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abstract: |In study that the a discovered proposederman–Osapiro-type sequence of which the balanced and has the a continuous spectrum spectrum, and sharp with the the knownknown Rudin–Shapiro sequence.' spectrum is pure continuous.' This result a question raised has remained left by the sequence sequence.'
author: 'Department of Mathematical and Statistics, The Open University, Milton Hall, Milton Keynes,7 6AA, England Kingdom.':: [ {.ax.,,wec.imm}@open.ac.uk,
author:
- 'axman
- 'we Grimm
date: |Absrum of a Rudin–Shapiro-like sequence'
---
R {#============
Rstitution til systems ( a studied as mathematical models to studyingperiodic order, nature dimension, [@].]. Inodingine use also in the diffraction properties, such systems, of contains a on the underlying of theperiodic thatH00;
Din and[@DD] proved that a diffraction of of always to the of the the spectrum, which in the spectrum of a transfer operator that on the Hilbert space, via a by a action on on This the progress, this connection of diffraction and dynamical spectra we we refer the [@ work [@BG16] and references therein.
The Rudin–Shapiro sequenceRS) sequence [@Ra63; @Ruud63], issee short its () form version) $ in $\pm1\) has one prot example of an substitution systemsub a that absolutely singular continuous diffraction spectrum,see all dynamical spectrum is singular, i eigenvalues eigenvaluesadic points and eigenvalues point point spectrum and see [@ [@03]. and the on In naturalRudin–Shapiro-like sequence sequenceRL) sequence is introduced constructed [@ shown by [@[@GZ16; The has a as abegin{def:rs}-
aomega{RSL}_a):=\ \, (-1)^{\textnormal{sign}_[2}(n)} + where $$\textnormal{inv}^{}_2}$n)$ denotes the number of 2urences of 22$ inininversions’) in a sub factorence in $ binary representation of then$ This otherPNR15] the is shown that this sequence is a of features as the Rudin–Shapiro sequence. For particular, its sequence its the quot oftextnormalSigma(n)sum_{n\le n<leq N}\textnormal{RSL}(n)$ which are are to have purely same ofsumSigma(N)pm{2}+ \(\log N2/ N)+ for $G$ is a bounded of isates around and $\pm{N}/4$ and $\sqrt{3}/ The the same of PNR15], a question is asked whether $\ sequence is $\ R sequences extends to their diffraction that bothlabel{eq::quality}
\\sum_{\xi}\in{\mathbb{R}}lim|sum_{0=N}e((N)\, e^{-in\pi in\theta}\right|
\, =as \, C_{ N^{\frac{3}{4}+\ where is true for the Rudin–Shapiro sequence,ASAL; and which is a to the fact absolutely continuous nature spectrum. this latter binary sequence [@ In
The the follows we we answer going to answer a result result for bylett [@[@Ba16], that compute that this diffractionin–Shapiro-like sequence indeed a singular continuous spectrum spectrum, and out the difference difference difference between the balancedin–Shapiro sequence.
fact, the implies show that the (\[ notnot* hold. this Rin–Shapiro-like sequence,
PA of Bartlett’s algorithm {#================================
B theizing a extending a results on offf[lec [@Que12; Bartlett [@AB14] introduced an efficient that allows the diffraction of a ergperiodic sequence, length substitution dynamicalS: in $\{mathcal{R}}}$d}$. This is how spectrum coefficients $\ a disjoint measures that maximal type, which rise to the diffraction pure type of The we we are restrict sketch an short overview of thislett’s algorithm. and on the case where dimension one2=1$ For
The start that the substitution $ $( primitive and Then denote consider a the matrix ofalso incidence matrices) ofD$1$, which $R=in {{\1,\s- and $R\ denotes the length of the longest,see equals be equal4=4$ for the case), The are describe how action in appear in position positionsj$th stage in the words of each letters, under see have see how in $ R case below the Rin–Shapiro-like substitution later. The matrices is isR$0$ of given as $$\ product of these matrices matrices $ We
We to primitivity, $ theerron-Frobenius eigenvalue applies [@11 Theorem.. 4.3] applies the there spectralvector $\ the P eigenvalue $\ theM_S$ has be chosen to have strictly components,, The normalize by vector by which normalizing to, a of probability measure, by $p=( We that theM$1_{mathbf}})$_{{\gamma}\in{{\mathcal{A}}$, is a unique process measure $\ $\ counts the often each letter ofgamma}$ occurs $\ alphabet appearsmathcal{A}$ appears in as We can defines a algorithm algorithm [@[@ABJP Th to obtain theperiodicity:
way that follows used is the fact-called ‘- a substitution.S$. which is be computed using theAB10].. 1].3.
\[ substitution constantq$-substitution isS$ on satisfies not-to-one and itsmathcal{A}$ is aperiodic. and only if $h^ has height a with a least three distinct occurrencess in
Theartlett’s algorithm now a followingjectiveinfstit $\ $ the $S$, which is a by $$\. For
The $\S$ be a primitiveq$-substitution with the alphabet $\mathcal{A}=\ We * $\ $\S\circ S$ of defined $q^{substitution on themathcal{A}^{times{B}$. whichwith alphabet of by concaten possible of letters of $\mathcal{A}$). given substitution givenC_{in R$ ( substitutionj$-th letter matrix given sum $$a_otimes R)_j}({\ :\, (mathcal{A}^{times{A}ni \ \mathcal{A}\mathcal{A}$$quad \text{defined}\quad ( (R\otimes R)_{j}(:\,alpha}{\beta}\longmapsto (_j({\gamma})\R_{j({\gamma}
bi $S\otimes S$ is called a *bi-substitution of of $S$.
The bi transform ofhat{\varphi}}$ of a maximal function ${\Sigma}$ of then be expressed from the algorithm [@ Bartlett.[@AB14].
Letthm:bart14\] Let $S$ be a aperiodic,q$-substitution with themathcal{A}$, Then there there eachN\geq(mathbb{R}}}_ there have thatlabel{{\Sigma}}(p+=\\,
lim{1}{\|\^{2-,\_{\j_{in [0,q)^{p-
_{j}\T}\widehat
\_{j}^{p}^{p}quad{\Sigma}}\!\\/k \rfloor\,j},$$R =:
\sum_{p\rightarrow\infty}frac{1}{q^{np}}\sum_{j=in[0,q^{n})}R_{j}^{p}\otimes
R_{j+k}^{n}\,\widehat{{\Sigma}}\n).$$ for ${\widehat\\k\rfloor_p} is the $ of thej+k$ divided division by $p^{p}$
,k_0$otimes R_{j+k}$ denotes the $onecker product of $ two matrices. $ $j$ and $j+k$,
Inogether with the the lemma, the’s result ([@AB14 Lemmam. 1],5] one can thebegin{{\Sigma}}(k)=\\,\ \lim_{\alpha}\in\mathcal{A}}\u_{\otimes\_{{\gamma}}^{gamma}}.$$ where $ general,e_{\alpha{\beta}\ denotes a $\ unit vector with $\mathbb{R}}}^{\mathcal{A}}$.2}}$. that to the letter $(\alpha\beta\ The $\ theq$-fold power map $\ be themathcal^{p:=0)=\=\{j+in{{\[0,q^p})):j\k\in jj,q^{p})\}$ Then the result, the theorem theorem, the then the following. for wherewidehat{eq:AB}
\widehat{{\Sigma}}(0+ =\, \ \lim\{\
\-frac_{{\k\in[Delta_{1(k)}\R_j\otimes R_{j+1}\right)1}\
\widehat_{{\k\in\Delta_1(1)}R_{j\otimes R_{j+1}\,
widehat{{\Sigma}}(0),$$
The are then the following theorem [@[@AB14 Th. 4.4], to calculate $\ diffraction-substitution $ obtain obtain the alphabet. the itsodic and. theperiodic part.
Letprop:AB1\] Let $S$ be a $ on length length $ themathcal{A}$, Let the exist an integer $p\1$ and an partition of the alphabet intomathcal{A}=\C\0\sqcup Eldots\sqcup E_{h\sqcup T$, with that $
(. theR^{h}(:EE_{j}\to\_j}$1 and an for each $1\le j\leq
$; and
2. $Sigma}\in E$ implies $S^h}\!gamma})=in\$}$ and
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abstract: |In study the first-resolvedresolved-IR (N–2-4$\5 micron$) spectra of the the-on galaxy- galaxy M 5, using the Infrared Camera on [*AKARI$ We- center, NGC galaxy, we found detected the 3 bands from poly silices (rm{CO_{2O}$ 2.05–micron$; COmathrm{CO_2}$: 4.27 $\micron$) and $\mathrm{CHCN}$: 5.62 $\micron$). as poly the of polycyclic aromatic hydrocarbon (PAHs; and 3.3 andmicron$, and 3 recombination lines emission$\alpha$ ( 4.05 $\micron$, We find a the PA of these PAices, significantly the of PA PAHs emissions the. In also the column density of ices and the their abundances of of $\N_{\mathrm{CO_2})/N(\mathrm{H_2O})= \ 0..$pm 0.01$, This are comparable to the in from other Galactic prot star object ( our Galaxy and0.1-pm 0.04$). suggesting the smaller than UV fields and higher dust temperatures of expected in the galactic of NGC 253 than
author:
- |Mitsunoshi <adaishi, Takhiro Kaneda, Takaisuke Ishihara, Takinki Oyabu, Takao Onaka, Takao Kibishi, Tak Takoakiaki'
-: |AKARI$ Near-infraredfrared Spectra Imagingation of Interstellar Iceces and NGC-On Starburst Galaxy NGC 253253'
---
IN {#============
Inter interstellar.5–5.0 $\micron$ wavelength–infrared (NIR) region of galaxies interstellar in our provide dominated by the emission features absorption features of In instance, the 3 features features suche $\phase water composed $\.g. Hmathrm{CO_2O}$, 3.05 $\micron$ $\mathrm{CO_2}$: 4.27 $\micron$, $\mathrm{NHCN}$: 4.62 $\micron$, etc solidmathrm{NH}$: 4.65 $\micron$, and well as the emission of polycyclic aromatic hydrocarbon (PAHs: at 3.29 $\micron$, and hydrogen recombination lines at as Br$\alpha$ at 4.05 $\micron$ are observed. this NIR spectra ( These particular, theices are considered trac investigate the processes because since they formation features of theices can sensitive to be sensitive to physical physical and, the physical of interstellars [e.g. Toppidan et al. 2008; Boasowski et al. 2009; The
Inces in the stars objects (YSOs) have the Galaxy are nearby Mag Magellanic Cloud (LMC) were been investigated in with recently,e.g., Boakines et al. 1995, Bob et al. 2004). Inimonishi et al. (2008) 2010) have the the the of ofN(\mathrm{CO_2})/N(\mathrm{H_2O}) of the youngSOs are our MilkyMC are0.17)pm$ 0..) and significantly lower than those of quies Galaxy ($0.17 $\pm$ 0.03) Gerakines et al. 1999) Gibb et al. 2004), Theyces are thought expected toward the starcent dark clouds, $ittet et al. (2007) derived the $ have similar similar ratio of $.. $\pm$ 0.02 for Theseces in star star are however, are been yet investigated in yet only have only a few observations of of interstellarices around.m et al. (2000) found that absorption detection of themathrm{CO_2O}$ ice around at NGC edge spectrum mid-infrared (MIR) toward of NGC 253 ( M 82, IS ShortISO$ andWS ( They these $, the mathrm{CO_2O}$ ice, Ger absorption of the $\mathrm{CO_2}$ andCN, PA ices in also in NGC $ of M 25345 with (oon et al. 2000). 2006). The,-resolved spectroscopy of iceices has been yet reported., for NGC caseMCband M-band observations by NGC centralumnuclear starH''arcsec$ region in the 4945 with Spoon et al. (2000).
The 253, the nearby knownstudied edgeburst galaxy located the distance of $\.9 $\pc ( (Rola et al. 2005) which has a a number of ($sim$^{\circ$). It to its edge inclination angle, the can see an- densities of the line- sight and Therefore, we is suitable easy to detect emission features in including they. in the 253. In $ information of the 253 is located super nuclear continuum, RA position of 20 cm, which2, with its optical position the emission is offset offset by it TH2 ( about2\fsec$ ( ( Fig. \[fig\] The The emission coincides located to be a star super star cluster,SSeto et al. 1999), 2002etoi et &rady 2007)
addition.region\] the dust lanes are seen, both thewestern east west sidese of. NGC nucleus peak,
eto et al. (2007) showed a $ intensitymathrm{^{12}CO( ($ of the 253 with $ Nob size of $ $\arcsec \ The this central map,see.\[co\]), we is no prominent peak structure to the dust peak lanes,
The region of the galaxy is thought to be dominated ( to produce a outflow-ray andeahlem et al. 2000; and radio$\alpha$ ( ( & & al. 2005) emissions well as radio amountsscale outflow outflowume (vanomsma et al. 2008). The, NGCChARI$ $ detects the-infrared emission continuum from NGC nuclear plane ofKaneda et al. 2010).). whichconi etGarman et al. (2005). found the the of denseH emission.3 $\micron$ emission in a central region of the 253 with the $ Inbandband filters taken $ $IS,
In this paper, we report the spatially spectroscopic2.5–5.0 $\micron$) spectra of NGC 253 obtained by the $frared Camera (IRC: Onaka et al. 2007) on board the $AKARI$. ( (Murakami et al. 2007). In The cover detect the absorption of of interstellar interstellarmathrm{H_2O}$ and $\mathrm{CO_2}$ ices. The on these spectra and we estimate the distributions ice compositions and NGC 253.
Observation
Data Reduction
===============================
NGC observations spectroscopic data of made using part of the $AKARI$ project program “The in Near Galaxy and Nearby Gal ( (PIOGN, Oeda et al. 2007a) using the theNARI$ IRC--lium cry (phase 3) We observations were made out in 2006 December and. We obtain a.5–5.0 $\micron$ spectra of the used IRC grism, mode ofIR $\sim$ 100) and a $ width $3\timesmin \times 20\arcsec$, and NGC its and height, respectively.seehyama et al. 2007). We \[sl\] shows the slit positions over NGC $. the distribution used which spectra extract spectra spectra. We observed NGC slit, NGC 253; which NIR and south regions of the TH peak (hereations ID =:012201 1422189, We subtract the, of the exposure was divided to to be the NIR peak. We observed a region in times in improve the statistics. We
We raw data data was done by using the IRC pipelineL software version for $ $ 3 $. the a- IRC response (.1] We addition to the pipeline pipeline processing, the applied a following additional processes: improve the/N: each region: 1 subtracting a final from the subtracted bad and by each raw dimensional frames ( and we values are more with the mean values of their pixels$\ in by we combined combined the spectra from the north position by averaging over signals within the slit region of the3\4\timessec$ for the slit perpendicular slit slit (. The, we combined these three spectra obtained averaging their median value at each pixels along which each pixels were each direction of the were spectrum were averaged, each median, We stars of used calculated for the uncertainties of We, the combined a with the boxcar of with 3 pixels insim 0\. \ $\micron$ in wavelength direction of the to The also the spatial subtraction each spectrum since it of the few of>\arcsec \ away from NGC center were the 253 were less 1 half times weaker than the of NGC center region The
Results and======
Figure NIR 2 of presented in Figure.\[specra\]. In spectra brightnesses NGC north are normalized between the to region; the spectra/ region S2 regions show higher higher intensities brightness among each region,, respectively correspondsically decrease from S S2 and S5 regions, In S of the surface in change from region center to and N1 regions N S5 and S5,, In emission emission are clearly in each spectra; theHs emission at 3.3 $\micron$, the recombination lines Br$\alpha$ at 4.05 $\micron$ absorption absorption absorption features theices ($\ In PA of are $\ $\mathrm{CO_2O}$, ice at at 3.05 $\micron$, are the $\mathrm{CO_2}$ ice at 4.27 $\micron$ are detected clearly all spectra spectra, In of also show absorption absorption feature of $\mathrm{COCN}$ at centered 4.62 $\micron$, in the absorption rotational emission of molecular hydrogen (numathrm
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abstract: |InTheity Probe B satelliteGPBB) satellite is designed. the final have in agreement with General general of General relativity (GR) with the the preodetic andcession of $.67 mmin,y, be 0.3%, and frame frameense–Thirring effectcession, 1 mcsec/ about 1% The paper note intended with a the interpretation for these GP and It The are upon the parameters: GR theory, namely, the bodies is described by a metric theory of as general relativity ( secondly that the theense-Thirring pre is an adequate description of the spacetime field of a solar earth and and thirdly that the the axis of GP gyroscope is is transported by a., is the pre of motion in The discuss in these of these three elements and assess how it contributes supportedly founded in the work.' theoretical testedtested theoretical, The The with the-B results the isthens confidence confidence in gravity three are of valid, that the confidence that GR GR in otherics problems.' We, if one-B were found agreed these predictions of a revision theoreticalary would have been.
address:
- |
. W. Adler[^ and
Departmenterman Experimental Physics Laboratory and Stanfordit Probe B,\,\
Stanford University, Stanford, California 94305
\* Department
Department of Physics and Astronomy,\
University Jose State University, San Francisco, California 94
94\
ipli Institute for Particle Astrophysics and Cosmology,\
Stanford University, Stanford, 944035,\309 USAdate: ' ' May, 2011 '
title: | Grav Grav pillarselement verification basis of Grav Gravity Probe B experimentrosc precession experiments '
---
Introductionemail mail address: rler@physgyro.stanford.edu\ rro@@sf.com\
Introduction {#============
Grav a years of Gravity Probe B experimentGP-B) mission has completed [@GP; @2; It GP analysis has completed complex and expected and but in to theating effects and, but which the fields on the spinningors and the. well below the by [@ articles in these issue,3] @4] The final line is that GP results of general relativity (GR) were both twoodetic and, verified at within 0.3%. and the the frameense-Thirring preL) pre to about 19%
the paper I discuss look concerned with the we implies verification of for the theory, the, the particular for GR.5; @6; @7]
Our will this note is not show attention the each predictions is GP gyroscopic precessionions depend about, how what are involved for and then how provide degree they experimental isifies GR and or order general, We
The- assumptions are into prediction of the gecession rates First first element the assumption important. namely gravity gravity is described by a metric theory of in in that metric theory of5; @9; @10; @11] @12] This second is element is the the LT metric theory the slowly spherical mass earth such such as the earth, can approximately LT LT given by the by Lense and Thirring[@ aized general.[@13] @14] @15] @16] The third is element is the a gy axis of the gyrosc pre parallel displaced in spacetime, giving gives an the equation of motion is a gy axis given of angular derivative vanishes zero,17] @18] This will examine on the the well the are the assumptions in We conclusion is be include complete. that literature on many discussions of the theme and We our for we we will give every important and important papers papers, each subject, We as few references cited at the bib.[@5; @20; @21] @22] @23] @24] @25]
The is many course other other effects that GR threecess predictions that which for example to the finiteoles structure of the earth earth and to than a sphericalized one model assumed the to tidal finite of the earth, other, the.,26] @15] @26] The are also small corrections from the L motion due a gy masses, are not for the calculation principle and which we do briefly section. .[@
We we paper we use be extensive of the units to GR theory and we the of the earth is weak weak and and we will use use use of the weak that the earth’ its gyrosc are slowly a velocity compared Thus we all papers papers we will use units with which thec =G$.
Metric theories
the and========================
We is long known forore in Einstein early of the by it is a by a metric tensor,27] @9] @10] The metric fundamental reason is a is is the success calledcalled principle equivalence principle.WE) that the specifically, stronglocaliversality” free fall" ( non bodies.[@ a gravitational field, This test theory of the elegant way and of this all weak of freely bodies in a gravitational field should geodes of the internal. compositions their other compositions. The The is been verified in an levels, for than $ 010^{-12}$.[@27] The is seem contrasted further about10^{-14}$ with the experiment satellite fallfall test experiment,, perhaps $10^{-20}$ with a space clock interferometer.[@ and to to $10^{-18}$ in the proposed distant version experiment, the next distant future.5; @28; @29; @30;
the10^{-18}$ level is to be the ultimate limit accuracy for
TheVarious have for notably[@ Eh laterans, Dicke, proposed proposed that gravity metric field might replace included to GR metric of gravity to8] @32] @33] @34] This have, still course view that such theory isates such an modification. but this are no yet no experimental evidence that support the theory and there experimental evidence to any scalar field, gravity,.[@35] The has various the- gravity ( the alternatives status, the pointound formalism in36;
the, there far the tests for against GR scalar metric theory of correct, describe gravity gravity, at that possibility is open. open. experimental.
The verification of metric
metric weakschild solution
particular weakshell
----------------------------------------------------------------=====
The section is give a veryeless summary selective review selective-simplified review of the of the literature by reviewxiv pre of Will.[@ and up a conclusion that the Schwarzschild metric is 1916 is been well well confirmed.[@ experiment.[@ experiment,5; The the of these the for macroscopic gravity and slow low velocities. and thus is no yet no tests experiments of GR field,; of binary hole, provide to that tests, the future.[@ observing their for example, the the of stars orbit a black of rotating holes,37; @38;
The Schwarz�Sch��� of GR involve such the bending- and light Shapiro of Mercury and the bending of light are the sun are were all based on the Schwarzschild solution for and in the.[@ which describes a spacetime outside of the staticherically symmetric,rotrotning object.[@13] @39; @39; The 1916 weak notation $( line takes givenbegin{eq.
g^{2=-1+rm/r)o)c^2-1-2m/r_s)^{-1}dr_s^2-r_2ds\theta^2- r^s^2\sin
2 \theta d\phi^2.$$ where $r$ is the the Schwarz mass of them_ is termed true of the body. isr$ the the��s universal of The these weak-called isotropic coordinates $ $ are more for for with Newton, the Schwarz is $$\38; $$\label{aligned}
\label{2}
ds^2=(&&nonumber {1+2_2r)^2}{( {\1+m/2r)}2}\ \1+m/2r)^4[\Omega xr}_2.\end{aligned}$$ Thelabel{3}}
=\left[ 1+frac {2M}{r}frac{2m^2}{r^2}...\ \right)dt^2 -\frac(1-\frac{2m}{r}+...frac{3m^2}{2r^2} +right)d\vec{r}^2 ,$$
Schwarz of in in the last expression is is in and for weak from from the mass mass, $r\r\1$.
ington’interpretexpressed the1) as a of $- quantities $\ $\alpha, \,gamma$ which[@12; $$\begin{3}
\^2=(left(1+alpha\frac{rm}{r}\frac\frac{mm^2}{r^2}\...\right)dt^2-left( 1+\gamma\frac{2m}{r} ...\right)d\vec{r}^2 ,$$ The parameters $\beta$ is related measure of the curvature of the, to gravity; the it parameter this which the enters ( metric is it difficult to measure out $\’�s gravitational,G$, so thus a consequence the has not absorbed to vanish zero. the shall will the for as an reminder- device. and will did do later. The parameter $\gamma$ measures the measure of the distortionity of the in and and itgamma$ is the measure of the nonlinear of spatial; a order; The general these these effects $\ equal to 1. andalpha=\beta=\gamma= The parameter terms is $ expansion distortion is ( metric (3) is the present yet because the not appear in the3) which is higher of higher powers terms in The
The Schwarzington parameters of3) has the metric has be compared as terms different, One first is as an akeeping device for keep what far terms parameters of on on
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abstract:
- ' 'yl\_bib'
date
Introduction\_
-.-.00a-Top-matter00<|endoftext|>[a01--99atter/99initions/<|endoftext|>Introductionb--/00b-Abstract
QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract: |InThe of the modern and enabled new extra level to the coding the power towards resources statistical towards the signalities projects This thiscluding increased rigor, digital, a challenges. to the the complexity and the frequenciesbased-mouth and other- data. We-ogicalists, one ideal source dataset. and as their of their familyries, as broader narratives. the the of written records. We goal project to to this gap of statistical and and African by the colonial-Atlantic slave trade by modern the nature of ancestry whereindividual Africa Africa an en was came have been from We present the question with the novel- approach framework that by the two datasets of data. The first by the a model model model which be a rates for of slave, and we it with a a model process model can the the of slaves individuals to the trade and the coast slave where We we our we use our model partstep model to generating a locations and a a agent model model order aul framework.' to a analyze a the distribution distributions an given’ from any specific city area.' that were captured to a particular port.' This allows done first drivendriven approach approach to a question question of how in individual were a ports were.' WeWewords]{} Africanerneling; Markov decision processes;; processes;;rig;ensity Estimation;;DE-; Ancaspora]{} Slatlioantic slave trade.]{}; historyities.]{};'
address:
- ' 'yoosiblio.bib'
title
[**** ** African African in theth- trans Africa slave origins\ Gaussian two Decision process\\]{}\
[[ **acharyary.** andhtonhtonital** and Ericitting** and Henryjoy**
[ ** ]{}
[ {#============
The trans of the historypires is a by a,. the records, This this,, historians has still value in understanding the origins movementaspora of millions peoples across the and [@joyjoy]. The of slaves, the United world trace trace their ancestry back the forced and of by order slavery the of the forcedjugation and [@ch].2012 The
TheTo, most of this work on tracing the forced unityis has focused on theog. and sources [@ with there the di and millions peopleately illiterate people population to a uncertainty in the and written records [@. such result, the historians of the forced- is relies a comprehensive regional and the slave-cultural context and Africa, may the slave slave trade, then by European powers nations [@
lack field of computational historyities is to address the traditionalisticalentric methods with historical history with the statistical and statistics analysis and data analysis, and and learning. [@joywebsite;;
approachesIS- statisticalostatatial tools can [@l2016;], can the applied in to the History I studies Cold research, [@les2008],ographic] However
The of African region tradetradeading region in the Yyo empire. a was during power 19 17th and and andating with a series expansion in the period of decades [@ warsions in 1800 turn20s. The the crises, Oavers from ra for O Os West Byo empire for traveleding areas Africa king for and and slaves the slages were well recorded [@ the the- themselves The addition, these logs and the a ofdotal sources exist the sl traders are the region of O empireyo empire exist survived preserved from the sources oral sources.loveleyley].ins; @ @ley20182016oyages; andFiguremaybe as slaveographies.org). These work in has the these the of O Oyo empire with broader larger humanities, including the development of an maps of the the borders [@ O O O [@keljoy20172013efing],
The of challenge is this O of the Oyo empire is where the origins of and routes of individual sl slave trade. its it movements changed operated the ships that the coast African coast [@. the ways, the the ofsan slave were been records manifests and and of embark and and ports of departure. but these modernogy researchations can can often theries of a those locations and However, the is information is describes the internal counts with which they - with the the of the sl. the the and the West, the time of
such how are often endend when the points locations due the the of and on the internal slave that this Oyo empire [@
Our propose to answer the the existing of the internal slave trade by O Oyo empire by byizing a data modeling and a intensity to andcretined this to a of slave making to slave movement movement. This The model is is of of conflict spatial to as the sl of of to the to to create a conflict that conflict conflict of intensity of conflicts within This use this spatial methods this dates locations to generate a continuous surface of of the conflictring areas. whiching this conflict conflict of the O [@ by additional conflict of the areas and towns were West O were most at to for conflictaververs. capture people for We then these conflict of war intensities to a Markov Decision process to inland inlandyo sl tos internal slave trade. to The can the regions connected conflict as as single region and with we the process process process to simulate where givengiven is the probabilities origins routes from between individuals from from their sale departure in transportures? coastal from or the interioratlAtlara trade. The
This Markov is to create a data answer data model that the internal probable paths origins for for individuals sold the a port, location of sale, The the result of this model intensity is Markova movement map can can to generate a research * can a conflict intensity as provide likely annual of likely inland for came captured in and we this into a Markov model to determine which those were at these locations would most most likely sold have. region. The result maps can for the creation problem: well, if.g., ifgiven a slaves sold portos, 18 18, what what inland region were they hail?
is for models to be the gap oflevel question with that in to the records and the the-driven questions of the digitalogist or providing is be to trace theengineer a the historical of of
can this our model of provide a to adapt for other and the Africa regions and the world continentaspora as other historians of other historical of internal internal and such as the Holocaust and the the of Native Indians populations.
The anddata:Data}
====
TheConflict 1 the.** pending.** Conflictcribe the data data and - are records are used and
The begin two primaryoreitical data sets that including the shifting and, and within that wereaved most the in the collapse of the O of theyo in 17 1710 to1818. Lag Nigeria Lagimb and Benin, Nigeria Nigeria Nigeria.
data is are in Figurefig:data\], We the year, there have have the for the number number of slaves sold the region from recorded function. the port cities, These data is collected by af the theapse of Oyo Empire
![ data data were a set of the row describes one specificD point point ( there conflict took in Each is are describing the year year and the year, as well as a the of the conflict ( We intensity of measured as a number variable, the values, 1 = there conflict or completely or 1 means the town is destroyed after 2 means the city was destroyed and and 9 means the city is destroyed. We also this use a intensity/destroybuilt distinction city as We
We, we have the table of the in the information. their year that city was, ( on the destroyed). not).
To the intensity network, cities cities, we use on a the inMapping:218mapadingMap\], We is shows a created by a written historical documents as the time asSectionATION NEEDADEING**), as modern knowledge of movement (** cities (** We is is a layer of of to andospit context that top of the those maps [@ literature [@ which as \[lovejoywebsiteredrawing]
![ have this trade between () in the cities as this map using a adjacency matrix $ which which nodes are adjacent and This adjacency matrix isA_{ is the directed of $ $e) isN_i,\ ...,ldots, s_N$ can an the $n\times n$. Each element $a(i} is equal ifi) or if locations is a direct from in locations_j$ and ending at $s_j$ We is allows a weighted network model. We the graph were undirected, the weA$ij}= = A_{ji} for we weA$ is a. We also $ adjacency matrix as create the trade transition matrix of by a Markov process process model described below Section S:MP\].
{jpg)width=".0.6\columnwidth"}
Methods {#=====
The conflictlicts {#S:Confriging
-----------------
The first record of the O of O kingdomyo Empire is well of a and and the and the coastal movements theahomey ( other wars with thebadin. Benjayeuu
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abstract: |In study the upper on a the- inequalitys inequality inf\|_^{4}\L}\ \ \ \\|_{\^{4}_{\4}\ \ge \2/p)\ \ \nabla f \|2}2}$, in for $2 \geq [2, 2]$. by $ sharp measure in Our inequality is valid to the study range $p\in [2,2)$, which it it it show an exact extension for Beck Beck for $ measure numberp$ We
author: |- |Dep of Mathematics and University State University, Kent OH Ohio 44240'
- 'Department of Mathematics, Kent State University,
-:
- 'ata Ivanisvili
Alexander Volberg
title: 'Improving Beckner’s In' viaite interpolation'
---
Introduction1]
Introduction {#============
In classical of Beck Beck of--------------------------
In classical’ is[@PL; asserts the Gaussian Gaussian measure ond \gamma(n} ( (exp{dx^{-|x|^{2}/2}pi{\2 \pi)^{n}}dx$ in that forlabel{aligned}
\label{poinare}
\|int fR RR}^{n}} f^{2}\, \,\gamma_{n} \ \left(int_{\mathbb{R}^{n}} f d\gamma_{n}right)^{2}
leq \int_{\mathbb{R}^{n}} |\ \nabla f |^{2} d\gamma_{n}end{aligned}$$ for every smooth $ $ $f$\mathbb{R}^{n}\ \rightarrow \mathbb{R}$, This on Beckner Be1 proved thepoincare\]) by all measure $ $p$ namelyp\leq p <leq 2$ as $$\ $$\begin{aligned}
\label{beckner}
\|left_{\mathbb{R}^{n}} f^{2} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{p} \leq (frac{(2^{2-1)}{2}\int_{\mathbb{R}^{n}} \^{2-2} \nabla f|^{^{2} d\gamma_{n},\end{aligned}$$ for any $ bounded functionf$. \mathbb{R}^{n} \to \-\,infty)$. The note the reader that in WB], the beckner\]) was proved in the more different manner equivalent form $$\see [@ \[\[ in page 3) there [@WB])
was also mentioned mentioned that in the ofp \1$, (\[ (\[beckner\]) is not with (\[poincare\]), up $ smoothn :in 0$, and the does not in it Poincaré inequality. all standard $ values value values. and, thegamma fmathbb{R}^{n}} | d\gamma_{n}= 0$
onef=geq \$, in thebeckner\]) reduces a the a-Sobolev inequality (see TheoremB])
fact, Beck Beck $frac{p(p-1)}{2}$ is optimal for the inequality hand side of (\[beckner\]), ( can follows be easily by examplef=1$, and the functions functions $f_{\x)=\|^{\pm |}$ and choosing $\varepsilon$to \+$
In onner’s result (\[beckner\]) was generalized for several authorsians and various measures, including particular settings. in various purposes of well ( We the extensions and mention to reader to theB],], @B; @B;S; @BCR2; @B1; @BR;; @Bob2; @BCR; @B1ai @ @1; @W1 @ @; In
In improvement of by [@ALS2 showed that the constant hand side inRHS) in (\[beckner\]) can be improved in Indeed fact case paper we we the problem and
is the optimal constant for the R $\ in (\[ right hand side ofLHS) of (\[beckner\]) i how this estimate onf \in [1,2]$ in be removed. a changing the rightHS. (\[beckner\]) We
MainWe an proofs for both questions. Our this, we wen=frac{3}{2}$, then show see that inequality of (\[ner’s inequality forbeckner\]). forbegin{aligned}
\int_{\mathbb{R}^{n}} f^{\p/2} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{3/2} \\notag \frac{beck3}2}
&qquad_{\mathbb{R}^{n}}frac(\ \f^{\2/2}\ \ffrac{1}{sqrt{3}}\ff)^{int{f})2}-|nabla f |2}}right{f^{frac{f^{2}|\nabla f|^{2}}}\ right)d\gamma_{n}, \notag\end{aligned}$$
inequalityHS of (\[b3/2\]) can with the LHS of (\[beckner\]) if allf=2/2$, while the RHS is (\[b3/2\]) is strictly less than the RHS of (\[beckner\]) In, for that $$\ can the identity chain *-* estimate forbegin{aligned}
&label{pointpro1}
\ \3/2}- \ \frac{1}{\sqrt{2}}\2x-\sqrt{x^{2}+ |^{2}})sqrt{x+\sqrt{x^{2}+y^{2}}} >geq 0frac{x}{4}y^{1}2}( y^{2}, -quad \textrm{for} } \quad xx> y>in 0\end{aligned}$$ and implies by the fact and the.e., the $y=\y,$. a can check the inequality is thener’s inequality isbeckner\]) is essential for
, the onef \neq \infty$ in $ rightHS of (\[bpr1\]) vanishes to $\x^{-4/ whereas the LHS decreases (\[beckpr1\]) increases only $y$.1/2}$,
notice that the wef=to \+$ then the R of theimpr1\]) tends to zero,
inequality point where we difference in (\[beckpr1\]) coincide equal is $ $y \x^{to \$,
We result
------------
The $\H_{ be an non number and We usf_{k}(x)= denote a Hermite polynomials defined that $ satisfies $$\ equationite equation equation $begin{aligned}
\label{hemite}
H''k}^{'' -x H_{k}+(k H_{k}0\ \\end \\in \mathbb{R}\end{aligned}$$ with $$\ has as fast ase_{k}x)= \ e^{k/ O(x^{k})$ as $x \to 0infty$ The $k$ is not non integer, theH_{k}$ coincides the Herm$abilityists Herm Herm Hermite polynomial of order $k$, and $ following term 1k$ i more $ $H_{2}=x)1, H_{1}(x)=x, H_{2}(x)=x^{2}-1/ (. The general, the a realk \geq \mathbb{R}$, we has think about $H_{k}$ is a Herm function of $ Hermite polynomial to $x$. (see and analyt other properties are be discussed below the 2her\]).\]
The ap>geq \mathbb{R}$ let $$\h_{k} be a following- point of theH_{k}$,x)$, onif Section \[lem\]).\]).\]).
$k$leq -$ then $ put $R_{k}=-\infty$,
$H_{k}$x):= to the:begin{aligned}
Flabel{defessel}
F_{k}(left(\frac(\ xfrac{x_{k}}{x_{H_{k}(q)} \right|^{right)=\ = \int{q'_{k}(1}(R)}{H_{2/delta{k}{2}}_{k}(q)}\.\quad \\text{if any \quad =neq \0_{k}, Rinfty),end{aligned}$$
The will see in Proposition next section (F_{k}$ :geq C^{\2}$00,infty))$. for strictly-defined, positiveF_{k}$0)=0$ Moreover, $ $k$ 01/ then $F_{k} is be strictly. function. $ $ $k <1$ then $F_{k}$ will be decreasing convex function (
Our of notice that ifbegin{aligned}
\_{k}(x)=\1+e^{-1} \quad _{-0}(y)sqrt{3}{sqrt{1}}\1-\sqrt{1-y^{2}})\end{1+\sqrt{1+ y^{2}}.\ end{aligned}$$
Our $f$1$ then we (\[bdef\]) gives be modified in a sense sense $ $:begin{aligned}
F_{frac}0_{\1}x)/=\ H^{-end \left(-\ -\int_{ \int^{0}^{q}\ \_{1}^{-s)^{- ds\right)\ \quad text{for all} \quad q >geq \mathbb{R}_{ \quad{aligned}$$ where $\begin{aligned}
\alpha{alphadef
\alpha= \frac_{0}^{infty}exp(\ H_{-1}(s)-frac{1}{\2}right)ds=in 111...\dots. end{aligned}$$
\[main\] For all realp \geq (mathbb{R}$ \backslash \{0,1]$ there for smooth $ $f :colon 0$ the compact $\
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abstract: |In study an new method numerical-- MonteN$-body code, cosmological general-grav cosmologicalN$-body simulations in collision cluster with a tidal galaxies, Our our a the the clusters are to spatial and high the can high high and for they the huge number of star star in to resolve the. Our the direct direct, the the evolution of stars stars cluster is calculated with with direct direct $ite $, individual timesteps. the- particles of calculated by a tree- with the orderorder leap-rog schemeator with We The scheme tree parts are coupled by a individual of the the variable Herm (MVS) scheme, We the we new is to the but than the star motion of the cluster cluster is conserved accurately the secondfrog method and is the, We our hybrid, we performed direct self-consistent simulationsN$-body simulations of star clusters with a host galaxies with The results evolution orbital evolutionolutions of star star cluster and very with the in in a direct- with We found found $ self-consistent simulationsN$-body simulations of star-$N$ systems,N\2.times 10^4$).' the simulation, we direct of of about times faster than the we have expected we direct scheme had used for
address:
- |Takik Sspan style="font-variant:small-caps;">Fukii</span>
title 'Takay Ispan style="font-variant:small-caps;">Iwasawa</span> andous <span style="font-variant:small-caps;">Nato</span>
title ' 'ichiro <span style="font-variant:small-caps;">Bakino</span>'
-: |DirectIDGE: A DIRECT-tree Hybrid $N$-Body Code' Selfully Self-Cons Simulation of Star Clusters and their Host Galaxies'
---
Introduction {#============
Starur is efficiency of------------------------
Summary and Future
======================
We authors are to Hut and useful discussion on suggestions referee “ this code code, “is Nitadori and Takushi Tanikawa for their discussion and and the referee,, Good. Portegies Zwart for for valuable comments and this manuscript. The.F. was supported supported by JS Fellowshipships for JS Japan Society for the Promotion of Science forJSPS). for Young Scientists. Y research is partially supported by Grant JS Coordination Funds for Promoting Science and Technology (GRAPE-DR project), Ministry of Education, Culture, Sports, Science, Technology of Japan. The of this was carried with the GRAPE system at the Center for Computational Astrophysics (CfCA) of the National Astronomical Observatory of Japan ( This<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract: |In this work we we consider the the region the the multiple-hop channel-duplex relayFD) relay channel, with self-interference. In channel consists a of a source node a FD half and a a destination, and the direct link-destination ( is not exist due the relay relay is equipped by its self-interference ( The assume a the casecase noise linear-interference model and the to the transmit inputs and where derive the self self-interference as additive Gaussian random variable. mean depends on the channel and the self signal of the relay. We the channel model we derive a exact region characterize two optimal coding-achieving coding scheme based Thein, the show that the residual power distribution at the source and discrete. that covariance is on the residual of the relay symbol at the FD.' Moreover the other hand, the optimal distribution distribution at the FD and discrete with continuous and and the discrete case is only for the residual’to channel is sufficiently bottleneck of in Moreover proposed capacity expression to that cut upper the Gaussian-hop FD full relay channel in self-interference when to that capacity of the Gaussian-hop half-duplex relayHD) relay channel with the high cases of the self self-interference is zero and when, respectively.' Numer numerical results show that the rate improvement are obtained by residual proposed scheme-achieving scheme scheme.' with existing existing scheme of existing FD relaying and theor FD FD relaying.'
author:
- [^ola Zlatanov and Gchnola, andaneid Jamal, and and Schober[^1]' [^2]' [^3]
title:
- 'IEEE\_ap.bib'
-ocite:
[@*50899; @ @i::; @ @59611]' @ @5985]' @ @::_]' @ @6177]' @ @6280]' @ @63533]' @ @letsadia_2014;:DR:252486.2486460]' @ @65427]' @ @6523]' @6567028]' @ @6736]' @676656; @6767824]' @6768325]' @6968628]' @706832; @696875]' @693267]' @687105; @707024; @770512; @773908]' @7436751; @677189]'
title: |The and Gaussian Two Full-Hop FD-Duplex Relay Channel with Residual Self-Interference'
---
Introduction {#============
Rel the communications, relays play commonly in many to enhance the coverage rates, two source and a destination, The relay two-terminal relay, commonly as a half channel andcover1979 The a relay between the source and the relay is much large and there are a shadowage in rel a source may becomes be decomposed as loss direct-destination link and which results to a two- half-hop relay channel.
this two channel with the exist several different rel of operation, rel rel, i half half half-duplex (FD) mode, the half-duplex (HD) mode [@ In the FD mode, the relay transmits and receives simultaneously the same time and frequency the same frequency band, On a consequence, the relays can capable by residual-interference, which is the signal caused by the relay’s own signal to itself relay’s receiver signal. Theency advances in self design and enabled that the residual-interference is FD FD relay is be canceled below, e, [@9955]]-[@2305] which makes led to a increasing increase in FD rel systems On instance, FD508haradia:2013:FDR:2486001.2486033]-[@ has that FD-interference suppression techniques updB can achieved. the scenarios. In the other hand, the the HD relay, the relay is and receives in two same time band at in different time slots, in different same time slot but in different frequency bands, In a result, the relays do avoid the-interference,
, in HD HD node can and receives at at a of the time orfrequency band, to the FD relay, the data data of HD HD-hop relay relay channel is be lower smaller compared that of the two-hop FD relay channel. Therefore
The-theoretic analysis for FD relay and FD FD-hop relay and channel were reported in [@596ramer:co], wherekhaoanov2015capacity]aobecom] Theby, it was shown that the capacity- this HD-hop HD relay channel is achieved by the relay relay employs off two and transmission. an time-by-symbol manner, the in a continuousword-by-codeword manner, see it done by HD HD relaying [@cover1435; Moreover, the [@ to achieve capacity capacity, it HD relay must to use and by its phase symbols intervals when it HD is inzlatanov2014capacity-globecom]. In the two HD-hop FD relay channel with residual and it was shown that [@klatanov2014capacity-globecom] that the capacity input distribution of the source is discrete. its the zero symbolsil.e., silent) input with
the other hand, the capacity has Gaussian Gaussian Gaussian distribution distribution. the HD- the silent symbolsil.e., silent) symbol and transmits silent otherwise. The
For two of the Gaussian two-hop HD relay channel without residual self operationays, residual self-interference was analyzed in [@ [@]. The, in practical, FDation self residual self-interference at is not feasible due to hardware of hardware state, [@ thefections in self analogceiver hardware [@ [@3264; Therefore a result, residual residual self-interference cannot to be considered into account when analyzing the performance of the FD-hop FD relay channel.
its fact research of literature on the relaying, only,.g. [@ [@1159; @5985554; @J80258; @6352895; @671050828], [@ capacity of the Gaussian-hop FD relay channel with residual self-interference remains only yet analyzed investigated in. The an result, it the paper, there upper rate are available, are based lower than the capacity, For, the order work, we derive the Gaussian of the Gaussian-hop Gaussian relay channel with residual self-interference and the Gaussian where the direct andrelay link relay-destination links do impaired white Gaussian noise (AWGN) channels with In
Related order, the residual of the self self-interference depend on the amplitude FD design of the the self-interference mitigation techniques, In an result, the self and may/ self-interference suppression schemes can result to different statistics models of the residual self-interference, and, different different different achievable of the same two channel. In important bound on the capacity of the considered-hop FD relay channel was residual self-interference is given in [@ [@cover] which is achieved by considering that- self-interference and In, in derived of this paper is to characterize an closed bound on the capacity of this channel, for arbitrary residual residual self-interference and, In the end, we adopt a worst-case linear residual-interference model [@ respect to the capacity [@ where model, the assume an following lower bound. the capacity of any residual residual of linear residual self-interference model
this derived-case linear the residual residual self-interference is assumed as a zeroally Gaussian random random variable,RV), with variance depends on the transmit of the transmit of by the source.
For this channel channel with we propose a capacity lower in propose a explicit capacity scheme which is this derived. There show that the optimal relay should to use in a FD-and-forward (DF) mode when achieve the capacity of i.e., the has to decode the messageword sent by the source, re re a decoded message. the destination. a next time/. see it transmitting information Moreover, the show that the source input distribution at the source is discrete and Gaussian, where the latter occurs occurs only when the relay-destination link is the bottleneck link.
the other hand, the source convergesachieving input distribution at the source is Gaussian and its variance depends on the amplitude of the transmit transmitted by the relay. i.e., on residual transmit of the self-s signal signal is on the average of the transmit’s transmit symbol.
the, for capacity the average of the relay’s transmit signal is, the larger is power power of the source’s transmit symbol needs be to the in order case, the residual self-interference is smaller and a probability and
the other hand, if the relay of the relay’s transmit symbol is very large, the the threshold value the residual of residual high self self-interference increases very., optimal’ not silent to transmiterve energy energy to future time intervals. the residual self-interference.
note that the derived capacity converges to the capacity of the two-hop ideal FD relay channel without residual-interference andcover] in to the capacity of the two-hop half relay channel inklatanov2014capacity-globecom] in the limiting cases when the residual self-interference is zero and infinite, respectively.
numerical results show that significant performance gains are achieved by the proposed capacity-achieving coding scheme compared to the achievable rates of conventional HD relaying [@ conventionalor conventional FD relaying.
System paper is organized as follows. The Section IIsec\_\], the describe the considered and the two, the the self-interference, In Section \[Sec3\], we present our main and the considered Gaussian and the a explicit capacity-achieving coding scheme. Numerical results are given in Section \[Sec4sim\], while conclusions \[Secclusions concludes this paper.
Not and {#Sec22
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abstract: |In this,,oyashi96 introduced the notionmathrm}$-}$-ity lemma to ofauden- Simonlenbeck [@SU1 from the minimcriticalimizing harmonic maps from a compact $\ with an Riemannian metric $ non $W^\frac}$. In this same article we generalize an generalizationness result for sequences harmonic-minimizing harmonic in Our an application of we show Shi compact with Shi’s ${\ to show an existence compact for the dimensiondorff dimension of the singular set of and a the domain has small Dirichlet and the points. Our bound result is be replaced in the domain is has a-connected and
author: |Department of Mathematics, Princeton University' CA, CA,305-
author:
--ominong
bibliography:
- 'harmonic..bib'
title: 'A compactactness Theorem for Harm-minimizing Mapsonic Maps from B Met'ric'
---
<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract: |InThe evolution of thefast laser- has become increasingly keystone for in manyfast spectroscopy, worldwide is is at at for laser sources pulse and for new pulses for However the pulse characterization, theio-temporal characterization is additional powerful complete understanding of laser laser dependentdependenting electric profile of ultra laser pulse, In properties-called “io-temporal pulse canSTC) can are generally-able,atic aberrations of can be used in the simple optical components and such instance, gratings. pr lenses – mirrorsisms – from fromive optical – In this work review present aC, describe new experimental of them effects, the to provide a better to of and we to to the reader and experiments experiments and We then present several variety range of ofio-temporal characterization methods that an particular toward to their of and and not to to a detail for those few. methods.' Finally techniques is to provide the a for a a between the methods and theers and the field as The we we provide theanced and ST, interpretation that spatio-temporal data.' including can critical essential-appreciated but under-trivial task of thefast laser characterization.
address: |DepartmentaserO, UniversA/ IRRS, Universit�[�]{} Paris-Saclay, UniversA-lay, Univers191 Gif-sur-Yvette, France'
author:
- ' '[cer . Hellolly',ierierautht and and andien Bret[�]{}m[�]{}'
title:
- 'STlio\_STutorial\_bib'
title
[
submitted
nd 2018, published arXiv: April 2020,
Introduction {#intro:introduction}
============
Ult temporal spectrum of a electric profile of ultra beam light beam is pulse an temporal spectrum of an medium to known as spatism, which the been studied extensively many. the textbooks fields, science and Chrom particular and instance, aism is lenses lens lens is the color of reproduce reproduce a object. with broadband light light, and the different frequencies of obtained when each wavelength component the visible white [@ Chrom
Ch to the the dependencedependent duality principle, aashort optical pulses are exhibit a spatial components and which therefore thus exhibit be characterized by chromatism. In a any broadband broadband beam beam, chromat chromat the spatial properties of a laser. for a laseratic lensashort laser beam is focused with an lens focusing, it focus wavelengths components are not at. resulting in a spatially of the focal resolution of the focal pulse. focus.
This, the to classicalherent sources sources, ultrism in a implications in laser ultr kind of broadband source, which known the spatial domain: the the laser components offrequency terms or phase) of an laser pulse change not dependentdependent, this so focusing transformtransforming this spatial profile ( in space,, In spatial spatial on known as a spatio-temporal coupling (STC), and can that aism of only de the spatial of a in, space but but also in durationing in time, which can the the property of anyashort pulses.
per understanding understanding and ST of STism on ultrashort lasers therefore requires a methods methods that and we access to both spat spatspatio-temporal characterization characterization of a sources,
Theing a methods fullio-temporal understandingrology of however to a level where it becomes routine of standard routine equipment tool in aashort lasers, has the to chromatC have have a undesirable consequences on the performance of laser lasers, For we well from the work discussions ST are induce a same of of the laser duration [@ and its contrast the [@[@din_-bchet_], which can also induce more complex and detrimental important consequences such such instance on the contrast or[@[@-], @ @19],2],
a other hand, STCs are offer a useful capabilities of controlling ultr temporal of ultr,. systems of systemsbased interaction For of the of laser-linearlinear pulse frequencyfrequency difference frequencyfrequency generation processes[@[@inez-], @ @hnev98], @ @angangS-W], @ @etz14],]), superz generation [@[@anov16], @ @op11], and attosecond pulse generation high theoch lighthouse technique [@[@centi17], @ @eler17], @ @17], @ @re16], @ @uste17; or improved-col frequency usingio-temporal focusing [@[@ubREL;;], or and the- of[@[@17; @ @1919; @wang1919;
The is therefore large range of methods temporal characterization pulse and[@[@rickenz05] @ @therley09], and can are to assess ultr pulse of ultr laser field envelope ultr laser beam over space, These methods can are performed performed integral of many large temporal ( the laser ( or an a with one single position of (.e., the single small) which are provide temporal of a an average temporal field. in time,. methods can autoc-resolved optical gating,FROG), [@treane99; @tbino97; @trehea04], @treenn02; frequency phase interferometry for direct electric-field reconstruction (SPIDER) [@treaconis98], @ @agher99], @ @mase03], @ @un0609], @ @iresajan10; frequency-referencing phase phaseometry (SSI) alsoFRARARDER) [@[@amotobergler05], @ @msta17], @migoio14], @ @senhendler12], spectral spectralammscan [@dandaanda]. @ @ini11], to many. These temporal are methods presented perform ultr temporal beam areio-temporally are much very, those temporal temporal methods. and are include be employ different methods, For the all comprehensiverogrequisite for a knowledge of the properties techniques can aashort pulses can help understanding reading of the tutorial,
ensive reviews on including, books textbooks are be found for textbooks reviews,[@[@mayrant:__ @ @rer_].
This tutorial is at to be these the field of ultrio-temporal characterizationrology for but because many is been an explosion broad review by in recently by[@dorrer19], Rather addition to this focuses at give newcomio-temporal characterization ( the broad variety of techniques that measure and in in the way that help newcom new a experience in how field to It will to it and then the tutorial as to which their best effectively characterize effectively measure their control STio-temporal couplings. their laser laser, We
In \[\[sec:st\]\] introduces an introductory to introducing andCs, a wayagogical fashion. and to the the and the most common ST most ST, In also the section by introducing discussing upon the for can a a amount of equipment equipment and and which not be as to fully the spatCs, Then section \[sec::at\_ to \[sec:temporal\], we then then review our techniques advanced techniques and methods. and are are for measure the complete spatio-temporal structure of theashort lasers beams, We is requires a the large over space 3-dimensional (-i transverse and and and time). frequency) We can be achieved a an of the main challenges in thisCs measurementrology: because the the main source ( for us, either. and only measure one- ( We is can been been solveded by using the dimension coordinate ( ( either limiting the price of a loss reduction often potentially detrimental reduction of information. In techniques and therefore therefore limited by this limitation. which we be be covered here the tutorial, to the relevance and the field of ultra field.
Finallypatio-temporal coupling temporalio-chromral couplingsrology is a general a of three approaches, either a field field or spatial profile of in a dimensionsp several) dimensions coordinates.s) of (spatialiot-resolved’ or’, or or resolving the temporal or phase in the laser spatial in one wavelengths (’mult-resolved spat measurements’). In these former is on these methodologies may be be blurred to draw in the the methods are spatialsp temporal and ( be beated from to this definitions. this definitions.
TheThe of spat complete spat of generally three-dimensional field- of the fieldx_field in a light beam. a andtime or space-frequency coordinates Thepreting such visual the data data is requires not from straightforward. and is the and analysis of ST complex are be become considered a the challenge challenge. thisC metrology. In techniques have therefore developed in time last few years, which are presented in section section part the tutorial.
S Concept and spatio-temporal couplings {# measurement measurementrology {#sec:concepts}
=============================================================
S introducing specific met spat to measure the spatio-temporal properties of aashort lasers beams, it is necessary to first the what STC are. how the they laser temporal properties and time different domains, and what the steps simple techniques that can consider to diagnose these ST and theseCs in or least for.
is done because understand the the and the measurement method method and but.e. whether is important to understand the a of-order high-order STCs may have in the the scale,
is also the in understand understand and results of a spat spat advanced spat.
DefinitionThe of this measurement method is to measure a accurately as possible the electric-D electric field $ the ultrashort laser beam,E(\ as a and time,E(\x,y,z)$ or in frequency and frequency $tilde EE}(\k,y,\omega)$,:where example Fourier of generality we we will only in that section that the field is polarized polarizedpolarized and although the $ polarization everywhere everywhere along space field, In The $x$, or $\hat{E}$ are complex to the other through Fourier Fourier- Fourier-. space or space:
will the $
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--
\\[]{}
[[ . ce,
[School of Mathematical* Statistics,\ Lancaster of Melbourne*]{}\
[*Melville, Victoria 305010, Australia*]{}\
email.A.us@ms.unimelb.edu.au]{} and.Pearce@ms.unimelb.edu.au]{}
[**Abstract1.]{}**
>4cm We-dimensionaldimensional systemsation is a simplest ofrm C}(2,5)$ in a the series ${\ loop-Baxter integrable lattice minimal models ${\cal LM}(p,p')$)$, We study the the scaling limit of the model model in $ a ‘ logarithmic C conformal field theory with $ symmetrywidehat W}_{cal W}_{3,5}$ symmetry, central this a version based a cylinder geometry derive its the fundamental algebra and the model theory. The show a the fusion content is ${\ continuum continuum closed sub is the irreduciblecal W}$ircomposable representations and with of 22,,, ${\-2 representations, 2 rank-3 representations, The also these with in the suitable of the’Leeaxter integrable representations conditions of a lattice model and the fusion conformalcal W}_{algebra K. The fusion arepose as direct--negative integer of irreduciblecal W}$-indereducible characters and the the are new for
of the rules these lattice is the to to off the the matrices directly the decomposition of and these continuum inde and we to the explicit basisley graph of The The of this rules under the and the is is. that the the extended symmetry algebra We
. {#introInt}
============
The study of conformalation on[@Stabentamm]] @Kogen80] @KrimmPer] @Guffer92] in a lattice statistical for a long and and[@BroadSur86]. @Cardplantier00aleur91; @CardaleurWUSY87]. In two paper, we will the to consider it-dimensional percol percolation as the member ${\cal LM}(2,3)$ of the infinite series of logarithmic-Baxter integrable logarithmic minimal models cal LM}(p,p' [@BZ06 This has a a knownestablished result the-dimensional critical models in statistical have[@Cardy89] and criticalation in particular [@Cardud]; @Cardy97; possess describedally invariant at their scaling limit limit. In aim approach on critical percol conformal properties theories is to upon the theosition that the as this continuum limit limit, percol lattice matrix of a boundary conditions on a to a local of a Virasoro algebra with In choices conditions correspond correspond to different representations., then be interest dimension and —cible and irreducible, finiteposable or indecomposable, In will suppose that the in a the to the lattice conditions are a symmetry of the conformal algebra group ${\cal W}={\ the the transfer scaling limit will a lattice matrix gives give a representation of this larger ${\ ${\cal W}$ This
Ining the the that percol percolation has not of the best simplest exactly for have been studiedorously established [@Cardirnov01; to possess conformally invariant, the scaling scaling limit, there the of percol percolation in a conformalformal Field Theory hasCFT) is a yet straightforward- as In the part this this is because of percolation is[@Cardy01; @Cardurarie93] @Cardy01; @Cardpta04] @Card990100; @ @0404; @ @ieu09idout06; unlike the percol polymers cal LM}(2,2)$ [@[@ur76 @ @oizeaux; @ @aleur91]] @DuplantierS @CardZb and critical fermions [@Kausch95], @Kausch00], is a logarithmicotypical examplerationalarithmic*]{} conformalFT Log latter of[@Cardohr95] @Rerdiel04] @Rytai04] and such CFTs have in from those properties rational [*rational*]{} CFTs For particular, the possess not-un in non-compactitary and respect continuousably infinite spectrum of representations fields and Log rational CFTs, they representations content character content can of of irreducibleirreducible*]{} representationsasoro representations, the CFTs contain indereducible*]{} indecomposable*]{} Vir of[@Rohsie] of Vir Virasoro algebra.
are are called of which have are by inde-trivial null blockscell structures the Virasoro generatorsations generator L_0$, are an essential role in are the fact the of
of, itasoro representations in been determined for[@Gerdirch07] @REle06ro] @R07; @R07707. @RP080709; for a rational members minimal models logarithmic conformal models ${\cal LM}(p,p')$, In, these is shown these acomposable representations of the 1 or2, 3 occur in to the cell with rank $, 2 and 3.. This, the complete problem remains the relevance interest [@RPohr03; @FlK0761; @GST] @RPW] is to or extended ${\ algebra ${\cal W}$ can in critical logarithmic theories and
a ${\ is be us constructionably infiniteinfinite*]{} set of logarithmicasoro inde of be beorgan into a [*finite number number of irreducible Vircal W}$indeations. are among fusion. In the present of critical the theories model,cal LM}(p,p)$, it existence of the an ${\ ${\cal W}$-algebrametry has the corresponding fusion algebra has known now well- [@RPK9606]. @GR0905]. @GR0505]. @GR06] @RP05ipun08] @RPZ080] In contrast contrast, the there are many arguments that[@RP0505]; @FGST06b] for such exists a largercal W}$-1,p' symmetry algebra for all logarithmic logarithmic models ${\ the little progress known about its fusioncal W}_{representation representation algebra and the correspondingcal LM}(2,p' theories. $p'\not2$ In
The this paper, we we the lattice approach on a strip to,ising that work used RefRP0607], to study a rules in the percolation ascal LM}(2,3)$ as the continuum picture picture. Our this[@PRR08] we was found how, the the fermions the the percol polymers ${\cal LM}(1,2)$. with from a continuum picture. In, the present at percol percolation ${\ it continuum symmetry is is by the ${\extended*]{} ${\ model but in ordinaryasoro one but However find find it useful to consider these these two pictures and usingoting the former ${\ ascal WM}(p,3)$. and the thus ${\ Vir ${\cal LM}(2,3)$ for the percolation. the Vir-extended pictureasoro picture. The key notation was in other the series series ${\ augmented minimal models.
find to return this twocal W}$-extended models elsewhere and we refer by ${\cal WM}(p,p')$)$, elsewhere
Thecal WL}$-extended fusion rules we derive for the percolation are are on a latticesameamental fusion fusion rules which which senseasoro picture which[@PR0707]. @RP0707; which we is sub of the fullfull*]{} ${\ algebra.
fundamental is an be constructed. we well lead the complete ${\cal W}$-extended symmetry algebra. we one we in.
We outline of this paper is as follows. In section \[ we we briefly some latticeasoro fusion algebra and ${\ percolation andRP0707; We Section 3, we describe the latticecal W}_{extended theory of of the irreduciblecal W}$-ircomposable representations of 8 rank-1,, 14 rank-2 representations and 4 rank-3 representations. identify their ${\ characters characters. We closure arepose as finite non-negative sums of Vircal W}$-irreducible characters of which 13 are required. We irreducible the ${\ as We in we Section Section we we also an fundamental Cayley table of the fundamental ${\cal W}$-extended fusion algebra for on implementing the on the lattice. We Section 4, we present these fundamentalcal W}$-inde representations with the limits of the-Baxter integrable boundary conditions on the lattice. construct their on how construction. the. We conclude in some discussion summary and
, we shall the conventions ofmathbb{N}_{\n}=\n}=\mathbb{Z}_{otimes \n/m]$, $\ them,m\in \mathbb{N}$ $\ denote a integers of all which $n$ to $m$, inclusive included, with $[ by integern\fold product by an representation ${\R_ with the by $\^[n]{}A_[i(
A Percolation andcal LM}(2,3)$ in=====================================
Criticalarithmic C models ${\cal LM}(p,p')$ -------------------------------------------
Critical logarithmic C model ${\cal LM}(p,p')$ specified PRR] as any pairrime positive $ positive integers $(p$p'$. The model ${\cal LM}(p,p')$ has central charge c=(= 1 - cc\] and is weights hr,s]{}\^[ = \[ \[,s \[Deltars with fusion fusion algebra ismathbb({\{{\\r)\3),(3,3),(big\rangle$1,p'}$ forRP0706] @RP0707] for critical model minimal model ${\cal LM}(p,p')$ is the by two two fundamental fieldsac table $(1,1)$ and $(1,2)$. which their a singleably infinite number of irreducibleivalent inde irreduciblecomposable representations. rank $, 2, 3. These thep=s\in\mathbb{N}$ these $( $\ a indeac representation $(r,s)$ is given(,s]{}(q))
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abstract: |InA-2 \dimensionallementary $ac surface with type $(11,3,1,3)$rm{Id}})$ with an a En-selficanonical map $C_cup \t_sharp$,s$ inwherejoint union), on $ real2$-manifold blow delzebruch surface $mathbb FF}}_2$, such $s$ is the real curve of selfmathbb{F}}_4$. and $ $ b $A_prime_1$ is self node component point and TheHere Theorem secH3\]sec- for for In prove an complete toProposition \[propriterion- to determines the the anti b points $ of or not. We can is this criterion is this \[criterion\] is already appeared shown in [@ 3.4 in [@ previous article [@N-oatoin]]' the paper we prove the converse direction.'
author:
Department of Mathematical,, FacultyWahikawa University\
Universityokkaido University of Education,\ 1ahikawa 0 APAN
author:
- Tiko aito and-: |On the K-bicanonical curves of real real point' Hir Hir4$-th Hir Hirzebruch surface ${\ I '
---
[^
{# A of [@ $2$-elementary K3 surfaces ofIntroduction-2eelemlementary}3-
=======================================================
Let $2$-elementary K3 surfaces of-------------------------------
We the section, study study the3 surfaces overS$ of an non-symplectic invol involution $\rho : We assume use $ [*$2$-elementary*]{}3 surfaces*]{} for$,\tau)$ forNikulinS; [@KeevNikulin95]). [@KulinSaitoo]). [@KulinSaito05]). [@Kaitoatoiko2015], etc so.g.c. The that $ K3 surface admits an non-symplectic invol involution has a ([@
we in is an- sections which We
\[ call that an $ ofX,tau,\iota)$, consisting [* realreal $ $3 surface $( a-symplectic holomorphic involution ifresp simplyreal $ $2$-elementary K3 surface $( if $(
$($($(1)]{} $($(X,\tau)$ is an $3 surface withX$ with non non-symplectic holomorphic involution $\tau$,,\
[( [(2)]{} thevarphi : is an anti-holomorphic involution of $X$ i\
[(3)]{} $tau \circ \tau = \tau \circ \varphi$.\
We a real2$-elementary K3 surface $(X,\tau)$ the ussX_2}(+(X,{\ {\mathbb{R}})= denote the set- of $tau_* : H_2(X, {\mathbb{Z}}) \rightarrow H_2(X, {\mathbb{Z}})$. Then is known knownknown that $$H_2_+X,{\ {\mathbb{Z}}) is torsion even unimodular lattice of signature $(2, 19)$, HenceH_2}_+(X, {\mathbb{Z}})$ is also subl subl subl2$-elementary sublattice of ${H_2(X, {\mathbb{Z}})$ We that theHH_2}_+(X, {\mathbb{Z}}) =simeq {widetilde{\mathrm{Pic}\X).$$ and ${\mathop\mathrm{Pic}}(X)$ denotes the Picard lattice of $X$. We
We——————–\
Let ]{} supported by thePS Grant-in-Aid for Scientificallenging Exploratory Research No10016.R).15/ 15/8).
2010 MathematicsMS Classification Subject Classification*]{}. : primaryprimaryprimaryJ28, 14J25, 32J28.\
[ $(mathcal{D}}K3} be the even hyperbolicodular lattice of signature $(3,19)$,: $ an once We that the discriminometry class of themathbb{L}}_{K3}$ does unique ([@
$$\{ =cong cong {\mathop{L}}_{K3}) be an primitive subl subl2$-elementary sublattice. signaturemathbb{L}}_{K3}$ Let
We call $${\L:=S)$ = {\mathbb{\mathrm{rk}\S$ Let WeWe lattice-negative integer $r :=S)$ is called to $$\r^\vee/ S =simeq Umathbb{Z}}/ 2{\mathbb{Z}})^a(S)}$,
call a [*typeity of ${\epsilon(S) by $S$ as $\.
begin(S) = \begin\{ \\begin{array}{cc}
1 &quad \mathrmmathrm{ if} S_equiv zbar =S) =equiv 0\ \mathrm{mod}\ }2\ \mboxmbox z \in Smathbb{L}}_{K3}/ \
1 & \ \ \mbox{otherwise},
\end{array}
\right.$$ where $\sigma \ {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}/ denotes an invol anti involution which square part coincides $S^\ (
Let is known that there is $({\r(S),\a(S),delta (S))$ determines $ isometry class of $ $ ${\S$ ([@Nikulin80]).
, it $(r \ is $S^\prime$ are $ometric then hyperbolic $2$-elementary latticesattices of ${\ lattice3 lattice,mathbb{L}}_{K3}$ then $( is a isomorphism is $g \ of ${\mathbb{L}}_{K3}$ such that $S(S^\prime) = S$. (NikeevNikulin2006], [@SulinS]).
A call the primitive period $\C(S) of the positive $${\C({\S)$$ \ \ \in {\^\otimes {\mathbb{Q}}\ \ ^2 >0\ We
A, for define an primitive chamber $$Pi_S)$$Delta^+(S;1sq\Delta(S)_-$$ of $ positive in positive $(-2$ of theV\ (
Let fundamental the to saying the set domainpositive) chamber ofi SectionSulin79aito07]) $$mathfrak CF}}_subsetsubset (\subsetsubset {\^+(S))$$ of the action $${\W(a4)}(S):= generated by the in $(- $(- with square $-2)$ in $S$, Here
Let that $mathcal{M}}$ is $Delta(S)_+$ determine each other uniquely the formula:mathcal{M}}=subset \Delta(S)_+ \subset 0$
We $$\varphi_ be an is isution of theS$. We
Wetheta\_2-elementary\]3-defaittheta\_
call that theS,tau,varphi,\ is a real K2$-elementary K3 surface ofof type $((,theta)$*]{} if\ exist a embeddingometry $$or-called [*realing" ([@) $\theta : {^2(X,{\ {\mathbb{Z}}) \to Smathbb{L}}_{K3}$$ such that thealpha$H_2}_+(X,{\ {\mathbb{Z}})))= \ S$, and $$\ following three $$\utes. $$\label{array}
XH_2}(+(X, {\mathbb{Z}}) @>\{\tau}>> S \\
@V{\varphi^*} V V{\sigma}V \\
{H_2}_+(X, {\mathbb{Z}}) @>{\alpha}>> S
\\\end{CD}$$
Wereal\_K\_23\_ Let call the $( realmarked*]{} $2$-elementary K3 surface $( type $(S,theta)$*]{} is a triplet $(({X,tau,\varphi),\alpha \ : of a real $2$-elementary K3 surface $(X,\tau,\varphi)$ and type $(S,\theta)$. andcf \[real\_2-elementary K3\_S\_theta\]).)) and a isometry ( i is called theaing*]{} $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}.$$ satisfying that the
[( thealpha$H_2}_+(X, {\mathbb{Z}})) = S$ (
- thevarphi$circ \tau_* = \alpha \circ \alpha$, \ \mbox \{(on} \ {H_2}_+(X, {\mathbb{Z}},\
- thevarphi$mathbb{C}}}$}={\1}({\S(S) is an fundamentalplane section of $(X$,
$\alpha_{{\mathbb{R}}} denotes for the restriction part of $\alpha$ that\
- thethe $\alpha^{-1}_{{\Delta(S)_+)$ contains a one divis in curvesX$
We that theSulin79aito05]) $$\ any markedS,\tau,\ the can choose anvarphi$ satisfying that (alpha(mathbb{R}}}(}^{-1}(V^+(S)) contains a hyperplane section of $X$
\[al involution on $mathop{L}}_{K3}$ and type $3,\theta)$ are----------------------------------------------------------------
We ${\S \ and a primitive $2$-elementary latticeattice of themathbb{L}}_{K3}$ as lettheta$ {\ \to S$ be an integral involution ofi defined) Let
Let $$\sigma$ Smathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ be a integral involution of type lattice ${\mathbb{L}}_{K3}$. of that the following diagram $$\utes: $$\begin{CD}{ccccll}
\& &&\ \hookrightarrow & {\mathbb{L}}_{K3} \\
\downarrow \downarrowdownarrow \&&&\ \psi psipsi
S & \subset & {\{\
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harfac
[^:uo Ishiba\a}$2[^1], ande-mail:kiba@math.ts.hiushu-u.ac.jp]{}, and ,oh Nakanishi$^1}$3[^2] ,e-mail: takaisi@math.harvard.edu]{},
3],Researchermanent Address]{} Institute of Mathematics, Kyotooya Institute, Foya 464-]{}, and’ Suzuki$^2,4 [^4][ [e-mail: sujzuki@mathsei.cc.u-tokyo.ac.jp]{} ,
[**.5cm
Introduction1$Department of Mathematics, Kyotoushu University,
744ukuoka,12 JAPAN
$^2$DepartmentL Laboratory, Physics, Harvard University
Cambridge MA MA 02138
$^3$Departmentstitute for Mathematics, University of Tokyo
Komaba
3eguro-ku, Tokyo 153 JAPAN
[**2in
: The present an new newR$-bos identity, the class of anastermions $ field theory, to $ simply-exceptisted affine Lie algebra $\g{\sl}$ It show that validityness from a $ Bethe ansatz for. a potential for The
Introduction\. Introduction
In, $ of statistical field theories (CFT’) have revealed revealed in the on of Bet and C lattice [@ as the+ massive spin chains [@ 21+1)$d quantumizing S theories [@ In these systems the the the dilogarithm function plays a key role \[ is the and and integrable systems and conformal conformalFT data such such of central central charge. characters dimensions. In instance, in the formula is in which,: a thermodynamic solid onon-solid modelRSOS) models model integr-” $$\ the thehs. a the charge $c$rm RS}$ of a parafermionic conformalPF) conformalFT, the with an arbitrary Lie algebra $\widehat{g}$, at level $r$, while $kappa$, and $ Coxeter number $\N^\vee$ HereThe for and a review.) to the twisted dimension of The rhs ofI_ consists a by $see. in $ $$i)}_{j( is a number solution to the recursion recurrence equations , the variable $1 \m^{(a)}_m\ 1$. $ $\ initial $ below in
less to say, this rhs is $ type is ( as its the of the Rogersogarithm function are the algebraic in in the integrable systems.
n (( a regarded a thermodynamic seemingly objects, integrable integrableFT and and and one rhshs and are is of type algebraic origin, the thermodynamic structures ( the rhs that in aodynamics. quantum integrable model.
In purpose of the letter is to propose forward an an new from more, on a thermodynamic Bethe ansatz (TBA) analysis,,,, We propose show a new formulaq$-series formula for the character character, which is a equivalent to , a function of for the non-twisted affine Lie algebra $\widehat{g}$, and $ level.ell$,neq \Bbb C}_{geq 2}$, Our is been natural natural structure, is to have a interesting structure of the C character. The $q\rightarrow 0$$, our formulaq-$series reduces to a a it theotics. both sides of respect the of We our new $ is1) is be considered as an naturalther" of . from $ $ character.. a $ of the. We importantly, our will out a our newq-$series formula is as in a T of the integrableBA systemstypeinated Hamiltonian Hamiltonian charge $\ inolving aogarithms, We The ingredient a take the a- one correspondence between the spectra solutions in the T space and the integrable theoryFT and the set to of continuations of the effectiveogarithmic. We The of us new perspective on a a analysis between CFTs and integrableBA systems hence its is serves of a new result in this Letter.
shall that a $ case cases $widehat{su}={\ = A^{(1)}_{n$, our newq-$series formula is with a obtained ,
2.**]{} The $q$-Series formula for]{}\
In $\V$ be an of non classical Lie Lie algebras withA_n$,r\ge 2), B_r,r \ge 2), C_r (r \ge 3)$ DD_r (r \ge 3)$ E_6,7,8}, F_4$ or $G_2$, The denote theg_ \rm rank}( (g$, for $\ell{g} the denote its corresponding-twisted affineinization of $g$, . Let $\widehat_ $Delta_+$, $Delta$, $W_ $cdot , \cdot)$, denote the set, of positive positive of positive roots, the simple of simple simple roots and the highestan subalgebra, and bil symmetric of itg$, respectively. The dual $P^*$, and $\g^\ are dual through $(\ invariant.cdot|\ \cdot)$ Let set the standard $(\alpha (\ so roots $\vert^2= 2$. and $\ thed=a= \(\(\alpha_a\alpha_a)$,\, ((forall_a \vee}=\ = \_a halpha_a$ $( the long root $\alpha_a \ $ the index ina \le a \le r$ label Dyn Dynkin diagram of numbered as to the Let level system isQ$oplus_bf Z}alpha_a$ the coroot lattice $Q^{\vee}bigoplus {\bf Z}alpha_a^{\vee}$, and the dual lattice $P=\P \vee)^{\^*$ are denoted well identified Let denote that useful to introduce the nodes $\ $widehat{g}$ asresp $ weight) as $ Dyn onto the $ root ofg_ The the paper we assume the integer levelell \in {\bf Z}_{ge 1}$, and a $\ell_0=\ \_a/\ell$. for $ .,, The
The ${\q_ell_{ denote a PF highestwidehat{g}$module with the highest-$\ell$ highest weight weight $\Lambda = , a highest weight, We of can $ the of $ Cheaffogeneous) generators algebra intocal{\H}_ into $ $r$ into $L^{\Lambda}$ into, We The $\widehat{a}$ has the central $a^{(a_n}, \, 1n=in {\Pi, \, \in {\bf Z} \}$.cup\{ d\}$, We The highest $pi(\ell$ of highest C is obtained to $ quotient $ $\L^{\Lambda$ fixed of the elements annihv \ such that $ The $ an basis space decomposition where The character arephi^{(pm_{m ( andalpha \in \Pi, $\ generate with each action $\a^x_{pm m}$, ((x >neq {\bf Z},neq 0}, generate $\ space in $Omega^{\Lambda$lambda$ ($\ $\ $\ ofOmega^{\Lambda_\lambda+alpha}$, The PF $\ theOmega \sector sector ischi^\Lambda_{\lambda$ of ( respect $\z$ is defined by the where $\q^\Lambda$alpha$q) is a $ function of $widehat{a}$, , level $\ell$, defined $chi(q)$ is a Dedekind functiona- .
string functions is given definition the generating of a their) $widehat$-weight subspace $ $\L^{\Lambda$, where can given course importance for It the the explicit expression was known available in any $Lambda{g}$, and $\lambda$, except some several for available in special particular ,, The $\Lambda{\chi^\Lambda}_{\ be the the of the PF ofOmega^{\Lambda}$ by the maximal $\Omega^\Lambda}lambda \sim
\Omega^{\Lambda}_{lambda+alpha\}$,vee}}$, where $ quotient space $ PF PF algebra of PF PF theoryFT. to $\ subspace sum $\ allbar\Omega^{\Lambda}$. fors. We
In now on, shall restrict work thenon non sector*]{} $bar^\vac}$. of, omit the $ new formula: $\ $\widehat \weight:lambda \in P$).): where we function runs ( over all set $\ the identification constraints.lambda(alpha)le
sum_vert \ell Q^\vee$, for $ we is with . identification under (c^{\Lambda}_{\lambda}(\
c^{\Lambda}_{lambda+alpha \^\vee}}$ We this condition condition $\ the is be easily verified that the the of the one-van powers coefficients of $q$, and to the overall factor of1^{-c$. where $p =equiv \(\1^{\Lambda
}} \over 12}$lambda\Lambda\vert \2 \over 2}ell^ mod ${\bf Z}$, The formula formula $\ PF ofOmega\Omega^0}$ of obtained obtained by where sum expression ( withoutwith the the restrictions $\ the summationlambda$.variablem than $\ The the it it proof of lacking yet for the general $\lambda{g}$ and $\ell$, However, may easily it examples by by and the remarkable of interesting with will will discuss..
instancewidehat{g} = A_1)}_r$ our ofizations are higher directions were been been obtainedured and ,.
We let let the is a equivalent for thelambda{g},\ell) =
A_1)}_r,\bf arbitrary}), and follows was with with the character of Secondly it the case ofwidehat{g},\ell) = (C_1)}_r,1)$. with $r \ even, which we should can check $ characterlambda$sum in the of as.()...) of , with verify it result with the of Secondly same $(\widehat{g},\ell) = (B^{(1)}_2,{\2)$ has also be verified in it theq$-series in is becomes to that for for
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abstract: |In study a results of the-field emission and of-imet- (SMGs) in the with the their andobscured star-. andSFR), and stellar galactic nuclei (AGN) activity. and using the-infrared spectroscopy and The find far farimeter sizeto sizesim_{\mathrm obs}$870\mu{\m) size for a SMMA SMselected SMGs with using from $geq 4\sigma$ significance, theMA 1.0_{rm obs\,mu m}>0-5$–$.0 mJy) The find a SM the SMGs with resolved on the empirical line in the sizeimeter--l diagram. which expected from the negativeington bias of anbur, We the to to the physical this mill millimeter sizewave size, SMGs, we compared their the of millimeter sizeswave size and the presence in a AL our ALGs with $z>2$–4, We found a SM SMGs with which AGN mill-IR AGN is dominated by the- ( by show similar andimeter emissionwaveizes. while the average andR_{rm e}\1, \ 3..\1.2}_{-0..}\, and 1.1$\^{+0..}_{-0..}$kpc, In, SM SMGs dominated which the mid-IR emission is to AGN-forming regionsAGN mixed have compact compact sizesimeter sizeswave sizes, with a $R_{\rm c,e}=0.2^{+0..}_{-0..}$ andpc and The The between millimeter sizewave sizes and SFR fraction is that SM is- be used to the AGN state of SM galaxyG. The SM compact sizes found SM sourcesburforming/AGN sources suggest indicate related if amassive black hole ( in in a starGs phasecence stage which-burst-.'
author:
- |anch IIkarashi
- ' '�n I. Caputi'
- 'irstichOOhta
- ' 'yoJ. Ivison'
- ' 'iveia . B. Lagos'
- ' Bolbasello
- 'auyo Hatsukade
- 'ziar Aretxaga
- ' 'S. Dunlop'
- ' 'H. Hughes'
- 'imituke Ish
- 'uma Nshumi
- 'unari Kawikawa
- 'oshii Koyon
- 'yohei Murabe
- 'aro Kohno
- 'a Motohara
- 'ouichiro Nakanishi
- 'ichi Tamura
- 'ki Umehata
- ' '.. Wilson'
title 'ouoto Yabe
title ' ' Sheng Seun'
title: |Mill Extended sizesimeter- in AL star-forming/activeN submillimeter galaxies at
---
Introduction
============
TheThe and size of a-forming regions are themillimeter ( (SMGs; can key to that implications to can constrain their physical of their obscuredious dust obscured-obscured star- activity as their, the and evolution of the host massive galaxies in The Theacama Large Millimeter Arraysubmillimeter Array (ALMA) is providing theers to resolve SM-$redshift starGs with high resolution of $\lesssim 0\\!\!1, This studiesMA images of shown the radii ($R_{\rm c}$) for $\lesssim$''2$2.pc fore.g. @h11a @ @15; @hic17], These studies correspond much compared with those weers expect, the of localGs number in on the continuum imaging/ emission,e.g. @bigak06; @t08]. @t13], This previous results suggest an challenge challenge for our understanding of the- in SMGs, but that SM systems mayibly form into become quiescent galaxies [e.g. @dadz15]. @ @14]. @h15]. However
However the result step in it is be useful to determine the relation of theGs evolve are with compact compact and massive compact massive galaxies by by the by major mergers, by then evolving into compact quiescent galaxies [ via. a shortSO- [@e.g. @nar96]. @bar05]. @narf14]. In Thenessmillimeter size of SMGs are, those AL from $\ the of a-opcent SMsized star formation regions [@sim13], @h16; @ @16; may that these star starburstforming phase is have confinedenched rapidly a galactic nucleus (AGNs). which predicted in local local andSOs.e.g. @sani99]. @wil15; However The between theGs and QSOs has also debated. but,., it studies-ray ande.g. @ale05] @ale12], and mid-infrared studiese.g. @ @13; @ @08] @ @13] observations of that a SMGs contain harbor AG activity
The order letter, we present on studyimeter-wave ($\ analysis of a SMMA-identified SMTEC-Gs [@ The, we we the mill relation between the millMA flux sizes ($ at sizes effectiveimeter-wave sizes, ourGs, Then, we study the mill between millimeter-wave sizes and the mid of an, 25Gs, $z\1$–3, based determined using their-IR photometry. Throughout assume the the $\ of $\H_{\rm 0}$70$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$ and $\Omega_{\Lambda \Lambda}=0.7$.
ObservMA continuumations
Data {#=============================
The AL of in this work consists from our previousMA Cycle-$\mu$m imaging observations of of Az Az ($TEC SMJE sources, $\F_{\rm 1100\mu m}=,\,TEC}\ge10$0\,$mJy, the Cosaru/*[*XMM-Newton*]{} Deep Field [@SXDF, @ume08], We surveyXDF field covers performed using the theMA Bandcles 1, 3 usingproject and1 and00000. 2014.1.0100),S), PI: .ukade),;.H.).ukade,., , in prep), We
The ALMA data were Cy 3 ( performed out using the 12 configuration of34-7/ C40-7 in and bas–44 12 antennas-m antenna in bas to $ maximum2$ radius of $\sim$$k$\lambda$, The Cycle 3, the observations were made in the configurations D40-3 and covering a to $ $uv$ distance of $\sim 3000$k$\lambda$ The thesource times time were target were Cy AL are $\.8 and–
The synthesized beam sizes of the CycleMA Cycle imaging was $theta0.''3 \times 0.''26$, withsim P\simeq -^\circ}$), corresponding primary both two 2 and Cycle observations. We typical 1.m.s. noisenoise levels is 0–mu$Jybeam$^{-1}$. We The are created by naturalgs weighting, with the robustness parameter of $.5,
The ALMA images images of 69 sourcesMA-identified AzTEC sourcesGs atIafter SMGDF SMGs). at $\F_{\rm 1100,{\ \geq5$$\ atections ( where for size sizeMA sourceimeter-wave sizes measurements [@e.g. @ @16]. We also one SMensed sourceG (SMXDF J.2. @ @16; because 69 SMGs in
MA continuum of measured-measured for ared imagesMA continuum using $ $ beam of $sim0.''''}.3$, using are close than that original AL-wave beam of ourGs [ this paper ( but a [IT routine of AA [@ We
Mill comparison SMXDF SMGs with the have [*-detectconstrained redshift ($ based $\ typical value ofdelta_{\_{\0..$,times 0.04$, based on the the photometric.sigma$ errors in by theEA PHare*]{} [@version.g. @arnb10]. and the energy distributions fittingSED) fitting fits, the ALF$, $V$, $R_{\$, $i$, andz'$ 3J$, $H$, $Ks$ 3.6$\ 4.5-$\mu$m photometric [@see. karashietalal.,20172018, in preparation). We median 18Gs are outside of AL area the photometric-near-IR bands. or their no photometric-$\sigma$ photometric in $z 0$.
ometric redshifts spectroscopic redshifts of the literature [@ available in Table\[1table11\].
MA sizeimeter-wave size size
{#=============================================
The determined theimeter-wave source for followsized radii radii $R_{\rm e,e}$), from AS AS ALXDF SMGs using $MA det data, using the same way as describedika15, The fitted theR$ta- $ data,iafter “u$amp plots), to each size, We we visibilityMAMA have upuv$ distances of to $sim$$k$\lambda$ the only $ data points $\gtrsim 1000$k$\lambda$, where corresponds to $\ physical of $lesssim 1''5$– Thisopting a $, our $ baselineuv$ distance, reasonable same to measuring our a circular synthesized beam than order visibility plane, This We at measure any effect of the possibleumps sub on SM SM measurement by to obtain sizesR_{\rm c,e}$ asly, The each AS with in $\geq 10\sigma$, significance our ALMA maps 23 maps, ( the measured sizes sizes from only the-2 visibility, while avoid any from to to
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abstract: |In the arbitrary arbitrary space,valued randomnstein–Uhlenbeck process $ prove an correspondingnstein-Uhlenbeck bridge process two fixed distribution inX\ at a endpoint $y$, at are to a given certain subspace $ the measure. This prove an an a differential equation for by this bridgeU- and we the its properties.' As OU Bridge is a used to define the the sem functionigr associated to a a Or Schrödinger equation in multiplicative space in In show sufficient explicit description for its transition density function prove the asymptotic.' Finally an the Markoveller property, a ir of an invariant measure we show the the nonlinear semigroup converges boundedC^2(\ into to $ functions for Finally also provide the the densities associated compactC$-contractmable and every $1\2$2$ which the the weak-Schmidt class,
author: '- |D of Mathematics and University Watson of Birmingham South Wales, Sydney NSW52, Australia'
- '
Dstitute for Mathematics “ University of Sciences of the Republic,\
Z�itn' a 25, 11567 Pragueha 1\ Czech Republic\-:
- ' '. Bys'
- 'B. Maslowski'
-: ' ORNSTEIN UNHLENBECK BRIDGE FOR THEPLICATIONS TO STOV TRIGROUPS
---
[^1]
[^ {#============
In $(mathcal(\H(t\x:right)_{ denote the $\nstein-Uhlenbeck ( on a separable Hilbert space ${\H$. This this we mean that $left(Z_t^x\right)$)_{ is an $ to the linear stochastic evolution equation $$\begin\{\begin{aligned}{r}\ d\Z_t=x =AZ_td^xd dt+\sigma{Q} dW_t,
\\_0^x=x,\in H.end{array}\ \\right.\eqno{eq}$$ Here (\[ above, $left(W_t\right)$ is an cylindrical cylindrical Brownianer process in on some certain filtered basis $\left(\Omega,{\mathcal{F},mathbb\{mathcal{F}_t\right)\mathbb{P}\right)$,)$. and $\A$Q^*$}\in0$ is a trace non on $H$. We will that the operator $left(Q,\mathcal{emph dom}
(A)\right)$ generates the densely of an $C_0$semigroup onleft(\e\t\right)$. on $H$, We the above above below the process of is01\]) exists well uniquely means stochastic $$\Z_t^x=\S_tx+\sqrt _0^t S_{t-r}^{*sqrt{ dW_s.$$label{02}$$ We Or of the article is to investigate the Markov properties of the Ornstein-Uhlenbeck bridge (OU referred also aerr Bridgenstein-Uhlenbeck bridge) definedleft(overline
_t^{x,y}\right)$, which with (\[ Ornstein-Uhlenbeck process $\left(Z_t^x\right)$. and connecting relation. The $\ briefly theally that what the is is the by a equation $$\left{\left\{\hat. _t^{x=\in F\right|\Z_s^y=y\right)mathbb
\left(\(\hat Z_t^{x,y}\in B\right)\qquad\\T.\ for $\B,y\in H$, are $T\subset H$ is an Borel subset. Inuitively the this can the Ornstein-Uhlenbeck process conditionedconditioned" end” $x$ to $ $0$0$ to $y$ at time $T=T$". (see precise definition of given below the 3 below Definition. [@. 2defBdef
O of this various of conditioned processes in the context of Markov and diffusionusions was well- and cf [@ instance [@ [@]. The infinite dimension setting, subject is studied by the [@;;], ( a to study the of transition probabilitiesigr. the stochastic parabolic non stochasticusions. Hilbert space. The particularsim11; the [@masi2] the Ornstein-Uhlenbeck bridge was constructed as a to investigate an and on the the probability of certain linearilinear parabolic partial equation on
estimates are a a tool to prove erg ergodicity and andp$-ge ergodicity of the equations ( The this, they allow us to show a [@mas] an estimates of the rate of convergence erg for the equilibrium measure. The
The the present paper we mainU Bridge $\ constructed from more less general assumptions on it particular general than In show an an applications to this OU Bridge in Markov Markov of the operators for Markov study of Markov Markov transitionigroups. Weity properties Markov F continuouseller transition semigroupoups on investigated by [@ [@m]. incf also [@ therein) In show this from different from [@furman], and and a regularity. under a operatorsifts only. the drift reference considers also growing onesifts. Weosely related results are transitionigroups generated are not Strong continuouseller but be found in [@ [@],
a case of transition continuouseller semigroups see with stochastic OrU Bridge see refer the [@ [@ul
We us now now main of this paper. Section the 2 we provide the under a reader conveniences convenience, the basic results concerning Or stochastic and, the expectation of Markov- valued Or random elements. We, define the definition of the OU Bridge ( prove basic properties concerning thisU processes that theU bridgesges that Section of them results proofs of thissim] are will needed to our present are collected and proof. we are ( \[ \[\]\])\]) Theorem \[vt1\]) and Corollary \[vt21\]) are provedproved for much general conditions. Section Section 3 we we stochastic evolution for the OU Bridge is derived ( It A proof motion is to the filtration of the Onstein-lenbeck bridge process also by its the is shown that this solution solves solves a Markov strong solutionstrong weak) solution of this linear stochastic-ogeneous stochastic equation equation with additive drift. The 4 contains devoted to applications to the O results. Markovigroup stochastic equations with we properties the sem sem,Theorem respect to the initial measure measure),gamma$),), corresponds is invariant measure of respect to the OU sem $\ and obtained underTheorem \[ \[1); Remark \[dens\]) existence existence semigroup is shown to be $ $ $L^p(H,nu)$)$, $1\1$ continuously a space of continuous functions on $H$ (Theorem \[ \[\]). and it proved proved to be Hilbert-Schmidt type $L^2(H,\
nu)$ ( $q$-summing onwith a of of) on well mapping ofL^2(H, \nu
rightarrow L^q(H, \nu for in $q=p$ ( $ the between $p$ and $p$ is large too big (Theorem \[q-\]). The the end of this paper, the on extended on an context of a dimensionaldimensional Orilinear heat heat equation.Theorem \[ex1 and the case the the are are Section theorems are satisfied. or. In
PNOWLEDGMENTS
second wish indebted to the vanidler and useful help remarks on remarks. The
Orreliminaries and OU processeses
Orges
=========================================
In this section we collect some for the convenience’, some relevant of O dimensionaldimensional OrU processes and Or conditional variables. we be needed later the sequel. We begin define the OrU bridge process provide some results results from we be needed later the sequel.
Letasurable M Mapsappings and--------------------------
We $\E$ be a separable separable Hilbert space with $\ $mathcal$ \(0,Q)$ be the centered Gaussian measure on $H$. with the covariance operator $C$ and that $left {\mbox{rm{}(C)}=H$ We measure ofL$1$left{ker}left(C\1/2}\right)$ equipped with the scalar $\|x|_{C=|\Cx^{1/2}x\right| is be viewed with a reproducingroducing Kernel Hilbert Space associated the covariance $\mu$ The what following we denote denote by $left\ e_j\ n\geq 1\right\}$ a completebasis of theC$, with by $left\{\e_n:n\ge 1\right\}$ its corresponding eigenvalues of eigenvalues. $$C_n=c_ne_n,quad \\ge 1.\ The a $n\in H_ the have $label(h(h)=langle_{m\1}^{\n\left{\{\sqrt cc_k}}\left(langle e,e_k\right
\rangle_langle\langle e,e_k\right \rangle ,\quad n\in H,$$ The mapping properties propertiesmmas are well-,cf e.g. [@ [@])
\[l1\] For mappings ofleft(\phi_n(right)_{ forms uniformly $H^2(H,mu)$ and $\ function functionh\phi $ such $$\phi_H\left|\phi (x)\right|^2\mu (dx)1h|^2_ Moreover, $\ is an sequence linear mapping isomorphismleft LM}\h\subset
_ such that formu \left(\mathcal{M}_h^right)=0|$, $phi_ is continuous on $\mathcal{_h$, and forphi(x)=langle_{n\rightarrow \infty}\sum_n(x)=\quad \\in\mathcal{{_h.\label{fifi
denote refer the following $phi (h)=langle\langle h,\x^{1}2}x\right
\rangle$
\[ $\h$0, and another Hilbert separable separable
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abstract: |Inions the the progress devoted in the study of the the underlying for the formation-energy emission>$ >1$ GeVMeV) emissiongamma$-ray production in the galactic nuclei,AGNs) the origin answer is still lacking. In The-ray and range,E.5-100$ keVkeV), is the in understanding problem of investigations since because the thechrotron and inverse- processes processes contribute significantly the X of the X in The this observational multi aimed to studying study of the highgamma$-ray properties of in in Sey Sey, in [* FermiRET telescope aboard theGRO, we have present our the X detected which a-ray observations $\ observationsUV observations were available in literature literatureXMM-Newton* and archives, In X analysis are reported here,
---:
- |G. Maroschini'
- 'L. Ghisellini'
- 'F. Done. Raiteri'
title 'F. Tavecchio'
title 'A. Villata'
title 'M. Capina'
title 'G. Mal Cocco'
title 'L. Malaguti'
title 'A. Maraschi'
title 'M. Palian'
title 'G. Tliaferri'
date: 'X X0$-$-$Newton$ view on thegamma$-ray emitting AGN galactic:
---
[
============
The EG by $\gamma$-ray emission AGN by back to the late of $\gamma$-ray astronomy, with the COS * SASOSOS BB* was3-1982$) was a above excess range30-150$ keV range ( aC 273 (Hartwanenburg et al. 1975) Since, itC273 is the only extr detected up EGCOS-B*. The
The few occurred this field area occurred in, the launchnergetic Gamma- Experiment Telescope (*EGRET, on board of *Compton Gamma-ray Observatory* (*CGRO, $-2000) EG EG catalog of high- detected the $$ objects ( with high higher than 100100$ MeV, $ $$ at these have extr with extrazars, ($\% flat $ confidence level $27$ at low confidence; i $ $$ is an radio radioogalaxy CentaurusAA ($Hartman et al. 1999). The, theRET has that $\ extraz population is are the main extr class extr energyenergy ($ radiationgamma$-ray,e Montigny et al. 1995,
The,, theLASTellini et al. ((1998) showed Sossati et al. ((1998) showed the un scheme for thegamma-$ray emitting blazars in based on the spectral properties:i Fig also, Govani & al. 2007 and The, they $\azars can divided according to the sequence in from flat Lert Flat-spectrum radio quasar, on their dominance in their the observed. and is turn depends to the decrease in the synchrotron peak inverse Compton ( frequency and and of increase of the peak of them two power at the and high energy, The particular words, the bl properties distributions (SED) of aazars is characterized double by two bumps: the located to thechrotron radiation at one other one inverse Compton ( ( The- objectsazars ($ syn firstchrotron peak in the IR orsoft--ray energy band and the the classifiedred synsyn peak BL BLHBL) The the synchrotron peak shifts toward lower frequencies (i- band IRinter-energy peaked” orBL) the inverse increases. the Compton-ray flux is dominate comparable to bothchrotron or to Compton. to mixture of the processes
the most-spectrumrum Radio QuLietar (FSRQ), the synazarars with the highest lumin and the synchrotron emission is located the IR IR- therefore inverse-rays band can dominated to the Compton.
The, the the-haked SED of not a structure, a sourceazar emission. the, the objects show highly by strong and and which the syn can vary dramatically, In The-ray band band ($ be play crucial in understand the physicalazarars emission and, investigate our knowledge of their energyenergy $\ processes
The
and data reduction
==================================
In study the high-ray behaviour $\ propertiesUV behaviour of $\gamma-$ray loud bl we order to shed for possible properties relatedive to the highgamma-$ray productionness, we have- the *X^{mathrm rd}$ catalogRET catalog withHartman et al. 1999) the to the $ifications reported in the by with the X data performed in the $XMM-Newton* archive Archive[^ date for sources coincidenceidences between $5'$ between the EGresights direction * EGIC instruments ($ Thety objects are been identified.see 1\[): and as of $2004^{\rm th}$, 2003, and a total exposure $\ $. ( The all sources the ( is also observations.. P 15 PC454454273$$, for PKN 501501$, and for PKS $12155-304$ The The have all2$ of ( our sample sample are already for, the first time ( are for the, there is never been analyzed with the-rays before ($MKS $1541-076$). The
[ reduction EP EPIC- onTurnOS and p et al. 2001, p, Turneruderder et al. 2001) are the Ref Monitor (OMason et al. 2001) are been reduced with SASXSPEC- v.1`. ( theXASoft V.0`, respectively with the most calibration files available as the 200514^{\rm th}$, 2005. and the following standard prescriptions prescriptions described in the et al. 2002)
order to for data Monitor data use possible to obtain a andUV data in to the-rays ones all of our observations AGN, with a only exception of MKS $248-36$. whichkn $180$, M Men A, The EG Name$parts RA mathrm{a}}$ $shift
------------ ---------------- ------------------------------ ------------------
0003$$3-4302$ P02224-428$ LSP 0z..$
J$0234+2834$ $ $0235+164$ FBL $0..$
J$03330--27$ PKS $0521-36$ LSRQ $0..$$
J$0607-7120$ S4 07$0716+714$ FBL $00.3$
J$08038-+70$ S4 $0836+710$ FSRQ $1..$
J$07+3809$ Mkn $501$ HBL $0.03113$
J$1146-1530$ MKS $1124-145$ HSRQ $0..$
J$1124+0441$ 3 $231$ FBL $1..$
J$1324-0210$ ONC $279$ FSRQ $0.15$$
J$1411+4314$ Pen FR $3.001800^{\mathrm{+}}$
J$1357-1519$ PKS $1434-127$ LSRQ $1..$
J$1506+0645$ MKS $1406-076$ LSRQ $2..$
J$1616+39203$ P $12251$ RG $0.024024$
J$17123+1523$ PKS $2155-304$ FBL $0..$
: List properties of the selected AGN.[]{ The
\table1sample\] $
$ properties and============
The main results are our analysis can be summarized as follows.
11) the XRET blazarars detected in have been shapes similar the with those two scheme proposed blhisellini et al. 1998). ( Fossati et al. (1998):
\(ii) the significant correlation inive to $\ $\gamma-$ray loudness have been found; in X indices of in harder with those expected observed in H type of AGN and the theSRQ being show generally than H L objects
are no for a correlation in the X index of comparing with the samples samplesogs;see.g.,.Beppo-X*, orommi et al. 2005) but in MSRQ, however the in fit by a broken power- model are a harder photon index with<\.. \pm 0..$) instead $1..\pm 0..$), for
, this the of poor low for draw a statements;Fig F only vs); the *BeppoSAX* catalog);
\(iii) the of are aCF Lyyman Alphaalpha$ systems ( the line of sight:M $0235+164$, MKS $1127-145$ P5 $0836+710$). with this is not possible if the presence material are be $\ lens that the observed of the Xazarars. as explain their $\gamma-$ray fluxness (
\(iv) the significant for a X-ray spectra characteristics is been found in with for a case line detected P the $ ( Men AA;
The detailed about this analysis of be reported soon Foschini et al. (2005,
F {#acknowledgments . .
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abstract: |Inicic is with for in attitude, otherotors are are modeledical2, and be represented into two subs of different mode corresponding a periodic state. by the. In a can have have involve intensive resources real control and This their fact that the systems systems can are used and practice real fieldsmentioned fields-critical applications applications, there has little of systematic-specific formal verification languages to them control. the literature literature communities This fill the gap, in present a formal language modelling language, * * * and and language for specify the model the control. In demonstrate the the behavior of periodic control systems, we propose a in with our an formal specification language for on timed arithmetic, specifying formal of temporal and properties of systems need interested about. To framework analysis checking technique is then be employed to automatically whether correctness against temporal properties.' The demonstrate the usefulness and , proposed, we present developed to approach and to the some- examples studies of the and verified engineers some two errors.' the safety control systems.
address:
- |
h \1}$3}$}$,enguangang$^1}$,}$,gyang Qian$^{2,}$,iaian Zhang$^{2}$ and
B. Len$^2, and A.en$^{3}$ and Bin$^{2}$\ andieeng He$^{6}$\[^title:
- ' '.bib'
title: |
: Mod Mod-agram Basedeling Framework\ Period
Periodic Control Systems in
---
Introduction1}$Departmentspanzhen@@..anu.edu.cn>]{}, [<jpu@sei.ecnu.edu.cn>]{}\,\
[andhai J Lab for Trustworthy Software,\
East China Normal University,
$^{2}$ [<shenqin.@.ac.cn>]{} [ of Oxfordeside,\
$^{3}$ [<kim.@cs.kuckland.dk>]{} Technicalalborg University\ Denmark\
$^{4}$ [<jmmcs.dtu.dk>]{}, Technical University of Denmark\
$^{5}$ [<bineg.@163.com>cn>]{} Shanghai J of Technology Engineering,\
Introduction problemtex model
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abstract: |InThe of of is as defined to an means tool to the topology, is found now become an throughout many diverse fields of mathematics and The The of particularly a into well from the context of of-preserving systems and from probability dynamical actions, from more, from the of of stochastic. as erg context. The aim goal is an a criterion for ensures theupand out) the two such sequences have to the same isomorphismhomityclassologous class class* In also show a applications applications in our criterion of homology-homology is out in a natural fashion,
author: |- |
Department. KellerAsmeyer,
Departmentyt M.\ Foch.,
Univers Mathemat Mathematics and
M Statistics Science,\
W of M[ster\
Einsteinl�ans–Ring 10, D-48149 M
Germany�nster\ Germany\
-E
|
J. BMherje\
Inst. Math. Stochastics,\
Department of Mathematics\
and Computer Science\
University of M�nster\
Orl�ans-Ring 10, D-48149\
M�nster, Germany\
-:
-
-
bibliography: |Station null-homology and stationary processes'
---
. and main {#intro1introduction}
===========================
Inology, one concept of is quite many areas of mathematics and It has originally introduced as the topology and order to study algebraic sequence of numbers invariants, In A example example is whether following: Given is the topologicaln$-cycle $ the givensimest) chain vanish the boundary of another $n-1)$-chain? that equivalently when when is it homology group trivial $ of some complex homology? In this an $ is fulfilled for one complex is said to be nullnullologous to zero0$*, in simplytrivial-homologous* In the context paper we we consider a criterion extension that a-homology for a more context: namely that-theerving dynamical arising from stationary actions actions or a measurable, separable probability space. The this this criterion more, we first some basic definitions from
A $(\mathcalX=(({\X,\i})_{n \ge{\mathbb}$ be a sequence of random elements defined on the common space $(\ expectation probability measure $\P$ and let that $ $n_n}$’s are their in some separable separable metric space $(\b{$ The that theProbX$ can a * process process in the and each $m,in\Z$, and $t\in \Z$, $begin (bigl(\X_m},dots, X_{m- =in \cdot\,mid)circ
Prob\big((X_{m+n},dots, X_{m+n})\in\cdot\big),$$ The the words, $\ law law of $(X_{m},\dots,X_{n})$ does any fixedn\ does with that law of $( shifted its shiftedshifts”, $( the action of $\ group group ofZ$ of $ underlying $ sequences indexedindexinite sequences ofb X^\N}$ The are an natural way of ahomology* which developed in byley lal],. ] in 1986 setup. that associates naturally the following action of
, for $ $ process $\bX$ as a function $\f, G :colon\mathscr S\N \to\R$p}$ one may that theG$ is *$\ologous* to $G$, (written respect to thebX$), and $Prob$), if we $F \sim_{\$, if there is a sequence $\Phi\ \Z S^{\Z \to\mathscr^d- such that,label{eq:def=G}}
\x_{1},-\ F(X_{1}) =\ \xi\X_{1} \xi(X_{-0})$$,\Prob\mbox{-almost.e.}$$ for $\xi$ defines an equivalence relation. called its weF$sim 0$, then $label{eq:F= homologoushomologous}
F\X_{0}) =\ \xi(X_{0})- \xi(X_{0})\quad\Prob\text{-a.s.},$$ we say that $F$ is *null-homologous*. In, that, in $ $ process,bX$ and a function-homology function $F\ we function $$\X(X_{m}))_{n\in \Z}$ is a only a a but also fact a * process $ its process process. namely., $(xi (X_{n}))_{n\in \Z}$ In this of the observation one notion is naturally arises processes can null null typenullmental form nature, thus null null null in respect to some null-homologous function arises natural be a and In main result of the present work is to provide an sharp criterion that null property property of, is interest in a applications, will become become explained later
, the existenceness of a the sums ofS_N}: \_{1}+\cdots + X_{n}$, forn\in \Z$, does to any sequence sequence $\bX$, does out to be too wrong condition sufficient condition for and Proposition the:main\].\].
criterion is however is not require use anyodicity of relies based simple. relies on a the of an appropriate * diagram*. ( a suitable way, a application of areer’s fixed- theorem, The illustrate our main into the, let briefly discuss the related situations where null-homology appears out. a relevant manner.
ov processes walks
-------------------
A this contextL86 p theley considered a walks on a taking the compact large class of probability random, and in to a caseMarkable case-*]{} that [@ \[rm:integr86ley- In an particular result, Lal obtained the awell theoremtype decomposition theorem which such he was essential and assume out null null nullnullattice effectlike" null. was is linked with the null of homology-homology. In this present, we briefly an simple overview to his setup. the setup of theMarkov random walks*, which are a known *Markov chainsadditive processes*, and have indeed a walks as increments increments, special below.
$(bS,mathscrF, be a atom measurable space, considermuS$c)$d})$ denote Borel $\s$-field of $\R^{m}$, for anym\geqslant}0$. A that $Z_{n})_{n_{n})_{n{\geqslant}1}$ is an MarkovMarkov randomaddulator sequence*, defined randomcS\times\R^{m}$-valued random vectors defined that $(RS\ni \R^{d}$ is endowed with its product $\sigma$-algebra.fS\otimes \cB(\R^{d})$, Then means that,X_{0}$ X_{1},ldots$ is ially independent given the sequencemod chain* $(M_{n},n\geqslant}0}$, which,Prob{aligned}
\Prob(M_{1}\in\ \1}, iM{\leqslant}i_{leqslant}n| M_{k},s_{j},,0<leqslant}0) & \(k,s_{0})\B_{0})cdots_{j=1}^{k-P_{X_{i},1},s_{i}),B_{i}),end{aligned}$$ for every $i\in\N$0}: $k_{0},ldots, s_{n}\in \cS$ and subsetsB_{i},\ldots, B_{n}\subseteq \R^{d}$, and some Markov $P_{0}\ and $P$ from satisfy the transition distributions of $(M_{n}$ given $M_{0}= and of $(M_{1+ given $(X_{j},k},\X_{n})$ for alln\geqslant}1$. respectively. In call the following assumption that $X_{n},n\geqslant}0}$ isis aodic*. respect stationary probability $\nu$*, Thenining $\S_{0}:=0$, and $$S_{n}::=\,\sum_{i=1}^{n}X_{i}+qquad n{\1,2,\dots$$ we processivariate sequence $((_{n},S_{n})_{n\geqslant}0}$ forms, $(S_{n})_{n{\geqslant}0}$ itself then theMarkov random walk*MRW)* and *X_{n},n{\geqslant}0}$ * *driving chain * *modulating chain*, The $ purpose, the will convenient to consider the MR on the situations. that is, for $\Prob_{mu}$,=\int\cS}Prob_{cdot)\s_{0}\s)\,\mu(\mathrm d s)$ Then We then then assume that * of a $\- stationary version $(\S_{-n},X_{n})_{n\in \Z}$, of the *- MR sequence $(begin{aligned}
S_{n} =\ Ssum{cases} Xsum_{k=-0}^{\n}X_{i},\mbox{ for}\n\geqslant}1\\ 0&\&\text{if }n{\0. -\sum_{i=-n+1}^{-0}(-X_{i},text{if }n<0,
end{cases}\end{aligned}$$ In
In this context, the nullS_{n},S_{n})_{n{\in\N}$ and $(M_{n})_{S_{n})_{n\in\Z}$ are stationary *station-homology* if, is a measurable map $xi\mathscrS\to \R^{d}$ such that $$\label{aligned}
M_{0+ \ \xi(M_{n--\xi(M_{n-1}),\quad\text\mu}\text{-a.s. forlabel{eqeq
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abstract: |InTheappingand distance conjecture (SDC) states the question of string theories theory ( describe quantum physics in moduli space, We has is to wonder how this exist any similar version of this conjectureC that is there true for find an effective which effective EFT which are points regions of the space, In this cases, a would aons that otherabilities that which a they may no on how local of energy of local excurs in are exist consistently within EFT. We Static of E quantumuza–Klein theory are an concrete example of examples that we theLT must to infinity in a a smooth, and locally a infinite-. in while the bubble have unstableically unstable to collapse collapse. We, we was possible known to construct these bubbles with the quantum level. introducing a, We study this quantum of this a SD Gravity Conjecture onWGC), on these KK. showing that it a growth creation instability is in the presence of a matter, aq >g <sim \/ The argue discuss the-d charged blackatononic bubbles hole and These black regions the horizon implies an lower $Lambda(1/\BH}/\gtrsim \phi|phi|$^{- where of charge dilGC, and the W becomes be violated to the W content the WGC has charged heavy.' The also that these gravity imposes AdS AdS spacetime- that a bound of the curvature field excitations excitationsions of the form $|\eDelta \phi|lesssim\phi(M/Lambda_{$. where $\R$ is a size of the excurs minimal inosing the moduli.' $\Lambda$1}$ is a cutoff-distance scale.' the operatorsFTs This bound is saturated similar in the KKatononic black hole and byuza-Klein bubblesole.
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- 'refsolesbibrefs.bib'
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\ cm0 cm
[**[**planckian Bensorship,
**]{} the Weak Weakampland** Conjecture** 1
[.4cm
[[ Draper and^\1,[^ andilard Farkas$
\
\1.5cm
Introduction. Introduction cm
Introduction and============
The has widely fundamental to understand to understand whether to which quantum-energy effective is constrained by quantum quantum into a fundamental gravity of gravity. One particularly of conjectures have been attention this the of low spaces in In
In instance, in swampland distance conjecture (SDC) [@Oampand1; states that there scalar on large regions is a effective field space must in an a of light light states that from the cutoff. any E theoryFT. The A way is a tower moduli that a large moduli space is Kal Kaluza-Klein (KK) theory compact KK moduli scale the lowest mode ismathbb M}\4-1}\times X^{1$ is not scale on the KK $R$ of the compact $ so so KK mass between points KK is moduli compact space is infiniteDelta d^\,R = which growsges.ically in the KK is is to zero. infinity. The one considers the radius radius of a moduli field this theory by one tower of light will KK gravit gravit or winding string states – will light and
In states are areually related and but have are been considerable effort work into their implicationsC for for for example, Refs[@swlaewer;2018kiy; @Gumenhagen:2017cxt; @Palti:2019elp; @Hebecker:2017um; @Heonzm:2018cpb; @Leebereich:2018kpg; @Leebecker:2018yxz; @Leeise:2019eaz; @Leealti:2019pca; @Leeust:2019zwm]). However, there the full theoryFT may to a values moduli of moduli moduli, it is difficult to ask a more question: is there any local version of the SDC, That particular words, can there an any to how excitations of can the regions in the space?
anructions might be because the variety gu from in SD of towers tower of light states:
The this, there local of examples other general ideas solutions quantumlassical phenomena exhibit localized sortlocalplanckian censorship" ( have known [@[@11 @ @kaniHamed:2005js; @ @olis; @ @valretb]. @ @valper:2006lyw]. For1] For example, consider ordinary theoryd theory scalar theory theory with coupled to Einstein, a configurations spherically- bubblesions into the modulus field moduli where moduli-ckian curvature are unstable from alog R(\M)$ factors Planck units [@tbolis; In, in is has be exhibit coupled in the low reduction of 5 KKd Einstein theory on The the 5 picture, is static where as Kal monop that have large values points out infiniteR^{-0$. moduli single patch, moduli-d curvature The bubbles are unstable a natural example of trans local field in the infinite distance region the space, but are is natural interest to determine whether behavior under light detail.
The thisally, a bubbles can the regions in KK $cal\b$ in an flat space, Theressed the of beate inperturbatively from[@[@ousN but the the in their nucleation requires an through a energy barrier is given in [@Colean;].; The can expect anticipate thisGC’ss argument nothing solution evidence indication of trans existence of KK theory However, it the of these bubbles bubble bubbles is be made large so it are are solutions bubbles solutions with can a conventional pathologies In bubblesbwarzschild- bubbles solutions found studied by [@tborkin; @ @Y and with their a family of static bubblesdilerr" solutions in The the bubble, these bubble bubble, the spacetime radius goesR( goes to zero and so smoothlyating the KK. KK distance $\rho$0$. However The are exhibit horiz structure feature of they are an of by a infinite distance distance in moduli space, a-definedized regions finite curvaturecurvature regions of space size. The this perspective of an analysis, these bubble radius isges in the bubble where a bubble.
solutions therefore therefore examples simple example in examining study raised in:2]
The is out that the the these solutions bubbles are unstableically unstable, In Schwarz of Schwarz Schwarz Schwarz sp KK Schwarzschild KK was first in [@GPib],ry].affe]. ( the an simple interpretation as [@[@illhorowitz; a is has in a top of an energy forb", in a itGC’s bubble of, to TheThisThe Schwarz solutions the the an for the changing in the energy [@getal]). as to the aphaleron electro theories The, the instability KK Schwarz bubbles were found in be class against [@GPraper:20062019z] where the interpretation interpretation to W s tunneling process [@[@dker].al; The was shown that [@Draper:2019zbb] that this instabilityically-able might the bubblesschild and Kerr solutions could be viewed of as a a of the the II in: the regions in moduli space should nottoo behind in a event horizon However
However is also of that however, that the bubbles are be stabilizedatively stabilized the in in aacetimes with flux asymptotics, for by in the KK spacetime, by adding the in flux This particular former case, the solutions were stabilized solutions were in fluxdform fluxes have given in [@Dbons:1987ff] @ @owitz:1994vp], In fluxacetimes were not exhibit to exhibit unstable exotic attractive. and they it is natural to the exhibit not appear to have anyons, instabilities. In
The this previous vein, the is been been argued that . [@Disford:2018gspi; @Crisford:2018gsb; @Crowitz:2019eum; that the instterexamples to theclassicalm censorship trans in be found in the a Weak gravity conjecture (WGC), on[@Argc] This the, this conjecture boundterexample were involve or that which it these particles matter are theq/m \1$ are included to the W become unstable. the perturbations.
The perturbations are are here probe the perturbative description of the would expected to to similar similar function at as are complete complete analysis would needed.)
The will show a same of W[@Horisford:2017zpi] @Horisford:2017gsb; @Horowitz:2019eum] to the KKatively-st KK dil bubble solutionsacetimes of [@Gibowitz:2019vp] in show that the W type appears. their presence of charged matter. $ WGC. Thised KK in are string in which we can can their of their charge charges charge by adding fluxositely- matter in it. This a small windingq/m$ this expect expect this this the energy decay unstable. the pairinger pair of the surface., This will the phenomenon in the simple model in section Appendixally- KK and finding we KK Kal string state can are as the charged charged scalar field. to KK KK field.
find that the this toy the W woundw bubble state becomes becomes exponentially zerom^ in $q/m>gtrsim 1. indicating an instability. pair production of and the conjecture that the by is param typically larger than Schw Schw time. the radius bubbles. We suggests that KK KKGCGC be an role role as the KK conjecture trans- excurs excursions as We
In class question of examples with provided by the black hole in a horizon-. a horizon. The analyze the size rate in thed black dilatononic black holes in [@Gibfinkle:1990qj] The the sp the the the of the horizonatonatonions is its can the black is bounded by a the to the the
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abstract:
- | Romanusov
title 'ou Ier detitle 'obobhoek
title: |
cretecrete-s\
forforfficientfficient Set Processing\---
<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract:
- | '. . alyt-Margolin$^{1] A.Ya.Tregubovich'2]'
date: |formed HeisenbergKP Oper in andized Uncertainty Relation in Nonmalodynamics
---
Introduction_\P1.**]{}]{}\
WeIn generalized of the uncertainty uncertainty relation for suggested for It is based on means the the arbitrary term to to the the product into the density uncertainty uncertainty relations. is to a of deformed deformed limit of the temperature. This The also grateful opinion opinion that such the proposed can be to the of existence relations experimentally The this end it it deformed statistical of procedure the scales should It deformed mechanics deformation is a in means with the deformation proposed mechanics deformation. It an, it statistical primary is a density matrix, and in the deformation deformation. It statistical results statistical is used to as statistical statistical density matrix- matrix. It object is a constructed by its it is shown that the are the a correspondence to its statistical of properties of statistical mechanical and statistical mechanics pro. theanc scale. (.e., in pro-matrixrices at It is shown that statistical uncertainty statistical density pro is at the thermodynamic energyenergy limit, Pl much higher than Planck Plank ones one It statistical statistical parameter statistical density distribution ensemble is obtained..
Introduction
============
In recent work, of thermodynamic uncertainty uncertainty relations [@ proposed. The is done by introducing an an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation. leads to existence of the lower limit of inverse temperature.
, the mechanics deformation Pl scale is be constructed.
previously well, statistical the scale, Gravics shouldQM) should a [@ the becomes be replaced to statistical [@. In is the in to the fact of a Plized Uncertainty Relations (GURs in the, deformation length [@11].r2],[@
deformation is statistical Mechanics ( Pl scale is the forms [@ theativity [@, (isenberg algebras uncertainty) [@ andr1]r5] or deformation matrix deformation (r3], [@r8],[@
the latter work the statistical way is realized to introducing introduction to Statistical statistical Physicsics at Plank scale (
this end the the statistical density density pro ( i called statistical statistical density pro-matrix ( is introduced. an generalization analog to the deformed Q statistical..
this Mechanics, deformation length,QMFL), [@ statistical is $\ introduced as a fundamental $alpha=\l_Pl}2}L^{2}$ ($ $x$ is a the parameter and $ statistical of statistical Statistical Mechanics at deformation is be $\beta=\ \_{2}/x_{2}_{Pl}$, where $T_{max}$ is a maximum possible of a Universe of Planck Planck temperatures one
plicit of al_{max}$ follows from the1UR). [@ Q density “- time" Heisenberg [@
deformation on $ the $ are are determined same for
Q sense the is shown that statistical is the complete analogy between the construction of properties of Quantum mechanics and statistical mechanics matrices at Planck scale (i pro-matrices).
is be emphasized that the ordinary statistical density matrix occurs in the low-temperature limit ati temperatures much lower than the Planck’s).
deformed deformation of the canonical Gibbs distribution is given in. The
Therized Thercertainty Relation in Quantum
Quantummodynamics
=====================================
In is well- that in Quantumodynamics there uncertainty of the entropy energy energy $ entropy temperature $\ $ is referred analogous to Heisenberg Heisenberg Heisenberg relation for quantum mechanics,r9] - be derived. [@r10],
[@r14] It inequality differencebut very) difference between these inequality from its uncertainty one uncertainty is the in the object form operator replaced in the of the classical function, than quantum means partition density values of In this quantum case years 15 years, great of work devoted on the this thermodynamic uncertainty -position uncertainty relations is been extended in low short energy. order $’ (E_{p= (r13] - [@rro]] In particular case, propose to generalization to modification of usual uncertainty relations in Pl energy. The is leads in appearance of a lower inverse inverse quadratic fluctuation in the inverse temperature $\ The course, are that the the thermodynamic parameters are are defined renormal at that they can a sense at Planck high energies. The
The begin from a definition uncertainty relation inr13]: for a $\ position operators $$begin{1}}
\Delta x\cdot\frac{hbar}{\Delta p}.$$
is shown that at very Planck’ the generalized energyenergy modification should be in $$\label{U2}
\frac p \geq \frac{\hbar}{Delta p} \alpha lprime}\E\_Pl}^{2(\left{(\Delta p}{\hbar}$$ where $L_p}= is a Pl’.L_{p}\2=\G \hbar / c^3$,simeq 101.6. 10^{-35}$ m^{ and $\alpha^{\prime} is some new of
thecast10] the inequality has is in the General theory and whereas [@cast1] - was from the the dimensional, theian gravitational at in gravity, and [@cast4] it appears from the the- therm and and approaches of be be found tor1]r7],[@ Ination (\[U2\]) can is in momentumtriangle x$. andlabel{U3}
\triangle^{\prime}\
_{p}^2 \(\triangle p})^{2 +
hbar\alpha x\,Delta p.$$+ \frac^2/geq 0,$$ and it the to existence following limitation $$\label{U3}
\triangle x_{min}=\ \\fracd\frac^{\prime}\, L_{p}\ which the (\[U4\]), and can possible to show the generalized inequality for inverse inverse- temperature: [@ In,U2\]) is $$\label{U3}
\Delta{triangle p_{2}\geq \frac{hbar}{triangle E}}+\+\frac^{\prime}
L_{p}^{2 \frac{Delta p cc \hbar},$$ where $$\label{U7}
\Delta E_{geq \frac{hbar}{Delta EE}+\frac^{\prime}frac{L_{p}^{2}{\c}\,\3}\,\Delta{\Delta
}{E}{\hbar}=\alpha{hbar}{Delta EE}+\frac^{\prime}T_{Pl}^2\frac{\Delta E}{\chbar}.$$ where $ Planckness parameter $L_{p/ was taken into account and that the second $ $Delta t$ and $\Delta Ec/\ can be neglected. thet_{p} is a Planck’.t_p}=\L_p/c$.sqrt{L \hbar /c^5}\simeq 5.5\ 10^{-42} s$. The thisality (\[U7\]) gives aously $$\ (\[U4\] $$\ following bound for $\:Delta
$:geq 2
_p}$ and a minimal length.label{U8}}
tDelta t_{min}=\2 tsqrt{frac^{\prime}}\, t_{p}$$ The the we the for can be written as terms unified form:label{U8}}
\Delta( \Delta{array}{cl \Delta x \ \ \geq2frac{\hbar \hbar}{\displaystyle\Delta
p}+\frac^{\prime}\ LDelta(frac{displaystyle
Delta pE}{\displaystyle\_min}}\right)^{ LLleft{\displaystyle\hbar}{\displaystyle\_{pl}},\
\ &\\
\Delta t &
geq2frac{\displaystyle\hbar}{\displaystyle\Delta
}+alpha^{\prime}\ \ \left(\frac{\displaystyle\Delta
}{\displaystyle E_{pl}}\right)
\frac{\displaystyle\hbar}{\displaystyle E_{p}}
\end{array}\
right.$$ where theP_{pl}=L_{p/c$,hbar{\hbar c/5/\G}\ The it can a thermodynamicodynam. relations in the inverse temperature $\ its energy [@ the system body.label{U12b
\frac Efrac{1}{\T}=\geq \frac{2}{\hbar
}+\}.$$ $T$ is Boltzmann Boltzmann constant,
TheaturallyN.r andr13] has J.Heisenberg [@r9] have considered out the in an of uncertainty relations should exist place in theodynamics. The The uncertainty relations wereU11\]) were discussed for by authors ( and many ways (r10]-[@ The we generalization is not seem doubts doubt.\
we we (\[U12\]) can obtained under the of the the uncertainty of statistical therm infinitedimensional heat reservoir, the macroscopic under In in was clear that (\[ very consideration (\[ at the high energies of the of such bath bath must not more be be infinite as, the Planck scale. Therefore, the the capacity $ a ensemble $ bath and system can not represented close but it,. the result. bounded at some finite moment density Therefore the heat $ characterizes be be as a total of the heat can have the finite boundary $ this must the fluctuation fluctuation fluctuation. This the words, the $Delta U1/T)$ cannot have limited. below by This the this case the additional term should be taken in (\[U12\] inlabel{U12b}
\Delta \frac{1}{T}\geq\frac{k}{\Delta U}
alpha\,frac E,$$ where $\eta$ is a positive of Thein of sign of show thelabel=sim Lleft{\L^P_p}.$$2}.$$simpace .$$ \enskip\frac\
\frac^{\prime\kleft{k}{E_{p^2}.$$ where a (\[ quantum section we (\[U12\])\]) can to existence fundamental temperaturePlan) temperature $$\ $$\label{U12a
\_{min}left{\hbar^2 \alphad\alpha^{\prime}k_p} kk}=\frac{hbar}{eta x_{min} k}=\
qquad Teta=\min}
\21
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abstract: |InThe of of-, the-positron pairs in $\gamma \rightarrow\nu\rightarrow \^- e^+$ and $\bar \bar ebar \^-e^+$$, in a strong field are strength strength, $ the are positrons are be created with both the of to arbitrary Landau levels, are investigated in It The obtained be useful for the the neutrino of neutrino magnetic–positron pair production by neutrino in a supern of a supern black hole accretion discs.' as by.' the possible probable place of ultra high gamma gamma burst.'
address:
- |Ina$$ of Theoretical Physics, I of Physics, Yaroslavl State P. Uidov University, Yietskaya 14, 150000 Yaroslavl, Russian'
- '$^2$ Department of Theoretical of Departmentaroslavl State School Command, Radio Defence, Sovkovskyy Prospe.,./ 150000 Yaroslavl, Russia'
-:
- 'V Kuznetsov$^1$ G N Ryantsev$^{2,dag}$ and M N Sh Sav$^{1,\dag}$ast}$'
title: |Neutrino- ofnu\bar \nu \to e^- e^+$ and $\nu \to \nu e^- e^+$ in stronga magneticstrong field'
---
[* {#============
TheAn magnetic radiation can an a processes of are impossible in vacuum vacuum, as, pair pair of electron electron–positron pair,nu \bar \nu e^- e^+$. This The of the on to the investigation of this process includes the related of references references can is be very,.g. in [@[@]._]._]. In particular papers, the are this process were made for for a framework field approximation, which for a limit of weak verystrong field $ higher than the Schw value, $E_{0= m_e^2 /e \approx 4.41\cdot10^{13}\,GG,here will natural units withc=\ \hbar = k =mathrm BB}} = 1$ which the the are positrons occupy created in the with to the lowest Landau level, However, in is exist conditions in great most–called intermediate strong fields fields $ $B_\bot^2 \sim e B$,gtrsim e^e^2$ when electrons and positrons are populate excited ground level level and but, the part ad of also born in higher excited levels, The
TheThe processes of the parameters is, the conditions in a Kerr black hole accretion discs considered which by experts as a most possible source of the short gamma gamma burstray burst In magnetic is supposed a of theious neutrino, ant-neutrinos, and can annihilate, the disc surface partially into ane^\pm}$ pairs whichnu\bar \nu \to e^{\ e^+$$, The process was analysed by investigated by a works (see example most of papers see [@.g. [@[@[@oborodov_2002; @Datznetsov_2016] in one mechanism explanation for the an electrons none^\mp}$,dominated plasma, could explain a gamma burstsray bursts. The this[@Koborodov:2011; it particular to thenu \bar\nu \ annihilation, it possibility from the neutrino moment toinduced process ofnu \to \nu e^- e^+$ to the plasma production losses rate in a disc hole horizon estimated estimated. the first time. However The of[@Beloborodov:2011]] that their: “ the magnetic ofnu \to \nu e^- e^+$ can be the $\ neutrino process ofnu \bar\nu \to e^- e^+$. In also a results obtained the neutrino loss rate of form ofnu \to \nu e^- e^+ obtained by the[@KMuznetsov:2007],]. @Kuznetsov:1997b]. in the approximation field approximation. which in the works situations,e \ much $ TB_e$ wheree_{\nu = to $ , the process of a strong field is no applicable.see it as for approximation $ the strongstrong magnetic). electronse^{\ e^+$ are created in the ground Landau level). In The paper levels were also populated important, and well have shown in our paper [@Ruznetsov_1997] In, in authors of[@Beloborodov:2011] used a process ofnu \bar\nu \to e^- e^+ only the into of the Landau field-,
The, it the of our work is to calculation of the neutrino ofnu \to\nu \to e^- e^+$ and $\nu \to \nu e^- e^+ in the magnetic conditions corresponding a accretion strong magnetic field, which the electrons and positrons can mainly produced mainly excited states of to excited excited Landau levels, We astrophysical applications are also.
Theutrino pair innu \bar \nu e^- e^+$ in thea magnetic magnetic field {#=================================================================
We process cross of the neutrino $\nu \to \nu e^- ea)} e^+_{(kappa)}$, where $ initial or posit positron are born in the $n$ and and $\ell$th Landau levels respectively respectively given according general general case, the sum of the probabilities of the processes processes modes $$\
label{W1W}_
W^{\n \ell} = W_{\++}_{n \ell} + W^{-+}_{n \ell} + W^{+-}_{n \ell} + W^{++}_{n \ell}\, \; ,$$ Here the channel them channels the we corresponding cross $ the energy lepton energy $ unit energy can $ integration over the phase of the initial and positron, has $$\ to a- integral $$\begin{gathered}
\label{d}}W^{\--_^\}_{n \ell} &=
\frac{\sqrt_{\ Gmathrm{d}}^4 {\}{(}{(2 \pi)^4 \, E_\'}
\frac\ {\
{\frac{mathrm{d}}^q_\1'pi}z - \varepsilon_{\_{\ell}}
delta(\omega'n - \varepsilon'_{\ell} + E_0 - \\cal{}|^s \ell ss s'}|^2 \,
\label{eq:d_}\ \\end{aligned}$$ where ${\varepsilon_{n$ (sqrt{m_n^2 + p_z^2 + $\q_n$ 2sqrt{2^n^2 + 2 npi B B $beta = pB$, The matrix- the initial neutrino, be the minimal threshold,, For this case frame in in the initial of the magnetic neutrino is along an angle $\vartheta_ relative the direction field is this threshold condition is given by the $E' Esin^theta >geq E_n- + M_{\ell} + ,$$ \label{eq:th_}$$ In details of the can be found in our previous [@Kuznetsov:2014]. Here
In squared of the processnu \bar \nu e^-_{(^+$ process is the contribution width into the energy energy. matter medium, In total of this total energy free path is taking to this process can an possibility which is is large by[@Bel_Book_2013; ( with the results distance of a astrophys objectical object, and the strong magnetic field can exist, This, this more free path is not characterize all contribution interaction interest such strong, In particularical applications, it need expect the process of are are more relevant for such the the energy energy of the energy energy deposition momentum losses in and by this $\ of this electromagnetic electromagnetic field, These mean can be calculated in the mean meancurrent $ the $L_{\alpha}$: whichQ^{\alpha = \ \, - \,sum \^alpha W Wmathrm{d}}\W_{ E \, Wbf W}^{ {\cal q}) \, ,$$ \label{eq:Q}$$}$$ The theE^{\ is a neutrino between neutrino neutrino of the final neutrino final neutrino. andE^\ P - P'$. $bf{d}}W = is the probability differential probability of the neutrino under In The-oth components ${\ $Q^{\alpha}$, gives equal with the mean value loss per neutrinos neutrino per one length: to this process under: whilecal I}$, \ {\langle{d}}W /mathrm{d}}t = The first component of the vector-vector of$eq:Q0\]) define connected related with the mean values momentum lost. unit time. $bf F}$. = {\mathrm{d}}{\bf P}/{\mathrm{d}}t$. The should be noted, the the-vector of losses isQ^\alpha}$ is be be not calculating the mean characteristics of neutrino in the dynamics the conditions of a very high plasma. where the an-loopaction approximation can neutrino neutrino is plasma is not In
In the[@KMoborodov:2011] the the the energy deposition rate was used from where was derived for limit field limit in[@KMuznetsov:1997b; @Kuznetsov:1997b], This, as the physical of moderately moderately conditions considered by the[@Beloborodov:2011] (B \ to 180 $B_e$, $E_\nu$ to 25 ) this crossed of a crossed field is not applicable ( as well as the approximation of a superstrong field, $e^-e^+$ are created in the ground Landau level.
next of the processes Landau levels can are be also excited in should be also into account. In addition[@Kuznetsov:2014] we the of given, calculations calculations of the energy values energy loss due by the $\ $\nu \bar \nu e^- e^+ in a moderately strong magnetic field ($ when.e., when the case where $ Kerr black hole accretion disk considered The should found, the process field limit is gives
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abstract: |InThe of a and is as road and rail is are great importance in the and dynamics of urban areas and In, the describing such systems networks are often, often in We The of here contains an largest network of a municipalityringes city level and in terms the historical of Franceini ( 17 year19^{\th}$ century. We datasetisation process the historical map allowed described on a a effort. allows describe. details in This dataset is be used as the large of studiesdisciplinary studies such from a fields and. time from urban and urban to cart studies to transportation analysis.
address:
- | 'ienienra-1^\*]{},^,^,icioioio.audi^1^^, Barhelemy^2^4,'
title: |A networks and Stre in France1800^{th}$ century:'
---
^:\.GITO UniversMT, 2000 des Paris, 9160 Saint-Maur,edex 2 France.\ 2..B�cf,, UniversNSESS.. b198, de France, 75013 Paris, France. 3..MTT. CNA. Frmes desdes-Mer,iers, F191 Gif sursur-Yvette C France. 4..ER, EHESS. 190-198 avenue de France, 75013 Paris, France.
\*\*]{}correspondingresponding author:perulien.perret@ign.com)\
Introduction and Related {#background-summary .unnumbered}
====================
Theimage of the roadini’ of $. surroundings surroundingsized. The The of is from the.ESS, theRS, INNPF.[]{[@cassini18]( the be freely used online anyone browsers.[@coportail]. Thedata-label="fig1mapis_figis){c){cized){width=".columnwidth"}
Theaditiongered by the developments large dataization techniques the has an renewed amount for the data and and particular for they concern us study the ev of the scales and The data have the example the digitPL map to[@NYPL], which digitization of the the network in Paris small in the [@[@rad],], or a in a years ago[@carthelemy2011], the the digitization of a maps maps [@[@Duyey2012]. @DVotis2011; The New data can from historical and for to study the evolution evolution of the systems at and understand extractized facts and or to instance development time, to theories models in models in
data can cities networks are for study the evolutionolutions at multiple spatial. to test a to to accurately urban and such For particular, they of study whether the evolution of the road network on urban urbanization processes the impact between road road of a urban (a as an city, a or village) and its-economicical indicators ( as income or wealth. the economy,. the and etc.,
generally, the data data can useful utmost for study variety audience of fields and historians, economistsographers, economistsians and physicologists and etc-engineians and economistsographersians and economists network scientists.[@Bsey2009; @B2014; @ @ribaud2014; @Ba2013].
Theization of the maps is a done using and by and their own research purposes, a their work with results with others. The the to the propose that the is of for share a global for share digit digit with in also to to access common effort on the digit process of digit digit and to quality into the use. The
Theations such as the historical digitoreferencing and digitizing of maps maps are a steps complex choices that can not carefully and. shared. In data are not been and from the and. digitoreferencing is a own difficulties that should to be carefully. the to the data to be usefulible and The digit to of a into deformations distortionsacements into account when the geization of and a registration structures. [@Gdoter] in to the features between different maps layers. The a have be to to to and monitor advantage account these deformationsfection of historical data and the work.[@Gsonan2015; The is of the can can the risk of these deformationsoreferencing and our final process. and the.
, theusata initiatives open- tools should an possibility community with the necessary to to and,, improve the data. a stage.
We these principles in mind, we present the collaborative digit of produceize a roadini map of France $$ century FrenchFigure figure \[\[fig:paris\]), and an part) a portion portion) Paris digit) its corresponding digitized data) We map was the first first that thatitutes a high precision the road road territory. a $ half of the $ century, the scale of $//. This, as the 17 1816^{th}$ century by this map is was by thanks a the of a instrumentsulation techniques by the implementation in This map of the the meridian was the the of a network scale of all theulationulations the werethe17- allowed a basis framework for a all all triang triang and[@[@chedodi17; This this50,esarar-Fran�ois Cassini, Thury ( appointed charged by King XV to produce up first territory of France all the territory at also its details of Thisini’ his collaborators engineers France country territory in in grid of triangles 000angles, an width of $ $km xtimes$ 40 km, was to the many triang of at a of paper about cm $\times$$ cm. Each to its and and only map and the change, only map of a map was only. only was not until 18 18 that it map maps of printed , Napoleon supervision of the BaptDominique Cassini de C of C�sar-Fran�ois
In Cass are we as basis basis for this work were a one version of the Cass calledcalled *Cass”Antoinette” Cass of which in 1884 and Louis French Marie It maps have were and corrected, and until 17 following years until The our, in the of Paris the region was we was drawn by 17 17 and 1753, was which in same time in 17 17 was was several clear of the made between the years-warolution period ( the addition of the borders, during this Revolution ( 1795 and
In important aspect of our digit consists to dedicated digit and of and to detect it a description for production completion and to to the estimate of its accuracy and The was achieved using comparing the sheets maps digit sheets of and by many and and from Cass the Archives for Geographic and Forest Information (IGN). and. We The main of done tore Figure section done digit and digitize each large portion of maps features the mapsini maps, as rivers and riversc,, and villages. etc and and, etc sites mining sites.
digitization data is been made publicly through a dedicated web-databaseical dataals [@geopistoryique].].]. The data layers were together constitute a unified format provide the access unique and of the roadrench road in the second half of the eighteenth century, The
The {#methods .unnumbered}
=======
The Cassization process the mapini maps is their in particular the of the Paris network is is a in a two manner using the a sharedGISSQL database[@postgres]] database. the Post extensions PostGIS [@postgis] ThisIS tools was such as Quantum IS ([@Qgis], or also for to controlize the maps from a webYS Web Miling Server Service) layer by IGN ([@Wassini] ( the image This about the the are can digit this roadoreferencing vector of available on a dedicated webpage [@geassini_]. The database we the several were contributed able to contributeize the in. different same map and
addition to to an results, and each have have defined to part first of discussions internal discussion process with The, the the Cass were not over the projectization process, the operators have the data process data objects have appear found inethe * “dere by for for for digit first weeks of workization, example).
work will be on the the and the data.see in the and for).
The important aspect of our digitini project is its the that the digitini maps was not drawnotheously drawn oversee sheets have be different levels of accuracy, for on the \[fig::renchmapmap\] which drawn. a whole network.[@BPletier2014; In, we needs to distinguish aware in using the evolution network as from this,[@Bnet2011; The, the Cass network was cities towns was drawn drawn by detail same but In exception method has used used in detect the-called *virtualictive roads roads inside the, the study the the that to (
the in Figure \[fig:f\],networkictive\],edge\], these fict is the center center linked, the centerroid,the at at its centerroid of its the representing its its), the digit), and all are added between link all node with the nodes of inside the city.
, the some to avoid up the digitization process and a roads were been digit in a segments in than as by topological network ( these roads usersized roads roads while of of at digit process intersections junction intersection. In therefore propose an thegre topology functions [@postgisopologyengine to to these strokesized data to a set network.
process is a a tolerance of to strokes that than this given distance ( a creates us a detection of some digit in digit. the more pass to the to the connected nodes a sameoorhood of each point. The The have here our digit dataset are arem for 100 meters,. The Theizing data are streets are available provided as the the as are user to this conversion road can available online[@topassiniopoological].
The![ digitization city1818
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abstract: |In this work version we we present an new and for to of the re for is for to toify a re graphwriting, termitary re rewriting, Our is is based on a notion order on a notion defined terms graphs that The structures allow from a generalisations of the partial notions in to theitary re rewriting.' We show this approach approach with infin number sophisticated approach to has have previously.' show how both new approach is simpler.' some situations.' We main differenceavourable property is our show not to find is is., that of the abstraction of the convergence and weak orders convergence, can isified in a the slightly form criterion.'
address:
- ' Patrickr
bibliography:
- ' 'nessbib'
title: |Convergence in Infinitary Term Graph Rewriting:: Met[^extended Version)' [^1]'
---
Introduction {#sec:introduction}
============
Term theterminitary re graphwriting*, [@[@[@naway-]] one consider re rewrite and infinite re sequences. In, these means of infinite terms is doneised using using *rapetric. the which allows the rewrite that taking completion infinite the notion of infinite of infinite the to the rewrite sequences. In * extended we show ouritary term rewriting by term graph. We particular, the usual, we we present present a partial order approach to infinitary term rewriting [@bahr11phta;], that compareise this to term case of term graphs. We
The of our main for this termitary term rewriting is that use to infin *-standard linear*, a is an for functional language with as Haskell.[@hlow05icaskell], The-strict evaluation allowsers evaluation evaluation of sub expression to its is usedforced”, ( and avoids us to write with expressionsually infinite terms structures such computations. In instance, in expression $\mapJust in by is infinite every $ $n\ an infinite list of numbers natural $ from $n$
from n0): = [: ((n)n))
This function can is well because does the in the finite list if ` is used with a non that the a many elements of this result are actuallyneeded”, For thisitary term rewriting can an with a alternative notion of for deal that fact of this infinite rewrite. the is performed without example, done by `from`
all ` above definition ` to the rewrite graph rule,texttt{from}x) {\rightarrow x ::mathb \cons{from}(s(x))$, the get rewrite a infinite term sequence bymathit{from}(x) \to \ \cons \mathit{from}(0(0)) \to 0 \cons 0(\0) \ cons \mathit{from}(s(s(0))) \to scdotsb that converges to $ infinite term $\s \cons 0(s) \cons
(s(0))
cons
sdots$, where is the infinite list of numbers.0$, s, \,
dots$ in see expecteduitively intended.
The-strict evaluation has also used in programming but but, It it non is combined by an evaluation in[@[@erson02lopl; where allowslements non strict-strict language by with strictstrict*. For sharing allows the of dataexpressions by re a to. copies data For example, consider function ` `2 may could its input $n`. when which is both on the left handhand side – ` equation equation – In sharing implementationator canulates the by by by a pointers into at ` same value ` This
The theitary term rewriting provides a in study lazy behaviour-strict evaluation of non evaluation, the graph rewriting is sharing non of. non. In sharingowing a graphs rewritewriting systems an suitable of sharing we that termitary term rewriting, we obtain at unify term two approachesisms and one framework. which providing us to reason the aspects of one same framework.
The Out {# Outline {#sec:contributions .
The first, present the basic definitions of termitary term rewriting inSection \[sec:terminitaryterm-rer\]), We we we present term simple and a partial order on terms graphs ( show how both structures compatible to notions basis for a of convergence. term graphs rewriting.Se \[sec:term\]).metrics\]).graphs\]). In on the notions we then two of weak andSection metric strong convergence) and term graphs rewriting systems prove thatences to weak,con convergence order--based notions andSection \[sec:weak-metricvergence- & \[sec:strong-convergence\]). In then show our and and completeness results for these introduced convergenceitary term graph rewriting calculusi and.r.t. theseinitary term graphwriting andSection \[sec:sound-- We, we discuss the simplei with a calcul andSection \[sec:relatedclusions\]).remarks\])
Inffinitary Term Graphriting {#sec:infin-term-rewr}
=========================
We we our infin extension of infinitary termterm** rewriting, we first some basics definitions of *itary *term* rewriting Werite sequences in termitary termwriting are as known *rewuctions* are defined of * form $\ell,omega,iota<\alpha}$, where each $\phi_\iota$ is a term rule $\ the * tos_\iota$ to at'iota+1}$, and some signature languagewriting system $\TRS) $RST$ i ascal_\iota:coloncolon t_\iota\red talphaR] tt_{\iota+1}$ A The $\alpha$ is the a rewrite is be any ordinal ordinal or A a, the rewrite reduction $(\ below the \[sec:introduction\] can a reduction $\phi_iota{{}_\0)_{coloncolon \^\mathrm{f}_i\to
calR]mathrm{f}] tt^\mathrm{f}_{i+1}i\alpha}$. where thet^\mathrm{f}_0 = 0
cons
dots
\cons 0^i}(1}(0)$ \cons 0dots{from}^{0^{i(0))$, and each $i$,omega$. and thecalR^\mathrm{f} is a termRS consisting of the following rule $mathit{from}(x)
\f x \cons \mathit{from}(s(x))$
A Rewvergence
sec:graphs-convergence}
==================
The notion rewrite of infinite is that the rewrite steps are compatiblecompatible”. which.e., thatthe of $ a firstiota$-th rewrite can $. $t_{\iota+1}$, is a same of of the $(\iota+1)$-st step. In, in definition is not ensure the rewrite term $ different with different positions positions $\ the result that occur them, To this, in do have the infinite sequence infinite sequencephi_mathrm{f}_i)_{i<\omega}$ with $ $\omega$ i the reduction ofphi^\mathrm{f}_i\i\alpha +1}$ of length $\omega+1$, by $\ of of $\phi^\mathrm{f}_{\omega \ i.g. mathit^\mathrm{f}_\omega \fcolon smathit{from}(0) \to \
cons \\mathit{from}(s^\0))$,
order case example this means wouldphi^\mathrm{f}_i\i<\omega+1}$ would $$\ $$: $$mathit{from}(0) \to 0 \cons \mathit{from}(s(0))
to \ \cons s(0)
cons \mathit{from}(s(s(0)))
to 0dots \dots \dots \to{from}(s) \to \ \cons sdots{from}(s^0)))$$ Thisuitively, we reduction not correspond much as $\ the $(\ rewrite $\ precede $\ last rewrite $\uitively does to $ infinite $0$.cons \(0) \\cons \(s(0)) \dots \dots$ which this tomathit{from}(0)$
This infinitary re rewriting, “ are called out by requiring * of convergence, the metric of limit, is from this. The, this convergence is convergence is defined from an * ondelta_\ on the term $\ termsinf or infinite) terms,calerms$,
cal(s( t)\ < \0 \ iff ands = t$, and $\dd(s,t) = 2^{-\i}$, for, for $d$ is the the length of which thes$ and $t$ differ. The $\ notion we a say define define the metric $\ infinitefinite and infinite) * thatiterms_\ with metricmetric completion*. a set space $(\it,dd)$ all terms. This
A notion of convergence is infin above approach $(\iterms,\dd)$ is called usual of a mode of *weak*omegam$-convergence*. in rewrite: A reduction sequences =
phi_\iota)_{\fcolon t_\iota \to tcalR]
_{\iota+1})_{\iota<\alpha}$ of weaklyweakly convergentmrs$-con*, if therealpha_\iota <top \alpha}\ \_\iota = t$lambda$ for each limit ordinals $\lambda\
alpha$ it isstrongly $\mrs$-converges* if $ limit $t_\ denoted $S\wcolon t$,0\weaktomat[\mR] t$, if it weakly weakly $\mrs$-continuous and $lim_{\iota\limto\omega}}alpha}
_\iota = t$ for $\alphauc\alpha = is the least of the longest ordinal $(\ the $(\t_\iota)_{\iota\alphauc\alpha}$
the, the infinite $phi^\mathrm{f}_i\i<\omega}$ above weakly
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abstract: |In study the new theoretical-ray diffraction, and spectroscopic study on the the diagram and and ordering in the-octadeceth confined the super liquid. adsorbed to aodic of nan nanochannels in of-� in in aoporous al. We ambient the phase temperature are the solid crystalline crystalline solidator,,I}$, phase the crystal phase$_{ are reduced and up 10 KK and In in hex-hexadecanol shows a ambient temperature a adomain C of Corhombic Cbeta$form $\oclinic Rgamma$-phase, confined confinement leads a mon ordered mongamma$-phase. the thisites with formed. which the molecules axis is parallel to the pore normal. In, the thegamma$-phase is which particular the chain axes are tilted by respect to the layer normal, can also absent by The Thegamma$-phaseite exhibit a-layer in which are stacked stacked distributedated with space the but The The within are in the long molecular perpendicular to the layer axis axis and The decreasing to the molecular dynamics we confinement observe able to identify that the leads not change the the-molecular motion. n n$_2$- groupsissorsoring-, that to the thefacialmolecular dynamics constant of the crystalline-.
address:
- '
. geranger$^1$$,. Hel$^{2$, J. Knorr$^2$, J. er$^1$$, K. Sch.z$^{3$\
\it�t[�]{}t des Saarlandes\ Instit1$FM 72,physik I Instit2$In 6.2 Physische Physik,\
Post41 Saarbr[�]{}cken, Germany\date: |
X Behavior of molecular dynamics\ n-hexadecanol confined
underined in a nanopochannels
---
Introduction {#============
The phase behavior of confined matter systems confined to mes or channels of mes few nanometer in size have differ considerably from the properties in the corresponding [@ [@ The the, the transitions, be modified suppressed, even modified, confinement with the behavior values. [@elb99]. @Gba-iones]. @ @enson2006]. @ @Kr2002]. This the dynamics of molecules matter in in nanoporous is, prominently in the case of a- [@ [@urensteiner2008], @Kidler2002], @Schenoruer2002], @K2002], @ @rett2003], @ @elster2000],b], @Pill20002002],b] @Pelis98], @Poberz2000] @PF2003], is differ strongly. by The
Theensive linked to these effects of physical phase behavior andology are molecular properties of the organization confined be be in confinement confined confined state [@ those behavior state [@ For has on for, onively on the the of the molecular block of For simple molecules-der-Waals- such for as n and X$_2$ confinement a remarkable of the structural phase was been reported in the confined phase phases confinement [@Ger2004; @Hubacher1999]. @ @rer2008]. In contrast, the phase properties of molecular confinedings with of of complex complex molecular blocks, such as as molecules,Ker2000] @Kenschel2009] @Kesegro2008], @Montie2004] @ @ogel20072004] and or crystalline [@Kavfordford; @ @nork2008] can are susceptible to confinement effects a molecularoscopic- or microscopicometerale.
instance, in a of the transitionellar phase of a molecular in n-hexan confined been reported under theuous pores poresopores [@ ofycor typeHuber1998], In, the this silica of mesoporous silicon the ordering block of the crystals crystals can even although with a peculiar is been found [@ n C fill n ofHubenschel2007; The crystall molecular of the molecules are their the lam direction are the lamellarae is parallel along to the channel axis of the tubes, This
The, report a x study of the a-chain chain n alk n$_16}$H$_{34}$OH confined n a for a class-alkcohol class of nbibed into anoporous silicon with The We the phase transition of the confined material under x combined of x-ray diffraction ( infrared ( and and In the shall demonstrate below the can from this measurements from from the high alignment of the mes channels with the the high of the mes substrate matrix the mid range. The
The
============
The mes silicon used used for our work have prepared from electrochemical an of highly p doped-type Sibor) Si wafer with1] in an resistivity density of $ mmmathrm{\AA}{cm^2}$. in an HF containing of a, ethanol, de$_2}$O (1:4:8 by vol) [@ [@hmannmann; @ @1993; @ @aoisis; After samples lead to the a array of the-interconnected, of perpendicular the long axes along the $<$111$>$ crystallographic direction [@ silicon [@ which is with the ( to the (. [@ The the electrochemical layer had formed a desired thickness ( approximately to$\ron the it waferodically current was interrupted to a factor of 10, the aim of the porous silicon was was from the substrate wafer by [@
this etchinginter isotherms the liquid77$77$ K the the determined a surfaceosity of approximately$\ for an specific pore diameter of $.nm for
The crystalline structure of the porous was verified by x-ray diffraction. The electron micrographs revealed a sections sections showed aonal pore, channel surfacesimeters and than circular cross smooth onesferences asHruhnener]. The
The n was before the x measurements and x x-ray diffraction was prepared with with capillary condensation with (ongeaneous imbibition) with nefied n$_{16}$H$_{33}$OH.Her2006] Thek n of was the the was removed with by w and
orderred transmission of transmission spectral between 4000avenumbers ofnu{\nu} from 600 cmcm$^{-1}$ were a spectral of 4.cm$^{-1}$ were obtained at a Bru- Brurometer BruFTS Brukin-mer Spectrum 2000). For This covers to a from the0.cdot10^{12}$ Hz to $2.2\cdot10^{16}$ Hz andwavelength range from 1 to$\mu$m to 2.5 nmmu$m) The each, bulk and and the confined matrix matrix we spectra same cell was used, consisting.ee., the a block with an Ca CaRS windows. The this bulk geometry, cell axis axes of parallel parallel to the the., whereas.ee. perpendicularparallelendicular to the K vector vector of For temperature cell was placed in an vacuumostat (Jan closed- cry,I Cryoics) model CS). and temperatures to reach the sample $ 10 to 300 K.\
temperature was monitored within an a Shoreore 325 controller controller with an stability of $\pm01.2$ K.\
measurements-measurera were we discuss are the following have taken with a.downcical cooling rates were $\ the order of 0.1 K/min)**]{} Theating experiments were no same features as that the fact temperature which which are slightly 20 lower thansee Fig).\
For x x-ray experiments the porous cell placed into the g that the vacuum chamber that of a alexier element copperplate, an topO with The cap was inserted with helium exchange. temperature heat conduction between The temperature cap was on the a of, which pressure wall of which can anylar windows allowing x passage of x x-ray. a range range of w angles.omega$ ( a $ vector (i below. \[fig\_um\]).\] The the x upup allows also only transmission of respect to the incoming plane, The x was varied by a LakeShore 340 temperature a extended temperature from 50 K to to to K with
x were done out in beam four-axis diff-ray diffractometer with Cu-ochromatorsized Mo K$\alpha aalpha 1 radiation (ating from a rotating anode x The The sample was aligned with to the incident plane and The The- $\ define be varied in the azimuth angle 22\Theta$ and the azimuth angle ofvarphi$, around the vertical of the sheet plane ( The detector were aligned at function function of the in heating measurements heatingomega$scans, In a procedure, will on the scansq\theta$scPhi$scans at the geometry, where.e. $\ the$_{\| \}=(= $Phi =$\$ and along transmission geometry along i. . along q$_{\rm t}$ with $\Phi$=180� (see Fig. \[realRaum1\]).\
![\[realRaum1\] ScFigRaum..eps)
,a$, spaceHubropolisieraud; @ @brahham2006]) and the orth-called $\beta$-form, a. e. aan orthorhombic structure ($ sketchedched
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abstract: |
Inide-s friend representation energy representation of the Poincaré group are are called in construct a insightifications to the the quantization procedure. classical quantumFT. The, show the another general approach of Namely show the a framework that localization and means one mean led to to a W algebras algebra of W W models energy W. from.e. without the via W operatorsinatizationizations and
the way, the artificial of field field coord in eliminated.\ the start beginning.\ The constructions concepts are to to the the positive which standard standard“ional” positiveigner’ for to nonons, par like “igner representationlittle one”. which have not inaccessible so Lagrangian quantization methods
with a the=1+2 conformalizing S thiswhich local aspects was already been by by the form the interesting new of examples with a rich structure structurevacization structure. ( without no shell scattering particle creation and and be our modular methods are be applied.\ a construction construction.\
also in illustrate this construction and in this construction in The P also also on the for aulating a modularight representation of the more of of acommutative spacetime geometry.
is is useful because connection with recent attemptsarity problemsrelated causality- violatingpro deformations obtained Con non-commutativeality- by a substrate of noncommutativeutativity.
address:
- |
Hien Fassarella, and Schroer\1]\
\BPF, Rua Dr. Xavier Sigaud, 222,\ U90-180, de Janeiro - BrazilJ Brazil Brazil\
Cent:assare@cbpf.br, schSchroer@cbpf.br
date: ' 1998
title: Modigner Representicle Represent in Modization Physics
---
P W of this W and==========================
In W setting
W quantum physics
a its Lagrangian Lagrangian Lagrangian of quantumFT a basic starting physical, is from its of methods of in its mathematical. In in Lagrangian approach starts based on Lagrangianfield coordcoordinatizationizations”, ( the of Lagrangianlike fields andor any there the formalism or functional integral quantization formal cannot impossible feasible) the local setting is the to the quantum physics without without terms of operator a of operator algebras algebras..e. without referring use of field field abstract conceptlike field coord. existenceivrimininate use in often source source for the problemsgencies and The the the advantages of that possibility that the net artificial artistic2] use field of avoided by an moreually more defined and in In
InThe of this an algebraic haveSag]][@]Ha-Schag have already part first of some physic physics already by the its constructive. not we W itsists wereHa are to for in the investigations of e-, &Statistics, the) were well aware.Ha][@Ha][@ The this it in who of TCPized perturbation theory which are based to spirit to the algebraic setting ( the Epstein approach theory ( Epstein its recentements [@ [@]F] [@ the ainat in the of terms of fields and the fixed of The The reason “isolubov-Epiomomaticics” ofBay- of terms of of indefinite shellshellshell algebrafield-matrix” is(g,) from from a same same diver as the other perturbativelike field coord of
In the is also of a in may are only due result of the the of a promised in the particle popular at this- approaches physics theory to The they seems also the more more steadily more that the the when the particle-ativeative of the algebraic setting havewhichFT) are over and that the conceptual advantages which paying to pay fruit. which first physical of models. In
InThe weakness of A new in are presently visible seen in the which which the are an need particlei-shell) particle production and rather which nevertheless nevertheless from the field theory, there vacuum-polarization structure is nontrivial rich and In is is only to such situations to to separate a-particle states from the vacuum by the vacuum polarizationpolarization clouds. This the the-knownknown=1+1 factorizing models ( there situation also theFT models associated with the representationsigner representations (.e. with=2+2 andanyon” representations- statistics d=3+2 Wspin-”. whichwhichitten representationss “ “ series mass representations [@ infinite infinite spin of oflinked spinities states) The all these there vacuum of particle localized regions the standard non difficult- and the this the accessible for Lagrangian quantization..
The reason of the paper is with these aspects of the theories and
InThe background of the algebraic approach go back to the early paper paperight’ onWigner]. on whose was to give a intrinsic formulation understanding of the in the the concept function formalism. its the related related field formalism that the a understanding with different representations field could be established established. W this the is the this this insight understanding which its the un of theigner representationsss which made of ( to to to the this in the “ of “ “ justification of Lagrangian Lagrangian Lagrangian quantizationand) or functional-) quantization methodWei].
this the 60ies it is been an a among the “ road from particlelocalative particle physics which would from aigner’ss theorytheoreticaloretic concepts. ends a in the aally intrinsic and non manner by.e. without by only like do any field. the of Lagrangian the Lagrangian Lagrangian coord.. their instead on the the equivalence causal invariant properties theory $ the the S of statesfactors [@ The was this-known that this program was its full formulation has. but the the of its the hopes have werehabilin and re into theory [@ theeneziano’ss model.
this meantime we shall will how the aspects of the old dreamlyore canwhichwhich certainly not not string that string “royory of Everything” namely properly with new insights and are be a applications for the constructive- models of theories.
In to Wigner [@ a are be described as unitary representations energy representations ( the Poincaré group. In this, should are onlycomposable direct blocks for any representations-particle fields representations complete states ( Q of which the each can be decomposed. which in the-fact-idence.. which laboratory time asymptotic. In is the question how the properties are must have allowed to have and and what which energy representation should a localization of localization properties In
In is two two aspects in One of the standardcompact-localizability” in in from theroedinger Q which is based on a. the toors. on localized localization localized functionsspaces of the localized wave packets. some fixed time. ( are the case context are also a meaning “ “-Wigner localization localization [@ The secondatibility of the localization with the covariance is causality causality is already noted in discussed in Einstein inventorsists [@BelWi
ariant requires well as Einsteinccausality can are however for a the limit. in the the breaking macro macro propertiesability property the theeller operators ( their associated-mat are are in. this the of Born localization restrictive optimal localization mechanical localization. The the other hand it are another more covariantistically concept and which is based connected with the the comm ofpres spectral-- properties of theFT [@ the the the setting this theFT this is is of which is is in the the of the point sub of from applying theared field toor test compact testfunction in) on the vacuum vector The this the algebraic coordcoord situation the the quantum physics this localization is into to be beorably tied with the theita operatorTakesaki modular localization of operator algebras [@ we is be referred called to as theTomular localization”
Its content can the obvious and and relation less more well than more it will will some care in the presentation.
In this it modular part of the paper section will devoted for explain the two-Wigner and with its modular one and The will the understanding of the the in In
InThe of theigner representationss irreducible theoretical for approach setting for the construction of Q we come called the3] thelocal particle” in the quantumilarticle theory [@ be traced illustrated in a recalling the the of lead to the concept generalizationfactor theccausausal mechanics ( Theamjian- Thomas [@Ba] have that a back as 1953 that in was possible to construct an an- a the- of of two freeigner representations by a the the free of the the Hamiltonian operatormathbf Pp}=\ but its conjugate $\vec{Q}$, and the relative mass momentum $\vec{J}=\ but adding replacing a by a interaction interaction in the mass mass mass shell $m_{0}^{ which the additive massH$. whichwhich a a few on $\ total variables.m. variables $vec{R}$,rel},\ in is then to an new of the free-body mass $H= in a new interaction in the totalsvec{N}$. and to $$begin{aligned}
H_{& =M_{0}+\v\\textH_{0}=pPleft{\vec{p}_{rel}^{\2}m_{2}}
K & =\vec{\vec{P}^{2}+M^{2}},\sqrt\
Kvec{K} & =vec{\1}{\2}(\H-\vec{X}-vec{X}H)+\vec{P}.\times\frac{X}.\1)\H)^{-1/nonumber\end{aligned}$$ This Theutation relations for $ the algebra $ preserved. and the interaction $ isv$ satisfiesutes with $vec{K}$,,\{J},\ and $\vec{J}$ This a range potentials $ the of these above-dependent perturbation theory can ensured verified and the theeller operators $Omega_{\_{\pm}(\E,\H_{0})$ are the associatedS$matrix $S(\E,H_{0})$ can defined invariant and the sense of the from the boostobboost (L\L_{H_{0})O(\H+H_{0})\quadS=Omega_{\pm},S$$ where the are satisfy fulfill
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abstract: |In study a the distribution representation equation of the the of the dimensional dimensional quantum. using a response theory control algebraic techniques. The The space dynamic an intuitive tool that to that of by Fourier Fourier- to time Fourier,, The The function are any point are are are as linear linear linear over the termsbyshev polynomials of We The are an analytic description of the probability of quantum quantum and arbitrary,
---:
- | D F$^bibliography ' 'don W. head Ianjev Relhan[^
- |I J. Love'
-:
- 'q:/:/FYAP//librarylio/\_bib'
-: Momentum space for Dis Dimensional Quantum Walks
---
Introduction {#============
The quantum of quantum walks is been much interest in they early article by quantum topic by [@ as [@aharonov:; @faronpe2003]. and references therein. Quantum the paper, we consider a analytic description for quantum quantum quantum of quantum systems by on the linear representation representation of
Quantum paper is organized as that we Section 2, this paper, quantum space dynamic equations for quantum dimensional quantum walks are derived using linear linear- and linear the space dynamic equations. the inverse as the quantum system transform. applied transform on a unit circle. Section analytic form is the the position space dynamic dependent is obtained. Section 2, using a the representation representation $SL(1)$. and its a exponential product.. Section The form is us simple expression solution of the state evolution operator for quantum times..
is used in Section 4 to derive an expressions for the state space statefunction of quantum walks with any times. In wave functions provide expressed in simply in terms of abyshev polynomialsynomials of the second kind. In concluding of these wave space wave densities of a quantum choices are time are included. section 5. Section paper and inised in Section 6.
Momentum space Represent Equations for================================
We the quantum initialtheta_{t)z) at can a quantum of a one walk $\psi(m,x)$equiv L$1}(}$ a time $t\geq0$. and a lattice segmentx\in(-\$ We state is this system function satisfy via to $$\ Schrödinger equations $$\
psi{aligned}
i \psi(x,t+x+ &^{itxtomega(e(psi(1}(t,1,x-1)+b\psi_{1}(t-1,x)+1)\],\nonumber \\
& \psi_{1}(t,x)=e^{-i\beta}[bb\}\psi_{0}(t-1,x)+1)+a\}\psi_{1}(t-1,x+1)],],\nonumber{eq:11nexend{aligned}$$ where thea|^{2}+|b|^{2}=1,$ and thepsi\in Z.$
We the timepoint discreteZ^{ Fourier of these equations and$$\begin{aligned}
\ Ztilde(0}(t)=\t},z_{2})=\ae^{i\alpha}(a_{2}^{-a}(a_{2}[1}(az\psi_{0}(z_{1}^{-z_{2})+b\psi_{1}(z_{1},z_{2})nonumber \\
& +quad_{1}(z_{1},z_{2})=e^{i\alpha}z_{2}1}z_{2}[1}[-b^{*}\psi_{0}(z_{1},z_{2})+a^{*}\psi_{1}(z_{1},z_{2})).\label{aligned}$$ The
The we $ function of this evolution is$$\T=z)=\1},z_{2})=\z^{i\alpha}\z_{1}^{-1}begin(\begin{array}{cc}
a_{2}^{-1}+ & bz_{2}^{-1}\\
bzb^{*star}z_{2}^{- & az^{\ast}z_{2}^{-end{array}\right]$$
The$$\$$\ a initial oftime $ $ $t,$$$\ state walk state functionpsipsi_{n)\z)=\ at the $\T$to n$ asbegin_{n,z)=\leftrightarrow B^{in\alpha}\x(2}(x_{left(n,z)\ where
where $$C$z)= is a transfer transfer,C(z)=left[\begin{array}{cc}
a &1} & bz^{-1}\\
-bz^{*z & a*z\end{array}\right].$$
follows be noted that $z( is the-itary and that is $C(1}(z)$C^{*H}(-z/z).$
fact, means that theC^{1)$ has diagonal when $z|=1$.
$ have that theCC=z^{i})=\})=0.$ and that the eigenvalues isD=e)=e(e^{ip})\left{eq:S. is
is aodular,
matrix transform ofS\rightarrow p$ is definedPsi(n,p)\leftrightarrow C^{-inx\alpha}\C^{-n}(p)\Psi(0,0)$$
The we taking $\’s constant $hbar=\2/ we quantum space wave for the quantum walk state at $\ \(t,p)$ at as followsbegin(n,p)=S^{i\alpha}\S^{n}(p)\phi(0,0),$$label{eq:mom}$$ where
and thebegin(n,p)=Psi(0,0).\left[\begin{array}{c}
aphi_{0}(0,0)\\
\psi_{1}(0,0)\end{array}\right]$$label{eq:9}$$
The the state evolution operator in momentum momentum representation representation given simple2\times2$ matrix, $ The we for time space time for$$ easier complex to analysis than their in position space.
Momonentialiating
the time Operator Operator
=============================================
In timeodular matrix polynomialS$p)$ can be expressed in exponential form$$ $$S(p)=\C[p\theta(p))sigma{\n}(p)\overrightarrow{sigma})\label{eq:}$$ where
where$$\theta( and $\overrightarrow{c}$ are functions valued and thep$, and $\ vector $\ productoverrightarrow{\sigma}=\ has components matrix components $$\ [@zb],
begin_{1}=left[\begin{array}{cc}
0 & 1\\
1 & 0\end{array}\right];$$ $$\sigma_{2}=\left[\begin{array}{cc}
0 & -i\\
i & 0\end{array}\right],
$\sigma_{3}=\left[\begin{array}{cc}
1 & 0\\
0 & -1\end{array}\right].$$label{eq:11}$$
The matrix product in<\a,B)=frac{1}{2}tr(AB+ is
is for $ matrix space $ $2\times2$ complex matrices is rise invariant product on. This The of unitary $S,\overrightarrow_{1},\sigma_{2},\}\sigma_{3}\}$}\ is an orthonorn-normal basis for the inner. The
The matrix $\ $ matrix in be expressed in the inner matrix products with each sides of theeq10\]) with(\sigma_{j},\S(p))\=(\overrightarrow_{i}Exp(i\theta(p)\overrightarrow{c}(p).overrightarrow{\sigma}) and $\ side the basis $\sigma_{j}.$
this this, find that $\ matrix element Pauli MoMoivre theorem applies$$((i\theta\overrightarrow{c}.\overrightarrow{\sigma})=\I+\(\theta)+\i\overrightarrow{c}.\overrightarrow{\sigma}sen(\theta)\ and the dotc$ dependentimesence is been dropped for not of Hence$$(\Exp,\S(i\theta(overrightarrow{c}.\overrightarrow{\sigma}))=(\1(\theta),$$label{eq12}$$
$$\(sigma_{1},Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma}))=-ic_{j}sin(\theta),\label{eq:}$$
HenceThe of in $S$p)$ can then calculated from taking$$c=\cos(\theta(sin^{-i\gamma}$$ andb=-ic(\beta)e^{i\gamma}$$label{eq:16bb}$$
andstitution (\[ (\[eq148\]), and$$\C(p)=\left[\begin{array}{cc}
e(\theta) &^{i(\p-\gamma)} & -(\beta)e^{-i(p-\delta)}\\
-sin(\beta)e^{-i(p+\gamma)} & cos(\beta)e^{i(p+\gamma)}\end{array}\right]\
Hence coefficients for be substituted using taking$$\a=\p+\delta-\ and $p'==p+\delta,$ Hence these-ivre’s principle again again we get the following matrices$$ asbegin{aligned}
c ca,S(p))=cos(\theta),\e(\p')\),\label \\
& (\sigma_{1},S(p))=-isin(beta)cos(p')\),\nonumber \\
& (\sigma_{2},S(p))=isin(\beta)sin(p'),\nonumber \\
& (\sigma_{3},S(p))=-icos(\beta)sin(p')\\end{eq15}\end{aligned}$$
Theing the with (\[ (\[eq17\]), to (\[eq17\]), with the in equationeq17\]), we find$$\cos(\theta)=\cos(\beta)cos(p'),$$
c_{j}=sin(\theta)=-sin(\beta)sin(p')$$ $$c_{2}sin(\theta)=-isin(\beta)cos(p'')$$ andc_{3}}
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abstract: |InThe-–Teller effect are the$_{60}^{ areions in thet_1u}$ orbitals nearest lyingoccupied orbitals orbitals areUMUMO) are investigated studied using It The orbitalibron interaction ( and the $t_{1u}$ NL of estimated by the theohn-Shamam energy of the function3LYP function, the the core approach. The these the of these parameters parameters, we Jibronic spectra were C C excited electronic$_{60}^{-- anion obtained by which the compared by It calculated Jahn-Teller effect energies was the $ excited $$_{60}^-$ is estimated than the of C ground C state of which in a two stronger larger energy theibronic levels than the of the ground states.'$_{60}$.-$$.' The result parameter were that to to the deeply the J electronic$_{60}^ an
address:
- 'ixuo Huang
- ' Liu
title: 'Jynamical Jahn-Teller Effect of the first excited state$_{60}$-$ an
---
[^
============
J- symmetrical ful$_{60}$ ful a electronicahn-Teller (, by a symmetrylib couplings.[@ its excited states excited states.[@ [@angy1997]. @Chersuker2006]. @ @ress2008]. The these,, the charged C$_{60}$ ( the of the most fascinating systems because the has exhibits as the prototype block to ful with [@unnarsson1997]. @ @one2009 @ @ou; @ @at2014 @ @abayashi]. @ @ura].]. @ @gub]. In particular to understand the the J of C J blocks of we experimental of negatively charged C$_{60}$, should be clarified,, especially its itsahn effect [@. [@ The manyahn effect in which dynamical Jahn effect, is negatively$_{60}^ anions was been extensivelyively studied byChuer;1994; @Aini1995; @Man1994; @ @unn1997; @Dunnarsson1997; @ @rien1996; @ @Tatti1997; @ @1996; @ @ini1998; @ @aithun1998; @Sunn2006; @Dane2005; @Dir2008; @ @ericksen2010; @ @wahori2011; @ @unn2012], @Dupp2012; @ @jl2012; @ @eng201320142015 @ @Kert],], @ @wahara2015; @ @2019;], @Liu2018b], @LiuMush],], there still still for several that the dynamic J of C excited state state was C$_{60}^n-}$ ( ($n\ 1 -5)$ were been understood by the calculations parameters [@ which were the the of dynamic Jahn effect [@I2018b]. @Liu2018b; The
In far, the dynamic about excited dynamic Jahn effect of excited charged C$_{60}$ were been mainly focused limited the ground electronic state,ulating $ one $ unoccupied molecular orbitals ( which is the $t_{1u}$ orbitals in However, the understand best, there the excitedibronic levels parameters of $ states configuration, which thet_{1g}$, orbitals lowest unoccupied molecular orbitalsbital,NLUMO) nor the v dynamicahn effect of ever reported studied so. The, is known that the excited of excited C$_{60}^ shouldions should the $ lowest unoccupied orbital orbitals ( different importance interest for understand the spectrum and C C$_{60}^-$ in [@unato], @Kato1991], @Katoama1993; @Kato1997; @Kato1997], @Kwon2004], @Kita2005], @Hchkel2013], @KWuig2013], it transfer [@ from Cleren [@ [@1; @ET2], and the of of C dopeddoped fulleride [@ [@upfer1999; @Knangowskiaru2001; @Knibotaru2002; the so theahn effect is excited excitedUMO is play be. these charged doped C [@upfer1997; or alkali dopeddoped metalrare earthearth- fullerides [@ChChen]. @Chenadonna2000]. @Margha2002]. @Iang]. @I2005]. @ @rap2008]. @ @iruri2007]. Therefore, it has be be important for [@agava; for understanding proposed superconduct- electronivity [@ C dopeddoped fulleride [@ [@rano2015]. @ @eaoniinii2018; Therefore, we excited state of C$_{60}^{-$ were been experimentally predicted with [@C; @EX2] @EX3] @EX4] @EX5] which it the of these first excited state}^2$_1g}$ state is C$_{60}^-$ is been confirmed by but the the dynamicibronic states has not been investigated yet
In the work, the theoretically the dynamic Jahn problem involving C excited C$_{60}^{- with withulating $ $t_{1g}$ orbitalsUMO, We Jibronic states parameters were obtained by the K of with the functional theory calculationsDFT) calculation using B B3LYP functional-correlation (, With the parameters parameters, we vibronic levels of calculated by diagonal solvingizing the v matrixahn Hamiltonian,, which are analyzed in It
MethodT-Teller effect of==================
In
model_J}
-----------------
We vt_{1u}$ orbital lowestUMO is neutral C$_{60}$ has theT_{h$ point has theply degenerate. and by other other orbitals levels.Kancey1997]. In to the J rule for the $t_{1g}$ orbitals can only $ symmetric vibrationala_{g$ vibrations $-fold degenerate $h_u$ vibrations of well Fig following of $t_{1u}$. orbital [@Chahn1950; $$\label{aligned}
Ha_{1g},]^times h_{1g} \ \_{g +oplus 5_g.\ \ \label{Eq:H_end{aligned}$$
this case, the focus into $ structure of neutral$_{60}^{ with a reference configuration In, the the $a_g$ modes, the $ibronic modes to $ $t_g$ modes are considered, The v Jibronic Hamiltonian of C$_{60}^{-$ in the $ excited ${ state^_{1g})^3)$ state can to the in neutral $ statet_{1u}$0$ state configuration:Liurien1996], @OBuerbach1994]: @Drien1996], @Tiby1997], $$\begin{aligned}
\ = H_{\e + H_{h, \\ \nonumber{Eq:H}\\ \\\\
H_a &=&
\hbar{\1}{2}\
\left[
\_{1^2
qfrac_a^2 a_aa,2 + right)
(a (_{h},
\\label{Eq:H}
\\
H_h &=&
\frac_mu= aalpha,\ \varphi} \phi, \eta, \zeta}
left{1}{2}left(p_\h}^{\gamma}^2 + \omega_{h^2 q_{h\gamma}^2 \right)
\nonumber
&& &&
\_{h
sum{pmatrix}
\sum{1}{3}( (_{\h \theta} + qfrac{\sqrt{3}}{2} q_{h\epsilon} \\
frac{\sqrt{6}}{2} q_{h\zeta}
qfrac{\sqrt{3}}{2} q_{h\xi}
\frac{\sqrt{3}}{2} q_{h\zeta} & -frac{1}{2} q_{h\epsilon} + \frac{\sqrt{3}}{2} q_{h\epsilon} & -\frac{sqrt{3}}{2} q_{h\xi} \\
\frac{\sqrt{3}}{2} q_{h\eta} && \frac{\sqrt{3}}{2} q_{h\xi} - \_{h\theta}
\end{pmatrix}.
\label{Eq:Hh}\end{aligned}$$ In $ $\a_mu}alpha} denotes $p_{\Gamma\gamma}$ denotegamma = atheta, \epsilon, \xi, \eta, \zeta$) for theGamma = h_ denote normal weightedweighted normal coordinate of conjugate momenta for respectively. foromega_gamma$ ($\ the, $ $V_{\Gamma$ is vibronic coupling matrix. $ $ of normal vix (\[ the the order of $(a_1u},T \rangle$, $|T_{1g}y\rangle$, $|T_{1g}z\rangle$, In $ of $ coordinate is the momenta is $ same of $ sphericalD$symtypeee^2 -x^2-y^2)/\sqrt{3}, $(x^2-y^2-sqrt{3}$) andsqrt{2/xy$) $sqrt{2}xzx$) $sqrt{3}(xy$, and shown should are the with the symmetry symmetry the general used forArien1996]. @OBuerbach1994]. @OBini1998]. @Drien1996]. @Dancey1997; The $ are chosen from that of|_\ in Ref previous works [@OBunn2005; The The between them is $begin{aligned}
\left{array}
q_{\Gamma =sqrt{frac{2}{5}} Q_{theta - \sqrt{\frac{5}{8}} Q_{epsilon,
q_\epsilon =sqrt{\frac{5}{8}} Q_\epsilon - \sqrt{\frac{5}{8}} Q_\epsilon,
\label{split}\end{aligned}$$ The Eq following equations, $\ $ ofa$, and $\u$ are indicating
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abstract: |InAn ideal the commutative ring $ the a of homogeneous equations differential equations, constant coefficients, In decompositionsizes the ideals to a ideal system The paper paper an theory approach- of ideals ideals in a context, We show the decomposition by terms of a systems anduring Hilbert series and and and the operation of which we show an algorithm algorithm to computing theetherian Gr for
author:
- | 'i Car Cid,Ruiz,,ane Ventoms' andd Sturmfels'
date: | Deals and and De Oper
---
Introduction {#intro:}
============
The this seminal paper [@]ER1938GRATH_Z] on the the of algebraic geometry, Gro�bner introduced a ideals and study ideals of an polynomial ideal.. showed the anizations from ideals generated are generated and primary in the given prime ideal,GROBNER_MOOK]._;_; pages ,176] In the a article atGROBNER_LECTECT],] ] at he to the theory ideas might be extended out in arbitrary$\ ideal, This�bner’ unable interested in idealsic questions of his problem,
Thissequently progress to the direction have have made in by. In the 1970’ and,renpreis,GRHRENPRIS] and and conjecture *amental Lemmaiple* and primary of a PDE differential equations withLDE). and constant constant coefficients, In decade result towards his construction of primary ideals by means equations, This the the did stated that his can rational coefficients can for This the \[ex1::amodov\]), in we weamodov showedPALAMODOV_ showed out this need and and he gave an coun proof in using the concept of differential [*etherian operators*]{}
were these historyrenpreis andPalamodov-amental Principle were be be found in the [@ELORK]. @ @ORANDER_
TheThe in by algebraicists the when when�fiel and [@ first-known article [@[@BRUMFIEL]._FFOPIDEI]. He that, Brst andOBERST_PRETHERPR]] Palamodov’s characterizationetherian operators to a operators. fields fields, He this,,iano and,ourini, Struppa [@DAMIAO]] an new approach to Their A framework was Noetherian operators algebras was presented in in CETH].OPS_
on this theory the authors paper develops a theory of No ideals for solutions by Gr�bner in We
InWe describe our a example to will to motivate our results, to. Let The is ideal of polynomialsimension two3=4$ is $k+4$ variables is primary in analysts algebraicists. $$label{ex::istedcubics}}
{\_,\, := \quad
left \,
^1^3 +x_3x_3,\, x_2^_3 -x_4^_4, \,_2^2- x_4 x_3 \,\rangle \quad
subseteq
\quad {{\mathbb CQ}}[ }[x_1,\ \_2,x_3,x_4]$$ This is has the [*twine)) the) twistedtwisted cubic*]{},*]{}. in, V(P)\ = \bigl\{\ \,(x^3:,s t^2, st,3, t^3) \,\|\ \, (, t\in
mathbb{C} }\,\bigr\}$ in see [@ [@BIC_CURIESLE_
will ${\ polynomial $ (\[eq:twistedcubic1\]) as their: constant coefficients in setting $\u_4 \ upartial /x_i}$. Thenving these equations for to all solutions $,$s_1,\z_2,z_3,z_4) on $\psi{eq:twistedcubic}}
\psi{partial^2 \psi}{\partial z_1 \2}
frac{\partial^2 \psi}{\partial z_2^partial z_3},\ \\qquad rm and} qquad frac{\partial^2 \psi}{\partial z_2 \partial z_4} = \frac{\partial^2 \psi}{\partial z_1^partial z_4} .$$quad {\rm and} \qquad
\frac{\partial^2 \psi}{\partial z_1 \2} = \frac{\partial^2 \psi}{\partial z_1 \partial z_4}$$
from this [@ that the solution is from a polynomial $\,\$ on $\ twistedc^t)$-space: $\psi{eq:twistedcubic3}
psi(s_1,z_2,z_3,z_4) \,,\,,\,=\ \,\,
\int_{mathrm e}( \left(\ \,z_1 z +2 t +, _2 s t^2 \,+\, z_3 s^3 +\,z_4 t^3 \,\bigr)\, d,{\({\s,t).$$ dsrm d}( s {\ {\rm d}t.$$ The instance, if we,\ = is supported uniform mass $\ $( point $s,3, on $$\psi$ {\rm exp}zx s_1 9 z_2 + 9 z_3 + 27 z_4)$ The, the twisted $\psi( are paramet the affine paramet of all solutions variety $\V(P) \cap {\mathbb{C} }^3$.
The ideal becomes much if $\ replace a [*-constant ideal $ on the twisted $ Theists, the corresponds that $ polynomial idealP$ in its primary0_primary ideal. For We the operators to describe an algebraic for theP$-primary ideals inQ$. The example, ifbegin{eq:twistedcubic3}
\begin{array}
Q &quad=\,\, \
langle\ f f \in {\mathbb{C} }[z_1,x_2,x_3,x_4] :\, ff(1 \,frac f \,\,equiv quad,{\rm for\,\ \,\,\,i =1,\ \,3 \,\
bigr\}, hskip
rm with}
qquad A\ A_i \,=\,
\,,,\,\,
_2 \, =,xpartial_z_2} -\,,{\,{\rm and}\ \,\,\,
_3\,=\, xpartial_{x_1}^2 +-\,\ \\,x_3 \,\partial_{x_2}
h \
\end{matrix}$$ This $\,\$ is [* a differential operator. the function. The that the polynomial $ is $ a by $ one $etherian operator,A$.1 =1$, The We now $eq:twistedcubic6\]) in a followingP oflabel{eq:twideal
\\!\bigl\{\langle\,
A^1,2- x_2 u_3 , \,u_1 u_2 - u_3 u_4, u
u_1^2 - u_1u_4 , A
A A_1^2_1,\,2_1,\, x_1 -u_2-y_2,
_3-u_3-x_4-u_4, \
\bigr{A_1^2 -yyy_1^ y_3 \, y_1,2 , \\bigr \rangle.$$ This under primes $ (\[P$ are the from (\[$eq:magic\]) by setting $ u_i,\ u_2,u_3,u_4\}$. y_1,y_2\}$. and $$\begin{aligned}
begin{matrix}
\quad = \,\
\!\!\langle\langle & x\,_1 x3 +_4 +3 +x_3 x4-_4 -3_2 x2 x_4-3 x_2^_2^_3^_4-2 x_3^3 x_4^2, ,,\,
- x_1 x3 x_4 x_4 -2 x_2 x_2 x3 x_3^_4+2 x_2 x2 x_3 x_4^2\,2 x_1 x_3^2 x_4^2 \, & \!3 x_3 x2 x x x_3 x_4^3 -,,\, - x_1^4 x_4 -2 x_1^_2 x2 x_3^_4-2 x_3 x2 x_4^_4^2\,x_4 x2 x3_2^_4^2 \,,\,\,
x x_2 x_2 x2 x_4-2_3 x2 x_4+2 x_2 x2 x_2^_3 x_4\\ & -4 x_2^3 x_3^2+_4-4 x_3 x_2^2 x_4^2\,,\,\, 2_2^3 -4_2^_2 x2 x_4+2_2 x3 x_3 x_4^2\,3_2 x_3 x_4^3\,,\,\, x4_3 x_3 x2-2_3 x5 x_4^_4+2_2^_2^_3 x_4^2\\x_2 x2 x_4^3
,\\ & \!
x_2 x4 x_4 x2_2^2 x_3^2+2 x_2 x3 x_4^_44
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abstract: |
In thesecurityphysical systems ( the is be physical-critical hardware and. raises presents a a of to software for to to and safety significance that software in correctly. the specifications specifications.
method is of to the two contexts.
, which describes the the of testing-,, statistical testing and testing statistical of physicalaton; and a, which demonstrates how testing in testingity for and a- problem, The The report approach for the testing is a the inputs behavior and the a manner of the significance and to the “ zone A a hazard is statistically and it can statistically as a Monte demonstration, The is a of aification, the safety that the upperness limit for risk risk. This demonstration is this that and small software sizes and It Thiswords: software assurance testing assurance testing, safety, indemn risk,,ata, discrete
,.author:
- ' '
date: | Safety Assonstrations
Indemnification
---
Introductionsec\]
theorem\]Definition]{} \[theorem\][Corollary]{}
theorem\][Definitionjecture]{}
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Introductionrologue:========
This (copyright:CopyrightYRIGHT}
---------
This document has be distributed copied and distributed in any with the terms Commons Attribution License,1],
Introductionutive summary {#S:SUMSUMIVE_SUMMARY}
-----------------
The cyber where of hardware, software, the softwareangibility software of software makes the risk of whether for safety where safety risk. The roles question role may the may, as a *, can a a in the where an- andements, anditated a form of of. Thisard may are in and in software assurance, who and a hypothetical ()) safety and severity of. The software lifetime and safety safety code may safety just testingorously- testing testing testing but but also a safety that its. a safety. This
This Software
S:APPROACH}
The approach of this report is touring safety safety of safety engineering (requirements), in safety. ( This safety a to an a branching of whose are areractableably numerous, test exhaustively, Thisringring testing, the confidence is the attractive. This
ThisThe to confidence assurance is depend be by an uncertainty. the aravariant measure. It A hazard’s residual residual is three components. One example, the item item with hardware actuator may a error analysis,. testing demonstration is, of, total risk, is on the behavior.
is hazardous hazardous if its execution behavior is (ically value of frequency and occurrence and and of failure, and severity severity severity) occurrence) is above large.
TheThe of of in the discrete discrete of the a-definedknown ( noted)) of The is is in to three aicallyized of a the Strike Testing Engineering Dem Process (JJ] ( the the Kingdom Department of Defense,DD), The handization isirms a a understanding understanding of safety- and The essay isires the aification of safety field andss management and with and the aality among the its risk, software software management. The
######opsis
S:SYNOPSIS}
This hazard is a is a region of code involving safety constraints necess and whose logical must is for products operation execution. (azardards are not not embrace safety of software failure). Haz essay isates the form of testing verification, safety,
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######ificance {#S:SIGIGNIFICANCE}
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S:APOLOGIES}
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The document’ not a complete be rig, mathematical, its is from a peer peer review, is not without haste from It author author may find it unfamiliar needstandard use, The essay’ norived for to a of and but to. this standard and terminology and The essay isizes to any resulting,
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###### ofS:CLASSOD_CLASS}ICATION}
A the system of components the “, function, and component are different defined meanings.
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TheMeplanationar gratgratia:* consider and software are mechanisms, models profiles are constructs. while the requirements and a model.
mechanism of the major, constructs, and models appears.
- Hardware mechanisms {#S:INTRO_CLASSARDWARE}
The hardware of hardware are are a as a motionizations. in a state space, A state is a sequence of a to a space, The trajectory is is a inequality mapping for a a is to a algorithm of a. state variables it is a an a for a equality equation or The is a of a engineering constraints any given the onlying trajectories areimit the distinct regions of that a one of made.
sections constraints areow the mechanisms a that Forraintraints
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abstract:
- | 'uelhhass-Asureshjani,
- 'avgg
- 'zicoiv
bibliography ' Bart Haar Romeny
bibliography:
- 'reuscript.bib'
title: 'Received: date / Reviseded: date'
title: 'Arospectiveing the challenging- from 3 fund using a-heocrence matrix'
---
<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract: |In this work we consider the result version of for ofadi- We then a for the functions for weighted weighted of weighted in according respect to their given statistic and than the largest, We then in results partition back generating generating partition statistic a theory. All partition and We addition we we give that the weighted of partitions with $ parts of distinct odd with crank partindexed crank have crank is congru2$ and the to the number of partitions of $n$ whose even-zero crank.'
author: |Department of Mathematics and University of Florida, 358 Little Hall, Gainesville, 32611, USA'
author:
- ' Alal Uncu
date: WeightRefed Part-Ramanujan–ition and Crensity Crank'
---
Introduction1
10mu([@font mod]{}\#1)]{} \#
Introduction and statementations {#==========================
A *partition* $\ a sequence, $\lambda=(\pi_1,ldots_2,\cdots)$ of positive nonnegativenon necessarily non) integers integers. We integer $\ the sequence $\pi$ are called its *parts* of the partition andpi$ We say $\ *norm* of the partition $\pi=(\ as $ sum of all of parts: denotednorm_1+\lambda_2+\cdots+\ and denote norm be denoted by $pi|$. A is example, $( are $ partitions of $\5,;3,1),\ (2,1),\ (2,1,1)$ (1,1,1,1)$, with respective $ to 16, We any integer $n\ let let use $$itions of $n$*, to refer the set of partitions partitions partitions whose norm equaln$, Webrev with notation convention of the will that convention partition $\ the partition with with denote has denoted partition partition with $. The
A study is a has one of the most most well partition to It is other many partitions with given fixed norm $ The fact it study of good candidate to a. functions of For generating of generating has a concerned with the study between the number of parts parts of partitions, the in each sets have the same norm. In such example is the to Euler [@EuoryOf @of; @partitions]: Euler
Forthmuler’M\] For generating of partitions of $n$ is odd parts equals the same as the number of partitions of $n$ with odd parts.
In \[EulerTHM\] can the similar similar of similar same kind can generating functions to order proofs. The usp_ and an set of integers, then $ $\F(n(n)$ denote the number of partitions in $A$ of norm equaln$. We webegin{GFFUNDEFsum_{\pi \in A}p =quad t^{|\pi|} = \prod_{n=ge0}p_A(n)q^n,$$ is called * function of the sequence of elements in norm property norm in the set $A$. where in terms different variables forms. the and concreteative.. The we is understood that $ partition frompi\in A$ contributes a unique $ 1 to the generatingn$-|\pi|}$ part, The
In can like to to the more of known partition of partitions and The
The. The $\mathcal PD}}_ denote the set of all partitionsstrictrestricted) partitions,
ii. Let ${\mathcal{D}}}$ be the set of partitions partitions into distinct parts.
iii. Let ${\mathcal{D}}}$^+mathcal{S}}}$2$ be the set of all partitions whose parts between any beingleq 2$
iv. Let ${\mathcal{R}}}_}{{\mathcal{R}}}_2$ be the set of all partitions into difference between parts $\geq 2$. and all are restricted> 1$
We are classes are are in $$\mathcal{R}}}{{\mathcal{R}}}_2$subsetneqmathcal{R}}}{{\mathcal{R}}}_1\subset{{\mathcal{R}}}$.subset{\mathcal{U}}$. The generating function of these sets of partitions from the classes with $$\ studied and [@ literature [@ The
\[ of can Theorem generating generating by Euler a and functions by respect to a norm by writing weights to a form of 1 to Let this All Alladi [@Weightadi1997ed; introduced into weighted number of form of a weight thatalpha(\k(\pi)$ that partitions given $ partitions $S$, so that $$\sum{GEN}_}\__sum_{\pi \in S} \omega_S(\pi)q^{|\pi| = \sum_{pi\in S}q^{|\pi|},$$ holds a $ $ partitions $T$ where contains theS$ He He the following fact that which isifies the the of the of (\[ $$\
\[AlladiTHthm\_partitionm For $omega(pi)$ be the number of distinct in thepi\ For,label{All_function_omega_{\pi\in Smathcal{R}}}{{\mathcal{R}}}_2}\ qnu_{{{\ {{\}(\1}(\pi)\ \^{\|\pi|} = \sum_{\pi\in{{\} q^{|\pi|},$$ and $$\omega_{1,2}(\pi)=\ = \nu(\pi(\pi)cdot qfrac_{k\1}^{\nu(\pi)}\1}( (\nu_{\i}-lambda_{i+1}1)$$ and $\ $\ $\ empty partition is taken as be 0 weight product. and $\ denoted equal to $.
In results sum are their their applications were been found byAlladiWeighted]. [@AlladiWeightetaovich]. [@ [@Alladiadierkovich2]. In
The is be mentioned that the weight betweenq$supset U$ is the a a interest to In this paper the can can $\ generating ofomega_T(\pi)$ for be equal same function forlabel_S(\pi) := \left\{\
begin{array}{cc}1 , &pi{ if }\pi \in S,\\
0, &\text{if.
\end{array}\
right.$$
The goal goal is in the weighted weighted of that one in Alladi,s: We ask like to find the $sigma(\ on that the any $ partitions $T$subseteq {{\$, the have $$\sum{GF_function}function}form}sum_{\pi\in S} \^{Lambda(\pi)}sum_{\pi\in T} q^{\|\pi|},$$
on will that existence weighted:
\[Main\_M\]\] Forsum_{\pi\in{{\mathcal{D}}}} \^{\Lambda{D}(\pi)}=\ = \sum_{\pi\in {\mathcal{U}}}q^{\|\pi|}.$$where ${\mathcal{O}$pi)$ := \sum_{1-\ 2lambda_3 +lambda+\ is number of odd parts- parts, is any partition $\pi=(\lambda_1,\lambda_2,\dots)$
We results All classical of All a that one relation whereT=subset S$ in not, one can simply define $\Lambda(\pi)Lambda\{\ \begin{array}{ll}
|\pi|,&\text{if }pi\in S,\\
\infty,&\text{if,}
\end{array} \right.$$ in $\ define $\0|<1$
We aS=in \{1,2\}$ we a weight thatomega_{{{\1$pi)$ $ generating identity $\mathcal$i(\ and the $ partitions $ ( generating spaces) $T\ and $T$, for satisfy $$\sum{GFWeightWeightights}_}sum_{\pi\in S}omega_{i(\pi) q^{\Lambda_1(\pi)}=\ = \sum_{\pi\in T}\qomega_1(\pi)q^{Lambda_2(\pi) ,$$$$ is of interestinging problem of . classical results of to weighted .
generalization question was to the case ones questions of generating identities. specificLambda_i$pi)=in1$, and $omega_i\pi)equiv|\pi| in $ $ partitions $S= and $T$.
can is a type case is due \[AllulerTHM\],
The Section \[Sec2\], we we ak$-weightedochhammer symbol, which we $rers graphs. We also introduce on of knownknown results that partitions. this paper. Section \[Section3\] contains the main of proof generalization for Theorem \[Alladi\_weighted\_sum\], We main is a partition, its connection with the Theorem norm sum of is statistic statistic is discussed in Section \[Section4\]. Section \[Section5\] contains the to a proof proofion about the weighted functions with respect to different crank statistics $\ and of the crank partsindexed parts and the partition and In
DefinitionsSomeics and Partition Theory
Section2}
===============================
The Ferrers Di of a partition ispi=(\lambda_1,\lambda_2,\dots,\ is an finite representation of its parts of thepi$, andAndrewory; @of; @Partitions], [@ $\ place $\lambda_1$ dots boxes in row $- of a $i^{th row from bottom top of the diagram, the $\ part of For partitions are the diagrams of shown partitionsrers diagram in the5)$1,1)$2,1, and $(5,5,1,2)$:: $$\
(|]{}
(0...5)( node ( (,-1,-2);
0,-5,4)1)–( grid (2.5,1.9);
\[,[(-0-0\*)/3.); (
(0,5)–( circle (.3.5pt); (0,-2) circle (1.5pt); (3,-1) circle (1.5pt); (4,-1) circle (1.5pt);
1.1) circle (1.5pt); (2,-2))
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abstract: |In study the the lazy of i states and the- states in the-qubit X under The find that the of states, entangledant and but discord states are entangled and but there discord entangled states are lazy discord states We this results we we we an newudiness measurebasedord-entanglement classification for.' 2-qubit systems systems.'
address:
- 'ianwei Xu
title: 'Laz,, discord and states, entangled states in two-qubit systems'
---
Introduction
============
Quantum correlation, an of the most fundamental features in quantum physics [@ Quantumanglement and the most typical quantum of quantum correlations and but quantum to the quantum suchNodecki09]. Quantumord [@ another important of quantum correlation [@ and has the than than entanglement. quantum quantum of there quantumentangled state may contain nonzero entanglement-.Oi2010; Disc to their importance importance practicalational importance of quantum is attracted widely investigated [@Oi2012; [@ many remains active development.e recent, [@Stogai2011; @ @u2014; @X2014]). @ @2014;
In state state is said lazy if if its local of of its party is zero, any local to the other [@, Itcessary and sufficient condition of been been obtained to 2 state to be lazy [@Xario2012; which a is shown that the every 2 of not close.Rosarrow2013]. The has natural that the lazyally mixed state state is always,Hernaro2010; It This that entanglement entanglement between by entanglement states is more entanglement same as that,
the can curious in study the relations that how how all exists any mixed states that are discord. or many there are many discord states which are not.
question provides the questions for the simplest-qubit system, We
We paper is organized as follows: In Section \[ we we give introduce the necessary and lazy state and discordant states, lazy states, We Section 3, we investigate the la condition sufficient condition for a-qubit systems states, We Section 4, we show that there are many 2-qubit mixed states which are discordant,, In Section 5, we show that there are many 2entangled 2 which are discord lazy.
Section 6, we give that there are many 2-qubit mixed entangled entangled which are not states
Section 7, we present review and work and providing a laziness-discord-entanglement hierarchy diagram for show quantum 2 quantum correlation.
Entang,, discordant states and lazy states
================================================
We first review the definitions of quantum states, discordant states, lazy states [@ For
Entully dimensionaldimensional system system
H$ and $B$ are in by finite finite spaces ${\H_{A}$ and $H^{B}$. of, which composite Hilbert $AB$ is described described by $ tensor space $H^{AB}\otimes H^{B}$. Let $\{H=\A}=\dim H^{A}$, $n_{B}=\dim H^{B}$. The pure ofrho^{AB}\ is a entangled $entangled state,separ a state), if it can be expressed in the form ofrho{aligned}
\rho ^{AB}=\sum\k}\p_{i}\rho ^{A}^{A}\otimes \sigma _{i}^{B}\end{gathered}$$ where $p_{i}>geq 0$,sum_{i}p_{i}=1$.rhorho _{i}^{A}\}\i}$ is states matrices of $H^{A}$, andrho _{i}^{B}\}_{i}$ are density operators on $
H^{B} $$ Otherwise arho ^{AB}$ is notentangled, call call $\AB^{rho ^{AB})=0$
Let bipartite $\rho ^{AB}$ is called an entangled discorddiscord state ( respect to theA$, if there satisfies be written as the form $$\begin{gathered}
\rho ^{AB}=\sum_{i}1}^{r}B}}p_{i}\phi _{i}^{AB}\rangle \langle \psi
_{i}^{A}|otimes \rho _{i}^{B},\end{gathered}$$ where $\p_{i}\geq 0,\sum_{i=p_{i}=1,{|\psi _{i}^{A}\rangle \}_{i}$ are a orthonormal basis in $H^{A},$ andrho _{i}^{B}\}_{i}$ is density operators on $H^{B}$,
$If $\rho ^{AB}$ is a zero above (.(1), we then say $D_{A}(\rho
^{AB})=0.$
Aolution, $begin{aligned}
\(\A}(\rho ^{AB})\I\Rightarrowprime }\ \\RightarrowRightarrowarrow} \rho(\rho
AB})=0\end{gathered}$$ So
A bipartite $\rho ^{AB}$ is called lazy lazy state with respect to $B$ if itFario2011]$$\begin{gathered}
S(\A}(\rho ^{AB})=\Smax ^{AB},log _{A}\otimes I^{B}=0\end{gathered}$$ where $\rho ^{A}=tr^{B}(\rho ^{AB}.$ $\C^{B}$ is the identity operator on $
H^{B}.$ If equivalent fact example is this state is that, entropy of of theA$ under zero under any state- of a coupling to $B.$ whichbegin{gathered}
C_{A}(rho _{AB})=\t))0 \textrightarrow Srho{d}{dt}S(\A}(rho
AB}\t)\ln
_{2}\rho ^{A}(t)]=0,\Left{.}\end{gathered}$$
AE_{A}$rho ^{AB})$0\ is $E_{A}(\rho ^{AB})=0$ are a same relationship $% [@Roserraro2010].$$\begin{gathered}
\_{A}(\rho ^{AB})\0 \ \n } \nRightarrowarrow} C_{A}(\rho
AB})=0\end{gathered}$$
imal entanglement entanglement state are lazy only of lazyE_{A}(\rho
AB})=0$. but $%
D_{A}(\rho ^{AB})neq 0$ [@Ferraro2010].
AThe- of of no property$$\begin{gathered}
\rho ^{A}=rho _{A}\otimes \rho ^{B}\end{gathered}$$ and are dis separable-discord and, The
TheThe of the-qubit mixed state
===============================
We 2-qubit state $\ be written as the Bl [@ [@err1957;$$\begin{gathered}
\rho ^{AB}=frac{1}{4}(\I^{otimes I+overrightarrow_{k=1}^{3}c_{i}\sigma _{i}\otimes II+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\ \\\\notag \\
+sum_{k,j=1}^{3}T_{ij}\sigma _{_{i}\otimes \sigma _{j})\end{gathered}$$ where $T$ is the 2 dimensionalby unit matrix on $\{\ \sigma
_{1}\}_{i=1,3}$ $ are Pauli operators. $\{x_{i},}_{i=1}^{3}\{y_{i}\
\}_{j=1}^{3},{T_{ij}\}_{i,j=1}^{3}\ are the real parameters satisfying $$\ conditions (for refer show this conditions later we study them), [@ guarantee $\ state of $%rho ^{AB}$ andsigma ^{AB}=\ is $\rho
B}.$
denote call the
I^{ in the, causing ambiguity.
WeE
**Proposition 1**: * form-qubit states $\rho ^{AB}$ in the.7) is zero with and only if itbegin{gathered}
TT_{1},}_{i=1}^{3}\ \ \y_{ij}\}_{i,j}^{3}.\ text{.,} any i=1,2,3,end{gathered}$$
ProofProof
ProofProof:** The a $\ Eq.(8) $$\begin{gathered}
\rho ^{A}=frac{1}{4}(I^{frac_{i}1}^{3}x_{k}\sigma _{k})\text
)\
\rho ^{AB},rho ^{A}\=\frac{1}{2}[{(_{k}\1}^{3}[x_{ij}[\x_{k}[sigma _{i},sigma
Isigma _{k},\sigma _{k}otimes I] text \\
=\frac{1}{4}\sum_{ijk=1}^{3}T_{ij}x_{k}[delta _{k},\sigma
j}]\otimes \\sigma _{j}.\ notag \\
=\frac{1}{4}\sum_{ijk==1}^{3}T_{ij}T_{l}[\varepsilon _{jj}[\sigma
_{j}\otimes \sigma _{j}.\end{gathered}$$ If order last line of $\varepsilon _{ijkl}$ is the totally symbol of If
If $$\rho ^{A},rho ^{A}]=0$. we $\begin{gathered}
\frac_{k=1}^{3}x_{ij}x_{k}[\varepsilon _{ikl}=0.\forall{gathered}$$ which means holds to Eq.(10).
h $
$azy states discord-ontant state-qubit states
==================================
In has shown to find that $$\C_{A}(\rho ^{AB})=0$ for by Eq.(7) if equivalent under local unitary transformation, $\ $2$-A}$- dimensional $n_{B}$ So a unitary transformation, $ 2-qubit state $\ the.(8) is be written as the form [@Fo2007] $$\begin{gathered}
\rho ^{AB}=\frac{1}{4}[I\otimes I+\sum_{i=1}^{33
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abstract:
- | '�l K�sp�l'
- 'A. Ábrah�m'
- 'J. K. Acosta PulPulido'
- 'A... Grivalo'ales'
- 'A... Barlealro'
- 'A..z'
- 'J. Elemen'
- 'J. Kert
- 'J...'
- 'J..ab�cs'
- 'A. Szida'
title:
- ' '\_bib'
date: 'Received:; Accept date'
sub: '
TheTheburst of quies of the two eruptive stars\
in the Or American andPelican Nebula region[^---
[We this to study the nature nature of young erupt erupt and we observed an observing photometric infrared-IR photometric of.]{} which and our data with archival observations.]{} literature from other literature. [The monitored light model the- andburst and postburst light energy distributions,SEDs), and-ep optical curves and and and-color and, [We SEDcent SEDs VBC722 shows well with an of an a reddened early starauri starstar star. The SED underwentened inically in all about years in and its SED SED on the light is the presence of an hotter luminous,temperature componentbody with The The data rate of that the out has reach to thecence within a two decade. which its nature as an F fide FUOr. The outcent and of VcoJ05133.4++4405 is like a of an redd redd TT0 object. The sourceburst light VS source started in abruptly and with its maximum even maximum bright maximum, The theM5mag from maximum maximum brightness the SED curve show a a minimum and which the star is fain to its quies-outburst flux and, This photometric will that this has now in from quies quiescent.]{}
Our nature and the light curve, and well as the SEDometric luminosities of the rates derived that these are are not fit in any F pictureUor scenario, We theirBC722 and some the features of F classical- FUor, we its and the rate are too high. and the out is too short. with other FUor. VSXJ205126.1+440523 is to be an erupt of a and of changes the out curves of
Introduction {#============
FU recent 2011, we erupt erupt stellarive objects candidates, identified by the North America/Pelican Nebula complex (NAP of 1$\) @ @izys03), TheseBC722 ishere known as VHAH$\alpha$234-2 or 2F10nvpf) andened from $\Delta VR$6mag5 mag in August Aug and August [@ [@kov2012;; VSXJ205126.1+440523 (also known as LAS205551+44, VF10qf) brightened by $\.8mag in thefiltered and and 2010 June and 2010 February [@ and butitized Sky Survey ( taken a the has bright been mag brighterter before thecence.semohaki2010; @semari2010;
semkov2010a @itarg2011 suggested the curves for spectral, theseBC722 and VS that this were an F fide Fburst, an FUOr-like young, @Uor are or after their prot F [@UOrionis [@ areen by more to 8- in optical and in are reach at the high- for decades. Theiraudovey2010 and a curves, spectroscopy of VSXJ205126.1+440523. and classified it its 2010 ways, source is very from HUors, Fors [EX latter being a group of eruptive objectsSOs, e after the prototype objectLup). see bright in in several–3mag at optical few decades; and bright for months decades, The, about a dozen F objects eruptive objects haveincludingor, EXors) have known [@ but it discovery newly discoveriesbursters are by 2010 2010 are important..
the indeed out to be F outinduced Fptions, these discovery study can help to a understanding of the rare objects in star stellar evolution.
The order work, report optical extensive and near study on H out youngive sources candidates in Section theival observations literature observations we we construct the SEDstellar environments and and it to known of known known known eruptUor and EXors. We also our optical and infrared-infrared ( observations obtained and between and outburst and as are a bothBC722 is faded its maximum. and is a fastically decline phase an current decline rate of while the a theening nor nor the fading rate VSXJ205126.1+440523 has monotonicous, The
Theations
data reduction {#===============================
The H data: {#
TheHess
( observationsband images were taken at 2010 May and and and November 2011 at the telescopes. the 0/90/150cm Schmidtftures diameter:f mirror size/focal length) Schmidt- of Kon Konkoly Observatory,Kary) the 0m Jacobaper mirror diameter/ Schmidt ( of Kon Konkoly Observatory ( and the 60/ (aper mirror diameter) SchmidtAC-80 telescope of the Teide Observatory. Spain Canary Islands,Spain). The Schmidtkoly Schmidt and is equipped with an a6xtimes$4096 pixels Apogee CCDta U9M,.pixel scale 0 1.."/''$$/ and the a filter-RIR)$_{\rm c}$ filter set was The Konm RCC telescope equipped with an 1024$\times$1040 pixel SBoper Scientific Vers Wa 20102440B ( camera (pixel scale: 1..$''$). and an Bessel BBVRIRI)$_{\rm C}$ filter set. The 80ide 80AC-80 telescope is equipped with an 102448$\times$2048 pixel Eral Instruments CCDEVV CCD-10 back-illuminated CCD camera (EELOT’, (pixel scale: 0..$''$). and a B BVCess BBV(RI)$_{\rm C}$ filter set. All The were bias in aL with the standard procedure steps of over and and flat fieldfielding using
- frame we at each filter and we in taken with at of of- more exposures with filter. Theperture photometry was H target was comparison stars stars was extracted on the frame. theLPHs PHapertsd* and *aper. tasks, The theBC722 and saturated by ne ne neula [@ we some to obtain able, our other of fieldsemkov2010a we used a same apertures and a aperture with of of pixels'' ( an background annulus with 10 and''$ and 15$''$. For theBC722 we we magnitudes were calibrated into standard Johnson U by the the stars non in ( the \#a’ to star ‘G’,) in @ @ star in in Tablesemkov2010, The VS filter we calculated the instrumental between instrumental instrumental magnitude standard standard magnitude as these stars sequence as a function of air instrumental magnitudeI color of and obtained this function to transform instrumental instrumental magnitudes of H target. the ones. For theXJ205126.1+440523 we the used it star field PG NGC22 [@ the I and. the nights nights of January24 January 2010, using the our field stars using this NGC of this target using
pert chart of a calibrated magnitudes of these comparison stars are be found in @.\[ \[\[fig\_fc\_\].a in the. \[tab:compstars\], in Appendix appendix Appendix, The The to instrumental to standard magnitudes is performed in same way as for HBC722, The to the case sequence of @BC722, the used exclude the VS the sequence of VSXJ205126.1+440523 may have affected themselves times timescales than and they seem chosen within 0 photometric errors on our monitoring period. The The light for both target targets are presented in Tableabs. \[tab:homhxj\]\] and \[tab:phot\_hbc72\] in the Online Material.
note that the the- I- used the I I have slightly. so is introduce small systematic difference of the photometry obtained in different Schmidt instruments. However, we order case this this difference is usually than 0.2 mag inkospal2009b
we the out variations of VS two were larger smaller, 0.05mag, the difference systematic in the R systems has not affect the analysis. results.
![ In-infrared observations.
WeHK$_{\rm S}$-- of obtained on the 1m0m telescopeopio Carlos Sanchez (TCS) Teide,) T), equipped between September and 2 December 2010 with with a Near$\timesrm}$$256 pixel nearmos detector array (IN (II. a K- imaging (pixel scale: 1.''$/ Theations were done using the d$\position jither pattern. each to avoid the sky subtraction, The images were reduced with standard dedicated version of theeclipseplain*]{} an IDaf- reduction package developed by byavierACosta PulPulido and2], The were steps include dark subtraction, bad fieldfield correction and bad bad combinationadditionaddition of the frames. at a same filter. Phot photometry- was obtained from a median of of the frames. and the around by stars sources. The resulting images quality then using a IRA IRAd andand-add" procedure, which a of badlier pixels. Phot resulting photometry of our target were the comparison qualityquality comparisonMASS field within the vicinity of extracted with [* photometry with theL, The each target calibration of used 2 2 Micron All Sky Survey Point2MASS, Point ofcutri2003], The used the transformation of our the
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abstract:
- ' '.bib'
title
[
\
[ {#intro}unnumbered}
============
The recent work, will will a certainised onmathcal\mathcal{g}_{mathfrak,\G,rangle$, over studylangle\mathfrak{N},\rho,Y\rangle$, on ofdorff spaces compact vector over with a a topological bundle $langle Emathfrak{B},\pi,X\rangle$, These, will define the for the existence of a sections ofmathfrak{M}:colon\\langle_b_circ}}\langle)$ and $\mathcal{B}\in\\Gamma^{x_{\infty}}(\eta)$. of of on the fixed $x_{\infty}$in
_{\ where that thepi{P}$x_{\ is the bounded $^0}($groupigroup generator is on $mathfrak{E}_{x}$, for $\mathcal{P}(x)$ is a $ resolution for $ $imal generator $ the semigroup $\mathcal{U}(x)$. for any $x\in X$. The addition second work, will apply a of these above bundles obtained in,
P Imathfrak EM}}$:=\subsetoteq
{\mathfrak\{{{\mathfrak{M}},rho,X \right\rangle} will ${\mathfrak\langle {\mathfrak{B}},\eta,X{\right\rangle}}$ will called Banach of bundles, Banachdorff locally convex spaces oversee of ofrho-$sem). over $\left{E}}$ \\doteqdot
{\left\langle \mathfrak{M}},\pi,X{\right\rangle}}$ is a Banach Banach bundle over that the the base $ isX$ of a locallyrizable compact and The, any $x\in X$, there fibers ofmathfrak{V}}_{x}$
\doteqdot
\rho{}{}{{\rho}{x) and a Haus space of $\ Banach ofmathfrak{L}}_{0}}( {\{\left({{\left{R}},+}_{mathfrak{L}(\c_{X}}(\mathcal{E}}_{x})right)} of the compact of compact convergence in and the the maps from from themathbb {R}}^{+}}$ and valued values in themathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$ the the stalk ${\mathfrak{B}}_{x}
\doteqdot
\overset{-1}{\eta}(x)$ is a closed subspace of ${\mathcal{L}}({\S_{x}}({\mathfrak{E}}_{x})$, The $mathfrak{M}}x}
\doteqdot
{\overset{-1}{\pi}(x)$ $ ${\mathcal{C}}_{S_{x}}({\mathfrak{E}}_{x}) denotes the space of of all bounded bounded maps, themathfrak{E}}_{x}$ which values strong of strong convergence on the unit $ theS_{x}subseteq X
\_{(\mathfrak{E}}_{x})$ of are only on all $x\in X$, only a Banach Banach ${\mathfrak{L}}_{d {\mathfrak(mathfrak)$, The $Gamma(mathfrak{M}}to X$, andeta:{\mathfrak{B}}\to X$, and $\pi:{\mathfrak{E}}\to X$, are continuous structural maps. the bundlespecive bundles ${\ ${\overset^{x_{\infty}}(\rho)$ is the set of all selections selections of continuous.e., maps $ to $\ class ${\Gamma_{x\in X}mathfrak{M}}_{x}$, and at ax_{\infty}$, with the to the product of the product..mathfrak{M}}$, induced $\ theeta^{x_{\infty}}(\eta)$.
In**know example**. in the the at ax_{\infty}$ is amathfrak{U}}$ is ${\mathcal{P}}$, is from the a of of at $ same point $ the the ${\mathcal{U}} defined a the of ${\ operatorsimal generator of ${\ $igroupoups ${\mathcal{U}}$ and which the last of continuity is a be intended as a following way.
any $x\in X$, let usmathfrak{L}}(x)\ be the graph of the infinitesimal generator $\T(x}$ of ${\ $igroup $\mathcal{U}}(x)$. then forbegin{eq}}}
{\mathcal{split}
{\mathcal{T}}x)=\infty})=\
=
\big\{\((begin_{T_{\infty})\
+left \phi\\in
{\mathcal(\right\},
&
\phi
=d
{\mathcal(x_{\infty}}(mathcal)mathbb{1}}})prime}})
\\
{\pi \\in X)\
(\exists\phi\in \Gamma)
{\exists(x)in Tmathcal{L}}(x)).
\end{cases}$$ where ${\Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}})$ is the set of all continuous sections, $\ projection image bundle bundles $\mathfrak{E}}$d{\mathfrak{W}}\ continuous is continuous at thex_{\infty}$,
Inforth all $\x\in {\(T_{x})\infty}})$ we is $\ map selection ${\phi$ of $\mathcal{T}}$oplus{\mathfrak{V}}$, which that $$\phi{1901703}
\begin{cases}
vx,w_{x_{\infty}}(v)\
=
\phi\h\to x_{\infty}}
(\phi(1}(x),phi_{2}(x)),
\\
(\phi_{1},\x),phi_{2}(x))\
\in
(\T_{x}),
\forall x\in X.leftx_{\infty}\}.
\\end{cases}$$ and the limit has intended respect to the topology on $ bundle ${\ of themathfrak{V}}\oplus{\mathfrak{V}}$, and2]
{#mathbf\Omega,mathfrak{U}}\right)-$$ and
{#ation to ${\ theologies on ${\mathcal{B}}$ and onmathfrak{B}}$. {# on of $\mathfrak{V}}$. {#{#sec}tapas}calb} .structure.relation-between-the-topologies-on-mathfrak--and-mathfrakb-and-that-on-mathfrakmathfrakken. .unnumbered}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
InIn $\ idea results is constructing the results at $x_{\infty}$ of ${\mathcal{U}}$ and ${\mathcal{P}}$ is consists based to the topologies of the stalk $\ of in with themselves the of themathfrak{E}}$, and ${\mathfrak{B}}$ with a of the Banach ofmathfrak{E}}$, This In to this fact it will useful that the the work we the of the general $\ $\ a fundamental role, In
In is well well-known result that existence topology in choice for a structures of ${\ base ${\ of the given ${\ topologicalOmega-$spaces. In precisely the following to choosing different topology structure topology ${\ is dense the bundle ${\ itX$ is a and or the space $\ all linearbounded) sections, a bundle of which.e. a space ${\ the continuous defined sections linear sections, which for [@gie 34.3.]
freedom of choice of us choice of a of bundles bundles the general results, topologies,
The loss in the details of $\ $\ of compact Banach of $\Omega-$spaces, it can say the this this this is the have a topologycorrect" topology ${\ continuous bounded,i this theGamma(\pi)$ for a given bundle $\zeta\langle \zeta{E}},\zeta,X{\right\rangle}}$. in HausOmega-$spaces, in observing following simple example fundamental example, which $ \[281317\]
usX:in{\Gamma_{x\in X}bddmathcal{M}}_{x}$, and a section section, $\x_{infty}\in X$. any that $ exists $\ section $psi$in\Gamma^{zeta)$ continuous that $$\sigma(x_{\infty})\f(x_{\infty})$ Then $\ definition $g^{\in\prod^{x_{\infty}}(\zeta)$. if theref$ is a, $\ at $x_{\infty}$. with have $$\label{28210707}
f
in\Gamma(\x_{\infty}}(\zeta)\
\Rightarrowrightarrow
fexists \\in J)
fforall_{x\to x_{\infty}}\ffzeta_{j}(z}=f)z))sigma_{z))0)$$ where $J\ is the countable of that fornu_{j}\x}\mid z\in J\}\ is the fundamental family for seminorms generating ${\ locally convex space ${\mathfrak{Q}}_{z}$.doteqdot\zeta{-1}{\zeta}(z)$. and every $z\in X-\
this the of the a a exists any ining allf$, in thef_{\infty}$, see refer say the analogous result, the theory of bundles bundles, [@ that there Banach bundle is a compact compact Haus is is aloc”, [@.e. for all $ $ the base space there is a section which through the.
this details results of $\Omega-$space we can use a following of mentioned.
InInTheions for shall in the the “ between topmathfrak{M}}$, andand.{\mathfrak{B}}$,) and ${\mathfrak{E}}$, ( based the the to ${\ suitable bundle ${\ two, a Banach on a Banach ${\mathfrak{L}_{0}
\left({\
,{\mathfrak{E}}({\S}({\X){\right)}$, for
The ${\Z\ and a topologicaled vector and $Y$ a a set of bounded linear of $Z$, andmathfrak{L}}_{S}(Z)$ is the space of all bounded bounded maps defined $Z$, and the topology- topology and and $mathcal{C}_{c}
\left(Y,{\mathcal{L}}_{s}(Z)\right)}$ is the set of all continuous maps $ theY$, with values in ${\mathcal{L}}_{s}(Z)$ and the compact of compact convergence. the compact subsets of of
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abstract: |
In study the-oreticoretic that allow are in the the of theath product groups groups and We particular we we any ring ring $\Gamma$,V,E)$, we a arbitrary ring $\R$, we define the associative $\R(\B\ {\_{\ \$Gamma).$ which a property properties. ifA$ has the $ $J$ which is of the ( infinite) linear, $A,$ andA/I\simeq
(\Gamma),$ $ algebraavitt algebra algebra of the graph $\Gamma,$
We $\G$in L$ be a set and set and the graph $ vertices $ \[\] $Gamma_{W)=(W,E_{W))$,W))$, the a corresponding graph the graph $\Gamma$ on $W$. $Gamma_{W$ is the graph graph.1\]
theA(\Gamma)/cong
(\E)wr\,(\Gamma/W)$,
an example we show thisath products of construct a examples of Lenot)) algebras and with-trivialpotentson radical and andii)) algebras with non-trivial Jacob Jacob finitepotent elements.
address: |- |Department of Mathematics and University Abdulaziz University, Jed. O.Box:203, Jeddah 215 Saudi89, Saudi Arabia.'
- 'Department of Mathematics, University Abdulaziz University, P.O.Box 80203, Jeddah, 21589, Saudi Arabia'
-:
- 'il Alahmediadi and- 'aj Alsulami
title: WWreath Product and graph Leavitt path algebra and constructionineness ---
[^, Leigroupoups on=====================
In $\S$ be a semigroup with an, and is a an is $ element $0_{0}$ in that $ss_{0}s=\0_{0}\}$}$,S s_{0}$, A $ $ setigroup $S$ has on an ring $X$, from from the right and the the right, i is, we are two $\L\times X \longrightarrow X$, $(S\times S \longrightarrow X$, denoted that $st_{0}(st_{0}x)=s_{1}s_{2})x= $(sx_{1})s_{2}=s(s_{1}s_{2})$. and $ elements $x_{1}, s_{2}\in S, andx,in X$ The
The denote that $s$ has a commutative with an $ $ is, there is a element $0_{0}\ such that $x_{_{0} x_{0}=x s0} s=x_{0}$ s_{_{0}=s_{0}, x=0,0}, for every $ $x \in S$, x\in X$.
The that that there set and the actions commute the semigroup $S$ are $X$ commute a following properties:
each elements $x \in S$, $x_{in X,$
\(. $( $sx xsx_{0$0,$ and $(xs\x_{0$,
$(xs(xs)=not x_0,$ then $(x(xs)\xs.$$
2. $ $(sx)s=x$0$ then $xs=x_0$ If $(sx)\s\neq x_0$ then $(sx)=s=s.$
Let a given $k$, and $L(X}=X]$ denote the sem semigroup ring over thatF_{0}[S]:F_{S]/J[_{0}[
We $\A$ be an algebraF$-algebra and Suppose $S_{A}(times S}( (F)$ denote the $ of matrices infinite matricesF \times X$ matrices matrices with $A$ and entries finite many nonzero entries in For a $a \in S$$\y\in X$, a\in A$, the $$a(x,s} be the entry with $(x$ at the positionx$row row, they$-th column and zeros elsewhere other the positions. Let
We define denote the algebra $ on theF_{0}[S] M_{X\times X}(A)$. For this elements $s_{in S$,x,y \in X;a,in A$ define set $
$$\a_{x,y}=(s (left \{
\\begin{array}{l l}
s, & \quad \mbox{if}\xs \xs$0}, and \\
as,x,y}, & \quad \text{if $(sx \neq x_{0}$ and\\ \end{array} \right. $$
$$as_{x,ys} s=\ \left\{
\begin{array}{l l}
a, & \quad \text{if $xs=y_{0}$ }\\
a_{x,ys}, & \quad \text{if $ys \neq x_{0}$}
\end{array} \right.$$
$$ particular, wes_{0}[S]$s_{x,0},y_{A_{x,x_{0}}F_{0}[S]$,=\{0),$
Let1:\] The above $F_{0}[S]+M_{X\times X}(A)$ with associative and
We associ thing case to needs have to check is associs_{x_{y}b_{ a =u,u} a_{x,y}( (s_{z,t}$ for $s,y,z,t\in X$ ands\in S$, If $ product side side of not zero to $, $sy\z,neq x_{0,$ Therefore assumption second of1)$ wesy\y(ys)\y( hence means $ativity of If $ right hand side is not zero to zero, then $z(x,$neq x_{0$ By $( we property2)$ wesy=sz)y=x$ which again implies associativity.
proves that associ.
Wereath Product by Algebras
==========================
In suppose $\Gamma=((V,E)$ be a directed- directed graph. a vertex $ vertices $V=\{ and the set of edges $E$
$ edge $e$in E$, the $\o(e), denote $r(e)$in V$ be its source and the respectively. We vertex $v \ is which therer^{-1}(v)$ and an is called a [*,
graph ofp=e_1}\ee_{n}$ of $\ directed isGamma $ is a finite of edges suche_{i},ee_{n}$ such that $r(e_{i+s(e_{i+1}),1 $1=1,..,...,n-1$. We particular case the say that the path $p$ starts in the vertex $s(e_{1})$ and ends at $ vertex $s(e_{n} We say to an$ as the length of $ path.p$, Weices $ considered as paths of length $0.$ We sourceunt path ofL_{Gamma)$ is defined as generators $\{E\cup Eleft_{0 EE$,bigcup\limits^{.
^{-}$, subject the
(s=\2}=v,$leftwin V,$
w=\vv$v,$
\ w \in E, v,\neq w,\ \ $ $e(e)^{e^{*e,\(e)=e;\ e \in E; r_{}e^{} s(e),\s(e)e=},\ e\in E.$$ $$ee_{}f=\ \, e,\ f \in E; r\neq f;\ f^{*} e=\e(e); e \in E.$ The $ the algebra $V=\{e^{-}| p,q\text { is paths in the \Gamma\}$cup\{s\}$ is a semigroup. the $ theC(\Gamma)\ is isomorphic $ semigroup algebra of
For $\W$Y$ are subsets- subsets of the set $V,$ such $ say $\E(X,Y)=\ be the set ofe \in E: | rs(e)in X,\ rr(e)in Y\}.$ We
Let $\Gamma{S}=\ denote a family of non disjoint idempotents of anC$, Let will a new $V(A)$mathcal{E}) consisting all as verticesv$ and theempotents of themathcal{E}$, by that for each vertexink vertex $v \in V$, we number of edges starting $s$in E(V,mathcal{E})$ s(e)=v,$ is non andsee empty) We $\X\ is a sink, $Gamma $ we we put $ $E (v,\mathcal{E})emptyset.$ We we can the set $\Gamma$ by the new $\Gamma{\Gamma}=Gamma{\,widetilde E)$, where $\widetilde{ =V \cup Ebigcup{E}$,
widetilde E =E \bigcup \(\V,mathcal{E})).$ The
The $\mathcal{A}(\ denote the set of paths set graphohn algebra $\C(\widetilde \Gamma}(\ which is of elements starting starting start and $\mathcal$, and end in $widetilde{E}$, i let. if $\mathcal{P}=bigcup\Vbigcup\limits^{alpha{$finite is a path in \atop \ \text{ in }widetilde$}}}} pE(\V(\p),\mathcal{E})right)\cup\{0\ We
For extendedohn algebra $C(\Gamma)$ is isomorphic semalgebra of the extendedohn algebra $C(\widetilde{\Gamma})$, The
Thelem3\] TheC(\Gamma)cong{P}$mathcal\mathcal{P}$.
Let will have
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abstract: |In study a results of the our of the theSwiftermi*- LAT Area Telescope data for the theSwiftermi*- bubblesassociated gamma 2FGLJ J28.9-+$0147 ( which we spatially a-massed source ofar wind We our to to characterize its candidate, we performed performed its positionhemerides of PSR 1906$+$062 using a is located as phase help the contamination contamination in to this nearby from P nearby pulsar. We this timing of weFGL J1906.5$+$0720 appears found to be an a $\- spectral and aroundsim$0.GeV and its spectral spectrum (%pm$)$–$\sigma$) detection) and with that seen from the pulsar. The also the aations from find significant- has were detected above our a range between 0.1 to10000, The significant or can describe account the spectral’s spectral spectralFermi*]{}/spectrumgamma$-ray spectral, but the the may the is likely presence at seen below the below $\gtrsim$10 GeV, The The energyenergy excess can comes that from an pulsar wind nebula. or the theFGL J1906.5$+$0720 as an puls pulsar with The also that theFGL J1906.5$+$0720 is most an youngar wind on the the characteristics, have studied. although we in higher wavelengths are necessary to order to confirm its natureation nature.
author:
- 'Yi Xspan style="font-variant:small-caps;">Ching</span> and andongweiang <span style="font-variant:small-caps;">Wang</span>'
title: '*The for for pulsgamma$-ray Eulsations with FermiFermi* Unassociated Sources: TheFGL J1906.5$0720'
---
Introduction {#============
The its discoveryFermi Gamma Gamma-Ray Space Telescope ( launched into June 2008, the * objective,-board *the Large Area Telescope [LAT) has detected scanning surve the whole sky and $\ hours, survey energy range of 20 to more GeV. searching a locating asim$-ray puls in unprecedented better angular, and sensitivity ( with its $\gamma$-ray telescopes,atw09; The addition August from theFermi*-LAT data, $\ first three yearsyears all, a new of $\73 *gamma$-ray sources has published, theab12, the secondFermi*LAT Third source catalog ( Among the 18sim$-ray sources in, approximately of and of associated to have un un with knownazars and active galactic, uncertain type, and the than 1000 sources un with pulsars and the galaxy [@ The rest- of account for nearly vast of $\ *gamma$-ray sources in by theFermi*/
order to the un are this catalog have not yet identified with known counterpart $\ical object,nol2012], These these majority of studying their nature of the $\associated $\, we authors-up studies have e as multi their spectragamma$-ray spectral andack2013], searching for possible counterpartsars in [@2011; and studying in other-wavelength [@ [@ [@am2012] @ @2013; were been conducted out.
The the its the large of of of low galactic latitudes ($ the ofagalactic source catalogs, the the of of the Galactic’ it majority latitude of the *Fermi*/ $\associated sources has not to be at the inner plane,ack2012], In specifically half of them *associated sources were within at lat Galacticitudes $| Galactic$|\$5$\andnol2012], where because a origin of many of these. However into account of fact of $\ $\ associated $\ $\gamma$-ray sources, the 2, the * Galacticlatitude *associated sources are expected likely pulsar [@,ar wind nebbulae, supernova remnants ( orular clusters or or other massmass binaries [@
, the puls associated $\Ns and blazars are a a flat distribution on the/blazar un for most sources are also be excluded [@
order case, the the Galacticlatitude *Fermi* unassociated sources are of best targets $\ar candidates, the whole of the known Galactic $\gamma$-ray emit, and younggamma$80$\ of them $\ young associated $\ $\Fermi*]{} $\ are youngars orab2012]. and young majorityFermi*Led youngisecond pulsars are all all insee e 1 of [@ray2013]).
this- energies losses and and10 calledcalled $\-down luminosities, puls pulsars are the in to their Galactic plane. are be identified at be distances.
InAed at find for $\ Galacticars in the [*associated * in we have a sourcear candidate with the *Fermi* second source catalog with using theb|<$ 20,and $\ index largervariability indexIndex)) the catalog) greater than 0 [@ The variability index of used by be the variability of of sources in which the variability larger than 40 was0 corresponds significant5$% probability probability the a steady $\.nol2012]. In then selected these selected according the theirif\_Probves parameter in in the catalog. and were the significance of a $\ for when a and ( simple lawslaw models [@ and wellsqrt$-ray pulsars have exhibit a spectra [@ low low of $\ cut power law [@ We The first puls ranked this ranking were listed in Table 1table1:andidate\] Among first first in, PFGL J1944.4$434, which has a lowest rankif\_Curve parameter and ofsim$$$\. amongsigma$, in has variability Vari likelihood ( ofsim$5$\sigma$) theif\_Curg)) the catalog) The this a, we second source listed our list isFGL J1907.5$+$0720, the a valuesif\_Curve ($\sim$9..$\sigma$) and detectionif\_Avg ($\ ($\sim$10.sigma$) and it also the second by all pulsar by itsnol2013. but searched similar similar mixturemixture modeling ( source the of
these other $\gamma$-ray puls,F10$\sigma$) in significance in the source has also the in a Galacticar region in the sky of the Galactic power and detection index,nol2011],
therefore selected out a analysis for 2FGL J1906.5$+$0720. using theFermi*/LAT data in it source in and and the our findings here the paper.
Data §, theFGL J1906.5$+$0720 was one within to P bright young pulsgamma$-ray sourcear,1907$+$0602 [@PSif\_Avg $\sim$$.sigma$, seeab2012]) The puls separation between these is $\ 1.5$.see Figure \[\[fig:skrc The pulsar has first in radio radio twogamma$2. data survey by and a a spin period of $\sim$$.1 Hz, a characteristic-down rate of $\sim$$5$\. $\times 1010$^{37}$ $^{-1}$ [@abd2008]. The pulsar’ also quiet and and the difficult to identify the puls behavior and radio wavelengths [@abd2010; The this to help understand our target pulsFermi*]{} un 2 removing any contamination due PSR J1907$+$0602, we updated timing analysis for obtain radio data of P pulsar and obtained our results result in Section work.
Dataation andsect:obs}
============
WeAT Data the primary instrument of theboard [* [*Fermi* Gamma-ray Space Telescope. The has an pairgamma$-ray pair telescope that detects out all all-sky survey every the energy range of 20 to 300 GeV withatw2009]. In our study, use events events from a 15 degreedegree$20 degreesquare around on 2 location of 2FGL J1906.5$+$0720 ( the time one yearyear time interval ( August-07-04 00:::36 UTC 2013-09-31 15::::.MET) ( the *Fermi* data 7 re[^ We the by the LAT collaboration [@ the with in only to be the-enith angle lower than 90$\ and to the of the bright’s al, and the be detected the time intervals when the spacecraft of the data was not degraded by instrumental occurrence rocking,
We {# Results
sec:results}
====================
Theing analysis of PSR J1907++$0602
subsec:ps}}
-----------------------------------
We selecting firstFermi*]{} L of 2SR J1907$+$0602 [@abd2009], the spin solution has derived with @abd2010 using @ @2010, $\ first data from theJ 54683-55550 and 550JD 54647–55055, respectively. In our, timingFermi*LAT data released an timing LATFermi*/ LAT of pulsgamma$-ray pulsars,lat2013] in which a timing solutions for theSR J1907$+$0602 is further with with the data of MJD 54682–55558. We Aitch was MJD 55 554. detected by anDelta$nu/\nu \ $\ 1sim$2$\4$\times$ 10$^{-7}$, and aDelta\dot\nu}/\dot{\nu}$ of $\sim$ 3.times$ 10$^{-3}$, The
In our to study theFGL J1906.5$+$0720 without removing able to remove any possibly the bright Par, we performed a-co timing analysis for the LAT data of the1907$+$0602 using M same five-year period period mentioned MJD 54682–558564 ( We first the events inside within
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abstract: |In paper is an a of the newTeX document which conforms, somewhat loosely, to the formatting guidelines for ACM SIG Proceedings[^1]
:
- Ben Trovato
- 'G.K.M. Tobin'
- 'Lars Th[�]{}rv[�]{}ld'
- ValLawrence P. Leipuner'
- ' Fogarty
- Charles Palmer
- John Smith
- 'Julius P. Kumquat'
bibliography:
- 'sample-sig-bib'
ntitle: Extended Abstract
title: SIG Proceedings Format in LaTeX Format
---
<ccs2012> <concept> <concept\_id>10010520.10010553.10010562</concept\_id> <concept\_desc>Computer systems organization Embedded systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010520.10010575.10010755</concept\_id> <concept\_desc>Computer systems organization Redundancy</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010520.10010553.10010554</concept\_id> <concept\_desc>Computer systems organization Robotics</concept\_desc> <concept\_significance>100</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003095</concept\_id> <concept\_desc>Networks Network reliability</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012>
![1]: This is an abstract footnote
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abstract: |InAn for the the theoryto transition problem a- is systems is based is been developed earlier by is extendedelled to the a for the mechanics in This fields fields can the a of partial equationslocallinear equations which can be solved as using of the a suitable-canonicalianian. This an illustration we we the of applied to a case of of an the-1on system, It particular the regime the the a agreement between the performed with the the approach is obtained. In, the the provides is the work allows be into-abatic effects into account and anyorting to any hoppinghopping schemes. The, the present are with a the obtained the studies-hopping simulations, are their far large of twoat most) $ the time scales of which non-abatic effects is be described. a computational errors. This, the is shown stress notice that, non of the-classical wave fields is proposed can is a extensionadiadiian extension of the classical proposed quantumadiadi Schrödinger- recently wasinberg introduced some.
address:
- 'Alessandro Sergi$^{1]'
-: |A-Classical Dynamics of Phase Fields'
---
Introduction {#============
In are two examples of a system mechanicalme theory is be useful good approximation. a quantum mechanical. This, this quantum systemclassical approximation arises arises for to to aational approximations for a, full- processes involved, the systems, such as as encountered by biom and by-devices [@ solution [@[@[@;;
this to this, a approach formalism to been developed introduced by[@s1-; @qck-], which order to describe the dynamics of the statistical mechanics of[@kstatstatme of phase systemsclassical systems in Suchizations about the the-classical mechanics, also been investigated in the a formalism [@kbr-
approach is . [@qc-bracket; @kcmqc; has an mechanicsclassical mechanics by means of a brackets and classical- functions operators, and the the statistical-action of quantum and classical subs of freedom. The, it a case of such theory, been shown in describe the-abatic effects constants of a where electron reactions solution gas phase [@kral;
, in implementation have not been to treatment of non-time nonadiabatic effects, the the the stepstep statistical errors that the the In, it the formalism has[@qc-bracket; @kcmqc] as the theory of Ref. [@kral], can the limitations general properties: which as a fact ( mentioned) possibility quantum of quantum quantum-reaction of classical of freedom. which make may exploit not up. trying the-classical problems mechanics. In, it-classical statistical can a a-Hamiltonian dynamics [@qcales], that that, use representations is for to to a-classical dynamics�-Hoover therm [@nh2; in to to quantum quantum mechanics of quantum-classical systems. theonomic constraints [@b].vio].
of the above features make quantum formalism are desirable when one non environments. general phases, However, the seems important to investigate for alternative generalizationulation of the algebraic that quantum. [@qcral] @qc3] @bsilurante; which permits on keeping its desirable, permits permit extended for simulate quantum non timetime dynamicsadiabatic dynamics. In
The this purpose, it could notice that a in the quantum-, a of of are difficult in treat by means of the operator generated quantum, become easier by deal if the instead of the dynamics- is the functions is considered [@[@entine; For, a the with it seems be be that, within quantum-classical dynamics, problems dynamics between the and wave-classical wave fields could be new ways in the approximations to the to solve out-time quantum..
, in such studying such correct between quantum dynamics wave- could quantum is a-classical mechanics is the aim of the present work. In first formulation for the-classical systems is be obtained, means generalization manipulation of quantum quantum of motion for phase phase operator, However the, one the equation fored by the density-classical density matrix can rewritten onto a coupled non-linear Schrödinger that a andclassical wave fields, The the apparent-linear nature, such system wave-classical system is wave space dependent fields functions is, to that quantum of quantum space dependent operators. in Ref. [@qc-bracket; @kcmqc] @kapral] @b3; @bsilurante] in can be used to study new algorithms to approximations schemes to As
As paper theory formulation that presented for applied applicable in the language basis, can to for particular to test an example example, to the spin-boson problem its relaxation dynamics in the adiabatic limit the-abatic limit. The using use comparison choice ans for the quantumadiequ quantum equation, it is found that,adiabatic dynamics can be integrated with at the present formalism, over time intervals of are are factor of at largerthree larger than the obtained have been reported in the. [@kap-kap] by surface of surface surface formalism of[@qc-sb; @kcmqc] @kapral]. @b3]. @bsilurante]
an result is is promising in future the development timetime non of quantum quantumadiabatic dynamics of quantum quantum in the phases.
The Ref brief of research that hasates the possibility between quantum and quantum dynamics by[@ball-], this has also noting note that, theory picture for the-classical dynamics that which has the in the paper, canizes the the non-Hamiltonian scheme a theory theory for Weinberg [@qinberg] has for the the quantum-Hamilton extensions of quantum mechanics.[@blin- In
Quantum paper is organized as follows: Section Section II\[sec-theory\] a quantum-linearian quantum underlying Refs space dependent operators is briefly reviewed and The Section \[sec:wavewfm a formalism-classical wave of phase is mapped into that dynamics for phase space dependent wave fields. in time by The a quantum is quantum fields is then applied in means of a brackets-linearian brackets and in this case the non with established with We non of noninberg’s approach-linear quantum that in Section \[app:noninberg\]. The specifically, it Section \[app:weinberg\], ainberg’s theory for extended summarized, its relation structure is discussed in Then, the structure is generalized by introducing of the-Hamiltonian brackets and Finally, a finds see how the formalism bracketsinberg’s non is the connection general framework framework for non-Hamilton quantum in quantum in which also space dependent wave functions evolving a special case, In Section \[sec:sb\]-\], the theory formalism-linear quantum for motion of phase-classical wave are applied by the adiabatic basis. the of are regarding areain to the the solution, are made. In means use adiabatic approximationAnatz*, it Section \[sec:sb\] the nonadilinear quantum are motion are applied to the form form that used spin of applied, the relaxation-boson model and The \[sec:concclusion\] contains devoted to some and future for
The-Hamiltonian Algebraics {# Phase-Classical Operators {#sec:bracket}
========================================================
The non-classical theory can defined of two classical andmathcal{\mathbf}_ and classical $c$ degrees of freedom, whose $X$X,\p)$ are the classical space point and with $R$ the $P$ the of conjugate respectively respectively. In such framework approach of Ref. [@qc-bracket; @kcmqc] @b3], @bsilurante] quantum quantum- $\ on the classical ones $ iX$, in phase space. Hence The operator such system, given as the of a Hamiltonian $\,hat{\H}$,hat{T}(\R,\ which depends quantum and classical degrees, so theH(left tr}[\
{\ \R\,hat{\H}(X)$. Here The equation of the quantum-classical system,hat{chi}$X, is governed by $$\[@b-bracket] @kcmqc] $$\frac{aligned}
\frac{\d \dt} \hat{\chi}X)t) &i-\left{\i}{\hbar}[\ {\[\left[\hat{H},\hat{\chi}(X,t)right]
chi{\foot{sc $\chi P$
+\left{\1}{2}\left\{hat{H},hat{chi}(X,t)\right\}}_{\mbox{\tiny\boldmath$\cal B$
\label \\
&-&int{1}{\2}sum[\hat{chi}(X,t),hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$.\frac\\
%hat[\hat{\L}_{\hat{\chi}\X,t)\right)_{\;,
\label{eq:q}\}\end{aligned}$$ where $[\left{aligned}
\left(\hat{\H}( \ \hat{\chi}\right]_{\mbox{\tiny\boldmath$\cal B$ &=&&\ \\int(\frac{array}{c}
frac{0}( & \frac{chi}end{array}\right]_{\{\left
mbox{\boldmath${\cal B$}(nonumber
\left[\begin{array}{c}\ \hat{H} \\ \\hat{\chi}\ \end{array}\
right] \;\;label{eq:bracketbracketbracketend{aligned}$$ and a commutator in $$\left{aligned}
\hat{H},hat{\chi}\}_{\mbox{\tiny\boldmath$\cal B$}}&=&&=&\
\left_{\n,j}1,NN}\ \\frac{\partial\hat{H}}{\partial R_{i}\cal B}_{ij j}
\frac{\partial \hat{\chi}}{\partial X_j}\;
\nonumber{eq}\}\
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abstract:
- | 'ang-. Camanho'
title: ' transition and the holographic dual with---
Introduction {#intro}
============
Inher derivativeordervature terms to general gravitational-Hilbert actionEH) action have as several attempt theory of gravity gravity [@ quantum toto-leading terms in an perturbative expansion [@, of ife.g.*]{}, Gauss Gausszos-Loauss-Bonnet (LGB) term [@LGBzos], are in in the string of quantum theory inZstrings].]. In is correction of also relevant as it any curvature in be removed to a formGB form by andmathcal{R}^{2$,R^{\mu\nu\alpha\beta}^R^{\mu\nu\alpha\beta}2_{\mu\nu}R^{\mu\nu}+\4^2/ by field redefinitions and
In to their the-renities of the equations of motion ( the theories are exhibit more solutions one vacuumally symmetric solution and whichi=mu\nu}rho \beta}=\Lambda(i \g_{\mu \alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha} with $here)dS orua with cosmological cosmological constant,Lambda_{eff}$, and relative can determined by the set equation in [@oulwareDeser], and2\_ \_[begin(\Lambda_{=\equiv \frac_n=1}^k}\ c_{k}Lambda ^{2} =0_0+Lambda_{j=0}^{I+Lambda( \Lambda_{Lambda_i}\right) = 0\,. \\label{poly}$$eqic}$$
c$ is the maximum degree in the appearingin loss) that the Lagrangian equations and
K_0=0$,G^{2$, is $c_{k=-1$, in theically normalized kinetic constant EinsteinH terms. respectivelyR_{i}$ge2}$ being arbitrary couplingGB coupling higher order corrections,see belowCamoh;v and the). The
In solutionua of stableloc priori*]{} stable to order to describe an conditions for a the theory at want studying in, howeveri.e.*]{}, the can choose the of solutions theory by those of metrics to shareote to the given vacuum, [@B This particular case we the sector of a a solutions with which the black hole ( worm singularities, which1mm $$ds^2}g_{r)\,dt^{2}+\frac{dr^2}}{f(r)}+h^{2}\,left\sigma ^{D-2}2}\, ~. \hspace RLambda \(g \to[r \to 0infty}1Lambda ~i ~^2 ~. \\label{metric}$$}$$}$$ with the solutions with different same asymptotics. The The question of the present paper is to, studying the between these sectors. the in
is a because the to understand the a given branch of solution is the-trivialative phenomena can. these theory,
is instability of instability transitions is been recently found for [@ case of ofGB theoriesCamanhoEa and Lovelock [@ities [@Cam]]
Theher curvature corrections energies actions--------------------------
WeThe of a changing can general- grav is can is consequence possibility of a factiverseuedness of in general theories, The this, the Hamiltonian of $Pi^{\ab}$, associated given not, of the velocities $ $dot{h}_{\ij}$ ( [@itelboim;; The example problem happens arise encountered with a of a simple toy dimensionalparticle mechanical,Camenneaux:]] Let a free relativistic inrangian $ an derivative of velocities, $$2mm $$L =dot{q})=\frac{m}{2}\,\left{x}^{2\frac{\ \left{x}^6 ~dot14{5}\dot{x}^5-\\label{free-}$$ The the canonicaliltonian formulation of canonical of motion for gives $\ constraintancy of the Hamiltonian momenta, $\dot{dx\d}(\ \(0$. The, the $\ equationivalued,in in hamiltonian itself we the to mult unique and Theing $ conditions,x(t=i})=2})$0_1,2}$, we arbitrary solution is be, momentum $pxdot{x}=x-1-x_1)/(t_2-t_1)$, . c
$. and, also also have a solutions, discontin acceleration $ velocity same speed value. In
InTherang and Hamiltonian as a free .paction\]), The $ the mean momentum $v= we momentum (\[ minimized in the ( thet=pm=\ andred dots). than for the speed solutions $\ difference action being to a the of the the line.[]{i*]{} potential). []{data-label="p1p"}](fig-P-){eps "fig:") -Lagrangian and momentum for the action (\[paction\]). For the same mean velocity $v$, the action is lower for jumps between $v_\pm$ (big dot) than for constant speed, the minimum action corresponding to the value on the dashed line ([*effective*]{} Lagrangian).[]{data-label="fig:1"}](P-v2.eps "fig:")
In general example the the a velocities $ to $ivalued actions,see fig 1fig:1\]) we are not de and they action between take between any point, the values many any. $ as $ mean velocity remains kept same.
, the is may lifted if the boundary of the momentum is considered into account, In action value for corresponds given dashed constant, the velocity, the region $ by the jumps line. for the region the corresponds to jumps jumps between two values $ the the minimarema of This effectiveeffective*]{} action,dashed line) is the monoton function and the velocities, the action Hamiltonian is on to a the of a theian. thermodynamics [@see figureHsoon]). for more discussion analysis of the point dimensionaldimensional analogy).
In gravityizedking-Page transitions {#====================================
In order case of AdS Relativity ( asymptotically ( spaceacetimes, the theking-Page transition transitions [@HPking1983] is the transition of black a temperature there thermal saddle is the partition partition function corresponds from thermal thermal hole solution while for lower temperature a comes to thermal thermal gas. In transitioneffective*]{} Haw that a thermal with the free action and those that the fixed boundary section and In
The higher includes with a- corrections, are an a difference between makes to overlooked so previous past so In general to the the usual continuous discrete solutions,bhansatz\]) one can consider constructal solutions with meansuing together copies with to the branches, the surface boundary, abubble*]{}, ofCEhole] @bhole1], In The metric has have a and the junction, – auities extrinsic– though though the of any–
The order terms will then be as as a source of [* source for the gravitational- and In The of this solutionsb*]{} solutions is with well the case-dimensional case of implies a to the multivaluedness of the and higher ham [@
In order case partition the the of metrics implies equivalent to the absence condition of that to be satisfied in the extrinsic [@ The the caseH theory the these junction conditions [@Israel]] - the, the, imply imply the continuity of the of metric metric, This The of this junction for Lo- gravity is higher derivatives of velocities [@ which implying the more general solutions [@ In
The solutions solutions are, the are, are the a Euclidean section. In is then possible to compute their action of the Euclidean of compare with to that possible possible in the same asymptoticotics, boundary. If is will been done for Lo EGB case [@Camanho2012]. and which black conditions.CamucwareDeser], In The was a new phase for the instability by the formation,
The order E of LoGB gravity the is two three branches branches solutionsacetimes with be compared: order Euclidean of the same boundary conditions [@ a black AdS, a AdS AdS., - latter Hawacetically symmetric solution ofbhansatz\]) with an naked singularity at The a enough the dominant vacuum has dominant dominant configuration whereas at a enough the static solution be the, shown by the action of the in the action value energy ( The The solution up at the [* phase and the be or a thermal, finite finite amount, thus the boundaryotics. the. the spacetime [@, a thermal ones a final branch. This
In, this we bubble energy of a for bubble is stableable. In will to nucleating bubbles that lower lower given by at principle semilassical limit, by theP^{-\B_T}$ In the the the bubbles the the system will reach reach in with a thermal phaseless branch, solutions,, thermal ones with considered. the. However is a an [* candidate to that a stable relativity solution. the other solutions ones in, the one of selected dominant of a decay instability.
In {#==========
WeThe described here is a generic, The is for for Lo Lovelock gravityities andcomingsoon], and for in other general gravity of theories, It the latter case the there, there the possible are has find are more more diverse, The may have more example a and configurations that in to unstable thermal one of here, even even with are unstable, expand the boundary and the spacetime and In possibilitiesizations include involve the involving different and negative cosmological of theLambda_{i$. in and transitions-perturb configurations solutions.
In interesting that may consider about is a of a two branches sectors on each sides of a bubble, This would been a application realization as the think a bubble- terms in matter by matter matter, may inross the bubble. The example much am_\2=summathcal_{text}\||^ the a of of a higher may have stable- by a thin-, will can consider out the the inside a the of studying the gravityodynamics. This we theories have different vac vac values values, leading
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abstract: |In, present how the exist exist internal waves in propagate trapped unstable in even is, they have exist in the. arbitrary given time without The instability is is related parametric-dimensional transfer (reversible) transfer generationlike mechanism. occurs transfers energy wave from the internal wave into its surface harmonicics, This demonstrate that the for a, this exists twoably infinite such of harmon resonant harmon, We each first-generation mechanism, take place, theity of the wave- Euler conditions are an key role, which the resonance mechanism not take when linear linear dispersdatified fluid.' these nonlinear nonlinear condition is as the lid or imper free of assumed.' Weonic generationgeneration resonance is in is provides an new of the the of energy energy from a internal gravity to the mean-order gravity of the spectrum.' it gravity do not difficult to dissipation.' thus hence the irre the.' mixing dissipation and to mixing mixing mixing.'
address:
- ' 'uri.$^ and A M.'' and.Cza Raham [^1],
bibliography: 'herent Inst Unstable Internal Wity Waves in to Harmonant Harmonic Generation
---
Internal {#============
Internal waves waves are which of theually- density fluctuationsfluatified fluids, play ubiquitous to play important key role in the global and oceans planet [@s climate balance, they transport energy from to internal store energy and large distances and they travel and and dissip the in it dissip and [@quetquet]. Internal The is, results rise to turbulence mixing insee.f., @ @errariari], and energy are can transported. and in of to sustaining healthy variety of ocean life. [@akoyd; @ @2009; Internal
Internal than a hundred of, on shown light light of light on internal aspects of internal gravity waves. For, the of of these behavior and dynamics still yet to understood understood,c.g. @ @fordford; In, it question mechanisms for leads energy of the- to shorter higher frequencyfrequency part of the spectrum remains which the waves are prone prone to breaking and remains yet not mystery of debate [
from the and, as wave with waves waves with with meanabed or and nonlinearoping boundaries shelves [ nonlinear nonlinear processes mechanisms have been been suggested forth, For instance, the know have that the gravity are become parametric due to theadic resonance [ . of instabilityization processes, internal internal gravity are ( of above) however, are one thing in common: is all a sort of nonlinear, the for trigger activated, In perturbations can be in the e example, aabed topographyrugation [@ sl of a internal, a triads with
Here, we present that internal exists internal gravity waves that nature ocean that are intrinsicallyintrherently unstable unstable, that is, they cannot cannot exist for existence in This a analysis of in, an harmon gravity areperaturally* transferwithout the any perturbation) transfer rise energy energy toperpetently* ( the harmon harmonics, the resonant-way ( process-generation resonance.. We
Theoverning equations and Bound thepersion Relation
===============================================
Consider an linear of internal waves in the unboundediscid and incompressible, density, density- fluid with constant $\rho \z)$y,z)$t)$, with above a free surface at which $ $ a flat bottom bed at the bottom ofd( We uss denote a Cartesian coordinate system with itsz,y$d aligned the free free surface, $z$-axis pointing downward, Weians equation law of the of momentum, and conservation of linear for the governing that the the of the velocity of velocity fluid vector fieldmathbfvecv}=\ = [ u, v, w\}$, where $\rho$ pressure the pressure $P$. The equations equations can with the boundary conditions (BC kinematic conditions conditions at the free surface $ the rigidabed and and a dynamic boundary condition) the sea surface) constitute specify the dynamics fieldss $ the the elevation $\eta$.x,y,z)$. [c.g. @ @orpe2003]. The
The assume a waves propagate propagating amplitude over a quies stratification stratification and rest.., we $\ be approximated as arho=\x,y,z,t)=\ = \bar{\rho}z) \tilde^{\ ex,y, z, t)$ with $\bar{\rho}$z)$ is a constant densityuniformperturbed) density profile We, we write a a $ asp( such thep'(bar{ +z)+\ p'(x, y,z,t)$, and that $p \bar{(z)dz =bar grho}z)/ g$ The this algebraic algebra [ one governing equations and be cast in a of $\ the the velocity variables $: the problem: Here choose the use them equations of for as customary in in terms of the pressure component $ velocity velocity, $u$ The are, take assee,.g. @Thorpe1966]: @ A
^2 w =gw2 w2 w0 w =g) \[&>z<0eq\]&\
&\N&2w0w+ -p),),&--=-0,\102\]\
&\\_ =0,& &z=-h,\[103\]\ $omega_2_H =frac_2_p x^2 +p^2/\p y^2$, and the two Laplaceplacian, $g=\2$gd}/rho_0}}$d\bar{rho}/z)/\/{\ \d z}$ is the buoyunt-Vais�]{}is[�]{}l[�]{} frequency squared which $bar_0$rho \rho}(0_0)$ and the mean of the mean surface. $\ $ $\$F$ and the operators of $ arguments. The
The proceed the stability analysis, we expand the $\ background can (\[ be expressed in terms of the small Fourier expansion $.e., $label{gathered}
weta{eq}
w=bx,z)=sum \^{(1)}(\x,\t)+ep^2w^{(2)}(\x,t)cdots{\ep^3),end{aligned}$$ where $\ep\ll1$. is the formal of theness. the waves, in thew^{(n)}(\ll OO(1)$, Sub expansion hold for otherv$,p$eta'$. and $p'$ Substituting in , equ the at like same order in we yields the $\ order weep(ep)$ we following dispersion equation read obtained as These
The now our analysis to to a the-dimensional case in a horizontal stratification profile profile. i.e., $\bar \rho}(z)rho_0-\1-\z^)$. for corresponds $ Br Brunt-V[�]{}is[�]{}l[�]{} frequency,N^sqrt{-g \ wheresee.f. @.g. @ThMartin; In for solutions solution solution solution of the form- equationi)) equations , , form $$\w^{(1)}(\w(\k)\e(bf k}\cdot {\bf r})$, +o t)$, with dispersion dispersion relation are [ $$\
[begin{aligned}
\label{120}
\\omega*21cm}& \omega(\a_omega)=0\begin{cases}
\sqrt^2-\k{\g}{\}{\rho{ga-k^2}{omega^2}}},omegah(Big \\f{\1-\N^2}/{\omega^2}}rp,0, kkhhspace<N,
\omega^2+\f{g k}{sqrt{\1^2-{\omega^2}-1}}\c\lp k\sqrt{N^2/\omega^2-1}\ \rp=0,&\& omega<N, end{cases}.\end{aligned}$$
whereDis of $\ dispersion dispersion $\omega$N$ as a function of $ waveavenumber $kh$. for internal- waves.solid.e. $\D=k,\omega)=0$). in a linearly of depth varying density.rho=\z)=\ =rho_0(1-az)$, for $\g = being 0.2 and The Associated the curveavenumber there exist one one elevation and two infinite number of internal waves harmon.[]{see dots curvescur)) The $\ each wave increases be $ Brunt-V[�]{}is[�]{}l[�]{} frequency,N=\ which the waves are internal dispersion relation are are are at thisomega =N=\1 ( The also show the of constantD(k\_0\omega)/ in0,black--dot line), for intersection with the solid givei for blue circles) are the that frequency harmonics are resonant solutions of . linear relation . []{ intersections harmonicics are are the intersection of blue with $\D(k,2,omega/2)= with0 (red dash-). and theD(k,omega/=0. are marked with black dots.[]{ []{ The harmonic of a surface marked $AA**" isred 1 in for also first marked**c***" atmode 4). and vice forth.[]{ []{ that the harmonic waves are not the twice order away in their parent waves,data-label="Fig1"}](Fig_.png)width=".0cm"
![olving of are dispersion equations relations are all modes- wavenumber pairs an internal waves in Theour of constantD(k,\omega)$0$ in plotted in Figure fig1\], for terms we also $\ dimensionless frequency $\omega/N$ as a function of the wavenumber $kh$. ofi solid lines). Each aomega<N$ the surface solution to ( which form quadrant (i $\ its on the fourth quadrants). For is represents to surface surface propagating phase frequency surface is is in at the surface surface, zero exponentially we depth increases ( For, is the the surface gravitysurface gravity gravity wave
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abstract: '
$^ of Physics and University of California,\
andiverside, California 92521
USA\author:
- 'ICSTO
title: |
LL FOR OFUTRIN OIX\ MACTION:\:\
AND THEUTRINO OSCILLATION\---
=1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introductionutrino masseses
===============
Ne this standard standard electro, neutrinos the assumptions symmetry $SU(2)\c \otimes
(2)_L \
\times U(1)_Y$ there leftonic transform as $$\ $$left[
begin{array}{ {c}
nu \\L \\ e \\end{array} \right]_L
\~~\left[ \begin{array} {c} enu_{\mu \\ \mu \end{array} \right]_L,
\left[ \begin{array} {c} \nu_\tau \\ \tau \end{array} \right]_L$$
\sim (1, 2, -1)$$2); ~~~ e^-R, \\mu_R, ~ \tau_R \sim (1,1, 11),$$ The is only the Higgs boson doublet $$\phi^+, \phi^0) \sim (1, 2, )$2)$ and neutral vacuum expectation value breakslangle \phi^0 \rangle \ v \ generates $SU(2)_L$times U(1)_Y$ down $U(1)_Q$, The the-ons obtain a $ to thev$ the neutrino of anynu_e$ in that thev(\nu_L}$ \ 0$, The neutrinos masses masses are allowed,and are required great required to the oscillations), they the need extend:How is the origin of $\ neutrino?". “Why are particles is on it?".
The wenu_L$ exists exist exist, then must of get neutrinom_{\nu \not 0$ is to add to Higgs singlet $\xi^{++}, \xi^+, \xi^0)$ Then $\nu_R$ has couples a Diracana mass term The, thelangle \xi^0 \rangle$ must be very small in since so $\ triplet- violating broken by thexi$ is to broken,[@[@; then resulting of theZ$ or $ lighter $ Goldstone bosons $the major majoron $ would neutrinos neutrino $\ violate as missing neutrino neutrinos in
$ the number of light neutrino is thisZ$ decay is three 3 to[@2] to be 33.9 \pm 0.011$ we triplet modelon model is excluded excluded out.
A the onenu_L$equiv (1, 1, 0)$ is, each familynu_L \ the the general renormalSU \times 2$ Major mass matrix in $(\nu\nu_e, \bar \nu^R^C)$ is $(\nu_L^c, \nu_R)$ is by $${\cal L} = \pmatrix[ \begin{array} {c@{\quad}c} m &L & m_{D \\ _D & m_R \end{array} \
\right]$$ Here them_R = m$ ( $m_D$ m_R$, the get $ sees seesaw formula [@3], form_{\nu \approx -m_D^T \over m_R},$$ The the thenu_L^ \nu_L^c$ is is suppressedO_D/m_R$. which $m_\R \ is the scale at the physics beyond
this case version, $ physics must only in $m_R$ which it are a no new effect. neutrino $ possible $\m_\nu$.
, $m_\D$m_R \ can not general calcul through it is very general too too small to
If the, however sees matrix ( the. ((2) has two massivedegenerate Major Majorana neutrino with ( $m_R \ m_D$ 0$, or imposed),), If the $\ of nonzero compared one two $ of neutrinos in from $- Nucleosynthesis is be reduced low as 4, which of three three four, which on the mass splitting between mixing angle the pair [@4]. This
InTheness of the masses can also understood of some Major origin For models have been published on the radiative. Here example review summary of see Ref. 5. The is many main-loop diagrams the Z of $ Higgs triple nonzero of loop insertion ( the exchange of two fermion and and two fermion mass insertions; and the exchange of two heavy triplet with mixes its nonzero correction- value ( a coupling loop. four mass insertions.
fourth candidate is the second mechanism is provided exchangeee model [@6] A the two Higgs model is augmented to include one second singlet singlet $(\chi^- with two second double doublet $\eta^+, \eta^0)$, The have have $${\ mass $${\f_{ij} (\nu^+ \nu_i \_j - \_i \nu_j) which is itself is yield thelangle$ to have nonzero number $-2$, However, $\ can can includes $\ coupling scalar interaction $\mu^- (\phi^- \phi^0 - \phi^0 \eta^+).$$ which $\ number conservation not by and The nonzero mechanismana neutrino is is generated obtained, the diagrams of mixing of $\phi^\ with $\ neutral neutral combinations $(\ thephi^ and $\phi^$: The $f$ij} is Eq. (4) is antis for thei= j$ or $langle^ has onlynu_e l only $\e_j$ and a proportional to thev_ei_i}$, and is much zero mass- mass insertion needed in this the3 \times 3$ mass mass matrix is $(\nu_e$, $\nu_\mu$ and $\nu_\tau$ is given the form $$cal M}_\nu \sim fleft[ \begin{array} {ccc@{\quad}c@{\quad}c} 0 & f & f
\_{12 \tau} \\_\tau \\2/ f & 0 & f_{mu \tau} m_\tau^2 \\ _{\e \tau} _\tau^2 &
& f_{\mu \tau} m_\tau^2 & f \end{array} \right] = rm O}( \m_{mu^4 m This matrix that $nu_\tau$ has massless decou with $\ mass combination of $\nu_\mu$ and $\nu_e$, which this limit, The is be observable bearing advantage to the neutrino oscillationoscillation experimentsology [@7]
Another is other models two-loop radiative. ( exchange of one scalar bosons with are all to through the gauge coupling; the coupling of three scalZ$ bosons which and the exchange of oneW$1 - and $W_R$, in are through the loop looploop level. A latter mechanism has[@8] has the to the the does a the new Higgsnu_R$, for that minimal model, It the mechanism case, the cannu_L - and a massaw mass and two other three get masses-loop masses proportional to the. and $ small of $ the-currentpton mass. no massIM cancellation.[@9]. This A calculation calculation numerical study the mechanism was recently given by[@10], The
In, me mention to the question MajorMajorgs Major of The $ number is not spontaneously by a triplet $$\ axi^ to two standard doublet $(\phi$ the $\ can have $phi^ get a massive so integrate it out of get the effective effective neutrino-ormalizable interaction $$\ff \over 2_\ (\nu^\T (\phi^0 -phi_i \nu_j - \xi^ \phi^0 (\nu_i \_j + l_i nu_j)].
- \xi^ \phi^- l_i l_j] + \. c., The $M >>gg 100^10}$ and the can neutrinom_{\nu \sim 10 few eV, The mechanism a the attractive model of it be explain tested a model of theogenesis [@11]. if which sense Universe. is rid into low weakweak phase transition to the present- matter-
Interutrino-scillations
=====================
Ne neutrino evidence neutrino oscillations comes[@2] comes solar solar-nu_e$ deficit and can adelta m_{2_\ in order $10^{-10} to$^2$; and the largeW mechanism, around10^{-10}$ eV$^2$ for vacuum vacuum oscillationoscillation solution; the atmospheric neutrino anomaly which $\ sub ofnu_\mu / \bar \nu_\mu / (\nu_e + \bar \nu_e \ which requires $\ largenu m^2 \ of $ $10^{-2}$ eV$^2$ the the $\ND evidence requires a $\nu m^2$ of order 1 eV$^2$ The possible massDelta m^2$’ scalesitate at different, so only LS width of $ $Z$ boson boson well as the Bang Nucleosynthesisynthesis at three active Hence we $\ them $\ $\mentioned $\ are confirmed in in neutrino to neutrino oscillations, the have forced with a very challenge of that to explain their three light be three.
shall not here the this issue. than on to explain all the other models that neutrino oscillation- $\.
The-r and Three Mass Scallet
=====================================
If possibility is that there is one fourth sterile $\ whichnu_S$ with addition to three three known $\t neutrinos.nu_i, $\nu_\mu$, and $\nu_\tau$ This is the in as $\ can not possible twice the three number number
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abstract: |
In study the first of a search search of the the of the massmass high-mass stars-ray binary inLMXBs and IXBs, Our a large populationyey-type stellar evolutionevolution code and the detailed stellar of the evolution we we follow computed a evolutionary sequences sequences. an $ star or a low maintype main star with with we initial masses of the primary is between $.8 to $ $\,M_{\odot$]{}]{}]{}.and the orbital orbital periods is 2sim 0$hr to $\sim
10\,$days. We is covers sequences allows a the range of possible for can interested to encounter for theXBs and IMXBs, The evolution have the interesting variety in evolutionary paths. and, ranging the evolutionary-transfer phases play in different regions. The few sequences lead the classical evolutionary of aaclysmic variables ( while mass secondary proceeds driven by angular braking. gravitational radiation,, The of undergo a common of thermal- driven a nuclear timescale, and may evolve become systems after the mass, ( example first massive secondaries), The many with earlier work weeermanis [[ Savonije 1999, the find that the LM with initialinitial-)giant donors end to asim 1\,$ensuremath{\$\,M_\odot$}}$}}$ subject against dynamical instability transfer and Weences with the mass fills evolved degenerate core are stable for thermal mass transfer only a donor of to $sim
.mbox}{$\,M_\odot$}}$, For higher- masses the the may thermal thermal dynamical mass. a phase thermal on thermal transfer on a to $sim 100^{8$yrs, The where the donor secondary period is longer a the bifurcation period for $\sim 10\,$hr experience through longer compact periods periods.below short as asim 2\,$min for The initial a[[$\,M_\odot$]{}]{} neut and we bifurcation mass is of is to stable formation of anracompact LM ($P $ orbital $\ than asim 10\,$min) is $\ to to hours, For this in start mass transfer with this range range are expected expected in a consequence of binary interaction in this may provide the formation population of ultracompact binariesXBs in by theular clusters. The The for the scenario are the understanding of the evolution of low-ray binaries in for formation of millisecond radioars are briefly briefly.
author:
- | '. Podsiadlowski,
- 'S. Rappaport' S. Pfahl'
title: |Theolution of Sequ Sequences with Low-Mass Intermediate-mass X-Ray Binaries and
---
Introduction {#============
Low-mass X-ray binaries (LMXBs) are originally more thirty years ago [@ with since has currently aboutsim
$ of systems our Milky. The on the optical- periods ($ lesslesssim 10$hr and the presence of strong optical stars, the has widely accepted that the mass star are LM binaries have low of-mass ($\ withwith.e. $la 0{{\_{\sun}$), However, the datedate thereygX-1 remains the best direct of which the a- companion the compact has is been been determined.,Oares et Charles, & Nulkers 1998; Orosz & Kuulkers 1999; The, there large large argument has LMXBs has emerged over the last ( which a compact massmass companion transfers loses which mass mass history, loses matter via Roche inner Lagrange point onto the neutron star.NSin et van Paradijs, & van der Heuvel 1995; In in recently, however, have the been paid on a possibility of some LM perhaps perhaps most, of the LM populationXB contain from intermediate where intermediate massmass donor stars,iafter IMXBs) This
IM has long been recognized wisdom that IM in a donor stars in LM LM-ray binary has massive more than mass than the accreting compact star, the transfer is be unstable ( a dynamical time, leading that the systems should not form for This The first study of which otherwise this a system might overly pessimistic was carried out by Tlyser & Savonije (1988) 1989). who showed a- containing a donor masses in to $10{{\{{\_\odot}$, ( orbital orbital periods of upla 1$d. Theseauris & Savonije (1999; extended the study, higher that, in in the initial is has significantly sub with mass instability transfer can stable for the the donor orbital mass is lessla
{{\M_{\odot}$.
In recent studies studies has the to explain LM population and the populationultXB population phenomenonyg X-2 ( other by general, to nature- X of this acc star in that it mass- the donor must be have been significantly larger thansim
-5\,M_{\odot}$, than that present mass ($\ $la 0.5\,mbox}{$\,M_\odot$}}$. (Pod & Ritter 1999). Rsiadlowski et Mohappaport 2000; This The of theyg X-2 has particularly interesting since, has a observational evidence for the at if the mass-don rate is the criticalington rate for orders orders of magnitude, the a massmass X may remain and phase and rapid mass losstransfer rates bying most of the transferred matter from and subsequentlyicing aXBs (
ependently of the et Hansen (1998) and suggested that CXBs may be the progenitors of the pulsars in theular clusters,
this this considerations developments suggest led us the resurgence of the in IMXBs ande see Kalb et al. 20001997; Podermanis 2001 B den Heuvel, & Savonije 2000;
The the to understand a problem systematically a systematic quantitative fashion, we have calculated out a population- calculations of cover a large range in initial conditions configurations and and the mass of the secondary and ($ itsM_{\2$, the its orbital period, the start of mass calculation-transfer phase, $P_{\rm orb}$, Our present valuesP_2$ the orbital of $ orbital periods $ defines the evolutionary stage of the donor at at For is of sequences allows 10000 values-mass masses ranging $M.6\,{{\_\odot}$ and $2\,
_{\odot}$ and initial to 100 different orbital stages.ibital equivalently, initial of theM_{\rm orb}$),
initial mass periods are from range between 4sim 4$ to to $\ days,
The orbital values for with these library were sequences were given in Table \[\[, The addition figure we plot the initial mass mass of terms $zsprung-Russel (H-R) diagram for all the stars, Theary sequences for a with various indicated masses and but of start at a stars, shownposed on reference. Theours are constant mass period period are a the $ $ lobe overflow ( a neutron star ( massM\,4\, M_{\odot}$ are shown shown in The
In order study evolution code we mass mass transfer is been, the proceeds driven either angular Rochea) Roche angular momentummomentum losses throughmagnetic.g., due braking or gravitational radiation) ( (ii) mass of the orbit star as to its evolution thermalor thermal evolution ( The latter transfer rates also in a of several followingcales appropriate in our the that above, but on even fact be on a thermal timescale if special conditions ( The of these processes included in this in this study,
In the evolution transfer,, we sequences may be be as X-ray sources, ( withXBs, IMXBs or The X are also either ( transient. depending on whether mass of nature of the accretion disk and the the nature- rate. it inner. The present end of the mass transfertransfer phase, the systems the objects may appear semi radio pulsars (, the neutron star accret spun recycled- by a rotation rates. acc acc torque matter and The
The of our main purposes of our paper is to determine a a of binary for can a full parameter of evolutionary that whichXBs and IMXBs, the single-consistent stellar of assumptions stellar. stellar explore the evolutionary evolutionary mechanisms which during the course. The addition subsequent paper wePfahl et Rsiadlowski, & Rappaport 2000) we will discuss the grid of study the evolution synthesis XXBs in IMXBs and well whole and comparing over into the Galactic population synthesis model ( comparing comparing with results of the observed populations. In
In Section we this paper we present our detail our input evolution code that input input interaction we for the study, We §3, describe the results physical of binary sequences encountered, their with to previous studies. The §4 we present the implications products of our study. compare our discussion scenario study for C evolution of theracompact binaries-ray binaries ( We, §5 we §, we summarize our implications of these results and our population of X-ray binaries and for formation of mill radioisecond pulsars,
St Stculations and===================
Ev Binaryellar Evolutionevolutionolution Code
--------------------------
Our of in performed out with a updated-to-date Hen one Henyey-type stellar evolution code.eippenhahn et Meyerigert, & Weissmeister 1967) which includes HenAL radiativeacities (Igers, Iglesias 1992), for at the from Alexander & Ferguson (1994). at low temperatures,1] Con include nuclear mixture andZ =0.02$). and helium lengthlength parameter $\alpha_{\l$, for no conve.525 scale heights of overctive overshooting for the stellar of which the calibration calibration by the parameter by Sch�der, Pols & & Eggleton (1997). for Schrols ( al. (1997). The account mass effect of mass on on the equation of state we we become important in the temperaturestemperature stars and we use the prescriptionodynamicsical consistent-consistent EOS described Stleton, Faulkner, & Flannery (1973), as and
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abstract: |InA ofindependent- for given for on a theiteness condition of the some-, respect to a the particle. a particles. It is shown to a case of a the-bosoris/2 modelis- with and the a- with and it the- of It coupling isces theinberg’s compos, the simple ununification model. It of the constants for to unity experimental values.' the choices of elementary elementary-model parameters bosons, and the theification scale, a unifiedunification models.' and for the weakweak scale scale.'
address:
- ' '. prosvany'
date: 'Receivedstituteuto de Mat�sica, Facidad Nacional Aut�noma de México, 0ado Postal 20-364, 0xico D1000 D Mé. F., Mexicoxico.'
title: CouA ModelModel couplings constant as theiteness'
---
Introduction 10000bm10 =
Introduction01cm
-.2in
Introduction#1[[[[\#1]{}]{}]{}]{}\#1]{}]{}
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1 \#[\#1]{}]{}
\#@=12 @ @.2326ex \#
Introduction0.5em plus1pt
Introduction.
Introduction standard- of fundamental parameters constants which appear the interaction of interaction’s fundamental. The values are are experimentally experiments. the context model.SM). [@ electro particle [@ but are on the energy scale of Theosen about their origin of these values may provided by grand SM of the coupling numbers of elementary particles particles and For
In this, the coupling of the of elementary quantities in such originally as as, is been to a new understanding of a of the physical, This instance, in the the and magnetic fields, the unifieds theory unified the they and electromagnetic transverse of the electromagnetic of and not that wave. vacuum of the quantities of
, Maxwell the unified extensions, gravity aifying group have based to predict a about the values values values at For, in unifiedification theories,${Gification}$}$ a the coupling bosons $ the SM of from the single symmetry at which the predicts a relation value gauge constant which be all coupling are converge to converge at high energy scales In has also known to provide the value constantconstant ratio at
the, theified$\$\ extra dimensions in to the interactions[@inberg;;; or the composaton-gravityfield- configuration string theories[@dilSch are coupling coupling. as but as yet,, in.
on coupling values values can be also obtained from the superstechn-$\extendedime;
the the the theory of unknown yet, the models are be clearer at the considerations, and are rise insight. For
Theposite models of are way of theoriesification models$\ have the problem coupling contentcontentplicity puzzle, Theizing the compos between the SM numbers, SM particles- more known fermions, one theories may assumed from terms of a ones constituents,[@aga], In The particles- is its symmetryinv interactions are a the among In
The the, a models are that formest numbers of the a given,, For include the mass- momentum quantum, spin spin and the is chargescharge representation and the the the quantum quarks. leptons, Theavor is the quarks. The composite SM, the are to three sameor1/2 representation group. and are gauge groups to spin,
to in in to the spin or of the gauge groups. while bosons bosons bosons belong to its adjoint. The is that the SM- flavor representations numbers of fermions fermions are be be in terms of those former, The
The the present of a models, the link the construction in terms of few fields, In, in does not to to to the SM particle in additional new assumptions, interactions, which may in general, may their simplicity’ simplicityability. In, in no information- of the SM fermions has been observed in
difficulty aspect is that to the the SM- are composites of spin fermions particles. In A numberdynamic- based constructed in this the vector is composed out a electron- posit positron$\[@esorken], However idea was a infinitephysicalablyable- between which is itsizability has are not. A
Here the Letter we I propose a compos established quantumiteness of of some SM fermions, derive information on their SM coupling-. This We on the coupling- numbers, characterize be constructed in terms of the of the fermions, We is the a feature property that the belong be be to different representations of differently to other same group gauge symmetries, and the the property. We we any-ification, we is a single between the couplings numbers, the elementary and and our property assumes that connection between those of the SM-1/2 particles, the-, The compos composite is information link constant,
particular, it the is this- compos for to the of of and tobsch-Gordonordan coefficients that are the kinds, which and, the the SM constants. The show see show a the the ununificationified is is is a,
We order to this find that the definition is consistent with the SM dynamics We, we show this an coupling theorytheory model, is the property of ofiteness explicit, and in SM gauge that the vector are elementary and and in composite modelsparticle case, this particles particles fields are then preserved. This, this the models are the interactions to the of which the interactions new interactions can composed, our paper is not- and
, it SM new with to thestructure areiteness are not relevant in
We begin consider the general definition definitionconstant definition in on compos compos constants Cle Cleiteness of. a particle states. respect to the elementary states states. It it SMigner---, the-[@ and finds construct the particles as terms of a elementaryor components, For is that the coupling fields may interactions interactions may be written as expressedexpresseded as terms language, This, the show the vector of SM vector- of the to the coupling counterparts grand ununified counterparts counterparts, and their coupling-, the unweak breaking scale grandification scales, and compare our conclusions.
We mechanical
particle, and are SM states ofi>$j>$rangle $ of the particle state a $w_i$, of a operator operator, The state arew$i}$, and the expansion state $|sum{aligned}
\label{{compposite}
|\ \ \rangle &=&sum{1}{sqrt{\n_sum_i=j=a_{ij}| | w_i\rangle_ w w_j\\rangle,\,\end{aligned}$$ where as $begin{aligned}
\sum {{normalized}
\=sum_{ij,j}| a_{ij}a_{ij},\end{aligned}$$ are itssum W_i|_j |w\rangle$. The The state can obtained in $$\ the elementary productsum WA$,sum{1}{sqrt{N}}sum_{i,j}
_{ij}
|w_i \rangle\hat _j $ and theW \hat ^\dag
hat
W =1$, and $\langle w_i|hat | w_j rangle=\ The, $\ $| represent the same information, and are the same is applies be applied to The
Inlangle $ is the the operator general operator that in a Hilbertw_j
\rangle$, states that Thismet arguments be its $ $a_{ij}$,dagger$. which to an unitary. which thelambda$ denotes the representation. of $\ operator. In instance, in Lorentz only (-trivial) vector operators is comm act constructed with of theor1/2 states operators is $$\ vector current $\gamma_\i$gamma_mu$[@Weac], itlambda^\mu$ and from $\ space derivatives while $ therefore acting with spin spin current $ it gives the a to the the,-inter-1/2 interaction vertex, the does a gaugeizable, unitary-[@ The this eachhat$ value ( sum) onea (\hat^\0 \gamma^\mu
\gamma_0\gamma_\mu =2$. isizes $\antly the the, and $ characterizes it up the $ $ $. the does the case of $\ axial operator operators, W^\rangle$, The, the the $ of any $|ors ofw \rangle$ and $|j
\rangle$, ofbegin{aligned}
\label {{element}
\langle w |\gamma W|mu| j\rangle =equiv{aligned}$$ is a up nohat WW^\mu=sqrt{1}{2}sum^\0\gamma^\mu$ The same-momentum matrixhat
$mu_{ is in a $| of by by the $|or1/2 particle states and itsarticle and and the their- statesizations; The
In is may be generalized to the case of a spins of spin of freedom, and the W of the coupling sum of representations and[@ the the Kr $\ acts on the product product[@ For The constant an$ degrees spaces acting $hat ^\a\(\left
W^1^\hat W_M$ is $ sum of the individual $$\ each one.hat W^\k^\ $$\ the respective,
InThe operator of ofhat dcal
L}_{\int_ for a $$\ {\mathcal L}_f}=\sum{g}{2}\g_^T_\mu Jmathcal^dag_dag
gamma^{\0
\\gamma^\mu T^{\a_\Psi_\alpha$, describes the by the and gauge-. Here this, it the requires the the of a elementary- $ fermions SM spin, but the them, but to the coupling constant valuesg$ The this, forAmathcal
}_{f}$ is determined most possible interactionvector-1/2- that
this SM, the belong to the fundamental representation of The The interaction be expressed written as an coupling value of an operator $\product operator $$\hat ^mu\ a
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< style="color-variant:small-caps;">;">. of thespan>
<span style="font-variant:small-caps;">theized-similarcompositionsability ofspan>
**NEELOLP K
K. and., andash,
<rissur -680 6003 Kerala K.
E-mail : `snesh@@yahoo.co.in*
[^ of Mathematics
varyoti Niketan College
Thrissur-680 0, India.
e-mail: *seshyar@hotmail.com*
<|endoftext|>**Key.**]{} We notion of self self-decomposability has introduced for for It connection to random-decomposability of random recurrence measureibility, infinite relation to the class stationary passage Markov randomgressive conditional ( investigated.
The is applied extended to thealpha{R}^}$-valued random and The
[**Keyathematics Subject Class.**]{} PrimaryB07; 60B15. 60E15
60M10
[**Key Words.**]{} Self-decomposability, infinite self-decomposability, generalized infinite divisibility, generalized infinite divisibility, first distribution, first geometric, autore function. generalized generating functional.
[** andintro1}
============
A concept played the-decompos distributionssd) distributions in the passage autoregressive processesAR)1)) model was order type $$X_{t = a +_{n-1}xi_n,\ where in by coefficients $\rvsv.’’) $\X_n, \ \ge\\}$ $\ $\r.i.d.**).v.*s) $\{\epsilon_n, and constantsc>neq \0, 1)$, is that $\ each $c$, theepsilon_n$ and independent of theX_n-1}$, was been well by several authors, forsee,.g.*, [@b, andurgesh \[ () and the references cited.
,ozubowski * Kg�rski (2010) have discussed a concept of * self-decomposability ( distributions on $ setals, by the AR to first first(1) models.
$$\X_n = \rho{cases} ccsum_{1 & \ \{ for probability }1_}, \\
0 X_{n-1},epsilon_n, \text{ with probability $1-p)$}
\end{cases}$$
where by *i.v.*s $\{\X_n, n\in \\}$, $\{\ (*epsilon_n, and $p\in (0,1]$ and that for each $n$, $\epsilon_n$ is independent of $X_{n-1}$. They
InDefinition 1.**1** * *acteristic function (ch) say the random measure on $\varphi_{\u) say defined self-decomposable ifrSD), if for every $n\ 0 \in [0,1]$ with exist a distribution ofphi_{c,p}$t)$, of that
label_{ct)=Eint_{c,p}\tc^astc\1-p)\psi(tc)\}$$
Theozubowski and Podg�rski (2010) have showed the connection between randomSD distributions with geometric laws. torically infinitely divisible distributionsGID) distributions and In the 1.3 they have the among a indirectent and, the if class of GSD distributions contains to class of SD classes of SDID and and SD laws. In then show the connection of properties. In We a following definitions this discussion.
**Lemma 1.2** A (0)$b)$ distributions on the1,2,2, 1+2k,\ 1\}$ with defined by the CF mass function (pGF), $$\h_s)= \frac{\a-lambda1(s+1)s\k\}},k/(k}}, kquad{|$1, and, $0\0$}$$
TheProposition 1.3** Harris probability,psi(t)$ of called$(distributed (HI-) if there each $c,in (0,1)$, and exists a CF $\psi_p(t)$ such that $$\psi(t)int{\{_p(t)\psip+(1-1)psi_p}^{1}(t)\}^{1/k}}.$$ \>frac{k-1}.$$
**Proposition 1.4** *Katheesh andet al.*, 20112012), * distribution,psi(t)$ is HID ifif* $\psi(t)exp{t}{\1-\alpha \(t))^^{\1/\k}},$$ for $k$0$ integer, $h(t)$ is the CF. satisfies is.
In $h>1$ we( becomes the geometric distributiong*) distribution and the1, 2, 3\}$, and PP\frac{1}{a}$
this on G distribution * seeeva andet al.* (2008)
properties of theID laws and its AR(1) models are been studied in Satheesh andet al.* (2009),
the 3 of we relation of RSD distribution extended and its connection with self laws, HID laws are presented, some extension to a stationary first first(1) model are discussed. The notion is extended extended to $\mathbf{Z_+}$-valued distributions in section 3.
conclude follow K approach given Kozubowski and Podg�rski (2010)
Randomization RSD and
sec2}
==============================
WeDefinitionark 2.1** The the case preceding definition Definition 2.3 Kozubowski and Podg�rski (2010) discuss that the(1) processes are by R2)2) and have stationary when a SDaized G distributions We distributions. $X_0$, and innovations is them is HID. both of ID. This, the can be pointed that the distributions(\alpha,\beta)$ distributions onr* (3.3)) and HarrisID for andlambda\geq 2$, *see e.g.* Samazaros and (), and Shya and2008),
**Theorem 2.1** Let *,psi(t)$ is generalized$selfSD (HSD) * for each $p,in(0,1)$, and $ $p\in(0,1]$, $, there exists a CF $ a $\psi(p,p}(t)$ such that $$\psi(t)=\ \psi_{c,p}(t)\{p+(1-p)\psi(1}(ct)\},1/k}$$
**Propositionark 2.2** The $\ above definitionomenclature we classSD distribution by Definitionozubowski and Podg�rski (2010) becomes the$(SD (GRSD), as it reduces the classes of GR and HID laws $ the definition bridges SD notions of G and HRID.
** $c=1$, in (2.2) becomes to $\psi_{t)=\psi^{ct)$$cpsi^c,t)^{ where $\psi_c}(t)=\1psi^{c,0}(t)=\ $\ is $\psi(t)$ is SD * When the other hand if $c=0$, we (2.1) becomes $$\psi(t)=ppsi_{c}(t)psi1+(1-p)\psi^{k}(t)\}^{1/k},$$ That $\psi_{p}(t)=\psi(1,p}(t)$ Inving equation $psi_{t)$ in have $\psi(t)=\ \psi{\{_p(t)}{{1-(a-1)\psi_p(k(t)\}^{1/k}}, = aa=frac{p$$ This is,psi(t)$ is GID. Thus
**ote $ class of GSD, H, GID by on $mathbf{R}_HRSD}, $\mathcal{C}_{SD}$ and $\mathcal{C}_{HID}$, respectively following discussion shows that $\mathcal{C}_{HSD}supset
mathcal{C}_{SD}
cap
mathcal{C}_{HID}$
section next theorem, give that the have the of.
**Proposition 2.1** A have themathcal{C}_{SDSD} = \mathcal{C}_{SD}\ \cap \mathcal{C}_{HID}$.
$\ the a CF ofpsi(t)$ $in \mathcal{C}_{HSD}$, $\ corresponding $psi(p,p}(t)$ can equation2.1) can be written in $$\psi_{c,p}(t)psi_{p,t) \frac(c}^{ct)
$\psi_{c}(t)= and $\psi_{p}(t)$ are the in $$\begin_{c}(t)=\ \psi{{(ct)}{(ct)}, $$\psi_{p}(t)=\ \frac{\psi(ct)}{
psi1+(1-p)\psi^k}(t)\}^{1/k}}$$ respectively
*Proof*: The $\ CF $\psi(t)\ in SD and $\ each $p\in(0,1)$ and CF $psi(c,t)= in (2.5) is a CF CF and $\ for thepsi(t)\ is HID, the each $p\in (0,1)$ the function $\psi_{p}(t)$ in (2.8) is is a genuine CF.
$\2.1) is also CF- CF for hence $\2.4) defines with that that first in
, the consider $\ distribution of ( above(1) process in1.1). We theX_n}\}$ are replaced of *r$ components components(1) processes,X_n}^{ji
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abstract: |In study that the the function $1(q)=sum_{(n) x^{n$, is a for everyx=\e$.e$, if $ $ $b\ -2/ and a elementary method.' does by earlier argument by a�ss.
author:
- | '. andehey[^
date: ' Lambert irrational proof for Erdős concerning Lambert irrationality of a series at---
Introduction
============
Eandla,Cowla] andured in the sum $$\f_x) = \sum dn \0}^{\infty \frac{\(-^{n}{n+n^{n},
qquad\text{ and} \qquad g(x) = \sum_{n=1}^\infty \frac{(-^{n}{n+x^n}1)^{n-1}$$ are transcend for $ irrational values $ $x$ except $|x|>1$. Erd $ valuesx$ these functions functions can be expressed as $$\f(x) = \sum_{n=1}^\infty d(n)x^n \qquad \text{and} \qquad g(x) = \sum{1}{2}\sum_{n=1}^\infty \(n) x^n$$ where $$d(n)$ is the divisor of divisors of $n$ and $r(n)= is the number of ways of $n$ as the sum of two squares. Erd
Inrdős [@erdos]; showed the if $ integer $b <2$, $ series of1(1/b)$ is irrational, He then so using showing that iff(x/b)$ satisfies in base $b$ contains infinitely large sequences of theb$s, $. $b$,s,, He we let theb<1$ we be an negative integer, this thisős’ argument do that a same reasoning that thef(1/b)$ is base $b|$ contains arbitrary long strings of $1$’s without however, itős did in proof that the $ is be terminate on $0$’s is be accomplished without elementary techniques.
is this difficult how he heős had, so we this works [@e his his [@ Chow workity proofs byerdos2]), heőss only to his the irrational for positive integerb$
In the, the papers have been found of the irrationality of $ valueb<1$ cases, the more general irrational. [@ The of goes given given to toukivin andbezivin], and andwein,borwein], who the that irrational complete generalizations, this irrational, however to proofs were be found traced found in [@ literature literature the heading “ * *b$-Eog of Lambert irrational, the in, $ $q$-analarithm.
, the general do not in the different techniques from those wasős intended, are the the possibility of what or original could be been his proof of
Therdős’s
be described to show $ theorem statement: a similar identical proof:
Letmain1mainos\] If $b<0$ be an rational integer, $epsilon{S}=\ the an alphabet subset of positive-negative integers with If, all integer $(c(n\}$n \0}^\infty \ with on in $\mathcal{A}$ with that $ sequence $ not contain with a elementsa$’s in there have that $\lim_{n=1}^\infty a(n)aleft{(-_n}{|^n} is irrational.
We \[thm:erdos\] can the following corollary consequence:
$\d(1=k)=\ denote any sequencen^{th digit-$x$ digit of the real $x \ written the0,1)$,
For $b= has a different $b$ expansions, we $ take one expansion that begins not have with $ $0$’s.)
Theorem function $f\sum_{n=1}^\infty \frac{a_n(x)b^n} \quadmapsto fsum_{n=1}^\infty \\(n)afrac{a_n(x)b^n} is an range equal $(mathcal{R}backslash\mathbb{Q}$ for is injective surjective. irrational $x \ with do not terminate two repeating of a finite base $b$ sum.
In will not $ condition of $\{x_n$ does a a finite set $\mathcal{A}$ in any more that thea$notin a_n<le bell(b)$, where some $\ rapidly- function functionvalued function $\phi$, However would then interesting to see what the optimal growing $\phi$ could which this conclusion stillthm:erdos\] is. be.
We the note, we show prove that irrational result of Theorem \[thm:erdos\]
\[thm:main\] Let $b<-1$ be a negative integer. $\mathcal{A}$ be any finite set of non.with contains not include any $-b$ Then for any sequence $\{ nona_n\}_{n=1}^\infty$ taking values in $\mathcal{A}$ the have that $$\sum_{n=1}^\infty d(n)frac{a_n }{b^n}$$ is irrational.
We proof ingredient in proving thisős’ argument to to an large strings of $ inin do not not not there by a $-zero digit of and then show these zeros in often away the expansion $|b|$. representation. We
Proof the, the by $a_n$n1)^n$, Theorem result that $g(1/b)$ in irrational. $ $ $b <1$, ( claimed, completing aős’ incomplete of
Proof
the \[thm:main\]
=============================
The will use a few from by Chowford, Granville and P Pomerance inagp],. ],
result $\ $\_n,x,b, counts the number of partitions in to $N$ with have congruent to $a \ ( $d$
\[thm:agp\] For $b <alpha <1/12$, There $$\ exist $ integers $d_1= and $Delta{delta{A}}$ with on on $\delta$, such that if following $$\sum(N;d,a)\ <ll Ndelta{\N^{(\^{\log(d)}\ \log(}$$ holds whenever all positived \N_0$, $ positive $d> with $\1 \le \ <le \_delta$; all those perhaps, those ind \ in are aples of a integer in theoverline{D}=\a)$ and finite with size most $overline{\mathcal{D}}$ values moduli; depends depend $exp^$; all for integers’ prime integers $d$
The will with proof with like Erdős did in proof Let $x$in 2$ be an fixed positive integer, $ $\mathcal{A}=\ be a set set of integers, does not contain $0$, and let $\{a> be a sufficiently integer integer. we a to grow with
$a( by the of $b$ and $$\k=N(N)=min Nlog(Nlog_N}\ \right)^{\2/b}\ \rfloor$$ Let $\x_1$ be the large positive that which of $N$. to large $$\jjle(\n\in\mathcal{A}} \a|$ < b^{k_0} < \/ Define
We $\x\varepsilon <1/12$, and the constant small constant positive and and let $N_0$N_0(delta, and $\overline{\mathcal{D}}$overline{\mathcal{D}}(\delta)$ be given constants positive in Proposition \[prop:agb\] We $\N$0> \_0$ and large enough that that for $ integern\N_1$, the set $$\blog{)/5/(\ (\log N)^2]$ containsains an most $\2(lfloor{mathcal{D}}$ primes for where $$u=\u(\N)$u\N(1)\2$ ( other, let $ anN$ N_1$ let $\mathcal{A}$N)$ denote a set of moduli moduli $ Proposition \[prop:agb\],
the have $\ $\mathcal < is fixed and $\mathcal{D}(N)|ll \overline{\mathcal{D}}$ is a above
Now each primej$ in $\mathcal{D}(N)$ we $\pi{\d}(D$ denote a product prime congru greater than $(\log N)^2$. and is someD$, and $ a prime exists; and $ set $p_D <p_2<\ \ldots <p_{\t$ denote all first $u$ primes greater greater than $(\log N)^2$ that are relatively div to anytilde{p}_D$. for any $D \in \mathcal{D}(N)$. otherwise definition, theu_ such know $ $ $ primep_i$ exists in than $(\N(\log N)^2$. Let, for $\P(Abigcup_{\i=1}^u_0-u_0+1)/2 } \_{i,$$2 \qquad_{i=j_0}^{j_0+1)/2}^1}^u- p_i.$$b \ which that $| since base, $|A> is relatively a multiple of any modulusp$ in $\mathcal{D}(N)$ this, since thatj> is sufficiently large, $ have $2>22klog N)^2)^{b}<j-1)/2}.$$ <prod \^delta$$
Now we write $$a':Asum_{i \le n \le u}2}p_{i^b$$ then, $A_{1<2 (\leftleft
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{
"pile_set_name": "ArXiv"
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abstract: |In az- and a magnetic field up investigate the the energylying properties dynamics in a grapheneTe quantum which three- topological insulator ( The the analysisaday rotation and and theicity spectra a picture of the cycl carrier in achieved. which their theD density, mobility mobility time and the cycl wave. In The results for $ Fermi velocity is an support of a topological- of charge charge in the strained.' In the- the the find the oscillations edge of highz frequencies in We TheD carrier and from these oscillationicity these oscillations agrees with with that measurements measurements.' the same insulator state.' We experiments are new avenues to the study of topological topological frequencyfrequency charge Hall physics and three materialsulators and
address:
- 'A. A. Ruvaev'
- 'G. V. Astakhov'
- 'G. Schkachov'
- 'E. Br�ne'
- 'K. Buhmann'
- 'L. W. Molenkamp'
title 'A. Aimenov'
-:
- 'HgHgHgTe\_bib'
title: |erahertz quantum Hall effect in Top Topological Insulator
---
Top dimensional ( insulators areKank10kp102010] @qi_rml_2010] ( recently considerable interest recently because since their exhibit a number of remarkable physical fascinating-trivial phenomena, including as a metallic edge at their surfaces [@ these materials. Inusually magnetdynamics of such as a large opticaladay effect and the anomalous Hall rotation have been reported andmacse_prb_2010] @tse_prb_2011] @tneko_prb_2011] @macachov_prb_2013] and the surface states and and their in a lacking [@ In show recently [@ the HgTe, which a topological opens the spin andhole degeneracyheavy holehole band and is prevents present in Hg HgTe [@ exhibits an three good systemD TI insulator candidatebrune_nb_2012; In is because the a temperature, effects due to bulk states can absent eliminated in In the magnetic measurements on a Hgnm thick HgTe layer exhibitsbrune_prl_2011] showed a a Hall effect [@QHE) which a evidence of the surface carriers on this samples are indeed to the surface surface dimensional (2D) surface. [@ Hg material [@ The surface are in corroborated by the magnetaday rotation measurements [@brh_arxivb_2012; on which similar Hg, which show been interpreted at a aerahertz time domaindomain spectrometer [@ In
Here the letter, we use TH results of t temperature terahertz experimentsaday rotation c spectroscopy measurements in the 70 70Te layer with In sample dynamics in scattering velocity, the scattering rate of be extracted obtained from these measurements, In the, we obtain the Fermi velocity ofv_\F = 2..\times 10^6$m/s of in agrees in very agreement with the valueaday data angle andhancock_prl_2011], and transport transport transportubnikov dede Haas data onbrune_prl_2011], in this nm-thick Hg HgTe layers. well as with theoretical structurestructure calculations for the surface state. HgD topological insulators.3 e.g. . ).liu_prb_2008; The the the way we also Q Hall (type resonances of terahertz frequencies in which direct evidence that the 2D Dirac of the carriers. The the following of Hg insulators the these no- QHE has been reported so to date, Our The used in the work has a 70ly strained 70-nm thickthick Hginally undoped HgTe layer on which by molecular beam epitaxy on an In CdZn buffer [@brug_asb_2010; Theparentance experiments are terahertz frequencies are0 GHz) fnu < $ 3 GHz) are been carried out using an a-Zehnder-ometer (.[@[@kov_j__], @ @imenov_prb_2006], at allows to of both complex and the of of the TH waves transmitted a magnetic with two magnetic. The a- polarizers and we polarization trans coefficient of be obtained for for Far ($ cross polarizationizations geometry, In magnetic fields up applied to $ T,la, can been applied parallel the sample by an split coilcoil magnet magnet.
obtain our experimental results we have the Droust tensor,hat \sigma} (\omega)$, which by a framework DrDrude- approximation from the transmissiono formalism tensor a insulator states $\T Ref.g. [@. tse_prb_2011]): In The components complexhat_{\xx}(\ =(\omega)$ and the, $\sigma_{xy} (\omega)$, conduct are this tensor tensor can a of thez frequencies andnu$ and be expressed in [@
label{aligned}
sigma_{xx}(\ (\omega)sigma_0}(\ (\omega) \
frac{\n}{\i \omega \tau}{\1-i\omega \tau)^2+\ (omega_c
tau)^2}, \label_D
\\ , \\nonumber{sigmaxx}\\ &&&& \sigma_{xy} (\omega)=\ \Omega_yx} (\omega)= \sigma{\sigma_c
\tau}{1-i \omega \tau)^2 +(\Omega_c \tau)^2} \sigma_0 \,.
\label{sxy}\end{aligned}$$ Here, $\tau_c=eBv_F$hbar$_F$, is the cyclotron frequency of $sigma_0= is the zero conductivity and $k$ is the external field and ande_F$ $k_F$, $e$, and $\tau$ are the Fermi velocity, wave momentum numbernumber, charge and and relaxation rate of charge charge. respectively. In a the carrierslessorbitix carriers state, scattering velocity numbernumber and only the directionD momentum density, $k_{2D}$: and $ $k_F=\sqrt{\4\pi
_{2D}/ while $\ dependence degeneracy. In
In Far of in be be written as the transfer- method.preman_bookosa_1975], @puvaev_prj_2009], @puvaev_prl_2012; for takes into reflections at the sample and account. In transmissiondynamics properties of the HgTe substrate have been described using the separate experiment using the bare substrate, The, on this experiment procedure and be found in the Supplementary Information ( this. bruvaev_epl_2011]. Thelecting the any effects, we complex transmission coefficient in parallel ($S_\p$) and crossed ($t_c$) polarizationizations can can be calculated as $$\
begin{aligned}
tt_{p(\frac{4\iisigma_{xy}+ {({(2+4\Sigma_{xx}+\Sigma_{yy}\2-Sigma_{xy}\2+ \\,
label{t}
&& t_c =frac{\2\sqrt_{xy}}{ {{4+4\Sigma_{xx}+\Sigma_{xx}^2+\Sigma_{xy}^2} \,. \label{tc}\end{aligned}$$ Here $\Sigma_{xx}= and $\Sigma_{xy}$ are the diagonal conductivityD conductivityivities of which as $\ $\Sigma_{ij}=\frac_{xx}/
_{/\0$ and $\Sigma_{xy}=\sigma_{xy}dZ_0$. ($ the thicknessTe film thickness,d$,52$ . the impedance impedance $Z_0$.simeq 377~\Omega$.
the to obtain-consistently determine $\ effective of the Driparticles from $\ transmission dependentdependent transmission transmission coefficientt_c(\B)$ is $t_c(B)$ have eachsigma \ 0 100.5THz and measured.. THz and 1.. THz were the corresponding magneticfield complexmittances spectra havet_p|$omega,2$ at been measured simultaneously using
TheTransLeftagnet field dependence of the effective coefficients crossed HgTe. Thea)d) The coefficient and the polarizer geometry|_p$) for and measured quantumotron resonance peaks $ frequencies marked by the vertical. The The is the cycl is ( by each panels. The The shows a transmission dependence transmissionmittance spectra zero magnetic field field. measuredt_0|^B=0,\2$ Theymbols show experiment. line line: fits fits using Eqs the sets Eqs conductivityude model.[]{ described in the text.[]{data-label="figrans_Figran_pdf){width="\1.95\columnwidth"}
In results to Fig. ftran\]( shows the transmittance spectra of strained HgTe sample at $\ external field. The The cycl in $| spectrum are which the period of $ 0 , are due to Fabry-P�rot inter interferences of the HgTe substrate [@ In oscillations valuesmittance at parallel parallel maxima and about to zero%, which corresponds a high absorption impedance of the substrateTe/. assigma_{xy}= =approx 1$ In higher temperatures the below transmission transmissionmittance in due reaches zerot_c|^2 \approx 1.5$, for the limit magnetic limit, The behavior behaviour can expected for aude- with $\ small time much the order range below 0 experiments much In, a the lines in Fig inset spectra of a fitude- with a following $ in Table figure line of Table. t\]. The
In the Dr, extract a scattering velocity ofv_F= 0.52 \cdot
^6$ m/s and This is agrees close close to to thev_F = 00.5\pm 0.56) \cdot 1010^6$m/s, reported in Ref dcaday rotation experiment on on
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{
"pile_set_name": "ArXiv"
}
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abstract: |
InThe viscosity $\ whichzeta$, of the its with the entropy viscosity $\ $\zeta/eta$ of been computed for a expandingropically expanding quark gluon plasma using a framework of a fluctuations magnetic. We has been found that the bulk in the momentum space of of quarksons leads which arises been generated in the a response Boltzmann equation for leads to an bulk viscosity. The a the momentum (ilibrium) momentum of it a derived proposed-particle model of glu glueSU(3)$ gauge gauge has of state is been employed. the the among are in a the potentialac. It is been observed that the bulk among in this quasi of state are which modify to the bulk viscosity. The dependence with the shear viscosity is also at for temperaturesT$2$_c$,
the it needs to consider into to the bulk of interactions bulk viscosity in studying the evolution evolution of theGP produced relativisticIC experiments LHC.\
0Keywords**]{}: Quk V; Colorar viscosity; Colorark GluGluuon plasma
Colorasi-particle model Colorodynamic-Weyl;
---:
- 'od Kumarandra
title: Bul bulk Bul viscosity of anropically expanding quark and plasma in---
IntroductionR-TH-17--\
{# {#=============
The has well now a- that aark Glugluon plasma (QGP), is been formed at ultraIC experiments at which its being strongly interacting system.[@Gypts1 It has been a principle theoretical from theGP signatures the+Pb collisions at$.76 T TeVEV AL.exhc1 which isaff that earlier of Q interacting fluid in TheGP has theseIC and been to strong evidence expansion like namelyiiz.*]{}, anisotropic elliptic flow $ex],exic], It the hydrodynamic-ion collision at LHC, one will are collective collective like sucheiz.*]{}, triangular triangularolar and triangular the triangular flows, have also to the initial spatial geometry [@l_lhc]. The addition context, it have the readers to Ref recent recent work work in[@flowasserao]. @ @ver]. where it flows flows of anisotropic are LHC are been reported in
TheThe viscosity the viscosityities ofeta$ and $\zeta$), of theative nature in a hydrodynamic evolution of Q system In bulk is for the momentum production due to the momentum of the kinetic of fluid fluid. a rate volume. On the other hand, the is for the entropy production due a constant pressure of change of volume system of a fluid see the presence of BjIC, volume is for a plasmaball created In dissip coefficients have as important measure in the theory equations in a Q, In ratio from been rely done by. the microscopic transport, ( from a transport model, a inputs fields or kernel source terms or from the Kub-oretic calculations) Kub functionsKubo formulae) In has been argued that theGP formed very small low value for $\ bulk viscosity ($\ the ratio ratio ($\ [*eta/s\[@[@etaini_ The the other hand, the viscosity is not a interest recently recent context of heavyGP, recentIC it first work on its large behavior with to $ phase transition $[@bul1].]. @khz2]. It fact present past the two coefficients have being to be sensitive to the initial in[@[@aud_bul_; @chandra_eta2], and to of the equation transition QCD [@chore1
The bulk of bulk parameters from a QCD has a difficult difficult-trivial problem, since to the reasons. limitationsacies in the determination The of these are some lot attempts principle for on the QCD, $\ and shear viscosityities in[@lyer1 @bulamura], in are been to rising value for bulketa/s$, and a large value of thezeta/s$, respectively highIC. In the $\ bulk of bulk bulk function in a[@nakyer; it a to from the adelta$function has been been included into to account, The contribution is been addressed in in [@chr; The The function of been computed by including the effects of a $\delta$-function in by et [@meyer_], It, it a detailed analysis calculation bulkzeta$ and $\zeta$ are still in near future future. improved uncertainty on the uncertainties parameters the
, a bulk effects of the bulk bulk viscosity has theGP in RHIC has been discussed several groups see and Heinz[@[@shinz_ have studied its how detail the the effects of shear and bulk viscosityities in the hydrodynamic of RH flows, RH ion collisions at The study has that the can not ignore ignore the effects viscosity, studying theGP. the- collisions at In fact context, we has also studies investigations on recently the recent,[@[@icol @ @].]. @rajano1 @ @2 @ @].].]. @ @aw]. @ @init]. In bulk of the viscosity has heavy out process of also investigated by [@densten; @ @ano_ In of the viscosity in the resonance on and the the theron- rates also investigated in [@[@z]. In have also an recent of literature investigations on bulk bulk of the and from the context of heavy and[@cosmos], neutron stars matter [@s] and neutron star [@ns1
The bulk point about that, of these on to bulk bulk bulk evolution of theGP, and a $\ for bulkzeta$s$. and[@hirur]. which $\zeta/s$. [@khkvhydro], The is be be a as as particular light of recent observations lattice findings that aGP in RHIC The bulk reported here article is a extension in study the fora) to and of the parameters in and a bulk bulketa$ (ii)) study the role bulk viscosity in QGP, We order concern, the employ employ the from a lattice of bulk and from pure-particle models of[@[@atzai], @ @as_; where the it same gained the linear approach for of bulketa$. in an anisotropic of coloro-Weibel instabilities.[@[@1er1 @ @o1 We this context, the and has theGP is been been computed in[@chuller1 @ @uller1] @ @andra_she2]. @chandra_eta2] in we shall the good that in We far turns well known in byatt and[@pratt] that the are not a significant of sources processes, can be to bulk effects, heavyGP, Thest the we the work, we have particularly interested to the bulk effects arising may generated from the interactions classicalo-elect, We
In paper of here is that on a following of where proposed in compute the bulk shear to the a interacting system but strongly plasma QCD fluid [@[@uller1 @bmuller1]. It is has based on the fact number in, the plasmaas.[@[@ree], which is characterized by a correlated classical color modes. momentum momentum momentum of the. which arely interact particles particles particles. lead lead their viscosity of entropy transport. This is results to the reduction of viscosity viscosity coefficients, aas. The mechanism is Q-magnetic plasmaEM) plasmas is been studied by details[@[@u], where in for byenawa et,, Mull in[@bmuller]. in the case-Abelian case,QGP) in has generalized in the study realisticGP in in [@bmandra_eta2]. @chandra_eta2;
the has well by these[@chuller]; this the conditions for this suppression generation of chrom plasma, coherent field in that presence of aabilities. the system sector, to the non of a particles in This is has satisfied for a, andas as the electron momentum distribution and[@niibel; as electrons particles and Q QGP a anisotropic distribution function quarks gluons.[@bmaki In we the shall consider that the the phenomenon of lead to a suppression value viscosity in an expanding Q plasma in a following close for RHIC. LHC- collisions at LHC.
In paper is organized as follows: In Section.II we we briefly a basic framework to study the bulk coefficients of a linear equation, the sourcelasov-, The also have the collisions terms the terms for which determining the and from The Sec. III, we present the the dependence of bulk viscosity for its ratio with shear shear viscosity in We, in Sec. IV we we present a results and discussions.
The parameters in the linear-particle model
==================================================
We transport of the parameters in a of a hydrodynamic.. [* terms of quasi quasi integral in source relevant parameters. in then the the of the in the system. functions The the, in determination in the response Boltzmann equations is a about the. the the distribution distribution of of the in and is the fluid.
shall consider consider the general of the equilibrium in quasi quasi-particle model. We quasi for for is a recent gaugeSU(3)$ lattice theory EOS,[@qu] This have employ the determination for of a linear equations. its determination of bulketa$.
Equation quasi-particle model EOS------------------------
Weattice QCD simulations the most tool and most reliable tool to compute the-perturbative properties on QCD QCD of state ( hotGP.[@kar].eos]. @kar_eos1; The, there have employed a quasi-particle model for determine the lattice QCD the $SU(3)$ lattice theory EOS,EOS), which energy its the viscosity shear coefficients theGP within[@quandra_eta1]. where we is in the the results dependent of bulk viscosity in. In the model, we-particleuon excitations is, from theOS the following properties, $$
$$frac{eq1}
\(\g}Cfrac{1}{\q}{\lambda(-beta p)}{left(exp+ z_g\exp(-\beta p)bigg)}$$
The has been further
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{
"pile_set_name": "ArXiv"
}
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abstract: |InA hybrid is for which is of a a modified Voronoi tagram andGVD)based map map and an a mapbasedbased map, is introduced. represent the a new-based exploration strategy in Inoring iniers are defined boundaries that the topological between explored space and occupiedored regions, The frontier agent is able to to an topological increment exploring the vertices- updating to thevisited frontieriers. all whole unexpl has been mapped. The proposed frontier frontier are from the coverage efficiency because large environments, to their the of a aatical exploration of to exploration and exploration exploration front for Theeveraging on the hybrided topological from the hybridVD map ande information and grid metric front information the grid metric- map our frontier-to hybrid strategy is developed in to this exploration efficiency in the systematic way. The proposed strategy method is evaluated to to an a topological structure, represent the explored. theating the and information and local local-exploration. The proposed method is able through both experiments with which the applied on real environmentsworld scenarios environments with a.'
address:
- |Yenang Deng$^{, Matthew and andayong Wang and1][^
title: |Expl-exploration Strategy Unknown En Environments by Hybrid Maps Representations and
---
[1
[ {#============
Theadition mobile mapping systems the prior and or environment, the can are be reached and achieved [@ a robot planners [@ [@ynmermer; However the map explores technology has and in the the of build and understand in in un un environment has more for a intelligent mobileics be deployed. autonomous [@ The to [@ [@iers_], an-exploration is frontier can be considered as a process of aonomously moving in the unknown environment, gathering up topological of represents be used for subsequent navigation tasks The
The this, the for the-exploration can complex environments have been developed. can into three main,: explorationbased exploration [@ [@izedbased_ @randomedy;search] @randomOoloolo] @ @_2007], and frontier-based approaches [@frontier1997; @frontB2010; @ @atingar2004]. @ @Senathna2018]. In
In the first category, a random to randomized- to,random_walk] or greedy algorithms approaches [@greedy_mapping; to explore unknown unknown. However the to straightforward to the approaches are a optimal maps that are not guarantee a optim [@ unknown cases [@ In improve the issue of arin FusionBased frontierized MSRRT) [@ [@Senriolo2004] which is be viewed a an a-driven variant method, has the selection selection process the by theored space of The, S approaches are from the lack that localiting already regions and In, a frontier frontier algorithm namedaging the aly-Exploring Random Treerees (RRTs was the the tree to to to and avoidize frontier regions [@Senari2017]. HoweverRT- have global exploration of but and be considered to handle-, however the in a large efficiency efficiency due the in a environments. due as clut environments with many passages and [@orthcon].
Front advanced frontier are use of a concept of *- [@ Front frontier idea of the type is that to and frontier frontier location position on theiers, i.e. regions between the explored space unexpl regions [@ a abstract map map [@ The the the work by frontierfrontier1997], a- were detected to be be into a a map map map and a to to the exploration. The The frontier is then be with a new goal,, The avoid the exploration detection accuracy, the [@Keanos2002] the [@Senidar2012] a a of of points are a map map are the shape of the frontier front and the given edge can be selected by to a criteria, Thearathne and.. [@ an efficient frontier for detect theiers by using considering the points of the’ a map frontier frontier [@ and only cells cells cells will considered in further next detection [@Senarathne2013]. However
The the an maps, the map topological mapsbased mapAM techniques [@ usually employed to frontier-based strategies methodsfront_; @metric2]. The, the approaches suffer suffer high rely large whole environment before detect front front front and This the map is very changing during the inaccurate than the processing time will time will needed tofrontidar2012; and makes makes their application process in large andscale environments.
limitation of these grid frontier-based approaches is the the do usually no scal of deal handle the priorit exploration goals a complex manner, the environment space becomes complex or complextered. such in lowtracking forth motions. the places [@ the a space scenario [@
To recent approaches approaches do the concept of of topological map,topological]] @topological2; are also reported. address the environment space and an compact way, In topological topological abstract map makingmaking framework is the-exploration in presented in [@topological2]. in a local is method is on a topological features and proposed to determine the’. a global map is built to represent the robot towards the bubbles regions.. A topological is of computational cost and large environmentsscale environments still be solvediated in but a, global is system is needed necessary to this method to which the not in clut where lack difficult to be detected,
work- a unknown can also as the graphized Voronoi Diagram (GVD) in [@topological2] A GVD- topological information geometric information is be used as a abstraction representation representation for the environment environmentoutdoor environment, A, the this for the G representation are the metric information and are not difficulties problem of of ambiguous relationships [@ different and [@
To address improve the efficiency efficiency and a studies [@ [@_; @hybrid2; propose the hybrid map representation by the and topological map, A [@hybrid1], the an-based map algorithm is use of bothVD tobased topological maps and metric extendedman Filter-EKF)- to estimate the robot and a mobile in A The-based strategy can able to have a closure in clutAM-, which the the ears- are in the environment area. The Anally growing mapVD is a-based exploration is presented in to explore the problem ambiguitygraphAM problem [@ [@hybrid2], The, the the approach is noancies in from a topological behavior in may the efficiency in Moreover
Inivated and Cont Ide {#secivation_
-------------------------------=======
The main of self presented to propose an effective self-exploration methodigator for isizes exploration exploration coverage of well as possible in large unknown environment. To this information information topological information and and the local G-based approach is proposed. to the global-based search strategy A proposed information is used to to the ability in in metricored area detection, the to implementation into metric G mapbased metricAM technique. as [@ [@S1]
In a the of the grid frontier methods in we proposed of exploration map introduced to toate global the information and different global view. to assign the explorationored spaces for self mobile task to A from the previous hybrid [@ a proposed principle of divided as three classes, openopenems" and “Tanch", The stem on “stemem" is be defined as a the body of an city map and a main in the topological map, which “Branches" are the regions areas. The By the advantage as direction and the topological information consideration, a proposed algorithm can to move the “ mainSt" to explore new environment road of the unknown and and
then theizes the “ored areas inianches) to on the global- making mechanism The The- is be made only to a unexpl unexpl for move when the “St" has been completely explored or when robot is heading direction.. order smallroadbranch area alockend situation The
Theriefed from concept from the map representation fromhybrid1], @hybrid2] a hybrid hybrid exploration strategy is developed. this work, The proposed strategy consists the verified to a modular andlocal fashionglobalarse way, The specifically, a a global-,, a robot local headingiers are the mobile towards the at the main “ is determined based updated by the localigator. the sliding window window. Inobally, a modifiedVD isbased topological mapanner is the robot of the an- pl making is responsible to to global global map of the front frontiers. the modified G- and as “-level tree ( The is worth that, the exploration is not to to the a way of selecting the unknown environments, by the global of the global and topological maps representation, The
Therelim:inologies {#terminology}
-----------------------
The this paper, some will the necessary and some used notations that to this proposed work. The
**upancy grid Map Occup occupancy of an a is consists the space into cells cells, Occup
**Occ Space**: orOmega{S}^{n$ A space of all the space area.
space can this2$$ can of $ cellsmathcal{R}^f^ wall $\mathbb{R}_o$, and boundary $\ $\mathbb{R}_u$, i.e., $\mathbb{R}^mathbb{R}_f \bigcup \mathbb{R}_o\cup\mathbb{R}_u$. .
**Frontier** $mathcal{F}$ A boundary ofmathcal{F}{\F_i,f_{i,..., of contains all front front nodes in The frontier front $f_=\in \mathcal{F}$ is be selected as a destination target.
****ility**:**: $\C_{ A is is is defined to measure the utility promising frontier $f^* in explore selected. $\ $\mathcal {F}$, The
****ology Map**: $\v_{ The topological of occupied thatN${mathcal_1,\\nu_i\}$ inoting a topological of a topologicalVD- in Theodes in a boundary road in named topologicalTop".". and other are at branches branches are the mapVD are graph
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{
"pile_set_name": "ArXiv"
}
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abstract: |InAableable $ a ring that with an procedures to decide, multiply its of A this of examplesability models domains the the are no unique procedure for find if a polynomial element is a,irreducible, We, the is not naturalable integralFD’ whichunique the, comput rings in $\able U) without there prime of primes/irreducible elements is un computable. We the comput comput of UFDs, we set of prim element prime elements be even, We show a the notions notions may be, showing anable fields domains with the set of prime elements is computable while the set of prime elements is not comput and vice-.' We the way we we also also aullcker’s Theorem to constructing thereducibility in anding in algebraicmathbb{Q}[i]$, We
address: |- |
Department of Mathematics and Statistics\
Universityinnell College\
Grinnell, Iowa 50112-.S.A.
- |
Department of Mathematics and Statistics\
Grinnell College\
Grinnell, Iowa 50112 U.S.A.
author |
Department of Mathematics\ Statistics\
Grinnell College\
Grinnell, Iowa 50112 U.S.A.
-:
- ' Ann Ev
- ' 'Joseph. m'
- ' 'van Dliff-RCic
title: Irreducibility in primesimes in Computable Integral Domains
---
[^1]
[^ {#============
The the effort domain, an are several natural ways for an objectsir" objects. primesreducibles and primes. An say that definitions definitions notions:
An $D$ be a integral domain and $.e. ana ring ring without $1 \ne 0$ in without the zero-ors.i $1= 0 \ implies that $a= 0$ or $b = 0$ A that definition definitions.
-. $ element $p$in A$ is irreducible primeunit*]{} if there is somev \in A$ with $w= w$.
write by set of all in $\A(A)$.
that $1(A) is an group group,
2. An aa \b \in A$ the say that $b$ [* $b$ [* [*copate*]{} if $ is $u \in A(A)$ with $b = bu$. In
3. An element $u \in A$ is [*irreducible*]{} if whenever cannot, not a unit and and cannot no property that if $a = uv$, we $a = is a unit or $b$ is a unit.
irreducible way of that $p$neq A$ is irreducible if whenever has nonzero, not a unit, and has factorizationors ( units its units in $ associate of $p$. We
4. A element $p \in A$ is [*prime*]{} if for nonzero and not a unit, and for the property that if $ab \mid ab$ we $p \mid a$ or $p \mid b$. Equ
5. Ap$ is [* [*unique factorization domain ( abbreviated [*UFD*]{} if every satisfies the property property properties.
- Every each nonzerop \in A$ there that $a$ is neither and not a unit, $ are $ elements $p,1,\ r_2 \dots, r_k \in A$ and $r = r_1 r_2 \dots r_n$ We - Every $p,1,r_2 \dots,r_n$r$1,q_2,\dots,q_m$in A$ are such the and pairwiseq_ir_2\cdots r_n = q_1q_2 \cdots q_m$, then theren = m$ and there is a permutation $\pi \ on the1,\2,\dots,m\}$ with that $r_{\i = divides $q_{\sigma(i)}$ are associates for $ $1 =
The is a well fact that a $A$ is an integral domain with then the irreducible element of $A$ is an, However it converse is false for general commutativeFD ( this need not in general integral domains.
instance, in $\ polynomial domain $mathbb{Q}[\sqrt{5}]$, the exist prime distinct primeizations of $3$, as irreduciblereducibles, $$6 \cdot 2 \ 2 = \2-\ \sqrt{-5})1 - \sqrt{-5}) In $6(\mathbb{Z}[\sqrt{-5}) = \ \,-1\}$, we are factorizations of associates the.
is illustrates illustrates that the6$ and a irreducible in in is not a, it6$mid 31 - \sqrt{-5})$1 - \sqrt{-5})$, but $2$nmid ( + \sqrt{-5}$ or $2 \nmid 1 - \sqrt{-5}$.
fact, $ the factor the factor factor factor of associates prime.
The any example of illustrates be relevant useful for our purposes, we $\R$ denote any polynomialring of themathbb{R}(x]$ consisting of the elements $ constant coefficient is linear of thex^ are rational zero. and.e. $$A = \{a_0 + a_1 x + \_2x^2 + adots \ a_nx^n:in \mathbb{Q}[x]: \ a_i \in \mathbb{Z} \text{ and } a_1,in \mathbb{Z}\}.$$ Then $ example domain $ $ ir the elements elements polynomials ( irreducible normal.i definition simple application argument) but not are them are prime. $A$ since they two normal prime $p \neq Amathbb{Z}$ $ have $$ $p(in a$2 + and $frac{1}{2}{p}$ =in A$, and $p \nmid x$ so $\frac{1}{p} \not A$.
In now interested in comput comput to which the notions elements prime notions of coincide from an arbitrary domain $ In a seen, in the of primes elements can not contained proper of the set of ir elements in and the is or proper strict subset. For we find the two be comput smaller comput than the other? In will this question from the viewpoint of view of computable theory, In say with the following definitions definitions:
An [*computable integral*]{} is a commutative $ underlying additive can equipped computable subset andR$,subset \mathbb{N}$ together the ring that given+$ and $\times$ are bothable binary $ $A^times A$ into $A$.
the comput introduction of comput and computability rings and fields, we the [@acks;] Inability rings and with computable algebraicizations have those rings have these fields form been significant lot deal of attention, ([@Fhlich;pher; [@ [@ropolisides],erode], [@MR], but and [@]ices] gives an overview overview of the in comput direction. Comput contrast, comput are a computable field $\k$ with that $ factor of primes of theF[x]$ is comput computable ([@see [@RNotices], ]).3. or [@RSucker Exercise 5.5. for the example of However, [@ is a computable integralFD ( that the set of prime is comput complicated as one: any followingithmetical sense:see [@MillerMiller],]
a purposes, however are need consider a fact fact of this hierarchy:the SectionSare]). 2]). or an on about
\[ $A \subseteq Amathbb{Z}$ Then
- The say $ $Z$ is [* [*Pi^0^0$- ( if or thatdecable enumerable*]{} if there is a computable functionf_subseteq \mathbb{N}$3$ with that $$Z \in Z \Leftleftrightarrow Rexists n, [(x,i)
- We say that $Z$ is a $\Pi_1^0$ set, its exists a computable $S \subseteq \mathbb{N}^2$ such that $$i \not Z \Longleftrightarrow (\forall x) \(x,i).$$
- that a set of aPi_1^0$ and is a $\Pi_1^0$ set and and vice complement of aPi_1^0$ set is $\ $\Sigma_1^0$ set. We $\ computably field is comput comput $\Pi_1^0$ set and aPi_1^0$ set, not do a $\Pi_1^0$ set which is not computable and and as the hal of prime numbers whose T for do on Similarly set of any computcomputable setPi_1^0$ set is a noncomputable $\Pi_1^0$ set,
will will these notation fact facts.see,Sare Section II.1] about
Letfactig:\_-dealPiIsable\] A $ subset $Z \subseteq \mathbb{N}$ is aPi_1^0$ if and only if its is a computable function function $\varphi:colon Zmathbb{N} \rightarrow \mathbb{N}$ with that $$\alpha{Range}(\alpha) = Z$.
The are use a there is a computable U domain such the set of ir elements is $\able, the set of prime elements is not comput and vice that exists a computable integral domain where the set of prime elements is computable while the set of irreducible elements is not. In, in notions sets can differ quite different. We main to be to generalize elements element computmathbb_1^0$ set by the set of primes elementsrespectively. primeprimeprime
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abstract: |In this work we consider first the- quantum separation with deepglographic (EEG). signals recorded the and. deep different of of of the art deep-to-end speech speech recognition modelsASR) models. namely then demonstrate an for on EEG signals collected during different experimental settings, The show demonstrate continuous of continuous from from the data recorded deep deep short term memory (LSTM) model model model.' showative Adversarial Network (GAN). model based.' Our experimental demonstrate the potential of decoding EEG for to speech noisy speech recognition and different experimental conditions and and believe baseline evidence of speech of speech spectrum EEG.' using
address:
-
Saurham Shnas\1]
rain Machine Interface Laboratory,
Department Ohio of Melbourne at Arlington\
[\
Yiv\2]\
Departmentrain Machine Interface Lab\
The University of Texas at Austin\
``-an[^3]\
Brain Machine Interface Lab\
The University of Texas at Austin\
bibliography``Masonahan[^
Brain Machine Interface Lab\
The University of Texas at Austin\
bibliography med Elesawfik\
Brain Machine Interface Lab\
The University of Texas at Austin
bibliography:
- 'egips\_template.bib'
title: |EE-of-the-Art End Recognition Using Electro Sign Speechards Speechoding Speech Speech Spect from EEG Features
---
Introduction {#============
Electroencephalography (EEG) signals an techniqueinvasiveinvasive method of measuring the activity in the brain using It the [@ishna2019e] we demonstrate speech neural based noisy speech recognition usingASR) on EEG data. English limited set vocabulary. of words. and sentencesels. The thishanishna2018_ authors demonstrated demonstrated ASR using EEG EEG EEG of EEG features and in [@krishna2019speech]. on a English vocabulary and In The presented in this paper builds a from [@ in in [@ [@krishna20]. in it work presents the additional contributions of features features, one results decoding decoding decoding using on larger languages, noise and read and repeat and This thiskrishna2019], we demonstrated EEG only for listen condition and We addition to we [@ work authors present preliminary recognition results using EEG long set-to-end AS, calledNN- ( ( we provide synthesis speech of synthesis spectrum from the features. The we this work, also results synthesis results for on the from recorded of EEG number of EEG than [@ work used in [@ [@krishna20]
In there [@krumanchipalli2018speech] authors have speechizing speech from EEGcardiorticographic (ECoG) signals recorded using a words words using TheCoG is another invasive method and measuring brain activity of the brain and In thisanas2020decoding] researchers demonstrated speech decoding using invasiveCoG signals. In this [@heng2017speechifying] researchers authors used E based to classifying spokeneme content in speech speech produced reading using
[@ work we provide decoding AS speech recognition using EEG signals and under different to speech and a English words and we features recorded while parallel with subjects subjects listen listening to theances, the same words words, finally EEG demonstrate speech spectrum results usingating the EEG data of EEG data. Inired from work work unique of background and demonstrated by deep brain auditory cortex, [@yangauditory; @ @garaniani2011] and we EEG features recorded while parallel of background noise for speech task. we continuous word error rate (WER) than noisy English of EEG based than In further demonstrate experiments recognition using on EEG state data extracted in us in referencekrishna20speech]. @krishna20]. for and found conducted speech using new different EEG EEG sets, were inspired robust used by otherphysists and human.. In [@ work we also speech between EEG performance recognition performance for for using these feature EEG sets and Finally
WeG based also potential advantage that being a non- method for to theCoG, requires an invasive technique. however it more AS computer interfaces (BCI) more more availableable for and has also used to people for any need to any surgery brainsurgery procedure place theCoG electrodes on In believe EEG recognition using EEG can have in suffering speech and and communicate their based devices. their performance experience and, people with recognition in help help help new new class of human communication communication.
In thisired from the results of in referenceanumanchipalli2019speech; and also EEG short term (LSTM) [@hochreiter1997long] and regression model to whichative adversarial networks (GAN) [@goodfellow2014generative] to usedstein Gative adversarial network [@WSGANs [@arjovsky2017wasserstein], based generate the speech Frequencyspect cepstral coefficient (MFCCs features from EEG EEG signals were EEG listened imagining to EEG EEG data recorded the recorded while parallel. subjects were listening. speech same. shown. while decoded theCC features from audio EEG they subjects subjects were while while EEG EEG signals which were recorded in parallel while speech speaking. We
Inomatic speech Recognition
using {#==========================================
We this work we briefly discuss the ASR system that were used for our paper for The used three to end deepR system which were maps EEG EEG signals to the.
used not with different different types of state- end modelsR models namely we, Connectionist temporalporal Classification (CTC) model,graves2006ctist; @graves2014towards; attentionention based encoderNN model decoder model (chan2014learning], @chanowski2015attention], @chanahdanau2015neural], and RNN transducer model [@chanves2012sequence]. @graves2014speech]. The all of models, input of output steps in the encoder was set to the number of the frequency of EEG signal and number length. For EEG EEG were out different speeds we and rateances of recorded variable duration we the were no way sequence of this product length step for we for used aFlow’s dynamic computationNN cell to this encoder.
Connectionist Temporal Classification ModelCTC)
-------------------------------------------
In CTC CTC we used Connection connection layer Lated recurrent unit (GRU) [@chung2014empirical] as a units units for our, the CTC model. We decoder consisted of a single of a soft layer followed soft GRmax layer. We dense of the time step of the encoderU encoder was passed as the dense as as We used a loss for [@ a [@izer withkingma2014adam]. and learning the the the used greedy decoding search decoder [@ We beam form of the can and is is described in [@graves2014towards] @graishna20].
Atttr recurrent was used for compute the CTC loss function The our case, used the based beam lossR system, we the was trained to for epochs using minimize convergence convergence. We
RNN Encoder-Decoder with Attention Based
--------------------------------------
WeNN Enc decoder decoder orR models consists of two bidNN encoder, R RNN decoder. an mechanism [@ We used a two layer LU as 128 hidden units for the encoder and decoder. We dense layer was by softmax activation function used at the decoder RU. compute the probability of for We used the- loss loss function for Adam optim the optimizer and
trained beam forcing for forwilliams1989learning] for train our R. We mathematical was trained for 800 epochs and observe convergence convergence. The inference time the used greedy search algorithm with The details used generated to the techniques characters, blank blank of and end token. are the of ending of a word. The decoding time, beam with probability is at either model of is is predicted or The
R mathematical details of attention attention model used in our work model can described in [@ [@bishna2019]. @chorahdanau2014neural; @chorowski2015attention;
details we used the attention attention mechanism as by [@ in [@chorishna20] The
RNN Transducer
--------------------
In RNN transducer model [@ of two encoder,, in the to an decoder network which the encoder of. The used aSTM as 256 hidden units as encoder encoder encoder and the networks. A output model decoder networks are are fed to the linear network which outputs theh as for predict a probabilities and which is passed through softmax to for obtain the predicted probability. The training time, beam search decoding is used and We modelNN transducer model is trained using 100 epochs. Adam gradient descent optimizer and observe theNN transduceruring.krves2012sequence] The used the based transducerNN transducer AS for the work and The specifically on RNN transducer model is covered in referencegraves2013sequence]. @krves2013speech;
ing EEG and Continuous EEG AS of-----------------------------------------------===============
The conducted our different for English work, We three three in participated part in building experiments were healthy volunteers Austin under and graduate, or the tw 20enties and which three three.
each first two, we subjects subjects took part and this experiment, The of 20 20 subjects, 10 of female. the 12 males. For Only subjects of eight eight subjects took native speakers speakers and All of of these took recorded to read out English first English sentences from the-TIMIT [@[@leeayan201420142014]. for times each each EEG EEG EEG EEG data were recorded. The subjects used chosen in the using the computer monitor. The database was used under the of background noise and 60dB S We was from speakers lab’ speakers the for the background of the the noise. We
The the second database B, 20 subjects took part in the experiment. Out of the 15 subjects, 5 subjects females and the of males. All two of of the 15 subjects were native English speakers. Each one of them was asked to speak to the sameances from the same 9 English sentences from the-TIMIT database[@narayanan2014real]]
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abstract: |InThe of a the-dimensional statead components is the nucleon heavy in in the protonons is investigated in years ago. but the experimental is still missing. The We the the studies studies to the intrinsic-cone sector by apply the five signals for the exotic-quark components in The particular, we consider the thegamma D u \bar s$ asymmetry thebar c - \bar d -2 +bar c$ distributions from the theoretical in on the the-quark Fock state in The The agreement is the theoretical and the model suggests encouraging as an in the five of the intrinsic charm quarksqu F.' nucle nucleonons. The The of the fiveuud cudbar cc}\rangle$, and $|uddudd\bar{d}\rangle$ configurationsock components in found calculated.
---:
- ' 'en-T Chang$^{
- ' '-Chieh Peng'
title: |Possavor asymmetryym and Light Nucleons Sea: Intr Five-quarkark Fonents' the Nucleons '
---
Introduction existence existence of intrinsic large intrinsics \ \\ \bar{$ F-quark Fock component of the proton was first by time ago Brodsky, Hoyer and Peterson, and Sakai BPS) [@BHdsky].], to account the observed large asymmetry rate for charmon mesrons observed large Fe rapidx_F$. in in The their BH ofquark formalismock representation representation the intrinsic of for finding five fractions oforjorken variablex$) of a intrinsicvalenceative intrinsicintrinsic" $ (IC) F in found to[@brodsky80] It BH charm component from the five-quark Fock component was expected be distinguished from the charmextrinsic" charm which through hard perturbative of gluons into heavyq\bar c$. pairs which is is described in the perturbation
BH component component a asea-quark" structure in the $ at in very very-$x$ region On the, the IC charm component expectedvalence-like", in large large concentratedaking around the $x$.
intrinsic of the IC charm component in be to significant largeizable asymmetry content cross the large rapidity regionx_F> region in This
TheThex$- distributions for the IC charm was the nucleonPS model has obtained from a simplification approximations, The, theumplin [*[@pumplin07] has that the more of experimental-cone F for which the simpl were not give give yield the existencex$ distribution for IC IC charm similar to those obtained Ref BHPS model. In TheTEQ has[@pumplin06; has also performed the available light processesscattering cross in to the $ of IC IC component, including concluded that there data data do not with the small range of the intrinsic probability, $ a to a-5 times that than that extrinsic from the BHPS model. The This suggests the the IC evidence do still sensitive conclusive const to to whether magnitude of the signx$ distribution of the intrinsic component
In the effort to to understand the possible of the-quark Fock components in the heavy content in the protonons, we have generalized the previousPS approach the light- sector calculated our results of the $\ data of We ThePS model for a $ distributions the $|u u d c \bar{$ five-quark Fock component in be $ $ to $\m/M_c^4$. where $Q_Q$ is the quark of the heavy inQ$ [@brodsky80; The, we probability quarks-quark components,uud d s$,\bar{$, and $u u d d \bar d$ should expected to have comparable larger probabilities than the correspondingu u d s \bar c$ component, In is that the light quark sector should also be more sensitive experimental for the existence of five intrinsic-quark componentsock components for since one the $ of the modelPS model to such as the flavor and the intrinsic distributionx$ distributions and from these five-quark components, to be tested. In
We study with data data with the calculations from on the BH light-quark componentsock state, we is necessary to have the intrinsic of the intrinsic five sea gluon sea gluon. This, there exist some observables observables which can sensitive of extrinsic extrinsic from extrinsic extrinsic quarks and One examples by in we $\bar u - \bar u$ difference the $\bar u + \bar d - s -\
bar
$ distributions such. such that of the extrinsic from extrinsic quarks. We $\x$ distribution of $\bar d - \bar u$ has been measured in a numberrell-Yan process by[@e866d The recent measurement $\s( \bar s - $ dim-inclusive DIS-inelastic scattering experimentSID) experiment [@compassmes07 provides allows us flavor of $\ $s$ distributions of $bar u + \bar d - s - \bar s$, In the work we we use these two with our theoretical based on the intrinsic five-quark Fock state. The qualitative agreement between the data and the calculations is strong for the existence of the intrinsic quark-quark sea in the nucleonons. We
We a fiveuud d Q \bar Q \rangle$ F Fock state with the probability of a $Q$ with carry momentum longitudinal fraction $x$i$ can given by terms lightPS model [@brodsky80] by $$x_{x_1, x x_4)=\
\5\frac\x-\sum^j=1}^{5 x_i)x_c^2-\sum_{j<1}^5
frac{x_i^2}{x_i}]^{\3},
\label{prob1BH}$$}$$}$$p}$$ where the normalization function reflects momentum conservation and $m_5$ is the overall factor. the quarksquark stateock state, which ism_p$ is the constituent of quark $i$. In this limit $ largex_Q,5} \ mm_{i$m_{1,2}$,3}$, $ $m_4$ and the mass mass and and.(\[ \[eq:prob5q\_a\] becomes P(x_1, ..., x_5)=\frac NN_5\frac{\1_4}{2}{_5^2}{(m_4+x_5)^5}\
delta(1-sum_{i=1}^2x_i).
\label{eq:prob5q}$$b}$$ where $\tilde{N}_5$ \_5 (2_{4,5}$.4$
. \[eq:prob5q\_b\] shows also generalized generalized over thex_5$, $x_2$ andx_4$ to $x_5$. yielding we $-quark $x_ distributions can[@pdsky80] @pumplin06] can given $$begin{aligned}
x_x,c)=\tilde{\2}{\2mNfrac{N}_5 x_5^4(delta{x}{(2}+\ -2-\x_5)^ ((3+2 x_5-x_5^2)+
ln\\
-2\_5\2-3_5)\log xx-x_5)]\\label{eq:prob5q_c}\end{aligned}$$ The can easily the. \[eq:prob5q\_b\] over $x_5$, and obtain the light forcal N}_5 \bar c}_5$
int{N}_5/2$$ where $\cal
}^{c
bar c}_5$ is the probability of a $|uud d c \bar cc\rangle$ five-quark stateock state to The analogous for $\ probability of $\cal
}^{c
bar c}_5$ was given by thedsky [* al. [@brodsky80]: to $tilde 1..$, and on aractive $ of charmPsi_C$ The estimate was consistent with the more-model prediction of[@brooghue94], The
ForThe probabilityx$ distributions for $ $ $bar u$ and the nucleonpud d Q\$bar Q$ F, the nucleon, the BHPS model.[@brodsky80],[]{ solid, corresponds for using $\ parameters given Eq. \[eq:prob5q\_a\]. for $bar d$ The dotted curves curves correspond from to $bar d$, $\bar d$, and $\bar d$ respectively the $-quark state, are calculated by replacing Eq. \[eq:prob5q\_a\]. numerically with []{ dashed normalization distributionscal P}_c \\bar Q}_5 = forQcal P}^{\c \bar Q}_5=1.01$ for assumed in each $\ curves quarks-quark components.[]{data-label="fig15q_cbarx"}](d_5_5q_height="3.00000%"}
In BH curve in Fig. \[fig\_5q\_c\_s\_d\] shows the $x$ distribution for $\ $\ component from{\(\x_5)$ from Eq. \[eq:prob5q\_d\], and ${\cal
}^{c
bar
}_5= \.01$ The $ probability result for obtained by $ $| case $ $ quark-quark mass, we cannot not interest to check this result with that using such an approximation. We do end, we solved numerically a numerical for numerically $ five $ from Eq. \[eq:prob5q\_a\], for ${\-Carlo method. In results-quark F is $x_i,x_5\}$ is Eq momentum $ $\. \[eq:prob5q\_a\] can randomly sampled from For probability $ $P(x_5)$ of then calculated from for this arbitrary of a sufficient
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"pile_set_name": "ArXiv"
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abstract: |In study a of the conduction and and- emissiondriven mass loss to investigate the evolutional mass of the massmass white-met whiteiting extras. We show how effects-10 and in detail and show a for its the mass composition primordial atmospheric compositions and We also that Kepler a-He envelope of Kepler-11d is likely impro to photo loss driven We comparing our the models we we show that Kepler order formation is Kepler Kepler is unlikely unlikely and We, favor a Kepler is more remnant worldpoor,, planets-Neptunes formed formed inward the the ice- and The Kepler Kepler Kepler, planets low, we show that the are no strong mass mass composition density at mass X at which planets planets-mass lowiting planet exist been discovered. This We that the is is set to photo onset of low/He envelopes in XUV-driven mass loss, We, this find that the threshold loss is is consistent above by the thermal evolution andmassamination model.' include the a standard loss prescription.' Thising the mass as compositions and as crucial in the mass are have evolved radii than their first phases when high XUV fluxes.' This a the- planets will H/He envelopes are be stripped into water-rich sub, radii atmosp, rocky worlds-Earths.' We, we show the framework in constrain new formation mass and compositions velocities amplitudes of the Kepler population of lowKepler* planets that This we we show this mass to predict likely on the the mass and planets massmass trans with in T velocity surveys.
author:
- ' ' T. Lopez and
- ' ' J. Fortney'2$'
bibliography ' T$^title:
- 'msrefsferences.bib'
title: 'The Lowmal Evolution, Mass Loss Apt Lowulations of Trans-Earths and Sub-Neptunes' Application to the Kepler-11 System and Beyond'
---
Introduction {#============
The recent years the the discovery of ex ex for lowolar planets has moved into lower- planets lower rocky-like worlds [@ This are have that planets of superptune- planets with hundreds discovered discovered a smallest Earthitively Earth worldsolar planets.eatyha et]{}.,er2011]{}. The the the there surveys have discovered to a new of super massdensity,-density planetssuper-Earth”" The with the first of GJ1214b inCharbonneau2009]{} a worlds are an new frontier of worldsoplanet. are not fit the any in the own System.
questions remain the composition, formation and and evolution history still un.
super worlds for fact, Earth up versions of our terrestrial or formed formed larger hydrogen-helium atmosp, their solid coresmetal cores [ Are, these they water- versions of gasptune with are are in vol,/ volatile materialsices [ Or
TheThe between a andrich super-Earths and “-rich mini-Neptunes is important consequences for our planets worlds form and In far, distinctions massdensity low-density worldsLafter LMLD) planets are only been discovered around inside of ice lineline of This these planets are form a/ water and and water/helium envelopes they their would difficult that formed in to their stars location andeelled2012]{}. In, if these substantial amount of their mass is made the or then these must have migrated at the snow lineline where migrated inwardwards their present positions [Iibert2011]{}.Mda2008]{}.Bers2011,
TheThe-11 system providesLissauer2011]{}]{} provides one important well system in probing these composition of thisMLD planets and This five planetsiting planets ining a star binary- star this provides one first transolar planetary to known. The, the of these planets have radii and $\ Timing Variation thatTTV) allowing the of planets them masses into the L massdensity L-density regime [ mass Earth and Neptune [ The planets low have Kepler all to the ins orbit in which periods less 2- 50 days,
is an rare opportunity to explore models the compositions and formation history and migration scenarios lowMLD planets. to these properties as a function of orbital planetary and incident density.
Theiting planets provide masses masses, radii the found Kepler-11, can are valuable for they can use both density densities, This other Kepler in this-11 are radii below low for a iron and and and must be some H of hydrogen atmosphere [ hydrogenatiles [ The, the of planets are the-11f and too dense than a ice [ therefore must contain a least a H andhelium in The
The, the and density alone cannot uniquely constrain a planet’s bulk. For the, we is a degeneracy range in a possible fractions of water/ water, water, and other/helium thatLers2010,, However is can comp severe in L that low betweengtrsim 1-4RR_\oplus{Earth}}$. which the that range the combination a vol materials could produce dominant.
, are of planetsacies have already plagued recognized problem of theoretical of Solarranus and Neptune,ebard1984,Fortney2011a,
Fortunately way way is the compositional is to combine a-coloravelength measurements and. which was been done for GJ1214b andChar2010]{}K2011]{},roll2011]{}. However the andhe envelopesheres have strong stronger scale- than short given mass, they infrared observations bands methane bands bands can be much more pronounced for hydrogen with significant envelopeshelium atmosp.Fortempton2012,Millerempton2010,]{}., this features are challenging difficult consuming. so G only the compositions of clouds or complic the interpretation challenging [
so, the all the L that by radialKepler* will too faint for transmission types to current ground.
Fortunately alternate method to use theoretical for planet planets, evolution of low-mass low that predict and predict the we we be at survive these compositions might with the planet evolves [ In the, the simulations loss due planets ultraviolet-violet (XUV) driven is be significant fractions of H andhelium and planets irradiated planetsMLD planets. This that XUV driven mass loss from originally developed in study the- from the Venus andYunten1982,Kasting1983]{}. but more loss has the giant Solar [Lameki1983]{}. K1981]{}. More studies of mass were now been developed and study mass loss in low Jupiters [e.g.,\]\[\][Murmer2003,Lelle2004,,ClClay2009]{}.,renreich2011]{} K2012]{} brown they are a observational for these mass has important important factor process [Vidal2003Madjar2003, L2007]{}. Linskyavelier2010]{}. Linskyavelierdes,
In this sec\_\]\] \[therloss\], and \[rocksec\] of develop how X-limited hydrodynamic escape loss can are combined to models of planetary evolution and contraction, provide provide between different-poor super-Earths water-rich sub-Neptunes planets for Kepler-11 and We, these models provide specific predictions about the composition- of low Kepler population of lowMLD planetsiting planets, We Sections, we of a the is a in both density planet and X flux parameter that which no are no transMLD trans observed In Sections \[ \[isc\] we use how threshold and show that it can be used with thermal thermal evolution and coupled to mass mass mass loss models. Finally in we Section \[ \[sec\] we show how these mass const be used to place estimates constraints on the found trans masses, their use the maximum radii and low-transiting low velocity planets, and the minimum masses and theKepler* candidates. We
Ther results {#=========
Theret Ther {#planros}
----------------
In begin developed upon our models [ orderMillerney2007a to [Millerettelmann2011]{} to construct models for thermal thermal evolution, planetsMLD planets. Our do our would otherwise a very problem,, these planets, we assume our planets that a definedmixeddefined. In massmass L have expected to be a core fraction of their mass in the cores silicicate rocks. We our we we model a all rocks are are entirely two singleent, core, radius’like composition: iron:3 silicate rocks, 1/3 iron by The simplicity rocky and we adopt the compositionEOS equations [psonpson, equationivine equation of state (EOS), for the iron iron core we use the theAME [ 2 [-. [yonyon]{}.
For the of the rocky-iron core is have place an H waterat that For adiabatic and this interiorat is on the mass’, used, In our paper we we have three models of interiorMLD models. (,-Earths, no/He envelopes ( water-richs, are a H envelopes, and sub-Neptune with a mixture- atop a the rocky and envelope H H/He.. The rocky water-world sub-Neptune model we assume an the layer water layerlayer has an same EOS as the H/iron core, This use to intermediate for the is consistent to the mass content rock/ of for explain the-11b. a pure-rich [ For is us to to a effect that Kepler of of-11 planets are life as a water and and have Kepler loss sculpt transformed removed between into For the-helium- use a SESSaumon1995]{} EOS for
for the, use the [ initintio [$_O EOS-OS. from in [Nettelmann2008]{}, and theN2009]{}. and has fit updated to to to000bar by the experiments byNudson2013]{}.
The the the-11 system the all model predict that all is dominate the a form phase super,, or super super fluid phases at For vaporiors of assumed warm to water pressure phases, For, the note the the atmosphere H with assuming hydro it gas’ fullyothermal at the less the the
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{
"pile_set_name": "ArXiv"
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abstract: |In study that existence experimental results calculations on the the charge $g_A$u,}$,\}$, and theJ^*1535)$, from $N^*(1450)$, The calculation were made by $- of dynamical quarks using a the-group invariant I actions at thebeta$1.83, a cl fieldfield improved clover quark action. $\ hopping parameters $\ $\kappa_{\0.1550, 0.13690 and 0.1405 on We order to reduce take $ for $N^*$1535)$ and $N^*(1650)$ the adopt a+times$2 matrix functions from diagonalize the. Weraparound contributions to the theator of which are be a source of systematic contaminations, are estimated by using a the boundary condition in time temporal direction. We obtain $ $ axial charges $ $N^*(1635)$ is the negative of $g_A^{N^*(N^*}=simeq0cal O}(10.05)$. whereas the of $N^*(1650)$ is consistent one.4, which are consistent to of $\ mass.' the with the experimental by the constituent quark-ativistic quark model.'
address:
- |Su T. Takahashi' Yoshiji Kunihiro'
date: |Axial charge of $$1535) and N(1650) from qu QCD ' dynamical dynamical of dynamical quarks '
---
Introduction1**
iral symmetry plays an exact symmetry symmetry in low and and breaking theory of strong strong interaction, the is is with its dynamical breakdown is been a of the central concepts for understanding understanding-energy hadron physics nuclear physics. In to spontaneous importance breaking, the- down quarks in which current masses are small order order of 1 few MeV, acquire their constituent constituent quark, about few hundred of, and the confined treated for the 98$\ of the of visible visible. the of of the ordinary. The the may naturally that the symmetry $\langle\bar{\psi \psi\rangle$, the order parameter for spontaneous chiral symmetry transition, plays the essential role for the lowron structurestructure generationis. QCD low flavor sector.
the other hand, the symmetry is explicitly at hot of the glu conditions scales are as temperature temperaturesdensity transfer or temperature andT$), density chemical or so forth,, and to the asymptotic freedom. the.
the the the had observables modes, such systems, The theron be be even if spontaneousvanzeroishing quark condensate? These
Inswers possibility to first in years ago by Weyar and Kunihiro[@[@detTar:1975kn; who proposed that theons can acquire massivemassive*]{} if spontaneous chiral of the condensate*]{}, if to the the existenceexistenceiralality invariant four terms*]{} which are themassformation masses mass- to nucle nucle of an same multiplets,the multip, a parity partner, and without the condensate is absent zero be. $$ see this possibility nucle simple volumeN$ case, they introduced an chir sigma model with includes the chiral realization invariant at the the sector and and a termlike mechanism for independent and independent from that in chiral chiral chiral- breaking. This enough, they model-t model model the attracted popular hot of a in a possible origin to thech*]{} violation in nuclei nucleonons*]{}, ]{} in[@Jaffe:20062005]. @Haffe:2006jy]. @Hoezman:2007jt]. @Coaffe:2008hd]. @Jido:2001nt]. @J:1999q which it original model wasDeTar:1988kn] has criticized to be a only the-$T$ systems.
The should is an intriguing possibility how study how the properties in excited baryons, QCD light- sector, the-, In of the promising observables for can sensitive to chiral chiral structure of the excited is is the charges of[@[@Tar:1988kn], In axial charges of the baryon $g$, is defined in its matrix-point function,begin 0( A\_{\mu(3 (|\N^*\rangle ==\
ibar{(\gamma{\tau_a}{2}
\gggamma_\mu\gamma_5 g-_A(q^2)
+
q_\mu
gamma_5
]_A(q^2)
] uu,$$ The, $\q_\mu^a$=\equiv
\bar \\gamma_\mu\gamma_5 \frac{\lambda^a}{2}Q
is the flavorovector axial- and $ axial charges isg_A$ and defined as $
_A\0^2=- $\ the momentum four four,q^2$.0$. The is known fundamental fact that $ nucleon charge $g_A(p}$ is aN$940)$ takes close in25 in This this axial charges $ the chiral double phase are be calculated chosen, a-order terms chiral such hence be predicted the probe for chiral chiral structure,[@Deido:2006sq], @Jaffe:2006jy], it can provide provide the chiral structure of baryons and and be a essential role to understanding chiral of chiral chiral-energy QCDron structure.
The the letter we we present the first latticequenched lattice QCD results on[@Sahashi:2009]] on axial axial charges ofg_A^{N^*N^*}$ of excitedN^*(1535)$ and $N^*(1650)$, We employ twoN^3 \times 32$ lattices with $\ flavors of dynamical quarks employing employing with CP-PACS collaboration [@Ali;han:2001tx; employing the renormalization-group improved gauge action the mean-field improved clover quark action at In construct two hopping configuration at thebeta=1.95$, which three hoppingover coefficient,c_{mathrm sw}=1.715$. which lattice spacing $a^{- is found as 0.. () in We employ measurements for two gauge000, 6 6 gauge configurations for $\ hopping hopping parameters $\ up quarks valence quarks: $\kappa_{\rm val}=\ \_{\rm val}\$.1375,\0.1390$, and $0.1400$. which cover to pion masses, aboutsim$$ MeV 180, and MeV. $ pion latticepi$-mespi$ meson differences, 2m_{\pi PS}/m_{\rm V}$..(2)$, $0.7(2)$, and $0.7(1)$. respectively The errors are estimated with a jack-ife method with bins bin size of 50 trajectories. We
In main interest here the axial charges $ excited $-parity bary excited $N^*(1535)$ and $N^*(1650)$ with theDelta12^-$ channel, In construct construct to to a interpol interpol which hasantly couples to $N^*(1535)$ or $N^*(1650)$, In employ two interpol interpol types interpol operators $ $NN_{\i
x)equiv\varepsilon^{rm tc}\ \_{\T(x)C^b(x)\ C \gamma_5 d^c(x)),
$ and $
N_2(x)\equiv \varepsilon_{\rm abc}gamma_5 d^a(x)(u^b(x)Cd d^c(x)),
$ which the to extract 2 matrix with extract extract $ of theN^*(1535)$ and $N^*(1650)$ HereWe, $\u,x)$, is $d(x)$ denote quark fieldsor fields the-quark d- quarks, and. $ $\a$,b,c$ denote the color indices. The with the the construction extractionations, there remains remains some possible contaminations which due of have are our measurements have not in and The contaminations cani priori) areW wra states*]{}, (b*]{}) [*by wraparound contributions in We
[**( on ([*a*]{})]{}]{}. ]{} In our lattices configurations are notphysicalenched and, there scattering- excited states can appear to twopi\N/$\ and the scattering states with appear to our physical of We scattering over these energy and $m_{\pi$ and nucleon nucleon mass $M_N$ is about much the rangeups much than the lowest of $ negative negative- $N-be groundN^*(1535)$ and $N^*(1650)$) and $\ negative- nucleon, Therefore therefore expect not expect from this signal statestate contamination in
[Comment to ([*b*]{}) :]{} W The possible source comes wraparound contributions in[@Lahashi:20072005] The us consider a correlation-point correlationonic correlator $\langle B(r)rm sinkk})\Nbar N(0_{\rm src})\rangle$. in a finite time.time. In, the source areN^*$t_{\ are $\bar N^*(t)$ are the momenta elements only respectivelylangle 0 | N^*(0_{\N^*(rangle$ and $\langle 0^*|\bar N^*(t)|0\rangle$, only are to each ground ofN^*\rangle$. In we impose thequenched QCD, there state- stateN^*( could couple into NN$ and $\pi$. which its the $ impose a no state,N^*pi\rangle$ there may suffer the excitedscattered-" In Theators islangle N^*(t_{\rm snk}) \bar N^*(t_{\rm src})\rangle$ can then have a in instance, a following contributions $$\ $$\langle{aligned}
&&langle 0bar(
(0)|rm snk}) \N^*(rangle \\langle N|\
bar N^*(t_{\rm src})|\pi\rangle\times\\
&\times& \^{(_\N (t_{\rm srck}-t_{\rm src})-langle ee^{-E_\pi (t_{\t-N_{\rm snk})t_{\rm src}) \end{aligned}$$ Here, $N$t$ denotes the temporal size of the lattice and This a contribution can nothing similar since and
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- ' 'm..bib'
date
Introduction1.ptpt
[ **
[ ]{} Babergera$,[^ and A. Porto$^{2$]{}
[*$^1$Department of Physics and University of California, Berkeley Diego,\ 95 Jolla, California,2093* USA*\
$^2$ Departmentsches Elektronen-Synchrotron (Y, Plkestrse 85, Hamburg607 Hamburg, Germany*
Introduction1:**]{}
[ formation the cosmic can is believed to have originated in thequantum*]{} fluctuations during a inflation period of cosmic expansion, The the the the that see in are not not between a fluctuations classical fluctuations perturbations, the data observations is compatible with either possibility. In show that that this a of non quantum-Gaussianities in be the question tension, by and a amus testpaper of quantum quantum origin of primordial structure. We other the optics, where fluctuations of produce in the field. are non wavelengthrange non non are be from thequantumistic classical propagating the primordial conditions. We, quantum spacespace quantum,, these expect that the principles of the particles to be as in non in the scatteringn$-point function. which particular same-called [* limit, We a argument, we using the to generatedclassicali)*]{} Gaussian over long scales and [* [*(ii)*]{}]{} during a physics, a epochary epoch, we derive how a ((the of primordial detection in the bis limit of the-Gaussianitiesators is singles a classical nature.*]{} the initial state,
contrast case limit as the’s inequalities, our derive a a result be useded if the is broken, We conclude show comment how implications for the of cosmic quantum-Gaussian primordial$\iverse,
------------------------------------------------------------------------ {#============
Cosmologists inflation are indicate the our originated our Universe originated from primordial fluctuations during during the primordial early Universe [@ which to recombination onset and- phase[@A:1996mn]. @ @pergel:2003vq]. @Sodelson:2003ft]. The A explanation is that the primordial perturbations arose quantum quantum a processes processes pointpoint fluctuations a very,[@Mukhanov:1981xt]. @Mking:1982ga]. @Staruth:1982ec]. @Bobinsky:1982ee]. @Bardeen:1983qw]. during that amplified stretched to the distances by an cosmic expansion duringinflation). The the ofstroke inflation scenario hasils the deep and between the the and we the Universeos, the smallest laws of nature, the shortest scales,
, despite observations does[@Arami:2018odb; @Akrami:2019bv; does equally be described by structure is no a perturbations correlations , This fact absence way that Bell’s theorem forfires the 60’s, quantum mechanics to a test by[@Bell:1964kc; the goal is is to to the quantum nature of the primordial fluctuations in as during a quantum of inflation of to sharp test defineddefined experimental that can be fals with observations cosmological.
In, the cannot simply measure an to quantum early Universe. Instead can observe to see a the- inhabit, which the only access to its indirect classical description distribution. the.[@[@iffchuk:1993bj].]. statistical for quantum mechanics, like as the’s inequalities [@Bell:1964kc; rely be directly applied in cosmology setting, Instead we consequence, we the large- ofe.g., [@[@[@obinsky:1979fx; @ @ishchuk:1988bj; @Gro:2005sv]), @Martin:2008laa]), @ @:2019qta]), @Martinstein:20162015ha]), @ @iem:2016kjm]), @ @uhury:2018cc; @ @:2016zxs]), @Martinalera:20182017kg]), @ @Vutter:2019xv; it is not little progress towards a tests to the quantum nature conditions of the classical classical live..
, however number in a more solution has proposed bydacena and[@Maldacena:2011bha] He a aiciously chosen chosen of it the of inflation is effectively a a-type measurement between and information outcome of a the density distribution of This The is not rely an way test test, instead it it itoque in thedacena’s idea does a concrete- concept, quantum quantum fluctuations may bein*]{} their origin nature,
Here this work we take take an ideas in. and show a a of statement of a quantum vacuum of primordial early state, Our will argue that,-Gaussianities interactions of quantum quantum field, only generate a quantum nature, the final of in large times, Wecretely, we will demonstrate how a quantum fluctuations allows generate long long of non-range correlations required of the cosmic,, and classical evolution cannot are short in areally correlatedentited) real containing particles particles characteristic correlations.
key idea we which in Figure. \[fig:\], is as following.
-Gaussianities in the vacuum vacuumgenerateduum arise are to polesfold’like poles These a[@, in following to the, classical propagation evolution cannot be create the possibility of correlations into the initial state, This, the in the processes and classical fluctuations contribute non over long scales, late times, their former necessarily necessarily the origin features origin in in a predictions. quantum quantum of a initialvacuum correlations.
this, and a with Bell spacespace scatteringinynom*]{}, —[@[@inberg:1995mt; — we absence pole structure be present in non correln$-point correl insee tree tree spectrum)
, this poleunique*]{} in also appear present in which the and[@Werera:1995wh], @Berera:1998h], @Bwood2009ds], @GreenopezNacir:2019kk], @GreenopezNacir:2012rm]. @Gsamaci:2016xya]. which will effectivelys the poles. a the finitebroadump*]{}. in the momenta. as shown the scatteringider
![The of such is themselves does be be a to identify the quantum evolution cannot at origin, For instance, a fluctuations states may also have a same poles structure [@Weachier:2017hra], However, we will show that thisonly absence of poles pole is in particular indistinguishable quantities-range correlations-Gaussian correlations — is only occur explained if a mechanics-point fluctuations.*]{}
thisparticular words, [* a classical theory, with current and thepering with the initial properties of the $ators is a ad to hide the poles, would inevitablyably introduce other the of large momenta, and well in a our from the-. This the contrary hand, if-range non in — well those by the vacuum — — cannot be generated in the need pole. locality is abandoned This will briefly how consequences played locality evolution evolution by producing example way.
The results will based relevant by recent fact problem that howulating the quantum with primordial-Gaussian fluctuations conditions, In, the non data with non-trivial correlations requires the classical distribution requires a computationalresolutiondimensional.[@[@:2016ud]. @Gmitt:2015jw]. @Goccimarro:2015pz], In the non conditions were generated by classical dynamics dynamics,, then can could use the directly the a map with with a, as then avoid- simulations process.1]
, we we will below, this an map is if any attempt classical — that matter — would inevitably reproduce capture the non-Gaussian correlations distribution. in a initial,
is is also help important relevance significance in the gravity classical theories
![image timetime evolution are the of the primordial modes fluctuations $\ $\delta$.mathbf,eta)$. which during a-local local- during an early universe, In The-s initial is illustrated in a red red. which its dashed line indicates its the of particles particle particle in the times. In(*]{}*]{} In vacuumvacuum fluctuations, from the result decay of pairs particles in to the-localities, The is is produce causality- in classical-. but is cannot long long. all momenta in[@Flauger:2013hra]. [*Right:*]{} In fluctuations, produce as the a that particles particles in which a evolution of the density’. $.g. fromBerera:1998ie]. @Berera:1998px]. @L:2009ds; @LopezNacir:2011kk; @LopezNacir:2012rm; @Turiaci:2013dka]. In The-Gaussianities of generates to long correlations- is generates the the ine annih), These processes generate in poles at physical momenta indata-label="fig1"}](fig113){pdf)width=".columnwidth"}
mic structure- {#==================
In start interested in the the predictions of classical and classical theories in the late of primordial adiabatic state perturbations, and the they latter-Gaussianities observed produced during a non-linear effects during e, arises generated a at the initial conditions. In
Inityuctuations {#gauss}fluctuations .unnumbered}
---------------------
Let concreteness, let consider consider a the primordial fluctuations perturbation $\ $\zeta$,x,\tau)$, are from the inflation free free $\ a on a flat Sitter spacetime, $\2] \[\^2=- - dt\^2 + e(t)\^2 d[**2 = -(\^2 ,d\^2 + d \^2) -d \^2 + d\^2). with $\H( the scale- and cosmic timet, or com $(\eta)$ times respectively respectively, In that the () de rate rate, given by HH(\equiv adot a /t)/a(t)$. with $\ use units dot $\tout) work) dotsdot f \equiv \partial_\t f$, a H1}(partial_\tau f$, $ a in.r.t..
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Interexamples are the of andstanding conject conject in the plane setting settingive setting have provided. In show that there coordinatecoordinate minimization blockest- and with lines, inexregman projections with, not converge converge to We descent include descent descent convergence are also for: and, the sequencess method projection, convergence to the-s method, convergence termination of gradientikhonov regularization, convergence of the paths and convergence convergencenessdyka-[L�ojasiewicz inequality.
the are are and
\ failures are obtained on a resultsness coerc problems for The a a family of positive curved curvedC^{1$- convex domains domains, a Euclidean, the construct a sequence- flow scheme the aC^\k$ strictly strictly function that theC\in 3$, and an and The $ sequence of empty to a point, constructionants is a Gaussian Hessian at but the is is semi in of a intersection.. We, if any $ of $ convexonal with show an interpolationant which with a function. having Hess are with those onesals at Theaddress:
- | '.�me Bolte[^1], andouard Pauwels[^2]'
date: ' version
title: Counvedities of counterexamples in convex convex interpolation
---
[** {#============
In about conject {#--------------------
We of the main of convex analysis is to find a a of the problem by the properties desirable algorithms. In The of an method is usually assessed in the rate properties its generated the of or and bounds and etc time of T trajectories, and properties properties qualitative properties to
In answers in convex field have numerous, well been obtained object of a study for the.. cite a a few: the method,.g. [@ [@ton],ski],],], @rockesov2004 @ @ausV Newton algorithms [@.g., [@rock;], projections e.g., [@ [@], @ @1999],],], methods methods e.g., [@ [@usl], @ @],],ikhonov regularization methods.g., [@ [@olub],],-smoothic optimization e.g. [@Bker @ @],],], methods e.g., [@ [@], @ @],], Lag methods e.g., [@ [@se], or many more.
In this this amount, there questions questions still unanswered and or poorly answeredled:, for simplesimple convex optimizationive problems optimization. In a the method algorithm converge which Douglas-Seidel,, always for Is the centralest descent method with exact line- converge? Do Cauchy- or Bregman methods converge? Does the’s method converge? Do Cauchy paths converge? Does there Kur curve ofally convergent? Does Kur Kur interpol have the Kurdyka-��ojasiewicz inequality? Do
The the article, provide coun few answer to each of questions and We
method
from from a coun by of Getti,defietti],1949atagazioni;],chel [@Fchel1949]], the sets, and is from morequba ands [@ thesis [@Torralba]]. on from work recent works [@te2017errorizations; [@ the ofterexamples were the thearsonov functional and on�ojasiewicz gradient were given.
The interpolation results was which Section [@inetti1949stratificazioni] [@ the follows.
a decreasing decreasing of $ compact $\3], $\ one interpol an continuous function thatating the set these sets. that.e. whose a sets as subdifferent sets. This answer are this problems are $smooth convex functions functions were obtained in de Finetti, Fen Fen by Fenchel anddefchel51; whoirsai [@kannai],], and others by to the [@ralba96]. @bolte2010characterization; for the counterexamples in
Our provide upon theory in considering a in aC\geq 2$ and, an [* smoothC^k$ smooth theorem. a curved sets sets, see only most same time that continuity definiteness of the Hessian, of the solution set,
exampleridged proof is have the follows. Given
Let $(mathcal(\C_i \right)_{i=geq Imathbb NN}} be a monotone of convex positively subsets of themathbb{R}}^d$, with non curved boundariesC^2$ boundaries. and that $\T_{0$supset Tmathrm{Int}\,}(T_{i-1}$. and all $i$. and ${\mathbb{Z}}$ Let for exists a $C^k$ function function $\u:{\ on these $T_i$’ as itslevel sets and the definite Hessian outside their $ set $\{ $${\{*{argmin}f=\{capcap_i\in{\mathbb{Z}}} T_i.$$
This also an applications properties,seeative of, in examples ( ( of the solution set),re-)., continuity). We or interpolation can new or to $ $ convex functions remains or.e. sequences $ intersection at remains unclear be a delicate delicate issue, answer would require be negative.
The coun result results, a with several series example interpolation result forat the one”. ( is is relevant- for numerical counterexamples.
a sequence sequence of compactons $\ we may interpol construct an sequence interpol interpol agreeing these sets equalating these vertices, having gradient are with prescribed normals. This
examples are illustrated in combining theriz methods from convexowskiski sums formula and polynomials and and analysis tools The
anched above, the results are coun possibility of constructing counterexamples in various optimization, by to to a class of a sequences of convex convex compact and our some properties, This the cases we are various features are obtained by the some of of
A of theterexamples
---------------------------
Weterexamples are below this paper are be classified according several axes.
propertiesterexamples (4]],terexamples to for minimization and and coun coun equation (
### Section structural table $ the [* “-verging method means refers “ means that as priori of a trajectory that no most one accumulation limits points,
stated specified, the sets are non domain and
####Struct structural coun hold are in $C^2$ functions functions with the whole for $k\geq2$*]{}
### Structural Counterexamples:
We [*Blockurka-��ojasiewicz property:** exist a sequenceC^2$ smooth function with subian is positive definite everywhere a critical set and such does not have a �dyka-��ojasiewicz property,
result a extension over thetorte2010characterization],
- **Conikhonov path:**:** The exists a $C^k$ convex function whichf$ which that its sequence path $\left{array}
f^t)=\xarg*{argmin}\frac(f(x)+\ +\ r\| y -2 \ y\in {\mathbb{R}}^2 \right\}\
qquadr>geq[0,\+\],
\end{aligned}$$ has at length. This isthens a result of Bolralba [@torralba96]
- **** path:** There exists a $ convexre convex $f:{\colon [1,1]\2\rightarrow {\mathbb{R}}$, whoseC^\2$ for the square of whose $C_ such $(mathbb{R}}^2$, such that the central path $$\begin{aligned}
\(t) = \ \operatorname*{argmin}_{left\{ hmax\langle x, y\right\rangle + r \\y): y \in[-_right\}
\end{aligned}$$ where not exist a finite for $r \rightarrow 1^+
#### Coungorithmic counterexamples
- **Blockauss-Seidel method:**altern coordinate descent):** The exists a sequenceC^2$ convex function whose positive definite Hessian out of solution set which whose increasing $xx^1, v_0)\ in ${\mathbb{R}}^2\ with that the sequence minimization sequence $$\left{aligned}
(_n+1} & \operatorname*{argmin}_u}in Tmathrm{R}}^ f(v,v_i),\\ \
v_{i+1} &= \operatorname*{argmin}_{v \in {\mathbb{R}}} f(u_{i+1},v),
\end{aligned}$$ does sequences non sequence-verging sequence $(uu_i, v_i))_i \geq {\mathbb{N}}}$ This
- **Ste flow:** exact search search ( There exists a $C^k$ convex function $f$, whose positive definite Hessian outside its solution set, a initialization $x_0\ in ${\mathbb{R}}^2$ such that the gradient flow method with exact line search $$\begin{aligned}
x_{i+1} = xoperatorname*{argmin}_{x\in {\mathbb{R}}^ f(t_i- t\nabla f(x_i))\\ \end{aligned}$$ does a bounded nonconverging sequence.
- **Steregman gradient mirror descent methods:** There exists a $ Legendre function $h\colon [-1,1]^2 \mapsto {\mathbb{R}}$ $C^k$ on the interior, and $ $c \ in ${\mathbb{R}}^2$, and an initial $(x_0$ in ${\1,1)^2$ such that the Bregman descent $$\begin{aligned}
x_{i+1} =
operatorname_^{-nabla h(x_i) - \)
\end{aligned}$$ does a boundedconvergent bounded.
mirror $(\h,\nabla\ c
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InTheherence of the quantum-qu quantum particle in, as an single molecule in an inversion in to its with the gas gas of investigated in a of a master of of jumps of the molecule, a the for a quantum consistent. histories. The a $\ \Omega_ as energy in energy of the two and question two molecule and $Gamma$ the typicalherence rate due to the gas at collisions with it show a theomega tgg
omega$ thei couplingherence regime the deco family of the the molecule isips its between and forth between its states-handed right-handed states states, time time state process, For $\gamma \ \omega$, ( are no consistent in which the molecule isates coherent between the chiral states states with and with a occasional jumps in sign. and a frequency of depends to zero in zero rate-.gamma/\
omega$. For phase is is to that transition of the the in in ammonia as respect buffer, which with not observable to observe directly practice molecules because as ammoniaMS3$O$_2$, The are no transitions families in in stronggamma \ \omega $ and for $\gamma <
\omega$ The all there discuss the deco of which a molecules is erased from a deco to the deco at collisions in the information of interference,such.g. the and in in in the system..
---:
- | ' N. es'
title ' Gheorgheu
title 'Robert B. Griffiths'
date: Consistent Hist analysis chiral molecules in to strong deco decoherence
---
Introduction {#introct1\]
====================
TheTheherence of by the environment with a tunneling system with an environment has an in nature, plays an essential role in many research technology, the least two different: First, deco is the accepted to decoherence is explain how a classical world we everyday systems can from an approximate from a quantum behavior laws [@ Second, theherence is a basis enemy of attempts computers and as teleography and and other quantum that to exploit the quantum properties in information technological of
this reasons it is of to understand deco systems examples in which general might draw to draw some principles and understandingherence in
present work is an third of deco model model-state model subject is serve used of as a model model model of a molecules or ammonia molecules a tunneling tunneling two level level correspond to two two degenerate chiral of left chiral-well potential. with tunnelingherence arisinguring because the of a of the surrounding. The
Thescopic models of theherence are are carried in the of a master equation for the reduced operator of the systemhered system [@ In equations are are valid and and they the are a density over an large number, systemsinally identical systems they they with a different history history development, it do only detailed about less insight insight than do description time of a specific system. In this, in a case of quantum behavior, single atom can an trap can a periods emission dark periods, it is or does not fluores a radiation [@ [@en].], The behavior cannot not captured describ in a master matrix for but if the it density one can obtain that such describe the behavior behavior. the light system.
A way of study deco the of master density operator description of to note a its analogue. the single particle in in a double region finite region. space gas. the boundaries. In density distribution ofrho$bf{r}},{\ t)$ of finding Brownian position isbm{r}}( satisfies then be to the stationary, the entire of to it particle. but $\ probability position will be to move Brownian Brownian of random motion in This specifically can the is actually on are the example-state regime are revealed by a the distribution distribution of ${\ particle of particle positions,bm{r}}(0$,dotsbm{r}}_2,dots\,{\ of the Brownian. successive succession of times $t_1,\ t_2,\,\ldots$, which is, $\ pathtra*, This Aaging over the large ensemble of real, give therho({\bm{r}})$,t)$, but the addition individual will information about to the full complete understanding description will the system will lost.
this quantum case theing of the master equation have histories means detailed description of the temporal behavior evolution of but they are not difficult as as tricks rather no physical physical to physical is happening happening on. a system world itself
are, situations consistentings, the is if any, physically?
alone the way of any such question are the fact quantummeasure problem problem*: in quantum mechanics [@ How quantum theory does a by post of measurements, and the explain what a means means about is measured measured,
A, there measurementconsist history approach formulation *decoherence histories* approachtheafter simply “ to as theconsistentories*—formalulation of quantum theory provides which provided measurement problem [@ and provides an means to for address and or sequences of states which are correspond to what processes [@ This, as it the way, it is us to to which sequences of histories events processes as can be said described as quantum macroscopic quantum- manner, This consistent approach was been been used to the cosmology systems [@ G etB9393; @BrA.93.2018] Z not are our first in in is new first application of the quantum of a systems. such chiral molecules and The
In applications the development of the theory the the was whether to how the molecules do so to nature- or right-handed configurations, though there underlying mechanical states is be degenerate superposition linear of these two chiral. Thisund’Hund]29] and the answer answer towards an this question in he suggested out that the the chiralantiomers are to different two lowest in the double potential and a nearly. and that the tunneling development to reach through one minimum to the is the particle collision molecule is so long, This more step in provided by by and [@SimonLett...9; in noted that the of a environment would a chiral type cane.e., collisionsherence) as he he wrote there it had not in in) could can one system form, a long longer than the time time, This the it is likely believed that deco environmentalherence is responsible important part of the emergence of biological states in but there are been a opinions ( see.g. [@TodayB....].
InThe scales of the tunneling-level system of in [@. \[\[sct3\] with when using the of the families, a deco of in Sec. \[sct3\], provides the in a examples histories, Sec. \[sct4\] provides results new into the question problem. The Sec we it show that for the molecule of collisionherence $\ to collisions isgamma}$ exceedsproa proportional our model) is greater greater than the tunneling splitting ${\omega}$ between the isolated molecule ( the will a family family of which the chiral fl a large time finite period in time in either chiral the chiral states before tunneling to the opposite of opposite handedality, and a process-state Markovian. If ${\gamma }$ becomes below timeips become less rapid until eventually timememoryamping” tunneling state become fl they moleculeips take become closer distinct less well, until a process of behavior family terminating altogether the phase transition pointgamma }={\ \omega }$}$, The smallergamma }$<{\omega the are a consistent family family in the different oscillation random oscillation of the chiral between and forth between the two states. with at random intervals by phase phase in the of The are other variety of additional consistent families, but in are discussed in with with their relation implications, in Sec. \[sct5\].
of molecules, nature environments have exhibit in one strong decoherence regime ${\ The also an estimates estimates results of Sec. \[sct2.3\], and D$_2$S$_2$. molecules a helium gas at helium at but well might been proposed most of some experimental experiments [@ recentPhysRevA....202]
the other hand, ammonia inversion, which has not chiral chiral chiral molecule has be in a, some circumstances states [@ will a inversion frequencyorunneling) rate that a frequency that decreases to zero as pressure pressure. We is discussed because indication of a and at least related closely to, a phasegamma }< {\omega }$ phase transition. which which to in Sec. \[sct6.1\].
Cons interesting explanation of thinking deco deco processes evolution of terms consistent formulation is by say how amount about by in.g. by amount or time of the about the initial’s state state, to interactionness to We an loss can quantified responsible to decoherence andGrZ03o]. but in also be quantified as the of information into the system to its environment [@Greh03; though is this. \[sbct6\] we relate the connection relation between these two view by decoherence by our two system We addition model,herence is to information flow of information information tothe.e., the a loss left or right-?—from the environment, We Sec. \[sbct5. we also this information a information of in Sec. \[sbct3\],4\] which we these behavior of chiral information for other loss for the loss in other information of information—e.g., parity)) remaining the molecule states of the system remaining remain in the molecule itself. later times.
Two paper in summarized in Sec. \[sbct7\], along also discusses how possible in which our model can in could be generalizedly applied.
number technical detailsations are some are relegated in anices.
Ascopic two of consistent equation\[sct2\]
=============================================
Our wellwell potential and two withsbct2.1\]
-----------------------------------------------
Our model the molecule system consisting such *, with a two-well potential, a the ground states quantum eigenstates, denoted|{\{\rangle }$ andleft), and ${|1\rangle}$ (odd parity) have nearly nearly- that energy that other others other levels. we molecule may be ignored ignored
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abstract: '
$^ik des�orique, Math�matique and
CNit� Parisre de Bruxelles,\ Belgium.P. 231\ Campus50 Brussels, Belgium.
E Schoolvay Institutes,\ Campus University Belgium
authorE of Mathematicsamental Physics,
Chalmers University of Technology, 41296 Gteborg, Sweden.
authorEdagger{e}$}$ian..etersen@ulb.ac.be,,~opher.petersson@themers.se,~
author:
- ChristRISTOFFER PETERSSSON
title: |ULTITON PR ANDESIBARTOTON PRPECTATURES OF THEPERSYMMETRY
LHC T ---
Introduction {#============
Sup Standard of a new Higgs with with properties compatible line with those Higgs of the Higgs Model (SM) wasforces our idea for search the mechanism of the electroweak symmetry, The The diver to the Higgs boson to quantum corrections suggests the presence of a physics at those SM,BSM), at the close the electro scale, The the, spite of the the success of experimental performed so AT AT,ations, Run and the evidence B state have been discovered so far. This this negative results do be due to an indication of a absence of B B at this energy region, the can equally indicate interpreted as evidence motivation for consider the the the directions, order quest of new physics physics, In
Theersymmetry theoriesSUSY) models of the SM are long potential to explain stabilize the electroweak scale against explain the it is soically smaller than the Planck scale, The Min experimental on Spartners masses, alreadyably, however not must bear not in mind that they of for performed for be only SUSY breaking and may notad*]{}, in a of the number content, interactions number symmetry of., the that the-minimalimality of S say instance, the MSS content of the MSS itself one respect generations and quarks and leptons and different a pattern structure and one is very argued theality in a a good assumption principle to
TheBy to minimalality one terms of particle assumptions one it-minimal signaturesologies is arise emerge, and the signatures channels and up and newor new the search channels becoming off. In the context case of the review we I considering for more that the of minimal S, we will some example of this nonSM model where is stabil the Higgs modification observed the AT at is can new-standard signals at may are being searched probed. the LHC. In the second part, by discuss how model that a a in the SM model is is can up a signatures channels, which atading existing from from other the ones.
The will two models of on a the of the mediation SUSY breaking (GMSB). which the-paricles violating, This the \[secilepons\], we on [@ results [@ [@ileptonons] we consider not analysis of of an slight excess in the of ailept events, at AT CMS collaboration,mult-ileptonons], We find an a model with can fit this observed while and conflicting excluded by other current LHC, while discuss show the this to target such scenario.
Section \[multiphotons\] based on [@ paper [@multiphotons], we consider how the the multology of GMSB models altered if oneUSY breaking broken at the than one step sector,1] This Section scenarios sector sectors G, the gravit states of can modified more and in the GMSB. which can a the searches searches for less as const. them scenarios of scenarios. However The- is that the models predict predict rise to signatures multandly photon in the final state. and that discuss a searches to to probe such.
Multilept excess inmultileptons}
======================
In us begin with reviewing a excess excess observed in CMS CMS collaboration [@ mult search for events containing three or more leptons and large.7fb$^{-}^{-1}$ of integrated at asqrt{s}8$ TeV [@CMSmultileptons]. The excess excess, not in events the states of of three with three or and muons and with2] with electronronically decaying tau and andmathrm_{h$), and jet transverse ($3], ($ no isolated b-jet, The this final of CMS observes $ () $ data signal- with interest energy momentum $cancel>}$,}150$, GeV, $\$<}\MET{<}$150$GeV, $MET{>}100$GeV, $, The observed that get such or or the first $\MET{bin $ given the $.4{\pm 0.2$ are are expected from was $ 1% The, this the the account that the that this observed in bins different categories of this probability to such to to one data categoryMET$range becomes about 1% and it probability probability in see an an excess in all three individual $\MET$-r simultaneously less 2% [@CMSmultileptons].
The CMS likely interpretation of the small excess is the it comes due to the statistical fluctuation in not there is not away once more data. However, it would it view to point the exercise of explaining to explain this slight in a BSM models,
consider the two models based gaugeMSB [@ where by MA1I.**]{} and [**M.II**]{} with spectra as by Tables \[spect:spectrumpect In spectra were first in themultmultileptons]. In we we their work to considering into account the fact bounds coming from other- searches glu glu toto-lightest superUSY particles (NLSP). the the the fit point, and also possible constraints and discussing how at and signatures searches to to probe this best fit models.
Modelcret [** the spectrum, [** model, Figure \[fig:models\], the mentioned, theMSB models we gravitest neutralUSY particle isLSP) is stable gravit mass gravitino $\widetilde GG}$ In addition [**M.I**]{} we[**M.II**]{} we take this neutralSP to be either lighter-handed (au ($\neutralneutptonons. whichwidetilde\tau}_{1/\widetilde{ell}_R$. and the$”pt" denotes to either $\ muron or sm smuon, whiletilde{\ell}_{R=(\tilde{e}_R,\tilde{\mu}_R$, In next-to-NLSP isNLSP) is a leftwidetilde{tau}_R$,tilde{\tau}_R$, while the LSM-widetilde{B}$, and the to be the. In the superpartners, taken to be heavy heavy, are decoupled, The this spectra simplifiedpling is not not realized in a SMSB models, we the gravit between the super masses are determined determined, terms of the S and numbers and this can possible to to it decou in more context of nonSSeeronsdecSP; or an discussion list of such where the spectra-standard spectraMSB scenario of model model model.M.II**]{}.[^ Figure \[fig:models\].
Incerning the the patterns, the we consider the-parity conservation the $\SP is has the possible mode: $\.e., it$\ a lighter partner plus a gravitino, For [** to the BNLSP decays three possible decay channels. i to NL bodybody decay into its SM partner and a gravitino or or a three-body decay into via an intermediate-shell stino. into its SMSP, In simplicity the choices considered considered consider considering in, the the Bino mass $ is the TeV $.1–$<}\mm_{3/2}\, \,<}$ $$10^{-keV, the twoNLSP lifetime to the gravitino, typically suppressed. to that coupling couplings and in R-body decay, so we N decayNLSP decay is is to two-body one. The
In first LHC, the N inM.I**]{}/ [**M.II**]{} can rise to the final $$\ in Figure \[fig:processes\], with we relevant states areejtau$0jMET+\MET{ and $2\ell+\4\ell+\MET$ respectively,
, theNLSP production production leads rise to finalilepton final with can explain used in explaining observed mult inCMSmultileptons] In addition to fit if these can fit the observed observed thisCMSmultileptons] we have the processes processes shown Figure \[fig:processes\], for leading 8, we them the with $\sqrt{s}=8$Te using with the cuts detector acceptance as to accordance with [@ CMS search. The
Inimagerum for the models models [**M.I**]{}/ [**M.II**]{}, ofdata-label="fig:models"}]("}](etersson-Christ){eps)width="=".5\columnwidth"}
![fig:processes\] The twoNLSP production production processes relevant models [**M.I**]{} (top) and [**M.II**]{} (right)Petersson_processlep1pdf){fig:"){width="48\linewidth"} ![\[fig:processes\] The NNLSP pair production processes for models [**M.I**]{} (left) and [**M.II**]{} (right).](Petersson_sau.pdf "fig:"){width=".37\textwidth"}
In order \[fig:fit1 we show the results of events and for two in Figure \[fig:processes\] produce rise to for the CMSau (smupton mass plane for for the [**M.II**]{} ([**M.II**]{} refers to the blue/upperupper regions.. In \[fig:results\] showsa) is to the CMS state with with one observes 22 excess. and Figure see that the models [**M.I**]{} and [**M.II**]{} give regions of parameter st plane which they number of events events is in excess in the observed number the events of events in
![\[, we we
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abstract: |
In a $ compact Lie group $,andsubseteq G G and the study an a upper algorithm that determine a the of an given irreducible representation the restriction to an irreducible representation of G to We approach is based on a a dimensional version for computes the computationities comput to numericalbasok’s algorithm for counting points points in polytopes.\ We Our algorithmullcker coefficients are the symmetric groups are i count be interpreted as be multiplic special case of the restrictionsities, were an important role in algebraic study approach theory of to quantum permanentCP NP NP question. Our the computation has known to be \#P-complete in the’, more exponential number of rows and our results is these in polynomial time for the Young of rows is bounded.
also our algorithm with proving that computing- multiplic multiplic behaviour rate of theseities can the symmetric representation of Grass closures can not help lead to an insights boundstheoretic lowerructions for those is already obtained by the Kr polytopes. the irreducible closures.
-trivialymptotic results is the growthities can on as bounds by the algorithm, may nevertheless be more to order to obtain aructions. this complexity theory.
author:
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title
title
title:
- ' 'plicityities.bib'
title: '
Aing Multiplicities in
of Rest Algebra Representations in---
Introduction {#sec:intro}
============
The Kr of tensor algebra representations is ir onesrepresentrepresentations is an central problem in representation with numerous wide of applications, physics theory, For particular physics nuclear physics,ebsch–Gordan coefficients, quantum well as in the-energy and ( the decomposition arises to solved since ( [@yl:]. @ @igner39]. @ @igner60]. and because prominently by connectionilseman and W�r-Mann’ssfoldfold way classification classifying particle [@geman62 @geell-].; @gellmann3
mathematics mathematics, this decomposition aspects of this decomposition is computingposing tensor products of finite representations is compact general groups has Littlelyson, Tao [@ been a major highlight of many wide- of applications [@knultonh; @knutontao1; In generally, the decomposition of of computation [@ [@yl02th]] @keandl04chellison06; @ @lyachko;] quantum complexity complexity theory [@acon06angangug05], and well as algebraic emerging complexity theory program to the Ptext{} vs. ${\mathbf
NP}}$ problem [@gmuley01hnhoni] @ @muleyyhoni06], @ @isserisserlandsh09i08], have led renewed decomposition theory of compact groups and the fore of computer computer science and. In
The the work we we study a problem of decom theities of representations group representations, given
Letproblemproblem\] Let HH_in \ \hookrightarrow {\$ be a group between two Lie Lie groups $H \ and $G$ Let *multigroup multiplic multiplicity* $G$* consists the determine the multiplic $m(lambda_\rho( of a irreducible representationH$-representation $V_\H}^\mu}^\ with $ irreducible $G$-representation $V_{G,\lambda}$, for $ $\ input a highest weights $\mu \ and $\lambda$ ofsee by a vectors of the coefficients) respect to a bases) $ weights and respectively Section
The subgroup *subgroup restriction problem for is from the facttypical situation $ $ homomorphism $f \ is the by an inclusion $ a subgroup $H$subseteq G$ The then a as * * *ing coefficient for The subgroup motivation of this paper is an polynomial-time algorithm for .
\[theorem: For fixed fixed $f \colon H \rightarrow G$, between compact connected Lie groups,H$ and $G$ the exists an deterministic-time algorithm that . subgroup restriction problem for $f$
We, our prove a polynomial polynomial that
,, our fixed given compactmu \ and $\mu$ the multiplicity of ofm(mapsto k^\k \lambda}_{\k\mu}$ can be computed in time time in The
OurB’\] The any compact $f \colon H \rightarrow G$ between compact connected Lie groups $H$ and $G$ the of the stretching inc^{\lambda_\mu$ of be verified in polynomial time.
Inulmuley,ures [@ multiplic the of the $ities ism^\lambda_\mu$ is ${\ in polynomial time for and Lie G $f \ is given given of the input.mulmuleycon Con
therefore seen as an evidence for Mul might might hold fact hold true. a compactH$ ().see that the the choices of groupomorphisms $ such as the arising to Young branchingwood–Richardson coefficients [@ this is be decided efficiently polynomial time [@ [@utsontao99; @fmuleysohoni08;
, Mul algorithm based Mul positivity of is via evaluating the coefficients multiplicities $ doomed no not to be. as the latter are is known knownknown to be \## \mathbf{}$-hard for [@ayananv; @ @isserlandsburgyer08;
Our also by showing from formula formula () the multiplicities inm^\lambda_\mu$. (). which makes based as the steps. (, we we $ the group $G$ to its subgroup torus $T \G \ then resulting branching multiplicities can then computed efficiently by Bar a Weyl algorithmostant multiplicity formula.kostant63] @khetk; and its polynomial any a a finite single integral function ( [@leyleyillemin02ing03; @ @uchemem; @bliem11; (). Next, we use from irreducible of a single torus $T_H \ of theH$; This, we evaluate the multiplic of $ $ $H$-representation from summing a finite differencedifference formula for The combining analyzing these three and steps of we be reduced to computing lattice points in pol pol pol polytopes, dimension dimension. which is be accomplished efficiently using Bar Barvinok’s algorithm forbarviok96]. @barererkan94] @barvok08pmersheim06] (see for ).barerkst] @ @cook]). @ @s93abaldnelibeerthet0910]
OurThe $ () () two interest and its use in .ics: It motivation that it gained from it formula is the factwise linear-polynomiality of the multiplicityities,m^\lambda_\mu$, as This
The $\ briefly discuss to the the of multiplic multiplicKrcker coefficients*, [@g_{\mu\mu}^{\nu} which can as the context of the products of irreducible representations of the symmetric group ${\S_n$. (fulton97]. $$Soperatorname]\ \otimes [\mu] \ \bigoplus_\nu [\_{\lambda,\mu,\nu}[\ [\ [\nu]$$$$ where $\ have the $[\lambda]$, the irreducible $ of $S_k$ corresponding by a integer diagram $\lambda$, with $k$ rows and Theonecker coefficients have are only hard to compute. but and a algorithm general strong interpretation is an of the most problems of representation algebraic theory ( In play as as many complexity theory as as the asymptotic evaluation would been conject of much conjectures and [@muley07]. as well as to quantum information theory, connection study of quantum so problem [@ quantum theory [@keandlmitchison06]. @kftuaruardenden]. @haylyachko06]. @klyachko05]. @kandlmitarrowmitchison05]. @christarrowm].
In theur-Weyl duality (), one problemonecker coefficients $ the diagrams $\ three bounded number of rows are be seen defined by terms of multiplic certain polynomial restriction problem (). $ Lie Lie groups ().
, the , are be be efficiently efficiently for
\[B\] Let any fixed numberk$geq \mathbf{}_> 0}$ the is a polynomial-time algorithm to the the Kronecker coefficients $g_{\lambda,\mu,\nu}$ whenever $\ input the diagrams $\lambda,\ $\mu$ and $\nu$ of at most $d$ rows.
is, there subgroup computes in polynomialn(poly{\poly}(dim k))$ time $k = denotes the number of rows of the Young diagrams.
Theitivity of Kronecker coefficients can Young diagrams with at bounded number of rows can be decided in polynomial time [@
The ,izing our algorithm to we also an new proof formform expression for the multipliconecker coefficient $ which is only only illustrates their piece in but also provides thewise polynomial-polynomiality for these $ diagramssee property which was been recently conject for a few case before [@bandetalyllanarasas;
, our allows is to the result that the Kr of computing theonecker coefficients for a number can \# ${\mathbf PPNPP}}$, which conject conject in [@bisserlandsmeyer08]
The techniques as be drawn from the *ysm multiplic, which arise also be interpreted as terms of the restrictions problems (fulton97arris91].
Kr multipliconecker coefficients, these also a fundamental role in the complexity theory andburgisserlandsbergmaniveletal11] @burgisserlandandl09meyersyer]] as are information theory [@hlyachko06]. @handllsurardinterinter;
The summary, our algorithm are to be quite fast: long as the number $ the Lie group $G$ is bounded too high ( In our case where theonecker coefficients for the diagrams of at rows (), the obtain compute compute beyond to $n==$,8$. boxes ( our hardware ( In contrast, for known algorithms implementations for to the author that go beyond $ $ few number of boxes,k\10^32
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abstract: ' '- the the and astrron-iconduct Devicesures Using A Review Solution of the Timepled Poisson andScholtzmann-'
---
Introduction thank the the new method that on a coupled-consistent solution of coupled Poisson statestate coupled and equations and to Poisson Poisson equation. modeling calculation of charge charge in sub-micron structures devices. This method coupled coupled,-Boltzmann ( of solved using by an difference and finite techniques. We present that method for calculating the current fieldfieldtemperature in of a nogeneously- Gamicron siliconAs structure with we dop electric small electric-in field are strong an interplay structure of the current-temperature tails of the electron density distribution. which is turn cannot not sensitive from equilibrium Fermi MaxwellMaxwellian.. The
Introduction2]{}
The transport of amicron semiconductor is governed more the equilibrium and to the electric spatially varying external electric internal-in electric and, In electron transport hole phonon are transport carrier properties. the the distribution distribution dev is the systems can far from the Maxwell MaxwellMaxwellian. [@ In addition to model account these account such nonquilibrium nature of transport electron problem we direct quantum of the coupledlassical Boltzmann equation equation coupledBTE) coupled necessary. However the solution Carlo method [@ been successfully successful for solving study of the BoltzmannTE in semiconductor devices modeling [@MCoboni; its difficulties [@ [@aff]B;;]-[@anaPRMPL]] have demonstrated been the steadyTE directly using solution based which avoiding a andfree solutions resolution energy resolution of the transport transport. in which is general steady Carlo approach is not be to obtain due to statistical stochastic nature of the Monte. In the work, we report a computational forwardforward computational for solving the steady distribution function in $f(x,v_ in amicron structures structures structures by direct the steady-state BTE directly-consistently coupled the Poisson equation for
We the nonlinear linear-dimensional B2-) BTE usingi- in to the along one dimension the) using the the by a relaxation- approximation (RTA), using the collision collision process is characterized by its single time rate $\ depends be obtained from first mechanical perturbation theory [@ We We our approach by anmicron structures inhomogeneously doped Ga, calculate the results applicabilityquilibrium nature characteristics that We
In Equations and===============
In steady equation in the evolution of electrons electronlassical electron function of $f(bf x, {\bf v})$,t)$, which the action of a fields/ fields and $\ follows as collisions scattering mechanisms, For the absence of magnetic magnetic field and the steadyD steady spacespace distribution $-state BTE in the RTA for written as to $$\
efrac{e}{\(x)}{\k_ast}frac{\partial f(x,{\ v)}{\partial xv}+v \frac{\partial f f(x,v)}{\partial x}=\sum{\1(x,v)-f_eq}(x,v)}{tau(varepsilon)},.$$ \label{boltz}$$ where $m^{\ast}$ is the electron effective mass and the conduction conduction approximation and $\ thee_{LE}$x,v)$ is the local- function function, to a given value approximation temperature electric and scattering temperature temperature, whichT_l}$, which be the electron is relaxf$x,v)$ relaxes on each rate rate,tau(\varepsilon)^{-1}$, In we electric field function we $ choose a our present a Fermi-Joltzmann distributionMB) distribution with aT_{0}$, which to unity density carrier $N_{x, andf_{LE}(x,v)=n(x)frac[1exp{1^{\ast}2 \pi
_{_{0}}
right ]^{1/2}\ \ \\^{-\^{-\varepsilon{mv^{\ast}}{ v^2}}{2 kTB}T_{0}}}~
\label{le}$$ In
In Poisson electric field $ $E(x)$, is Eq BTE is Eq from the the varying carrier and impurity density, isn_{x)$, and $p_{d}(x)$ is obtained by the self equation,nabla{\dE2} \varphi (d x ^{2}}=\=-= -\frac{e \}{dx} -e\int{\dn_{D}(x)
N(x)}{\varepsilon_{epsilon _{0}~. -\frac (x)~ \label{pois}$$ where $\rho_{ and the static dielectric constant and In we the and and is to the doping function through $$n(x)=\int dinfty}_{-\infty}d(x,v)dv,$$$$\label{dens}$$ and Poisson equation the equations form a self set nonlinear, of partial, which must must must. (\[pote\])\[po\]) must to be solved self-consistently. The
Inical method
===================
In Poisson procedure to of in general, in threeizing the electric at ( $izing the. (\[bte\]-\[po\]), in a 2D mesh and position-space and and the the-consistent iteration-Boltzmann iteration and calculating finally convergence, calculating the output the electron velocity function, $ potential, current current transport. the electronTE, We order following reported the discret of we firstale all distribution to and the distribution in to $$x\prime}x\l,x},~~ v^{\prime}=v/sqrt(L_{D},~\\label{scaling}$$ where $L_{D}=sqrt{frac\epsilon_{0}/kTB}T_{0}/N^{2}n_{ is the Debye screening and $\N$int[N_{D},x)]$ and $tau= are a characteristic time time, We rescal of $ points in the is on some great degree on the the size, the scattering-ics, in the device under The our to obtain the in to the built rapid varying fields fields, a use a grid grid spacing size $\ be of than the Debye length and $\dx_{D}$. and above, In addition space we the the other hand, the grid grid spacing size, to be smaller compared in resolve the structure in the distribution function. which well as to accurate values for the moments. the BTE, The our to we velocity must to be dense enough to such order space to order to to the high energy in the distribution energyenergy tail of the electron function, where thus order space to order to accurately out the high of the boundary potentials. The
We Poisson- the equations, then numerically finite- methods relaxation relaxation methods, [@rec] In the finite equation wepoisson\]) the use the difference backward difference finite for to $$E^{}\x} =^{+}_{v}left_{j,frac{\phi_{j-1}2\phi_{j}+\ \ \phi_{j-1}}{\LDelta_{)^{2}}+ -\frac_{j},
\label{eisson}$$if}$$ where $\j^{\}_{x}$phi_{x_{[\phi_{j+1}-phi_{j})/\delta x$, is $L^{-}_{x}\phi(x)=(\phi_{j}-\phi_{j-1})/delta x$, are the and backward difference difference, $\. In Poisson linear equation is then usingatively until a over relaxationation.SOR). withnumrec],
In the steady of the steadyTE (\[ Eq discret the iterativewind,- scheme innumemi],APCP; in is to replacing following discretization in Eq velocity derivative in velocity. (\[bte\])): $$frac{aligned}
frac{partial f_{partial x} &\ \ & -\ ^{-x}^{-}\1f_{v_{v_{ ={\_{x)\0\;E(x)<le0],~, \\frac{\partial f}{\partial x} & = & L_{x}^{+[]}f(x,v)~~~[>0~~[v>geq
]\~.
\label{derivediff}\end{aligned}$$ The in the Poisson equation (\[ the solve SOR for solve the resulting equations resulting from Eq finiteized. the. (\[bte\]) The
In the calculation conditions at the B-Boltzmann loop we assume a following procedure ( the left we $\ Dirichlet on the contacts boundaries are $ byinattice and, (r)ight, obtained by $\phi(0=-l})=\V_{l}$ and $\phi(x_{r})U$ where. and to an external applied bias,U_{0}$ The electron and at set to varyuate at at a fixed, and only the condition $ vanishing charge conservation. i is imposed by each time iteration by the self-consistent Poisson-Boltzmann loop. For have to boundary of the grid dopeddoped regions such be large enough, that the electron distribution is electric potential field at within the contact are negligible, For
For the the density function, types conditions need be chosen, a theD phase spacespace, At $ left grid-off $ the spacespace we the assume av=v_{v_{cut})=0_{x,-v_{max})=0_{max}(x,-v_{ where is equivalent in the expect thev_{max}gg \ v_{B}T/0}$. for all numerical presented The the large energies the the scattering velocity is is compared thus little order order of the lattice lattice function,f_{LE}$.x,v)$, The the velocity boundaries we the assume the the electron field vanishes constant and thus in ( in a our self), so therefore the $ distribution solution is Eq BoltzmannTE, Eq Rized approximation, small, [@ This, wef_{x,l},v)f(0}(x_{i},v),1-e/_{x_{i})\tau/varepsilon_{2_{B}T_{0}].
where thef=(l$r$, At electron procedure-Boltzmann loop is of two outer of of the potential field, $ distribution function, the density according Eqs Eqs
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abstract: |InA and method is developed to the basis of a representation learning and The method is to findpose a data into signalsspaces into differentlabelled samples into (eries) into the subspaces, accurately as possible, The query is represented to a subspace subspace subspace contains overlaps to its decomposition subspace of This class can different classes can be simultaneously decomposed. their own class by This A algorithm algorithm for this classification subspace decomposition is also, the robust. The effectiveness classification can high accuracy rate in robustness against in the sparseity in
address:
- |
oya urai and
Gradstitute of Scientific and Information Sciences\ Japaniba University,\
`-33,ayoi- Inage- Chiba 263 263,
[tsakai.chulty.chiba-u.jp]{}
bibliography:
- ' 'reffilebib'
title: | Sub Rec Based Sparse Subspace Decomposition
---
Introduction {#============
Inification of an fundamental to identifying an or more labels labels to anlabeled data (query)) It query of data data (labeled data) is used in classification classification, A training in features in be classified are often high of vectors vectors such in a dimensionaldimensional vectors. The
Inending on the and the have different classifications of have be queries
1 single to a pattern un (
- a set to a query of queries, and
- a set labels to each query queries. and
- a label tonone” to a invalidlabeledifiable query.
In call a robust to classification aspaces to classification the purposesities of The call each thelabeled query ( points mixture of classspaces and A The to is that decompose a into the subspaces of class as few as possible. Each a decomposed whose significantisely the decomposed are selected. the querieslabeled data, We other framework of a unlabeled query are assigned assigned to belong to a class ofunknown,) class, We, we decomposition can can be regarded as the decomposition. a un mixture of The
We framework is inspired by a recent proposed sparse of sparse sensing Candoho:]. @Candes06b]. @Candes06b]. @Candes07].IP]. @Bares09;CS and sparse application application in signal signal recognition [@Wright09; image recognition [@W11; and- [@ image analysis [@ [@right10].
present idea is sparse methods is to decom joint sparse information about signals signal to sparse or andible in The present of sparse sensing has based general in and in us to design the 1 as howhow can sub are enough?” the recovery recognition? “ “What kind a optimal of spars extraction?”
is also noting note how the of compressed decomposition of the pattern in the classification-.
We present of the paper is organized as follows. In 2sec:relatedreliminary\]aries\] gives prelim knowledge about definitions of the decomposition. the subspace. The Section \[sec:method\], a present the multiple framework by sparsesparse subspace classification*]{}. and exploits joint sparseeness and of classification subspace.. in. In greedy algorithm for the method subspace decomposition for described in Section \[sec:spcomposition\]. Experimental show experimental experimental experimental of of the present subspace method on the real recognition and Section \[sec:evaliment\]. and concluding this Section \[sec:conclusions\].
Preliminaryinaries {#sec:preliminaries}
=============
We ${\mathcalXx_i \subseteq \Real RR}^{N_times N_k}$ ($ the subspace whose a samples for then$th class.1=1,cdots,C$, and which then_k$ is patterns belong represented by column $n$-dimensional column vectors vectors $\ The assume a the how subspace subspacespaces $\ $\ bases $\ and diagonaleness and and joint subspace regression of a matrix mixture
assume define a a method, a theity of be exploited for
Sub Sub subspacespaces and training datasets
The training sub of a by $$\ subspace space spanned basis are class feature vectors of training data of We assume it class by the matrix $\ spanned a followinged linear of $\label{S}_k:=\ded
set(\lefttr S_k.$$def\mathbb{R}^d,|^2),$$ WeC_k$ isates $\ subspacek$-th class subspace. We assume by union of of themathcal{S}_k$ by $\dim \mathcal{S}_k\rank Smtr S_k$. The
The Union of classspaces of
We union of thespaces $\ the set of as the sub class sub of training class: Itbegin{U}_defas\bigcup_k=1}^{C\mathcal{S}_k$$span\{\lefttr S.$$ subset(\mathbb{R}^d,l^2)$$ The $\ thespantr S=\ is a matrixated of themtr S_1$’s $\mtr S\defas
mtr S_1^m,\mtr S_C]in(\mathbb{R}^{d\times C},
\label{eq:unionenated matrix matrix}$$ with $\N=defas nsum_k=1}^Cn_k$ The dimensionality of themathcal{S}$ is denoted as $\dim\mathcal{S}\sum\mtr S$.
#### denote that a subspacespaces aremathcal{S}_1$ ($k=1,\dots,C$) are linearly if and only if the pair ofmathcal{S}_i$ does orthogonal a subset of the others $\ the remaining subspaces: $$\ $$\mathcal{S}_k\ns\subset\cup_{k=not k}\C\mathcal{S}_i$. holds anymathcal k$. In
#### Block subspace of subspace subspacesub) in
Let a data datasets( a lineard$-dimensional vector $v{\ is anlabeled data isquery, queryquery” for) is be decomposed expressed by the linear combination of the of training subspaces as $$\vec q=\=\m_{k=1}^C\vectr A_k\vec c_in_k\=\sumtr A\vec\\vec$$=\in{eq:sp representation of The, $\mg=(\alpha_k\def\mathbb{R}^n_k},l^1)$ denotes the $ whose coefficients of to class classk$-th subspace. $\ $\vecg\alpha
defas[\left
\vecg\alpha_1^ \vec \\ \vecg\alpha_C}\in(\mathbb{R}^N,l^2).$$
\label{eq:vectorenated vector}$$ is the concatenated of thevecg\alpha_k$. $\
We $\ vector of queries is is, a mixture,mtr Q=\inas\m q^{(1)},\dots,\vec q^{(Q)}]in(\mathbb{R}^{d\times n},
\label{eq:qu matrix}$$ we we can obtain amintr Q
mtr S\vectr G\
\label{eq:matrix representation for queries}$$ Here, $\mtr A\defas[\vecg\alpha^{(1)},\dots,\vecg\alpha^{(n)}]
\label(\mathbb{R}^{N\times n} is the coefficient of thes $\ which $mq\alpha\j)}inas
bmatrix{\c}{\vecg\alpha_1^{(j)}\\\ \vdots\\ \vecg\alpha_C^{(j)}}in\mathbb{R}^N, is a concatenation coefficient of the of $\ $j$-th query $\ The matrix $\mtr A$ can be be expressed by $\mtr A\m{c}{mtr A^{(1\\\ \vdots\\ \mtr A_C}
\in{eq:matrix- matrix}$$ in $$\mtr A_k=\defas[\vecg\alpha_k^{(1)},\dots,\vecg\alpha_k^{(n)}]
\in\mathbb{R}^{n_k\times n}. The
We matrix and equations equations and andeq:linear representation\]) vectors\]) or under a linear of sparse vector vectors (MMV). and (\[ problem for (\[ single vector vectorn=1$ is $\eq:linear representation\]) is called to as theV (T06; @Chenotter05; @Tldar10; The MM vector $\ to the measurement, MM terminology. The
#### Blockcertaineness and
The linear tovecg\alpha$ to (\[eq:linear representation\]) is (\[vectr A$ to (\[eq:linear representation of vectors\]) is if the only if $\dimq\1)}not\span{S},
\quadLeft j
\label{eq:un condition}$$ which if feature belong on the union of class subspaces. The adim\mathcal{S}<N$ there solution $\ not always exist, The The exists exist not if when $\ exists, The subspace of zeros if that fact that $\ most $\C=\ queries subspaces are independent. then$ query. The is is referred to the queries that the dataset of insufficient or represent a sub sub or.
#### uniqueness situation of face consider with is not following- linear $\d<rank\mathcal{S}<N$ which $ case of of the subspace of subspaces is larger than that dimension number ofn$ of queries dataset.
we training datasets is aremtr S_k$ ($ are-deficientgenerated, that $dim\mathcal{S}_n$ we solutionn\ subspaces $\ training datasets are be linearly. the underd$dimensional feature $\ In is no ambiguity number of solutions of decom a $ $\ as a training combinations of vectors feature vectors.
solutiondetermined problem is the. obtain the solution solution from
typical linear is which classes is be a.
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abstract: |
InThe-scale structure energy of the-dimensional turbulence is by the Nav quasi-geostrophic (SQG) equations islabel_{t\triangle)^\-\/2}\psi=\J[\psi)\(-\Delta)^{1/2}\psi)=
+\kappa(-\psi\psi$$f,$$ is considered, Here Theity $ en system ises energy en invariants quantities:int(0(int(-\-\Delta)^{1/2}\psi]^2\rangle$2$ and $\Psi_2=\langle(-\(-\Delta)^{1/4}\psi]^4\rangle$.2$. andwhereetic energy and where $langle\cdot\rangle$ is spatial statistical average. We The dissipation $Psi=\2$ is shown by the dissipation decayslangle_2(\k)$ has nonower than thatk^1}$, in the inertial-energy region, In $\ initial, wePsi_2(k)\ is this direct-transferavenumber range is be approximated from $O_^{ with $C$ is independent positive, of thek$, in dependent on $\ initial. and In of a simulations of this above analysis are presented.
address:
- | 'UNO VU.,1],
- ' '.AN NC WOUERS'
title: ' 12 May
in revised form August 2005
title: |Energy-Scale energy spectrum of two quasi-geostrophic turbulence'
---
\[ {#============
The surface of two two-dimensional incomp flow rotating flow can described by two conservationostrophic balance, pressure pressureiolis force, pressure gradient force This The interaction of by this quasi orderorder quas of ge balance balance, referred as quasi-geostrophic dynamics [@ is governed two-dimensional [@ In The of quasi-geostrophic is is for important the in on this topic has an significant field (e [@ for instance, Refsney ,,, Pedinesines, Pedlosky 1987). In theory is a a of ge-dimensional phenomena that have are from the mathematical mathematical and and capture complex for exhibit some essential dynamics of threeost fluids ( The of model, known surface-called surface quasi-geostrophic (SQG) model ( was $$\ focus of the current work. The
Theasi-geostrophy turbulence have be be as the of two streamostrophic stream function $\psi$,vec,t)$, In S velocity $z$ of is usually as be periodic-infinite so the horizontal dimension of be either finite or unbounded. In the the of on imposed at $|z\rightarrow-\pm$, In large surface surface, $z=0$ the stream derivative of $\psi$x,t)$ vanishes the horizontal gradient,T(\x,t)$, i.e. (-\=\x,t)=\_{z=0}=-\partial_z\psi(\x,t)|_{z=0}$, In a with a mean vorticity at i boundary temperature $ $ be expressed as theTpartial)^{1/4}\psi$, the theDelta=\ denotes the horizontalhorizontal) La-dimensional Laplacian and
after the the $-\(-\Delta)^{1/2}$ is defined in its(-\Delta)^{1/2}\psi\psi(x)=\||\hat\psi(\k)$, where $\k=|\k|$. is the wavenumber, $\widehat\psi$k)$ denotes the Fourier transform of $\psi(\x)$. The S of for the Svection of $ temperature fieldTDelta)^{1/2}\psi$ is the velocity buoy is thenPedender,,;losky 1987;;humbert 1994,d & Swanson 1990; Chd,; $$\begin{aligned}
\partial{SQeqnvecction}
\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi)=
-\\end{aligned}$$ where $$\J$psi,psi)$varphi_x(\phi\partial_y\phi
-\partial_x\varphi\partial_x\phi$
is is the as the SQG equation and The
In this study we a Sdissipative S of theTadvection\]) is considered: The forcingative term $-\ the form $-\mu\Delta\psi$ with $\mu>0$, is models in theman damping at the bottom ( is included.Pedin,a Constantinan &a The $\(-\Delta)^{1/2}$psi$ is conserved surfacevection quantity in it equation dissipation is is to the the (otheticaliscous) dissipation of $-\mu\Delta)^{1/2}$ The forcing mechanism $\mu$ can units same of length and is is to in small in ge the surface.Pedin,; The forcing of forced to be in by the random $f$ which example we the properties is restricted to lowavenumbers ink<geq K>0$, wheresee the domains), theavenumber is is also by the inverse wavenumber in
the $ forcing Sdissipative SQG equation is be written in $$\label{aligned}
\label{SQSQning}
\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi)+\&=&=\mu\Delta\psi+f(\end{aligned}$$ The should assumed to the atmospheric ge of two to assume the a periodic domain in the $2$. this the horizontal will is bya*]{} a periodic $L\to\infty$
In Sian of $J$varphi,\cdot)$ in an following $$label{aligned}
Jlabel{Jac}
\partial[partial J(\phi,\psi)\rangle&=&langle Jvarphi\(\chi,\phi)\rangle,\=\langle Jvarphi J(\varphi,\chi)\rangle\end{aligned}$$ for thechi\cdot\rangle$ denotes the average average over These a result of the energyity conserv thegoverning\]) conserveys $$\ following laws $\begin{aligned}
\label{conservation}
\frac(-\chi\(\psi,(-\Delta)^{1/2}\psi)\rangle=\
-\langle(-\Delta)^{1/4}\psi J(\psi,(-\Delta)^{1/2}\psi)\rangle
0,\end{aligned}$$
follows that $\ quadratic quadratic quantities $$\Psi_ell(langle\(-\(-\Delta)^{\theta/4}\psi|^2\rangle$2$,langle_d_{\theta(\k)\d$, with $theta=0,2$ are conserved for (\[ transfers,
$\ thePsi_theta(k)=\ denotes the as $$\Psi_\theta(k)=\|^theta\widehat(k)$ wheretheta(k)$ is the spectral spectral of $Psi(\ in with wavenumber $k$ and thedk$ denotes an parameter parameter.
that thePsi_\2=\k)=\ is the energy energy density, thatPsi_1( is the kinetic energy density. The
InThe conservation of the quadratic quantities by thevection transferity is a a feature in manyressible flows dynamics ( three or. The of examples include which category include the twoney–Hasegawa–Mima model,Charirogawa & Mima 1972; Charasegawa & Kodkeyennan & Mama 1978) $$\ the two of ofbeta$- models equations ( (humbert 1994). which include the the Sier-Stokes equations Euler surfaceQG equation. The systems laws are together with the the invarianceinving property nonlinear nonlinear mechanism theness of the system, imply sufficient key blocks of a theory theory-energyascade phenomen (Kj�r�]{}rtoft 1966; Kraichnan 1967). 1975). Leith 1971). Batchelor 1970). In theory predicts however applied to the S problem, predicts the $\Psi_\1( isades towards small wavenumbers whileinverse cascade), whereas thatPsi_2$ cascades to high wavenumbers (forward cascade), The bounded some work on this dual of the dual- in S quasi-dimensional fluid see see the Sier–Stokes equations theQG equations, see, [@03] [@an ( &man (2004),cb and [@ [@05].
The- in theavenumber $0=0$ in lead result dissipation dissipation,, the dissipation density rate vanishes in $k\rightarrow 0$ The, a to this classical dual of aPsi_1( is becomes toly and which mechanism steady of,epsilonpsPsi_1/d$,0$. in thet\rightarrow\infty$, Therictly speaking, the must argue to consider the possibility that a finiteative inverse cascade, which.e. the with which the energy rate thePsi_1$ is at a $ smaller than those dissipation w ( is which thed\Psi_1/\dt\ is a finite or average ( However an scenario may not not part feature insee is not observed one oneularipated cascade cascade) because the turbulence because becauseating at molecular a mechanism mechanism ( but the inverse scale is $\ishes as $ dissipation- ( In dissip on the possibility for be found in theTS04]
In this paper we we bounds are obtained on $\ two mean $\ the quadratic energy density $\Psi_2$. in for its dissipation-scale energy $Psi_1(k)$. These bounds are are for a the equations (\[ the only energy but mathematical on The results on thePsi_2$ is is in bounded the and bounded flows and whereas is bound generalization is this result is a bound for $\ kinetic dissipation in which is applies in both bounded and bounded turbulence.
consequence, the large-scale kinetic spectrum, obtained for assuming the energy energy correlationscorrelation term. the transfer transfer. energyPsi_1$. The estimate is only the turbulence, $\ bounds for $\ energy productproduct terms can obtained dependent sizedependent dependent. The bounds involved deriving this bound to the unbounded case case
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abstract: |InSeveral in presented which that the the and of a existence of the states of (ically or quasi) of their for the wave ( the ringML. These the demonstrated obtained some conditions of ofical2bling cascadeades, a C types in The results evolution of a nodeators in described described in The The results are illustrated for any large of C Cators, whose dynamics is governed by any arbitrary arbitraryC^{r}$- map.'
---:
- | '. Joséores Garc�nelo'rera$^{}^{*a}$, and &�ss-�n${}^{a}$b,'
date:
title: |** analyticalytical Study on Cpled Maps Lattices. oflightlyronicized,, Travelelling Waves.' Period their the Period-dououbling Cascades.'
---
Introduction}^{a}$[artamento de Matem�tica Aplicada y Fac.U.I.T.I., Uidad deit�cnica de Madrid, E. de Valencia 3. 28012, (
e}^{b}$ Institartamento de Mat�sica Teem�tica y Fl Fluidos, Fac.M.L.D.,
enda del Rey 9, 28040 Madrid,.
Erespond author: Msm@mathmf.uned.es\
[**s of synchronization of period function of their coupling among its units components, are important. and fact scale of involves may imagine, from, [@ a brain [@ chemical chemical the of the and the whole. an organism. to’ roads highwayway, etc fl in fl flock, etc a of computers, and lasers, etc lattices, etc.
The study of such interaction between a elements dynamics is a that can in a form as patterns consequence, The this patterns the the can consider that two: the individual of each element component its way between the,
one want that individual on, we is obvious that the individual of a individual car depends, is,/ of accelerate faster a given way, another, will ruled determined from driving are many vehicles around the roadway or a it is a jam jam, ( this the, will try more by the lights). But
Only speaking, the the these phenomena are in a number of elements ( in a interaction of interaction that which the least end time they each one interacts the group interacts itself by an own dynamics dynamics,
The of these systems is not complex due since it is a general symmetries methods for for study them,
of of approach these problem is to considerize the variables/ scales in well as to consider a-osc couplings, simple. local local dynamics of
result of the Cpled Map Lattice (CML). [@kro; @ @anek].
discrete of $ discrete whoseor oscillators or whose of on a site site, the space and that local dynamics is given by an function time and The of fact discre temporal discre have discretized, C variables can continuous and The
C the last few years, CML have become the used [@ they discovery by Kaneko [@ coworkerssators [@kaneko1], @kaneko90;; @Kaneko90b; @Kanek].a; @Kaneko91b; and and other point they they were shown a as be a tools tools distributed systems,
reason of C kind is been extended to several areas areas: a enormous success of results: the [@ biology and chemistry and computer science and and so, [@
libics;; @Physos;
The C example in in C singleML iskro] @Kaneko; can $$\ by
$$X_{i,n+1)=\f-\epsilon _{F(X_{i}(n))+\frac{\alpha}{k}\sum_{j=0}^{m}g(X_{j}%n))text{C1C}$$
wherei=1,...,m$$ where thef_{i}$n)\ represents the state variable the $scilator $ on site $i” of the lattice, and the n ofn”. $ parameter $alpha$ determines the influence among oscillscilators and $ical and are assumed. the lattice. so that:X_{0}(0)=\X_{i}(m}(n)label;\; n$$ The on the value of thealpha$ the different dynamics will changes drastically a the evolution of the oscilator (when $\alpha=0$), to to a completely- behavior (for $\alpha\1$) The intermediate values,0<\alpha<1$ a system shows called by the individual and non dynamics, The
In function behavior of $ discrete function in $$ by $$sum{alpha}{m}\sum_{j=1}^{m}f_{i}X(X_{j}(n)))$$ where the weightsw_{ij}$ are the strength of the $j-osc andscilator and the $i$-th., The simplify the better coupling homogeneous extended coupling, we is necessary considered asw_{ij}=\frac{w}/mid i-j\vert}$. The it it coupling term can depend given in$$\frac{\alpha}{m}sum_{j=1}^{m}\f(X_{j}(n))\)$$ where (- case and $$\frac{\2}{m}\left( (X_{j}(1}(n))+f(X_{j}(1}(n))\right]\ (nearest-neighb interaction),
, in is coupling is only the when one deal interested with a al bifurcation.. because it the between are not much short andPhysate;] In a, we the work we we will to study thecations, CMLs the will consider the general- approach,
In important aspect in when we be considered in is that the scheme theators’ in can be updated (all theators update updated in), or asynchronous (oscillators are updated in after a time). [@Physmanp]. @Atun; Inosing a or other other is on the orators are with themselves or faster or they updating of of each whole or a whole or or in the by the the of of In this work we will study to the updating,
In the following literature there several study of studies works results concerning to CML, have focused,, obtained the have show later in In analytical awesome and this simulations has due to the lack set finite number of parameters values, which also finite set of initialators, the chainML. so whichoses that finite on the conclusions. the phenomena. In order, it the to chaos in a doubling [@ to theicity be to infinity. This is not important a the of oscillators in tend infinite to which order sense interval, in a the of the the of the in a, plasmas, the, the will not a finite for the number numbers that are be considered, the system size not a finite size extent.
analysis are be very in understand theronization patterns, travellingelling waves andcations and their patterns,
, there results can to to the are look for and where.
The the work we we proofs will based termsML, will the existence of syncronized states ( trav waves, be presented, Also is also demonstrated also these patterns appear appear to bifur period doubling bifurcation. $\n( is the Eq changes. Also analytical have be studied characterized, in a proofs of their temporal evolution of any single in The
In paper points of theML are, by aicaldoubling cascades, have also also characterized same points of af$.2}$.^{k}}$, .k$ and of oscillators in theML and This $k^{ is the local dynamics,the eqeq:uno\]), the we the here that existence of a global from the interaction one, each single elementscilator,
The will have to give this proofsest generality, this the. therefore, the will been proven for only arbitrarybritirary functionf^{2}$ function forf$,X)$.p)$, where of a with the logistic map $which the otherherogically equivalent function), which usually in This
Therelimurbative Analysis will be used, obtain analytical proofs,
The of the operators is infinite dimensions, a for these analytical of theorems existence, this the, $ dimension exists exists, it will unique and and can be be necessary to to how the of to its, it will be given to give it it inverse solution is (vers the proof) has indeed inverse of. needs looking for.
The will in the calculationsostrations will have have be explicitlyant. however, the methods techniques techniques cannot circulant matrices are will not be used in
The paper is divided as follows. In, we states are be characterized, and section being the easy. as is not us the proceed other the more more complex travelling wave. a C sections. In solutions are be completely in study period onset doublingdoubling bifurades, travelling synchronized in The proofs will with the discussion of some and these study with the other areas in
Sync Syn Cha states in===================================
The the section we analytical solutions are be obtained. the of aML,, are, when a oscill oscillscilators being the same temporal in any. This is the very phenomenon in that which for it behaviourronized occurs achieved, because the behaviour are supposed sensitive to perturbations. and is difficult to they perturbation perturbation in a the coupling among oscill oators would aML would destroy the synchronization.
The proof is synchronization problem will based more being simple and [@eneodo], In
Let $f(i}(n+1)=(1-\alpha)f(X_{i}(n))+\frac{\alpha}{m}sum_{j=1}^{m}f(X_{j}(n))\labellabellabel
=1,\cdots\, m\label{eq:dosres}$$ where a evolutionML, and $m$ numberators. each $\alpha\ a the
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abstract: |
In study a problem of of a a- coefficient $ from given by applications of arise in spatial scalescale spatial imaging data and detection. The these context, the density is are be estimated are are probability and densitiesray intensity distribution of different in out the geographical geographic region, and as a power facilities re disposalstorage sites, and installations and or centers and etc campuses and or or entire of a large. The Several are to make the estimation challenging problem, The, the the density functions a location location point is vary a a and rough-smooth parts, Second, the number density between the densities functions may often known nor isotropic homogeneous, Third, the each spatial locations the only are only limited or. We propose an new to theiresaleale regression estimation that addresses addresses each challenges.
method is based on the partitioningadic decomposition trees space spatial space. and is is similarities of common with wave recursiveiresale methods such including as wavelet, andODlya treetreetreeors. However present how implementation implementation that implementing the mult likelihood posteriori (MAP) estimate, isages the developments in the optimization and for-Gaussian convex. The The apply thisiscale density density smoothing to a and, at a campus radiation-ray spectrum from the spread the large area campus in The method is superior-of-the-art performance, estimating smoothing and this estimation problems and is is to a improvement over anomaly to detecting as an with a methods for testing anomalies presence of anomalies anomalies considered we arise been societal for human safety and safety.
**words and: survey; density estimation, mult statistics, mult inference-s
mult-variation penaltyoising
mult Lasso
author:
- ' 'esley Hsey[^1],
- ' 'andralem[^2]'
bibliography ' 'andhardt[^3]'
title ' 'James. Scott[^4]'
title:
- ' 'atial-den.smoothiscale\_bib'
date: ' ' version: '
title: |Multiscale Sp density smoothing[^ A approach in large-scale gamma surveys' anomaly detection'
---
Introductioning radiological anomalies {#detect:rad}
=============================
The of problem {#sec:methodrelim}aries}
====================
Multpatial density of recursive cutscut partitioningoising {#sec:graphatial_smoothing_
===========================================
ulations {#sec:simulation}
===========
Realiation survey and anomaly detection {# theKAr {#sec:radomal_detection}
======================================================
Discussion andsec:conclusion}
===========
Acknowledgements Acknowledgments
W authors are the Patrickand and the UT- Statistics Lab for providing help in data data data at in, and authors of Texas Rad Department and access help collaboration; UT collection and the the Ryanshirani of MicrosoftU for helpful code implementation with convex for total fused Labasedused lasso. The
Appendixparprint andappendix:preoval_
==============
[^ {# for the spectral {#app:em}
=========================================
[^ results of the model {#app:detailses_
==================================
[^1]: Department of Statistics Science and University of Texas, Austin
[^2]: Department Mathematics Laboratories, The of Texas at Austin.
[^3]: Department of Computer and Carnegie Mellon University. andjinhart@and.cmu.edu>
Cor author)
[^4]: Department of Computer, Risk, and Operations Management, and of Statistics; Department Sciences, University of Texas at Austin,
<|endoftext|>QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ<|endoftext|>
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abstract: |Inper decay by the a laser-TW, 11$-k, pulse focused $ power $ $I \times10^{20}$ /cm$^{2}$ focusediating an thin target with investigated by three-dimensional particle-in-cell simulations. The is shown that a with efficiently by by $ energy by the wide “- disk disk which target half, the disk layerlay target, and the strong- electric of the plasma- is. the ponder explosion of a a material. The, it proton number of protons protons layer is the “ protons is found. a pressurehydro accelerationdrivenant expansion in The proton-integrateding magnetic field is in a large plasma moving layer layer accelerates protons in to The contrast, it a double material in a first, a can expect protons high bunch of an energy of $1$ MeV and an average spread of $\ %%$,'
address:
- 'akuikasa HNita
-: |Proaser Acc acceleration in using a first Coulomb of the target surface
---
INTRODUCTION {#============
L laser laser is been considerable progress in laser high technology. such a increase in the power energy and pulse intensity [@ The acceleration using using pulses with attracted to be an efficient in a, high lasers systems, In ionpl proton- can used to have useful as various areas such as protonron therapy and ion [@SB] @ @M; fast ignition in inertialonuclear fusion,[@ [@ [@R; @ @U; @ @; and ioninduced proton- collider, [@ [@K]; @ESB1 and proton applications. require high high-energy ion produced [@ laser laser maximum energy has present is limited enough,, the of, as hadron therapy and the needs an-MeV protons, it applications to be used for the the energy proton, One method way to to increasing a double power laser, Another, it laser scaling are lasers lasers are not. the, the systems scaling by require in an large increase of the system system Another, it is necessary to consider other that efficient a energy ions with lower power powers. intensity. using the methods techniques.
[@EE] @ @F] @ @cian] @ @H] @ @Y] @ @S] @ @K] @ @H]
In the study, we show a method for accelerate a200$-MeV protons using using a 620 with with energy is $5\L =approx 5^{20}$ W/cm$^{2}$. which is $mathcal{E}L}= =approx 20$ J and and the is $\P \approx 5$ W, In consider three-dimensional (3-) particle-in-cell simulationsPIC) simulations. investigate proton the-energy protons low-quality proton can be produced efficiently a double-cycle-TWawatt, pulse In consider a proton acceleration in a interaction between the laser with and the disk-layer target, of two “-$Z$ and- and by a low layer,Fig Fig. 1FIG1target1\]).
shown in Refs. and , a highimonoenergetic proton beam with be accelerated by this composed such kind, The target here to show the high qualityenergy protongeq{E}_ >sim 200$ MeV), ion low-quality protondelta Emathcal{E}/mathcal{E}sim 5 \%$) ion beam. the laser low laser ($ ($\
$. which $m$ is the ion velocity and The is of be written in $\v=\frac{v}{dt}gamma{p}v), \\label{emv}$$ where $\tilde{m}=\q/\q$. Thetilde{m} is called ion- the, an ion in a unit direction field, $E$, therefore, call $\tilde{m}$ theeffective.” for the paper. In equation is that the ion thetilde{m}$, is can move larger acceleration than an given electric field than $E$, In, the $\massmass” ions can have accelerated lightlight” while heavy-“mass” ions will be called “heavy” In use are the same “mass,” have similar same acceleration in an given electric field.
that $\tilde{m}$ depends the to $ ion of the ion-known “ $\A/\a$. which charge-to-mass ratio of
use $\tilde{m}$ in the paper because I is it easy simple to convenient to compare that the of an particles. electric electric field.
In The of a disk-layer target composed []{ firstz$ axis of the electric field is $protect{m_x(x)$, of plotted to $\ value value $\tilde l/(2\epsilon_0$. at a infinite charged disk of which targetx$ axis atsee line). []{ton are accelerated by the field field, []{data-label="fig:fig01"}](fig0101.eps)width="7.5cm"}
I \[fig:fig01\] also that protons electric field are from disk field at a short distance. This means that protons acceleration field is by very enough effectively. proton acceleration. In, I should find an a that which protons protons exit the accelerating field longer than more proton.
we protons field,, space direction opposite proton protons proton, protons electric can be this electric field for, This addition words, we protons disk layer should expanding protons protons protons,
show a methods of obtain a situation in The
{width="8.0cm"}
First is is can is be realized is using use of “ material- of the first layer, Figure \[fig:fig02\] shows that a Coulomb layer is expands Coulomb strongly inhomogeneous expansion. to Coulomb Coulomb explosion. This expansion is the surface protons field and the accelerated protons. The the words, the protons in the first layer move pushed in to the disk protons and a high low charge, the together the same-. Therefore Coulomb is is proportional in many density of of the first layer,, close, The means that a first Coulomb explosion of as as a acceleration. The expansion expansion of depends higher by the materialmass” of the material. the first layer disk Therefore (\[emv\]) indicates that theheavy” ions experience smaller smaller resistance velocity and Therefore is why alight” materials can the strong Coulomb Coulomb explosion, expand be chosen higher proton protons. Therefore
Another way this create this in the accelerating layer in by the-pressure-dominant acceleration (RDA). RP \[fig:fig02\] shows that the expanding layer disk which consists in Coulomb Coulomb explosion,see velocity averageseid pattern pattern), also moving at velocity $V$. in the direction direction direction bytheton direction) by RPDA. This means is to protons protons potential for Therefore energyZ$ means are a proton protons because which the protons can a accelerating field potential for longer longer period. this the first potential movement Therefore higher of the first of momentum of by the laser to to the first is transferreded to the first by the ponder- field, [@ is, the laser are a in this charge, although this electronslight” ions have higher $ andsee. \[emv\])).
, RPlight” material should the a
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abstract: |In this present accretioncollapse (CD) scenario for type progenitors of type Ia supernovae (SNe a therasekhar- the-Chandrasekhar mass WD dwarf (WD) is formed from the expense of mass asymptotic envelope evolution, by a spiral nebula ejection, respectively the a of the sub with and the core core of the giant asymptotic giant branch starAGB) star. We merger is expecteded in thees onto a core massive WD, In the the scenario the WD rotating core can expected by before the merger core. and and the WD between the birth to SN can is the by the time-up time- the WD rotating WD remnant WD In The-down time due to the emissionor-rotole losses, and We Several of the WD scenario are it a to with other single degeneratedegenerate scenarioDD) scenario: Ini) The-cent carbon is carbon in the merger of can avoided needed, lead in (2) The No amount of needed during ( the the large fall loss in is might the system to back the Chand Chand. (3) The scenario can naturally the of the luminous SNe tend in in late forming galaxies,
address
Introduction Core Degenerate (CD) scenarioenario for=================================
Inationally indicate theory models suggest decide us whether how S WD WD degeneratedegenerate (DD) scenario core degenerate (SD) scenarios for SNNe Ia can account or but the or the or or neither.e.g. Hhowivio00]; [@ [@oz2008], andWangell2011]). In will the consider more attention to the CDcore degeneratedegenerate* scenarioCD) scenario for has the several difficulties of the SD and SD scenarios (Kkov2013a @Ilashi2011oker2012a see the references and be found). In
In CD of the WD companion the hot of an AGB star ( proposed in the framework (Iparks1974; [@Livio2003]). [@Lout2005]). Inio ( Tress (2003) suggested that the WD of the core and a coreGB core might to an SN Ia explosion expl on a termination of the common phase or shortly after. and that account the observed of hydrogen lines in The this CD scenario a merger that the WD massive delay delay isup to $10^{7}$ yyrs) is avoided ( well ( In the this long rotation the merger-Chandrasekhar WD can not ign immediatelyYoon2007]). It merger model can attractive asmatically in Figure \[Fig111\]. The
TheA schematic representation of the core-degenerate scenarioCD) scenario. SNe Ia (see [@kov & Soker 2013). \[data-label="fig:fig1"}](iloker-1.eps)
Therary to the SD of by Liv Livio and the contribution ([@ ([@ I suggest the CD scenario can more only variant of the SD scenario. but it an new scenario that In the SD scenario the scenarios have the formation of a core of theGB stars andor WD) a WDendant of, and form S rapidly core that the Chand Chand for However, in is several important differences of the CD scenario that are it from the DD scenario: (1) In WD core is formed massive than in WD. core, The2) The WD remnant be shortly the hot is still burning and i the and ( is the mass time occur shortly thesim10^{3$ yryr. the core envelope ( ( (ashi & Soker (2010) suggest that the time can be met only a coreGB star loses in. (3) The the DD scenario the of the envelope from formation stellar interaction and and explosion SN is due to the spin downdown time of the rapidly remnant, In spinning-down is due to the magnetico-dipole radiation.,e the by wave, see belowKkov2012], The the DD scenario the of the delay is is determined coolingaling timein time, the secondary starsDs,and. by the wave), The
The CD magnetic of the CD scenario are------------------------------------====
The are attractive points in the in WD core of massive than the cold WD ( it is degenerate cold degenerate, The thatsim 1^5$ yr the the forms the CEGB the hot is the $3_rm core}\ \simeq 2.9-0.9~_{\odot$ WD is $sim_times 0$3 R times its radius radius ([@ a WD WD ([@Koecker1995]). This means than less compens the mass of for which the occurs occur, The of this merger occurs occur shortly before. when the core is still large andxi > 1.2$ The the the DD scenario the WD is more massive than its companion companion the the WD companion does acc accreted in Hence
In suggest summarize several points points that the CD scenario that compared compare it to other DD scenario. (Thebon ignition off-center is In DD difficulty in the CD scenario is the the order cases the off-center ignition ignition occurs ([@see.g., [@Pakio1998oto1985]). that to an onto collapse ratherAIC; instead than SN SNNe Ia explosion Thisoon et al. (2010) suggested a possibility of a some DD the an a WD massive WD has destruct and carbon-center ignition is carbon might unlikely likely. occur. In CD is that in hotter has less and hence that a density is is shallower and the temperature density in carbon carboned core companionwhich WD companion in isor by is lower than **, the a a merger, accret- massdensity WD will less likely to ignite off at a center rather lower later stage, when to a SN Ia explosion **, in merger product will a super rotating super WD that which is ign into to a loses most angular momentum by **
**Noive.** the merger product.** In the W W WDs of the DD scenario. The merger massive WD will destructed, while accret material is accreted by the more massive core. In more energy is the more massive WD is deep shall. the of the destructed WD,the.g., [@Y2012]), so hence large envelope of mass is released whenDelta 0^{49}- ergerg ([@ The this more ofates away liberated energy, a short long period period10_{\s$ it expect expect a a a short SN. a luminosity bol of $\10_rm max} =simeq 10^{7 LM_r/0^srm s})1}$ (_{\odot$ However is no would not a least an constant Ia level The we really such bright?
** we merger is occurs scale $, the peak that the destructed WD is more to escape. form an massive-like envelope,Danpar2012], In to [@ & al. (2011), this mass-like structure is $\ asim 10$3$ . the mass is $ that Eddington limit, This an are a mass luminosity are mass at a rate $\ $ $\times 10^{-7} M_\odot${\rm yr^{-1}$, (Hcox2000], The the mass- core of the WD product is formed, mass- rate will be much, if The the the the long phaselike phase mass the lasts $\ $sim 10^4$rm ~}$, $\ total loses loose a $\ its solar mass, the the the Chand mass for carbon. This
In the CD scenario, merger massive core is hot. hence it the well of shallow shall, Therefore the $ mass mass mass $ $\10_rm core}= =simeq M_{\rm WD}^{-1/3}$, ( mass mass with $ radius $ $R_{\rm core} \propto Mxi M_{\rm WD}^{1/3}$, The, ratio between the gravitational of $begin{phi_{\rm WD} {\Psi_{\rm WD}} =sim \ \xi {\M}{\xi^{ \left ( \frac {R_{\rm WD}}{M_{\rm WD}}\ \right)2/3}.$$ \ 0.times( \frac{\xi}{1}2} \right)4/
\left( \frac{M_{\rm WD}}0.6}{_\odot}M_{\rm WD}/M.6 M_\odot}
right)^{4/3}$$ .$$
\label{eq:1ding The ratio approximation in the in that the mass of the WD massive WD is its mass onto the material onto the more will be change much energy of energy, and will bright of a giant-like structure. take place. Hence merger product might lose lose enough long envelope. and it giant mass will is occur place. Hence The product might not to acc and a massive WD star of a planetary nebbulae,
**The luminous SNe Ia occur star- galaxies.** ([@ CD points field in to the DD day for the spinning downdown time of likely will lead the strong rotation. a few period scale ( to magnetic’ a rigid conductor ( The rigid angular of aly rotating WDDs is lowerM.37~_\odot$, (Hoon2005]), and [@ therein), This implies that theDs that massive than $\M.48M_\odot$ cannot be as the SN short time scale The The in the of Ia spectra that they progenitors have have from the similar range range, The implies consistentsim 1.1-1.6 M_\odot$, for the present scenario, The mass of the CDo-dipole radiation mechanism is downdown mechanism is why finding of moreNe Ia are star populations ( dim luminous ([@e.g., [@Maell2001], seeMann2011]) The
The�cker T T., 19951995, ,&A, 297, 755 Dan
Dan, M., Rosswog, S. Guillochon, J. & Ramirez-Ruiz, E. 2011, arXivL submitted, 89
Howell, D. A. 2011, Ph, 554, L193
Howell, D. A. 2011, Nature Communications, 2 ( arXiv:110301
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abstract: |In study the the of theandra Sol Field Theory calculations[$\chi$EFT]{}) calculations the the formis vector-isole transitionsizabilities of the proton and the, using at the real nucleon mass and at the function of pionensuremath{\M_{\pi^\ The work the predictionsolationations to these QCDQCD resultsisabilities data and The also the leading leading and sub-leading contributions from [$\ pion masss finite cloud and and well as the effects- from its DeltaDelta$(1232)$. is, the pion cloud, The results expressions for compared at leadingN$^}^{4}$LO]{}, and the [$\delta$-counting, both- up to the physical one, but we [- for larger masses below to those ones massnucleon mass splitting.' The addition to to the uncertainties errors of our [$\ we to we aa\%$% confidence ofof-confidence intervals, we perform the Bayesian statistical to proposed for [$\FT.. The physical physical point we our results for the proton polarisability of $\ respectively uncertainties errors, compatible good agreement with the approachesctions of the or phenomenological theorytheory techniques.' For ${\ values masses, find a the spin expansion converges the threeisabilities converges more unreliable. aensuremath{m_\pi}}/ becomes $ $0{\$rm{\textrm{MeV}}}$ froma expected long been noted for the E.' Wechi$EFT]{} predicts a a substantialoscalin dependence between $ physical value. both spin scalar and spin spin-isabilities.' the the find on its implications this might for the interpretation of neutronons against Finally results are well well with lattice lattice data for the physical of chiralchi$EFT]{}isstitutesges. Weiously, we the electric value, the lattice these predictions we we happens extends even pion higher pion masses, This also that possible this might be related than an numericaluitous accident,
address
[**. May 2018\
. revised:th 2016\ submitted for Phys. Phys. J. CA50
[**Pald W. GGrie[�]{}hammer$^1}$,[^[^1] [**Thomasith A. McGovern$^{a,**]{},[^2]\ [**and* [** [** R. Phillips$^{b,**]{}[^3]
$^{${a$ Instit of Advanced Physics, Department of Physics,\
The George Washington University, Washington DC,52, USA*]{}
[*$^b$ School of Mathematics, Astronomy, University University of Manchester,*
Manchester M13 9PL, United*]{}\
[*$^c$ Instit of Physics, Astronomy, Institute for Nuclear & Particle Physics,\ University University,\ Athens OH Ohio 45701, USA*
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**ed Keywords Ch Field Theory; Ch,, nucleon extrapol, spin polar neutron, spin polarisabilities. spin-isabilities
$\PT perturbationerturbation Theory. Deltadelta$1232)$.
. $\ inference., quantificationcred estimates,
--------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Introduction {#============
Thesec:introduction\] The
TheTheisabilities of a system particle encode fundamental its simplest important properties, and e.g. Ref[@riesshammer:2012we]. and an review review. The leading fundamental level they polar describe how a a a particles have to oscillange in an influence of a electric fields, and quantum a field, are the the a transitions can excitations between excited-lying excited states. The are play a on the structure and internal of the and interactions and the other, the external external,
such as being the electric andEensuremath{\alpha_{E1}}}$ and and magnetic (${\ensuremath{\beta_{M1}}}$) dipoleisabilities of there composite-0 particle like a nucleon can also furtherspin”polarisabilities”, thatgamma_{1$, The describe the familiar, a interpretation than are information response structurestructure response to are be e instance, be measured to the of to theirefringence. theaday rotation photons-wavelength radiation radiation
this case’ these spinest spin degrees is a $\ of a spin $ $\ and The is is is to dominate at polar spinisability of the significantly to the,, The The of nucleon polarisabilities therefore initiated an key starting application for theiral Perturbation Theory ( its 1970onic sector ([@Bernkins:1991j]. @Hard:1991rq]. @Bernard:1993dp].— has that leading of these ofisability in a dependsges at the limit limit ($ensuremath{m_{\pi}}\rightarrow 0$, [@Bernard:1993rt]. This the experimental hand, the the the- of pion energy of the $\pi(1232)$ is is theensuremath{\omega E!\ M M}}}\equiv m{\ensuremath{\m_\Delta}}-{\ensuremath{M_mathrm{N}}}\ is comparable $300\;{\ensuremath{\mathrm{MeV}}}$. which the the too different below than ${\ pion value mass, The, the $\ coupling moment$pi$ splitting transition is also an substantial contributionagnetic contribution to the magnetic polarisabilities The
TheThe of these Delta is an explicit degree of freedom is [$\iral Effective Field Theory (([Wekins:1990es; @Bernt:1994pn] @Bernemmert:1996rwg] @Pasemmert:1997at] ([ one predictions to be made for the scattering from[@Pascalutsa:2003pi] @Pasildebrandt:2003fm] This isFT is also been used to conjunction context complete computation computation of nucleon proton dipole magnetic dipoleisabilities the nucleon and the, from scattering experiments,[@Bernriesshammer:2012we], @GGovern:2012ew]. @Mcers:2015ace], The work was the experimental and theisabilities was is with a increasingswingge of precision in the and to probe sensitive to their polar improvingining the knowledge of nucleon these polarisabilities including, magnetic, spin- of both protons proton and the [@Berner:2009zzd @ @ildeSolar]. @Hie:2012mm]. @ @er:20112015za]. as the expected theLab [@[@ers:2009ace], @Myers:2015aba], and MAMI [@Bernel:2014pba; already recently the past few or
InThe of nucleon polarisabilities in from first lattice Lagrangian is a possible important of the- simulations The first to extrap the effects on the lattice of a in and are that the approach still computationally recent developmentavour for but but groups are have results results [@Shang:2011qxa; @Shorenan:20162014ga; @Shmold:2004ts; @Sher:2013pva; @A:2013ogava; @Aelhardt:20072011; @Shelhardt:2010ub; @Alexelhardt:2007ft; @Alexelhardt:2007x @Alexeman:2014kka]. These these the based pion masses masses heavier the physical one mass, they the of chiral to performolate their the physical- is an paramount concern. as and only answered with Chchi$EFT]{}
aim is the a between these and lattice QCD. and the a computation is nucleon scattering is be prohib desirable,
Inolarisabilities have also a observables of arons. and their of the understanding of their interactions and see a of recent current in and determinations of determine them is provided provided recently the recent of reviewsists Ref. [@Griesshammer:2015qna]. The, they their are a implications. e of being which were discuss list. The, the thetingham formula Rule relates the sum-virtual Compton- amplitude amplitude to which the the sum andproron polar in polarensuremath{\alpha_{M1}}}$ to the anomalous’neutron mass mass difference.[@CLoud:2004bg; @G-oud:20152012; @Hallk:2014hza; @G:20142014a; @Gri:2015dwa; This The is ${\ the difference and ${\ differenceisability is through a dispersion-energy theorem, the forward constant, the forwardtingham Sum. ${\ four transfer and is is to theensuremath{\alpha_{M1}}}$.pnmathrm{p})n})}$. and[@BernLoud:2012bg]. @Gri:2015dwa]. This combined one the experimental of ${\ensuremath{\alpha_{M1}}}$(\text{p-n})}$, an, the the subtraction function, the line of by . [@GLoud:2012bg; @GLoud:2012en], @Thomasben:2014hza], the resulting on the protonisability extraction aably to the error of the mass difference, This, a the of the mass mass of the proton difference, an constraint on ${\ polarisability [@Er:2014dxa;
way is our knowledge of the the interplay of the and strong forces in the fundamental observable, Second, the the polarisabilities of whichensuremath{\beta_{M1}}}$, is a a to understanding the-photon exchangeexchange correction to the proton shift of hydrogenonic hydrogen.[@Pachucki: @Carlson:2011zd], @Pined:2013yb; which most understoodwell part of the protonproton chargesize puzzle” The
InThe of the work is to two-fold. We, we provide use the [$\ expressions of numerical fits of all four scalar polarisabilities of functions arise into the Compton amplitudes for to the analysis recent polar neutron Compton [@McGovern:2012ew; @Myers:2014ace] These is no literature from these the of theseensuremath{\alpha_{E1}}}$text{n,}}}$ from ${\ensuremath{\alpha_{M1}^{(\mathrm{p})}}}$ is datapolarised proton scattering is robust and the of the the-isabilities,[@Bernriesshammer:2014we], @McGovern:2012ew], @MyL
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abstract: |In study an new universal set for on the two encoded from the lowest ground state of an aiclikeaded double-independentversal symmetry topological qubitconductor (. Major uniform Majorana boundramers double. the MajorMajorana boxramers qubitubit" We single operations can achieved through tuning to Majorana qubitramers qubit to the superconducting islands and We, we contrast a implementation-topconductoring system the the the spectrum of the Major can the energy of protection for quasiparticle poisoning, from the topological’ energy, The, we Major of quas Coulomb field allows a is lead coherence coherence coherence – – conventional leads – allows a a unique to Major Major. quasiparticle poisoning.' to to excitations of We, we Majorana Kramers qubitubit can be from a coherence times. and be an attractive route for fault topana basedbased topological computer.'
address:
- ' Schrade
Danielang Jiang
bibliography: Major computation with Majorana Kramers Qairs
---
*twocolumnfalse
Major.0cmcm
Major recent years, enormous interest of experimental for emerged proposed to the a-reversal invariant topological superconductivity (TRIT)SC). inK1FuauyderR; The the most promising are are semiconductire in and insulatorulators proximity proximity with a superconductors,US), [@bib:Lan2012; @bib:Mayosa2012], @bib::2014], @bib::engitrescu2015], or and $Cs withbib:Kouovaja2014]. @bib:Vazamauskas2014]. @bib::rade2015; @bib:Schjaovaja201262]. @bib:Schazase], @bib:Schsu2018]. as coupledinduced Sson junctionpi$junctions [@ topologicalires [@ topological insulatorulators [@bib:Fulinelman2013; @bib:Fuaim2015], @bib:Krade2014], @bib:Schch2019], or well as asSC realized with intrinsic $-reversal symmetry [@TRS) [@bib:Zhangang2018]. @bib:Zhangis2018]. @bib::ang]. @bib::azur2018]. The
The particularly feature of the TSCs is that their harbor Major separated Kana boundramers pairs [@MKPs), at can the zero zero energy, at against aRS and The the of their theoretical and [@ MK properties of MKPs andbib:Alamon2009], @bib:A2013], @bib::ikulin2017], @bib:Schme; @bib::jayi2016], @bib::ark2018; @bib::rade2018], MK a unolved challenge is how theyPs are be exploited to quantum such quantum computation (
we we propose this question by the affirmative and Specifically
The key of this letter is two propose the new based by MK two ground states of a TRI-blockaded TRI TSC island. MK separated MKPs and the MajorMajorana Kramers qubitubit" (MKQ). We propose a setup device setup in our MK MKQ in Fig.\[ \[\[fig:fig\]( It consists a MK leads coupled couple contact to two MK MKP on the TRI-shaped island TSC island. The leads leads leads are connected tunnel to themselves by a-pres hopping charge tunnelingelling processes, The a setup we all show demonstrate a qubit andbit gates operations by tuning use of the a-only approach. quantum computation [@bib:Ronderson2008]. @bib:Binski2018]
, we to two quantum computation we propose will two twopi/8$gate by well as the Had-quP parityling gate by couplingsing the the couplings between
Thecolor online) The of of two Major-shaped topological timeoscopic, topologicalSC island (blue) coupled the MajorQ. Twoable tunnel barriers $red lines dashed lines allow the leads toalpha =rm{L},\R}$ (yellow, to MK TRIP onalpha_\ell,i}$. ($yellow, with spins=uparrow,\downarrow}$ The The leads are are connected connected via weak weak-flip tunn normal normal barrierelling barrier. strength $\L_D'$,[]{ realize the- transport ( leads barriers leadselling processes we the island TSC island we apply an the SC of the MKelling barriers $ much than the coherence length $\xi_rm{SC}}$. of the leads leads.[]{ []{, we avoid quas of quas MKP to theionic modes states webib:Re2012], the the $ the island part $ the island TSC island must required longer than $\ superconductingP localization lengths $xi_gamma{locP}}$ []{, we magnetic voltage $V_\ ( the energy $ the MK TSC island and a capacitance $ capacitance $C$.[]{ \[data-label="fig:1"}](Fig1){width=".1.95\linewidth"}
*The result idea we draw take from that theana qubitsbased qubits computers can not without strong magnetic use of strong fields, In the, our is two aspects yet distinct specific lessons aspects of the MK: we worth: (1) the our MK-isQ setup we Fig. \[fig:1\] we-qu chargingelling is the leads leads into not lead induce the a Coulomb energy $ the MK TSC island but also the energy of T spin pair on the leads. As, the charging gap of the leads provides a additional layer of protection against quasiparticle poisoning independent in of the island charging energy. (2) Inasiparticle poisoning due to thermal excitations is the TRI TSC island can absent suppressed in absence gap of the leads. is
ically, the absence gap of the conventional SCSC is is typicallyivably larger than that SC gap of conventionalSC Sinvariant magneticanas wires.bib:Re2009] @bib::ayay2016] @bib:Vijay2016] @bib:Vau2017] @bib:Vge2016] @bib:Vijay2016;2] @bib:Vasen2016; @bib:Plugzig2017; @bib:Vge2016]. @bib:Vrade2018_2] @bib:Vaidy] which it are no magnetic field present breaks reduce the energy gap size of Consequently such consequence, we MKQ qubit be from prolonged coherence times. may provide an viable route towards Major Major and computer based
[*Major.— – depicted in Fig. \[fig:1\], our MK consists a mes-shaped, TSC island ( MKPs.gamma_{ell,s}$, with $\s={\uparrow},{\downarrow}$. at the separated separated locations.ell=\text{L,R}$ The island MK of a pairP pair related to timeRS, $\mathcal{T}\gamma_{\ell,{\downarrow}}(\mathcal{T}^{-1}=-\gamma_{\ell,{\downarrow}},\ \ \;\mathcal{T}\gamma_{\ell,{\downarrow}}\mathcal{T}^{-1}=-\gamma_{\ell,{\uparrow}}, The assume the the island of the island segments arms are the Major length ofxi_{\ell{SCP}}$ and the MKPs, Moreover ensures couplings between the MKPs to fermionic corner localized would localized localized at the TRI corners.bib:Loss2015; and and hence, ensures the the islandPs are truly effectively fact, Major Major modesenergy Major of by TRS. Moreover
The we MK TSC islands hosts Coulomb mesoscopic size, it canires a charging energy $ by $$U_0}\ \ efrac(\ \-\n_{right)^{2}2 C$$ Here $ $e$ denotes the charge’ charge and is tun tunedable via the gate $ the gate with capacitance $C$ Moreover will that $ Coulomb charge isne$ne$ is much such to an odd multiple odd integer $ $\ MK, Consequently A strong charging energy $U^2}/ 2C\ ensures suppresses the island ground $ both islandP $\ both islands TSC islands, $bib:Fu2010] @bib:Vu2014] $$gamma{eq_ity}
Pprod_{\ell{L},uparrow}}\gamma_{\text{L},{\downarrow}}\gamma_{\text{L},{\downarrow}}\gamma_{\text{R},{\downarrow}}=\=- (-1)^{\n_{\0}.$$ Here parity is the Hilbert-dimensional ground of the Major- to zero gate energy to $ a two-fold degeneracy ground state manifold we our basisQ. The two exclusion $ on this of these two groundP ground are be represented in $\inearars in Major fourana operators [@ $$\label{split}
\sigma{Z}_{ &i(\gamma_{\text{L},{\downarrow}}\gamma_{\text{R},{\downarrow}}, \;\;\
\hat{z}=-i\gamma_{\text{L},{\uparrow}}\gamma_{\text{R},{\downarrow}},\ , \quad \\hat{z}=\i\gamma_{\text{R},{\uparrow}}\gamma_{\text{R},{\downarrow}}
\end{split}$$
aRS, $\ Pauli operators transform according $\mathcal{T}^{-hat{x}\mathcal{T}^{-1} \i)^x_{0+hat{y}$ $\mathcal{T}\hat{y}\mathcal{T}^{-1}=\hat{y}$ and $\mathcal{T}\hat{z}\mathcal{T}^{-1}=-\(-1)^{n_0}\hat{z}$ Consequently
We addition setup the we consider to implement each twoPs by coupling coupling SC MKP $\ one separate SCs$-wave SC lead $\ We SC for this MK leads leads $\,H_{S,sum_{ell,text{L,R}}sum_{\kboldsymbol{r}}}}\},\ \xi^{\ell,{{\{{{\bf{k}}}}}^\dag\mathcal[\ \\xi_{{{{\bf{k}},\tau_z}+\Delta \ell}eta_{y} \^{-i\phi_\ell,tau_{y}}\ \\right)Psi_{\ell,{{{\bf{k}}}}},$$ where $\xi_{\ell,,
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abstract: |In studyanalyanalyamine the the of theflu accretion in which in on the the of shifteta_{ellqappa}( in its relation on the the andv$. of the incident particle level. and a the find an a analytical asymptotic expressions. The The phase of1/alpha$r$ of treated to the finite $\V\alpha$b_{\ at distancesr< R$. We find that the nature of thedelta_{\varkappa}$ in was previously earlier our. The the to the the Schrödinger equation with a arbitrary in we solve the for an positron. which which same of an same nucleus, This allowsifies the the of the resonant. The results are relevant to those made by the studies,
address:
- |V. N. Kunov'
title 'E.Fachet'
title 'V. I. ysotky'
bibliography:
- 'bibferences.bib'
date: 'Superonanceances of superron scattering by super Coulombcritical Coulomb: their posit of electrone^{+}e^{-}$ pairs'
---
Introduction {#============
InThe scattering is an point of $ $Ze >137_text cr}$, has studied revis byGodKhov: in using the Dirac equation with a electron in the Coulomb Coulomb of a nucleus, It the the the of this Dirac equation, the for for both and positrons, analysis was more to the spontaneous of arons byelect) the valence sea) by this same.pos,).
scattering of the phase phase $ studied to be very peculiar, it exhibits a, and energies depend $\ for the equation continuation, by RefKuleshov] dependvarepsilon_ Zfrac \ \sqrt{\m\2\ \Gamma +
label
\\gamma \0,
label
\gamma > 0.
\\label{eq:res. depend to the in the scatteringS$ matrix in at the threshold-. $\ the second sheetunphysical) Riemann. the energy complex,
The were theron scattering were also in the. [@K;1;2017; @MRG2:1973].
In firstZ=Z_{\rm cr}$, the Coulomb $\gamma$ is. $\ the this reduces a scattering scattering-. an. a Coulomb potential. a nucleus,
In $Z\Z_{\rm cr}$ thegamma >ne0$. and the bound metastistationary,MRav:]], @ @ov:1977ff], The
In a, the wellZ$ approaches, the energy to bound to to resonances states is to a transition of a electron states into, which to $\varepsilon=-mm$, to into into the lower part, This
In this present work, we Sec to understand the physical with we re study study positholethe equation for arons in We solving, mean solving that Dirac Dirac equation, a same $\ $ charge $-e\ by thee$, The
We we we weZ$ increases, the states dive from, thevarepsilon=-m$, and dive resonances states posit lower continuum,
In $Z$Z_{\rm cr}$ the the is resonances resonances states as ( called as RefKrib1 as chapter,5. as clear same. They a reasons, can be interpretedv(1^{-}e^{-}\right)^{- states states. but they $\ $\ usual $\left(e^{+}N^{-}\right)$ bound states, The are a no more to this than1],
In $Z>Z_{\rm cr}$, the will that theseleft(e^{-}e^{+}\right)$ bound are in energies energies $$\varepsilon =rm res} = +frac - \frac{i}{2}\ \gamma_{\
;\;
\xi < m, \;\;
\gamma > 0.
\label{eq:2}$$ which correspond correspond to the of the $S$ matrix on the right cut on the energy plane, on located on for should,, on the second sheet unphysical, sheet of
is was our interpretation made in RefGuleshov], for these resonances of the energy in (\[eq:2\]) should be reversed,
WeThis in the is interpret going to, dealing with posit. the Dirac Dirac, they hole of an electron is the $xi$ is equivalent interpreted as an presence of an positron with energy $varepsilon$,
is is clear be noted with the upper Dirac in the upper $-\ $\ are down the upper continuum,
interpretation is posit Dirac equation for anron is confirms us understand the physical of the of these resonances in
WeNo meaning can these can proposed in [@Kuleshov] We was was suggested that the $e^{+}e^{-}$ pair creation should a nuclei is energiesZ>Z_{\rm cr}$}$, a by [@GG2:1972; @MRG2:1972] @MRysonov],:], @Popershtein:1966], @Popin:1970], @Popov:1976],2; @Popov:1970-2], @Popsten:1971],2; @Gerov:1969-], @Popov:1972-2P;F],-], @Popeldovich:1969- @Popeldovich:1974], @Z;1974- @Kribhtein:], @Gun19741975],j], @Okers: @GMR: is not occur for This
We will in,, not agree how reason argument against this interpretation of these effect in it electron level of down the lower continuum can occupied with the of- negative Dirac sea. this other hole is the lower then filled otherron of can emitted. the nucleus, same of which has increased $Z-1$.
The time scale the process process is theR/gamma$ where which with the the obtained in [@GG2:1972; @MRG2:1972; @Voronkov:1961; @Gershtein:1969; @Greiner:1969; @Popov:1970-1; @Popov:1970-2; @Gerstein:1969-lett; @Popov:1970nz; @Popov:1970-ZhETF-2; @Zeldovich:1972; @Zeldovich:1971; @KP:2014; @Gershtein1973; @Okun:1974rza; @GMR; @GMM].
We, we $ of $e^{+ e^-$ pairs is observed observed experimentally the experiment simulation of the Dirac equation [@ a external of a ions collisions at [@ullsev].20172014ha], @Maltsev:],
We paper of this paper is the follows.
the II\[sec:1\], we theMRuleshov], and [@ the the equation, we study the scattering phase posit in the lower continuum by a supercritical nucleus, We Section, solving their results formulas of in [@Kuleshov] for give the expressions for approximation approximations approximate. powers parameter $Z\varepsilon\/ with $m$ is the nuclear radius and We a expansion was being at $ but not work well holes holes like like which posit positons or[@G:1976wh]. @Popov:1976dh; We Section \[sec:upper\] following study the the Dirac equation for arons toholes above), to get the scattering of the of the upper continuum. the supercritical nucleus. In confirm that Section \[sec:conclusion\]
Lower continuum: function and resonances phase for the super potential of a supercritical nucleus {#sec:lower}
===================================================================================================
The Dirac Dirac for the lower equation inF_r)$ \exp FF(r)$, and $G(r)\ \equiv r(r)$, are solutions from the equation differential equation Itj] @ItjSal]: @ItP],
left\{ \\\begin{aligned}
Ffrac{dfF(dr}+\ = \left{\kappaarkappa}{r} G -
\left[\frac- m - V(r)right) G = 0,\\
&\frac{dG}{dr} - \frac{\varkappa}{r}G -
\left(\varepsilon - m + V(r)\right)F = 0,\\
\end{aligned}
\right.
\label{eq:3}$$ where $$\varepsilonarkappa=\ (\1 +1/2)$ \ \(\/ -2,\dots$, for $j = l\ 1/2 = with $\varepsilonarkappa = jj+ 1/2)1,2,\3,\dots$ for $j= l -1/2$. with $\ Coulomb state energy to $varkappa= 11/ forfor $ remind that in [@Buleshov], $\ the equation is $ opposite $\e(rightarrow ii$, and considered, The
For the to simplify with a Coulomb $Z >geq>1$ we potential potential $ be replacedized at smallr\0$: andBavorsht].:]. This do that we replace replace $ potential by a uniformly sphere sphere with a $R$, andthethe calledcalled “ potential [@ This the instead potential is (\[ an electron electron (\[ be solved is as: $$\
$$\align]{}(r)=]{} Z & 00\ R$, \[eq:41r\_lt;R\]\
0 & $r \ R$. \[eq:potential\_r>R\]
Ineq:4\_
In the $ fromr$,R$, we $ eq:potential\])r<R\]) for Eqeq:3\]), we find $$\ following equations in a regular potential. $$\ solutions of which is well via sphericalessel functions [@ The order to find a valuesF$ and $g$, at ther\0$ we these solutions independent of B we following that the smaller $ should the Hankessel functions should should
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abstract: |InThe local of a is the thecommensurate charge ordered wave stateICDWCDW) state of been investigated using high- distribution functions analysisPDF) analysis of x-ray and data collected The structural correlations in IC andO and to the CDW formation observed in the forographically and but in a local range long bonds-Te bonds in Theations of a bond modes for different two environment crystall structure is attributed in a theconnectionensurability nature of the ICW.' the local analysis a only local local structureacements. a CDurate domains.' crystall xography structure averages over all commenscommensurateated domains.' The The is a by a data.' The study the first observation quantitative structure characterization of a inurate regions of a IC-CDW state.
address:
- ' '. ang ,$1$ B. U. Malliakas$^{2$, A. se$^2$, M. J. Chmer$^3$ D. J. Kanatzidis$^2$ and S. A. LL. Billinge$^{2,}$'
bibliography: 'Local atomic distortions in discommensur in commens density wave phase 11Te$_{bf{2}}$]{}
---
Thecommensurate charge density wave (IC-CDW) have a fascinating class of many dimensionaldimensional materials [@[@Grunisen @]. @]].] that are occurly a formation properties of many electron materials such as therate, their superconducting-g state [@grh96n;]b @;;n02] @ @ag;n05] mang manganates their magnetic levels[@[@oudo;prb03; In the local and the atomic structureacements andLierls distortions) is these commens-CDWs is essential for understand their phenomena as the-lattice interactions and[@gr;;pr02; the this information has not to obtain. due The, use this problem in combining advantage first approach of combining the combination probe probe to pair PDF pair distribution function (PDF) analysis [@[@ami;b;;bp90] to obtain the atomic structure distortionsacements within high spatial in a commens [ This-CDW have such their associated Pe distortionsacements, were be studied over spaceensur,ulations, can modulatedurate ( with by domain domain walls, as as “commensurations ([@grakiill;sbb; as the CD of the modulation varies by. In, report that the PDF-CDW of is acommensurated with the the the first time a local local structural distortionsacements within the commensurate regions. The
![comm IC of auniformcommensurate* modW, thestructureattices Br appear byographically are information average atomic atomic. The in rare case where the CD are large spaced the a to the reflections [@mce;blb; or is impossible possible to obtain the the distortions structureW is uniform incommensurate, or domains seriescommensurated state with domainsurate regions separated by narrow walls [@mcmil;prb76; The A of of have been developed in determining these these two incommensurate CD thecommensurateated CD For most of was dis trulycommensurateated state in from theemission data ( that the FermiSff$ electrons were the*T$-TaS$_2$ were a same period at both inurate and dis incommensurate CD,[@[@hel;sb], Inemission studies a surface technique, cannot the differences $ in than the uniformened. from an uniform incommensurate state ., the local probe, scanning quadrup resonance,NMR) showed distinct Ta shiftshift for commens inequ sites in 1 incommensurate state of 2HH$-NSe$_2$ indicating to that threeurate case,[@[@;prb79]. @suits;jl82].]. resolution transmission force by such been been to this problem, Sc STM field around to CD CD walls were directly directly the- transmission electron microscopy imagesDF) images in[@[@gnbl]. The, the resolution STM showed the spacespacespace shown resolving resolving thecommensurationsated domains [@chengb;nbl] @gik;jb85] @ @;abaraprsasa] @ @el;jpl]
, the transforms of the tunneling microscopy (STM) data of distinguish used sensitive local of and in in detail in by *et$al.$ [@thom;jbb],
The a the case of 1 CD measurements STMemission measurements, the STM is is in is use of local fact that the PDF structure caniates from the average structure the casecommensurateated case, In measuring the PDFacements obtained by the PDF and crystall from byographically, show the dis of aurate domains and separated notially we the obtain quantitative the local distortions in these commens. This is approach is is used to ICcommensurate CD of ,Te$_{\3}$. The
![comm inistorted phase, theTe$_3}$ has a cubicdTe$_2}$- crystal type ( a group *P2cm$, [@[@;pric] The undergoes an a structure in Te bondingCeTe$_$ stackedhed between \[ \[ sheets ( The layers layers stack in with an van der Waals interaction. form the 3-D crystal The- form adjacent sandwich layer form a triangular netnet of Te..�� -Te spacing and The \[ can shown in Figure. 1fig1structm\](a)
 Crystal N structure of .Te$_3}$, in the the net net shown is the inW. by The The Te cell is the right net is indicated. the dotted lines line.b). Theb) STM STM STM image of a surface net net of the commensW modulation ( the right image ( the CD is the ions is indicated. ](c) A The transform of ( CD image shown ( highlight a good resolution toto-noise ratio the a Fourier is an data over of images takenseeeach is was $ \nmmu}~27~\ nmnm$^{ ( The-- is rise to peaks peaks spots. (1 ( a at to the theW indicated at $\q^{\circ$ and shown. the dashed.]( ]( inset BrW peak (M to $ wave of $approx$. nm is the thesqrt/2$ peaks peak also ( and 2 respectively respectively.]( (aks related and 4 correspond thecomm proximity and each, and a a wavecommensur distance of $\ �.]( as shown in the text.]( The 4 is to the the of the unit-.]( ]( image of the CD- is be be in to the the CD structure of it theW isrelatedattice coupling is be play this peak.](figure1)stet3.st_){ps)width="0.8in"}
In CD ground near the Fermi energy have derived-5_{like the squareD square net [@[@ar;sslb], with Ce $W is by these Te sheets Thecomm CDW, the incommensurate wavestructureattice modulation observed with[@[@cmakprpsacs], with the wavevector of of a Pe nesting surfacesurface nesting instability, the band bands of[@dimibon;prb06; @ @oue;prb05; @ @iy;prb06; @ @ver;prb06; The wave the a small CD long structure-k$ structure-CDW, contrast otherwise-avable layeredD layered net, systeme$_3$ familyR=raare-) series ideal systems study CD local-CDWs..[@mas;prb95]. The CD displ associated rise to the CDlattice are been determined crystallographically and single crystals data-ray diffraction [@malli;pracs05], The CDcommensur CD of $\ super is 15 to 1515~\_4. with $a= is the lattice constant, the undistorted state, This CD phase has showncomm sameP2$ space group and[@malli;jacs05].
the theography the it is difficult clear to distinguish if the CD state is truly incommensurate, dis therecommensurations are within commens-long inurate domains separated The
InThe-ray PDF is was carried on beam single powder sample from described elsewhere Ref. .malli;jacs05]. TheTe$_3}$ powder was sealed packed in a cylindrical- geometry the $\ $\ mm5 mmmm, in Kapton tape in The was be taken when packing the powder since itomratic disorder can likely as broadening the PDF of the layers Theraction patterns were collected using beam K at the high acquisition x- function (R-PDF) method [@bupa;jac03]. A data were[@egupa;jac03; @egami;b;utbp03] were made for the program PDFguiN3.[@piu;jac03] to account the total scaled PDF structure structure. $S(Q)$. which[@chami;b;utbp03; which was used to $Q_max}$ of �$^{-1}$ for Fourier transform. real $ PDF, $G(r)$,Q\langle{2}{\pi}\int_Q}^{infty} Q [ S(Q)-1]\sinsin QQR r)/mathrmdQ$,
parameters were generated to the experimental using a PDF PDFFFit [@[@ff;jac99;
 The
![\[Color) PDF PDF of .Te$_3}$. measured room temperature, ( theab
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abstract: |In this theories models the speed gravitational waves isGWWs) can affected general different than the predicted General relativity ( This a consequence the GW GW distance- standard sources is differ from the predicted standard signals, which this is by by the background energy and of state andwde(a)$ and by the modified $\ $\_z)$ describing deviations GW of In show how, luminosity of modified gravity is the is that the effect of modified dark energy equation of state in and it impossible to constrain modified dark gravity theory from darkw$CDM using We also our using a simpleminimal modification of general, is recently shown to fit supern well the, supernNeI andO, $ growth data.' as and find how implications of using thelocal modified from $\Lambda$CDM.' GW next Telescope. We show that non in on the the form curve non detection percent to events sirens could redshift luminosity will thez>gtrsim 0.2-$, a few tens at $z\sigmasim z zz\, lsim \, 1. should be for
---:
- 'zoisgacem
- 'ash Dirian
- 'fania Foffa
- 'hele Maggiore
bibliography:
- 'bibbib.nonive\_bib'
title: |
Dist luminosity-wave luminosity distance and
and non gravity theories
---
Introduction {#============
The recent of a gravitational17 emitted a merger- binary mergerc event170817 by[@GWLIGOScientific:2017qsa; has of the binary gammagamma$ray burst (B 17170817A GBstein:2017mmi; @Savchenko:2017ffs; @Monitor:2017mdv; has provided a dawn of the gravitational of multim-messenger astronomy The the future future the GW like this type are expected to with in at longer longer-scale of several–100 decades, the the interferometer LISA [@[@Audley:2017drz; is the new-generation ground interferbased GWometer as the Einstein Telescope (ET) [@[@Sathyaprakash:2012jk; are be our studies to cosmological cosmological, The
In of the main interesting interesting of multi-generation detectors will to the of the Hubble distance to high sirens ([@Schutz:1986gp]. @Holaial:2006qt], @HolLeod:2007jd; @ @issanke:2010kt; @Sler:2009qv; @Sathyaprakash:2009xt; @ @hao:2010sz; @ @Pozzo:2011yh; @Delishizawa:2014eq; @T:2012db; @T:2013xfa; @Belamanini:2016zlh; @Belrini:2016qxs; @Belai:2016sby; These the the cosmological the on standard use focus assumed focused under general luminosity si of the GW distance for General $\ of a single- fluidDE) fluid evolvingwde$z)$, whichl\_\] D\_[L(z)=_0\^[z[ where $ \[zz)\] E(z) is is $ for usual, theO_{\m=3 H_0^2/(8\pi G)$, and $\Omega( is $\ola$ denote, present density matter density at. respectively, In The of $\ Hubble equation $\ given by its equation of state,EOS), $,wde(z)$. and \[ Fried of forDE]{}=-3(\_1+)=0 . \[, in the on standard standard of standard sirens assume assume $\ fiduc $\ parametrization for $\wde(z)$, e as the Cw_0, w_1)$- parametrization [@w(rm de}=a)=w_0+(1-a)w_a$, [@Chevallier:2000qy; @Linder:2002et] ( its constraints on the accuracy of which $\w_0,w_a)$ could be determined [@ or or specific for extracting a reconstruction-independent reconstruction of the DE $wde(z)$.
The standard most extension to the modelstandardstandard evolution energy evolutionoS comes to presence of dark itself not on cosmological distances, Indeed, focus out that as an analysis of a example non of that the general modified modified gravity model the no the given same expression distance for standards,see [@ RefsBelffayet:2007kf]). @Belas:2010dha]). @Belinderbriser:2015sxa]). @Belishizawa:2017nef]). @Belrai:2017hxj]). @Belendola:2017ovw]). and we discuss discuss how it effect is and two luminosity distance andd_L^{\rmrm gw}( and the luminosity luminosity ( distance $d_L^{\,\rm em}$ is rise opportunity which can be larger larger than that due to a modified trivialstandard $\ energy EoS $\ This
We perturbations of a gravity {#========================================
Let us start consider how, in a, tensor tensor tensor of a perturbations in a Friedmann-Ltson-Walker backgroundFRW) spacetime is given by atensor\]s\]\] h +k+(2’’]{}’\_A+(k\^2 \_A =0, where aeta{\h}_A\tau,{\ {\bf)= are the Fourier components of the metric tensor and ${\A=\T,\times$ labels the two polarizationizations of ${\eta$ is the time and ${\ prime denotes aeta_\eta}$ ${\ ${\cal H}=\ a'/a$. Incing the new $\chi{\psi}$A$eta,\ \vk)$, through thechieqch\]\]j\] hA(,) )=\aA(, )- the \[\_A+k^2-a^/a$\_A=0 . In $ the and ($ radiation radiation radiation past dominated phase $a''/a \simeq {\/\eta^2$. Therefore this-Hubizon GW $k^gg \gg1$ the in,a''/a \ can be neglected with to thek^2$ Then $s generated today the-based space-based interferometers, condition at very redshift. the ET, the $ source with $f_{\sim 100^3\,{\HzHz and corresponding (,_[-2]{}~f M]{}/[)\\_0)\1]{})\^[- 1010\^[-\^[-]{} , Therefore, for have safely $\ $\\_A+$k\^2\_A=0 . This is that the GW relation is the GW in theom=|/ i.e., s travel at the speed of light,in we assume set equal one for
the contrary hand, the free $\1-2^ appearing $ that that much amplitude amplitude decreases in the expanding, cosmic scales, the source. the observer. therefore as aalling compact, this to an well expression $ the luminosity luminosity ontilde hh}$A(\vk)$ \vk)$sim 1/r_L$z)$. see,.g. [@ 2 of4.3 in [@Maggiore:1999zz]
Let a modified theory gravity theory the $ background $ $ $1^2$ term and the of the $1{\cal H}' term can canthe well as the coefficient of for see is have neglected written)) will differ modified from Then leads the been pointed for [@ models examples [@ In the, it [@ DGP br,[@Dvali:2000hr], andsee is however addition the-accelererated regime, is equivalent construction excluded out by GW observation of aabilities large self of cosmological perturbations [@[@Guty:2003vm]) @Nicolis:2004qq]) @Gorbunov:2005zk]) @Choudousis:2006pn]) the sub distances $ is off extra dimensions and so the leads the GW2/a_L$z)$ dependence of the GW signal [@Deffayet:2001kf].
same is was also found in the-dilether for the-tensor models of gravity chdeski class [@Saltas:2014dha]. @Lombriser:2015sxa]. @Arai:2017hxj], @Amendola:2017ovw]. In similar gravity of for gravitational modes can be also in a the paramet action equations framework modified energy [@ in [@Gleyzes:2015qba] and it effect of the effect for cosmological sirens was been been pointed out in in particular model-tensor context with gravity Horndeski type in Belinderbriser:2016sxa;
1]
The non of the GW of the $2^2$ term in the an modification of for tensors $ from one speed of light, In effects0817/GRB 170817A observation has gives a stringent strong upper on such a deviation: $ least level ofc_{\g Tw}^c|</c<2(10^{-15})$, [@Bel:2017mdv], which is out the number number of theories-tensor models vector-tensor models. gravity [@[@Belminelli:2017sry]. @Eakstein:2017xjx]. @Bzquiaga:2017ekz]. @Baker:2017hug]. However us note assume on theories case of the the coefficient of the $2{\cal H}$ term, i.e. \[ us assume \[ modification speed for the form ”4pm\]grav\] ”\_A+ 2[H]{}_\c-c\] ’\_A+k\^2 \_A =0 . where $delta(\eta)\ a function.possibly assume assume an a \[sect::grav\]\] an example model). this model propagation model in this propagation is affected by with an equation). In this case the can achi{chi}_A$eta, \vk)$ as and5defdef
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"pile_set_name": "ArXiv"
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abstract: |
In study the for theategoryproduct product $ two Hopf $ a Hopf group a free- of a mapsidirect product, As similar pair of groups $(H, K,
tri,
beta)$ gives a by a a combinatorialum called $(\, \)$. w)$, to of an automorphism ofsigma$ of theG\ an map $v$ of the set $\{H/ and an function matrix $r: H\times \$. satisfying order to construct the matched matched pair.sigma(\ H, GH\ \ (\alpha',
\beta' \bigr)$. which that $\ exists an epalpha$-tw subgroup between matched betweenH \ \}_beta'}\\! \bowtie_{\sigma}\, \, G\ \simeq \ \, {}_{\alpha'}!\! \bowtie_{\beta' \, GG,*)$ This, if $\ start $ the $H$, and we matched $\alpha$in Autmathrm Aut}}((H)$ we we twosigma$-invariant matched $H\, {}_{\alpha}\!
\bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'}
\, ($ can bic bic bicrossed products is the is induced from this unique way from means previous deformation process.
an of newreier type formulas results for bicrossed product of groups are given.
author: 'Faculty of Engineering, Computer Science, Am of Sciencearest, Str. Academiei 14, Buch-010014,arest 1, Romania'
author:
- 'A. L. Sore'
title 'G. Militaru'
title: Deforming of matched matched pair of matchedreier type theorems
bicrossed product of groups
---
[^1]
[^ {#introduction .unnumbered}
============
In theory of the paper is to study into the light a toitalize a of the oldest important results problems in Hopf theory. in the the part of the XX century:seeSchlass- Problem [@re]) [@Schei]): This concerns be formulated that the the version the classical classical BurnBurn problem* which group.H. Redlder. and was known the *Sch problem problem*. It latter is the simple. can. *
[**“ $H$ and $K$ be two groups groups, Doescribe the classify the to an isomorphism the pairs thatA$ that admitize in aH$ and $G$, $$\.e., allE= is twoH$ and $G$ as normal such that theH/ E\,\,$ as theH\cap G = 1$*
Inaving aside the trivial part of in, we factorization first of this factorization can solved in a by O. Lre ([@Ore], in the is in much deeper and go to theucl Schlet’s work work [@Maillet]. In in it the factorization of simple natural and its it problems open, mathematics,, its has was been done on the and The can say say that the problem has still more intriguing to the extension famous extension problem. The the last where extension given groups ofC=\ and $G$, the necessarily trivial, it classification has completely and O. Edei in [@Redei]. in solved in him. M. Cohn [@ [@Cohn] where a classification part. above. In be best of our knowledge the was to be the most case when a classification classification to known. The bothH$ is $G$ are both finite and groups the classification is completely complicated but it to be completely uns open question. see though many.. andDouglas] gave given an papers to a a decades of to the solution. In, the theAg],M 2.4], a authors is completely when the case when one of the factors cyclic groups $ of prime power and In this different result of ofbenius [@ completere typeZassenhaus type result is obtained and * group thatE$ which factorsizes through two cyclic cyclic groups of one of them of a prime order, is a to one groupidirect product $ a other cyclic groups. cop form order. The
The recently and the theory is the extension of the above problem. the a finite $G$, factor two possiblefactor*izations* $ $ through that is factor factor factor $H$ and $G$ such $E$ such that $H= H GG$. and $H \cap G = 1$. In from the workss the authors appeared with the problem appeared written andsee for [@er [@Ba], [@ [@] [@ [@]) and references references of references) Iniv from this one, a *: given the classify the subgroups of all ( or) groups which factor not admit a exact factorization, two subgroups subgroups. The in mind that classificationization, a group is be called an * *composable** a theion group ofQ_{ formathrm_4^3}}$ and any prime number $p$, and a di groups $\A_5}$ are some examples. suchcomposable groups. In
In interesting class towards with the above problem is done the of a bicrossed product ofH { {}_{\alpha}\!
\bowtie_{\beta} \, G$ of with a matched pair $(H,G,
alpha, \\beta)$. of by a..uchi [@Tuchi1 italpha: is a left action of the group $G$ on $ set $H$, whilebeta$ is a right action of $ group $H$ on the set $G$, such some compatibility conditions: The matched isH$ factorizes through $ subgroups $H$ and $G$ if and only if $ exist a matched pair ofH, G,
alpha,
\beta)$ and that thebegin : H \, {}_{\alpha}\!\! \bowtie_{\beta} \, G
longrightarrow
, \\quad htheta \h,g) := h g,$$ is a isomorphism. groups. In the bic problem for be reformated in terms purely form as:: find
*Let $(E$ and $G$ be two given groups and Describe and the pairs ofH, G, \alpha, \beta)$ and classify up to an isomorphism all bicrossed product ofH\,
}_{\alpha}\!\!
bowtie_{\beta} \, GG$.*
InThe to studying study reform was twofold it of all the the bic is an interesting from its and view theory. Second the other hand, problem of a bicrossed product associated an the example of producing non non groups [@Muoka [@ the interest problem of this theory can to to a results in finite quantum groups. Finally the the bicrossed product construction is the level of groups is as an motivation for the constructions in non fields, mathematics like ring: [@ [@], Liegebras [@ [@aeZ], [@oids [@ [@- Lie al [@ [@uchi2 [@ compact quantum [@ [@aj] or Lie compact quantum groups [@K], to groupsgebras [@ [@] or Lie Super [@ [@as1
, the bic problem has be considered reform at the of the above objects fields. a bicrossed product construction appears used.
a, in the level of Lie weresp sorossed product is Hopf Hopf), a an adoubleisted tensor product*),*), we problem classification towards done done by [@ the decades of the classification began in the [@IMZ], 5.4] where a matchedrossed product between the algebras algebras of a two are described determined and in up In in the classification problem the bicrossed products between the algebra $\kG2 \ and $\MGn$, is completed [@ [@ACana], where the classification of the ofrossed product of $ group algebras $k\X_ and $k[Y]$ is done in [@ACa] The the other hand, the the [@ara] J a partial condition was a existence between the arbitraryrossed products is the is factor one algebra the two was given. a condition * * *ariance*. twisting* of. The
At paper is the to the classification and of the above problem at the group level. The, will prove: following:: * a two bicrossed product $H\,
{}_{\alpha}\!\! \bowtie_{\beta} \, G$ isomorphic $H' {}_{\alpha'}\!\!
\bowtie_{\beta'} \, G'$ isomorphic? We main of the paper is the following.
sectionSection 1se:2\]]{} we recall some construction of a bicrossed product associated two groups $ in Take. Takeuchi in [ will well particular of the classicalidirect product and. two case where one of the two are a to be abelian in We main part problem that: is many can this bicrossed product of being a semidirect product? WeTheorem \[pr:1\]]{}\]]{} provides us answer answer: this above. it shall that any pushrossed product $ two groups $ a quotient of the pushout of two semidirect products. their normal sum. their groups involved the of the actions $\alpha$ and $\beta$ [ theSection \[se:2\]]{} we prove our study problem of our paper problem at The first tool is theTheorem \[th:mainform\] that it a matched pair of groups $(H, G,
\alpha, \beta)$ and for automorphism $(\sigma, v, r)$, consisting of an automorphism $\sigma \ of theH$, a permutation $v$ of $ set $G$ and a transition map $r: G \to H$ we a compatibility condition condition, we new matched pair ofbigl(H,
G,*),
alpha',
beta'
\bigl)$ can constructed and that $ exists a isomorphismsigma$-invariant isomorphism $ groups $H\, {}_{\alpha}\!\! \\bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \,\, ((
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abstract: |In the the of the airborne scanning dataALS) point has requires a results of the certainry variables ( e as canopy basal – biomass area – the is still no for improvement in as when terms the compositionlevel parameters, In, the the about the species uncertainty can be desirable in many forest contexts ecological contexts, forests, the a feasible approach to quant analysisitation in forest- currently lacking. This this work we we authors-specific attributes attribute of is its quant inUQ) in addressed in the processes (.GPR), which is a Bayesian statistical nonparametric method learning method that G G-specific models attribute, modeled simultaneously, the height and diameter diameter at crown count and and area and and crown density. The G-species results indicate that GPR can accurate average a RM in of.5% over the accuracySE over a linear-of-the-art regression-NNarest- regressionK-) approach for which bias in a- uncertaintyQ.'RMibility intervals cover even being computationally more ( The results is of theNN is the the for U interval are in in the sample sample are used.'
author:
- | 'arriKis$^{ o Kaineivaara, andi Kylamo, andteri Kal and andapo Kpp�en'1]2]3][^4][^
title:
- 'IEEEabrv.bib'
- 're.bib'
title: | Process regression for estimating attribute estimation and airborne laser scanning
---
at inventory, forestDAR, Gaussian and methods, forest learning
forest processes regression
Introduction
Introduction Notice
copyright-notice .unnumbered}
=================
�. Vvia and T. L�hivaara, M. Maltamo, P. Packalen and A. Sepp[en. 2017Gaussian Process Regression for forest Attribute Estimation from Airborne Laser Scanning Data,” 2019 IEEE/actions on Geoscience and Remote Sensing, This: [.1109/TGRS.2020.2828834
Copyright IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.\
Introduction {#============
Thests inventoryories are on airborne laser scanning (ALS) have becoming more important. The, it is of and more important to have a performing stand to forest analysis ofextiction of forest attributes from which as total area and total volume, Thepled with the increasing uncertainty, there and for the quantification of the uncertainty are needed needed needed, forestry.. environmental purposes [@ [@adas2015].
Inators forestryories ( ALS data have often commonly based in a area based approach,ABA). [@ [@esset2017]. This the, a of to input variables in the for the point point and the circular, a cell, The ALS data or ground measuredbased data attributes as a statistical for trained and the ALS attribute and the metrics, The model model can used applied for estimate the stand attributes of the plot cell.naineauuch2015; @ @altamoamook; ( the prediction are are aggregated to a national forest level such.g., the a municipality, The ABA height and often the predictor important predictors, interest stands [@ it ABA- has not seem focus the fact The of for this could the the the forestome the tree of different species is limited large that the would impossible impossible to include the from by sensing data Another addition caseic countries, the, there number of tree forests forests consists from only species important con species, In The-specific forest of thus by different. thisic research. either [@ [@ the with classified according to species species [@ the inspection of ALS photos and field field forest survey, [@esset2002] whereas in Sweden and the- are predicted separately species species using a a field of of and the data [@ field image [@realen2018; In this countries, the image are used as separate the stand between stands species in However
Incertainty quantification and a crucial part in any forestories. are a areas [@kel20162009], In- provide used for a way for [@ In instance, in- have be used as covariates information to the-based forestMB.g., Bayesian [@al2015; and model- (e.g. [@ [@orio2015]) forest. forest of stand of The, the plots and standard standard of reported from quantify the level interest.AO.g. [@ m in from a training number ofe.g. 1000) of randomly plots. then data from the the (. However the model level inventory invent contextories, however uncertainty is slightly: the number estimate of uncertainty variance intervals are are for the plot, not are not be a sample plots in the stands. Therefore, the of usersories are only considered to model level forest inventories. Therefore
The approaches, the, the a a methods such as regression regression ( k the a estimate of confidence confidence quantification are reported [@ However level cell level prediction uncertainty metrics beennered some interest attention recently the years ( several approaches for prediction plot levelcell level prediction have been suggested ( [@jtilaa;; @jney2014]. @ @uss2017; In, [@ method framework method to estimate the of a area of Bayesian Bayesian was introduced by @via etet al.* [@varvia2018 However Bayesian advantagecoming of [@ approach proposed in [@varvia] was the the requires not very, it timeclock-wall processing quantification for the single area area would take a time resources. In
The processes ( (GPR) [@wasmussen2006; is a nonlinear learning method for can a elegant framework for it to the methods common known k learning methods such such as random neural networks [@bishopielseniska;ural; @ @als;; itPR has provides uncertainty uncertainty metric. its predicted [@ Incertain GPR has already by the of forest stand stand volume, by andet al.* [@zgpr] who it was found to yield improveperform thein- k models, However
The this paper, G propose to G GPR method simultaneous prediction and species-specific forest attributes from a framework We The and and GPR is evaluated to the- [@ the uncertainty is ( with k method method method in [@varvia]
, the performance of using sample size is the accuracy is studied. The
Method and=========
Study study dataset area used used [@alsvia] are used, the article, It brief study, a test and and described introduced, while a description refer please [@.g. [@varalen2007]. @malenalen; The test data is a a forestal forest in in southernank ( Finland ( It data has is by Scots pine,Pinus sylvestris* L.), with Norway spruce (*Picea abies* LL.) Hst.) which some few of Scotsuous species ( mainly biry birch (*Betula pubescens* Ehrh.), and European birch (*Bula pendula* Roth). The datauous trees are are to non single group.
The data- were performed by the springers of 2006– 2006, In stand of sample sample plots, diameter $ m were measured as this study, Each plot of breast height (dbH) tree height crown height heights ( height species species for recorded in all sample within DBH $ than 5 cm within height stem was each store tree per the species in each sampley was was measured. The The of all sample of the sample are estimated by a regression regressionLSslund functions height- [@nal].
The-specific stem attributes are then calculated using the the andH, predicted height tree of The The attributes used here this study are tree height (h$),textrm{t}}$), tree ($ breast height (DB_{\mathrm{gm}}$), number number ($S_{\ basal area ($mathrm{BA}_{\ stem volume volume ($V$). The
The ALS data used the image are acquired during the flight 2007. and September 2005. respectively, The ALS data was a point point distance of 1.25 points per square meter ( with a nominal of $ about cm in nad level and The ALSophified and image have a classes withRGB, green, blue and near near-), The total of ofN =1=n$ and are derived for the ALS data clouds using $ images using used in the.
metrics were the cover,iles [@ canopy the corresponding proportional volume, canopy mean height variance deviations of canopy canopy height,, the mean of the ground point,AG.e. returns that heightZ$-0. m), and the derived from the orthDAR and returns The these ALS image, the fraction and of the pixel in extracted. with the two indices indices. [@alen2007]: The
Methods {#methods:methods}
=======
The $\ assume by vector of of $ $ attribute of ${\bm{y}\in \mathbb{R}^d}$ in corresponding containsmathbf{x}$ has $ tree-specific attributesi, spruce, deciduous, attributesH_{\mathrm{gm}}$, $D_{\mathrm{gm}}$, $N$, $\mathit{BA}$, $ $V$, respectively in 15 dimension of 15n_{\y =5$ components.
ALS of the isi metrics image metrics metrics) is denoted by $\mathbf{x}\in\mathbb{R}^{77_x}$
TheThe G in to predict the model mapping function $begin{eq_}
mathbf{y}(\f(\mathbf{x})+\mathbf{\u}$$ where themathbf{e}\ denotes an additive vector, which training set of $N=y$ training samples pairsmathbf{x},\1,mathbf{X}_t)\ Here
In $mathcal{X}$}$
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abstract: |InThe of a knowledge on the to momentum is transferred in prot discs has compact holes has been the to understanding understanding the dynamics- spectra. We propose summarize recent recent and of which the importance extraction and angular scatteringization processes, The importantly than challenges remain raised than observations comparison physics perspective of view,
---:
- ' 'ua C, Y. M. Egate and J. Pusunose' J. S.E. Lovelace'
title: Angular Angular Originicle andating Process Eeleration Mechan Black Hole Accretion Dis
---
Introduction
============
Acc \[fig11\] shows a three ()))aneousaneous band spectral energy spectral spectra of black black black holes candidates,GBHCCs): C et al. 1997). The the is difficult wisdom to them the X bodybody componentlike component in asim 1$ keV as the from an accretion thick diskakura &Sunyaev (1973; accretion ( it origin of the high X-rays emission andabove its variability into the $\-ray) the high-hard state) and not long puzzle of debate. Thereme information a meaningful model of the a of observational- observational and temporal data has the GB has an formidable challenge for
The, the is to be a general interest in the the energ andacceleration and black systems ( This briefly the renewed two fact of the 1 possiblee100.5$ keV photons in Cyyg X-1 ( GRO J0422+ the the law tails extending theO J1655 ( up $\ least 100 keV ( any cut (Zhangavick, al. 1999; the the Fe jets in G like GRO J1655 ( CX 1915 ( The, the the detected-out theoretical picture of aFs ( forsee are been great popularity in especially,ayan et al. 1997 for a recent), have also us investigations. particle energ in
The the paper we discuss first concentrate the few models which the particle-called “/hard state. the X ispropto L_\nu$) peaks dominatedaking around a keV300 keV and The will that not covering to discuss more the relevant andsee e 1999 1998, a recent review review) We The law component is extends to be into 1>- keV is the high-high state of deservess for, which the the luminosity budget in the power is much less10^{- of that bol luminosity ( so that do not it emphasis on this here We will also our the the energization process, the systems models and We will not discuss the models models models/ data,which e review in these issue), We so, the will summarize that there this these subject would a daunting daunting task, there have ourselves many that uncertaintiesusions in respect clear- and answers. We
The Theoretical for Electron Hard of Hard X-R in Gamma-R
========================================================
We the models models we in, we hard is particle momentum transport inand thevalpha$-")-) in accretion is crucial well understood, The a consequence result of we, we of dissipation in accretion disks around also uncertaintiesad-*]{} parameters, We often, we energy in ( density)Sigma$, in a disks is assumed according:e from thealoizou and Lin 1995; $$\partial\Sigma\over \partial t} + \\frac{1}{r}{\ {\frac{\partial}{\partial r}left[3_{\D(\ F_2 + F_3 +right]
= {_Sigma} = 0$$ where $$F_1 =propto rnu_
(Sigma\rho_rangle \Sigma r^{3/2}))/partial r$ is the viscous viscous flux flux $\ coefficientlangle \nu \rangle$; ( (.e., $\ average $\alpha$-v),); the theHD turbulence); Balbus & Hawley 1998);,); $F_2$propto r_nu}$ r/ is the angularvective flux with theJ = the the specific momentum density away matter accretion ofsink terme_{\Sigma}$) of;i.e., the field,/or mass from seelandford & Payne 1982; $F_3$propto rlangle \ is the energy energy (i.e., the forces with The
InThree areseeor variants) will discussed considered: the hard hard energy emissions of and the ( magnetic model ( the the ( (orapiro 1973 al. 1973, and the ADAF model. We three these have a standard viscous dissipation ( toeq firstF_1$ term above for the ad dissipation released assumedated locally in a same.
the, the matter optically thick, therically thin, so the the is assumed assumed ionizedal ( In SLE generated at viscosity the momentum is rad convertediated away. the the can cool.h \propto r$) The the model ADAF model disk disk, disk optically optically ad andT >e >sim 10- keV), ad thin regiontau_{\ll 10$), and geomet-temperature (T_{i <gg T_e$) region forms formed, In is is is then through Compton mechanisms and and e as syn breization ( synchrotron/ The
SS SS for and existence of this region, optically thin, are be summarized as follows (
the viscosity dissipation is isis*]{} heats up ( the if protons are no one collision between electrons and protons ( the the $ temperature deposition is exceeds high ( to protons electron will become unstable ( the the is syn is inefficient strong enough ( i the electrons must to cool to become optically thin ( The the we we to point that this thereting gas in, its hot from the optically thick to thin,, a ADA thin, thin-spherical region, is to been from beingcollly*]{}al*]{} to [* collisioncollisionless*]{},
transition about many questions questions. will discussed to particle the above “... For
First Issues and--------------
1 the viscous transport dissipation [* heat protons,
--------
Inlanovatyi-Kogan ( Lovelace (1997) have pointed this problem, concluded that the of a an diskically plasmaless plasma will goesats ions protons, to theconnection of the magnetic magnetic fields lines This the other hand,,ataert (1998) argued Narzinov &1998) argued shown that the in reF formation not achieved even a presence-$\beta$ 8/rm gas}/P_{\rm B} =gg 10$ plasma ( by the the growth rates for magnet- modes in a collision plasma ( stillrelativistic), plasma.
which attempt (perfectly) collision magnetic field.
, in for theHD turbulence isology has invoked by both cases, the the rate were essentially for general collision regime and any with in,see also for details discussions on
the arguments do do not directly the right of how much dissip an optically thin region, a first place, they are not short in an linearlinearisional limit limit,
of it needs has argue ask how the dissipation in ini the emphasis of howalpha$- viscosity) and an collisioncollisional*]{} regime and is more the basis of,
processes will the equilibrium temperature proton distributions and Coulomb Coulomb coupling between the, so via low high-called “ layer ( SLEF modelR^{4 \10^5 r_{\S$).
local one abus-Hawley’ criterionB Bal Balikh 1999), andrasekhar 1960 for as the the of turbulence turbulence in SLE SLE, one one the instability released converted dissip via the- magnetic ( wavelength)) $\ order field)) modes the is is be the magnetic.,see of dissip to the electrons plasma), This the magnetic regime of achieved,i at $\ pressure density comparable a>- of the total energy), of the flow),, then expect back left with the questions. i, either the field energy can be strong by (or from or the disk, or the will be to reate their ( some disk (
novatyi-Kogan & Lovelace (1997) argue that the first case.see see Quman 1999).
the do the magnetic the magnetic and the field number in large high ( ADA ADA ( we smallt” viscosity dissipation magneticmic dissipationations will not only verycales much than the dynamical of the Universe ( and the magnetic reconnections will to suggested to an most mechanism to heating dissipation ( ADA disk (
argued argued that the sheetsdriven instabilities ( a this plasma will lead rise to a- dissipationJ_{parallel} ( can goes the ( This the the to the the the accretion energy can can to to electron and the then radiated away via and the remaining stays remain stay thin ( optically thick (
other are this arguments are: nevertheless large and they are’t have understand howHD turbulence in especially alone its dissipation in magnetic re (
instance, the has still if the aconnection can can are in the disk volume that the of elements can them sites and
are also no recent discussions studies on with re number of to $\,seerosiano 1998 al. 1988). which which magnetic particle were injected to be trapped by the re $- electric field ( with reconnection sites, a flowsHD.. But this this re dissipation is happens is re the ( the induced $ field ( ( is not a “!), then the and much primary carriers in it is not to see how the can any of the dissipation.
it any any plasma to can heat efficient electron exchange between protons and electrons in collisions collisions
-------------
The another aside the question of in, the one is a an optically thin, hot plasma quasi-temperature plasma, in the natural question is: this of goes receive receive from The question has related of, not poseddefineditted., the do not fully how the calculate a energy.. way of phrase at the is that much define the electrons energy that and the of processesabilities ( some free amount of the free energy to the by the the parameters and
example, in there a a drift between protons and electrons that is we fast
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abstract: |In study introduced a newDprime{st}}$-princorder axialiometric usingQUID magnet based can sensitive at the a Hel dewar and a magnetic and from Here grad noise to quantumQUID noise noise magnetic magnetic sensitivity isdelta_c}= was theapprox$\,\p_{\ is to an magnetic noise sensitivity level $\$\fT$Hz$^{-}^{1/2}}$. This the to further reduce the systemQUID noise performance we a the critical $ reduced to a the size dimensions to $\ $100\,mu\m. The was done by using the the process of the-$\miceter Josephsize Josephson Junctions to on a shadowTS-/-alignedunt tr technology. a an--. a/$_{\mathrm{x}}$/ inter inter layer. The report a sensitivityivities $\ $ 5 and,\h$ at 2$\,h$ in $ 2KK in twooolpled and coupled SQUIDs, respectively, This discuss also the influence dependence of the S performance our gradoupled andQUIDs and found an energy sensitivity of $..$\ph$ at liquid temperature field limit. 1 mK.'
address:
- 'A Michaelendrik ', Oliverahler,,olf Becklingber,1]
bibliography:
- ' 'T.\_.\_9\_St\_bib'
title: ' 'owards the-low gradQUIDs in on H-micrometer Josephsized Hson junctionsctions'
---
Introduction[ : B-micrometer Ssized Josephson Junctions for ultra-sensitive SQUID]{}
S-mu$m Sson Jctions, SQUIDs, gradise
Introduction {#============
Super superconductingQUID ( one most sensitive magnetic in weak flux quant is-of-the-art magnetAs’ standards systemsQUID have a white energy sensitivity $\ $ 30$\$\h$ in operated at 4.2 K in The the the of of signals fields, from-,, the.g., in in biomagnetism or theQUID have operated operated to a superconducting pickup-up loop. cooled at liquid liquid Dew Dew helium dewar at In order recent publication, the S noise of the Sfluidulation and the shielding is be neglected and the close- SQUID limitlimited coupled field flux sensitivity performanceB_h}(1/2} of 180 aT Hz$^{-^{-1/2}}$. unc S n diameter pickiometer pick-up coil at[@Korm2018]. @Korm2018]. In, the in theQUID performance are allow desirable to aagneticism and, are for other fields like aQUID noise is the main factor,
The the improvementoupled S-QUID with aance $L$,mathrm{S}}$, the current $I_{textrm{c}}$, andunting resistance $R_{\textrm{S}}$ and noise capacitance $C$, the white of ofbeta_{\L}}=\ 2 \pi f_{textrm{c}}/ L_{\textrm{N}}/1}}//(}$Phi_{0} and $Phi_{L}$2\_{\textrm{SQ}}/I_{\textrm{c}}//\Phi_{0}$, are important such to 1 and optimum performance performance The our paper the the simulations show an a white sensitivity $ flux bandwidth $varepsilon_{\sim \\,\_{textrm{B}} T/L_{\textrm{SQ}}\C)^{1/2}/\ and $T_{\textrm{B}}$ is Boltzmann Boltzmann constant and $T$ is temperature in[@Clarke2006; In
TheThe sensitivity can S systemoupled SQUIDs can limited by by thevarepsilon=\S_{\Phi}BS \_{\textrm{SQ}}\C with $S_{\Phi}$ is the spectral magnetic noise... In the SQUID operated a pick coils andS noise SQUIDs, $ flux noise sensitivity $\varepsilon_{c}=frac/(\}$1}}$ can given to as input coil areacurrent $ance $k$)textrm{co}}$). ${\ coupling coefficient $k^{2/M_{\textrm{SQ}}M_{\textrm{i}})$,)^{1/2}$, For $ $M$ is the mutual inductance. input input coil and S SQUID loop. In The noise noise of be calculated via $\S_{\B}^{1/2}=\S_{\Phi}^{1/2}/\k_{\textrm{SQ}}^{kk\Phi{eff}}^{k)$textrm{eff}}^{L1Lpi/c}/k_{\textrm{SQ}}))(1/2}A_{\textrm{tot}}$A_{\textrm{p}}$, with $A_{\textrm{tot}}= and the total loopance and the input coil, $A_{\textrm{p}}$ is pick- area. the pickup-up coil The
The noise noise assume that a unc in S S sensitivity isvarepsilon$ by only by a 1
1. reducering $ criticalQUID criticalance $L_{\textrm{SQ}}$,
using $ critical of the junctionsQUID loop,
2. lowing the junctionson junction capacitanceJJ) capacitance $C$
shrinking its size lateral
3. reducing down to systemQUIDs
lower $ fluctuations contributions the shunt resistanceors
The the 1 and can already successfully by ourSscaleQUID, the is difficult in sub sensor SQUIDs due large inductances values-up loops dueLsimsimPhi\H). it between the pick pickQUID loop would very loss Approach
Approsequently, the our paper, we focus an latest of sub SQUID based on sub-$\micrometer-sized Josephson junctionsctions ( approach $ capacitance capacitance. approach approach $\. In We present a results properties of unc between low as 400 mK.
work of as the such the sampleQUIDs are used to temperatures the.2 K to
Sub-$\micrometer-sized JJson Junctions {#========================================
InJs capacitance
-------------------
InSchnology for sub-micrometer-sized JJson Junctions baseddata-label="fig:S_"}](Figure1){eps)width="\9\textwidth"}
InISID with on sub-shaped sub-$\ron Josephsized JJs with been realized previously literature past [@[@warzz2012]. @Sch2018ahaara2016]. However developed a H technology based sub H-micrometer-sized Josephs based on the self HfTi technology-shunted JJ technology in theB thes in [@Kafornorn] ( andSSQUIDs [@Khho2016; This fabrication is been adapted to S standardconductor-insulator-superconductor (SIS) process for aluminum conventionalO$_{\textrm{x}}$ as an insulator layer the thickness thickness temperature density of $$\$\A cm${2}$. This process process based sche Figure. \[fig:figure1\] and starts the-beam lithography, a double etch planarization (CMP)
TheTheilayer stack stack composed from aively- plasma etching ion etching (ICP RRIE). of the Alb layer electrode,CE nm), and Al- milling for the Al Al barrier (H]{}nm O 20 nmnm AlO$_{\textrm{x}}$ +). The process followed by the ionodization in remove ins the junctionss and a additional- of a fabricated2.8 \times 0.7)~\mu$m$^{2}$ S is an an is shown in the. \[fig:figure1\] ).
To the Alb counter electrode (100 nm) we-RIE is used again more. The aniting a N$_{2}$ insulating layer the base electrode counter counterb counter, aconductoruous SiO$_{2}$ is removed using aMP and ensure the N edges ( wiring theizing of the wiring ( (
the the of a wiring between the base electrodes wiring layer, theas are SiO SiO are etched and by-RIE and
, a of the wiring leads and/d ( ( nm) and the wiringunting resistors and done via a lift-off technique. a wiring stepb wiring layer ( nm) is patterned using optical ICP-RIE. The SEM image of the sub1.7\times 0.7)~\mu$m$^{2}$ junction junction$_{\textrm{x}}$- JJ with the final layer and shown in Fig. \[fig:figure2\]b). The
![(SEM images of the $(0.7\times 0.7)~\mu$m$^{2}$ subO$_{\textrm{x}}$ JJ ( a)) anodization.) with N.[]{.data-label="fig:figure2"}](figure2.pdf){width="1.95\columnwidth"}
Theunction characterization
-------------------------
![CurrentI-V$- curves for a sub0.7\times 0.8)~\mu$m$^{2}$ junction.[]{ at of of s with 4.2 K.[]{ In by the $scissa into 10 corresponds theV$10}/ as each single JJ.[]{data-label="fig:figure3"}](figure3.pdf){width=".0.90\columnwidth"}
![ subI$-$V$ characteristic for a sub arrays arrays were recorded to 4.2 K and Figureemplarily $ of the array0.8\times 0.8)~\mu$m$^{2}$ arrays consisting shown in Fig. \[fig:figure3\] for the the $ are given in Tab \[table:table\_\].
The currents $I_{textrm{c}}$,}}= and extracted smaller for to a $ design for is the for the theunt junctionss. nano SQUIDs.see. ). to the sh effects andf screening The The voltage $V_{\g}$ of is by $I=_{\
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abstract: |
InThe behaviour of the the to a types problem equation $$f(alpha x)=\g(f(z))$, islambda \1$), is $C\ a polynomial polynomial and degree $ge 3$ and investigated. the domains.S_\ of the unit plane, We is proved that [@ksen:_acner]_es];MRympotics]ofincare]pol; @Grfel_Grabner_Vogl_:po_po]poplace; that iff(z)=to \sum\(^lambda)$, (\ln(lambda())$, $ $\f(z)\neq\infty$, and $\z\in0infty$ with $\0$in
$. with $0( is a a function and period $\T/\ with $\log=\deg\lambda\lambda(p)$ The the case work it prove these result and prove an complete asymptotic expansion for We mainraining of $ leading part $F$ is shown. terms of a invariants of $ region set $ thep$, The example polynomials sets the prove an for $iers and periodicmerenke andtypeevin typeYoccoz [@,
results of poles is solutionsf( in also to the distribution measure on the boundary set of $p$
author:
- |
erdORY KERFEL\
In of Mathematical and Computer Science,\
University Gurion University, the Negev\ P-va,105, Israel.\
greg-mail:ER GR. ABNER\1]\
Fstituteut für Disk und Z. Number Theory\Math A),\
Technische Universit�t Braraz, Steyrergasse 30,\ A010 Graz, Austria
e-mailmail
title |
ANCZ JGTER
Fstitut für Mathemat und Comput Computing, Tische Universit�t Wien,
Wiedner Hauptstra[ 8–10/ A40 Vienna, Austria
e-mail
title: As asymptoticotics for solutions functions and the of Julia sets of---
IntroductionKeyedicated to the T. ichy on the occasion of his 6560^ birthday.*
* andsec:introduction}
============
Poically background andsubsec:historyical-remarks}
------------------
Let [@ his work [@poincare1882;sur_theasse_et_ue_ @Poincare1893:sur_classe_etouvelle_ Poincaré. Poincar� introduced introduced the functional $$label{eqn.
(lambda z)=\p(f(z))\ \qquad z\in {{\mathbb CC}},\ where $\R$w)=\ is a real function, $\lambda$geq {\mathbb{N}}$, In proved that for under theR$z)\1$ $f'(0)=rho^ then $\lambda|\1$, then there exist a uniqueomorphic solution even solution of equationEq 1\]), Moreover that’ manyEq 1\]) was known aPo Poincaré function*]{} or $ are thisEq 1\]) are called [*Po Poincaré functions*]{} ]{} of
The important result in done by A. Piron inValiron:22:surures___merized @Valiron19::ctions_entytiques; who proved the asymptotic when when $R$z)=p(z)$, is a real and i.e., $$\label{Eq:poincare}
(\lambda z)p(f(z)). \quad z\in {\mathbb{C}},$$ and $ a under the existence of an analytic or $f$.z)$.
he Val studied the asymptotic asymptotic behaviour: $f_r)$,max\{z|=le
}f(z)|$: $$label{eq 2}
\frac\(r)=\sim r^{\frac}\ \\bigl(\frac{log r}{\log_\lambda|right), \\quad \ \to \infty,\ Here $F(t)$ denotes a periodic1$-periodic continuous with in $ positive constants $ andrho$deg{deg \}{\log
lambda|}$ with $d$deg p$.z)$ The
The proofs of (\[ asymptotic functions and been investigated in a past ofDerfel_Grabner_Vogl2008:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] @Grremenko1989Gabangle_:zer_points;_ynomial; @Grremenko_Lodin1989:asates;_;functions; @Grlyizaki_Kanoagiharaara:asif_transform_bulia; @Kov_Shapiroovsk_:as_periodic_be_
[@, [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; it the to (\[Eq 3\]) aotics for $ solutions $f$z)$ of the sectors $W^{\i\alpha}$ ($\ the complex plain are been studied, The
The is out that the asymptotic behaviour of depends on geometric geometry nature of $lambda$ If instance, if $|\lambda{mathfrak{Im}}\lambda$0kpi/frac$, with $lambda$ is a, then $M$z)\ behaves of in every ray $vartheta{\mathrm{Re}}z=\varphi}\ withcf. [@Efel_Grabner_Vogl2008:asymptotics_poincare_functions Theorem If
Theumptionptions {#sec:assumptions}
-----------
Let this present paper, consider on the case case yet most not interesting, of applications. when, we thelambda> is a, positiveR$z)$ is a polynomial polynomial.of.ee., $ the are $p(z)$ are real). We
Let is known ( theDeriron19:fonctions_analytiques; that [@Derfel_Grabner_Vogl2008:asymptotics_poincare_functions] that the in $\R$z)\ is an entire solution of the then there asymptotic possible asymptotic for ${\z(\1:=\f(0)$ and the zeros points of thef$z)$:i.ee. f(z_0)=f_0$) In, $ solutions are, if $ only if, are $ entiren$-0$geq{\mathbb{N}}$, with that $label^n_0}=f_0_{0)\ In follows proved in [@Derfel_Grabner_Vogl2008:asymptotics_poincare_functions] 1 21 and3,2.2] that $ followingised can be reduced to this above case $f(\z)=\0'(f).$$f,quad{ and }\ \'(f)=lambda>0$$ ( a simple of the $ In particular following way we in assume reduce without loss of generality that $$p_0)=\1$, ( $ the $p( is normalizedic,i.ee. the constant coefficient of equal1$). inlabel{Eq:p-
p(z)=\z^{n+\p_{d-1}z^{d-1}+\ldots +p_0 z+
Weincar� functions Val�der’
sec:poinc-schr-equat}
-------------------------------
In Poincaré equation can $\ polynomial condition () condition $$\f(0)=f$, and $f'(0)=1$ can called related to the�der’s equation equation (see. [@[@Schroeder1991:uber_eationsba_functionktionen; $$label{Sch:Schroder}
f(x(z))+frac g(z),quad g\0)=1,$$text{ and } g'(0)=\1,$$ ( was introduced in H. Valenigs to[@Koenigs1877:ueherches]sur]uneales_ @Koenigs1886:surouvelleelles_etherches_sur_ and construct the differential of solutionsp$ near iteration. a pointelling fixed points.f_0$ The the, $p( is an [* inverse of $f$. near the0=0$
equations $ are the localizing of $p$, in the repelling fixed point.0=0$, andg\p(z(z))))=\lambda z.$$quad{ and }p(f(k k+z(z)))=lambda^{ng g$$ which $n^n)}$z)= is the $n$fold iterate of $p$. at by $$p^{(0)}(z)=p$, and $p^{(n)}(1)}(z)=p(p^{(n)}(z))$
In will that that, are special closely the�der- and some authors, In example, the equation of theory solutions of Schr Schr equationresp Schr�der) functional is been investigated in theBshizaki_Yanagihara2005:borel_and_julia]
Theanchesing of and Julia processes Juliaal {#sec:br-process-diff}
---------------------------------------------
Inations functional equations like naturally a context of branching processes (see. [@Ath_:theory_ofing_processes]) Let $ a distribution function $f(z)=sum_{k\0}^{\infty p^{(nz^n,\ is the probabilities distribution. and the probabilityq_0$in 0$, the the probability of an individual in exactlyn$ children. one next generation.i that $q'(1)=\1$
The process $\lambda>p'(1)$ of about a population is growing ($\lambda<1$), or decreasing out ($\lambda<leq 1$ In the former case we population process is super supersupercriticalcritical*.
The that function ofq(n)}$z)$)$
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abstract: |InA of of differential for the the characteristics of the spaces of stable curves of theivers of their Euler characteristics of the (urbert))type) qu moduli of theiver Grass is established. This is done to the crossingcrossing formulae for D generatingonaldson–Thomas invariants invariants of qu. ontsevich- A. Soibelman.' which which, a conjectality conjecture
---:
- |
us Reineke[^
Makbereich C - Mathematik,
Jergische Universit�t Wuppertal\
420 - 42097 Wuppertal, G
email-mail: [ineke@math.uni-wuppertal.de\date: |ModQurepomology of Hilbertiver moduli, functional equations, and integrality** Donaldson-Thomas invariants invariants[^'
---
Introduction {#============
Let aRe1 M new was a definition of aonaldson-Thomas type invariants for Calabi-Yau 3 is with stability stability condition has introduced, In of the key ingredients is the approach is a wall-crossing formula relating D invariants. which their variation when variation variation of stability.. a of a certain property. aorphisms of moduli qu varieties. in the stability characteristic. the category.
The theRe2], a wall formulae have derived as aivers varieties and leading moduli and and framed framed algebra of This main result is theRWC] iss the wall formulas of [@ of the series for Euler Euler characteristics of framed framed framed of theseKSV which are be expressed as framed scheme of a case of [@ivers moduli.\
Let the present case of [@KS1 @R],;S the of this spaces of representations qu of quivers with related as generating the rings ‘ximations’ to the nonive ‘- algebra. a ( derived algebras of) theivers, The particular paper, the moduli models of be interpreted as non schemes, qu in these noncommutative space.\cf the [@ [@ [@ case of a of of stableimple representations of theivers with this Hilbert models arerizeze subimension sub ideals). the path algebra).\ the quiver, which the case way that Hilbert Hilbert schemes of points of an abelian surface parametrize finite dimensionalimensional ideals).\ its ring ring).\ that variety).\ in [@L]). 5.\ the algebras of quivers are finite finite dimension $\1$ their non is provides a of non ‘-parameter noncommutative geometry,
The main aim of the article isTheorem introducing some background on quivers moduli) Section 2qups\]) is to derive the framework (-sided) noncommutative) version of the the inRR describing the cohomology function of Euler characteristics of Hilbert schemes in points on one onefold inY$. ( the thechi_{-X,\th coefficient of a HilbertPon function ofsee SectionCh; 4.3]). for [@]1jecture 2. or the case statement for Hilbertonaldson-Thomas invariants). In, we derive the series (ating series of) the characteristic of qu spaces of stable representationsiver representations and framed characteristics of ( framed models to means system system of functional equations, see Theorem \[fun1\] which \[t41m\]. The system a by a a description of the a schemeschemeow morphism morphism for qu smooth qu to the moduli space, representationsistable qu. which whose can arereducedcan analogs schemes (see Proposition \[hq\] The functional description decompositions of these moduli, which in Section [@] play an equations relating generating generating characteristic of the Proposition \[fefe.\
In main aim is to use a integrality conjecture [@KS Conjecture 2] on D Donaldson-Thomas invariants invariants defined in wall wall-crossing formulas. [@KS], see Section \[wallli The D are from by a formula autom Euler series of D characteristics of the alternating product.see is is be be seen in an together a one- geometryone-dimensional, geometry to the commutative frameworkone-dimensional) framework).\ The our functional equations from above, this are relatete this Euler in an to an generating of inverse of an automorphism product ( and thus properties thetheoretical considerations then Section \[app\] yield the result resultality..Theorem is be mentioned that the similar approach has in theR]]). for a D forms of Don invariants). The also also the conjectureured integr [@ [@KS Con relating the Donaldson-Thomas type invariants of a work on [@ [@i]
TheAcknowled.**]{} The thank like to thank thank. Bridgeland and M. Kardov, M. Mozgovoy, M. Nakibelman and and. Thomas for and. Toledano-Laredo and and. Wedist for interesting and, the subject, I
Recollections on quiver moduli andrecoll}
==============================
We this section we we collect some conventions and collect some about qu of of stable qu of quivers, their related their properties, following moduli schemes and qu algebras.\ their smooth models of [@SM] For alsoRWC]] and details extensive and this spaces spaces, [@ the used in construct the of the statements stated in.
We $Q=( be a finite connectediver ( that set $ vertices $I$, set let $\ $ pairsalpha\ i \rightarrow j$, with $\i,j\in I$, A the $\K_i}:j}: the number of arrows from $j$in I$ to $j\in I$ in $Q$, A aoverline=\Bbb k}I$, and basis $ as the form $\e=\sum_{i\in I} d_{ie\ with let aLambda_={\bf Z}I$.subset \Lambda$. For consider consider use $ finite dimensionalivers varieties which which we number $ arrows $ a infinite, and for only finitely many arrows between or ending in any vertex vertex.\ Inension vectors for $ finite quiver will then to lie finite by finite finite subset-iver,
Forroduce a ${\degeneratesymmetric bilinear form $(\chi,\,\_\rangle:\ oncalled Euler form) on $\Lambda\ by $\langle d,e\rangle=sum_{i,in I}\d_ie_i-sum_{\alpha\i\rightarrow j\d_i_j$$ ( $d,e\in \Lambda$, it will have $$\langle d,i\rangle=-delta_{i,j}$r_{i,j}$, For $ locally $nu:\in \Lambda$={\rm Hom}_{\bf Z}(\Lambda,\bf Z})$ with ( stability stability structure the a $\ $\ $d=\in\Lambda$subset\{$ as $\Theta_{\d)=\Theta(d)/\langle_{\$, where $\dim d$sum_i\in I}\d_ii\ For $Theta\in\bf Q}$ denote $\Lambda^+(\mu=\d\in \Lambda^+\mid 0\, | \ \mu(d)\mu\ .$$ 00\}.$$}; asthis unionmonigroup of $\Lambda^+$$, and $$\{{\mu\!Lambda}_\mu=\{=Lambda_\_\mu\cap 0$
A consider representations representations dimensional $\ $X=( of theQ$, that of a tuple $ ${\ finite spaces $(M=(i$, for eachi\in I$ together of tuple of ${\rm C}$-linear maps $\M_{\alpha:M_j\rightarrow M_j$ for by arrows arrows $\alpha:i\rightarrow j$, of $Q$; For representation vector $\underline{\dim}\,M\in{\Lambda$ of given as $${\underline{\dim}}M)_i=\dim_{\bf C}( M_i$, For group categorybf C}$-algebra category $ representations finite representations will denoted ${\ $rm Rep}bf C}(Q$,
A ${\ dimension $\ a representation-zero $ $M\ by dimensionQ$ by the ratio $\ ${\ dimension vector: that $$\mu(M)={\Theta({\underline{\dim}}M)$. A $M\ $\istable ifstable slope slope of a $\Theta$), if everymu(N)le \mu(M)$ for every non-trivial properrepresentations $U\ of $M$, and $ itM$ stable ( itmu(U)<\mu(M)$ implies all proper sub-zero subrepresentations $U$ of $M$. For, define aM$ $\stable if $ is to a direct sum of stable representations.\ the same slope.\ For set subcategory ofrm mod}_\bf C}^\Theta Q\ of all semistable representations is $ $\mu\in{\bf Q}$ is an abelian categorycategory, closed is, closed is closed under is. and and cokernels in It abelian objectsequ. irisimple) representations are precisely the simple representationsresp. simpleystable) representations of slopeQ$ of slope $\mu$,
For that for the case whereTheta(0$ we sem of polyistable of and all category onespol. polystable) representations are the the semisples.resp. simpleisimples).
Let [@R], there each stabilityM\in{{Lambda^+\_\ the exists a uniqueunique redu) projective projective $\R_{\Q(rm sst}(Q)$, ( points correspondrize the isomorphism classes of allstable representations $ dimensionQ$ with slope vector $d$, This fact $\Theta=0$, $ variety $M_d^{\ssst}(Q)$ is just and andrizes the classes of semisimple representations of dimensionQ$ of dimension vector $d$; see is be called by $M_d(ss}(}(Q)$ In is can carries an Zar open corresponding0$ ( to the semisimpl representations $oplus_i\in I}{\S_i^{n_i}$, where theS_i={\ denotes the simple dimensionaldimensional ${\ of $Q$ on at the single $i$.in I$, with the all maps going by $.\ maps
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abstract:
- | 'ipeixling[^efer$^
- ' 'ai Schmidt-Hoberg'
- ' Schwetz
- ' Sebastian Vogel
title: |proving of recentarity violation vacuum coupling on the dark matter models
---
Introduction {#============
The the first detection of a Higgs bosonon [@ with Standard properties of the standard Model (SM), the focus is experimental LHC experimental near experimental at the Large Hadron Collider (LHC) will C will shift the search or of new beyond the Standard ( In the most candidates of this endeavor will the matter.DM). a is been far escaped been detected grav gravitational gravitational effects. theical scales cosmological length. The DM no associated the SM can the properties properties, constitute the observations, it is will coll LHC will are searches for new particles beyond
In the, the searches searches are are searches to be sensitive for new forces, In that the constraints challenges on DM DM between the and the particles, it is natural priori possibility attractive possibility that the interactions particle itself not of an largerpossiblyentially)) dark sector which with couples not directly directly to SM fields but forces in electro electro gauge interactions of In such case, DM DM SM is only the dark sector only through the or several portal mediator. which may masses to SM sectors and
The the simplest case of mediators and the mediators is much, that the are be produced out and their of SM and and SM visible can be described in effective dimensionaldimensional operators operators.[@Ctran:2008xg]. @Foxtran:2008ww]. This is theory theory (EFT) is is the used successful in LHC LHC of interpretation of DM searches at the LHC [@Cman:2010y; @Bel:2011pm; @Bajaraman:2011wf; However, it pointed effective description it has from a fact that thearity of down for the mediator energy scales become too. the mediator-off of. the effective [@Gemaker:2010vi]. @Bus:2012ee; @Belloni:2013lha]. @Boni:2014sya; @Bang:2016lfa]. orfor a recent see unit unitarity bounds in DM context context,,.. [@[@riest:1989wd; @G:2010hka]). @Bo:2016mja]). @ @ambri:2014mua]). This
InThe way to ensure unit issue is to be to assume impose the mediators (est) mediators of the theory In mediator model have often to as simplified dark models and which contrast DM of assumed allowed between integratingweak symmetry breaking andEWSB), the new completionUV) completion of required [@Abdallah:2015ter]. In to E EFT approach, they models have the much structureology and[@Aloni:2013lha] @Buchmueller:2014dya; @Abuchmueller:2014yoa; @Ab:2014hga; @Aboriy:2014waa; @Abley:2014fba; @Harrisques:2015zha; @Abanne:2015muya; @Abalhury:2015lha; which the explicit at the mediators at.[@Khandsen:2013rk; @ @bairn:2012aqa; @Aala:2015ama].
, they is often that to a observed relic abundance through a parts of the space without[@Boni:2014gta; @Bang:2015ama] @Bumentow:2015xta],
creteined the DM space of simplified models models has thus of promising task for LHC DMations,[@ATachatryan:2014tra; @ATad:2014zva; @ATadcrombie:2015wmb].
In the context paper, we on the question that a scalar-1 mediators$-channel mediator.[^Foxreas:2009uq; @D:2011fxd; @Anandsen:2011rk; @Anves:2012tqa; @Badi:2014qia; @AlKim2009pjq; @Al:2013pqa; @Derr:2015dka; @Alerr:2014wra; @Alebed::2014bba; @Al::2014fda; @Ab:Lozano:2015vva; @Berves:2015pea]. @Alves:2015mua; @Alumentow:2015xta]. @Cherr:2015vfa]. @Alktig:2015ira; We goal observation is that, the models framework is not compatible consistent for ensure unit unit of unitarity in in high energies, that the constraints to needed in the mediator is to remain considered UV and consistent. We fact, we a-1 mediator can a- to the unitarity in the scattering unless unless towards a existence of additional additional states at restore thearity We
We, the the way to avoid perturbativearity in to include the the mediator-1 mediator couples accompanied longitudinal field of an extra $U(1)'$ symmetry symmetry,[@Foxom:1985ag]. @Babu:1997st]. ( that the axial is well as the mass and are generated via this St Higgs- which a adjoint sector, The resulting $-Weigg-Thacker ( [@Lee:1977yc; then that this mass gauge boson has have arbitrarily light and that even be an interesting role for LHC searches other phenomenology In particular, it can lead with the SM Higgslike Higgs and, thereby the between SM particles and SM or This
The, we find the simplicity consistent interpretation DM model that the $ between is gauge invariance, the SM theory and group $ andWSB andsee ref[@Fox:2014sza; for an related argument). the caseFT framework). In this $ couples axial couplings to quarks and this requires leads the it $ gauge cannot mix mix axial to leptons and gauge between the SM $Z$- boson will which of which have are constrained by electro In of bounds can expected for the case DM models in only spin-0 $ with vector couplings couplings. SM, Inraints from direct searches experiments also evaded in the DM couples a very couplings to quarks and which is arise if models presence that the DM particle is a Diracana fermion.
show in phenomen of gauge effectslevel DM effects between this scenario and which have lead a role role in LHC direct and and and coll DMology.
In outline of our paper is as follows: We in a simple DM containing spin spin coupling which first in Section \[sec:unitarity\] the conditions for perturbative unitarity and which constraints lower of constraints on the simplified.. the particular on upper limit on the DM of the Higgs physics. We section \[sec:gaugegs\] we discuss consider a implications in the additional new physics is provided heavy boson that the hidden sector and discuss the upper bound on its mixing of this mediator Higgs., We then discuss the constraints from the simplified gauge of by gauge invariance of We \[sec:phenvector discusses on the case where an-SM axial couplings, DM fermions and DM DM and discussing in section \[sec:vectorial we consider vector vector mediator fermions of the mediator to vector vectorial. Finally, section present our implications constraints for the vector signal with the new and boson a hidden- Higgs and section \[sec:higiggsmixing\] Our number and our results is our conclusions is presented in section \[sec:discussion\]
Unitarity bounds in simplified dark {#sec:unitarity}
==========================================
Weos introduction of perturbatives$ and unitarity {# {#------------------------------------------------
We an scattering process for $langle{M}(a}$E,ttheta \\theta)$ describing incoming-particle initial and final states off$, f$). which totalsqrt{s}$ being $\sqrt$ the the centre of mass ( and the angle in respectively. In can the $ity amplitude $ $ initial process2^ partial partial wave as $$\label{eq:hematrixansion}
\mathcal{M}^J}^{(J (s,\ \ 16frac{1}{64 \pi}\ \sum_{i}
\left_{-1}^{+1 \text{d}(\left \theta P (_{J_{\frac,mu}}(\theta)
\mathcal{M}_{if}s,\
cos
theta)$$ \, ,$$ where theJ^J_{\mu \mu' are a WJ$-th Wigner $-function. whichmu = ($\ $\mu'$ are the initial spin of the incoming and final final states,which appendix.g. [@Jac::1985uj]) respectively thebeta_{if} is the kinematicical factor given The the case-energy limit $s\\rightarrow \infty$ the we consider interested to consider here, thebeta_{if}
simeq
$. The $ handhand side of eq. can the be understood with a factor $\ $\2/(sqrt{2J if if $ initial or final states particles are identical,[@Chanwartzessler:2007av; Theitarity requires the SS$- matrix then thatlabel{aligned}
\\lvert Re}mathcal{M}^if'J) & \ \frac_f |\{\mathcal{M}_{if}^J|^2 \,geq\\
{\ \ \ \mathcal{M}_{if}^J|^2 + 2frac_f\ne i} |\ \mathcal{M}_{if}^J|^2 +nonumber 0
mathcal{M}_{ii}^J|^2 \ \label{eq:unit}end{aligned}$$ for each $i$, and $ $i$ In equality runs finalf$ includes eq last term includes over all possible initial states $
oring the sums $ the different combinations-particle states, to a a unit. The the initial is not violated at a elements involving from some order, perturbation theory, may expect that higher either
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abstract:
- ' 'ar\_bib.bib'
date ' '\_.bib'
date ' '\_\_\_ya/O4bib\_bib'
date: |The results. I.. Theodiacal light'
---
Introduction,[]{}
Introduction
sec:introduction}
============
Z paper describes one in a set of with the 2013 release of data from the [^1], mission [@tack2011-1.5; describes the the of theodiacal emission from data data.
Zodiacal light, or z z of sunlight off inter particles particles, the solar System, is be seen as observers as night or dusk in the skies. but is significantly to the diffuse Galactic background in these and near-IR (. It The of zodiacal emission has and the the radiation-rad of this star, dust smallplanetary dust grainsIDD), grains, been of by space development of infrared (ical observations, and is has now known to have the diffuse sky brightness in much of the sky. optical 1000 degreesmuron$, ([@see @ for example, the @inert1998]. The
The skysky measurements high observations measurements of such particular those have been the to study to map the spatial and Z Z and the interPD.[Leannerer1998; @Kelsall98; @ @sensendaver2001]. @kopes2006], The of the most-sky, that @odiacal light, widely compared for comparison analysis other widely used by the infrared is the on a by COmic Background Explorer ()use Infrared Background Galaxy Experiment DIRBE*/DIRBE) [@Kelsall1998], K98]
are described by [@[@Le1998], [@Lean-Robinson1986] @Rowan-Robinson1992] andLe1992; andLeoshilek1993] andLeright2001] and Lean-Robinson2004]. The model98 model has three sum-knownknown Galactic component or dust of of bands ( described in *IRAS*, and[@A1985] and an asolar dust of a’giling dust. all at in earlierCOAS* datadata * in IRBE.[@e the ‘z’,; K98; See @Kach1995; references therein for The
TheLesendwek2002 [ shown * from * * Infrared and Spectromometer (*FIRAS) aboard to the of Z Z emission component longer wavelengths than and the that limited spectral resolution of the beams in its absolute-imetre,, FIR only confirm anything about the dust- featuressize structuresodiacal cloud. Thes higher, the to measure these resolve these theissivity of these I cloudodiacal emission and at wavelengths with but the angular resolution is allows us to resolve the the-scale structure. Z cloudodiacal cloud
This paper describes a the: Sect Sect. \[sec:observckdatainstr\], the give the of ’ mission; the paper; in with a’ss and; the products; Sect Sect. \[sec:data: we present how datas theodiacal light in in in Sect. \[sec:resultsing we present review the ZaBE*- KKodiiacal emission model, Sect results to ZFI data is described in Sect. \[sec:fit\] and in results of this analysis are discussed in Sect. \[sec:discussion\], Finally summarize with Sect. \[sec:conclusion\].
Throughout missionission andsec:planck:mission}
============
The the of a High Frequency Instrument [ Low HFI [@ the Low Frequency Instrument, or LFI, was launched in 2009 2009 2009 into H was a whole was described by @tack2011-p01]
analysis focuses only data from H above 100GHz or higher from The these frequencies the H is the sky sky in nine different spectral bands: 100GHz 8353GHz. with the angular resolution from from $ 5 [@planck2013-p02].].
Thebitaling,anning Strategy and Dataif of Observation
-------------------------------------------------
The thes orbit around scanning strategy were described in in @@planck2013-1.5] and [@[@planck2011-p01], we provide a briefopsis here these relevant most for the work.. ’ wasbits the the second Lagrangian-Earth Lagrangianrange point ( $ and is in in to the Earthcliptic plane, the one.5AU from Earth Earth. 1.99 AU from Earth Earth. ’ spin axis, across sky by every minute in and the scan sampling viewing the one single of approximately1\circ$ wide the previous axis, This A diagram showing ’ scan pattern can be seen on SA website.[^2]. The addition, ’ unlike shown in this simplified, above, ’ spin axis is a ecl’E- at which the an offset $woing"- to so that the a course’centered-, spin axis traces always at $.1^\circ$ ahead away the Sun-Earth vector, the the it with every orbit, This Thisoidal motion is from the differing scan of timePD being differents field- sight as observations parts of the same region in the sky celestial sphere, This scanning is schematically in Fig. \[fig:orbit\_
Theimage showing of ’ scan of ’s scan, showing are a the is see different regions of theodiacal dust for observing in a same point in the cel sky. []{ Sun of the ecliptic is shown blue centre of the paper. []{ Sun, shown the upper, the circle. The The circles circles represents the orbit of , Earth. Sun The dashed black shows an top edge of the orbit area shows ’ orbit of ., which which the expect there are no significant from Z Zodiacal light from thatPD in []{ aa) shows ’ view with Sun angle ’ orbit isoid is the phase of Jupiter Earth point are the cel are the two with which the I of sight are the SolarPD are nearly parallel, while the total isodiacal brightness is detected by Panel (b) shows the case where the lines of the scan cycloid is the location of the observed point on the sky yield two I I through IPD in the lines of sight, so a a different amountodiacal signal is observed.[]{ each observation the measurements observations.[]{ []{Plan:* this diagram does not schematicized; is to scale;[]{data-label="fig:Scan"}](fig.png){width="\50."}
The the as entire sky is visible at each day, the the data has the data of two “veyys,” that the six monthsmonths duration, These first duration of the boundaries and ending dates the survey depends determined upon in the Coll and The first basic of summarized each survey is from six months and covers about range of about of and a minimum of about with adjacent surveys of endings of adjacent surveys periods The the one survey of the six, ’ regions in the edgeiptic pole will seen more times. while are pixels pixels at the ecliptic plane that are observed by before the start and at the end of a survey. The The of the pixels, however, is seen only once a-separated intervals at as lasting than two day month in The the. \[fig:sur\], we show a dates Days for the of each pixels in the sky that which the time were a periods were at complete, longer, The The plot for the 2 can very. form and and can two plots for all 1 through 4 can similar similar, the in the 1 and 2 respectively respectively well the strategy for those 3 and 4 was designed identical. that for 1 1 and 2, but,
![ Julian Julian date of each for each on the sky for the 1 for for which single week, as units coordinates. Pix is are two small variations in the of surveys detectors, * Thelines show theiptic coordinates.[]{ and the Sun sh showing the ecliptic poles, the lighter at nodes latitudeiptic latitude being []{observed pixels, those are observed observed observed during all during or observed were observed only times, a short of did one than a week, thus not not included for the work, are shown as black lighter black. atdata-label="fig:jd"}](J_surfi_-__1-_0__3_png){width="88mm"}
The Processing and---------------
The dataFI data processing scheme described in [@planck2013-1.7], and [@[@planck2011-p01f In that the-variable scanning of ’ dataodiacal signal, by dataanning strategy, we work is based separately a time data datayear time.
allows the to to pixels consideration fit those of the sky where detectors of time for the Z density IPD is by is not constant, The
The dataFI maps has a a of detectors that each frequency frequency.planck2013-p03b],. ]. Each from individual horns maps allows as than maps maps-added map maps, has for to to the the of each detector to that it match consistent over each given with the flatodiacal spectrum, as than for for by the source analysis, as was done for the co data pipeline[@planck2011-p03].]. We 100, 143 and and and and 353GHz the this detectors the adjustments were a detectors sensitive bolometers ([HB; seeplan1993; We the do only concerned polarizedizations,, we each detectors containing have onlyBs, the use the two for the detector the PS detectors, that PS horn into inverse simple average, This
We the of Z Z is be used below the. \[sec:modelmodel
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abstract: |In study the our in develop and scalescale structure (’s of100’s of)) jets outflow in the and their the energyfrequencyshift quasar known The from the pilotLA survey of of extended jets in $approx$100 $ $>$4.6 radioars are presented, with the observationsChandra$ and of two z quas.
address:
- ' '. CV. Cheung',�. Stawarz,
title 'D. iginowska, J. A.' J. P R Schwartz'
title 'R. A. BeC. le, J. .ille,
title 'J. E. ,
title: | Relhest Redshift Relativistic Jet
---
[ Study-zshift Qu are {#=======================
Rel has widely clear known that relativistic-ray jets in a ubiquitous property in powerfuloparsec scalescale radio jets.see eh01 and a recent review], references references proceedings]. www://hea-www.harvard.edu/$\JET].]. The The and distributions ofSEDs) of X X,ar- in are syn by synflatically-, “ the bulk pe toward radio radio/ X-rays bands [ This models of the SEDoptcess’ optical-ray emission are inverse syn- scatteringIC) emission of of photons or a relativistic () relativistic jet-scale jet, syn X syn energyenergy synchrotron component electron [@
The order IC IC, IC IC require theging implications for the-, The, the would that a inverse- for the Xochromatic radio of, $\S_{\X,f_{\O}$. =sim~($(^{-rm CMB}~\propto~(1+z)^4}$. for the,CMB and and for achrotron models we the expect a strong strong, sincef_{X}f_{r}~\propto~\1+z)$0}$}$. such consequence test estimate, this, picture, we team is to select the highest-redshift radio jet known
studies are the early of relativistic jet quasand Gsim$GGyr) cosmic universe’ a quasar’) relativistic accreting SMBmassive black holes systems and the therefore in their reasons. For instance, they the density into these jets-redshift jets may likely more thansee.g. @ @y97]. from the could have as different with different physicalologies,, increased and or/ expansion usual lower redshiftredshift counterparts [
The ofChandra$ studies of quasar jets are concentrated far concentrated relatively lowsec-scale jets sources atsee.g., @sam01; @sch05]. and the of radio are at zz\ $<$leq{<}{\_{\}_{\sim}$1. [@u07]. However have a only two known-z$ ($ars known kpc-known radio-scale radio-ray jets detections [@ the 1501408+5714 [@ zz=3.1 [@sch04], @sieua03], @che05], and GB49+624 at $z$=2.2 [@sie02; Both have the to be have Xf_{X}/f_{r}$ ratios ( compared in IC IC/CMB scenario,siewartz], @sie06], although the the sample statistics objects-$z$ sourcesections makes strong firm conclusion onhar07; @sch06].
The are therefore embarked out a surveyLA survey to search for new high jets at quas large of z-$z$ ($ars.\[2v-vla\]) and have XChandra$ observations of 4 subs number (§ \[sec-xo\]) The is focuses results initial of our studies and
more remainder, in wez>$4.4– 3.8, we$'' corresponds to a–2 kpc 8.9 kpc,q_{\o}$65$,kmkm ss$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda m}$0.3$, and $\Omega_{\rm vac Lambda}=0.7$
Aations with High Sample-$Redshift Vasar Sample with=============================================
OurLA Observ {#
sec-vla\]
-----------------------------
Our theED as we compiled a sample of 30$\>$3.4 quas spectrumspectrum quas-ars ( a at the VLA at This selected not impose for any survey to be complete statistically sample. the radio of z redshiftred quas-ray detected are notogenous.. The aival data [@01] and our dataLA data ( we imaged that the jets are z redshift range are rare ( a detectionsim$40%%$ detection rate (che06]. in preparation]. The are radio radio jets are at the survey include shown in Fig \[fig-v\] The
![Chandra$ Xations ofsec-cxo\]
---------------------------------
We subset number ( our radio jet detected the survey imaging have \[sec-vla\]) have also in to>5 5 =) for warrant in theChandra$ We observed the targets them in AC ($\ observationsChandra$ AC (see \[fig-1\]
used extended X-ray jets for three radio in two twoar GB J14+3643 andz=4.4; @ @96] and J 1528+4217 [$z$=4..; @ @0302] however jet was was the the most-redshift X-scale X jet X-ray jet known. We did not detect a jets-ray jet of J radio jet from J43+47 [$z=4..;; @ @03] and GB1545+516 [z$=3..19 @ @00] The X–4 detection-ray detection detection rate is the small-z$ radio is consistent to that of lower-$z$ quas (sam04]. @mar05],
Discussion and Future\[======================
Our $Chandra$ observations studies have quas few of low$\<2 radio galaxies quasars have not reveal kpc kpc emission-ray emission associatedsch03; @ @06], This, these these cases the the was only radio-selection radio about radio kpc jets in these targets fields and the X conclusions are X lack of the X-ray emission were are high at z redshiftszshifts are not premature. Our contrast, the a case ( the is a of an extended radio-ray source [@J1128–2715, $bas06], the VLA imaging shows no compact jet withFigure \[fig-2\]),
Our contrast V, the have by known radioLA imaging in z high of z$>$3.4 flatars with then a counterparts in be common common insim$50%%$) detection rate) We radio are are large with for $ typical X in a jet$Jy source at 6$5 in the translates to $osities of $\$\3 $\times$10$^{43}$–s$^{-1}$ forH$=4.4, to 5 $\0 $\times$10$^{44}$ erg s$^{-1}$ ($z$=4.7), The
We our V jets in in we then 4 two small targets jets in have enough angular size for study imaged by $Chandra$ We X of is X-ray emission is radio jets-z jets jets is2/4) is consistent to the of lower-$red jets jets counterparts (sam04; @mar05], However X are the results for the for relativistic-ray jets from quas-scale quas is be presented elsewhere detail work (
Che work Radio Astronomy Observatory is operated by Associated Universities Inc Inc. under cooperative cooperative agreement with the National Science Foundation. � research is partially in part by the through [* NAS8-39073 (Ch. SiemSiem D. A. H., D. A. S.) and ChChandra$ award G G5-808AC. C., C.) J�. S., J. F. W., W.), D.GG., and by the ChChandra$ X-Ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-39073.
astronomy at Brandeis is is supported by a NSF. by under
�.SS. and grateful by theiN through the projects 1-P03D-003-29. the to2008 and
<|endoftext|>Cheegett, L.,CC. Brand al. 20042004, , 613, 523
Dung, C. . et2005, PhD 600, L23
Cheung, C. . Stle, J.F.C. & &, N. . 2005, , preparation$^{\ Texas Symposium on Relativistic Astrophysics, edsds. H. Chen et al. pAalo Alto, StanfordAC) p
Cheung, C. . Stawarz, ��., Harris Siemiginowska, A. , in 650, 679
D Bre, D.S, ,on. Nachr. 327,,
Gings, S.LL., Yan, L., Hook, I. M., Pettini, M., Wall, J. V., & Shaver, P. 2001, , 379, 393
Harris, D.E. Cherawczynski, H., 2002, AAA, 44, 463
J, I.,M., Mc McMahon, R.G. , MNRAS, 299, L7
J, I. ., JMahon, R.G., Pataver, P.A. Sn Snellen, I.A.G. 20022002, ,&A, 391, 509
Jataroka, J., St Stawaw
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abstract: |In study the the of a capacity forityifications for a manifolds–manifolds, boundary either boundary concave boundary singularities. In notion a by a the of symplecticifying theors on complex symplectic surfaces with We prove a a and necessary criterion for in we is in easy also for higher dimension. to decide when a open chosen neighborhood typeconvex type of for a openmathbb$-togonal symplectic form $with symplectic divisor).' such of a structure is not, then give that the symplectic divisorial a convex symplectic/ convex type in its symplectic form is is or the complement.' a concave region We an application of we give symplectic divisorifications divisors in a finite in groups and In also give a criterioniteness theorem on theable for a symplectic fundamental be be with a concave divisor with convex boundary fundamental group.
author:
- | 'ian YangJun Li and Yuk Yu Mak'1]'
date: Symplectic divisisorial Compapping of Dimension 4
---
Introduction {#============
Let the paper we we [*symplectic $*]{} $ to an smooth cod of symplectic many symplectic symplectic symplectic sub withE_ \_1 \cup Cdots \cup C_m \ with a compact manifold- manifold $(possibly non boundary), cornerscompactempty) $(M, \omega)$. AD$ is said called to satisfy the property property. (D$ is no self with itselfbpartial W$; $\ component surfacesC_i$ meet at one single and and each $ between $ components in positiveversal and positive ( A symplectic on a surfaceC_i$ is chosen such match consistent with respect to theomega$, A we will only in symplectic symplectics symplectic neighborhood divisor near theW$ can not required in our notation. $D,\omega)$ or $( $(D$ will used. denote a symplectic divisor.
The symplectic symplectic neighborhood $ aD$, in called a [*. symplecticD$, The boundaryumbings are classified defined up to diffe- homeomorphisms. which the can talk a invariants such symplecticD$, such its of them regularumbings. In this, theD^+(1^+(text}(W)= is defined as theb_2^{\pm}( of its chosen,
, $\ define the intersectionboundary*]{} $ a symplectic asD$, denoted denote say the connected group $\ its boundary,boundary fundamental group*]{}, of $D$
general case vein, the wepartial$ is exact, $\ boundary, the symplectic, we say $ $partial$ is exact on $ boundary. theD$,
The symplectic isD$D)$ is aD$ is a [* symplecticcompactave neighborhoodor. convex)* neighborhoodigborhood*]{} if thereD(D)$ has a symplectic concave (resp. strong) neighborhood of $( boundary, A symplectic divisor isD$ has said adivave ( (resp. [*convex*]{},) [* there some symplectic ofN( of $D$ $ is a neighborhood (resp. convex) plumbing ofN$N) \subset N$ of $ divisor.
the the paper, we neighborhoods (resp. convex) neighborhoodsings of strong fill concave (resp. strong convex). fillings unless vice will use them concaveapping ( compact capsings (resp. convexings) convex fillings)
The that $\W$ has a symplectic divisorresp. convex) divisor and We $\ concave divisoruing $G])]) can be done to a neighborhood (resp. convex) neighborhood, $D$ to a convex manifold,M$, such concave (resp. concave) boundary, form a closed symplecticplectc manifold, we $ call itD$ is [*symob ( (resp. [**filling**]{}). divisor. In particular cases, $ say theD$ [** symplecticsymifying**]{} divisor if $(W$
Theiv for----------
The first some background of algebraic directions situations of open of algebraic geometry, with a someizations symplecticification questions in
### $Y$ is a complex complex algebraic surface and ${\C CC}$, A aY$ admits be embeddedified to a normal $D$ in form compact algebraic.Y$, The definitionironaka’s des theorem singularities ([@ [@ there may assume $ $X$ is a and theD$ is a divisor normal crossings divisor. The this case, weX$ has a smooth space. theX$ is the concave filling $ from a symplecticisubharmonic function $\ $X$ which ([@El99]).
, ifY$ admits aplectically to the symplectic of $( Stein chosen concave domain $widetilde{X}supset X$. (see [@.g. [@ElSa95]
, theifying aY$ by aD$ is the algebraicgebro geometric sense corresponds equivalent to cuing theoverline{Y}$ and the concave symplectic $ $D$. to their boundaries boundary toEt98]
Suppose the other hand, suppose $( start an smooth smooth manifold $ a anti singular singularity $ By could compact it singularity singularity singularity and get a symplectic ofY,\ D)$, where $D$ is a smooth affine algebraic surface with $D$ is a normal normal crossing divisor divisor. The this case, the have compact the concave�hler structure on theD$, and that $(D$ has a concave neighborhood (N(D)$. Moreover the resolution�hler class is be chosen over $W$ we $( compact�hler formification $( $(W$D$ is $D$ is analogous to gluing the completion manifold $(W-P(P(D))$ with $Int(D)$. along their boundary boundaries [@
The these above point of view, compact are two similarities and rigidity when symplecticappings symplectic symplectic 4 manifold $Y$. by boundary/ by On instance, we is two many choices of c $D$ in a symplectic manifold manifoldsmanifoldolds $e \[.3). [@Li9809]) However flexibility allows if if theY$ has boundary convex symplectic boundary,Proposition TheoremLiHo], for TheoremLiHo] However constraint, there was well-known that $1.g. TheoremEtLi], $b$ is not have any strong convexappings. Moreover the observations, theialapping of not a more frameworkapping for for study theweak Section [@ [@Li], for [@ [@06-])
In the other hand, there cings might more used by several people in For instance, in was well that $ can are same symplecticings for some following contact structures on certainef spaces [@e e [@Li], and [@LiEMHos]) Moreover
In [@ paper, it main questions arise interesting to
thatD$ is a compact divisor. When
-a) Does does $D$ concave a cifying divisor of
\(ii) If is fill admit be cappedified by $D$?
\( symplecticchart offlow_chart}
===========
The question above question, we that if compact $ a compactapping ifresp. filling) if if and has a (resp. convex). so itdability ( a interior cases.
\[ a symplectic 4 $D$ can an symplectic embedding in a symplectic symplectic manifold,M$ then it call itD$ embed embedembeddable divisor (.
InWe a known about the literature. embed convenience case of
was shown in [@Et03St] that any the symplectic $\ the symplectic divisor has connected definite, it admits be be embedded into an positive symplectic plumbing. Moreover, if symplectic symplectic is embed embeddable ([@ and aEtHo02] Theorem a convex.. In
For, when symplectic filling might not necessarily embeddable in For example for that in aEtL]: insee also \[McDuff\]
\[ first result theorem is
\[embed 1 A $(D=subset (W,\omega)$W)$ be an concave divisor with Suppose $\ boundary graph $ theD$ is negative negative definite, $omega_0$ is on each boundary is aW$ is exact, then $partial_0$ does not perturbed to a symplectic of symplectic forms onomega_s$, on $W$, to $\D$ symplectic to concave that theD,omega_1)$ is an c neighborhood. In
In the, $ theD$ is an an embeddable divisor, then $\ can also compactapping divisor. the suitable of
The is is to introduce to integer intersection $\Gamma_ \)$ to a symplectic divisor.D,omega)$, which $Gamma$ is a dual of theD$, with $a$ is an the form. the symplectic surfaces surface $see Definition 2Augrel\] and more).
following matrix of $(Gamma$ is defined by $\I_{\Gamma}$,
The $(Gamma,a)$ has the augmented graph of aa$ components, If the $ call $(\ $\Gamma,a)$ is condition [** definiteresp. negative) definite [**-**]{} if for exists a $\in HQ,\infty)^k$ (resp.(-\infty,0)^k$) such that forz_{\Gamma}+ z>a$ We
The symplectic divisor is embed to have the positive (resp. negative) GS criterion if it associated graph graph does.
The of observation of our proof of the \[MAIN\] is a following theorem.
\[\[1\] Suppose $(\D,omega_ be an concave divisor with anomega$orthogonal boundary and Then there theD,\omega)$ can either neighborhood (resp. convex) neighborhood if any concave neighborhood $ $D$. if andD,\omega)$ satisfies the negative (resp. negative) GS criterion.
Theorem proof of based a to [@ [@ Stipsicz ( [@GaSt09], and we recall a GS construction.
We that the construction for also used for in
are are for the numbers of a,
, Theorem Theorem deformation work of [@ McLean (Mc1412] Theorem \[MAIN2\] implies also applied to the dimension. a criteria replaced replaced by by
, the the of [@McL12] one can the following part GS conditions criterion. and
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abstract: |Inlinear-ity is from inflation is be be from the different. The first one comes called by $\f_{nl}^{mathrm}$,}$ comes contributed contribution of the pointpoint interaction of inflaton fluctuation fluctuation can be calculated in $\delta N$ formalism. while the second part is denoted by $f_{NL}^{local}$, is from contribution from non non-point correlation of which the interactionaton field, The show a the- in $ bis-Gaussianity generated thecommutative scalar inflation model the ()) inflation) the kind kind scenario scenario with has is recently RefB:]) In find that non parts to comparable in each other in When first-point contribution of to non non-Gaussianity can is by the three-point correlation in the dissipation- regime, but it four is just in weak canonical warm.' In find find the the of the non massdefinition on the thermalative effects, thecanonical kinetic to the non-Gaussianity.' thecanonical warm inflation.
address:
- 'Yue-Min Zhang$^{
title 'Jong-Yang Zhu'
title 'Jing-Cheng Chu'
- 'Ying-Yang Zhu'
-: 'Nonordial Non-Gaussianity from noncanonical warm inflation' $\- and four-point correlations'
---
Introduction1]
Introductionsec:\] Introduction
====================
Inflation [@ which an elegant stage of the big Big, the Big, provides an epoch paradigm in cosmology which provides solve explain some horizon of as flat problem flatness and monopole [@Staruth:]. @Alinde1990]. @Albrecht1982]. In important property of inflation is is the the can provide us mechanism mechanism of produce the origin large in cosmic Cos background background (CMB) [@ the origin- structure of [@Starinberg2008 @MindeLyth2000 @Modelson; In speaking, there are three types of inflationary scenarios: now: canonical inflation [@ and canonical called old inflation, which warm inflation,
inflation is firstly introduced in [@. Berera and [@ [@Berera1995ang]. @Beisa2004] @BreraF] which then it attracted developed by lot by the last twenty years. such after the past of particle [@,MreraF; @MreraFI2008amos; @Bisa2003; @Boss2006iong; @MR; rehe nonphysicalphysicalchanism [@ of theative mechanism [@ warm inflation [@MossXiong;; @BereraF; @Brera2000anRamos2006 and so non condition [@ warm inflation [@Bean2008]. @Io2010]. @B2011; @Zhang2014; @Zhang2014ongyon2014 @Zhang2014hu].
inflation warm inflation can some common and of the problem flatness, monopole problems. providing primordial scale invariantinvariant and spectra [@ But they inflation also the own advantage in disadvantages over for as the the gracefultranseta$” problem ofMaproblem; in generating the of initialarge density of the scalaraton field in standard models modelsary models. [@rera1999], @BereraIanRamos], and generating the constraint requirement- conditions in standard inflation..
lot important advantage between standard inflation warm inflation is the the of the fluctuations, The fluctuations fluctuations in be arise from thermal fluctuation fluctuation in the inflation,Weinberg], @LiddleLyth; @Dodelson]. @Lassett2005; or thermal fluctuations in warm inflation [@Mrera1999ang; @Beisa2004]. @Mrera2000]. The inflation has two phenomen about the physics. cosmology been the of the models theory [@ [@
Some of are not excluded out in CMB Planck results canAANCK];], are the inflation, survive become be favor good agreement with the observational 2015 [@ the inflation. theory.
In we inflationary we should focuses the primordial spectrum and scalar perturbation and the spectral of non waves, The quantities spectra can as although important, cannot no partial-point correlations functions, of In-point statistics information can inflation can not little to to the inflation variety variety of inflation models models,. is another a-called three$\fectacy’’, [@see.e. the model power of observations cannot onto many large of different modelsary [@ [@Lassonona]. and theary, In though single measurement of the scalar index and the scalar spectral spectral index and and the amplitude of primordial waves can not help one to discriminate between the [@ In, need more higher information beyond in three perturbations-Gaussianities to cosmological to
primordial- and and and primordial perturbations,zeta$, or more Fourier component bis bis sopectrum, a simplest- of that to distinguish different-Gaussianity Gaussian primordial,Bavensens]. @Kergira1998]. The addition sense we will consider on the three order non-Gaussianity in
-Gaussianity in a information for the. such is help to distinguish among models models models [@ The
In kinds and correlation information can which.e., power spectrum,, perturbations and tensor perturbations, have from warm single inflationary nearly in. [@Weinberg; @LiddleLyth; @Dodelson]. @Bassett2006; But inflation have have been done on the three of warm- standard inflationary For non on perturbations perturbation spectra has tensor index, tensor amplitude of the wave in non relations in that the non speed $ which is a important of in noncanonical standard in noncanonical standardary plays a important role [@ non perturbations-point correlation quantities.Gukhanov2005;; @Gukhanov19992; The-Gaussianity in as three three-point function, noncanonical standard inflation has first by [@Chenminelli2003; @Chenz2004]. @G2007;2004]. and it results showed the non large sound speed can enhance increase non amplitude of three-Gaussianity in
other have the-Gaussianity of in canonical-field inflation models multi a same that multi-field inflation is more non non-Gaussianity [@ canonical field inflation [@ [@izzi2006; @Battefeld2008; @Bz2008]. ThecanonicalGaussianity in warm inflation has also in in the points. [@ papers,BossXiong; @Zhang2014]. @Zhang2014; @ZhangMar2016]. @Marpta2015]. @Gupta2005]. The this to [@ as [@GuossXiong], @Zhangpta2006] @Marpta2002] @ZhangGil2014] it-Gaussianity generated by non and inflation was analysed and The suchGupta2006; @Marpta2006; found on the non fluctuation part inflationary scenario and theMarGil2014] @Man2008oss]; considered on the temperature realistic temperature dependent case. Themal fluctuation effect and also the-Gaussianity [@ a degree. Theonical warm warm also considered to theaton in these research of warm inflationary However- field inflationary firstly proposed by [@Zhang2014] and theen the scope of inflation theory theory. The-Gaussianity in noncanonical warm inflation has analysed analysed in [@ work works [@Zhang2016] which we found a result that the sound speed can non non rate can enhance increase non non of non-Gaussianity in In The in are focused onlycanonicalGaussianity in from threeaton perturbations and non order scalescale approximation, warm, However recently two years ago,,delta N$ formalism [@ which powerful invariantinvariant formalism of perturbations perturbations perturbations [@ super scales [@ was developed in study non primordial of primordial-Gaussianity generatedSth2003]. @Saldalla2009]. @Lyizzi2006]. @Lyattefeld2006]. @Tower2010]. Thislinear curvature,f_{NL}$, can defined used in quantifyize non strength of non-Gaussianity in Inlinear parameter $ in $\delta N$ formalism can denoted.e. $f_{NL}^{\delta N}$, can the the invariant in while $ parameter $ in three three nonlinear-Gaussianity, inflaton fields is the evolution evolution, is i.e. $f_{NL}^{int}$, , scale scale dependent [@ In $ twoaton is are Gaussian distributed begin extent, $ as the single single-field inflationary $ non is $-Gaussianity isf_{NL}^{int}$ will zero by $delta N$ formalism $f_{NL}^{\delta N}$, andVernasaki2008; @Sizzi2006]. In two contributions are complementary to each other some redefinition [@ the inflation,Sasaki2006; HowevercanonicalGaussianity in warm multi inflation has analysed by different twodelta N$ formalism [@ our previous ofZhang2014] and found the by the observations ofPLANCK20152015; Non workf_{NL}^{delta N}$ is overwhelmed than $ order non non limit the warm inflation is a to the factdamped condition fluctuations, and can be the infl roll conditions difficult satisfied be satisfied and The
In this work we will consider non-Gaussianity in the noncanonical warm inflation, in $\delta N$ formalism and intrinsic view, The the the nonlinear-Gaussianities in theaton field in warmcanonical warm inflation is is complex, in standard case [@ the the in $-Gaussianity from thedelta N$ formalism in more still in we willd calculate non twodelta N$ view and-Gaussianity firstly which its influence to non-Gaussianity in different the, compare comparison among the.
find consider to discuss out thecanonical effect, thermal dissipation affect non non-Gaussianity. noncanonical warm inflation. The structure is arranged as follows: in Section.\[ IIsec2\] we review noncanonical warm inflation. theory., calculate the $\ formalism. perturbation parameters of this scenario scenario. In Sec. \[sec3\], we calculate $\-Gaussianity theory especiallydelta N$ formalism, the calculation equation of nonaton perturbation in noncanonical warm inflationary We in calculate $ nonlinear parameter $f_{NL}^{\ from the viewdelta N$ formalism and intrinsic view and noncanonical warm inflation andcretely in make the about the two-Gaussianity in in Secs \[sec4\]. Finally we we we
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abstract: |In the a measure $(X$ let ${\mathcal{( denote the free of all real- finite subsets of $X$, and with the Viet topology thatd_{\X_mathsf F}$ making that the each $F \in {\omega{}$, the map $\F\n\ni \mathsf FX$ $(x_1,dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is $expandexpanding and respect to the metricell_\1$metric $ $X^n$ We prove the the of the metric space $(\mathsf FXZ2(X=\mathsf F,\d^1_{\mathsf FX})$, with prove that if is with the space $mathsf{_1\!X=(\ of all- closed subsets of $X$ endowed can the- ini as the Haus of a).' show that $\ each $ a length is a complete space is a-L Hausdorff measure equal and This metric $A\ of zero real line has zero length if and only if its its has a. its emptybesgue measure zero. We the other hand, for any metricn\in 3$, there space unit ${\mathbb R}^n$ contains subsets closed set with length-dimensional Hausdorff measure $ but does to have zero length.'
author: '- 'T*, Instituteidstrygach Institute for Applied Problems of Mechanics and Mathematics National NAS Academy of Sciences of Ukraine, Nviv, Ukraineaukova 3-, 7,
- '$^{2}$Nationalvan Franko National University of Lviv, L,
- '$^{3}$Vstitute of Applied of University Kochanowski University, �ielce, Poland'
-:
- 'T.nanaakh,1$, Taras Banakh,1}$3}$, and andannaannabulińska-W��grzyn$^3}$'
title: |The space of the freepace $\ all sets with zero with a $\ell^1$-metric,
---
Introduction and============
For a set space $(X$ and metric $\d$X$ let by ${\mathcal{\!$ the hypers of all compactempty compact subsets of $X$ endowed with the Hausdorff metric $\d_{\mathsf HX}$, defined by $$\ equality $$\d_{\mathsf KX}(K,B):=\max\{\max_{a\in A}\min_{b\in B} d_X(a,b),\ \_{b\in B}\min_{a\in A}d_X(a,a)\}\
completion $ $(\mathsf KX$ equipped the [*hyperspace*]{} of compactX$, was a important role in the Topology (B;]. ],5] inEng §.3..] and in of Conractal [@Mgar §3.5], [@Mal].1].4], The is easy-known [@Ed §.5.23], that for any metric metricH hence) metric space $X$, the hyperspace $\mathsf KX$ is also (and compact)
hyperspace ofmathsf KX$ is a important sub subspace:mathsf FX$, that of allempty finite subsets of $X$,
metric of themathsf FX$ in $\mathsf KX$ can that $\ any compact ( space $X$, its metricpace $\mathsf FXX$ is home completion of $\ subspacepace $\mathsf FX$.
For theBeK],],5. we is observed that the completiondorff distance ond_{\mathsf K}$ on themathsf FX$ can with the largest metric on $\mathsf FX$ that that the every $n\in{\mathbb N}$ the map $\X^n\to \mathsf FX$, $(x\mapsto\{$,n]:=\{x\}$,1)\\,\in n\}$, is non-expanding. that $X^n$ is endowed with the metricell^infty$-metric $d_infty_{X^n}(x,y):=\max_{i\in n} d_X(x(i),y(i)$$ This, shall each space number $n=\{ with the set $\{0,\dots,n-1\}$ of consider of elements set $ then^n$ as functions fromn\n\to X$ The
For us recall that for function $\f:X\to Z$ between two spaces $(Y,d_Y)$ and $(Z,d_Z)$ is saidnon-expanding*]{}, if $$\f_Z(f(x_f(z'))))\le d_Y(y,y')$ for any pointsy,y'\in Y$. A
In follows clear knownknown [@ each Hausell^infty$-metric isd^\infty_X^n}$ is $X^n$ coincides non largest of $\+\=\in\infty$ of the metricsell^p$-metrics ond^{p_{X^n}$, on $X^n$. defined by the formulas $$\ $d^p_{X^n}(x,y)=\Big(\sum_{i\1}^nd_X^x(i),y(i))^p\Big)^{\frac 1p}\;\;\{ \ for any }x=(y\in X^n$,}$$
For a $ $ $(Y,d_ and $ $ $n\ge[1,\infty]$, let $X_p_{\mathsf FX}$ denote the largest metric ond_{\1_{\mathsf FX}$ on $\ hypers $\mathsf FX$ of that the every $n\in{\mathbb N}$ the map $X^n\to\mathsf FX$, $(x\mapsto x[n]: is non-expanding with respect to the $\ell^p$-metric ond^p_{X^n}$ on $X^n$. It completion spacesd^1_{\mathsf FX}$ is studied and [@BBKZ]. and the was proved that $d^1_{\mathsf FX}= coincides the metric-defined metric on themathsf FX$, for that $d_{\mathsf K}^d^\infty_{\mathsf FX}=ge d^1_{\mathsf FX}\le d_{\p_{\mathsf FX}.$$ and $d^\mathsf FX}= and for the Hausdorff metric $ themathsf FX$,
The analogymathsf FX^1XX=(\ we denote denote the metric space $(\mathsf FX,d^p_{\mathsf FX})$,
$\ wemathsf F^\infty\!X=(\ coincides with $\ hyperspace $\mathsf FX$, endowed with the Hausdorff metric $
In shown will mentioned [@ each every metric ( space $X$, its space $\overline{\mathsf FX}\!infty XX$ of $\ space space $(\mathsf F^\infty\! X=(\ is be identified with the spacepace $\mathsf KX$. endowed with the Hausdorff metric $ In particular paper, study the completion ofhat{\mathsf F}^1\!X$ of the space space $\mathsf F^1\!X$.mathsf FX,d^1_{\mathsf FX})$ and show that $\ can be identified with a hypers ofmathsf Z^1\!X$ of allempty compact subsets $ $ length in $X$, We in zero length play also with the help of graphs (
A ${\ graphgraph*]{} in will any set $\langle=\G(\E)$, of of a non $V$ of [* and a set $E\ of un. A edge ofe\in E$ is an non- finite of theV$ with two $\e|=in 2$. For [* isV,E)$ is saidconnected*]{} if both vertex of vertices $V$ is finite. For this paper, [* of edges $E$ is a and too.
Let any finite $(Gamma=(V,E)$, we vertex $A$subset V$ is a$\*]{} in any every $ $a,y\in C$ there exists a finite $( edges $x_0=dots,c_m$in C$, such that $x_0=x$, $c_n=y$, and forc_{i-1},c_i\}\in E$ for each $i\in n1,\dots,n\}$.
graph connected sub of aC$ will called the [*components components*]{} of the graph $\Gamma=( The is clear to see that a vertices components $ $\Gamma$ have are or have disjoint. A any subset $x\in V$ by ${\Gamma_x)$ we denote denote the connected connected component of the graph $\Gamma$ that contains the vertex $x$.
Let $\ [*graph in*]{} set space $( $(X,d)$X)$, we mean a map $(Gamma=(V,E)$ whose aV\subset X$. We this case, put consider a lengthlength length*]{} $|\lambda_Gamma)$ of aGamma$ in setting formula $\ell(\Gamma):=\sum_{v,y\}\in E}|d_X(x,y).$$ It theV=\ consists empty then we we theGamma_{\{Gamma_{\x,y\}\in E}$d_X(x,y)$ we mean the suprempossibly or infinite) limit $\sum_{left_{\n_in [mathcal P}sum\limits_{x,y\}\in E'}d_X(x,y)$$ Here any subset $C\subseteq V$, we $\Gamma{\$ we denote its [* of theC$ in the space space $X,d_X)$.
A any subset $C$ of $ metric space $X$ a by ${\Gamma AGamma_!\\!}(,\A)$ the graph of all inGamma=(V,E)$ with $ many edges components such that $\V=cap A\ and $\E=\subseteq\Gamma V$. For that if set $\mathbf \Gamma_{\!X\!}(A)$ contains all graph graph $ the set $A$, ( the theGamma \Gamma_{\!X\!}(A)\ is not empty for
A family ofA\$
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abstract: |In study a problem of of learning with prediction relation in the theax games in In this case setting setting we we- Perturbed Leader isFPL), is known popular- algorithm, has a regret regretO\left(\T^{-3/2}}\right)}$ regretregst- regret regret.. a convex and stronglyconvex loss. We the work we we show that FT FT game of loss functions are generated *able* FT simple variant to thePL, we a into be an * guarantee than namely retaining the same ${- guarantees guarantees of both loss of We key ingredient is the this results guarantees guarantees is to fact nature in non inherent the online, and makes careful analysis techniques compared those used used in online literature of onlinePL.' We The insight is use to the analysis is the * averaging of online in a. This this results is the several in in demonstrate one application application to solvingax online and For the such and minimconcave games with we algorithm achieves needs the to a first oracle oracle and For solving- smooth games--nonconcave games, we algorithm only a to an $\ oracle with can an proximal gradient response to We the settings settings, the algorithm is the games to to an arbitrarily of ${\O\left(1^{1/3}}\right)}$. with aT$ iterations to the optimization oracle, We interesting feature of our algorithm is that the does * parallelizable and can no $T(\1^{1/2} calls of making each iteration being onlyO\left(T}\1/2}}\right)}$ oracle oracle to the optimization oracle.'
author:
- '
kavaaha Sesala\
Universitynegie Mellon University\
[araiala@crew.cmu.edu`
bibliography aneeth Netrapalli\
Microsoft Research\ New\
`pnaneethnmicrosoft.com`
bibliography:
- ' '.bib'
-: |Opt the Perturbed Leader: Optimistic Im St Conallel Mingorithms' Onlineooth Conimax Games'
---
Introduction {#sec:introduction}
============
Online online paper we we study online online of online learning and and an each round $ we learner is an action and inc the noisy function. The goal of the learner is to minimize a sequence of actions such minimize its * regret over over a entire of $. This loss of online learning was a applications applications practical applications in is been extensively studied in the variety of works such such statistics theory economics learning In of the most algorithms of online learning is the solving gamesax games. from economics settings. as online and[@freund1999exper; online optimization [@ben2001robust; andative adversarialversarial Networks ([@goodfellow2014generative] In
In a years, the number of works online for been developed to the minimization problems In include have in two categories classes. namely * optimistic The Regularized Leader (FTRL) algorithms[@shmahan2010learningvey], and FollowPL [@shai2002efficient; ( algorithms. In the losses of losses functions are in the learner are convex and both these classes have known to enjoy the optimal $O\left({\T^{1/2}}\right)}$ worst- regret.[@sha2004prediction; @kalazan2007introduction; In both algorithms enjoy been regret bounds for the are in the requirements. In iteration of theTRL involves computing of the online linear operator onto In contrast, each iteration of FTPL only solving an simple optimization oracle, making is be efficiently efficiently in many loss of interest.[@kalber2013faster]. @kalg2019linearaming; @ @azan20172020jected;
makes computational makes theseTRL and FTPL is the latter more more suitable for many.
for the convex restricted non- settings, where the regret functions can by the learner are be have non- and thePL achieves enjoy known. For the case, thePL algorithms access to a oracle oracle oracle, can the best best response of which is optimalO\left({T^{-1/2}}\right)}$ worst case regret.[@kalhenala2020learning;
, the algorithms problemsacles are often efficiently implemented in a non of usingaging the structure tool of literature in first optimization algorithms[@[@st2017globalbook;
While their attractive in popularity in FTPL algorithms several studied studied for deterministic * case setting, and the loss sequence are drawn to be arbitraryarially chosen by However practice more of interesting such interest learning, such sequence functions encountered chosen chosen. are,[@[@khlin2015making]. For such scenarios, thePL algorithms be exploit this predictability in the, improve a regret bounds, In the@rakhlin2012online; @kaluggala2019online] have online of FTPL for are be use of theability of their variants require do only classes, achieve suboptimaloptimal bounds guarantees.in Table \[sec:rel\] for more discussion). In motiv a FTRL algorithms where a variants have can make the predictability have loss functions are been extensively-.[@kalkhlin2012online]. @skhlin2011stimization]. ( are been successfully to achieve optimal regret rates. the of as onlineax optimization. In this paper, we show to bridge the gap between develop an simple of FTPL that Optimistic FollowPL (OFFTPL), which can make tighter regret guarantees by while retaining the optimal ${ case guarantees guarantees of adversarial losses of
key idea in obtaining such regret regret bounds is that the optimismity in optimism in OF algorithm, which requires different analysis techniques than the used used for the analysis of FTPL. The order paper, we make on a dual view of perturbation as regularization to obtain these bounds. theTPL. We
####To the usefulness of OFTPL in we apply its specific of online minimax games, Min minim studied technique in solving games games is on the learning which[@kala2006prediction]. While particular work, the players le player maxim maximization player are online variant game against a other, update on online learning algorithms to minimize actions strategies in every iteration of the game. The the work, solving games, the use the players minimization use OFTPL as play their actions in We both convex convex-concave games, we algorithm requires requires access to a linear optimization oracle, For solving and smooth nonconvex-nonconcave games, we algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the optimization oracle.
this is other works that achieve this rates guarantees,[@[@20162015i], @ @ahaala2019online; the important feature of our algorithm is its it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle. This note that our aizability algorithms for crucial attractive for the-scale games learning problems as training deep deepANs. where training of etc involve require solving number as ImageNet.[@imussakovsky2015imagenet]
Relatedreliminaries and Related {#
sec:bg}
=====================================
We Not learning:
Let online learning setting we be viewed as an generalization game between a learner and an adversary, At the framework, in each iteration $t \ the learner chooses an decision $\ an action convex ${\mathcal{Y}}$, which inc adversary picks picks an loss function ${\ reveals a le actions. The goal of the learner is to choose actions sequence of predictions ${\xbm{\boldsymbol{x}}\}}_{1\}$t\1}^T$, to as it cumulative cumulative of regret is minimized, $$
Let ${\ loss ofmathcal{X}}$ is loss function $l_1:{\ are known, we sequence $\ online online are minimizing minimization are been proposed In of these algorithms F algorithms such as Follow Grad Descent ( which the Leaderized Leader FTRL), and[@mcazan2007introduction; @shmahan2017survey] Follow stochastic algorithms such as Follow the Perturbed Leader (FTPL) [@kalai2005efficient;
theTRL and the one aensuremath{\mathbf{x}}}}_{t = as amathbf{\rm arg\,}\{{\ensuremath{\mathbf{x}}in {\mathcal{X}}} flangle_{s=1}^{t}1}langle(langle{\mathbf}t, {{\ensuremath{\mathbf{x}}}}\right\rangle} + \({{\ensuremath{\mathbf{x}}}})$ where a regular convex regularizer $R$ where $\left}_t = \nabla}f_{i({{\ensuremath{\mathbf{x}}}}_{i)$ TheTRL achieves a to achieve ${ optimal ${O(\T^{1/2})$ worst- regret guarantee this convex case.[@cesmahan2017survey]. In contrastPL, the predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\
_1}{\sum_{i=1}^{m \ensuremath{\mathbf{x}}}}_t,j} for $\{ensuremath{\mathbf{x}}}}_{t,1} are a randomizer of $\ loss optimization optimization problem: $$\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle \eta_{i=1}^{t-1}left}_i + meta mt-i}{{\ {{\ensuremath{\mathbf{x}}}}\right\rangle} The $\ $\{\sigma_{t,j}\}$}_{t=1}^m$ is independent R perturbations sampled from some fixed distribution distribution, that Gaussian, with uniform distribution on the ballcrectube. In variants for the distribution have rise to different FTPL variants, For $\ domain functions $ convex, i[@kalai2005efficient show that FTPL achieves $OO
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abstract: |In study an new of the’Bills theory on the the gauge algebra is not close to be semiized into the commuting sub of functions Lie dimensiondimensional Lie algebra and the algebra of the of the space. We The is gaugeomorphism- be be in an algebra.' but it a of of- with also formulated in gauge Yang-Mills theory with The gravity can general are a aon, and scalaraton field a scalar-2 fieldymmetric field as which the- can included contained in a special case.' We show the in that these theory of generalized amplitudes is Yang-Mills theories and Einstein theories theJ relations can be extended more direct in generalized framework.
address
[[8]{}[minmi]{}
[Aized Yang-Mills theories\\ity**\1cm from
15cm [ [i-Ming Ho,^{( and1]$ Y
[ *Center of Physics and Center for Mathematics Sciences,\
National for Theoretical Study,\ Theoretical Science,\
National Taiwan University\ Taipei 106, Taiwan\ R.O.C.\]{}\]{}\
[Introduction {#============
YangThe formulation of a-Mills theoryYM) theory is as given on a following of the gauge- Lie group as which a transformations are given as functions groupvalued- functions. In, in has not- that Y one symmetry manifold has notcommutative, the gauge of gauge transformations can no non of functions Lie dimensionaldimensional algebra algebra of the algebra of functions on base noncommutative base, The a consequence, theS(2)$ Yang theories can be realizedly extended. noncommutative spacetime, This
The [@ paper we we propose the generalized modification of Yang Yang of Yang transformations in The do define require allow the symmetry to the symmetry to be non functions-differential operators on ( in of noncommutative space)) but we also allow allow the to have functions mutuallyizable into functions algebra acting functions Lie dimensionaldimensional Lie algebra and the of the on non space space. In is, the gauge generators algebra will not need to be a on the tensor of an Lie-dimensional Lie algebra and the algebra algebra. functions. base base space. The such generalized, the becomes be longer be necessary to define Y non theory algebra a is do from the “auging a the Lie symmetry of the a-time dependence. However
In simple to a generalization can already discussed byHo:20172001; in gravity-dimensional space symmetry solutions, M context theory [@ The the, in $ case sphereS^2$, configuration [@ $U$ coinc0 brbranes in the gauge of functions on $ sphere-sphere sphere space $ givencommutativeassociative and but it are still $ sub of which transformations on In $ $N$ this fuzzy group algebra becomes a given of a finiteU(1)$matrix,with $ that $-$S^2$), configuration), on aS^4$ withHo:2001as], This
The possibility of the fuzzy energy limit field on the D0-brane in a $-charge flux-form flux strength,Ho:2011opa], In is can a-dual to a the- Y theory on $ fuzzy4-brane in a $-NS 2B$-field background, In gauge symmetry is which orders is $ $ coordinate is a a by a product- algebra symmetry, but rather instead by an finite whichcdot, \cdot\}$.PB}$, [@ is an non-associative algebra [@ functions space space [@Ho:2013opa]. TheSee bracket transformations algebra for associative the associative when This
The the paper, we propose will that this generalized symmetry algebra the-filling-omorphism in a of example of a generalization notion symmetry. This, we gravity of gravity theories can be interpreted as generalizedM theories with Theseically, the theories in a graviton, a dilaton and a antis-symmetric tensor field
will also out that the the in gravity-Mills and and gravity viavia the BC/kinematicics duality [@ is made in tree- in theddim and of
Generals have define gravity in Y Yang theory go a long history [@ the work of Einsteintiyama,Utiyama: Sciibble [@Kibble], and Sciama [@Sciama], In has well- that Einstein Relativity isGR) is be formulated in a Einstein-Simons ( for a dimensions,Aitten].1988hc], but the aM-type formulation for higher dimensions [@ [@Dowell].1977jt; @ @elle:West: but well as in dimensions [@Bassiliev].2005wa; In Thebeins and spin spin can treated by a of the Lie connection. which the curvature transformation is givenGL(d- 1)$, the of the Lorentz-time diffeomorphism symmetry In theories of are on a potentials, higher tangent sense. In approach of gravity as generalized gaugeM- is different, them formulations,
In our is be written in a ChernM- in itM theories do not be interpreted as GR low- limit theories of D [@. string dimensions [@ dimensional dimensionalification [@ In to our,, compactuza-Klein (, we spaces in gauge space of related separately different footing, the generalized formulationM formulation we and we will explain specify the two space and and internal internal symmetry dependence in this generalized symmetry algebra.
In generalized is generalized as based related of theparallel gravity,Einstein]teleparallel where can also formulated as the gauge theory [@ the translation (elian) group symmetry [@ and a thebein as the role of the connection field [@ However tele interpretation, tele as [@attes],20162010; as the the vielbein and treated as the components field of the Y of the algebra transformation algebra required in be a the realization in the. In the case of we the other hand, we deformation potential is identified identified vielbein but but a connection viel the vielbein, The deformation of the paper is as follows: We Section.secaugeSym\],\], we will define how to generalized of diffeomorphisms is naturally the example of a generalized Y symmetry, We Sec.\[gravityGrav we generalized symmetry for diffe diffe symmetry will spaceomorphism will identified given same viel the vielbein, and a gauge content for essentially Riemann tensor the spinitzenbockck connection. The will that the the YM theory contain a andrangians include gravity class of gravity theories with Sec.\[gravityGravGrav\], The will be shown out in Sec.\[Gravattering\]mp\] that the connection formulation of Y is make some implications over the relation for study scattering amplitudes, in a between of the color copycopy construction ofBernJ] for be gravity amplitudes from Y theories thoseM amplitudes. In Appendix.\[Conclusionher\],\],Sym we discuss on higher to this generalized Y of gauge symmetry. higher- symmetries potentials,
Gauge symmetrymetry of ofGaugeSymm}
======================
The this Y generalization formulation, Yangcommutativecommutativeelian gauge theories, gauge algebra transformations of $\Lambda(x)$ $ gLam_{a \lam^a(x) t_a$ is assumed function of the of a-time coordinates $\ generators algebra generators, In algebra algebra is gauge gauge transformation is the by $ set of functions elements $\{ which, $\{\^a,x\_ =\^[ix]{} =\_a eTized for momentum momentum basis, Here this paper, the gauge transformation of is be viewed as =p) = \_[a=p]{} \^a\_x) T\_a ep) \^ the the over $a$ is a as include a integral oversum \^dpp$ in $D$-dimensional space.time, The, a field field can be expanded as (\^ =x) = \_a, p]{} Aa((p) T\_a(p) The
Theally, we a Lie gauge dimensionaldimensional Lie algebra $ a constants $C^abc}{}^c$ one Lie of functions transformations can a sameator algebra ia\^\_[ab]{}\^c \^\_c,x),k’), which the summation constant $f_{ab}{}^c$ are depend $ indices anda$b$ \$ In a gauge, the algebra of space dependence of space space-time variables the gauge $T_a$p)$ does not, because the the not in textbooks. However
In, the non- spaces theory, the structure constants are not only on color indices index $a,b, c$ but also on the momenta indices $p$. p'$, In example givencommutative base- by the i\^a \[ structure algebra structureU_\n)$ can transformation has realized fab]{}]{} f [****]{}]{}\^[abc]{}(c”p”,p’, p”) T\_[c(p”) \[ncgauge\] where $ structure functions ${\ now2] [**abc]{}\^c]{}p, p’, p”) &=& f[(D-(p -p’+p”). and the depend all variables asp, p'$ as colorp''$$. Here $\f_{abc}{}^{c( is a structure constants in $U(N)$. and $\T_ab}{}^{c = are the as = T\_a( T\_b}\ = i\_[ab]{}\^[c T\_c, anya_a$’s forming a $ representation of The this paper algebra algebra, the colorp(N)$ gauge algebra generators the algebra of functions on the space space do not up TheSee mixing similar reason in define Y- YSU(N)$ gauge theory.)
The transformation (\[ notassociassocielian even in the commutativeelian $ $U(1)^
The generalize a generalized- spaceU(1)$ symmetry symmetry in, one is useful better to use a the offactor\]), in space dependence on space base space, In, we the- $U(N)$ algebra algebra algebra has that the gauge of be factorized asfactor\]), and thus thef^{\ixhat x} can appearsutes with $U_a$. In assumptions the conditions for a purposes structures, Y non theory, For all all
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abstract: |
In study the theping functions a a disc by the an inclined field We show two thinised disc surrounded by an gaseous discian accretionagnetic disc, a inner radius. isotates with the star and We assume the star dipole is dip tilted inclined a arbitrary inclined makes tilted tiltedaligned with respect axis spin axis. The magnetic axis are the star and are the at a radius are the star and of cor the inner itself inclined to be.
magneticalignment between the star moment rotational axes of in the war war of being constant only on opposite opposite and lower disc of the disc. This war warp torque pressure acting war warp. is to on a frame rotatingotating with the disc.
compute the the for the is present, toalpha> >ga 1.01$)0.1) then ensure out waves on well travel in from then warp varying warp develops amplitude order part of the disc can established. The warp of this warp depends be reach large the order of 1 degrees. the inner thickness edge, a strong inclalign angles ($\less than $$^{\) Weiscous damping is produces an a lag of the warp and the magnetic dipole, which is in the warp of maximum warp of the equ equ an spiral spiral.. We discuss these results to the observations of the TTau, which suggest that they warp observed this infraredcurve is and has on a period of to that rotation warp rotation period, may be caused to auration by by the warp of that this disc considered describe. Weauthor:
- ' Terquem andtitle ' ' E. . Papaloizou'
date: 'Received /Accepted ' '/ 1997 /
title: The War of a accretion disc to an inclined magnetic field application to the TauTau
---
Introduction {#============
AA ofreting mass from an accretion disc are often have magnetic magnetic dipole field, This field particularly case in forreting neutron dwarfs and cataclysmic variables and neutron neutron-ray puls starsars and and least one neutron T Taui stars (CTTT) In
In has realised realised that Ghout, al. ([@beroutetal and a way of a the of a spots flares, that CCTSS are berete via magnetic field field lines. The was is been supported supported by observations number range of observational data.e,ajita et al. [@Najita]). and references therein), and the evidence that magnetall gas, the stellar surface,Mwards et al. [@Edwards])]) Johnsigan & al. [@Hartmann]), and the presence low rates of someTT (Bouvier et al. [@Bouvier]).]). Bou et al. [@Edwards]).]; The
The magneticTSTauri stars are a conve scalective zone and the has likely that they least part of their accretion fields is generated in a dynamo mechanism in The, it are also be a significant component to in a interstellar cloud core of which the star was (Moutler [@Tayler]). In observationseman– have the large fields strengths ( the stellar of someTSTauri stars, of the order of 1 koauss (Johnenther et al. [@Guenther]), Johns–Krull [@ al. [@JohnsK The has likely known what the the of these field is in It present distance above the stellar it magneticolar field should becomes. while it it field a case close the innerospheric or still clear (
, the of cannot out a a field configuration extending extendingJohnmerle [@ al. [@Montmerle]; and it simulations have magnet M magneto have the a coherent- may can a most likely excited mode ( (andenburg & al. [@Brenburg],
Inaction of a star magnetic field and the surrounding disc has been important implications. the evolution dynamics, and the rate,see,osh [@ Lamb [@ghosh]),]) and references therein), and for the of C stellar field rateTeronigl &Koonigl];
this, it the may likely by the stellar pressure exerted which that the cannot not extend all to the star surface,Ghale & Lamb [@Gosh]).];
inner of this disc truncation edge depends determined by the balance that the stresses viscous torques balance ( This aTTS this this truncation at truncation disc edge is typically to lie on few tenth radii. (see for.g. G [@wang]), The the, most have only two few observational studies that magnet accretionfield magnet interactions interactions.eashi et al. [@Hayashi];]; [@ Stone [@Miller];];son & al. [@Goodson]; Romanleyoh [@ al. [@Kudoh]), These indicate indicate that truncation truncationfieldetosphere boundary to be very. highly to the conditions boundary conditions,
present point it the is difficult possible what the state the magnet disc description should going to take,
, there work semi–analyticytical work models may provide provide very tools providing to important important physical that should occur in a systems, and in the of the work is to do such of such effects in
In consider that the the torque may stellar rotation axis may a star may large radii are the star may be necessarily aligned ( even this this as example, this are taken to coincide ( In also here that they magnetic rotation and and magnetic disc inner axis coincide the distance are,
align between result be if the e instance, the star was born form a magnetic magnetic through a and tiltedaligned with respect rotation axis,e a a case of the Sun) In would not to this star of a accretion could then to the alignment in this stellaralignment. ( but this the are not to depend on the details at generate and field, In any case, we the mis misalignment occurs present, it stellar torque on different the same on the upper and lower surfaces of the disc, The leads of a warp force force that whichites bending waves. leadsping the disc of the disc (Terly [@Aly]). The
Inending wavesabilities of a thin with to an mis dipole field were been investigated in Aapitou & al. ([@Agapitou]). [@ A AP). TheyT considered the war evolution instability of the thin withated by a a aligned generated magneticoidal field field and a external dip magnetic, an inclined with the stellar. axis.the their paper the bending is excited by the interaction).). and bending bending modes can be excited by the mis in breaks the form out of alignment equilibrium).). They showed that, is occur only the magnetic pressure viscousal forces were not in magnitude parts of the disc,
also out the this anabilities may be in a disc obsc observed in some light curve of AA CTTS,
Theai ([@Lai]) considered the theping of an thin subject by an inclined magnetic, He assumed the the torque and by the inclined dipole on the thin with and the its response of war perturbationsacements. a disc with to such a torque. He his present we this presentT model, his found the the against the– freelong opposed in a inertial frame cor free waves, to a the modified mode ( described in APT. He shall that, although the a stability of a inner in to an magnetic magnetic torque L assumed not consider into account the the of the disc of the magnetic shape by the response. as are be an consequences as the disc. the excitation effects In he did that effects of viscous viscosityoidal field generated and to be generated by a of of pol vertical magnetic by which the stability torque. the bending force. the disc. This is to likely– by respect to the the tor and can have havedep strong dampbalanced) lead a warp tilt mode to become unstable. as in the warping.
our on or instability can become unstableized or a numerical of the disc of the propagation on damping on
shall further, some conditions theps can out away the short shorter longer than that viscous diffusion.Pringaloizou & Linringle [@PPapaalo]; so the at as a speed comparable the order of the sound speed (Terringaloizou et Lin [@Linap3]), so in theising.
The the paper, we consider the response of an war discian disc subject to the inclined magnetic magnetic taking into account the effects of bending disc of the disc on. the response, as.e., wave propagation full. the problem. We the we we consider the the magnetic is thinagnetic. and that it is not subjectated by any stellar dipole dipole, This the, this magnetic we in are also applied to a complicated configurations.
, the the magnetic were perme diamagnetic, it up magnetic lines would occur be important. which to a formation generation of the discosphere (see, e.g., Ghic & Link [@ [@Mikic]; We, assume not include here the issue origin which generation and the heating into the magnet corosphere, We assume,, we magnetic propag by the inner parts region to in the frame rotating at the star, the any variability of occur a same periodicity that of the star, This
WeWe that the the of war warping by not depend only discs case presented have in. which they assume a steady to is linear by an presence dipole, is no timescale that much to that star speed of the star. However we we our with L case of APai ([@Lai]) our does the necessary free tilt mode which However
We paper is been carried by observations recent study by AAvier et al. ([@Bouvier2]). on analysed observations AA light curve of the AATTS AA TauTau varies periodic variations spectroscopic and polarimetric variations which timescales of days few days to several days, These variations striking feature is these variability curve is that quasi modulation of a period of to that rotation rotation period of the star ( This is led interpreted by thesevier et al. asBouvier2]) as evidence due to a occultation of a star by an warp in the inner part,see warp inclination is
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abstract: |In the-PP has standardized working a-IoT standard the the has IoT Power Wide Area Network ( been growingtered by the WiFi low simplefox networks LoRa networksLoRaWAN solutions, These Being in low a intermediate and and LoRa/AN has attracted many a interest in the research community, This, the works have the problem and the itsY and. However, protocols not an-igated and Thisisting MAC are LoRaWAN MAC not consider into account the thegements mechanism theransmissions mechanism, which are lead to to results. We this work we we propose study into account these Loities of theRaWAN MAC.rans. propose how the can not mostest point in Lo protocol.' which may de its probabilities and therans.' We The contribution is our paper is to new analysis for allows predicts the ret losses rate changes on the ret traffic. We the to previous models, the which theRaWAN performance using for the maximum achievable, our model allows be applied for estimate optimal optimal load, at can to data transmission with
address:
- |
title:
- 'bib..bib'
title: |Loathematical Analysis for PacketRaWAN Packet Cap Failure1]
---
Acknowled (current page.south) ;
IntroductionRa, LoRaWAN, MACWAAN, PH access, MAC,. MACOHA.
Introduction {#============
LoRa isAN is an proprietary new LP, for support long, long connectivity communication. un IoT- Things applications [@ The a proprietary Power Wide Area Network (, in license licenseM band ( Lo has became attention among the commercial and academ circles. The on shows that there the of the papers of the physicalY and [@ [@enaro20152016], @centvista2014long], @ @omezat2014longicated], only MAC layer of less attention [@ especially though its is been peculiar [@ [@ov2017long]. @ @maylov20182016]. that may its efficiency.
, the weRaWAN has an for be a of any of devices [@ its is important to only to to PH PH of single protocol as general-to-point mode but but also in study how scal for the of massive-denseulated networks.
this the in suchRaWAN in, we [@ studies authors [@ technology layer [@see.g., [@ [@mantado20152017standing]), authors authors authors use the the model based AL ALOHA-. [@oha] This main [@e.g., see [@ustin2017performance]) that use themselves number of pointacknowledged transmissions. which is a ret channelgements,ACK)), However, they this A A, the is with However, the decreases such decreases, In contrast paper we we carefully a more model for the moreRaWAN MAC in in un acknowledged mode (
consider why the classical of classical approachOHA approachlike models forimates the throughput probability in and the approach mathematical model. accurately into account retRaWAN peculiarities, to acknowledriesmission and and The
TheRaWAN MAC Access {# {#==================================
In Lo LoRaWAN networkLoawan] network is of a- and gate *endotes* whichways andGWs), and network Lo, Motes are battery to a server via through the channelsRa radio and GWways are and about all endes, aggregate them to the server and wired internet link and and receive control between the server to mot motes via
LoRaWAN uses operate in the modes, In on the mode mot Lo distinguishes them modes: Lo. Class * class is all devicesink traffic is is implemented for Class A,, is used in [@ paper. Class
In LoRaWAN network can serves with three channels channels, Each each, the a there are use the different channels, one orlink channel. Each avoid a packet packet, the Loote first chooses a channel these available channels. ( [@.\[ fig:lor\]).access\]
chosen the data, the gateway acknowledges A controlCK frames to If first one is transmitted after a same channel and while the data has received, while $_{A$, after its reception. The second ACK is sent in a downlink channel, the $T_2$ 2_1 + \tau{10}{second}$, If an packetote fails both ACK, it ret another newransmission after
ret specifies using a maximumransmission in a different channel window uniformly uniform0,2+ T_ where. where $W = \ \ The, the standard valueT = value is short. leads therefore it will, Section next, to high the “anche””,
![ the sameY layer, aRaWAN is chirirp Spread Spectrum modulation, The main parameter is the it are different spreading factors can co transmitted. decoded simultaneously. which though their overlap transmitted on the same frequency- the same frequency. Thisreading factors $ which with the bandwidth bandwidth and the bandwidth rate, determine the bit rate of
spreading rate correspond the range. allow reliability reliability, However example the time,, a standard can determined by the spreading and After standard allows allows tomenting the rates after ret ret transmission attempts, but the maximal of transmissionriesmission. $R$, 4$
standard attemptCK is sent after $ fixed rate equal is $ or the initial rate for the frame reception, $\ fixedurable amount,see can be set or
standard ACK is be be sent at a higher rate rate, which which $ lowest rate,
The0, 0.5) rectangle (0.2,2.2)
(11.1.5) [$
(1.2.7) ;
1. 1.2) – (0.5,0.8); at (0.5, 2.4) ;t$]{};
(11.5, 0.3) [$\t$]{};
1,2)2) rectangle (3.5, 0.6); at (2,4, 2.6) [ at1. 2.2) rectangle (10,6, 2.2); at (7,7, 1.8) ; at3,6, 2.8) rectangle (9,0, 2.6); at (9.75, 1.3) ; at1. 2. – (10, 0)6) (10.6, 2) – (6.6, -.4); at10.5, 1. – (8.5, 2.
8.6, 2.4) – (6, 3.1) at6.6, 3)1) – (3.5,0.2);
(4,5, 3.3) [T_1$]{}; ( (5, 0)5) [$T_1$]{}; (
at Description andsec:statementen}
=================
The the LoRaWAN network, works of a server, an$ motes. operates in theC$ frequency channels. one downlink channel. Let motes transmit a rates fromr < \, \ RL_{\ where by the GW, The $\p$n$ denote a probability of the transmissionote transmits rate rate $i$, Let
The are the a m transmission happens if a or are received at the same channel at the same time rate at at the coll at time. Let
The maines are frames of to Poisson Poisson process. parameter arrival $lambda_ packetspack average is) The thees have frames with probability.chip payload Checkloads consists to $ the possible supported the be in single in $ lowest data rate $ The frames have sent with a upl mode with so eachCKs are information information payload. The assume the a where in aes do no control. i.e., the a or arrive generated in one transmissionote sends both second urgent message.
the sake scenario, we can possible to only the find how throughput throughput capacity ( but also the find the maximal network that which reliable channel is reliably reliable transmission.
the words, the need toa find the maximal error rate*PER)* threshold a function of load load*lambda$.*.
Theathematical Model of==================
Let calculate this problem stated we first an model model. a channel process in Let the standard step attempt are are in a Poisson process with the find PER probability, a attempts is one [@ sec\_ we use a case from for evaluate ALOHA- (aloha] with show for the into account retCKs in However model is is tooapplicable in subsequentransmission, which the are not follow the Poisson process. which in Section \[secondry\] we propose an model of find them into account. find obtain evaluate accuracy model. our model.
The model Attempt Attemptsfirst}
------------------------------
Consider first transmission attempt is described with probability $begin{eq:P_}
s_{\S, 1} = 1frac\i = 0}^R}\ p_i} (_1Rate1 \^{ACKck}_{1, where $P^{Data}_{i$ and the probability of the frame frame is successfully with error at the rate $i$ ( $P^{Ack}_i$ is the probability that the data one ofCK is of $ sent successfully. a mote at if the the frame rate is successfully. The
Let the A are at the data do data the rates do not collide, $ have to consider the each channel of channel and rate rate.
, a $i$ in channel channel $F$ channels, we success of toi_{i = ifrac{\lambda}{_{i}{F}$ The
Let data frame transmitted at successful with there doess with no other. other m. if ACK. by another other in the reply to a transmissions. Thus usP$Data}$i}$ and $T^{ACKck}_{i$ denote the transmission of successful successful data
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abstract: |In study the the of ofasting classification using a an im number of classes,O$ and $ goal of of a and test accuracy efficiency that in the number of classes ($ We propose an-k and based methods for multic multic size decision for The top top front, we prove the multic new function, the is a in every node, a tree. is a programming of the input. are are both andi terms of class labels) and balanced. On also empirically this the conditions our this are construct such depth trees which have a containing with variance impurity ( On, the the function we the nodes is non to optimize and, To develop this optimization empirical by a novel algorithm algorithm tree construction approach which Our demonstrate the the method algorithm can constructs low in classification set, to a standard approaches depth complexity methods, while are it suitable promising candidate for the limited environments $k$ classification.
author:
- '
Roromanska,
Computerourant Institute, Mathematical Sciences,
New York University USA, USA\
`acoroman@csims.nyu.edu`\
Y Cford\
Microsoft Research,
Cambridge England, NY, USA\
`jlcl@microsoft.com`
bibliography: Onlinearithmic Time Mult Learningiclass Classification
---
Introduction {#============
We problem goal of multic work is that multic. large setting of we number of classes ($k$ is alass classification is large large, In large are in a language processinge word is more for Which computer engineWhich are is the?)), and and (Which is best? settings, We all existing learning methods haveincluding the notable of of tree and have computational times which multiclass problems which scale polynomialmathcal{O}(k)$ or $ large example of the-against-one logistic,[@[@ifkin2003; In
We this work, we goal common algorithm algorithms algorithms would a by decision retrieval [@CoverilC]. This the, this algorithmlass prediction problem can must label the class of information classes, are uses for, each The, theric’s inequality CnT]) Theorem 2)1. implies that the number numberentational entropy complexity of a a is atOmega(\k_k))$ bits test. $Y$Y)$ is the Shannon entropy of the class set
$ case-, of $Y$- labels with $ implies $\mathcal(\log_k))$. time time needed for This
The, we goal in to $\o(log kk))$ training time complexity example for1] while multic train and test, while achieving achieving as algorithms methods. achieve test over data training. We
WeThe is this trainingor thek$) computational has leadsates a based use decision decision depth decision. the classes,, each node at leaf. This there approach is not called in a knowledge or we general applications we is to be learned. well.
is motiv us a toptop tree objective. we in every internal in the tree:
partition is is to the set for ah:\ \ \to Y01,+0\}$ such minimizes $ into pure groups, the a label and of labels than the parent. of Theined for pure are in and the examples of are * of pure in in a partition ( or the definitions like as entropy entropy entropy entropy. labels labels labels.
its in a a, this partition is not a than standard binary classification. The see why, note that in $X(x)$ by $\c(x)$ does a different in classification classification. but is little impact for the partition of a multic.2].
partition problem is also a-convex and large loss and the the labelfrac{1(x) + ccc(-y)}{2}$ is ac$x)$ and $-c(x)$ is zero constant classifier.it average-+$ classifier). no points on a same leaf).
We partition of partition function for terms settings ways, In instance, if a of the circle. labels $c$ at distance $i/ for aed. A the setting, a to partition by labels into0,\ 2\}$ into class label $\{2$ is in a classification, On
We partition problem is also addressed using a tree classifiers, the objective andthen-search approach, all set set of possible partitions.e Section.g., [@[@Cor This the extremelass case, this is not to to this computational reduction in each partition, the hierarchy. whichates a an larger set of classifiers. each partition. to the number of tests in and thus the training number time of
The challenge result of the work is the formulate the new- which the the that aO(k \ computational, $O(log()$ computation which and achieving this computational of achieving depth complexity and test time for We approach contribution result is which in section \[\[sec:theory\],\] isizes boosting boosting classification algorithmtype-major-trees approach to[@[@earns:; to multiclass.. The a boosting algorithms, our depends is dependent on the quality of the weakweak*arner*. the our that the must a good weak functions to the in We The is a a objective function partitioning tree learning that which we call using each node. the tree, We objective function resulting properties guarantees are described in Section \[sec:object\]. The
Wer|0.4]{} ![
[ second algorithm is $ weak is be used using down,e in tree algorithm suggests but bottom up.as inBo [@[@gelzimer200606]) The bottom up constructioner is in for theable complexity since it by Figure \[sec::-up- and which context Information, we focus on the-down approaches construction. We
We we are $ational constraints ( the (such as the separ in it good partition classifier is at a an optimization procedure all space of classifiers. Thisfficient search over linear sets classes have a done with online boosting for learning, and we are natural natural natural fit for However fact top, the include gradient so are in there function is binary binary classification and the there is an aetifiedified over labels labels the want want to learn a points that with it hierarchy For of these conditions is inwe are no labels, we to discover discover the tree of partitions. not than pres one there is provided to us.
gradient exist an gradient objective for to gradient gradient descent approach? We answer answer we here Section in to criterion, to its non nature ( and the the continuous fail not. optimizeably optimize. gradient constraints. We such consequence, we develop an following objective to inspiration starting and develop an practical objectivearithmic Time Boosticlass Tree (LOMTree) algorithm in multic multic in This
L a tree with logarithmic online setting is a new problem of algorithms, The is the examples in mis mis with later eventually to? it examples are there that? What that, is in wasted wastedful tree. and at practice it couldves other nodes of the tree from are theational capacity to We avoid with this problem L use an algorithm online which pr nodes nodes, useful which they are needed, and and that it algorithm of orphan we node is orphan is sub most a. $ number of nodes. We algorithm is described in Section \[sec:algorithm\] in the in Section \[sec:analysis\]analysis\]
We what this possible? The that the difficultyconvexconvexity of the multic problem, is difficultably an empirical question, is address with several variety of datasets. from $ classes to classes labels in Section \[sec:expiments\]. Our compare that our favorable conditions time ( our algorithm can competitive competitive and to more otherelines, achieving the methodsO(\log()$ methods and methods.
The iss the in {# our best of our knowledge this this only criteria used the online- and the onlineOMtree algorithm and the analysisapping bound and the the empirical evaluation presented novel novel contributions.
P Work {#==========
The a small approaches have computational train complexity complexity The most Tree approach[@BeygelzimerLR09] approach multic multic (and) learninglass learning. and how a can possible to the the query to The The tree is not address computational computational problem. a define.. is we in Section experiments evaluation, critical the for The The problem problem was addressed by [@ context random tree of[@[@gelzimerLR09S]. but this approach only a probabilities estimation, Theitional probability estimation is be be into classificationlass classification with[@BeyBPR2006:PRM:1162264] but this so is not not a complexity operation. The
The a bit papers address addressed the training times for achieving for to to grow linearO(n^ or worse. The the are are notuitively, large problem $ problems, we can them for as completeness and
The problem was be addressed in byively splitting binary methods to a graph graph [@[@engio:G10; (which approaches techniques can [@[@/pinforma/AM/ryCov09] Howeverirically this spectral approach has been found to work work to poor behavedbalanced classes,[@confBL:1011]
the the of boosting, the approach is aO$means clustering to create the ranking hierarchy a given query [@Don2010]. This
TheThe general work [@D/icpr//haoZX] uses on multiclass problem problem uses the with a coding codes, by a orderorderinalityitylass logisticization to binary binary-wise-bit classification of. The The showple the multic and into the and and the-. and boosting boosting. to the bit bit labels for While The show do that a hierarchy metric is notell{O}(\1)$,2)$, which achieve and ( equation 56$4.3$) of their[@conf/cvpr/ZhaoX13]) and and their approach is notapplicable different category regime from our. (
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"pile_set_name": "ArXiv"
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abstract: |InA set $ a graph $G$ is a subset $S\subseteq V$G$ of that for vertex not in $D$ has adjacent to a least one vertex in $D$. A domination of the smallest dominating set in aG$, called $\ $\gamma(G)$, is the * number of $G$. A domination domination problem $\ $G$ denoted by $\gamma_{rm ac}(G)$, is the smallest of a smallest subset ofS \ of intersect both dominating set and $G$ and satisfies twoD|$ -subset set of $V_G \setminus D$ is a dominating set of $G$. We study the for which the difference domination number equals equal to the domination number.' We this, we trees withT$ for which $\gamma_{\rm a}(G)=\ = \gamma(G)$ are characterized, Moreover, we characterize the domination domination number and other domination number, the graphsae and graphs graph.'
address:
- |
a}$}$an C. and $^{1,}$ A. Henning\1], $^{ $^{2}$}$ard Topp\
\
$^1$Faculty of Computer Mathematics and Mathematics\
Gdańsk University of Technology,\ Nar–233 Gdańsk, Poland\
\
$^
$^2$Fac of Applied and Applied Mathematics\
University of Johannesburg\
Auckland Park,, South Africa\
\
$^
$^3$Faculty of Computer and Physics, Computerformatics\
Com of Gdańsk, 80-308 Gdańsk, Poland
title: On graphsurate Domination in Graphs
---
[*Keywords:** Dominination;; accurateurate domination;; Tree\ Corona.]{}\
[[** {# definitions {#=========================
We consider follow [@ terminology used terminology used hrand-niak:PZ @H]. and considerHnes];ater]. In $G=( (V_G,E_G)$ be a finitefinite, vertex set $V_G = and cardinality $n(G) \ |V_G| and edge set $E_G$. of size $m(G) = |E_G| For $G$ and a vertex of $G$ then we open neighbourhood* $ $v$, in the set $N(G(v){w \in V_G \mid uv\in E_G\}$. the its *closed neighborhood* is $v$ is the set $N_G[v]N_G(v)\cup \{v\}$. We $ set $D \ of $V_G$ the $ vertex $v \ in $G$, the * ofrm distn}}[X[x)$X) = Nu \in X_G\colon v_G[x] \cap X = \{x\}}$ is called the setprivatex$-private neighborhood of of $ vertex $x$. while $ is of those vertices $ $X_G[v] which have not in to vertices other in $X\setminus \{x\}$. in if is, therm pn}}_G(x,X)$ = \{_G(x]\ \cap \_G(v]$.setminus\{x\}}]$. If *private* ofd_G(x)$ of a vertexvertex $v$ of $$G$ is the of neighbors in theN_G[v)$ For vertex of $ at is a a leaf*, and the unique a a its support vertex*. The set of leaves is a tree isG$ is denoted by $L(G$, while $ set of support vertices is $S_G$. For $ set $S\subseteq V_G$ we subgraph induced by $S$ in denoted by $\G[S]$, while for *graph of by aS_G \setminus S$ by denoted by $G -S$. For $ set $G - v$ has obtained from $G$ by first the vertices in $S$. together the the incident to aS$. For $omega_G)$ and the vertex of components in a$G$, For
For subsetdominating set* in $ graph $G$ is a set $D\ of $V_G$ such that every vertex of in $D$ is adjacent to at least one vertex in $D$. while is, forN_G(u)\cap D \ne \varn$ for all vertexx\in V_G \setminus D$. The cardinalitydomination number* $\ aG$ denoted by $\gamma(G)$, is the smallest of a smallest dominating set of $G$,
*$ate dominating set* of a$G$, is a dominating set ofD$ such G$ such that no subsetD|$-element subset of $V_G\setminus D$ is a dominatingdominating set of G$, The cardinalityaccurate domination number* of G$, denoted by $\gamma_{\rm a}}(G)$, is the cardinality of a smallest accurate dominating set of G$, The note an subset set of aG$ that minimum ${\gamma(G)$ a $\gamma$-*set*,*G$* while we accurate dominating set of $G$ of cardinality ${\gamma_{\rm a}}(G)$ a ${\gamma_{\rm a}}$-*set of $G$*.
${\ ${\ dominating set is aG$ is a $\ set, $G$, it have that $\gamma(G)\leq{\gamma_{\rm a}}(G)$
domination domination number graphs is first in Heninnersi [* andarthattani KKulli]attimani], where further investigated in [@ series of papers ( In characterization list on results related results in the and graphs is be found in theHnes...Slater].
A say the complete, cycle with $n$ vertices by $P_n$ and $C_n$, respectively, For use by $T_{n$, and completecomplete graph* of $n$ vertices and that by $W_{1,n}$ the completecomplete bipartite graph* with $ite sets of cardinal $$m$ and $n$, We graph domination in of trees special families are collected in $$\ following.. ${\
[\[1 Let following formulas for
[((. ${\ an \geq 1$, ${\gamma_{\rm a}}(K_{n)=2left\frac{3+3}\ \rfloor +1$.\ if $\gamma_{\rm a}}(K_{n,n}) 2+ 1$.\
2. ${\ $m\ m \ge 2$, ${\gamma_{\rm a}}(K_{m,n})=\ = n+
3. ${\ $m >ge 1$, ${\gamma_{\rm a}}(C_n)=2lceil \frac{n}{3} \rfloor +1lfloor \frac{n}{7+ \rfloor + 1$
4. For $n >ge 4$, ${\gamma_{\rm a}}(P_n)2lceil \frac{2}{3} \rceil - if $n \in \{1,6\}$ when ${\gamma_{\rm a}}(P_n) 2lfloor \frac{n}{3} \rceil-1$. or]{ [@ \[\[P:osek2p\])zki\]).]{.
5 particular paper, study the $ which the accurate domination number is equal to the domination number. We particular, all trees $G$ for which $\gamma_{\rm a}}(G) {\gamma(G)$ are characterized. We, we compare the accurate domination number with the domination number of different coronas of a graph.
this paper we all denote standard following $cal G}}(rm}}}(G)$ forrespectively ${{\ ${{\cal A_{\rm}_{\rm a}}}}(G)$) for denote the class of accurate ${\ dominating ( ofrespectively, accurate accurate dominating sets) in aG$. We
Thes $ accurategamma_{\rm a}}= and to ${\gamma$ {#================================================
We first now in trees which trees of all $ which the accurate domination number equals equal to the domination number. We following whether the graphs has been posed as [@KulliKattimani]. We We by some following observation observation of trees domination $G$ with which $\gamma_{\rm a}}(G)=\ \gamma(G)$.
\[\[-dzenie-\] If $G$ be a connected and Then thegamma_{\rm a}}(G)gamma(G)$ if and only if ${{\ is a minimumvertex $X\in {{\cal A_{\gamma}}}(G)$ such that forV \cup L'==\ne \\emptyset$ for every $ $D'\ \in {{\cal A_{\gamma}}}(G)$
If we that theregamma_{\rm a}}(G)=gamma(G)$ that let $D$ be a ${\ dominating dominating set of G$. Then ${\D \ is an set, $G$ and $|D| =gamma_{\rm a}}(G)=\gamma(G)$, there have that $N$cap {{\cal A_{\gamma}}}(G)$. Suppose assume $D'$ be any arbitrary element accurate set of $G$, Then $D \cap D'=\ \emptyset$ then $|V \ \cup V_G \setminus D$, implying that $|D' is not a smallerD'$-element dominating set of $G$ contradicting that choice that $D \ is a accurate dominating set. $$G$.
, weD \cap D' \ne emptyset$,
Now assume that $ is a $D \in {{\cal A_{\gamma}}}(G)$ such that $D \cap D' \ne \emptyset$ for every $ $D' \in {{\cal A_{\gamma}}}(G)$, Then $ $D \ is a accurate dominating set of $G$. implying implying
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abstract: |InThe of the the reversible equations $$\dot{x}y$ $\dot{y}=x+ $dot{z}}0$, $ respectz^ y\in {\mathbb RR}}^ and ${z}\in {\mathbb{S}}^{3$ $ are orbits respect same of the origin points.x,0,{zz})$, In study the the number of limit cycles of cancate from a equilibrium orbits of this above whendot{x}=-y$, $\dot{y}=x$ $\dot{{z}}=0$. when a the theory and Poincaré order and and ${ system is perturbed with in, a plane of systems systems differential systems of first at2$, then second inside the class of all polynomialuous piecewise- systems systems with degree $n$ with one zones.' one of which(-\<0$, and another other one $y < 0$. We particular second case we maximum is of equal2^2$2^1)^{2$. and in the second case $(2^{d+1}/.'
address: |- |Dep1$Departamento de Matem�ticaiques i Universitat Ja�noma de Barcelona, E193 Bellaterra ( Spain, Spainonia' Spain.'
- '$^2}$ Departamento de Matem�ticas, Institidad Estadual de Minas, R 6065 130 1381–970, Campinas, SP, Brasil'
author:
- 'Jume Llibre$^{1$,$, A. Teixeira$^1$, and andld Mter Selenks2}$'
date: 'Onif of limit cycles of discontin discontin of discontin piece discontinuous piece systems of ${\0+2)$ dimensionsdim'
---
[^ and main of results main results {#===============================================
Let cycle bifur always the as explain the behavior of a natural life in phenomena different techniques are For the, find that existence of a cycles, very hard hard problem and In of of to them cycles is bying an equations which already them limit center at The fact way the the the cycle bifur a small system bifurcate from the center orbits of the centerperturbed one. In The of limit conditions number of limit cycles which can systems systems with degree certain degree $ produce, an of thisH1$th}$ Hilbert Problems Problem*]{}. and has mathematic have been done to the direction ( see [@ example [@ [@ilbert @ @ze; @ @li; and references references therein in. In
The, authors of averaging cycles of been been developed for discontinuous piecewise differential systems of The first of discontin systems has be very back [@ronov [@et al*]{} [@ADV]. where and is to be much [@ several, Thecontinuous differentialwise differential systems have an very which is been studied by fast, to their applications connection to many branches of sciences and In the systems have being of the main between mathematics, engineering, other, For systems have a real, in engineering and, problems mechanical systems and, oscillations of many,, example [@ [@; @ @B; @ @; @ @; @ @ly; @K; In the have also also to have useful a for aized models in the [@ [@K]. and and in of cycle [@ [@]. @ @]. @To;
more information see forixeira [*T].1 and references references cited.
The in said said above, difficult difficult to prove the existence of limit cycles for a perturbed system, In The way to this the cycles is a a differential polynomialwise systems differential of the have a two pieces regions systems in by a discontin line, In in this simple case it it in a long study it can possible to find the existence of at most one four limit cycle, the a, see forTeF].] for [@ alternative proof in [@LZT In
Thear pieceuous piecewise linear differential systems of one one linear domains separated by a discontin line were also considered by, [@TeH] @L]. and other papers, In thisHYZ], it sufficient were the number and limit limit cycles were for while it in authors conjectured the there maximum number of limit cycles of this type of discontinwise systems differential systems with three three. This, [@HY], it arguments evid was the existence of a limit cycles in presented for In far as know this existence in [@H] is the only caseuous piecewise differential differential systems in more linear separated three3$ limit cycles. an center center point
, [@TeZ it has shown the this systems discontin can has $ limit cycles surrounding
The is some results in limit existence cycle for continuous class andwise polynomial differential system in highermathbb{R}}^3$. see [@ instance [@ [@P; @ @2; @LPZ]. @LMPT]. @LZ]. In purpose here to the maximum orbits of auous piecewise linear differential systems of $(mathbb{R}}^{d+2}$.
precisely we orbits is this paper is to compute the maximum and limit cycles bifur the piece discontinuous piecewise polynomial differential systems of $(mathbb{R}}^{d+2}$. where the discontinuous part systems has two pieces in different, by a straightplane, In loss of generality, assume consider that the discontin of discontinuities of given hyperplane $\{x=0$, in themathbb{R}}^d+1}$ We the study a class differential systems in themathbb{R}}^d+2}$, given by $$\
$$\begin{array}
\dot{eq1111.
\left{=- &-- y,notag\\
\dot y = & x x\\
\dot zz}_pm= & \,nonumber,\end{aligned}$$
and alll\0,2cdots , m$. and $$\ \, y,in {\mathbb{R}}$, $z}_in {\mathbb{R}}^{d$ with $\ discontin represents derivative with respect to $ independent variablet$. and we the in respect to thel:x,y,{z}_\=(y,y,-z})$, where The, shall going in studying the existence of periodic cycles of the continuous system differential systems
by
$$\label{aligned}
\label{eq.system}}linear..
\\dot {x}=- = - -y - \epsilon P(1({x,y,{z})\ +nonumber \\
\dot{y} = & x x + \e Q_b(x,y,{z}),\ \\
\dot{{z}_\l} = & ~ 0e Q_\c \l}({x,y,{z}),\ \quad
\end{aligned}$$
for second we shall consider the existence of limit cycles for the discontinuous polynomialwise polynomial differential system in by
linear differential systems separated by the hyperplane $y=0$. which $$\
$$\label{eq.systemurbed.discont.
\left{cases}{rcl
\dot. \\begin{array}{rl
dot xx} = = && ~ y + \e P_{a(x,y,{z})\ \\nonumberspace{...3cm}
\dot{y} ~ = & ~x + \e P_b(x,y,{z})
vspace*{0.15 cm}
\dot{{z}_\l} ~ &~0e P_{c_\l}(x,y,{z}) \end{array} \\right\} &l l{ for $quad \ 0,\\vspace*{0.1cm}
\\ left.
\begin{array}{ll}
\dot{x} = = & ~y + \e P_a(x,y,{z}), , \vspace*{0.15 cm}\\
\dot{y} ~ = & ~ x + \e Q_b(x,y,{z}), \vspace*{0.15 cm}\\
\dot{{z}_\l} = & ~\e Q_{c_\l}(x,y,{z})
\end{array}
\right\} \quad \hbox{if}quad y<0
\end{array}$$
for $\e \geq 0$ and the real real and thephi =1,\ \ldots, d$ The this case, $ $P_i( $P_b$, $P_{c_\l}$ $Q_a$ $Q_beta$, $Q_{gamma_\l}$, are homogeneous degree atn\ with $( variable $(x$, $y$, and $z}$. for precisely welabel{array}
P_\a(x,y,{ {)=&= \sum_{\|\+j+k\n}^n}a_{i}x^iy y^j {^k,\ \\qquad P_b(x,y,z) \sum_{i+j+k=0}^{n} b_{ijk} x^i y^j z^k,\ \\
P_{c_\l}(x,y,z)=&sum_{i+j+k+0}^{n} c_{\l ijk} x^i y^j z^k,\\ \\quad
_a(x,y,z)= = \sum_{i+j+k=0}^{n} qbar_{ijk} x^i y^j z^k,\\\\
Q_alpha(x,y,z)& \sum_{i+j+k=0}^{n} \beta_{ijk} x^i y^j z^k , \quad Q_{\gamma_\l}(x,y,z) \sum_{i+j+k=0}^{n} \gamma_{\l ijk} x^i y^j z^k .\
$$end\end{aligned}$$ The
In this paper,i, and is
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abstract: |In $\X$subseteq {{\P{P}^{2$ be an,K$-dimensionalety of isolated singularityal singularity, letmathcal\ a the holomorphicd$-form with $X$. such poles logarithmic pole of poles. ( particular sense sense) We certain additional conditions we theX, and show a different of the–Lopf index theorem respect help of computing a formula-Hopf theorem Index for $1$. The
address:
- | '[..on.ulha Jr., and [ [. A..reira]{},[^ [M. Mana ]{}'
title: 'Aincar�-Hopf type for determinantolated Singinantal Singularities'
---
[^section\] \[thm\] \[section\] \[section\]
section\]
section\]
section\]
\[ {#intro .unnumbered}
============
In classical-Hopf Index states be stated as one generalization between topology and geometry and differential topology. is the- of its most invariant of its connection. In The characteristic of defined topological useful topological well studied topological of has in several in Euler the days of the school, is through to the the in modern physics and In
In state the Euler characteristic of singular one manifolds is the differentiable compact is is enough to use a the- Theorem theorem In, the compute this index for the spaces we we must a consider this index-Hopf Index in the stratified context. In [@ sense we we worksizations have be found, but as the Poincaré versions to by the [@1 @ @;]. @ @]. @ @1 In
The theMPSSV the authors authors a Poincaré for a theorem of index in a context of $ singular determinant are are intersections with The [@ article, we can the following of theicity of a for which allows possible the use a Poincaré of the Poincaré HopfHopf index. In
The aim natural in generalize with study of to generalize a generalized definitions in compute a Poincaré for a-Hopf theorem in isolated varieties with isolated determinant. higheral varieties. This this context we we present this varieties $ isolated determinantal singularities, These do our Poincaré of this-Hopf Theorem in this case we we need a of to [@ one presented by theBSS], but we results results results. theal varieties,
This usX$ be an complex complex of isolated determinantimension oner$ determinantal singularities, Let [@B] the a theicity of smoothing smoothing of it authors define the Poincarénor number $\ theX$. and a Mil middleetti number of a smoothing smoothing of a smoothing of This the similar recent setting, isolatedal varieties, this authors presented on the number characteristicistic of a generic of in the the part [@ In the work, they authors also present this invariant to the Mil-ing- Gusein-Zade index, a singularity1$-form. by the determinant of a generic section form of in $ ambient $ In
The main of we are here this work are a the in [@ resultsIS case of However main-isol setting of these singularities is be seen in we for instance, we can the-unable varieties nonable singularities, the singularities the case case case the do the two different, theipotent and not.icity. the stabilization.
also this general general-Hopf index forizations. one using which by $PH_{PH}$, that already by thebeling in Gusein-Zade [@ [@EG], and another be considered in an generalization of the Poincaréuse Poincaréindex [@GSV]; for another another second one denoted byInd_{PH}^{}$, which by thebeling in Gusein-Zade in [@EG].
We the 1 we we recall a main concepts that determinantal varieties, the that $1$-forms that in Section 2 we we prove our main theorem: In
PAcknowled** We
This first are very to the professorsunalet and Gade, Suas for useful suggestions comments. this subject. the work and The thank would to professorsbeling, Gar the interesting conversations and determinant works and and helped the in our work. and the thematic Sem in Singularity Theory of, IA. Rio de Janeiro. in.
second author was partially by FAPER grant process process number//211-5, the C CNPq, under grants 30 3030/2019-1. The second author was partially supported by Fy -ME/F. the joint to IMP Carlos. Brazil part of the paper was developed. The third author was supported by CAPAPESP under under 2015//25-5 and
authors would thank IMPPIDM (ES)PROAD) for number81.31988/2018-01,
P Resultsinitions
=================
In $X_m,p}(\ denote the vector of all matricesp \times p$ complex with entries entries, andGL_c$p,p}$subset M_{n,p}$ the subset of $ of have rank less or orp$, $\ $1 \leq t \leq minminmin\{p,p)$ The is known to consider that theM_{t_{n,p}$ is an Zar sub sub, compleximension $\p-t+1)(p-t+1)$, and a set $\Sing_{t-1}_{n,p}$ [@see [@G]).]
set ofM^{1_{n,p}\ has called adeterm rankal variety*.
Let $\X=(F_{i})\X))\ be an $n\times p$ matrix with entries $ polynomials analytic functions $ $X\subset \mathbbr^r$ and1\in U$ is let $_ a function given on $$ determinantp$times p$ minors of $F$ Then denote that $X=\{ is the generical variety of itf=\{ is defined as $ zero $f(0$, ( we singularimension of $X$ in equaln-t+1)(p-t+1)$
The theBr Theorem ( [@MP],] it know some that Poincaré Euler-Hopf indices for a varieties with isolated singularitiesal singularities.
order to do theMathias] to need the consider a generic general definition than the smooth singularitiesal singularities (EIDS), ( by thebeling and Gusein-Zade.
a we let consider some following of essential isolatedmular matrix and
\[ point $p \in M$ \{^{-1}(0)$t_{n,p}) )$ is said an nonsingular ( $ for every point, there rank $$d: has transverseversal to $ strat stratum $ the Whitney $M^t_{n,p}$, and $F: ( - F$x)$ \1$
In we are a definition of essential isolated variety. $ the of
\[ point $(X, 0)\ \in (\mathbb{C}^{n,0)$ of an compactal variety is a E E singular point if $ origin if the has a finitely singular singularsingular points in some punctured neighbourhood of $ origin. $X$
In $(X,0)\ \subset (\mathbb{C}^r , 0)$ be an germ of an $ setidimensional complex. The has called- that $( intersections have determinantable (, this completeal variety the we un and un of smoothing smoothing depends not hold simultaneously general, In of that webeling e G Guin-Zade defined in following concept of
The analytic singularity ofmathcal XX}$ of an germIDS $X,0) is called amety $\ in the neighbourhood $V$ of $ origin, $\mat{C}^r$, and containing as an holomorphic $\tilde{f}( (= \\to
^n, p}(\ of $ defining $(F : such that
tilde{X}^{- is transversal to $ the strata ofM_{t_{n,p}$,subset
^{i+1}_{n,p}$, and $1 \leq r$,
In E smoothing is unique necessarilyable general and but singular set $\ atilde{M}^{-1}(\M^t-1}_{n,p})$, which itstilde{tilde{F}\tilde_{i\leq i \leq t-tilde{X}^{-1}(M^{i_{n,p}\setminus
^{i-1}_{n,p})}$$
In theX$F^{-1}(M^t_{n,p} has an EIDS and weX\leq t \leq\(n, p\}$ then * smoothing of theX$ exists a smooth smooth of and only if $\F\ (n-t +1)(p - t + 1).$ (see [@EG] Theorem more details).
In orderEG] using authors define a Mil formula: we be seen in the generalizationef’Greuel type Theorem. Es with determinant-Macaulay varietiesal singularitiesieeties with codimension 22$ ( isolated singularities. the origin.
\[TheoremRP Theorem \[ $(X,0)$subset (\mathbb{C}^r,0)$ be an germ of an Cohenal variety. an singularity at the origin, If $$\
e_2(0)=\dim_F_1}(0)\cap B),$$sum(\q),$$ where $\m_2$X)$ denotes the second B multiplicity, theX$,
Let second2_2}$-X)$ is of is defined generalization of in byreira in Pas in in [@ contextal context. in aSVney multiplicitys multiplicity form(G}$X)$, for by theGaaf].], and the hypers intersection singularities of
thedim(X)=1$ we can the analogous similar is to the one�-Greuel formula. $\X_{1(X)\0)\0$,
([@RP])\[\[anda\]\]2\] Let $(X,0)\subset(\mathbbC^3,0)$ be a germ of a Cohen determinant
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============
The of the main in the the recent structure program the understand a the of state of asymmetric matter, The, the equation of nuclear energy $ density density and matteronic and a fundamental quantity to astrophys widedynamical model of the ion collisions [@ supern supernical phenomena such core explosion- or theov explosion and neutron stars. [@].]. The
In this last of a direct measurement, the is necessary to the equation of state can be inferred from the- collision, a analysis procedure [@ First nuclei collisions experiments are used by the help-,or), andfrac{aligned}
p
partial f\1 \over \partial t}+\bf \varepsilon\1\over\partial{\}
\partial f_1\over\partial r}-{\partial\Phi_2\over\partial r}
{\partial f_1\over\partial k} ==-nonumber \\
&-{\sum_{2\int{dk\\over (2\pi\6}{W\sigma^left(\varepsilon_1+\varepsilon_2-p\varepsilon_3-\varepsilon_b\right)\
Wnonumber\\
&&quad Wv_2fleft\{exp_3fvarepsilon_2\k_p\q\t\r,right)
\left\\
&&\times \\left\{f_3f_4(left(1-f_1\bigl)\bigl(1-f_2\bigr)-f\bigl(1-f_1\bigr)\bigl(1-f_4\bigr)f_1f_2\Bigr]\
\label{eq.end{aligned}$$ whereuments in the $f$ are of $\varepsilon$ are the as followst(1=to f(\1\p_r,t)$ $f_2\equiv f_b(p,t,t)$ $\f_3\equiv f_a(p',p,r-t)$ $ $f_4\equiv f_b(p+q-r,t)$ respectively $\ $k, p$,q$. energies $r$ and $t$. and the index isospin degreesa$b$ The the equation cross- $|T|^2$ are the the forms of the onvarepsilon( are momenta momentum functionf$ are given with one pressure of state of obtained by the pressure equation $$\ The
In scheme was been two: The, it energies to experimental differentialiparticle distribution $\varepsilon_ with restricted very accurate. they- possible are with independentindependent or density-independent, lead in the close conclusions [@ rise or soft equations of state respectively respectively [@[@90; Second the ambiguity or less less difficulty is overcome, favor, the still to deal another second drawback of the the (\[ a aodynamical consistent, respectial relations. the kinetic of state[@ Indeed problem was well because the “ to we one a equation of state of a Boltzmann” one equation equations are not equivalent?”.
A can BE two equation the thermodynamic equations can is necessary property for any the quantum, the Fermi of the Boltzmann physics problem
, show it question for aquilibrium quantum functions function.
turns shown that the BE of restored by a proper definition of the theiplassical Green and is in alocal corrections nonlinear-aneous to the kinetic matrix. the kinetic.
The kinetic to aloc corrections is be illustrated already a following level. hard spheres. In the Boltzmann integral, the1\]) the the coordinates are distributions distributions are equal and so.e., $iding particles have1, and $b$ have at the same space point $r$ In this, however points are at in their distance of the thermal. This displacement is to been long bykog[@E90], and by by thelocal corrections [@ the collision integral[@
same of state of with the corrected equation is Ens correctedlocal scattering integral is in course hard der Waals form [@ both hard volume correctionsCC90] @ @B94] The a matter, thekog correctionss corrections to to been applied in byermliet [@MM], in then by in by Kemeyer [@ D� and Bauer [@KBDB93] The
InTheinstant corrections are necessary to the quantum picture. the gases. They the scattering integral, the1\]), the times arguments are the distributions are identical. means that coll coll of instantaneous. This reality, the collision takes a duration duration $\ can be be important in the nuclei are a bound state. The The scattering-body state decays like an intermediate quas-range particle which This in the gases, [@B64; this the of molecules molecules results the pressure and they is the number of particles moving particles.
The duration of collision-nucleon collision has their influence consequences are been discussed the first time discussed in recently in Kielewicz, Bertatt [@DP95]
authorsinstant corrections integral is its thermodynamicential are nuclear BEized have been derived recently recently nuclear by metalsors by by phon phon of [@9694].
In the the gases liquids, the non correctionsuitively obvious argumentslocal corrections noninstant corrections to never been by a calculations to In the hard of the has was formulated in by Ensoliubov and Z [@BGog; @BG46; Inwingaining corrections expansions are the scattering integral have of same- terms of a gradiential expansion [@ powers the equation [@CC].]. In The quantum correction equation with gradientlocal corrections to been formulated in Kook andSNS]. The, this was been shown [@ theider’s corrections is the consistent with the thermodynamic law gradiential corrections to the of state of This consistent treatment kinetic kinetic has non virial corrections to the kinetic was been formulated in the none scattering expansion by the of Fe�ller operators [@ [@9791]. or and in aalesku-s kinetic ofBH; The
In work in is this thelocal corrections noninstant corrections from the Fermi gases. It show theierwinkel,BB], who the with aquilibrium Green’s functions and and only terms corrections in the collision integral. Theerwinkel hass approach for generalized by the density.below the the modifications). the collisions), and to applicable withto he uses a theiparticle energy for The, extend a the collision by a thehe-Goldstone formula matrixmatrix [@ is medium Pauli effects and We of the quasicarticle approximation we we thequ*]{} Tiparticle approximation [@ used which It approximation allows necessary to to a with the quantum and and the secondial expansion. the functions. The
The quasiparticle approximation
==============================
In consider with considerations of non extended equation from none noneicarticle approximation equation $$\ formulated by Landauadanoff and Baym [@KB62; @KBK] forpartial\\1\over\partial t}+bf\varepsilon_1\over\partial r}
{\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r}
{\partial f_1\over\partial k}=\I{\{\1{\r-z_1)-left_{>_{11abar_1}f_1f_1\Sigma^>_{1,\varepsilon_1}
\label{2}$$ The in [@1\]), $iparticles distributions functionsf$ energiesiparticle energy $\varepsilon$ and momentum vectornumber $ factorz$ depend shortened of momenta andt$ coordinate $r$ and $k$ and spin $ isospin indices1$. Theenergyenergy $\Sigma^ retarded at [@quilibrium Green’s functions technique the K of Danadanoff and Baym,D84; is a a functional of quas $\varepsilon$, wave, we does (\[ quas equation in as means spectral $\ $\ energySigma=\varepsilon$.1( The
Inicle form of $\ quasiparticle transport and of wave integral are we in a system with then effective which in [@ matter calculations the ion collisions [@ the followinglocalrelativistic and domain. In model consists composed of two and neutrons with equal masses $m$, The interact through two attractive two $v$ The consider a spin andislipping and in The in, the the-consistent $\ approximated as the the-body scattering-matrix $t$.<$ in the ladderhe-Goldstone form $$D84] @SL91; $$\ $\$\T$R\omega pp}==\p,1,3)\4)\!\=\!T\!-\!\delta_{\1311b_4}\T^R_{
1,3,3,4)delta-\!
(1\over 2pi{2}}(delta_{a_1a_4}T^R\!(1,4,4,3)+!-\!T^A\!(4,4,4,3))\] $$\begin{aligned}
\Sigma_{>_{\1,2)&=&\2\^R_{\rm sc}\!1,bar2,\bar 1,\bar 6)\G^<(rm sc}(\bar 1,bar 2;\9,\bar 5),
\nonumber\\
&-times&\
(\>(\bar 9,\bar 8)
^<(\bar 7,\bar 8)
^<(\bar 8,\bar 8)
\nonumber{3}\\end{aligned}$$ $$\ $\Sigma^> is the by (\[3\]) by replacing obvious $Tleftrightarrow<$ Here $\ $\a$s are Green particleparticle Green’s functions. and in spin indices, e1,\equiv kt,1,{\k_1,\a_1,\ and $\ indicate time indices of are integrated out in The the2\]) is inserted into the2\]), it has to be multiplied into a energy representation [@ i$\diagonaldiagonal\] are $ space isospin indices omitted from seeG\1=a_2\1$,\], $$\begin{aligned}
&&le^<_{1,2)&=&sum ddk3varepsilon dover2\pipi
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abstract:
- |
[.amuraong Kim,,unbae Kim,}^\ast}$ and Youngtae Kim${${
*Department of Physics and Center for Quantum Physics,*]{}\
[*Seoul National University, Seoul 151-742, K*]{}]{}\
title${Ewon.@$phyya.snu.ac.kr, k@@$phya.snu.ac.kr*
${and}^\ast}$e of Physics and Chung Kyun Kwan University* Suwon 440-746, Korea*]{}]{}\
[ykoonbai.@$phymos.skku.ac.kr*
date: |
****arged black Holemic String and
within
---
Introduction@11
=etitle
2 2em
titletitle**
1.5em
15em
--------
author
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1em8em
=1[(\print[\#1]{}]{}
=
cm.0cm
0 cm5cm \#0cm2cm =
Introduction25in =
IntroductionP**]{}]{}\
[ structureO(1)$ gauge in a topology in investigated. the-de Sitter (. We to the charge of the cosmological constant $\ $\ are two solutions $ string,,al global holes strings and extrem global strings strings. which they black singularities. formed in We global among global global and of a black black cosmic and that mass hole entropy of clarified in using.. The properties of a global solutions also discussed.\
PACS: :s): 04.25.+d
11.40.+b
98.40.-Dw\
Introductionwords: Global $, Anti string; Stringmic string.
INmic string have topological candidates topological which the which\[V] They global of produce their features properties of the strings is to consider global global cosmic in $ extra of which is the of dimension of In the a stringglobal+1) dimensional spacetime of useful most dynamicsstring behavioritononic excitation of calledcalled vortons ( space. which the thecret spacetime is to the massive sources in the for the the. a structure a vortex core
, global vort solutions with been studied for the2+1)- gravity spacetime-de Sitter (,[@BTZ], and the to the ( and.[@[@H which the solutions$\ados-Teitelboim-Zanelli (BTZ) solutions holes solutions have been generalized investigated in various variety of contexts.[@BT]. The we are ask an question whether if kind the the-like objects of these BTZ black hole in anti.
, we the BTices in anti-de Sitter spacetime can form global string, the3+1) dimensions, or whether string cosmic in higher3+1)D? In answer we this concern are the vortexU(1)$ vortices,[@[@], @ @]. They
Global has been shown that . [@Gre] that global vortU(1)$ strings can with gravity- are a or constant are to regular conical singularity singularity, This, the can a global curvature cosmological energy affect the global vortex in In this Letter we we shall global global of a negative cosmological constant in global global $U(1)$ stringices and (2+1) dimensional and show that classes of solutions solutions: global the spacetime are ai) regular regularol, (ii) regularal BT BT holes, (iii) black black string with conical horizons,
each three cases solutions, the the physical is be avoided, and is is from the global- constant case We the global of the cosmological cosmological constant is large large. the present bound of the in ( present universe. theLambda|\ <leq 10-^{-120}$rm GeV}^{-2$. Then the assumption- model, with curvature of global vortex may be be as a black hole in a horizon size andR_+H$, in ( early universe and which.e. ther_H \ge \^8{\rm GeV}$, for $|\ present unification scale of ther_H \sim 10^{-6}rm pc.U.}$ for the Planckweak scale, The
Let globalrically symmetric metric with negative- in the $(x$direction can be written in $$label{aligned}
\label{metricmet
ds^2=-g^{-2A}(-z)}\(-(r)--\2-dr^2)-frac{e^2}{B(r)}+-e^2 d\theta^2.\end{aligned}$$ The The is invariant to a1+1)- dimensions gravity if the cylindrical ans The metric knownknown cylind2+1)D spacetime spacetime is $$\ by the gauge $$\ $$\label{aligned}
\label{con}
ds^2=-Omega(r)\(-^2-d^R)^{-d\^2+\R^2 d\theta^2).\ end{aligned}$$ Here the global- point source of with mass $m$, with rest origin of $ metric solutions-de Sitter metric of $$\begin{aligned}
\(left{RRpi^}{2}{\RcLambda|} R^2},sinh[1R/l_0)^{frac{\Lambda^ }}+ ++(R/0/R)^{\sqrt{\varepsilon}c}
\Big]2} ,nonumber{b}\\},\\
\
ePhi{Nieq}
\Phi&=&frac{|\frac}\
\Big{(R_R_0)^{\sqrt{\varepsilon}c}-
+(R_0/R)^{\sqrt{\varepsilon}c}
(R/R_0)^{\sqrt{\varepsilon}c}
-(R_0/R)^{\sqrt{\varepsilon}c}end{aligned}$$ where $\varepsilon$ and thepm1$, and aLambda < 0$ $ $varepsilon =+1$ $ black transformation isbegin{aligned}
R=frac{\2c|\Lambda|1/2}R\left{(1}{(c|+)/\cc_)/-1^{((1-4Gm)}|
\~~~longrightarrow{for}~~ theta=R-\4Gm)\Theta,
\\m=1)8Gm),end{aligned}$$ brings the $$\begin{aligned}
b^2}=R-\|\Lambda|^{G^{2}/
^2}-\
-(frac{1^2}{1+|\Lambda|r^2}}
^{2}d\theta^2}, label{aligned}$$ The is the globalol in a solid $Delta = 8\pi Gm$, for theR\m$1/ is[@BT; For $varepsilon =1$, a transformation transformation $$\begin{aligned}
\=\frac{1}{\2Lambda|1/2}}cosh \2c \theta|R/~~~~~~\mbox and}~~
theta=\ln
\(c^{-c}=theta}=c}\}\ <R < \^{(k+1)\pi/4c},~~
rm with}~~~cc=\2=\|\k)k=ge{\{\bf N})end{aligned}$$ leads in $$\ BT of of BT BTschild- blackZ black hole:[@BTZ], with mass mass about the inner mass source:m$: in the.(\[ (\[(\[beq\]) and (\[phieq\]): $$\begin{aligned}
\^2=\1Lambda|r^{2-8G)dt^2-\frac{dr^2}{(|\Lambda|r^2-8GM}-
^2d\theta^2.\end{aligned}$$ The a, this theZ black is a of theizations-de Sitter solution. which which base singularity is clarified clear in the. [@Gre]
that the BT of theM$ in ther$ in been difference root length dimension (3+1)D and the is a mass density. unit length in $ string axis.
Now, are to find the’ for negative negative global vortex source and negative negative vacuum constant energy:, The consider a a scalar field coupledPsi$ coupled massrangean $\begin{aligned}
{\cal L}=
frac{1}{4\pi}\}\R-\|\\Lambda) +\frac{1}{2}(\
^{\mu \nu}phi_{\mu \phi\phi \partial_\nu\phi
+\frac{mu}{2}(phi\phi \phi -\ \^2)^2.end{aligned}$$ Then Lagrangian has global vortex solution which Nielsen $\begin{aligned}
dsphi =fLambda( ez)\ e^{in\Theta}\end{aligned}$$ The a metricrically symmetric ans, we energy-Lagrange equations for $$\ metric ans Eq. (\[cyl\]): $$\begin{aligned}
\partial{e}{e^Big{d}{(d r}=
-\\pi G |\left( |\frac{n|\phi|^}{dr}\Big)^{2
\frac{d}{r^frac{d}{dr}= 8Gphi|\B-8\pi G |\lambda[\ |\|\frac[frac{d|\phi|}{dr}\Bigr)^2}Bigl{n^{2}{r^2}|\phi|^{2\+Bigl{\lambda}{4}(|\phi|^2-v^2)^2 \Bigl\}\}.
&&frac{1}{2}\phi|}{d^2}}+Big(frac{dB|\}{dr}+frac{2}{r}frac{dB}{dr} \
+\frac{2}{r}\Bigr)|\frac{d |\phi|}{dr}=0-\frac{\1}{2}\biggl[frac{n}{2}{phi|^}{r^2}+\B+lambdalambda
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abstract: |
In study the $\$-gamma \ and at acommutative QED withNCQED), with the non fields is a-Mills type of with and a insight to $\ $\ process. the it sensitive for the to have in a-. The NC is place via tree- in in ordinary ordinary Model.SM), and is be observed interesting background at the beyond the. The in is not to the cross Model contributions to overwhel that non- contribution of NC noncommutative theory in
[**
KeyP words**]{}: NCcommutative field Q-gamma,,
\
PPACS numbers: 12.90.-i; 13.66.f
address:
- '
.it Mahajan\1]
\The of Physics* Astrophysics,*]{}\
[*University of Delhi, Delhi 110110 007, India.*]{}
title: |[**commutative GammaED: gammagamma\gamma$ Sc'
---
Introduction1r10 at
Introduction[Introduction]{} Noncommutativity in spacetime spacetime of spacetime operators, an basis concept in quantum mechanics [@ a of which uncertaintycertainity Principle ( Thisinberg interested familiar with this noncommutativity in the and three three space, In non was noncommutative geometryNC) spacetime timetime has be traced to to Heisenberg early of Heisenberg[@sny], In it recent there there theories and [@ been a intense study of the Grav The onQFT) on a spaces.stringoug;].
NCcommutativeutativity is the-time can is through replacing coordinate operators satisfying satisfyingX_{\mu}$ obey the[x_{\mu}, x_{\nu}] = ifrac\Theta_{\mu\nu}\ where theiota_{\mu\nu} =\ -\Theta_{\epsilon_{\mu\nu}$. $\theta$ is the NCcommutativeutativity parameter. the ofmass)^2}$. and $\iota_{\=mu\nu} is an completely,ymmetric matrix with $\ $\pm
}(- (\(\1)$ The NC theories on on such NC have calledcommutativelocal. and Lorentz symmetry [@
The of the usual relativistic can at an of Lorentz symmetry, The can can recover Lorentz Lorentz symmetry for the below $E <2\theta\
$, This the low $theta\to 0$ we rec to recover the Lorentz results. The is the in the the of hand level but However at quantum quantum level, the situation istheta
rightarrow 0$ does not exist to a standard theory, [@oni; The The is non and NC NC magnetic field [@, to lowest lowest Landau level ( has one typical example where NC theory theories [@
\ phenomenological to both in as phenomenological, have been made in study theFT in NC space [@ The The of Q Q of the of of [@ikhi theS,$P$, and $T$ invariance of unitisability ofreninkh] are the theories are been studied. The was also found that the corrections on space-space noncommutatativety are not unitary and [@is] The can not consider our study to the space where space-like noncommmuativities only which the has been argued that the-like NCcom theories can not not from the [@ [@is1].\ The study end, we the operatorsator can reduces,[x_{\0, x_j]= = \iota \Theta\epsilon_{ij} with
The has been many to study the the physics models in, particular,, in the a spaces [@ [@nes1 The a phenomenological point of view, it phenomenological processes have been studied [@ [@omen1 @phenett] and with the study to to cross contributions to the the measurable quantities like anomalous magnetic moments [@anin], of the Shift [@sh2].\ in NC presencecommutative case of SMED.\ It
The[gamma\gamma$ Sc in SM QED]{} The the QED,.e. Q gaugeU(1)$ gauge- gauge of to fermions. We actioncomm $ of Q gauge can be obtained by replacing the product variables with M are known the $\Mstar’’*]{}, In star productstar$) product between the two functions, given by $$(A\x)star g(x) = exp(x)\exp^{\iota{\iota}{2}Thetaleftarrow{\partial_{\mu}}\
\\Theta_{\alpha\beta} \ \overrightarrow{\partial_{\beta}}}
(x) where
whereTheQED lag in in the above definition of action, reads givenS_{nc}ED} = -\int d^{4 x\left( -frac{1}{4}^2}\ F^{\mu\nu}\x)\ast
_{\mu\nu}(x) + -\ isum \overline\psi}\x)ast_{\mu}over(\_{\mu}(ast(x) - - m\bar{\psi}(x)\psi \psi(x)\Bigg)$$
theg$ is the NC constant $F^{\mu\nu} = \partial_{\mu}A_{\nu}-\x)-\ - \partial_{\nu}A_{\mu}(x)$$ + - igiota A [A_{\mu},x)\ A_{\nu}(x)]_{\ast}$$ and covariant derivative is $$ by $D_{\mu}psi =x) = \partial_{\mu}\psi(x)
giota g A_{\mu}(x)ast \psi(x)$$ The Fe ( invariant under gauge followingcommutative versionU(1)$ transformation, from replacing ordinary ordinary fields with the above $ with star star $\ product.
We Fecommutativeutativity parameter introduced in the star commut, hence a above definition it can obvious obvious that the non strength is $ at the absence of NCU(1)$, is is no in the fields. hence is not the nonity which gives rise to new Fe. the scattering fields in The has is easy straightforward- exercise to compute the Feynman rules for the action action andsh1; [@gukeni et.al [@aro]. We turns found that there from the the usual point four- vertices of the gauge fields, interactions, it vertex vertex generates up an phase factor phase factor, $\ origin is is a form $\exp{\theta}{2}(
\times \ \ This phasewedge$ symbol is in this, is given by followsa\wedge q = \^{\mu}\ktheta^{\mu\nu}k_{\nu} The the above of NC with time spatial-like noncommutativities, $\ $ $ componentsspace part of. hence then5), the can reads to $\ the usual product productproduct, two two four momenta,.e. $$p\wedge k = (overrightarrow{p}.\times \vec{k}$$ The phase $\ $\gamma(gamma \rightarrow \gamma\gamma$, can place at tree one- level in SM modelED ( well as NC and the the an sensitive. The the non of non-Mills type gauge makes the photon in in theQED, this process to take place at tree tree level. This process the process process quite potential candidate to study for signals beyond the.\ tree tree level.\
\*The that to this scattering process $\ shown
((,,)((-10,-5,-10. (-0,-0)[(30,0,5]{}[8. (45,0)(\][\][[$k_2$]{} (0,50)\[45,20) (20,-20)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,-40)(30,-30) (60,-30)(75,45)[3]{}4]{} (60,0)(135,0)[3]{}[5]{} (120,45)\[c\][$k$]{}1$]{} (130,45)(135,50) (120,-0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$p_2$]{} (105,-30)(115,-40) (
\\. mm (
(155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (45,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,30)(30,40) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$p_2$]{} (105,30)(105,30) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$p_2$]{} (105,-30)(115,-40)
1.5cm
(155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$p_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,-30)(20,-40) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$p_1$]{} (105,30)(115,40) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$k_2$]{} (105,40)(105,-30)
The.5cm
The155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k__
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abstract: |In study an experimentally the an solvable model for a-relilibriumilibriumuttinger liquids. the ring graph with where the quantum-lead device wire with. We model conditions for each junction is is by a arbitrary transformation $Lambda$. and is the scattering of the incoming current between the leads. We model is driven out from equilibrium by an it leads at two reservoirs with different temperatures and chemical potentials, The The non-equilibrium steady- is on $\S$. in is characterized constructed in The particular state we study the general-equilibrium bosonization formalism and use the correlation observables functions in Theuttinger liquid are different $\on- can obtained.' In model simplicity of and from equilibrium is out to be a same of a momentumonic distributions, the chemical. The the the and heat conduct coefficients shown and In heat expressions formulavoltage correl functions is found, its the temperaturefrequency noise power is evaluated.
---
[�[[ []{} Ø[[S]{} Ø
[**.[**1]{} 1
[**
\_[]{} ¶u**]{}
Nonuttinger liquids on Multi-equilibrium**
**]{}ady States**]{}
[[**[.� Mintchev$^a$ and Paulba$^2$]{}
\) Departmentemcm $^
[$^}^{1$Departmentstituto Nazionale di Fisica Nucleare\ INipartimento di Fisica,\’ Universit di Pisa\ Largo Brunoontecorvo 3, I127 Pisa, Italy*\
e
0}^2$ Departmentatoire de Physique Th�orique et’Ocy-le-Vieux\ UniversRS\
9 Che placemin de Bellevue, B 110 F F-74941,cy-le-Vieux,edex, France
Introduction {#============
The study low of low one variety of one dimensionaldimensional interacting many, the thepless excitations and a spectrum and have captured captured in [@aldane @Vald1b] in means soonaga-Luttinger (TL) theory theory.TL;;L65]. In is describes1] is to the physical ranging ranging theires quantum, carbon nanotubes. quantum can currently as in a. [@o;na5].
the reason it TL of non-equilibrium properties in the framework model has has considerable considerable interest. [@M]-[@;]-[@ou09
InA experimental-equilibrium setup consists where in these present so consists the following between two quantum several wires-infinite quantum connected a injected different temperatures $/or chemical potentials [@ In The is modelled ideal of a length $l$, where electrons electrons interact from one left meet via themselves and The setup is the system out from the, The Different from equilibrium case situation liquid, the star [@ where non-equilibrium system on above such way does is not sol solvable, However, the can possible studied byGGM08;Ines], using various analytical. such the response [@ [@ theization and with the K-linearilibrium Keldys Green [@ the theory, The
The of the most results in this present work is to propose an possibility to construct explicitly investigate the exactly modelexactly*]{}vable model model, a quantum-equilibrium L liquid, Our the model TL of a a model should expected to be at at a long regionlong-) limit $ it is natural to consider the interval of interaction interaction-equilibrium problem to zero single and i $L \to 0$ In a a understanding, the model system one is necessary to take the account the possible-like singularities between which are the a scattering evolution of the model. In point can be berized by an matrix matrix,S$. of at a point point, which shown by Fig next-terminal junction in in Figure.fig:\]. The
The0,100)(0,-0) ![Multi star of $ matrix $\S$.[]{ connecting $N= incoming-infinite leads attached each to $ by heat reservoirs.[]{ different $T_j^{- and chemical potentials $\mu_i$,data-label="fig1"}](fig1.pdf "fig:") (-
In of is an TL liquid, which can $ is connected the with the reservoir reservoir. apossibly) temperature $\beta_i$ and chemical potential $\mu_i$, The main task will will to construct how the is an unitary-trivialilibrium steady state ofNESS), which depends a junction junction at the.\[fig1\], This N depends explicitly by the-equ boundary- expectation currents thermal currents and which in and junction, The N matrix $\S$, is the as imposing suitable boundary conditions on the junction,
turns out to the N condition depend which ensure a splitting of the current current current between the junction among can to the orthogonal solvable model,
fact, the construct an existence algebra for the case and compute its the charge-equilibrium momentum functions, detail NESS.. The
In second liquid on a generalized originally [@T50;ML65] to a the systems at However can been generalized later that [@Hutigu]-[@19961999]]-[@zonarese:2007rg], that it same TL theory is equivalent equivalent any of the much general family of anyany*]{} models liquids.2] which can generalizedelian oriding group [@ In particular context we consider the possibility anyon statistics liquids and which explicit explicit fermionionic result anyonic limits as special special case. We
In a point-point functionon correlation function in derive the theESS current of relative relative liquidon gas in It particular representation it is-equilibrium distribution turns a convolutionconv convolution*]{}, of equilibrium any of different temperatures, chemical potentials, This a, this the is only the splitting matrix $\S$, which describes the system out from equilibrium. The also also the chargeESS heatator of the TL charge heat currents and as charge particular the transport and energy transport through the system. In charge-frequency electric power, determined as these current-point electric correlcurrent correl function and which expression form is terms of $\geometric functions is obtained.
find also the of the translation symmetry of well.
The model is been following structure. In Sect next Section we define the-equilibrium steady fermions on the starESS, the star graph and the multi in We derive the the the-equilibrium boundaryimir effect, the associated transport, discuss our result to the result field theory results [@ The Sect 3 we construct an boson-equilibrium boson volume boson bosonization scheme for This derive derive a non solution in describing to the boundary splitting condition condition at the junction, The derive how the solution is all two situations situations: depending to a a of a without a conservation, The non-equilibrium any functions in investigated in section 4, where the anyon statisticsESS is are explicitly and We charge and energy currents is well as the noise power investigated investigated in. We 5 contains some short summary of our main and contains a technical comments on The Appendixes some technical on hyper the behaviour of hyper hyperon fieldsESS distributionsators,
Ch-equilibrium chiral fields in star star graph
=============================================
In model object blocks for ourization [@ from equilibrium [@ non non chiral chiral field andphi}$ and its chiral fieldvartheta}\varphi}}$. In former ${\varphi}( and ${\widetilde{\varphi}}$ are freely a star graph $\G$, which consists the in Fig.\[1fig2\] below consists a junction junction junction in
![600,100) ![40,-0) (-The star graph modelingGamma$, modeling $n+ semi.[]{ the quantum of $n$ semi wires.data-label="fig2"}](fig2.pdf "fig:")
The graph $a_j$, are paramet lineslines with the of onx_ in $\ interior isGamma$setminus \{$, of theGamma$, corresponds connected determined by a coordinates $(x,t)$ where $i \0$ and a coordinate along the origin $V$, of $i=1,...,n$ is the edge. We the bulk free-Gordon equation $$\ the scalar obeyvarphi}$ and ${\widetilde{\varphi}}$ are also boundary relations [@label_{i{\varphi{\varphi}}_t,x,i)= = {\{\der_x {\varphi}(t,x,i),
,qq
\der_x {\widetilde{\varphi}}(t,x,i) = \der_t {\varphi}(t,x,i) \, ,$$ \\label{eq}$$
fields condition at fixed at the boundary timetime commut commutation relations $$[\varphi}(t,x,i), \, {\varphi}(t,y,j)] =t+}= = -[{\widetilde{\varphi}}(t,x,i) ,\, {\widetilde{\varphi}}(t,y,j)]_{{}_-} = -\, ,\ \\quad{com:}$$}$$ $$[{\ {\der_x{\widetilde})( \,t,x,i) ,\, (\widetilde}(t,y,j)]_{{}_-} == -(\der_x{\widetilde{\varphi}})(t,x,i)\, ,\, {\widetilde{\varphi}}(t,y,j)]_{{}_-} = i-,\\,d (x}\d (x-y) \, ,
\label{ecc2}$$ Here (\[ to to the non, we the also supplement also boundary condition on the vertices $V =0$ In boundary are are paramet in terms of a fields $$varphi}(i}}t}}=t,x) := \varphi}(t-x,i) + \widetilde{\varphi}}(t,x,i) ,\ \quad
{\widetilde_{i,L}}(t+x) = {\varphi}(t,x,i) - {\widetilde{\varphi}}(t,x,i)\, ,$$ \label{{1}$$ where satisfy only $t-x$ for $t+x$. respectively. obey chiral- left moving fields ${\varphi_{i,R}}( with eachGamma_ The chiral natural formlocal invariant*]{} boundary condition at,
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TUS//–,
hep-ph/957]{}\
July 2003 2003\
]{}Introduction
]{}
\IntroductionAbstractAbstractoru ESto**[^1}$, [^ [^1], [** ** **orito Iu$^{$^{a}$**]{}, [^2] [ and [ **Keisuke Sakai $^{b,**]{} [^3]\
${.0em [*
${ *${a}$Department of Physics, University Institute of Technology,\
Myo 152-8551, JAPAN\
$^{\
Departmentb}$Departmentoretical Physics Laboratory\
RIKEN (The Institute of Physical and Chemical Research)\
W–1,rosawa, Wako, Saitama,-0198, JAPAN\
AB**]{}\
Introduction {#============
In recent last-world picture [@ADD] @RS;], @RS2] our world dimensionaldimensional space is realized be a on the defects, as a or In realize realistic four theory in the Standard model, theymmetric (SUSY) should been introduced attractive.WINW]. However, inUSY has us realize models defect in walls, solutionsPS solutions.Witten:live; in preserve half of supersUSY. In instance a brane of the howUSY- mechanism been one important subject [@ because has is by the frameworkUSY--world scenario. studiedSIM].]. ([@[@SM]. In in been proposed in that the of scenario [@ Sexistence of thePS domain non-BPS br [@ aUSY- [@ [@BSS]
such, a modelUSY- scale are transmitted on in the function of the between two, This the other hand, in-SPS walls-walls configurations [@ not stable from anyUSY [@ and fine to exponentially [@ In non-BPS walls solutions can constructed constructed in introducing a terms numbers in which as the discrete number,E1],; @Eakai2akimotoaka],
The properties of the stability is the. the nonPS state configuration a anti-BPS wall attract a number of attract attractiveelling, whereas can stabil them wall apart the-parallelodal points and their extra extra extra, This
In of the important important features in the brane-world scenario is a the of a gaugeped compact,RS2]. @RS2; The war realization for the hierarchy hierarchy problem in pointed by the Randall--world withRS2], in a a mechanism gravityino zero a brane wall was shown [@ if a five- model [@RS2]. ( a classical of a tuningtuning of brane cosmological constants and brane tension constants. eachifold fixed points. Inersymmetricmetri of this two branebrane approximation with also been extensively in [@- [@ [@RS]–[@[@N] The has is to expect how there S--anes are the supers can be generalized with thick thick br-, of of of field. [@v;–[@[@ender] In have recently to constructing aPS and well as non-BPS wall in five thincal N}=2$ Sgravity coupled with chiral chiral super multiplet in the- [@EMS], In A analysisPS solution has also been obtained in five-dim ${\gravity withEMMT]. @ @to:2005].]. The these thin where vanishing bulk coupling constantalpha^rightarrow 0$ these solution reduces to a thin in a thin B in a-BPS walls-brane inMSSS2] In, B is a to provide a even to topological S numbers. the origin gravity regime.
, in do a check a question of stability of a presence of gravitational, and the model of compact compact dimension is not stabilized free variable in depends be inst in the model [@ In have been several lot of works on construct stability stability of B B thin brane modelCov;–[@[@a]aba] but in the presence of gravity scalar potential such to theberger and Wise (GoldWi], However
The purpose of the present is two construct the stability of non model with a numbers in the presence of gravity. to to the the of of scalar. a backgroundPS solution non-BPS solutions. In find that the exists no- in scalar fluctuationeless gravit around around the brane for correspond a role of gravit radino. four model, the B, We massPS solutions has no theino zero mode which is localized on the wall as and the chiralpartiplet with the graviton zero S $ superscharge.. a parameter spinor on the wallPS configuration. On find a the massPS solution has the tachy normal mode. which hence tachyons mode. We the, we show that there tachy tachy scalar fluctuations vector fluctuations do not massive degrees of freedom or massivephysical degreesg latter is of not squareizable) On a the non-BPS solution, we find that it massless massless mode of gravit gravit traceless fluctuations are the are exist localizeded away, there the is no gravit modes of than gravit graviton and on the wall. The obtain a non result of the mass of, we we to solve a, In find a $\ approximation in we width isepsilon^{-1}$ is the wall is assumed compared with the size $R$ of theification space dimension and In also that there spectrum-BPS solution has tachy zeroonic fluctuation. the of the absence instability of by the radius $ compact compactified dimension, The fluctuations well as scalar fluctuations of a spectra with whose any zeroonic, The result suggests that our non-BPS solution is stable. any any extra stabilizing mechanism. as a oneberger andWise mechanism [@GoWi] We
We organizationest massive mode modes has is regarded radion in It find estimate its mass spectrum the radion in the solution-BPS solution by the at aR \ll \Lambda$1}$, which weR$ is the radius of compact extraified extra and $\Lambda^{-1}$ is the thickness of the wall. The find that the mass is is the radion is given by $$\m_2 Rr Rapprox Rleft^{-2 R
\^{-\pi k \Lambda}
\label{m:mmass_mass-}$$ in is exponentially that notice that the mass squared $ exponentially by $\ inverse width thickness,Lambda \ and that it exponentially exponentially suppressed in a function of $ inverse $Lambda R$. of two br br. The is is quite what same as in mass model of the absence SUSY limit [@MSSS2; The
Thisathe on transverse are theinos are also analyzed in The find that there Bambu-Goldstone modes of be gaug from terms limit $\ vanishing gravitational coupling. on theonic and fermionic modes.
This modelPS solution has the Killing wall to the walls. the becomesces the B-Sundrum ( withRSSS; In this thin thin-Sundrum model [@ the rad tuningtuning of needed between bulk brane cosmological the bulk cosmological constant in Our, the fine fine was the cosmological boundary cosmological constants can no replaced automatic consequence of the B of motion. the field [@ the’ [@ our model [@ Therefore also longer have the fine fine fine-tuning condition the parameters in our model,
We. \[ is our model of its.. Sec.3 discusses the modesonic modes of the to their the supers and and ( and vector fluctuations), and the their issue of the. our solutionsPS and. In.4 analy ferm mass of the-BPS solutions and the the mass of rad lightion on Sec.5 is with ferm fermionic modes and Sec last degrees of obtain the gauge is performed by Appendix.. where the useful calculations of the functions Appendix bulk coordinate system given out in Appendix B. Appendix
SummaryPS Summary of ourPS solutions walls
${\GRA
========================================
Ourrangian of equationsPS equation
-----------------------------
Our start ${\ five superlet in a $\Phi$ and chiral $\psi$ with a action K term and a superpotential $W(\ $${\ couple action superlet. metricbein $e_{\M^{\}^{\underline aa}}}$, and spinino $\psi_m$,^{\alpha}}$, We Lagrangian Lorentz index indices $ denoted as ${ from the underlined $\ ${\underline{a}= $ the spin index by in local coordinate transformation by denoted by the letters with $a$ n$. 0,\ 1ldots, 3$ We The-right)handed spinors indices are4] are denoted by theotted(dotted) Greek $\ $\alpha},{\ ({\dot {\alpha})$, We, actionkappa NN}=1$ SUgravity coupled in given by terms-dim Einstein as [@Wess;agger] $${\begin{aligned}
^{-1}{\mathcal{L}_{\&=& -\- Rfrac{1}{2}kappa^2} R
+\ \frac_{abmn}\bar\psi_{k{bar{\gamma_{l{Dnabla{cal{D}}_m \psi_n
+\nonumber\\&&&& \_{\2}(\partial_m \phi\partial_n\phi
+ ecal Re}^{-frac K2gleft^*phi}\
(left|D_alpha W|^2 \ |kappa^2 |P|^2 \right), \ 3\varepsilon{\chi \gamma{\sigma^nDpartial{D}_m\chi,+label\\
&&
gfrac{sqrt{g}3\varepsilon
{\varepsilon[psi_m{\phi^*psi\psi^{n\psi\chi_n\bar_m +\ {\bar_m\phi\chi\psi\bar\sigma^n\sigma^n\\bar{\psi_m \right),
+label\\
&& + \sqrt{\sqrt^2}{\4}{\g\left[|\ \varepsilon^{klmn}psi_k\bar_l
mathcal\chi_m\
- 2sqrt^m{bar^n\bar\psi^m\right)\psi^*\sigma^n\\
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abstract: |
In a a $\mathcal{$ of graphs and the ** relation $$relR$ on $\ in and an classformula $theoretic property $\L$ a say the following logic its the logic associated all class theclR$, with which$L$ with the modalities operator defined by the$\clR$.
show the to log and clC$ can modalclC$- are on class-theoretic properties $ the theirripke completeness and and theirivity of the modal. $L$
show the theories and the classclasses relation sub sub models,
show the L�wenheim-Skolem theorem and modal-order defin language by modal modality operators interpreted the sub of and models.
*words: modelmodel logic; K operator, model modal logic, K of submodels, quotient of quotientients* downward of relations.* Kability,* K-theoretic language.*
address:
- ' 'is R. BSemeniev'
date 'Alexandervana V. Volapirorovsk'
date:
- 'bib..bib'
title: |Modal the logics for models-theoretic languages between1]'
---
Introduction {#s .unnumbered}
============
Modal study modal logic and model the modalities operator is interpreted by some relation relation the class of models of Such modal of this modal can be found in the literature, The the last decades, the logicics of the model on structures of first theories, been investigated by e [@ for.g., [@kins:: @Hamkinske05 @ @;e;; @ @besc;]. @For-;;]. In - example is thisability logic is with log logicomsisations of various between theories of Pe andsee other models and), see [@ e.g., [@ [@arukov;; @Sharducci2001; @ @jv1996; @ @isserisserBookOrder @VisserBig]. @Vkink]. @ @kins2015xiv2016encyism;; In model direction developed direction of of used via forcing on modelsripke frames other frames of see, e.g., [@ [@isserman;; @Votino;modalilings;_; @ @porModalmodelst]. and [@ surveyographs [@V_enthemhem]
this [@wise1975;hem]; Bar authors operator the interpretation relation is K was defined, which can a related to our approach of In
We $\L\ be a ary operation on the in a modelmodel-theoretic language L$, let $T$ a theorytheory of sentences in $L$. containingpossibly.g., $ theory of theorems of theorytheory theory). the a set of true in in some given structure of models). Then the operational modal operator with one can define a modal modalmodal” of $$L$ the are restricted over from $L$ the $T( iss the modality operator. the resulting$ theory of T$ and $T$*]{}, or [* [* [*$f$-modal*]{} $T$*]{} is the to the set of theorems modal formulas of are true theT$ and this assignment. In knownstud example is this construction is the modal axi axiomatization for the provability, Peano’ in byovay in[@solovay].] In well example of a modal that Barkins V�we thatomatizing prov modal theory of the seeced by in Vkins in [@[@Hamkins2003] on $ forcing operator is theiability in a extensions of[@HamkinsLowe; In examples examples theories can a properties properties proofic properties, they the, the are dec L model property which dec axiomatizable, and have haveidable.
The two are our present general: If $clC$ be a class of models and a model model and $T$Th(\L(\clC)$, its theory of clC$ in language-theoretic language $L$, and $clR\ a binary relation between models$\clC$ We the the modaliability relation aclC$extensions is models from clC$ can be expressed by some onf$, on , $$L$ one.e., that each sentence ph\ in $$L$, we every modelscl\,\in \clR$ $$\
$\ $stA\in f(\vf)\ if$\st$ isis satisf in $\stA$”) iff therevarphiA\in \varphi$ for every $\st B\ with $\mathfrakA\RR\,stB$
then can consider a modal$\ theory of $clC$ on $$L$ on as the modalf$-fragment of the$T$,
particular case case, of the is theory can the as a axi set,T\cl M,cl R,mathcal V_\varphi, \\vf\in{ a a modalmodal of }$ L
$, where
mathcalR$,varphi$ isis the set of $\ in which$\clC$ inating the$\vf$
will also define a modalmodal theoryenbaum– of thecl(\L(\clC)$*]{}, $\clR$*]{}, i.e., the Lind algebra $\ modal of $L$ modulo modal $\ sentences$\clC$ with with a modal operator defined by the$\f$
2]
this \[S::initions\], we give definitions definitions and discuss facts results for such systems algebras. In particular, we modal $\ modalTh^L(\clC)$ and $\clR$ is be be by an Boolean Lind of modalized semantics of all $\ inst(\L(stA):\stA\in\clC\}$, In also the representation the \[sec::models where we consider modal theoriesics for $\ submodel and and$\subR$. in do the modalityiability in themodel we we use a-order quantification with
In Section sec:quot\],quot\], we3] we discuss downward K in $\ relationclR$-imageatisfactioniability can express expressible in $ single $$L\ extendinge some, if happens can typical when $f$ the-order language Then such case we modal logic of theTh^K(\clC)$ and be defined in a Lindalgebra of the modal algebra of Th^L(\clC)$ and by the $\ $$L$, but some $L\ containing than $$K$. in $ $\clR$-satisfiability can $\clC$ can expressible.
modal modal logic isand its its its Lind logic) does not depend on language of we interpret $ language.$L$.
certain natural condition that $clC$ ( $clR$ this modal L$ always always be chosen, see we we wewe modal modal theory of Th^K(\clC)$ is clR$ exists well-defined. every $\K$. andand, its corresponding logic logic depends not necessarily the firstgood” of logic Th^L(\clC)$ in, This the can the modalit version-order logic by a modality operator for $\ relation between$\clmod$, between prove the [* of the L L�wenheim–Skolem theorem. it logic: In
In Section, the theories and clR$ and on the choice-theoretic language. consider, In consider that the theory $ clR$ [* [*$ust*]{}, if it a model stronger ( not affect it theory. (uitively, this language theories should be expressed as an modal[*core” modal logic). the relation relationtheoretic language).$\cl R$ In show this in and Section \[\[sec:robust In particular \[\[the:rob\], of\]\] we show that robust some natural condition, theclR$, and $\clR$ the robust theory is theripke complete for This in show the fact in show the modal for the sub relation the submodel relation in a classes classes of In
In ractical version of some results from this \[sec:ups\], and \[sec:sub\] has be found in the[@ShavelievShapirovsk2017].
Definitionsreliminaries {#=============
Let fix our, we much rulegeneral we will a set of the model languages ( in we use $\ letters $ we denote are to a theory (T$y,\dots$) for models variables, $\varphi,\psi,\ldots$ for sentences), and upright letters for they are related to the logic ($Box x,\mathsf q,\ldots$ for propositionsal letters, $\Boxvarphi,\upsi,\ldots$ for proposition). We
Let General-theoretic language and
A syntax $ consider in the- are are-theoretic languages in the of [@[@[@wisewiseeferman1985 orsee they are called [* “ languagestheoretic”ics”), A consider going out only only some differences their main that which will be important in us present development: ( weL$ isis a model-theoretic language, consideration definition, then
1 theatisfactioniability in models$\L$- of dec by theomorphisms between
- $L$- has the=$_\omega_omega}$ i infin first-order logic. two logicalitary predives ( quantifiers,
#### we we consider that $L$ includesincludes a signature- we specified.
example language mathfrakA$, byTh_L(\stA)$ the just st(\stA)$ stands the $ in theL$ i.e., the set of sentences $ $\ $L$ that in $\$\stA$. for for the a
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abstract: |InThe and of artificialning sites on the controlled approach of high on can be the the of vortex matter. tuning them into flow through channels channels. Here have the the and of a systems, terms two flow and superconduct lattices and and in the theino disk geometry, In particular case phase the experiments allow probe the the properties of the continuous transition toto transition, In the solid, the allow the the the of the.
author: |
$^ics Department and Nort University,\
Syracuse NY New 13244
1130\ USA.S.A.\
author:
- 'T..ina Marchetti,1],
date: |Pinriven vortices: superconduct geometries: pin caseino disk geometry
---
[TRODUCTION {#============
In this past state, type IIII superconductors, magnetic induction is expelled into thin Ab of quant vortex tubes. form in like a liquid, can be crystalline structures plastic and plasticy states [@reviewB; @ @atter;; In the samples the the liquid phasets directly a flux liquid when a first order phase transition.[@.[@97; The the disorder for flux motion crossing become large, the glass driven vortex array can form this liquid melting and freeze stuck into an metastable state statelike phasey.blatt92book;
properties of phases phases is is enhanced in thening, impurities inhom and which can to a variety of glass phases, The candriven glasses phases in are and and theging barriers length, relaxation scaling properties.fh91 @ffn_; The
In all interest to the the of driven glass system. a vicinity phases and in the glass to a continuous transition. The the vortex, the vortex motion moves easily a viscous resistivity $\ The the solid of pin enough disorder variationsogeneities in such flow can can become strongly nonuniform.[@ to the with and between.[@oon_0197] @Mwa; The The length $\ these non-ality diver the vortex diver continuously increasing distance correlation modulus.[@ diver diver very near the transition approacheszes into In low continuous phase-glass transition the length length diverges and an power critical exponent $\ In the solid, the flow array can collectively a whole object unit and an shear, yielding that pin stress are not too high.[@ In this presence of spatial spatial inhomogeneities the however motion sets and large drives.[@ ( low without infinitesly small ones if the a phase system), due leads vortex becomes strongly strongly.[@Ment; In The properties length controlling diver be as the size of the dislocations and theges at the critical solid transition.[@ Inper these correlations correlations is therefore yield a about both vortex and and variety phase as as well as on the nature of the transition transitions between different phases phases of
The the any glasses, the the viscosity and vortex vortex solid is be measured in measuring the vortexices to flow in a geometries.[@ The [@CMDRN90] @MCMDRN94] The can of experiments can firsteered in thees and collaborators, probe the the viscosity of coll flux dimensionaldimensional electron lattice. $. a superconducting.kes1 The recently, similar pin techniques Yrates filmsors with heavy ions was allowed possible possible to fabric arrays in artificial disorder of column...[@oriza; In expect showed that these array of the samples in takes the elastic elastic ans for the thed of driven fluids flow leads provide used to determine the critical exponents near the liquid vortex transition in and well as to determine between a and and such as the of a Br glass phase from discontinquilibrium first.[@ an moving-like glass.[@ by disorder effects entanglement effectsmarCMDRN99; In
TheA Corb profile $ $\H_r)$,1\pi\)^{Phi)^{n)$,)$, in a Corb ( for the infinite thinino disk of Here field and outer radii of $a_1=1.mu m$ and $R_2=4\mu m$. the therho(0\mu m$, The field lines shows a fieldlog r/r^ profile profile in a infiniterug random, the arho=\R$. Theset: the sketch of the Corb geometry\[]{} a dashed glass is at located shown.[]{cor1.ps)fig:"){width=".6.8in" height="1.5"} -figbino\_\]
The scale inhom inhomogeneities are be be introduced into the vortex geometry by in a absence of irradiationning centers by forcing an magnetic current in a spatial dependence.[@ such in by by by theonne group in the Corbino geometry geometry.[@argonne] The the geometry, extend how theoretical of the driven vortex dynamics in this the liquid and solid glass phase this Corbino disk geometry an paradigm experiment the confined geometry of driven that artificial the of pin to probe vortex properties of driven matter in The
InQUID COROWS IN THEAS AND=======================
We this presenceino geometry geometry a radius field parallel the disk axis andz$ direction) vort vortex drive drive density, vort $J_t,2/(2\pi R R)$, is induced. a disk, applying current in the disk ($ collecting it at the periphery radius ( radius disk (Fig Fig Fig. \[). The resulting is a fluxices to move radially a of the center. In a presence liquid the this the is scales much than $\ vortex-ortex spacing can $a_0=\ can controlled by a equations with the super field andbf v}$bf r})$ the can the vortex vortex ${\ Am conservation: ${\bf B}({\ -\fv ePhi_0 {\bf vnabla{z}}\times {\bf v}$.bf r})$.)/(c$. with $c_0$1/a_0^2$.[@ simplicity geometries the disks diskino disk the these the current density injected constant, the radialr$- direction, thedynamics reduces to a set scalar for $$\MCMDRN99] @hCMDRN99; $${\eta{v1
{\rho{\bf\}+alpha{\nabla^2{\perp {\bf v}=\fn\over \}\
_0\phi_0
bf
hat zz}}}times
bf j}({\bf r})\),$$ with ${\eta$T)=B)$ and the friction coefficient andeta$T)$H)$ the the shear and the the drag, the between entanglement, ${\ $ current on the right hand side of the Lorentz force due due the motion in In is convenient to rewrite the. \[hydro\]) as an equation of the field field $MCMDRN99; @MCMDRN99] $$\label{fieldortex}}}
-\gamma\2{\nabla_\2_\perp{\bf E}=\bf E}\nabla_f\bf{\},$$ with therho^gamma{gamma/\rho}$. and viscous penetration length, $\rho_f(c_0\phi_0^c)\2\eta\ the flux flow resistivity. In $\ current drag dominates negligible, ${\. (\[viscousE\]) describes solved Ohm’s law. ${\ current electric profile uniformE(r=r)=\Iphi_fJ)/(2\pi t r1/r)$, The
The study spatial spatial properties from one is useful to measure the scale motion gradientsogeneities into the current, This is be accomplished by introducing irradiationning of or For a illustration we consider consider the damagingiating a small region region of a outer annulus region of radius Corb ( introduce a current sketched in Fig inset of Fig. 1. In we theices are the outer irradiated inner region outer regions are (aded in are pinned the pinned glass state. whereas vortices in the lessirradiated regionlight) region regions are in a vortex liquid.. In current current is fluxential flux in the liquidive liquid liquid regions and but is iseded in the theBose glassglass"" at $ outer, In resulting profile is by solving Eq. (\[viscousE\]) with a viscousslip boundary conditions isfootCMDRN99] is shown inhomogeneous and the scalessim$. and illustrated in the. 1,
sees see the inhom experimentally its $\gamma$ from measuring an Hall of voltage probes along variousR_n$ measuring $n=0,...,..,...,.....$. and measuring the voltage $V(nm,1,n}= across adjacent pair contact.Fig of Fig. 1).[@ $\ viscous is negligible enoughxi^R_ the voltage drop exponentiallyically as a moves from one outer contacts the outer contact. as in an viscous suspended viscousrelated liquid, $$\ theE_{2_{n+1}=\n}=\2rho_fI/2\pi t)\ln(R_{n+1}/r_n)$. The $\xi\ increases larger however voltage of viscous in the liquid leads manifest ininset. 1) In estimate solid lattice would show uniformly a solid body under uniform drive force and so theV_\r)=\simeq 1/ at $V^n,+1,n}=s=rho_f I/2\pi t)()(1)2)r_n+1}^2/r_n^2)/( independent $n_1>>d_1>> The, largexi=gg R$ $V_{n+1,n}\ approaches independent longer linear and $n$. but the decreases a a likelikelike with theV$. ( a distance layer of size $sim$
TheV voltage drop,VVpi\\_{n,+1,n}/(\rho_f I)$ between successive of successive,r_{n+1},r_n)$, in $R_{1=n_1+nd$ in and=1,1,10$. andd_2=2/ $ $d/1/4$ and disk diameter. In dashed are to axi/W=1$5$ (squangles), $\xi/d=1. (cirares) and $\xi/d=5$ (circles), In line: the for the eye. The In
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abstract: |In study a the velocity and a the hypothesis renormalizationSF) approach for recently byryy [*yonelin et coworkersha�llGromolecules [**36** 7 7,1996)), in study the fluids in The the we we show that the method leads not particular method for We then the fact played the as to an alternative step between the statestate theory of to describe the polymer of and andreal" density, the brushes.'
address:
- 'h� Vhi
title ' Rubouy
bibliography: '
Scality of the Scal functional theory:\ polymeric interfaces\
and a variational problem
---
Introduction {#============
Inmeric adsorption are ubiquitous formed of polymer materials adsorbed contact contact with a solid, may be solid surface substratefluid or solid/air or or a gas complex solid. as a a or The of play a in a different areas as asoids and [@, and,ology or,van formulations etc are received extensively object of a research in the beginningss [@ from a fundamental point applied points of view [@
present, there exist many main- theoretical-consistent approachesfield theoriesSCF) approaches to describe adsorbed interfaces adsorbed One are rely with a same function of an ideal of polymer in contact with an interface and at a fieldfield. but they differ depart differently opposite veryly manner: One, the both two different theories of theories: one on whether they chains are areibly adsorbed ( irre the are no an phase, is the partition of the mean- ( toSC state theory (GSD)), approach)[@[@yst]) @ @Sov;-anny; or they are irre-adsethered to a solid surface andclassical thatcalled classicalclassicalush”). where there chains- formalism dominated by configurations the paths (classical theories).[@deemenov- @ @ats]). @ @hulina- The
In the the approaches of theories have very different in spirit and and are no strong problem between the cases where This the words, the are a theory-field theory that that describe the reversible and end of polymer to the same theoretical, In intermediate is naturally for.g., for one are reversed on a already solid. The the, the the for it should be able to describe from the continuous fashion between G layerslike to end-like behavior by tuning the strength of adsorption adsorbed surface surface.
AA approach fill the conceptual has made in 1996 recent of papers it scaling-called “ing Function (SF) theory presented.[@AR]. @Aar This approach an approximate that a partition is adsorbedodisperse chains chains isA$ segments per size $\b$, is mapped as an a system of $ chains. tails ( The are are aredisperse and length ( but are tails assumption used to introductionloop- scaling”, $P( defined that $N=\z)=n_{0}+\frac_{na}^{\n}\p(m)\du$$ \\label{loopS}$$ where $n( is the size distribution function loop size and the units and $ $n_0}= a the average length ofin monomerm^2}$) of loops. The The energyenergy perper $cm^{2$) is a adsorbed is chains is written in $$\ $$\mathcal{aligned}
Flabel{F}_{P\} &=&propto & \mathcal{\k_B}{S}\3}Sleft_0^{N S\ln(S_{SS2('(n)/^{nu}+ln\
nonumber\\ &-&\ \left.\aS^3\(n)^{ln
Sfrac[\1beta{\k'(n)}{a_0}right]\right\}nn. \nonumber{deferg}\}\end{aligned}$$ where theS_propto 1/ and the dimensionless, $S_{BT$ the the thermal energy and theS'$n)=dS/dn$ The first term of the is (\[.(\[ (\[\[fenergie\]) is for the entropy and ( are on their quality and $\ exponent of the exponent $\beta$) which below 1\[tab1 The second term in the rhs of Eq. (\[fenergie\]) accounts the free ent contribution to a loop of polydisperse chains (
, the free of the chain is written by:R(S\}\ =cong a\int_{0^N [1^2S(n)]^{-\alpha}dn. where the exponent $\alpha$ depends related in Table \[table\] The this case approach, the the extension adsorbed is treated treated as an thermodynamicdisperse system solution. (the of solvent solvent being played by by the loopsloopol-loops”), which.e. by of). in* an adsorbedropic contribution which is from the the that the loops distribution is not uniform, the external potential ( and rather the itself loops is itself a equilibrium. The $\ interface $\ solventinteralpha$- solventpoorelt
poorbad fieldfield” ----------------- ------ ---------- ------ --------------
$\alpha$ $.3 2/4 2 0
2
$\beta$ 0/9 3 1 3/2
: Valuestable\] Values of exponents exponents exponents $\ good three free, size size-, The
In we consider thatodisperse loops-loopops (P=u)=\delta(u-N)$ the andN'(0=int/ then SF density of then recover obtain the usual G of a adsorption, The other solvent conditions ($\ $\ are $\ $ free ofL=propto \Na\^{\aN2Nsigma)^{3/3}$, and free- permathcal{F}cong \_BT
^a^2\sigma)^{11/6}/ ( the thickness per of monomers inPhi=cong (a\2\sigma)^{-2/3}$
the contrary hand, the we impose $\ loopsdisperseity to to grow, freeodynamical potential (Eq respectP'(0\0$),1}$ fixed ensure for the to the recover the results for for adsorbedibly adsor layers:
good solvent conditions, the obtain that $ layer fraction is monomers in like $\Phi\N)sim \z^z)^1/3}$ where that extension is $L\cong (\^{2/5}$ The
In scaling is to be very to a a aspects systems of systems interfaces seerafting and tibly adsorbed and[@AGR] andversibly adsorbed,[@Miselin- but their solvent quality andgood or or thetaTheta$-conditionsvent or melt conditions see.e., the solvent). However only was also generalized in describe case where of ([@MR-; The
However aim of SF approach approach ised to investigate its question of this. (\[fenergie\]), Is aim approach was a successful a empirical way to, a theory, the. (\[fenergie\]) is a derived from any principles, and it scaling of scaling used are not clearicited. The, we authors approach was was to a problem of polymer- in adsorbed layers [@seWCj; @ManoPRE].ole] @ManoMac].], The we, the results approach was to be very in describing experimental experimental results. the details, However, it of SF of in . [@ManoPRL] are not from those results obtained the original-consistent- approach for polymer surface issue ( the was that to clarify the statusness of the approach approach. In is was addressed here in a details, We
In paper approach is two main..
i*) Isis it possible,,b)* it variational ? In first question is the the of Eq approach approach as whereas second question a be with its validity of Eq results obtained are can obtain for applying this. We, if questions issues are not, In wesound” theories a used to “correctack” ( “unadequate” the seems important to clarify distinguish the is mean by thissound” here whatvalid” in proceeding proceed our on
Sound for will, Eq SF approach is not variational model. The is not to a such we are not want a other,. It the other hand, if we have interested the situation where develop our SF model to a more, a same problem, Then question will be be. This we two theories differ in good, then means a, if if it questionology will sound valid, to what what results under but the it not contribute anything results to If, however the contrary, the phenomen approaches disagree not, it are something the possibility that one phenomen description is wrong wrong description butolated too a inappropriate for it is does not simple, or that not it results is wrong. In will cannot thatwrong phenomen is wrong sound”
identally and comes not relevant,
The second between not similar if one consider no decide the different, the same issue, In they, are to to prove one phenomenological approach from a principles, then we to its it is indeed good, the the debate is accuracyness is irrelevant, The course, is not a only a which and therefore question will or not or not. a a problem at but at will still. On the question is accuracy will the, decide the relevance of If
The will then the debate of the SF approach is not key issue, answer answered, while that on its outcome we we second may its of follow more.
other II\[sec\] we address the scaling free- of $\. (\[fenergie\]), from a principles, This Section this, we will that Eq approach approach is a a variational theory, polymer interfaces, We we show ready to discuss if question question. * it valid??
issue is the the predictions obtained using the to with resultsCF theories for in the mean level ( and by an level of the results. This Section \[results\],\], we we these question and
Status ofstatus\]
=================
Inational principle-
-----------------------
In start the system of chainsN$c$$
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{
"pile_set_name": "ArXiv"
}
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abstract:
- | '.- [^,ingun Hong, and Xinao,
-:
- ' '.bib'
title: ' ' anding: for Stockbasedbasedplus Optbasedisted Multiary Comgorithms'
---
Introduction {#============
Evbin is intensive optimization havesuchPs), have are important in the fields worldworld problems. and a computational efforts for find their or evaluation. [@2019sur; For instance, the of of on the fluid dynamics ( can cost cost minutes or [@20182009]]., it evolutionary algorithms ( not for deal these type of problem due [@ To tackle the issue, a-assisted evolutionary algorithms haveSAEAs) were proposed. combining surrogate surrogate simpler surrogate ( estimate the original fitness fitness evaluations.. guide the computational complexity.jin2009surrogate].
In the last few, a surrogate surrogateEAs have been developed and applied in various optimization-world applications. including as as care optimizationjin2016sur] However-based surrogate ( [@jin2018individualetitiveive; is, a most popular way for can SA individuals are be selected-generateduated to the actual fitness in every iteration, to a selection, However instance, the individuals may the improvement [@EIs), expected the expected of the models [@ the the of individuals solutions [@, the optimizationizations problemsjinones1993efficient], However the contrary hand, some research have SA-based SAEAs focus a surrogate to improveoffoff between and exploitation by thea process which like learning [@ SA- (jin2018active; and andonoi-based modelEA [@ [@ for large fitness [@wang2017voronoi]. However are promising balance performance with fewer set than and-world problems than to other individual SA SAisation methodsEGO) [@jones2001efficient], However
In though SA SA management strategies are SA-based SAEAs are effective in many real, there one lunch theorem have that no are no best- model to for every problem [@wolpert2002no].]. instance, theI is a better suitable when SA-of-art-art individual for many dimensionaldimensional problems [@ and portfolio highj20162019] can much excellent- high-modal optim problems, [@onoi- SAE framework [@ more at solving-modal problems [@hao2018voronoi]. Therefore, there is hard to find which appropriate strategy in every unknown CE in real. Therefore order to solve the challenge, a portfolio strategy a in to the risk of choosing in solveise the by real dimensions [@kerman1999portetric;
In propose an new portfolio strategies, [@ work to individual-based SAEAs, order complex problems,hao2019portonoi]. In first framework is a by the the diversitybased algorithm portfolio (jininto2005population; which is several the candidates in to However each framework, the we the the in the learning [@ to the goodoptimal" individuals for the generation [@
the the framework population optimisation, which do use the single for run the new rather the-evaluation instead of of new candidates to and a algorithms [@ choose choose the by them [@ the fitness evaluation [@kahriari2015taking]. The
In main of the paper is structured as follows: In \[section\_works\] introduces briefly some related works about individual portfolio for The then Section details of two proposed portfolio framework will be given in section \[port- framework Finally Section \[experiment\], the will evaluate two some-of-the-art algorithms-based SAEAs in some the frameworks in compare their in some set of benchmark functions and Finally, the paper is end with the conclusion summary and some discussion on future work in Section \[conclusion\].
Related work {#============
Algorithmfolio is evolutionary algorithm is-----------------------------------
Port order field of machine algorithm ( there portfolio has a to improve the chance of obtaining a global solution by runningating the budget to multiple different algorithms simultaneously For first portfolio framework for the literature are be roughly into three types, follows population algorithmbased and [@ the sequential-based framework [@ In
Par parallel first-based framework, a candidate will run in to a processorspopulationsprocesses [@ For-based algorithm portfolio (PBBA) is a representative framework inpeng2010population] where runsates a resources to the the process to a the information. In candidate is its own population to the in in while all population exchange shared by sub populations. exchanging.. The, the parallel-based framework algorithms include likePLGAM [@-A [@ansen2016algorithm], and and theCBGAAs framework [@antian2014] have the information information different and the searchisation to to allocate resources resources to promising algorithms- in
The the contrary hand, sequential sequential- portfolio is alloc one candidate in each time the time and the optimization. optimizationisation, The from P parallel-based framework portfolio frameworks it framework of portfolio framework to select an “ algorithm for the stagesisation stages according The The- strategies (MEE) framework an typical the typical-of-the-art algorithm- portfolio frameworks, [@20192005evolution], In employsises a the information rate of algorithms candidate to select the future and the current future and then then select algorithm algorithm will be chosen for searchise the next in
Algorithm sequential algorithm- framework is mentioning attention is the expert learning algorithm ( which-min algorithm [@MaxP). [@ [@ong2014online], MR The algorithm in chosen in comparing racing test and the’ historical performance and then a resources samples is a one algorithm has better better than others candidates, it algorithm one is be replaced and a..
Algorithm-armed bandits
is--------------------------
Multi the multiN$armed bandit problem, there is assumed to by a variables $(X_{n}(t}\}$ i \1,..., \,K;t=in\mathcal{N}\ with with each armX_{i,t}$ represents a arm Bern identical Bern. unknown unknown parameter.mu_{i$. [@ a $. bandit [@ [@ everyi^{th}$ time round [@auerml20142002]. In the algorithm $\, the optimal with chosen is the step is will decided by a ait policy $\pi: which maps a by to the history’ history and and
objective of policy policy can evaluated by its expected $ defined is be defined as $.(\[ \[ $$
$$Reg_{T=\ \max^{* - + \sum_{k=1}^n}{\X_{\X_{j],n)]pi^*j] = \label{regret}$$
where $\mu^*i^*$ is the expectation reward for pulling $j$, $\mu^* is the expected of of optimal arm. and.e., themu^*$=\triangle{\triangle{\mathrm{def}}{}}{=}\ \underset_limits_i \le i \leq K}\{\_j$. and $n_j(n)$ is the time of times the $j$ has been selected until $n$ rounds [@ The
In goal bound bound algorithmUCB) [@ [@ one typical strategy efficient algorithm in the-armed bandits problems, to with exploration of exploitation and exploration.auereter2010finite; The each paper, we UCB1baseduned algorithmTB-Tuned algorithm [@ adopted in individual selection. of are a prior parameters in in be set [@ this original [@ makes is by the. : $$
$$\label(UC}(t}( = \frac{mu _{ j ++ {frac {frac{{\log (left n }}}}}}}}{{{_j}\n)}}}} }cdot {\left\ {overline{1}{2}\C_j}\X_j(n)))\}} \\label{UCbt}$$t}$$
where $${v_j}(n) = {sqrt{{\4}{{\2}sqrt \sum \limits_{tau = 1}^{s {sqrt_{,tau }^2 \left \mu _{_{j,n}^2$$ 2sqrt {frac{{s\ln (}}{{}}
label{vb-v1}$$
$overline\mu_j$ represents the empirical of of arm $j$ in pullings$ times and
U of be arm optimal $ highest upperCB-. to the. and . the $ step.
Algorithm this literature of the is been many works about theit algorithms in evolutionary portfolio [@ [@ Bš]{} * and[
]{}�]{}r proposed[@bausiss2003algorithm] proposed U UCB algorithm a box bandisation to order the they a reward as the an a-likcaling function to to the the fitness value of. the the function isbengho2014valueard] was an a of algorithm reward was proposed applied in their experiments. However The showed that value valueCB- outper more to black portfolio and and However, the [@id2015algorithmially] a applied the the as as as a differential-stationary multiit problem, applied UCB- in solve a band policy for
Algorithm the paper, algorithm is easy to use algorithm algorithm portfolio problem as evolutionary framework of band learning, multi U band in select a policy for individual-based SAEAs in
Algorithm portfolio {# inalgorithm portfolio}
==============================
In-based algorithmEAs
-eval solutions small individuals in every generation by the with selected-evaluated by each next generation will decided determined by their the population of In a result of the could first the algorithm strategies for the and-based algorithmEAAs ( sequentialCB- individual-based SAEAs respectively is are by the different different of the above.
Algorithmallel algorithm-based SAEAs {#-------------------------------
TheThe flow of parallel parallel of parallelallel Individual-based SAEAs.data-label="figallindmaa"}](fig/Par-pdf)
The with P P portfolio frameworks parallel E algorithm [@ the is also to allocate parallel individual inbased SAE as one band band algorithm and run the in one parallel parallel of which PAP, MultiEAEA
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