]^{-1}$\\footnote{A resistance of 3.2 k$\\Omega$ was subtracted in the calculation of $D_{c}$ for $V_{bg}=-52.5$V to account for the non-top gated regions.}, where $e$ is the electron charge and $\\nu(E_{F})$ the density of states at the Fermi energy $E_{F}$.\nNext, we observe that the obtained spin relaxation times are higher than those on regular SiO$_{2}$ substrates ($\\tau_{s}\\approx$ 0.1 - 1 ns)\\cite{KawakamiPhys.Rev.Lett.2011,MihaiPhys.Rev.B2009} and in non-encapsulated hBN supported devices ($\\tau_{s}\\approx$ 0.1 - 0.5 ns)  \\cite{ZomerPhys.Rev.B2012}, reaching up to $\\tau_{s}$=(1.9$\\pm$0.2) ns at 4.2 K and $\\tau_{s}$=(2.4$\\pm$0.4) ns for RT.\nThese values surpass all previous nonlocal measurements of $\\tau_{s}$ in single layer graphene both at room and low temperatures \\footnote{2-terminal local measurements on epitaxial graphene estimated $\\tau_{s}\\approx$ 100 ns at 4.2 K \\cite{FertNatPhys2012}.} \\cite{FertNatPhys2012}.\nWe obtain a maximum of $\\lambda_{s}$ = 12.3 $\\mu$m at 4.2 K and $\\lambda_{s}$ = 12.1 $\\mu$m at RT.\n\n\nComparing $\\tau_{s}$ obtained in our devices with non-encapsulated hBN based devices ($\\tau_{s}\\approx$ 0.2 ns) \\cite{ZomerPhys.Rev.B2012}, we can conclude that the encapsulation of graphene on hBN significantly increases the spin relaxation times.\nNote that the non-encapsulated devices had comparable electronic mobilities which indicates that $\\tau_{s}$ is not linked to the momentum relaxation time in a trivial manner \\cite{K.NanoLetters2012}.\nBy measuring a region about 5 $\\mu$m away from the encapsulated part, we find $D_{s}\\approx$ 0.03 m$^{2}$\/s, $\\tau_{s}\\approx$ 0.3 ns and $\\lambda_{s}\\approx$ 3 $\\mu$m, in agreement with the previously reported results.\nThe increase in $\\tau_{s}$ for the encapsulated region can be due to several factors.\nThis region is protected from polymer remains or other contamination which can increase spin scattering.\nIn addition to that, the inversion asymmetry, which can generate an extra term for the spin-orbit coupling, is also reduced and controlled by tuning $V_{tg}$ and $V_{bg}$ separately as explained earlier.\n\nAs can be seen in Fig. \\ref{fig:figure-2}, $\\tau_{s}$ is modulated by $V_{tg}$, showing a dip close to the charge neutrality point in the encapsulated region (e.g. $V_{tg}$=1.1 V and $V_{bg}$=-52.5 V).\nFor larger charge carrier densities in the non-encapsulated regions, the modulation in $\\tau_{s}$ by $V_{tg}$ is smaller, although still present.\nFurthermore, the average value of $\\tau_{s}$ is maximum at low carrier densities in the non-encapsulated regions (large negative $V_{bg}$) and decreases with increasing $V_{bg}$.\nThis difference on the measured $\\tau_{s}$ for different carrier densities on the outer regions can be explained by simulations that treat the full device \\cite{GuimaraesNanoLetters2012}.\nSince the transport is diffusive, the spins can explore both the encapsulated and the non-encapsulated regions before been detected.\nWe take this into account by describing our sample as two outer regions connected by a central region \\cite{supinfo}.\nThe relevant parameters ($R_{sq}$, $D_{s}$ and $\\tau_{s}$) are set for each region individually.\nThe values for the outer regions and $R_{sq}$ and $D_{s}$ for the inner region can be extracted from our charge and spin transport measurements.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-3-alt.pdf}\n\t\\caption{Effective spin relaxation times extracted for different values for $\\tau_{o}$ and $\\tau_{i}$ (lines) compared to our experimental data for $V_{bg}$ = -52.5 V (dots). The inset shows a schematics of the simulated system.}\n\t\\label{fig:figure-3}\n\\end{figure}\n\nChanging the values of the spin relaxation time for the outer regions, $\\tau_{o}$, around the experimentally obtained values and changing the values for the spin relaxation time in the inner region, $\\tau_{i}$, we simulate Hanle precession curves that are fitted in the same way done for our experiments to obtain an effective value for the spin relaxation time $\\tau_{fit}$.\nWe get a reasonable quantitative agreement between our simulations and experiment at $V_{bg}$ = -52.5V for $\\tau_{i}$ = 3 ns.\nThis means that the spin relaxation time for the encapsulated region is higher ($\\tau_{s}\\approx$ 3 ns), but still within the same order of magnitude as the values obtained by analyzing the data using a homogeneous system (Fig. \\ref{fig:figure-2}).\nThe trend in $V_{tg}$ is also reproduced, which indicates that it is given by the ratio of the resistivities of the inner and outer regions.\n\nEven though the experimentally obtained value for $\\tau_{s}$ depends on the gate voltages in a non-trivial way due to the influence of the non-encapsulated regions, we can still study how the electric field affects the ratio between the spin relaxation times for out-of-plane to in-plane spins: $r=\\tau_{\\bot}\/\\tau_{||}$.\nAs explained in the introduction, this way we can get insight about the SOF in our system.\n\nTo compare the spin relaxation for spins parallel and perpendicular to the graphene plane we perform the Hanle precession measurements as described before, but increase the perpendicular magnetic field to higher values, $B>$ 1 T.\nAt such high magnetic fields the magnetization of the electrodes rotates out-of-plane and the injected spins do not precess anymore.\nThis is seen as a saturation of the nonlocal signal at high B, Fig. \\ref{fig:figure-4}a.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figure-4-altb.pdf}\n\t\\caption{(a) $R_{nl}$ as a function of $B$ showing Hanle precession at low fields and a saturation of the signal at high fields when the magnetization of the electrodes point out-of-plane. Inset: A cartoon showing the magnetization of the contacts: in-plane at low fields and out-of-plane at high fields. (b) The ratio $\\tau_{\\bot} \/ \\tau_{||}$ as a function of electric field $\\bar{E}$ for different values of carrier density $n$. \\textit{Inset}: The same data points for $\\tau_{\\bot} \/ \\tau_{||}$ used in the main graph, but plotted as a function of $n$ where different colors and symbols represent the different values of $\\bar{E}$.}\n\t\\label{fig:figure-4}\n\\end{figure}\n\nWe observe that different combinations of $V_{tg}$ and $V_{bg}$ result in a saturation of $R_{nl}$ at high magnetic fields at values always smaller than $R_{nl}$ at $B$=0 T, with the saturation occuring at 43-57$\\%$ of the initial value.\nGiven that the nonlocal spin signal is given by $\\Delta R_{nl}=\\frac{P^{2}R_{sq}\\lambda_{s}}{W}e^{-L\/\\lambda_{s}}$, where $P$ is the polarization of the electrodes, $L$ the distance between electrodes and $W$ the channel width, we can estimate the anisotropy in the spin relaxation times assuming that $P$ and $R_{sq}$ do not change significantly with field.\nDue to a large magneto-resistance at low carrier densities, our analysis is done only for points at large enough carrier densities for both the inner and outer regions \\footnote{At low $n$, $R_{sq}$ of our devices scales with B$^{2}$ leading to a background in our signal that overcomes the nonlocal spin signal at large B.}.\nWe can relate the ratio of the nonlocal spin signal and the ratio of the spin relaxation times by:\n$R_{nl}^{\\bot} \/ R_{nl}^{||} = \\sqrt{r}e^{\\frac{L}{\\lambda_{||}}\\left( \\frac{\\sqrt{r}-1}{\\sqrt{r}} \\right)}$, where $\\lambda_{||}=\\sqrt{D_{s}\\tau_{||}}$ is obtained via our Hanle precession measurements.\nIn Fig. \\ref{fig:figure-4}b we plot the ratio $\\tau_{\\bot} \/ \\tau_{||}$ as a function of $\\bar{E}$ for different values of $n$ where we see a clear decrease of this ratio, from 0.75 at $\\bar{E}\\approx$ 0 V\/nm to about 0.65 at $\\bar{E}\\approx$  -0.7 V\/nm.\nThe inset on Fig. \\ref{fig:figure-4}b shows the dependence of $r$ as a function of $n$ for different values of $\\bar{E}$ where no clear trend can be seen.\nThe value for $\\tau_{\\bot} \/ \\tau_{||}$ for zero electric field is similar to the values found previously on SiO$_{2}$ based devices ($\\tau_{\\bot} \\approx$ 0.8 $\\tau_{||}$) \\cite{TombrosPhys.Rev.Lett.2008}.\nIn the case of a inversion symmetric graphene layer with no extrinsic sources for SOF we would not expect any particular preference for direction of the spins, meaning that $\\tau_{\\bot} \\approx \\tau_{||}$ \\cite{ZuticActaPhysicaSlovaca2007}.\nThe fact that even at $\\bar{E}$ = 0 V\/nm the ratio between $\\tau_{\\bot}$ and $\\tau_{||}$ is below 1 means that even without an externally applied electric field there are probably remanent SOF pointing preferentially in the graphene plane.\n\nThe decrease of $r$ with increasing $\\bar{E}$ is in agreement with theories that dictate an increase in in-plane Rashba SOF with the increase of electric field \\cite{BarnafmmodePhys.Rev.B2011,FabianPhys.Rev.B2009,RashbaPhys.Rev.B2009,MacDonaldPhys.Rev.B2006,BrataasPhys.Rev.B2006,MelePhys.Rev.Lett.2005,FabianPhys.Rev.B2009a}.\nThe Rashba-type spin-orbit Hamiltonian for graphene is given by: $\\mathcal{H}_{SO}=\\frac{\\Delta_{R}}{2} \\left( \\mathbf{\\sigma} \\times \\mathbf{s} \\right)_{z}$, where $\\mathbf{\\sigma}$ and $\\mathbf{s}$ are the pseudospin and real spin Pauli matrices respectively.\nFor single layer graphene the spin-orbit constant is known to depend on the $\\bar{E}$ in a linear way $\\Delta_{R}=\\zeta \\bar{E}$ \\cite{FabianPhys.Rev.B2009,RashbaPhys.Rev.B2009}, where $\\zeta$ is the coupling constant.\nTheoretical values range from $\\zeta$ = 0.3 - 66 $\\mu$eV\/Vnm$^{-1}$ \\cite{MacDonaldPhys.Rev.B2006,BrataasPhys.Rev.B2006,MelePhys.Rev.Lett.2005,FabianPhys.Rev.B2009}.\nWe can roughly estimate $\\zeta$ by assuming a D'Yakonov-Perel mechanism for spin relaxation \\cite{DP1971,FabianPhys.Rev.B2009a} with different values for the spin orbit coupling for in- and out-of-plane spins.\nOur analysis \\cite{supinfo} results in $\\zeta \\approx$ (40$\\pm$20) $\\mu$eV\/Vnm$^{-1}$, within the range of the theoretical predictions.\n\nIn conclusion, we measured the spin transport characteristics of a single layer graphene device encapsulated with hBN.\nWe measured spin relaxation times up to $\\tau_{s}\\approx$ 2 ns and spin relaxation lengths above $\\lambda_{s}$= 12 $\\mu$m.\nBy taking into consideration that the non-encapsulated regions of our devices play a role in our measurements of $\\tau_{s}$, we estimate the actual spin relaxation time in the encapsulated region to be $\\tau_{s}\\approx$ 3 ns.\nFurthermore, we showed that the ratio between out-of-plane and in-plane spin relaxation times changes from $\\tau_{\\bot} \/ \\tau_{||}\\approx$ 0.75 to 0.65 with increasing the applied out-of-plane electric field.\nThis observation is in agreement with an electric field induced Rashba-type spin orbit.\nOur results show not only that $\\tau_{s}$ in graphene can be improved by improving the quality of the devices, but also that electrical control of spin information in graphene is possible, paving the way to new graphene spintronic devices.\n\n\\textit{Note added:} During the preparation of this manuscript we became aware of a work in which $\\lambda_{s}$ up to 10 $\\mu$m for single and few-layer graphene were achieved for a single gated structure which did not allow the study of $\\tau_{s}$ as a function of $\\bar{E}$ \\cite{arxiv-aachen}.\n\nWe would like to acknowledge A. Kamerbeek and E. Sherman for insightful discussions and J. G. Holstein, H. M. de Roosz and H. Adema for the technical support.\n\nThe research leading to these results has received funding from the Dutch Foundation for Fundamental Research on Matter (FOM), the European Union Seventh Framework Programme under grant agreement n$^{\\circ}$604391 Graphene Flagship, the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7\/2007-2013\/ under REA grant agreement n$^{\\circ}$607904-13 Spinograph, NWO, NanoNed, the Zernike Institute for Advanced Materials and CNPq, Brazil.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTurbulence is considered to be the last major unsolved problem of classical physics~\\cite{Frisch:1995tu,FalkovichSreenivasan}. It is usually studied in the context of non-relativistic incompressible fluid flows, which are described by the Navier-Stokes (NS) equations\n\\begin{equation}\n\\partial_t v_i + v_j \\partial_j v_i = -\\partial_i P + \\nu \\partial_{jj} v_i + f_i\n\\qquad\n\\partial_i v_i = 0,\n\\end{equation}\nwhere $v_i$ is the fluid velocity, $P$ is the pressure (divided by the constant density), $\\nu$ is the fluid viscosity and $f_i$ some external force. Due to the strong non-linearity of the NS equations, the study of turbulent flows has focused on the statistical properties of the solutions. There are strong indications that these properties exhibit some degree of universality but a full analytical understanding is still missing.\n\nSolutions to the NS equations are characterized by the Reynolds number, defined by:\n\\begin{equation}\nRe = \\frac{L v}{\\nu},\n\\end{equation}\nwhere $L$ is the characteristic length scale of velocity difference and $v$ is the typical velocity difference. Flows with $ Re \\ll 1$ are dominated by the linear terms and are laminar while flows with $Re \\gg 100$ are dominated by the non-linear term and become turbulent. One defined also the viscous scale $l\\sim \\left(\\nu^3\/\\epsilon\\right)^{1\/4}$.\n\nA statistical turbulent steady-state can be maintained by some stationary random external force constantly being applied at the scale $L$. One usually studies the longitudinal structure functions of order $n$:\n\\begin{equation}\\label{tur:structurefuncs}\nS_n (r) \\equiv \\left\\langle \\left( \\delta \\vec{v}(\\vec{r}) \\cdot \\frac{\\vec{r}}{r} \\right)^n \\right\\rangle .\n\\end{equation}\nwhere $ \\delta \\vec{v} (\\vec{r})$ is the velocity difference on displacement $\\vec{r}$. A very well known, and one of few exact results for the statistics of turbulence, is Kolmogorov's four-fifths law \\cite{Kol:1941a}\n\\begin{equation}\\label{tur:Kol45law}\nS_3 (r) = - \\frac{12}{d\\left(d+2\\right)} \\epsilon r,\n\\end{equation}  \nwhere $d$ is the number of space dimensions, $\\epsilon$ is the mean rate of energy dissipation per unit volume and $l\\ll r\\ll L$ ($Re \\rightarrow \\infty$).\n \nKolmogorov \\cite{Kol:1941b} assumed that in this limit the velocity statistics in the inertial range are scale invariant. This assumption and dimensional analysis then lead to the conclusion that:\n\\begin{equation} \\label{eq:univeral_scaling}\nS_n (r) \\propto (\\epsilon r )^{n\/3},\n\\end{equation}\nwhich is known as \\emph{K41 scaling}. Experimental and numerical evidence suggests that turbulent flows in the inertial range do exhibit universal behaviour, such that the structure functions scale as power laws:\n\\begin{equation}\\label{eq:anomscaling}\nS_n(r) \\propto r^{\\xi_n} \\ .\n\\end{equation}\nHowever, in $d\\ge3$ space dimensions, $\\xi_n$ is not linear in $n$, and deviates from the prediction of $\\xi_n = n\/3 $ obtained from the self-similarity (or scale invariance) assumption (except for $n=3$, for which \\eqref{tur:Kol45law} holds). The calculation of the anomalous exponents $\\xi_n$ remains an open problem.\n\nThe Euler equation, which is the Navier Stokes equation with zero viscosity, is a \\emph{dual scale} invariant theory -- it's scale invariant independently in both the space dimensions and the time dimension. The NS equations are of-course not dual scale invariant, and in the limit of zero viscosity the theory is still anomalous and the scale symmetry is broken to $z=2\/3$, as can be seen from \\eqref{tur:Kol45law} (which should be valid at $\\nu\\rightarrow0$ but not  vanishing). This breaking can be thought to be spontaneous \\cite{Oz:2017ihc} and in that sense, the study of turbulence has strong connections with the study of spontaneous symmetry breaking and anomalies in non relativistic field theories (see \\cite{Arav:2017plg} for a related discussion of non-relativistic scale symmetry breaking and anomalies).\n\nEling and Oz proposed \\cite{Eling:2015mxa} an analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence which is a Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation \\cite{Knizhnik:1988ak,Rhodes:2013iua} and is claimed to be valid in any number of space dimensions. In this formula the anomalous exponents are the K41 exponents \\eqref{eq:univeral_scaling} dressed via a coupling to a lognormal random geometry:\n \\begin{equation} \\label{eq:kpz_anomalous_exponents}\n  \\xi_n - \\frac{n}{3} = \\gamma^2\\left(d\\right)\\xi_n\\left(1-\\xi_n\\right) \\ .\n \\end{equation}\nThis formula has one real parameter $\\gamma$ that depends on the number of space dimensions. The use of the KPZ formula is inspired by the fluid\/gravity correspondence \\cite{Bhattacharyya:2008kq, Eling:2009pb, Eling:2010vr}, the fluid is thought to be coupled to random geometry. A derivation of \\eqref{eq:kpz_anomalous_exponents} was suggested in \\cite{Oz:2017ihc}, relying on the effective action for the spontaneous symmetry breaking pattern we will discuss in this paper.\n\nAs mentioned above, the turbulence problem exhibits a two-dimensional scale symmetry which is thought to be spontaneously broken to a regular one dimensional Lifshitz scaling. The above proposal and this symmetry breaking mechanism suggest we should study the spontaneous symmetry breaking of this dual scale, or ``arbitrary $z$'' scale symmetry and the Nambu-Goldstone mode that should exist in such theory. A particle with a logarithmic correlation function may serve as the mediator for the random geometry process needed to establish the KPZ relation, therefore, a logarithmic-correlated NG mode will be of a great interest.\n\n\n\\section{Dual Scale Invariant Effective Action}\nWe are looking for an effective action which would be scale invariant under an \\emph{arbitrary} scale symmetry, i.e., the whole set of transformations\n\\begin{equation}\n \\left({ \\vec{x},t} \\right) \\rightarrow \\left( { \\lambda \\vec{x}, \\lambda^z t } \\right)\n\\end{equation}\nfor arbitrary $z$.\nWe can also parametrize these tranformations differently by\n\\begin{equation}\n \\left({ \\vec{x},t} \\right) \\rightarrow \\left( { \\lambda_1 \\vec{x}, \\lambda_2 t } \\right) \\ .\n\\end{equation}\nWe are interested in the dilaton \\emph{steady-state effective action} which we expect to be a space(-only) integration of a lagrange density $L$.\nA natural guess would be\n\\begin{equation} \\label{random_geometry_lagrangian}\n L = \\tau \\left(\\nabla^2\\right)^{d\/2}\\tau\n\\end{equation}\nwhich induces an effective action invariant under a general tranformation $\\vec{x} \\rightarrow \\lambda \\vec{x} $, regardless of $z$ (or $\\lambda_2$). It would be nice if this was the only option since a particle with such action in $d$ space dimensions has a logarithmic correlation function, the problem is that by declaring the tranformation rule $\\tau \\rightarrow \\tau + \\sigma$ for $\\lambda = e^\\sigma$, we get that we can also have\n\\begin{equation}\n L = e^{-\\alpha\\tau} \\left(\\nabla^2\\right)^{\\beta\/2}\\tau\n\\end{equation}\nas long as $\\alpha+\\beta=d$\n \nNote that the original symmetry group had 2 degrees of freedom -- $\\lambda$ and $z$, or $\\lambda_1 $ and $\\lambda_2$, but the symmetry we required the dilaton lagrangian to obey was parametrized by only one of them $\\lambda$ (or $\\lambda_1$). This seems like a simplification of the requirement and we will see that although we don't introduce time dependence or time integration (by the steady-state assumption), there is a way to require both $\\lambda$ and $z$ (or $\\lambda_1$ and $\\lambda_2$), and therefore getting stricter requirements.\n \nNote also that usually when one deals with Lifshitz scaling with fixed dynamical exponent $z$, the dilaton transformation is defined to be\n\\begin{equation}\n \\tau\\left(\\vec{x},t\\right) \\rightarrow \\tau \\left( {e^{-\\sigma} \\vec{x} , e^{-z\\sigma}t } \\right) + \\sigma \n\\end{equation}\nsuch that the dilaton is able to directly compensate for space variations, by the $\\sigma$ contribution added to $\\tau$ which corresponds to the space tranformation $\\vec{x} \\rightarrow e^{-\\sigma}\\vec{x}$. However, we can suggest a different representation:\n\\begin{equation}\n \\tilde\\tau\\left(\\vec{x},t\\right) \\rightarrow \\tilde\\tau \\left( {e^{-\\sigma\/z} \\vec{x} , e^{-\\sigma}t } \\right) + \\sigma \n\\end{equation}\nsuch that the tilded dilaton $\\tilde\\tau$ is able to directly compensate for time variations since in this representation the added $\\sigma$ contribution corresponds to the time transformation $t \\rightarrow e^{-\\sigma}t$.\n\nBy moving to this representation, it is justified to require the space part of the dilaton lagrangian to be invariant under the two parameters tranformation:\n\\begin{equation} \\label{new_dilaton_transformation}\n \\tilde\\tau \\rightarrow \\tilde\\tau + \\sigma,  \\vec{x} \\rightarrow \\lambda\\vec{x}\n\\end{equation}\nwith no connection between $\\sigma$ and $\\lambda$, which are in fact related by the hidden (and arbitrary) $z$. In this representation, $\\sigma$ defines the time scaling, and $\\lambda$ defines the space scaling, so that it is clear that the lagrangian should be invariant under the dual scaling -- space and time independently.\n \nSuppose we are left with an unbroken scale symmetry with some specific $z$, so that $\\sigma_{1,2} = z_{1,2}\\sigma$, i.e. $z = z_2\/z_1$ is unbrokem while other $z$'s are broken. We should expect $\\tau$ to be invariant under the unbroken symmetry, thus, up to overall normalization factor,\n\\begin{equation}\n \\tau \\rightarrow \\tau + \\alpha\\sigma_1 + \\beta\\sigma_2\n\\end{equation}\nwith \n\\begin{equation}\n \\alpha z_1 + \\beta z_2 = 0 \\ .\n\\end{equation}\nIn our case $z=2\/3$ so we have $\\tau \\rightarrow \\tau - 2\\sigma_1 + 3\\sigma_2$.\n\nIn general, for when we have $\\tau \\rightarrow \\tau + \\alpha\\sigma_1 + \\beta\\sigma_2$ with nonzero $\\beta$, the possible leading order space only lagrangian is only $\\tau \\left( \\nabla^2 \\right) ^ {d\/2}\\tau$ because any $\\tau$ exponent would allow an arbitrary time scaling to ruin the symmetry since the exponent term would scale while all space derivatives wouldn't.\n\nWe can have higher order terms. Every $\\tau$ should be accompanied by a space derivative, and the total dimension (number of derivatives) should be $d$. So for example $(\\partial\\tau)^d$ is a possibility. Rotation invariance requires of course all derivative indices to be contracted in pairs, which is a constraint. This constraint seems to forbid all local terms in the odd space dimensionality case, we may keep $\\tau \\left( \\nabla^2 \\right) ^ {d\/2}\\tau$ but recognize the fact that it is non-local. For even dimensionality we have more terms the higher the dimensionality is. For example, in 2d we have only $\\tau \\Delta \\tau$ (which is equivalent to $\\partial\\tau \\partial\\tau$), in 4d we have $\\tau \\Delta^2 \\tau$, $(\\partial\\tau \\partial\\tau) \\Delta \\tau$, $(\\partial\\tau \\partial\\tau)^2$.\n\n\\section*{Acknowledgments}\nWe would like to thank Igal Arav and Yaron Oz for valuable discussions, and specifically Yaron Oz for introducing this problem to me. This work is supported in part by the I-CORE program of Planning and Budgeting Committee (grant number 1937\/12), the US-Israel Binational Science Foundation, GIF and the ISF Center of Excellence.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nAutomated methods proposed by the natural language processing and information retrieval communities often form the basic building blocks in an application. However, in this paper, we argue that such automated tools, even though they have achieved some level of maturity, are not enough for the needs of the end-users especially for\ndomains that require higher-level information assimilation and cognition like foraging and sensemaking over spatial information. For example,~\\citet{lai2022nlp} have recently used natural language processing (NLP) to understand context in the extraction and geocoding of historical floods, storms, and adaptation measures. They extend the state-of-the-art for low-level information extraction, e.g., named entity extraction and geocoding, but do not provide a holistic understanding of the story underlying these events.\n\nWe posit that our research community needs to ``see the forest for the trees.\" Sensemaking is an integral part of\ninformation processing, and tighter coupling between the lower levels (information extraction) and the higher levels (information understanding and sensemaking) can improve the state-of-the-art. Specifically, we call for the community to look more at ``higher-level tools and systems\" that enable end-users to complete tasks. Towards this goal, we study the case of extracting geospatial information from text using visual analytics (VA)~\\cite{andrienko2020va,yuan2021va} to perform tasks over the extracted data.\n\nSince text is unstructured data and the information within the text is often messy, the output from computational techniques includes associated errors and is not sufficient to explore mentions of movement in text without human expertise. VA can address this issue through human-in-the-loop strategies that enable analysts to work iteratively with computational methods that extract knowledge from messy data, cope with uncertainties in computational results, and improve those results over time~\\citep{endert2014viz,robinson2017geoviz}. VA is especially suitable for big, diverse, messy data that can be interpreted differently~\\citep{tapia2021spatio,angelini2018visual,ninkov2019vaccine,snyder2020,MacEachren2011a}.\n\nOur research objective was to determine if computational techniques and geovisual analytics can leverage \\textbf{large volumes of movement statements to enable an end-user to understand the movement described} quickly. If successful, research can then take advantage of the wealth of movement data found in written descriptions about people, wildlife, goods, and other things moving throughout our world. Text statements about movement can be used to understand what is moving, when it is moving, why it is moving, and how it is moving.\n\nFor our research, ``geographic movement\" refers to the movement of people, animals, objects, goods, information, natural physical processes, and similar things through spaces ranging in size from multiple buildings to the whole earth. We applied computational methods to identify and extract movement statements,\nand present them in GeoMovement, a human-in-the-loop\nweb-based geovisual analytics system for  identification, processing, and exploration of descriptions of movement. \nGeoMovement involves computational 1) cleaning of the messy text, 2) predicting the statements that describe movement using a machine learning (ML) model, 3) applying Geographic Information Retrieval (GIR) techniques to identify places mentioned, and 4) predicting statements that describe restricted movement or desired movement that is not possible (hereafter referred to as ``impaired movement\"). While there is substantial research on some of these subtasks, \\textbf{integrating these techniques with VA} and demonstrating its success in our chosen domain is the main contribution of this paper.\n\nWhile some progress has happened in processing descriptions of movement in text, there is very little work on detecting and understanding descriptions of impaired movement. For example, the COVID-19 pandemic prompted us to focus on impaired movement due to the importance of disruptions and restrictions in global movement patterns of people and, perhaps equally important, the movement of goods like food and medicines. Therefore, another important contribution of this work is that we show an adaptation of an existing approach developed to detect negation in picture descriptions~\\citep{van2016pragmatic} can successfully uncover impaired geographic movement in text documents.\n\nSpecifically, we integrated (and adapted or extended) many existing computational and VA methods to produce a system that supports information foraging related to geographic movement as reflected in text statements. GeoMovement is unique in identifying movement statements and filtering them by place and time.\n\nFigure~\\ref{fig:fig1} shows GeoMovement's user interface. Users can search and filter based on search terms, the statements' dates, and impaired movement status. The statements originate from three sources ingested into GeoMovement to demonstrate its capabilities and utility for investigating movement:\n\n\\begin{itemize}\n\t\\item 398 thousand News articles from August 2019 to May 2020,\n\t\\item 328 million Twitter tweets from February 2020 to May 2020,\n\t\\item 15.6 thousand Scientific articles from August 2019 to November 2020.\n\\end{itemize}\n\n\\noindent\nOver 520 million total statements contain diverse movement patterns, things moving, geographic coverage, and temporal differences.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure1.eps}\n\t\\caption{The GeoMovement user interface: A) Main mapping interface with hexagons selected for place aggregation B) Connection lines between co-occurring place mentions from an area near New York City selected C) Free text key term search D) Text data source buttons E) Normal movement vs. impaired movement buttons F) Geo-binning buttons G) Classification method for bins and connection lines colors and the number of classes chooser H) Statements that match the search I) Most common bi-grams that match the search J) Two-sided Temporal Bar Chart that shows the number of statements, by date, with a divided bar depicting normal movement on the right (green) and negated movement on the left (purple) K) Time range slider.}\n\t\\label{fig:fig1}\n\\end{figure*}\n\nExisting geographic movement research has improved analysis methods~\\citep{dodge2012similarity,dodge2016future,dodge2016movement,dodge2016traj,graser2021movement,graser2020massive,graser2020trajectory,soares2017,huang2017} and shown how these methods can derive valuable information about human movement and wildlife movement~\\citep{wang2020trajectories,dodge2014vultures,miller2019movement,zhu2021travel,li2021trans}. However, partly due to the challenges and because computational techniques to address them are relatively new, this prior research focused on geographic movement in precise movement trajectories from sensors such as GPS and mostly ignored movement described in text documents. The limited existing research to analyze movement described in text has only focused on narrow tasks like mapping route descriptions.\nThis work fills a gap in the research to support the study of geographic scale movement by developing and demonstrating methods for analyzing movement described in text documents at a broad scale. \nWe anticipate that our effort will spur more research to leverage this under-utilized geospatial movement information in text documents.\n\nWe present two case studies (and a third case study in a supplementary video demonstration) related to the global pandemic that demonstrate the potential of our approach and shows, despite the messy data and imprecise computational predictions, a human can derive meaningful and essential information about geographic movement described in text with GeoMovement. The case studies also demonstrate that GeoMovement can support multi-scale information foraging through vast volumes of messy text statements about geographic movement. Furthermore, the space-time concept\/attribute filtering methods implemented effectively narrow in on information relevant to an analyst's objectives.\n\n\\section{Related Work}\\label{sec:sec2}\n\nFirst, we review related efforts to analyze various geographic movement types described in text documents. Existing research into studying descriptions of geographic movement has primarily focused on reconstructing route descriptions. These routes (driving directions~\\citep{jaiswal2012geocam,drymonas2010routes}, hiking and general route descriptions~\\citep{moncla2014hiking,moncla2014itinerary,moncla2016itinerary,Piotrowski2010a}, and historical exploration routes~\\citep{bekele2016historic,blank2015historic}) form a constrained subset of movement statements that simplify and thus do not address many\nof the\nchallenges with a broader set of movement statements. Other research took the opposite approach of textualizing (convert routes to description) to take advantage of the benefits of text~\\citep{chu2014taxi,aldohuki2017taxi}. \nAdditionally,~\\citet{huang2020nlp} showed how geovisual analytics could improve the retrieval of trajectories in a search. But, their focus on text analytics was only on users' queries of the data (which was precise sensor-based trajectories). Furthermore, in addition to addressing narrow domains, most of this research has restricted the data to a small amount of text.\n\nThe GeoCAM project~\\citep{jaiswal2012geocam} created an application that identified, extracted, and generated maps of route directions found on webpages providing textual (often formatted) directions to reach a location. While the GeoCAM project complements the research we present here, their problem is\nsimpler since route directions \nare just a small subset of movement descriptions. Furthermore, route directions typically follow a semi-structured pattern that allows for simpler ML models and rules-based approaches. In a precursor to the GeoCAM project, \\citet{drymonas2010routes} used similar techniques to map route directions. These projects encouraged future work like ours to go beyond route directions to general movement descriptions~\\citep{klippel2008linguists}.\n\nSecond, we describe related efforts that use geovisual analytics on geographic movement described in text documents. The complicated nature of analyzing place in text documents, especially the need to represent spatial relationships best rendered visually on a map, has prompted other researchers to take a geovisual analytics approach. SensePlace was a system to analyze place mentions in text and pull information from other sources to aid analysis~\\citep{Tomaszewski2011}.\nSensePlace2 developed and applied geovisual analytics to methods that focus on the extent to which Twitter users' tweet location compared to the places they discuss in their tweets~\\citep{MacEachren2011a}. \\citet{Robinson2013b} performed a user study with experts that showed both the advantages, usefulness, and difficulties of such a system for crisis management. SensePlace3 extended this effort by advancing the geovisual analytics techniques and scaling the system to work on millions of tweets per month, thereby improving analysis~\\citep{Pezanowski2017}.\n\nThe SMART system complements the SensePlace versions' focus by providing a visual interface enabling human analysts to explore text's spatial, temporal, and topical components~\\citep{snyder2020,karimzadeh2019geovisual}. SMART implemented advanced geovisual analytics techniques, including a tweet classifier to filter semantically and a cluster lens to visualize keywords at a large scale. However, like SensePlace3, their system did not focus on analyzing movement. Finally, the NewsStand system~\\citep{teitler2008,samet2020} mapped places where news articles are written compared to the places they discuss but focuses less on geovisual analytics for analysis and more on correctly mapping the text.\n\nA few other efforts use geovisual analytics and mapping systems to analyze places mentioned in tweets~\\citep{Thom2012,Bosch2013,Bosch2011,felmlee2020women} and show the potential to take advantage of this geographic data source, albeit mainly focusing on tweets with a geocoded location that makes the challenge different. Mapping and geovisualizations have also been combined with topic analysis and network graph analysis to show similarities between cities~\\citep{hu2017extracting}.\n\n\\citet{jamonnak2020videos} combined location information associated with videos and the narrations of those videos to show their locations on the map and the topics and sentiment discussed at those locations.\n\\citet{xu2018reviews} created a system to explore Yelp business reviews in areas and their change over time.\n\\citet{ma2020local} showed that geovisual analytics is vital to understanding\ncritical local places that need immediate help in a disaster from 911 call transcripts and clusters of specific crime types from police reports. Although these systems successfully demonstrated geovisual analytics on text, they focused on particular topics and did not consider movement. Therefore, they could not be applied to our goal of analyzing wide-ranging types of movement described in text. Since impaired movement detection is not the primary focus of this research, we describe work related to it in Section~\\ref{sec:impaired}.\n\n\\section{Text Computational Processing Predictions}\\label{sec3}\n\nWe first acquired three different sets of documents consisting of 398 thousand news articles, 328 million tweets, and 15.6 thousand scientific articles. Our document sources and the keyword and time parameters used to obtain them are described in detail in Appendix~\\ref{sec:textsources}. We cleaned the documents using typical text pre-processing methods and applied computational techniques to identify places in the text, predict the statements that describe movement, and predict statements that describe impaired movement.\n\n\\subsection{Predicting Movement}\\label{sec:predmovement}\n\n\nIn \\citet{pezanowski2020geomovement}'s work, humans label the statements with a binary class as either describing geographic movement or not. Since they took this initial step and created a corpus to train a model, we can use this model to predict statements that describe movement. The prediction of this ML model is a probability value between zero (no movement) and one (movement). We set an arbitrary cut-off for GeoMovement to only show statements with a probability greater than 0.6 that the statements are about movement. This relatively loose threshold was arrived at by trial and error and is acceptable for all three sources.\n\n\n\\subsection{Predicting Geographic Location}\n\nWe used the GeoTxt system to perform GIR on our text sources~\\citep{karimzadeh2013geotxt,karimzadeh2019geotxt}. We chose this because GeoTxt performs comparably or better than other state-of-the-art geoparsers and performs best without case sensitivity, which is common in Twitter data (one of our data sources)~\\citep{gritta2018geoparsing}. \nA small evaluation corpus had an F1-score of 0.78 for geoparsing place names and an accuracy of 0.91 in resolving those place names correctly. Moreover, \nit performed even better on higher-order administrative places such as countries and states, which are common in our statements. For enterprise projects, paid commercial sources also exist from software companies such as Esri, Google, and Microsoft. We chose not to use these products because they do not allow for customization compared to GeoTxt, which is open-source software and\nallows adjustments to the software in the future.\n\n\\subsection{Predicting Impaired Movement}\\label{sec:impaired}\n\nThe global pandemic of 2020-2021 brought attention to global movement and how it spreads, and how the pandemic disrupted or prevented regular global movement. Because the pandemic highlighted the importance of analyzing disruptions to movement, we investigated potential strategies for detecting statements about impaired movement.\n\nTo detect impaired movement in our statements, we looked to adapt existing methods of negation detection in text. Negation detection strategies can potentially uncover statements about formal restrictions on movement (of the sort imposed by governments), decisions not to move taken by individuals for their safety, and impediments to movement created by limited public\/commercial transport such as canceled flights due to the lack of passengers or ill crew.\n\nIn this section, first, we review existing related research on negation detection and its everyday use cases. \nSecond, we describe how we adapted an existing approach from the literature that detects negation in picture descriptions~\\citep{van2016pragmatic} to our challenge of detecting impaired geographic movement described in text. Third, we show how we improved upon our initial attempt to detect impaired movement using our geovisual analytics methods to analyze initial mistakes in predictions and then modify the rules specifically to detect impaired movement.\n\n\\subsubsection{Existing Approaches to Negation Detection}\n\nAddressing negation in text has been identified as a problem in several existing works. For example, \\citet{Fialho} had remarked that ``when a negation was involved in a sentence, the classifiers found more difficult to return the appropriate label\" in the context of negation in sentences as identified as part of discourse representation structures. Negation detection in text is vital in challenges like automated summarization of medical reports~\\citep{Vincze2008,slater2021negation}, summarizing picture descriptions~\\citep{van2016pragmatic}, and as a hint in identifying sarcasm~\\citep{reyes2014sarcasm}. \\citet{hiremath2021sarcasm} show that sarcasm detection depends upon detecting negative sentences in positive situations and positive sentences in negative situations.\n\nMuch of the current state-of-the-art research on negation detection was influenced by a Workshop titled \\textit{Resolving the Scope and Focus of Negation}~\\citep{Morante2012}, which also produced labeled datasets that continue to be used in training and evaluating the success of new methods. Supervised ML-based solutions such as the LSM Network~\\citep{zhao2021sentiment} have learned negative terms while performing sentiment mining automatically from large-scale training data. The current state-of-the-art method, NegBert, is based on ML~\\citep{khandelwal-sawant-2020-negbert,khandelwal2020multitask}. Although this ML approach is the current state-of-the-art, the challenge in using ML approaches is the need for time-consuming labeling of large amounts of training data. NegBert was trained and tested on datasets designed explicitly for negation detection evaluation and therefore could not be used for our tangential challenge of detecting impaired movement. In the absence of training data, we show that rules-based approaches can still be used.\n\n\\citet{van2016pragmatic} and \\citet{van2016building} have used rules to detect negation. They had humans annotate Flickr picture descriptions for negations and define categories of negations. This annotation exercise produced simple clues for negation, thereby allowing their rules-based approach to be effective on picture descriptions. In general, rules-based approaches have been proven to work in negation detection when the domain is relatively narrow and,\nlike most rule-based systems typically provides high precision but low recall. Rule-based methods can work fine in our application, where a sample of the negative sentences suffices,\nbut having false positives can result in incorrect conclusions.\n\n\\subsubsection{Applying Negation Detection to Descriptions of Geographic Movement}\n\nDetecting impaired movement is similar to previous negation detection using specific key terms. However, what constitutes a negated word is ill-defined and varies\nfrom domain to domain and problem to problem \nsince terms, like \\textit{canceled} or \\textit{diverted}, would not always be considered negated. But, when applied to movement, they are. This ambiguity complicates the task. We investigated if we could adapt the current methods that are focused on the negation of words (ex. She does \\textit{not} have cancer. The alarm clock did \\textit{not} have the feature I wanted.) to detect impaired geographic movement (ex. Our flight to England was \\textit{canceled}. The fruit was \\textit{stuck} in Brazil because of initial fears early in the pandemic.). \n\nWe used~\\citet{van2016pragmatic}'s rules-based approach for negation detection. As discussed above, we adapted this rules-based approach, as opposed to an ML approach, to 1) avoid costly labeling of training data, 2) achieve transparency in how the results are obtained, and 3) because our needs in detecting impaired movement are relatively narrow, which the literature suggests~\\citep{van2016pragmatic,van2016building,slater2021negation} is a good fit for a rules-based approach.\n\n\\citet{van2016pragmatic}'s method tags part-of-speech in text (such as verbs, nouns, adjectives, etc.) and then searches for a list of negation keywords, prefixes, and suffixes to detect negations. Some negation rules examples are a) exact negated words like ``no\" and ``not,\" b) verbs that start with ``de,\" ``mis,\" ``dis,\" or c) adjectives that end with ``less.\" Overall, all rules are relatively simple and are easily understandable and reproducible. If the input sentence matches a rule, it containing negation.\n\nWe first applied \\citet{van2016pragmatic}'s exact rules for negation to our statements previously predicted to describe movement (as described in Section~\\ref{sec:predmovement}) to predict impaired movement. After 800,000 movement statements were predicted, we stopped and obtained a summary count. The model predicted about 28\\% of these movement statements as describing impaired movement.\n\nWe selected a stratified random sample of 50 predicted impaired movement statements and 50 predicted normal movement statements. The initial model did not do very well to predict impaired movement correctly. There were 23 true positives where the statement was correctly predicted as impaired movement compared to 27 false positives where the statement was predicted to be impaired movement, but it was normal movement. It did slightly better in correctly predicting normal movement with 42 true negatives as opposed to eight false negatives. Table~\\ref{tab:matrix} shows the confusion matrix for these predictions. These values equate to a precision of 0.46, a recall of 0.74, an F1-score of 0.57, and an accuracy of 0.65 on the stratified sample.\n\n\\begin{table}[htb]\n\\begin{center}\n\\caption{The confusion matrix for prediction of negated movement with unmodified rules from general negation.}\\label{tab:matrix}%\n\\begin{tabular}{@{}|c|c|c|@{}}\n\\hline\n & Actual impaired  & Actual normal \\\\\n\\hline\nPredicted impaired   & 23 (TP)   & 27 (FP)  \\\\ \\hline\nPredicted normal    & 8 (FN)   & 42 (TN)  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Improving the Detection of Impaired Movement}\n\nTo improve our detection of impaired movement, we either added new rules or removed existing rules. Since there were more false positives than false negatives, this hints that rules should mostly be removed so that fewer statements are predicted to have impaired movement.\n\nNot only did we analyze the errors in statement predictions, but we also consulted a list online of verbs common to movement see Appendix~\\ref{sec:A} and \\url{https:\/\/archiewahwah.wordpress.com\/2019\/04\/16\/movement-verbs-list\/}). First, we removed rules specific to verbs that had the prefixes ``de,\" ``mis,\" and ``dis.\" These prefixes could rarely negate the verbs we found related to movement. Second, based on an online list of adjectives related to movement (see Appendix~\\ref{sec:A} and \\url{https:\/\/dspace.ut.ee\/bitstream\/handle\/10062\/18053\/adjectives\\_and\\_adverbs\\_of\\_movement.html}), we removed the adjectives' rules beginning with ``a\" and ``dis\" because they are much too broad and did not make sense with those commonly related to movement. Third, we added the specific lemmas ``cancel,\" ``postpone,\" ``prevent,\" and ``avoid\" because they relate to impaired movement. Lemmas allow us to match many different synonyms of these words that all mean the same.\n\nThese rules modifications improved predictions on the existing stratified random sample with a precision of 0.74, a recall of 0.76, an F1-score of 0.75, and an accuracy of 0.79. Therefore, the F1-score improved from 0.57 to 0.75.\n\nSince we used the first set of 100 sampled statements to determine some of our rules, it would not be fair to rely on these improved metrics alone to conclude we improved our method. The rules may over-fit the sample. Therefore, we selected a second unseen test set of 100 statements using the same random stratified technique. On this unseen set, the original rules' prediction F1-score was 0.59, while our new rules were again much better with a 0.65 F1-score. Overall, our minor modifications to the rules produced substantial improvement. \n\nFinally, since we selected the two sets of 100 statements from new articles, as a final assessment, we sampled 100 statements from both the tweets and scientific articles using the same stratified random sampling technique. \nPredictions using our modified rules for impaired movement on the tweets resulted in an F1-score of 0.61 and an F1-score of 0.60 on scientific articles.\nTherefore, our predictions for impaired movement performed slightly less accurately for these two sources than news articles, but still respectable. This lower accuracy is likely because both sources contain language more common for that audience (i.e., slang and other informal languages in tweets and technical language in scientific articles). Based on our experience modifying the rules for news articles, we estimate that additional rule changes would also improve predictions for tweets and scientific articles. However, based on our experience and the literature, we surmise that given resources to label a large amount of training data, using an ML approach as in~\\citet{khandelwal-sawant-2020-negbert} would likely produce more accurate predictions. In summary, we are using a rules-based approach as a proof-of-concept prediction that is important to show the potential benefits of detecting impaired movement statements and use this as an attribute to analyze geographic movement described in text.\n\n\\section{Geovisual Analytics to Find Meaning in Descriptions of Movement}\n\nGeoMovement is a web-based geovisual analytics system that allows users to explore descriptions of geographic movement. GeoMovement serves as an interface between the human and the data described above and is summarized in Figure~\\ref{fig:fig1}. The statements' content, place mentions, date of creation, and impaired movement prediction are all searchable. The map visualizes place mentions in multiple levels of aggregate geo-bins. A geo-bin is many smaller places (e.g., cities) aggregated and displayed as one larger place that they all reside within (e.g., country). Geo-bins allow for a clear summary of location-based data.  By choosing particular places of interest on the map, the user can visualize the co-occurrences of places. The individual statements view completes the overview-first + detail approach. This workflow matches the information-seeking mantra of overview-first to gain an awareness of the information and details of interest on-demand~\\citep{shneiderman1996eyes}. This section is divided into three subsections that follow the information seeking mantra that starts with the overviews, then options for the user to search to filter to statements of interest, then detailed views of the filtered statements.\n\nFor detailed information on GeoMovement, Appendix~\\ref{sec:textsources} describes the sources and nature of the data sources and statements. In addition, Appendix~\\ref{sec:B4} includes technical details of the application development that allow for fast user queries on large volumes of text, thereby enabling efficient sensemaking. An important technical component of our approach is our use of Elasticsearch (\\url{https:\/\/www.elastic.co\/})) as the primary information storage and retrieval software for GeoMovement. Elasticsearch is a search engine that accepts many search parameters like free text and time, and returns matching results ordered by most relevant to the user's search. In addition, it can group results by attributes like place mentions in the text. Most impressively, Elasticsearch does all of this and returns results very quickly, most often in the matter of milliseconds.\n\n\\subsection{Overviews}\n\nThere are two primary overview means to explore the movement descriptions. The first overview is the map (Figure~\\ref{fig:fig1} at point A) that displays the number of statements spatially aggregated by their place mentions. Since we have a large number of statements from multiple sources, GeoTxt extracted over 98,000 unique place mentions from them. Displaying this large number of places on a map using points would likely be very confusing for users. Many places would overlap and quickly seeing overall patterns along and comparing quantities between places would be difficult. This is the primary reason for aggregating the statements place mentions. Coloring the polygon bins that contain places mentions a lot in the statements darker than those that are not mentioned frequently in the statements. Users get a clear overview of place mentions in the statements and a way to visually compare places.\n\nWe chose five shapes to spatially aggregate the number of statements by their place mentions. The geo-bins scales include continent, country, administrative 1 (the worldwide equivalent of a state in the United States), and two sizes of a hexagonal pattern. Each scale allows users to explore different types of movement data, like long-range bird migrations by continent, traded goods by country or administrative level, and detailed movement through the hexagons that do not adhere to political boundaries like the spread of disease. We chose hexagons as an aggregation shape since they tessellate and will distort values less than squares~\\citep{birch2007hexagons,esri2021hexagons}. The user interface provides an option shown in Figure~\\ref{fig:fig1} at point F where the user can choose the aggregate level.\n\nThe bin counts are divided into classes to ease visual comparisons between bins. The user has control of the bin count classification technique and number of classes, as shown in Figure~\\ref{fig:fig1} at point G. The user has five choices on how to classify the aggregate counts. Jenks Natural Breaks, Equal Interval, Standard Deviation, Arithmetic Progression, and Quantile are options, and they can enter the number of classes between two to seven. These classification techniques are well accepted statistical methods to make the data more understandable.\n\nWe used the geostats JavaScript library (\\url{https:\/\/github.com\/simogeo\/geostats}) to calculate the class breakpoints for the chosen users' classification method. Each classification method is valuable depending on the user search and resulting data. Darker colored bins represent areas that have a larger number of place mentions within the bin. The color scheme is a sequential color scheme chosen from ColorBrewer (\\url{https:\/\/colorbrewer2.org\/})~\\citep{brewer2003color} to ensure the classes of statement counts are easily distinguishable.\n\nThe second overview in GeoMovement is a Two-sided Temporal Bar Chart. Research has shown that there are differences in how people understand geographic movement such as the those who think more spatially than others~\\citep{liben1993space,ishikawa2016gis}. Based on this research, GeoMovement provides the user multiple views of the data. The Two-sided Temporal Bar Chart shows aggregate counts of statements grouped by the month they were published (Figure~\\ref{fig:fig1} at point J). The oldest month is at the top and the youngest month is at the bottom. The Two-sided Temporal Bar Chart allows for visualization of the amount and changes over time, both in impaired movement on the left side and normal movement on the right side. When hovering the mouse over the chart, the number of statements that match the current user search are shown in bars in the foreground while the total number of statements in GeoMovement are shown in bars in the background. Figure~\\ref{fig:fig6} shows an example of the number of statements that match the user search in the foreground once the user has hovered the cursor over the chart.\n\n\\subsection{Search}\n\nAfter the overviews give the user an understanding of the data, they can begin filtering it through search options. The five key ways to search the data are 1) free-text search (Figure~\\ref{fig:fig1} at point J), 2) buttons to select from the three sources for statements --- news articles, tweets, and scientific articles (Figure~\\ref{fig:fig1} at point J), 3) buttons to select impaired movement statements or normal movement statements (Figure~\\ref{fig:fig1} at point J), 4) a time-range slider (Figure~\\ref{fig:fig1} at point J), 5) and clicking location(s) on the map to filter by location. Multiple features can be chosen by holding the Ctrl-key on their keyboard and selecting the next feature with a mouse-click. After any of the first four searches are performed, the geo-bins and Two-sided Temporal Bar Chart update to show counts of statements that match the searches. After the fifth search, the detailed views appear.\n\n\\subsection{Detailed Views}\n\nThe detailed views of statements and their attributes include a) connecting great circles drawn\non the map between place mentions in the statements, and co-occurring place mentions in the same statement, b) the five most common bi-grams for the set of statements matching the search, c) and a table showing the actual statements that match the search. All connections between places are aggregated to show places commonly used together. The aggregate count classes adhere to the users' currently chosen map classification method and total class number in a sequential green color scheme that is color-blind friendly, also selected from ColorBrewer~\\citep{brewer2003color}.\n\nIt is important to note that the connection lines are completely accurate in showing movement between the places since they are drawn solely by the places' co-occurrence in a statement. For example, the statement below has three place mentions and therefore three possible movement pairs: Sydney\u2013New York, Sydney\u2013London, New York\u2013London. The first two are correct concerning movement references in the statement, but the third is not correct because there is no direct movement between New York and London. One could argue that the movement could be from New York to Sydney and then next to London from Sydney, but this is not probable. However, these connection lines to give the user overall patterns of interest that they can confirm through inspection of the statements.\n\n\\begin{quote}\nBut this will be the first time a commercial flight is flown from Sydney to New York, and just the second time from Sydney to London, Qantas said~\\citep{Garber2019}.\n\\end{quote}\n\nSecond, the ten most common bi-grams are shown for the set of statements that match the search and have place mentions in the chosen location bin. This provides details about the actual statements behind the overviews by showing the most common words and topics in the selected statements. Since many of the statements contain place mentions and the search location is an essential parameter for the matching statements, two-word place names are often in this bi-grams list. These place mentions may be valuable, but we also found that it was often more important to see bi-grams about topics and not necessarily places when exploring the data. Therefore, we allow users to double-click a bi-gram to remove the bi-gram from the list, and the next most common bi-gram appears, up to the 20th most common bi-gram. To re-populate the bi-grams list, the user can re-run the search. The user can also select multiple bins by holding the Ctrl-key and clicking another map bin. Users can then compare the most common bi-grams for both map bins to see differences between statements with place mentions in each bin.\n\nTo complete the overview first + detail approach to analysis, once the user found sets of statements of interest in the overviews and chose a map bin, the user can see the actual statements matching the search in a paged list. The displayed statements match all search parameters (when selecting multiple bins, statements can have place mentions in either bin). The user can scroll through the pages of statements. Each statement's published date is also shown. If there is a particular statement of interest, holding the Ctrl-key while clicking on the statement will open the original document on the Web in a new browser window.\n\nGeoMovement's tight use of modern search engine technologies and Information Retrieval (IR) allows for extremely fast human-in-the-loop sensemaking for the most relevant information on movement. Again, Appendix~\\ref{sec:B} provides further details, and our supplemental video showcases a real-time live demonstration of efficient knowledge discovery using GeoMovement.\n\n\\section{GeoMovement Assessment}\\label{sec:assess}\n\nFirst, we\nshow the challenge of interpreting movement described in statements without GeoMovement by discussing summary statistics of the data. Second, we present two case studies that show how our approach can retrieve information about movement from vast quantities of statements. A third case study is included in a supplemental video. It is recorded in real-time to show that a user can quickly extract meaningful information about movement despite the challenges \nposed by the large quantity of messy text, ambiguity in text descriptions, and imprecise computational predictions. \nThird, we show how these case studies also generated future GeoMovement needs. \nFinally, in Appendix~\\ref{sec:C}, we discuss the skill level and hardware and software requirements for GeoMovement users.\n\n\\subsection{Illustrative Data Summary Statistics}\n\nTo show the value of GeoMovement, we created summary statistics of the data to clearly illustrate how it is unreasonable to think a human can analyze and understand large quantities of movement statements without such a system. We chose three keywords related to our case studies and three prominent place names: one being a country, one a state, and one a city. The number of our statements that match these keywords and place names is shown in Table~\\ref{tab:counts}. To relate these statistics to our first case study in Section~\\ref{sec:cases}, we filter GeoMovement's 520 million statements (36 thousand of those contain the term \\textit{smuggling}, 275 thousand of them include the term \\textit{gold} and 201 thousand mention \\textit{India}) and efficiently identify important gold smuggling patterns around and in India. To produce these statistics, we used our GIR extracted place names to determine the number of statements that contain each selected place name. And, to find the number of statements containing each of the keywords, we searched the statements in a Postgresql database using a full-text search (\\url{https:\/\/www.postgresql.org\/docs\/14\/textsearch.html}) so that different variations of the same word will be matched (e.g. smuggling, smuggled, smuggle).\n\n\\begin{table*}[htb]\n\t\\begin{center}\n\t\t\\caption{Summary statistics of the number of statements that contain selected key terms and place mentions.}\\label{tab:counts}%\n\t\t\\begin{tabular}{@{}|c|c|c|c|c|c|c|@{}}\n\t\t\t\\hline\n\t\t\tsmuggling & gold & sports & London & California & India & statements \\\\\n\t\t\t\\hline\n\t\t\t36 K & 275 K & 336 K & 589 K & 165 K & 201 K & 520 M  \\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\nThis exercise to generate summary statistics is meant to show that without such a geovisual analytics system like ours, it would be extremely difficult, if not unfeasible, to perform geographic, temporal, and attribute sensemaking of the described movement. Although GeoMovement's 520 million statements are substantial, it is still a fraction of the accessible text available that could be included and analyzed in GeoMovement, given more development and computational resources.\n\n\\subsection{Case Studies}\\label{sec:cases}\n\nTo confirm our claim that the geovisual analytics interface helps users understand and make sense from the statements and multiple computational predictions, we provide two detailed case studies below from different types of (prototypical, fictitious) potential users. The case studies presented provide evidence of usability. A third case study, given only in the video supplement (due to space limitations in the text), adds additional evidence about the flexibility and utility of GeoMovement to explore the mix of text data sources from different perspectives.\n\n\\subsubsection{Understanding International Crime Affecting India}\n\nJennifer Lang is a college student who wants to write a class report about different types of international crime affecting India. She opens her web browser to GeoMovement and types \\textit{smuggling} to begin her search. \nShe notices that many statements involve smuggling are in October 2019, despite that month having fewer statements overall. She adjusts the time range slider to filter statements to that month and sees a hotspot of activity in England. She clicks the hexagon bin in England and is reminded of a significant human smuggling event in that month where many people lost their lives after being trapped in a truck that was smuggling them (Figure~\\ref{fig:fig2}). Although this is a significant smuggling event from a British perspective and also highlighted in the U.S. news, she decides to look for other ways to focus on India.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure2.eps}\n\t\\caption{By filtering the search results for \\textit{smuggling} to only show statements from October 2019, she sees many statements from a significant human trafficking event in England where many people lost their lives.}\n\t\\label{fig:fig2}\n\\end{figure*}\n\nAs her next step, she chooses to aggregate place mentions by the country level and select India. Figure~\\ref{fig:fig3} shows the results, and she sees that most results related to smuggling are affecting India from the neighboring countries of Pakistan and Bangladesh, and a few more countries. In the bigrams list, she sees many references to drugs and gold being smuggled into India.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure3.eps}\n\t\\caption{A search for \\textit{smuggling} shows important location sources for India and a spike of activity in October 2009.}\n\t\\label{fig:fig3}\n\\end{figure*}\n\nNext, after reading some individual statements, she finds out more about multiple cases of gold smuggling from both outside and within India (Figure~\\ref{fig:fig4}). To get a more detailed analysis of the movement, she types \\textit{gold} in the search box and chooses to aggregate place mentions by the state level. By clicking the neighboring state of Sindh, Pakistan, which has many place mentions in the statements, connections with many states in India are highlighted, including a state in southeast India.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure4.eps}\n\t\\caption{A search for \\textit{gold} produces many statements about gold smuggling and possible routes.}\n\t\\label{fig:fig4}\n\\end{figure*}\n\nShe clicks this state in India and discovers that the state is Tamil Nadu, where the large city of Chennai is located. From reading a few statements, she knows that gold smuggling in Chennai is arriving at the airport and through the Chennai Express train from Mumbai. The connection with Mumbai through the train is confirmed on the map by the strong connection with Mumbai's state, Maharashtra, on the west coast closer to Pakistan (Figure~\\ref{fig:fig5}). She can now clearly visualize and report on some of the prominent drugs and gold smuggling sources into and throughout India and read more detailed descriptions about individual incidents.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure5.eps}\n\t\\caption{While looking for more detail about gold smuggling, statements suggest a smuggling route from Mumbai to Chennai along the Chennai Express train.}\n\t\\label{fig:fig5}\n\\end{figure*}\n\n\\subsubsection{Examining the Impact of the Pandemic on Travel for Tourism}\n\nArti Reddy is a travel agent in India. She uses GeoMovement to understand the pandemic's impact on global travel and travel related to India. As of May 2020, like other countries in the World, India was dealing with a global pandemic. Since Arti had previously planned to advertise to potential customers traveling for sporting events, she chose this topic to investigate. She loads GeoMovement and enters the term \\textit{sports} in the text search box. With the geospatial hexagon geo-bins as a layer, she quickly sees a hotspot of discussion in Japan. She selects to filter statements for impaired movement. While mousing over the Two-sided Temporal Bar Chart, she sees that the impaired movement statements are more prevalent in recent months (except May, where there is less data), as seen in the Two-sided Temporal Bar Chart in Figure~\\ref{fig:fig6}.\n\n\\begin{figure}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure6.eps}\n\t\\caption{Mousing over the Two-sided Temporal Bar Chart shows the number of filtered statements for impaired movement (purple bars to the left of the axis) involving sports has been increasing since the pandemic began (bars are aggregate counts of statements by month, with the oldest month at the top). Filtered statements by the search are shown in the foreground, while all statements are shown in the background on the mouse hover.}\n\t\\label{fig:fig6}\n\\end{figure}\n\nArti switches to the country geo-binning, chooses to view both types of movement again, and moves the timeline from July 2019 to December 2019, and Japan is still a popular location. After selecting Japan on the map, she sees in the bi-grams list that the upcoming Olympics in Japan are prominent, and on the map, she sees that there are connections between Japan with many other places in the World (Figure~\\ref{fig:fig7}). There is much discussion about the potential impact of the pandemic on the Olympics. She is intrigued to see that there are connections between Japan and her home country that implies her business will be affected.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure7.eps}\n\t\\caption{Sports have been disrupted during the pandemic, including the Olympics, where athletes travel from around the World. Mousing over the Two-sided Temporal Bar Chart shows that the filtered statements for impaired movement have increased since the pandemic began.}\n\t\\label{fig:fig7}\n\\end{figure*}\n\nSince she is most familiar with India's geography, she changes the geo-binning to the state level for more detail and selects two states in India with much discussion (Figure~\\ref{fig:fig8}). A quick look through the statements shows that the Olympics are in jeopardy, and a closer-to-home event of the under-17 women's soccer World Cup scheduled to be in India was unfortunately postponed. Teams would have come from around the World for this event. This sad news prompts her to follow up on the story by clicking the statements to view the original articles in her web browser to see if it will be rescheduled and thus if there will be a future need for travel. From reading the statements, she sees that an auto show, the AP World Indoor Sporting Championships, and the Australian Grand Prix auto racing's China leg are three events postponed in China. Cricket in Australia was also severely affected, with teams planning to come from India, England, and many other countries to compete. Major sporting leagues in the U.S., like the NBA basketball league, were also interrupted. The disruptions to nearby events like cricket are particularly concerning given the sport's popularity with Indians and their potential spectator travel related to her business.\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure8.eps}\n\t\\caption{Sports in India have also been affected, including competitions where local teams compete against other teams from around the World.}\n\t\\label{fig:fig8}\n\\end{figure*}\n\nLastly, to see how widespread the pandemic's impact is on global sporting events where Indians may travel, Arti explores other countries and sees that the pandemic has significantly impacted professional soccer in Spain and other European countries. Madrid, Milan, and cities in Germany where prominent soccer clubs play all show up clearly on the map as having their games affected (Figure~\\ref{fig:fig9}).\n\n\\begin{figure*}[htb]%\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figure9.eps}\n\t\\caption{Europe is another example where the pandemic postponed many of their beloved soccer matches.}\n\t\\label{fig:fig9}\n\\end{figure*}\n\nThese use cases highlight the potential for information foraging and acquisition from statements describing movement. Geovisual analytics that combines multiple views of the data and overview + detail capabilities allow users to quickly identify important locations, connections between locations, time periods, and topics of interest. We showed how the map view could show critical hotspots, and the Two-sided Temporal Bar Chart can show essential time periods, prompting the user to investigate details.\n\nIn this section, our use cases show GeoMovement's current sensemaking capabilities. Moreover, the examples of computational prediction errors and suggested solutions highlight how geovisual analytics is well suited for allowing user interactions to either explicitly or implicitly improve GeoMovement's computational predictions. Improved prediction accuracy can make the sensemaking process more efficient by reducing the human effort to sift through incorrect intermediate results.\n\n\\subsection{Case Studies Needs Assessment}\n\nIn addition to our case studies demonstrating the effectiveness of GeoMovement in extracting information about movement, we used them as a needs assessment to propose additional functionality. In the first case study, Jennifer identified statements from an event that she was not interested in. Although she quickly found what she was looking for, other problems may require much further analysis, and future results should not include statements deemed not relevant. Therefore, a future addition to GeoMovement can consist of a mechanism to either mark statements as completely ``not relevant\" and not show them or as ``less important.\" If user accounts were added to GeoMovement, these preferences in statements could be stored. If certain types of statements will never be relevant to that user, their marked statements can be used to affect their future search results.\nSuch user input can be used as feedback to the system and in the future, statements that are very similar to those that are marked not of interest will also be filtured out; i.e., the system learns from the feedback.\nElasticsearch allows for on-the-fly criteria provided in searches that promote or demote statements or exclude them, allowing for these criteria to be personalized if user accounts were added and changed for each each search.\n\nIn the second case study, Arti found that impaired movement became more prevalent in recent months. It would be a fair assumption that this is because of travel restrictions from the pandemic. However, Arti may ask for more detail about ``How is impaired movement changing over time?\" \nTo do this, she should easily access statements by month. Currently, she would have to change the time slider filter to each month and select the impaired movement button. A straightforward way to answer this question would be to choose any bar in the Two-sided Temporal Bar Chart to update the time range and impaired movement filters. For example, clicking on the impaired movement purple bar for January 2020 would update all views to show these statements. Similarly, when Jennifer found a spike in smuggling activity in India in October 2009, she would likely ask herself why this is and want a quick way to filter to those statements.\n\nFinally, both Jennifer and Arti were interested in India. A question they both may have is: ``What other locations have similar problems as India?\" Currently, GeoMovement allows selection on the map of multiple locations to show bi-grams lists for both locations' statements next to each other. However, there are automated ways to give users hints on locations with similar statements. For example, Elasticsearch provides a ``percolate\" query where, after the user already selects a location, they could choose a user interface control and click a second location. The statements from the first location can be used in the percolate query. The result would be statements like the first set, and therefore other places like that place will show.\n\nOur evaluation of the sensemaking capabilities of GeoMovement includes a statistical summary illustrating the challenge in extracting meaningful information without it, case study demonstrations, a needs assessment, and a list of user requirements. We show that despite the large volume of messy data that is at times ambiguous, GeoMovement can quickly extract meaningful information about geographic movement.\n\n\\section{Results and Implications}\n\nAnalyzing a large volume of text describing geographic movement requires imprecise computational processing and predictions on already messy data that different people can perceive differently, resulting in some errors. We present the data in a geovisual analytics web application that follows the overview-first + detail mantra~\\citep{shneiderman1996eyes}. Users can find exciting patterns in the overviews and areas that need further investigation. They can also search the data and eventually drill down to the actual statements and articles. Results that appear to be erroneous or merely uninteresting can be hidden so that it is easier for the analyst to focus on more helpful information.\n\nOur approach tightly integrates geovisual analytics with behind-the-scenes computational predictions for statements describing movement from most text that does not, GIR methods to extract geography for mapping and analysis and a novel prediction for impaired movement. We show the value of GeoMovement through case studies where humans quickly learn about and visualize geographic movements. Furthermore, we demonstrate that a multi-scale system that applies a space-time-concept\/attribute filtering process can effectively make sense from large volumes of messy movement statements.\n\nBecause of the challenges of utilizing messy big text data for understanding geographic movement, previous work has either focused only on movement data from precise sensors or addressed narrow domains of movement described in text. However, the need for improved methods to utilize big text data about movement is illustrated in work described in Section~\\ref{sec:sec2}, and \\citet{hultquist2020crowd}, \\citet{hultquist2019nuclear}, and \\citet{Janowicz2019semantic}, where geography in text complements sensor data or stands alone to solve real-world problems. \\citet{maceachren2017bigdata} also encouraged such advances in his positional paper. GeoMovement 1) tightly integrates computational predictions and geovisual analytics, and 2) is built with modern web computing technologies that return results from user queries in milliseconds. These features allow users to make sense of and uncover movement patterns quickly. We show how this efficient and broad movement exploration is unfeasible without computational and visual means to assist a human. Moreover, this completes our research objective to show that modern advances in NLP and IR can leverage large volumes of movement statements to understand the movement described quickly.\n\nAlthough we stopped collecting text for GeoMovement, our techniques can be adapted in a straightforward technical way to accommodate a continually updating source given modest computational and storage requirements. In addition, all of our code is available with an open-source license to prompt future advances \\citep{pez2022geomovement}.\n\nWe anticipate GeoMovement or a production version to be beneficial for scientists, journalists, or even the general public to find information about the movement of humans, wildlife, goods, disease, and much more. Advances to analyzing movement statements can provide similar knowledge gains as movement recorded by sensors with precise geospatial and temporal data.\n\n\\section{Conclusions and Future Work}\n\nOur research shows that modern computational techniques can be combined with a human-in-the-loop geovisual analytics system to overcome significant challenges and identify, process, and explore large volumes of movement statements to quickly obtain an overview of movement patterns and forage for detailed information of interest. Future research should take advantage of the wealth of context information found alongside geographic movement in text documents about why, when, and how the movement is occurring that is not often present in precise GPS data. Our research methods can likely be adapted to analyze statements involving other attributes like time and more complex spatial analysis like correlation. In addition, future research should explore the integration of precise geospatial movement data with movement in text. One initial way is to link them spatially and temporally. For example, the following steps can involve connecting entities in the text to other information using the Semantic Web \\citep{bernerslee2001semantic} and linked geographic data \\citep{Stadler2012linked,janowicz2012linked}. Future research needs to identify what makes movement statements different from other statements.  \\citep{pezanowski2022movement}, took initial steps towards this goal by identifying vital characteristics of movement statements that humans use to differentiate the movement described.\n\nAlso, since we chose to illustrate the effectiveness of GeoMovement through the case studies and other assessments in Section~\\ref{sec:assess}, future work should include a more thorough evaluation of user needs, especially focusing on the additions suggested in this Section.\n\nMoreover, prior research shows that a geovisual analytics system combined with ML predictions can continually improve the accuracy of the ML predictions by having users correct machine errors~\\citep{snyder2020,andrienko2022va}. An extension of GeoMovement can make it an intermediary between computational predictions and humans. As more humans use GeoMovement, humans can iteratively correct any errors they encounter, thereby improving the predictions. As an example, in Section~\\ref{sec:impaired}, we discuss how an ML model would likely be superior to a rules-based approach detecting impaired movement but requires a large amount of training data. GeoMovement can show initial predictions of normal movement and impaired movement from an ML model that used a small amount of training data. A simple tool can allow users to correct errors in the predictions. Once many users are using GeoMovement, the training data set can proliferate. This technique has also shown success in commercial production mapping systems like Google Maps, where users reach their destination, and Maps asks about driving directions' accuracy.\n\nTwo overall ways that GeoMovement can improve the computational predictions are, first, explicitly asking for feedback on any incorrect predictions and second, through implicit user actions. For example, in GeoMovement's statements view, a button can be added to explicitly mark any statements that do not describe movement. Once a sufficient number of users mark a statement to conclude that it is incorrect, the corrected statement can be added to the corpus of statements used to train the model. As an example of implicit actions, the statements' list is currently returned in order by the search engine software that roughly corresponds to how closely they match the search terms entered and how frequently those search terms appear in the statement. Skipped statements can be recorded when users page through results to find what they are looking for. If many users skip certain statements, they can be deemed less valuable and given a weighting that lets them be listed lower in the order they are returned, thereby promoting more critical statements. There are many other possibilities to obtain either explicit feedback (users correct place mentions that were assigned to the incorrect location using the GIR techniques~\\citep{karimzadeh2019geoannotator}) or implicit feedback from users (identifying important locations to highlight based on where previous users navigated to on the map) to improve GeoMovement's sensemaking ability.\n\nAdditionally, none of the existing GIR systems have ideal performance metrics. There is often a trade-off between different geoparsers' false positives and false negatives and other ways to rank geocoding results on large datasets. Future work can allow user controls in GeoMovement to choose different geoparsing back-ends. Also, the open-source GIR GeoTxt can be modified to allow GeoMovement users to set a tolerance to allow more false-positives in situations where missing relevant statements are most critical or allow fewer false-positives in situations where the key is a quick overview and having every relevant statement does not matter. Our use of geovisual analytics for GeoMovement adds substantial potential for future improved analysis of movement statements.\n\nFinally, it is important to revisit that GeoMovement only supports English. Therefore, a valuable extension to our research would be the addition of other languages. The higher-level software, programming languages, and APIs (discussed in Section~\\ref{sec:B}) we used in our research to harvest, process, store, retrieve, and visualize data are all capable of handling many languages and character sets. However, incorporating documents in other languages into GeoMovement would require significant work on the three main computational predictions we used. First, identifying statements that describe movement would require a large amount of labeled training data in the added language. Also, for higher accuracy in predictions, language models like ELMo would need to be re-trained for that language like the research of \\citet{che2018lang} and \\citet{fares2017lang}. Second, our geoparsing methods to extract place mentions and geocode them to their correct location on Earth would need to be adapted \\citep{mandl2008lang}. Third, our negation detection techniques would also require adaption to support other languages \\citep{morante2021lang}. These changes are possible but need significant work as existing research is less robust than the equivalent in English.\n\nThe World is a dynamic place, and understanding geographic-scale movements of things is essential in many domains like business, public health, and environmental science. Leveraging information about movement found in text can complement precise sensor-based movement data. Also, movement described in text is valuable since precise movement data is often unavailable because of high costs or impracticality to deploy sensors. GeoMovement and the combination of computational and visual methods it integrates are steps toward that objective.\n\n\\backmatter\n\n\\bmhead{Supplementary information}\n\nWe include a supplementary video of GeoMovement to show how geovisual analytics helps derive valuable patterns about geographic movement from text documents.\n\n\\bmhead{Acknowledgments}\n\nWe would like to thank the Information Technology group in the College of Information Sciences and Technology at The Pennsylvania State University for providing computing resources to host GeoMovement. Specifically, Adam McMillen, a Systems Administrator in the group, provided his expertise in establishing a virtual server and deploying GeoMovement to it.\n\n\\section*{Statements and Declarations}\n\nNo funds, grants, or other support was received.\\\\\n\n\\noindent\nThe authors have no competing interests to declare that are relevant to the content of this article.\\\\\n\n\\noindent\nAuthor Contributions: Conceptualization: Scott Pezanowski, Prasenjit Mitra; Methodology: Scott Pezanowski, Prasenjit Mitra, Alan M. MacEachren; Software: Scott Pezanowski; Data curation: Scott Pezanowski; Formal analysis: Scott Pezanowski; Validation: Scott Pezanowski, Prasenjit Mitra; Resources: Prasenjit Mitra; Visualization Design: Scott Pezanowski, Alan M. MacEachren; Visualization Implementation: Scott Pezanowski; Project administration: Scott Pezanowski; Writing - original draft: Scott Pezanowski; Writing - review \\& editing: Scott Pezanowski, Prasenjit Mitra, Alan M. MacEachren; Supervision: Prasenjit Mitra, Alan M. MacEachren \\\\\n\n\n\\begin{appendices}\n\n\\section{Terms Related to Movement}\\label{sec:A}\n\nTable~\\ref{tab:verbs} shows the list of verbs related to movement used in our negation detection techniques as retrieved from \\url{https:\/\/archiewahwah.wordpress.com\/2019\/04\/16\/movement-verbs-list\/} on January 21, 2022.\n\n\\begin{table*}[!htb]\n\\caption{List of verbs related to movement used in our detection of impaired movement.}\\label{tab:verbs}%\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{ll}\n\\hline\nAdvance - move forward.                                    & Pootle {[}informal{]} - proceed casually.                 \\\\ \\hline\nAim (for) - go in the direction of.                        & Pop (in, to, over etc) - quickly visit\/pass etc.          \\\\\nAmble - walk casually.                                     & Potter - move in an unhurried way.                        \\\\\nAngle (towards etc) - turn one - s steps towards.          & Pound - proceed with fast, heavy steps.                   \\\\\nBack (out, away etc) - move in reverse.                    & Prance - move flamboyantly, with effected grace.          \\\\\nBarrel - move in a forceful, uncontrolled way.             & Progress - advance.                                       \\\\\nBeetle - hurry like an insect.                             & Proceed - go forward.                                     \\\\\nBelt - move swiftly.                                       & Promenade - take a leisurely walk.                        \\\\\nBez {[}informal, dialect{]} - zip around.                  & Prowl - move in a shifty or predatory manner.             \\\\\nBluster - move forcefully yet ungraciously.                & Race - move quickly, in competition.                      \\\\\nBolt - move swiftly.                                       & Ramble - walk far and wide.                               \\\\\nBounce - move with elastic motions.                        & Regress - go back.                                        \\\\\nBound - move quickly with large steps.                     & Return - go back.                                         \\\\\nBumble - proceed in a clumsy fashion.                      & Roam - proceed with no direction in mind.                 \\\\\nCanter - move fairly, like a horse. &\n  Roll - {[}literal{]} proceed in turning motions like a wheel \/ {[}figurative{]} move steadily. \\\\\nCareen - speed forward uncontrollably                      & Rove - wander far and wide.                               \\\\\nCareer - speed forward with little control.                & Run - proceed quickly, both feet leaving the floor.       \\\\\nCharge - move aggressively towards something.              & Rush - move with haste.                                   \\\\\nCrawl - {[}literal{]} go on all fours \/ {[}figurative{]} proceed slowly. &\n  Sashay - move in a confident and flamboyant way. \\\\\nCreep - move sneakily or slowly.                           & Saunter - walk arrogantly, confidently.                   \\\\\nDance - move rhythmically.                                 & Scamper - run like an agitated animal.                    \\\\\nDart - go swiftly.                                         & Scarper - run away.                                       \\\\\nDash - run quickly.                                        & Scoot - proceed at a fair pace \/ shuffle to one side.     \\\\\nDawdle - proceed slowly and reluctantly.                   & Scud - move quickly as if blown by the wind.              \\\\\nDive - descend quickly.                                    & Scuff - walk in a careless, friction-producing way.       \\\\\nDodder - move unsteadily, as if elderly.                   & Scurry - hurry like a small animal.                       \\\\\nDogtrot - move at a brisk, comfortable pace, like a dog.   & Scuttle - hurry like an insect.                           \\\\\nEmerge (from) - come out of.                               & Seethe - proceed like oozing liquid.                      \\\\\nEscape - move out of danger\/confinement.                   & Shuffle - walk slowly, without lifting one - s feet.      \\\\\nFile (in) - {[}of multiple people{]} go one-by-one.        & Skedaddle - depart in haste.                              \\\\\nFlee - run away from.                                      & Skip - proceed bouncing from one foot to the other.       \\\\\nFlounce - move in a flamboyant way.                        & Skitter - move hurriedly.                                 \\\\\nFlop - move loosely.                                       & Slide - move frictionlessly.                              \\\\\nFly - {[}literal{]} move through the air \/ {[}figurative{]} proceed swiftly. &\n  Slink - go smoothy\/sensuously. \\\\\nFootslog - march a long distance.                          & Slip - move frictionlessly \/ make an accidental movement. \\\\\nForge (on, ahead) - proceed strongly and steadily.         & Slither - slide forward like a snake.                     \\\\\nGallop - move quickly, like a horse.                       & Slope - as sneak.                                         \\\\\nGambol - proceed in a playful, energetic manner.           & Sneak - Proceed surreptitiously.                          \\\\\nGlide - move frictionlessly.                               & Speed - move very fast.                                   \\\\\nGo - basic movement verb \/ depart.                         & Split - depart.                                           \\\\\nHare - proceed extremely quickly, like the animal.         & Sprint - run at top speed.                                \\\\\nHasten - move with haste.                                  & Stagger - move unbalanced, unsteadily.                    \\\\\nHead (towards, for, to etc) - proceed in the direction of. & Stalk - move as though hunting.                           \\\\\nHie {[}archaic{]} - go quickly. &\n  Stampede {[}multiple people{]} - progess chaotically \/in agitation. \\\\\nHightail {[}informal{]} - move quickly.                    & Steam - power forward.                                    \\\\\nHike - go a long distance.                                 & Step - move with the feet.                                \\\\\nHop - {[}literal{]} proceed on one foot \/ {[}figurative{]} make a short journey &\n  Streak - move quickly, as if leaving a line of light behind you. \\\\\nHurtle - move quickly, violently and recklessly.           & Stride - walk purposefully.                               \\\\\nIssue (from) - come out of.                                & Stroll - walk in a brisk, leisurely manner.               \\\\\nJog - move at a medium pace\/half-run.                      & Strut - walk stiffly \/ arrogantly.                        \\\\\nJump - propel oneself through the air.                     & Swagger - move arrogantly.                                \\\\\nJaunt - go on a short trip.                                & Sweep - proceed swiftly.                                  \\\\\nJourney - travel a distance.                               & Tank - progress swiftly and forcefully.                   \\\\\nLabour - move with difficulty, requiring force.            & Tiptoe - proceed lightly, silently on the toes.           \\\\\nLeap - jump far.                                           & Traipse - walk a distance.                                \\\\\nLeg (it) - run (away).                                     & Tramp - walk a distance.                                  \\\\\nLimp - proceed unevenly \/ with an injured leg.             & Trample - walk without precision or care.                 \\\\\nLollop - proceed in ungainly bounds.                       & Travel - move a distance.                                 \\\\\nLope - move in large strides.                              & Tread - move using the feet.                              \\\\\nLunge - jump forward to attack.                            & Trek - travel a long time \/ distance.                     \\\\\nMarch - move steadily\/forcefully\/with purpose.             & Trip - proceed lightly, gaily.                            \\\\\nMeander - proceed in an indirect way.                      & Tromp - walk heavy-footed.                                \\\\\nMooch {[}informal{]} - go around in a skulking manner.     & Troop - march with effort.                                \\\\\nMosey - walk in a leisurely manner.                        & Trot - move briskly like a horse.                         \\\\\nMove - basic verb of movement.                             & Trundle - move arduously like a cart.                     \\\\\nNip (into, across, over etc) - quickly go.                 & Tumble - fall \/ spiral forward                            \\\\\nPace - walk steadily.                                      & Undulate - proceed in wavy motions                        \\\\\nPad - walk casually\/softly\/steadily like an animal. &\n  Waddle - Walk in ungainly fashion, from side to side. \\\\\nParade - proceed in an extrovert manner.                   & Walk - go by foot.                                        \\\\\nPatrol - walk around in order to guard.                    & Wander - travel without a direction in mind.              \\\\\nPatter - go with a light tripping sound.                   & Wend - travel by a circuitous route.                      \\\\\nPass - move beyond.                                        & Whizz - go speedily.                                      \\\\\nPelt - move quickly, like a hurled stone.                  & Wobble - move unsteadily.                                 \\\\\nPerambulate {[}formal, rare{]} - walk.                     & Zip - move swiftly.                                       \\\\\nPlod - move with heavy, laborious motions.                 &                                                           \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table*}\n\n\nTable~\\ref{tab:adjs} shows the list of adjectives related to movement used in our negation detection techniques as retrieved from \\url{https:\/\/dspace.ut.ee\/bitstream\/handle\/10062\/18053\/adjectives\\_and\\_adverbs\\_of\\_movement.html} on January 21, 2022.\n\n\\begin{table*}[!htb]\n\\caption{List of adjectives related to movement used in our detection of impaired movement.}\\label{tab:adjs}%\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l}\n\\hline\nIn the text, some nouns are qualified with adjectives (sharp, sudden) and some verbs with adverbs (dramatically, significantly).\\\\\nAdjectives used to describe movement include: slow, slight, moderate, gradual, steady, quick, rapid, significant, sharp, substantial, dramatic. \\\\\nUsed to show a small change: slight \\\\\nUsed to show a regular movement: gradual, steady \\\\\nUsed to show considerable, striking or unexpected change: significant, substantial, dramatic (both positive and negative change), sharp, sudden \\\\\nAdverbs are formed by adding -ly to the adjective, and sometimes one or two other letters change as well. \\\\\nDegree of change: \\\\\ndramatically, considerably, significantly, substantially, sharply, moderately, slightly \\\\\nNote that ``dramatically\" can refer to both good and bad changes. \\\\\nSpeed of change: \\\\\nrapidly, quickly, suddenly, gradually, steadily, slowly\n          \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table*}\n\\section{Detailed Description of GeoMovement}\\label{sec:B}\n\n\\subsection{Text Data Sources}\\label{sec:textsources}\n\nFirst, we purchased news articles from Webhose.io (\\url{https:\/\/webhose.io\/}). Webhose.io collects text documents from various Web sources and distributes them in a clean semi-structured XML format. To save costs and reduce the number of irrelevant statements, we used their web archive tool to obtain text documents that match the key term \\textit{travel}, that is in the \\textit{English} language, and that was published on a website categorized as \\textit{news}. We downloaded all articles in their archive between August 1, 2019, and May 10, 2020, that fit these filters (397,919 articles).\n\nSecond, we acquired and incorporated tweets using the GeoCov19 dataset spanning February 1, 2020, to May 2, 2020~\\citep{qazi2020}. This dataset contains hundreds of millions of tweets related to the Covid-19 pandemic. We harvested only the tweets in English, which resulted in 328 million tweets.\n\nThe third source for statements is the Covid-19 Open Research Dataset (CORD-19)~\\citep{wang2020cord}. This dataset contains scientific articles about the Covid-19 pandemic that were harvested and placed in a clean semi-structured XML format. The collection has over 50 thousand articles retrieved from PubMed Central (\\url{https:\/\/www.ncbi.nlm.nih.gov\/pmc\/}), bioRxiv (\\url{https:\/\/www.biorxiv.org\/}), and medRxiv (\\url{https:\/\/www.medrxiv.org\/}) that match search terms related to all Coronaviruses and is updated daily. We downloaded the dataset on November 14, 2020, and selected documents from that date back until August 1, 2019, which totaled about 15,600 documents.\n\n\\subsection{Application Development for Efficient Sensemaking}\\label{sec:B4}\n\nGeoMovement is an entirely web-based application. We built the web client using the React JavaScript Library (\\url{https:\/\/reactjs.org\/}) as its framework. We also used Material UI (\\url{https:\/\/material-ui.com\/})  React Components to make development efficient and provide a friendly and familiar user experience. We used Mapbox GL JS (\\url{https:\/\/docs.mapbox.com\/mapbox-gl-js\/api\/}) in the map portion of GeoMovement and Deck GL Libraries (\\url{https:\/\/deck.gl\/})~\\citep{Wang2017deck} to provide advanced mapping and geovisualization capabilities along with state-of-the-art rendering speeds for large amounts of data. The web client application is a visual interface for humans to search the server's data in Elasticsearch. Other software used to pre-process the data includes Postgresql (\\url{https:\/\/www.postgresql.org\/}) for data storage and PostGIS (\\url{http:\/\/postgis.net\/}) for geospatial computations. The result of this application design is a web application accessible in any major modern web browser, both desktop, and mobile that produces rapid results to user searches of hundreds of millions of descriptions of geographic movement. Almost instantaneous responses to searches enable users to more easily forage through data in the sensemaking process and identify important information. Figure~\\ref{fig:fig1} shows the user interface of GeoMovement.\n\n\\section{GeoMovement User Requirements}\\label{sec:C}\n\nIn Section~\\ref{sec:cases}, we address the utility of GeoMovement through two case study scenarios that demonstrate the ability of GeoMovement to support information foraging through the large volumes of text coming from multiple kinds of sources. The scenarios make specific assumptions about the required capabilities, skills, and prior knowledge of users.\n\n\\begin{center}\n\tUser Capabilities\n\\end{center}\n\nCurrently, GeoMovement works only with text in English; thus, users are expected to be English speakers. Given the focus on interactive maps and graphical displays, users need manual dexterity sufficient to handle interaction with the map and various controls and adequate visual acuity to read the maps and graphs.\nAlthough we did not test GeoMovement on many computer configurations, based on our internal testing, web development expertise, and established guidelines and recommendations for the development technologies, we can estimate that the user needs modest computing capabilities. The user should have a computer with a minimum monitor screen resolution of $1024 \\times 768$ pixels. A modern JavaScript-enabled browser (Chrome, Firefox, Safari, and Microsoft Edge) is required, and minimum computer hardware specifications to support these browsers. GeoMovement was tested and worked on a modern smartphone with a mobile Chrome browser; however, GeoMovement's sensemaking capabilities are restricted because of the small touch screen. Although GeoMovement's design and architecture provide swift search responses from the server software, performance depends on the users' Internet connection speed because it is web-based. Therefore, GeoMovement will function, but user Internet connection speeds under the typical broadband minimum of 25 Mbps will likely produce a lag in user search results that inhibits the analysis.\n\n\\begin{center}\n\tUser Skills\n\\end{center}\n\nGeoMovement expects familiarity with web browsers, graphs, and maps. We assume some experience with thematic maps (maps that depict data geographically), but no particular expertise is required. Thus, users familiar with the kinds of maps routinely found on news sites like those of the New York Times of the Guardian are sufficient. An ability to understand simple statistical graphs (bar charts) is assumed.\n\n\\begin{center}\n\tUser prior knowledge\n\\end{center}\n\nSince GeoMovement is designed for user-led information foraging, it does not provide suggestions to users about what to explore or specifically what to look for. The assumption made is that each user has some knowledge of the topic they are interested in, that they can apply, to leverage the visual interface and computational methods included in GeoMovement. That is, in fact, the point. This is a human-in-the-loop system intended for users who have domain expertise that is complemented by the ability of GeoMovement to process very large volumes of text and potential for interactive filtering tools to support quick narrowing in on items of interest through the use of that domain knowledge to decide what is interesting and what is not interesting. Beyond domain knowledge, however, users also need some understanding of the limitations of the computational methods that underlie GeoMovement. Specifically, they should understand that any system like this will generate some false positives and some false negatives; thus, it will show them some irrelevant information (which their domain expertise is likely to allow them to ignore) and miss some relevant information. Therefore, they need to understand that absence of evidence from GeoMovement does not necessarily mean the absence of the phenomena they are interested in.\n\n\n\n\n\\end{appendices}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}