diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdrea" "b/data_all_eng_slimpj/shuffled/split2/finalzzdrea" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdrea" @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\nIt is known that several deeply-embedded young stellar objects are driving highly-collimated molecular outflows with an extremely-high velocity (EHV) and jet-like component running along the axes of the lobes \\citep[e.g.,][]{Bac96}. \nSince the EHV molecular jets having the terminal velocities of 50--150 km s$^{-1}$ are concentrated within the narrow angles along the axes of the lobes and have large momenta comparable to those of the slowly moving (20--30 km s$^{-1}$) ^^ ^^ classical'' outflows, the EHV jets are considered to be closely connected to the ^^ ^^ primary jet'' which is responsible for driving molecular outflows.\nTherefore, studying the physical and kinematical properties of the EHV jet will allow us to understand the properties of the ^^ ^^ primary jet'' which provide clues to constrain the launching mechanism of the outflows.\n\nThe low-luminosity (7.5 $L_{\\odot}$; \\citet{Tob07}) class 0 source L1448C (also known as L1448-mm) in the Perseus molecular cloud complex (D$\\sim$250 pc: e.g., \\citet{Eno06}) is a spectacular example of an outflow with an EHV jet.\nThe EHV component of this source was identified as the secondary peaks of the CO $J$=2--1 spectra at ${\\sim}{\\pm}$60 km s$^{-1}$ from the cloud systemic velocity \\citep{Bac90}.\nThe CO emission in the EHV range was found to be confined to a series of discrete clumps, called ^^ ^^ bullets\", which are aligned along the axes of the outflow lobes and are symmetrically placed with respect to the central source.\nThe EHV bullets are also observed in several transitions of the SiO \\citep[e.g.][]{Bac91, Dut97}.\nSince the SiO emission has been barely detected in quiescent dark clouds because of its very low abundance (of the order of 10$^{-12}$; \\citet{Ziu89,Mar92}), the detection of SiO in EHV bullets suggests the presence of shocks that enhanced the SiO abundance in bullet gas by a factor of $\\gtrsim$10$^4$.\nAlthough the lower transition of SiO, i.e. $J$=2--1, was observed not only in the EHV bullets but also in the lower velocity component that delineates the tips and walls of the outflow cavities, $J$=5--4 emission was confined to a pair of EHV bullets located at the closest positions to the star \\citep{Bac91}. \nThis suggests that the excitation condition of the EHV jet varies along the axes, and that the jet gas is highly excited in the close vicinity of the driving source.\nRecently, higher excitation SiO up to $J$=11-10 has been observed by \\citet{Nis07}.\nTheir results have revealed that the innermost pair of bullets, labeled B1 and R1 by \\citet{Bac90}, have a density of $n_{\\rm H_2}\\sim10^6$ cm$^{-3}$ and a kinetic temperature of $T_{\\rm kin}\\gtrsim$ 500 K, which is denser and warmer than the bullets in the downstream.\nIt is also known that the innermost pair of bullets, B1 and R1 are resolved into two clumps, BI-BII and RI-RII, respectively, in the higher resolution ($\\sim$2\\arcsec) interferometric SiO $J$=2--1 observations \\citep{Gui92, Gir01}.\nThe high resolution SiO data exhibit the kinematic structure of the EHV jet near the source, which shows an apparent acceleration of the jet up to 70 km s$^{-1}$ within a region of 6$''$ ($\\sim$2000 AU).\nThe proper motion measurement of the SiO clumps carried out by \\citet{Gir01} suggests that the outflow axis is inclined by 21$^{\\circ}$ with respect to the plane of the sky, and therefore, the SiO clumps in the EHV jet are likely to be moving with absolute velocities of 180 km s$^{-1}$.\n\n\nIn this paper, we present the SiO $J$=8--7, CO $J$=3--2, and 350 GHz continuum images obtained with the Submillimeter Array\\footnote{The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.} (SMA) \\citep{Ho04} at $\\sim$1 arcsecond resolution, which is a factor of three higher than the previous SiO $J$=2--1 images \\citep{Gui92, Gir01}.\nA high angular resolution is crucial for studying the structure and kinematics of the jet near the base, at which the jet velocity increases up to the terminal velocity.\nIn addition, higher transitions of SiO and CO in the submillimeter waverange enable us to segregate the dense and warm gas in the EHV jet from the lower excitation gas in the cavity wall.\n\n\\section{OBSERVATIONS}\n\nThe observations of the SiO $J$=8--7 and CO $J$=3--2 lines and 350 GHz continuum emission were carried out with the SMA on 2006 December 5 in the extended configuration and on 2006 December 25 in the compact configuration.\nThe two array configurations provided projected baselines ranging from 12\\,m to 222\\,m.\nSince the primary beam of the SMA antenna has a size of $\\sim$35\\arcsec, two pointings separated by 17\\arcsec were observed in order to cover the EHV bullet pair closest to the central source.\nThe receivers have two sidebands; the lower and upper sidebands covered the frequency ranges from 345.5 to 347.5 GHz, and from 355.5 to 357.5 GHz, respectively.\nThe SiO $J$=8--7 and CO $J$=3--2 lines were observed simultaneously in the lower sideband.\nThe SMA correlator divides each sideband of 2 GHz bandwidth into 24 ^^ ^^ chunks\" of 104 MHz width.\nWe used the configuration that gave 256 channels to all chunks, which provided a uniform frequency resolution of 406.25\\,kHz across a 2\\,GHz-wide band.\nThe corresponding velocity resolution was 0.35 km s$^{-1}$.\nWe used Titan for flux calibration, and a pair of quasars 3C84 and 3C111 for amplitude and phase calibrations.\nThe flux calibration was estimated to be accurate to 25\\%.\nThe band pass was calibrated by observing 3C273.\n\nThe calibrated visibility data were Fourier transformed and CLEANed using the MIRIAD package.\nThe velocity-channel maps of the SiO and CO were made with a velocity interval of 1 km s$^{-1}$.\nThe synthesized beam size of the SiO map was 0\\farcs96$\\times$0\\farcs84 with a position angle of $-$84$^{\\circ}$ and that of the CO map was 0\\farcs96$\\times$0\\farcs86 with a position angle of $-$76$^{\\circ}$ with uniform weighting.\nThe rms noise level of the velocity-channel map at 1.0 km s$^{-1}$ width was 0.15 Jy beam$^{-1}$.\nA non-linear joint deconvolution, MOSSDI, which is based on the CLEAN-based algorithm, and is part of the MIRIAD package \\citep{Sau96}, was used for deconvolving the images. \n\nThe 350 GHz continuum map was obtained by averaging the line-free chunks of both sidebands.\nTo improve the signal to noise ratio, the upper and lower sidebands data were combined.\nThe synthesized beam size of the map made with uniform weighting was 0\\farcs93$\\times$0\\farcs83 with a position angle of $-$86$^{\\circ}$.\nThe rms noise levels of the 345 GHz continuum maps was 6.4 mJy beam$^{-1}$.\n\n\\section{RESULTS}\n\\subsection{Continuum emission}\n\nThe 350 GHz continuum map reveals a bright compact source at the center.\nThis source has a peak intensity of 352 mJy beam$^{-1}$, and is surrounded by spatially extended emission.\nIn addition to this bright source, there is a faint emission peak of $\\sim$5$\\sigma$ level at $\\sim$8\\farcs3 southeast of the center.\nRecent {\\it Spitzer Space Telescope} observations at mid-infrared wavelengths have resolved L1448C into two components, L1448C(N) and L1448C(S) \\citep{Jor06}.\nThe bright submillimeter source at the center corresponds to L1448C(N), which is considered to be the driving source of the highly-collimated molecular outflow and is also referred to as L1448-mm, and the southern faint source corresponds to L1448C(S).\nDust continuum emission from L1448C(S) was also detected by \\citet{Jor07} at 230 GHz and 350 GHz with the SMA, and by \\citet{Mau10} at 107 GHz with the Plateau de Bure Interferometer (PdBI).\n\nThe visibility amplitudes plot for L1448C(N) as a function of {\\it uv} distance (Fig.~\\ref{fig2}) suggests that this source consists of two components; one is from a spatially extended envelope that dominates the flux at a {\\it uv} distance of $<$50 k$\\lambda$, and the other is from a compact source that is prominent at $>$50 k$\\lambda$.\nThe visibility amplitude profile was fit by two circular gaussian components; one is an extended component with a deconvolved size of $\\sim$3.6\\arcsec and a flux of $\\sim$440 mJy, and the other is a compact component with a deconvolved size of $\\sim$0.3\\arcsec and a flux of $\\sim$330 mJy.\nThe peak flux values of the extended and compact components correspond to $\\sim$27 mJy and $\\sim$330 mJy beam$^{-1}$, respectively, per 0\\farcs93$\\times$0\\farcs83 beam.\nAs shown in Fig. 1a, the extended component appears as a bump in the northwest and a tail in the southeast (both of them are in the 3$\\sigma$ level) of the central source, suggesting that this emission extends along the outflow axis.\nSimilar emission feature along the outflow axis is also seen in the maps of 3 mm \\citep{Gui92}, 2.6 mm \\citep{Bac95}, 1.4 mm \\citep{Sch04}, and 1.3 mm \\citep{Mau10} observed with 1.5--3\\arcsec resolution.\nSuch a faint elongated feature was clearly seen in L1157-mm, and was interpreted as the edges of the cavity excavated by the outflow \\citep{Gue97}.\nIt is, therefore, possible that the extended emission in L1448C(N) also delineates the inner part of the envelope which is disturbed by the outflow, although the cavity-like structure is not clearly seen.\nAn alternative interpretation is that the faint emission comes from an embedded companion.\nA faint secondary peak seen in the 230 GHz map of \\citet{Jor07} implies this possibility.\nHowever, the position of the secondary peak at 230 GHz in \\citet{Jor07} is 1\\arcsec offset toward the northwest from that of our 350 GHz map.\nAs discussed in \\citet{Jor07} such a small difference in position between the 350 GHz peak and 230 GHz peak could be introduced by the extended emission from the envelope component sampled by different {\\it uv} coverages.\nSince the total flux at 350 GHz observed with the SMA (extended + compact components) corresponds to 43 \\% of the flux observed by \\citet{Hat05} at 850 $\\mu$m with SCUBA at the JCMT (1.737 Jy per 14\\arcsec beam), it is possible that the extended component that is not sampled well with the SMA affects the morphology of the faint component.\nAlthough the northwestern bump may harbor an embedded companion, it is likely that most of the extended emission arises from the inner part of the envelope.\nTherefore, the former scenario, which attributes the extended component to the envelope--outflow interacting region, is more preferable in this case.\n\nThe visibility amplitude profile at {\\it uv} distance longer than $\\sim$50 k$\\lambda$ implies that the compact component is not point-like but has a spatially resolved structure.\nIn order to exclude the contamination from the extended envelope component, we made a map using the visibility data with the {\\it uv} distances greater than 70 k$\\lambda$ (see Fig.1b).\nThe synthesized beam of this map is 0\\farcs70$\\times$0\\farcs50 with a position angle of $-$87.2$^{\\circ}$.\nIt is shown that the compact source has an elongation along the axis perpendicular to the outflow axis.\nA two dimensional Gaussian fit to the visibility data with {\\it uv} distances larger than 70 k$\\lambda$ yields the deconvolved major and minor axes of 0.37\\arcsec (90 AU) and 0.26\\arcsec (65 AU), respectively, with a position angle of $\\sim$70$^{\\circ}$.\nThe source position derived from the fit is $\\alpha$(J2000) = 3$^h$25$^m$38.873$^s$, $\\delta$(J2000)=30$^{\\circ}$44$'$05\\farcs35.\nThis position agrees well (less than 0\\farcs1) with the 3.6 cm continuum position observed with a smaller beam of 0\\farcs31$\\times$0\\farcs27 \\citep{Rei02}.\n\nThe 350 GHz flux of L1448C(S) was measured to be $\\sim$60 mJy, which is consistent to that reported by \\citet{Jor07}.\nL1448C(S) was detected in 230 GHz by \\citet{Jor07} but not by \\citet{Mau10} at the same frequency.\nThe non-detection of this source by \\citet{Mau10} is probably because this source is located near the edge of the primary beam of the PdBI ($\\sim$22\\arcsec).\nIf the response of their primary beam is taken into account, the 230 GHz continuum source with a flux of 12.8$\\pm$3 mJy detected by \\citet{Jor07} could be below the detection limit of their observations ($\\sim$8.4 mJy beam$^{-1}$).\nAlthough L1448C(S) is bright in mid-infrared, its sub-mm flux is more than ten times weaker than L1448C(N).\nIt is unlikely that such a small sub-mm flux is due to the effect of missing flux, because single-dish measurements at 450 and 350 $\\mu$m \\citep{Cha00} showed no hint of the secondary component to the south of L1448C(N) even though the angular resolution ($\\sim$8\\arcsec) was comparable to the separation of two sources.\n\n\n\\subsection{The SiO Jet}\n\nThe SiO $J$=8--7 emission was detected in two velocity ranges from $-$70 to $-$12 km s$^{-1}$ (blueshifted) and from 20 to 71 km s$^{-1}$ (redshifted) with respect to the systemic velocity of $V_{\\rm LSR}{\\sim}$5.0 km s$^{-1}$.\nFig.~\\ref{fig3} shows velocity channel maps of the SiO $J$=8--7 emission at 10 km s$^{-1}$ intervals.\nIt is shown that the SiO emission comes from the jetlike narrow region with its blueshifted part to the northwest and the redshifted part to the southeast of L1448C(N).\nThe SiO $J$=8--7 jet is partially resolved along its minor axis; after deconvolution from the SMA beam, the width of the SiO jet is $\\sim$0.8\\arcsec ($\\sim$200 AU) FWHM on average.\nIn order to estimate the missing flux, we have smoothed the SMA map to a resolution of 14\\arcsec and compared it with the SiO $J$=8--7 spectra observed with the James Clerk Maxwell Telescope \\citep[JCMT;][]{Nis07}.\nIt is found that 85--100 \\% of the single-dish flux is recovered by the SMA, implying that almost all the SiO $J$=8--7 emission arises from the narrow jet.\nSiO $J$=8--7 emission mainly comes from the B1 and R1 ^^ ^^ bullets'' identified in the single-dish CO $J$=2--1 map by \\citet{Bac90}.\nThe SiO jet consists of a chain of knots with a typical size scale of $\\sim$1--1.5\\arcsec.\nA comparison with the previous SiO $J$=2--1 maps with $\\sim$3\\arcsec resolution \\citep{Gui92, Gir01} reveals that the three pairs of knots close to L1448C(N) seen in the SiO $J$=8--7 map corresponds to the inner pair BI and RI in the SiO $J$=2--1 map, and the two pairs in the downstream correspond to the outer pair BII and RII.\nThe innermost knot pair BIa and RIa are located within 1\\arcsec (250 AU) from L1448C(N).\nThe high-resolution image also shows that the SiO jet is not straight.\nA close-up view of the high velocity component (Fig.~\\ref{fig5}) shows that the jet changes its position angle from +15$^{\\circ}$ at BII, $-$25$^{\\circ}$ at BI, $-$20$^{\\circ}$at RI, to $-$5$^{\\circ}$ at RII.\nThe kinks between BI and BII, and RI and RII are also seen in the previous SiO $J$=2--1 maps of $\\sim$3\\arcsec resolution \\citep{Gui92, Gir01}.\nHowever, it is more obvious in the higher resolution image.\nIn addition, it is clear that the jet axes in BI and RI are also misaligned by 5$^{\\circ}$.\n\\subsection{CO Jet and outflow}\n\nThe CO $J$=3--2 emission was detected in the wide velocity range from $-$77 km s$^{-1}$ to $+$79 km s$^{-1}$ with respect to the systemic velocity.\nIn the velocity ranges of ${\\Delta}V < {\\pm}$40 km s$^{-1}$ (Fig. 4a--4d), the CO $J$=3--2 emission delineates two V-shaped structures open to the northwest and southeast with a common apex at the position of L1448C(N).\nThe opening angles of the V-shape features become narrower as the velocity offset increases, suggesting that the CO emission in the V-shaped features comes from the limb-brightened shells.\nThe largest opening angle of the shell is $\\sim$60$^{\\circ}$ in the blueshifted lobe, while it is $\\sim$40$^{\\circ}$ in the redshifted lobe.\nIn the higher velocity ranges of ${\\Delta}V = {\\pm}$41--70 km s$^{-1}$ (Fig. 4e, 4f, and 4g), the CO emission comes from a narrow jet-like region.\nThe CO flux recovered by the SMA in each velocity range was estimate by comparing the CO spectra observed by the SMA with those observed by the JCMT.\nThe SMA map was smoothed to be a 14\\arcsec resolution so as to match with the beam of the JCMT.\nIt is found that the recovered flux is only $\\sim$20~\\% in the lowest velocity ranges (${\\Delta}V = {\\pm}$1--10 km s$^{-1}$; Fig. 4a), and is $\\sim$50\\% in the next velocity range of ${\\Delta}V = {\\pm}$11--20 km s$^{-1}$ (Fig. 4b). \nThe edges of the shells in Fig. 4a and 4b look very steep probably because significant amount of the CO emission from spatially extended component was filtered out by the interferometer.\nIn fact, previous examples of the L1157 outflow and IRAS 04166+2706 outflow revealed that the edges of the bipolar cavities were emphasized in the maps made with the interferometer data alone, while the cavities were filled by the diffuse CO emission when the single-dish data were added \\citep{Gue96,San09}.\nOn the other hand, in the higher velocity ranges with ${\\Delta}V > {\\pm}$20 km s$^{-1}$, 80--100\\% of the CO flux was recovered by the SMA.\nThis suggests that almost all the CO $J$=3--2 flux in the extremely high velocity ranges (Fig. 4e, 4f, and 4g) comes from the narrow jet.\n\nAs in the case of the SiO jet, the CO $J$=3--2 jet also shows clumpy structure.\nIn addition, most of the knots seen in the SiO $J$=8--7 map have their counterparts in the CO map.\nHowever, the innermost knots BIa and RIa, which are significant in the SiO map at ${\\Delta}V = {\\pm}$21--60 km s$^{-1}$ are barely seen in the CO map.\nThis is probably because most of the CO molecules in these knots are excited to the levels higher than $J$=3 because of high density and high temperature.\nSimilar feature with strong SiO and weak CO in the close vicinity of the protostar was also observed in the highly collimated jet in the HH211 outflow \\citep{Pal06, Lee07b}.\nAnother difference between the CO jet and SiO jet is seen between BIc and BIIa, and RIc and RIIa. \nThe CO map reveals the knot pair labeled BI-II and RI-II, while the SiO map shows only faint emission (Fig.~\\ref{fig5}).\nIn the CO $J$=3--2, the overall distribution of the EHV jet is more continuous as compared to the SiO distribution. \n\nAs in the case of the SiO jet, the blue and red axes of the CO jet are also misaligned by $\\sim$5$^{\\circ}$.\nThe kinks between BI and BII, and RI and RII are also seen in the CO jet.\nIn addition, the ridge of the CO emission wiggles along the jet axis.\nThis wiggling feature is clearly seen in the maps of the highest velocity ranges (Fig.~\\ref{fig6}).\nIt is likely that each knot has been ejected in slightly different direction.\nSince the typical knot separation, 2--3\\arcsec corresponds to a time interval of 15--20 yr, the observed jet wiggling suggests that the direction of jet ejection also varies in a similar time scale.\nOn the other hand, the CO emission with lower intensity that is surrounding the emission ridge tends to extend linearly along the axes.\nThe transverse width of the lower level CO emission component increases with a distance from the source.\nThis is probably because significant part of the CO emission comes from the outflow shell even in the extremely high velocity ranges (see the next section).\nThere is faint CO emission with the highest velocity seen ahead of the SiO jet.\nSince this highest velocity emission is spatially extended, it is likely that it arises from the highest velocity part of the shells.\n\n\\subsection{Kinematics of the jet along its axis}\n\nPosition-velocity (P-V) diagrams of the SiO and CO emission along the jet axes (the position angle is $-$25$^{\\circ}$ for the blueshifted part and is $-$20$^{\\circ}$ for the redshifted part) are shown in Fig.~\\ref{fig7}.\nDue to the change of the position angle, the outer knots BIIb and RIIb do not appear in these P-V diagrams.\n\nThe velocity structure of the jet is well traced in the SiO.\nThe jet velocity rapidly increases within $\\backsimeq$1\\arcsec from the star, and reaches close to its highest velocity (${\\pm}$65 km s$^{-1}$ from $V_{\\rm sys}$) at $\\sim$5\\arcsec from the star, which corresponds to the positions of the BIc and RIc knots.\nThe velocity dispersion is extremely large (${\\Delta}V{\\sim}$50 km s$^{-1}$ at 1$\\sigma$ level) at the base, while it narrows to ${\\Delta}V{\\sim}$20 km s$^{-1}$ at the positions of the BIc and RIc knots. \nIn the positions further downsteam, from BIc to BIIa and from RIc to RIIa, the observed radial velocity slightly decreases in the blueshifted side, while it increases in the redshifted side.\nThese radial velocity changes are probably due to the change of inclination angle, because the position angle of the jet also changes at these positions.\nThe velocity pattern shown in the SiO is similar to those of the atomic jets from class I source and T Tauri stars, in which the low-velocity components with broad line widths are located near the base and the high-velocity components with narrower line widths are located further from the source \\citep[e.g.][]{Pyo02}.\nIn addition to the global velocity structure, each knot shows its internal velocity gradient with its higher velocity in the upstream side and lower velocity in the downstream side.\n\n\\subsection{Kinematics of the CO outflow shells}\n\nIn the CO $J$=3--2, the jet shows a similar velocity pattern with a similar velocity centroid as the SiO.\nIn addition to the jet, the P-V map of the CO shows extended low velocity features (slower than $\\pm$15 km s$^{-1}$) and linear velocity features with the velocity magnitude increasing with the distance from the source (^^ ^^ Hubble-law''). \nThese Hubble-law features are obviously different from the jets, and are not seen in the SiO, suggesting that they are from the outflow shells.\nIt should be noted that the radial velocity offsets of the linear velocity components become larger than 50 km s$^{-1}$ at around $\\backsimeq$10\\arcsec from the star, and contaminate the CO map in the EHV range.\n\nThe observed velocity pattern of the outflow shell is different from that predicted by the jet-driven bow shock model, in which the P-V structure shows a convex spur with high-velocity components at the jet head \\citep[e.g.][]{Mas93, Che94}; it is rather similar to the velocity pattern produced by the wide-angle wind model \\citep{Shu91, Shu00, Li96}.\nIn the wide-angle wind model, an outflow shell consists of ambient material swept-up by a radial wind from a young star.\nA ^^ ^^ Hubble-law'' in the shell velocity is expected if the shell is expanding into an ambient medium with a radial density profile of ${\\propto}r^{-2}$ \\citep{Shu91}.\nThe shape of the shell is determined by the combination of the poloidal density profiles of the wind and ambient gas.\nIt is approximately parabolic for an $X$-wind type of wide opening angle wind with an angle-dependent density profile of ${\\propto}1\/{\\rm sin}^2{\\theta}$ (where $\\theta$ is an angle measured from the axis of the flow) expanding into an ambient medium with a ${\\propto}{\\rm sin}^2{\\theta}\/r^2$ density profile, which is appropriate for magnetized cores \\citep{Li96}.\nTherefore, we adopted the simplified analytical model of a wind-driven model proposed by \\citet{Lee00} to examine whether the observed morphology and kinematics of the CO outflow shells can be explained by means of wide-opening angle wind model.\nIn the cylindrical coordinate system, the structure and velocity of the shell can be written as follows:\n \\begin{equation} z=CR^2, {\\it v}_R={\\it\nv}_0R, {\\it v}_z={\\it v}_0z, \\end{equation} \nwhere z is the distance along the outflow axis; R is the radial size of the outflow perpendicular to z ;\n$C$ and {\\it v}$_0$ are free parameters which describe the spatial and velocity distributions of the outflow shell, respectively.\nThe observed outflow shell features and the velocity patterns of the redshifted and blueshifted lobes were successfully reproduced by the model curves with $C$=0.8 arcsec$^{-1}$and {\\it v}$_0$=5.0 km s$^{-1}$ arcsec$^{-1}$, and $C$=0.6 arcsec$^{-1}$ and {\\it v}$_0$=5.0 km s$^{-1}$ arcsec$^{-1}$, respectively.\nHere the inclination angle of the outflow axis with respect to the plane of the sky was assumed to be 21$^{\\circ}$, which is derived from the proper motion measurement for the SiO $J$=2--1 jet by \\citet{Gir01}.\nThe dynamical age of the outflow shell is given by 1\/$v_0$ and is estimated to be $\\sim$240 yr.\nThis number is roughly consistent with the dynamical age of $\\sim$500 yr derived from the extent of the shell ($\\sim$25\\arcsec = 6300 AU), the mass-weighted-mean radial velocity of the shell ($\\sim$22 km s$^{-1}$), and the inclination angle of the outflow axis ($\\sim$21$^{\\circ}$).\nThe model curves that delineate the outer boundaries of the lobes projected onto the plane of the sky are shown in Fig.~\\ref{fig8} on top of the contours of the outflow shells.\nThe model curves of the P-V maps are shown in Fig.~\\ref{fig7}.\nHowever, this simplified wind-driven model has difficulty in reproducing several observed features.\nFirst, the observed CO intensity drops sharply at the systemic velocity and does not extend to the opposite velocity ranges, while the model curves on the P-V maps predict the emission at $V_{\\rm LSR}{\\sim}$10 km s$^{-1}$ in the blueshifted lobe and at $V_{\\rm LSR}{\\sim}$0 km s$^{-1}$ in the redshifted lobe.\nSecond, the shapes of the shells in the different velocities cannot be reproduced (Fig.~\\ref{fig9}).\nThe observed CO emission shows V-shaped distributions in the velocity ranges close to the systemic velocity.\nOn the other hand, the model curves predict elliptical shapes (Fig. 9a and 9b).\nIn addition, the transverse width of the CO shell becomes narrower as the velocity offset increases, while the model predict the opposite trend.\nIt should be noted that the CO emission near the protostar in Fig. 9c and 9d arises from the jet.\n\n\n\\subsection{Kinematics across the jet}\n\nIn order to search for the signs of jet rotation, the velocity gradient along the minor axes of the jet was examined.\nThe P-V diagrams of the SiO and CO at the positions of innermost two pairs of knots (Fig.~\\ref{fig11}) show no clear velocity gradient at the positions of BIa, RIa, and RIb.\nOn the other hand, there is some hint of velocity gradient in the SiO at the position of BIb; the southwestern side of the jet tends to be more blueshifted than the northeastern part.\nHowever, this velocity gradient is in the opposite sense to the rotation pattern of the NH$_3$ core with its blueshifted part in the northeast and the redshifted part in the southwest \\citep{Cur99}.\nThis suggests that the observed velocity gradient is not due to the rotation.\n\n\n\\subsection{Physical properties of the EHV jet}\n\nThe physical parameters of the EHV jet were estimated using the CO flux measured in the velocity ranges of ${\\Delta}V$=50--70 km s$^{-1}$, in which most of the CO flux was recovered by the SMA.\nWe assumed that the CO emission in these velocity ranges is optically thin, and that the excitation condition of the CO follows the LTE.\nA fractional abundance of the CO and a mean atomic weight were adopted to be 10$^{-4}$ and 1.41, respectively.\nThe kinetic temperature of molecular gas in the EHV jetwas derived to be 500--1200 K from the far infrared lines of CO, H$_2$O, and H$_2$ \\citep{Nis99,Nis00}, and $>$500 K from the millimeter and submillimeter SiO lines \\citep{Nis07}.\nHowever, it is uncertain whether the CO (3-2) emission arises from the same gas component that contributes to the higher transition lines in the far-infrared.\nIn fact, \\citet{Gus08a} have modeled the multi-transition SiO data observed by \\citet{Nis07} using their face-on C-type shock models, and obtained much lower temperature of 70--90 K.\nThis implies that the most of the gas in the EHV jet is not so warm as $>$500 K. \nTherefore, an excitation temperature of $\\sim$100 K was assumed for the calculation.\n\nThe mass for each of the blueshifted and the redshifted jets is estimated to be $\\sim$10$^{-3}$ $M_{\\odot}$, which is a few times higher than the bullet mass estimated by \\citet{Bac90} using an assumption of $T_{\\rm ex}{\\sim}$20 K.\nThe momentum and kinetic energy of the bipolar jet are estimated to be 0.3 $M_{\\odot}$ km s$^{-1}$ and 5.0$\\times$10$^{44}$ erg, respectively.\nHere, the calculation was done using the de-projected jet velocity under the assumption that the jet axis is inclined by 21$^{\\circ}$ from the plane of the sky \\citep{Gir01}.\nThe dynamical timescale of the jet derived from the length ($\\sim$ 20\\arcsec = 5000 AU) and mass weighted mean (de-projected) velocity is estimated to be only $\\sim$150 yr.\nBecause of the rather large mass and short timescale, the obtained mass loss rate is also large, $\\sim$10$^{-5}$ $M_{\\odot}$ yr$^{-1}$.\nFurthermore, the high velocity of $\\sim$160 km s$^{-1}$ (corrected for the inclination of 21$^{\\circ}$) brings extremely large momentum supply rate of $\\sim$2${\\times}$10$^{-3}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$ and mechanical luminosity of $\\sim$26$L_{\\odot}$ for the jet.\n\nIt should be noted that the derived total mechanical luminosity of $\\sim$26 $L_{\\odot}$ is a factor of 3.5 larger than the bolometric luminosity of the central source (7.5 $L_{\\odot}$).\nOne possible reason for such a discrepancy is that the mass of the jet was overestimated.\nSince the CO $J$=3--2 emission comes from both the jet and shell, the high velocity part of the shell may contaminate the CO flux that was used to calculate the mass of the jet.\nHowever, this effect should not be significant, because the CO flux from the shell does not dominate the CO flux at ${\\Delta}V >$50 km s$^{-1}$ velocity range.\nThe second possible reason is the excitation temperature.\nThe assumed excitation temperature of $T_{\\rm ex}{\\sim}$100 K is much lower then the kinetic temperature of $>$500 K derived from the far infrared measurements \\citep{Nis99,Nis00}.\nHowever, the assumption with higher excitation temperature increases the mass and dynamical parameters (i.e. momentum, kinetic energy, mass loss rate, momentum rate, and mechanical luminosity).\nFor example, the assumption of $T_{\\rm ex}$= 500 K increases the mass and dynamical parameters by a factor of 3.8, and brings an extremely large mechanical luminosity of $\\sim$90 $L_{\\odot}$.\nThis implies that the bulk of the molecular gas in the jet is not so warm as 500 K, and that the warm gas which contributes to the far infrared emission is not the major component.\nOn the other hand, the lower excitation temperature of 40--50 K reduces the mass.\nHowever this effect is limited to a factor of 1.5 at the most.\nAnother possible reason is the CO abundance, which was assumed to be 10$^{-4}$.\nThis value can be as high as 4$\\times$10$^{-4}$ in the chemical model of protostellar winds proposed by \\citet{Gla91}, in which molecules such as CO and SiO are formed via gas-phase reactions in an initially atomic protostellar wind.\nIf the higher CO abundance of 4$\\times$10$^{-4}$is adopted, the mass and all dynamical parameters are reduced by a factor of 4, and the mechanical luminosity of the jet becomes $\\sim$7 $L_{\\odot}$, which is comparable to the bolometric luminosity of the central source.\nHowever, this value for the mechanical luminosity of $\\sim$7 $L_{\\odot}$ should be the lower limit, because it is derived under the assumption of the optically thin CO emission.\nThe physical parameters derived using $T_{\\rm ex}$ = 100 K and CO\/H$_2$ abundance ratio of 4$\\times$10$^{-4}$ are given Table 1.\n\n\\subsection{Physical parameters of the outflow shell}\n\nThe mass, momentum, and kinetic energy of the outflow shell were estimated using the CO flux measured in the velocity ranges of $\\pm$1--40 km s$^{-1}$ from the systemic velocity.\nThe CO emission is assumed to be optically thin and in the LTE condition with an excitation temperature of 40 K, which is derived from the observed peak brightness temperature of the CO $J$=3--2 line.\nSince most of the gas in the shell component is considered to be the swept-up ambient material, the canonical CO abundance of 10$^{-4}$ was used for the calculations.\nThe dynamical parameters of the outflow shell are summarized in Table 2. \nSince significant fraction of the CO flux is missed in the low velocity ranges, Table 2 show the parameters with and without correction for the effect of missing flux. \nThe inclination angle of the outflow axis is assumed to be 21$^{\\circ}$ as in the case of the EHV jet.\nThe dynamical timescale of the outflow shell is adopted to be $\\sim$240 yr, which is derived from the modeling descried in the previous section.\nTable 2 shows that the dynamical parameters of the outflow shell are comparable to those of the EHV jet listed in Table 1.\nIf the effect of the missing flux is corrected, the outflow shell has a momentum supply rate of $\\sim$1.8$\\times$10$^{-3}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$, and a mechanical luminosity of $\\sim$8.6 $L_{\\odot}$.\nThe mechanical lumiosity derived here is comparable to the bolometric luminosity of the central source and the mechanical luminosity of the EHV jet.\nIt should be noted that the dynamical parameters for the redshifted outflow are affected by the contamination from the L1448C(S) outflow (see the next section).\nHowever, the effects of L1448C(S) outflow to the dynamical parameters are not so significant, because this outflow is seen only in the low velocity ranges.\n\n\n\\subsection{CO outflow from L1448C(S)}\n\nIn the CO maps in the low velocity ranges such as Fig. 4a and 4b, there is a blueshifted component in the redshifted lobe.\nThis blueshifted component shows a triangle shape with its apex at the position of L1448C(S), and extends to the northeast direction with a position angle of $\\sim$40$^{\\circ}$.\nIt is likely that this blueshifted component is related to the activity of L1448C(S).\nFig.~\\ref{fig11}, which provides a close-up view of the L1448C(S) region, shows a redshifted counterpart to the southwest of L1448C(S), a compact component at $\\sim$1\\arcsec southwest of L1448C(S) and another extended component to the southwest of the V-shaped shell of the L1448C(N) outflow.\nThis NE-SW outflow from L1448C(S) is also highly collimated.\nThe opening angle of the lobes is $\\sim$40$^{\\circ}$, which is similar to the redshifted lobe of the L1448C(N) outflow.\nHowever, this outflow is seen only in the velocity ranges slower than 15 km s$^{-1}$, and has no high-velocity jet-like component.\nThere is no SiO $J$=8--7 emission either.\n\nThe redshifted part of the L1448C(S) outflow overlaps with the western wall of the L1448C(N) outflow.\nAt the place where the two outflows are superposed, the CO emission is significantly enhanced and the wall of the L1448C(N) outflow lobe is bending.\nTherefore, it is possible that the two outflows are intersecting, although three dimensional geometries of two outflows are uncertain.\n\nThe physical parameters of this outflow were estimated assuming that the CO emission is optically thin and that the excitation condition of the CO is in LTE.\nThe fractional abundance of the CO was adopted to be the canonical value, 10$^{-4}$, because the V-shaped morphology and low velocity suggest that the bulk of the L1448C(S) outflow is likely to be the swept-up ambient gas.\nA mean atomic weight of the gas was assumed to be 1.41.\nThe excitation temperature was assumed to be 20 K, which is same as the rotational temperature of the dense gas surrounding L1448C derived from the NH$_3$ observations \\citep{Cur99}.\nThe mass of the blueshifted component is estimated to be 5.3$\\times$10$^{-4}$ $M_{\\odot}$ and that of the redshifted component is 2.7$\\times$10$^{-4}$ $M_{\\odot}$.\nSince the CO $J$=3--2 emission is assumed to be optically thin, the values derived here are the lower limits.\nIn addition, the mass of the redshifted component is likely to be underestimated, because we exclude the region where the two outflows are superposed.\nIt is also possible that part of the flux from the spatially extended component is missing.\nThe dynamical parameters of the flow are summarized in Table~\\ref{table3}.\nTable 3 gives two kinds of values; one is corrected for the inclination effect and the other is without this correction (uncorrected).\nHere, the inclination angle of the L1448C(S) outflow from the plane of the sky was assumed to be 32.7$^{\\circ}$, which corresponds to the mean inclination angle from assuming randomly oriented outflows.\nWe also assumed that the outflow lobes have a size of $\\sim$20\\arcsec (5000 AU).\nThe actual size of this outflow is not clear because of confusion with L1448C(N) outflow and its sidelobes.\nHowever, there is no counterpart of this outflow in the single-dish CO $J$=3--2 map \\citep{Hat07}.\nThis suggests that the spatial extent of this outflow is not large, or the spatially extended component is fainter than the sensitivity limit of the single-dish measurement. \nIn the later case, the faint component would not make significant contribution to the dynamical parameters even though its spatial extension is large.\n\n\\section{DISCUSSION}\n\n\\subsection{Compact disk around L1448C(N)}\n\nThe compact component of dust continuum emission is partially resolved by the 0\\farcs7$\\times$0\\farcs5 beam.\nThe observed structure is elongated perpendicular to the outflow axis, suggesting that this component traces the disk surrounding the protostar. \nIf the beam-deconvolved major axis, $\\sim$90 AU, represents the diameter of the disk, the disk size of L1448C(N) is comparable to those around the youngest protostars such as HH211 \\citep{Lee07b} and HH212 \\citep{Cod07, Lee07a, Lee08}.\nIf the measured flux (330 mJy) comes from the region with 0\\farcs37$\\times$0\\farcs26 size, the brightness temperature corresponds to $\\sim$35 K.\nThe spectral energy distribution (SED) of the compact source from 8.3 GHz to 350 GHz is shown in Fig.~\\ref{fig12}.\nThe measured flux densities are fit by a single power law with a spectral index $\\alpha$ of 1.98 (solid line in Fig~\\ref{fig12}), which agrees with the previous result ($\\alpha$ = 1.84) of \\citet{Sch04}.\nThis spectral index is smaller than $\\alpha$ = 3.4 derived from the photometric broadband measurement including the contribution of the larger scale envelope component \\citep{Fro05}.\nSince the emission at cm wavelengths is likely to be the free-free emission from shock-ionized gas,\nthe contribution of the free-free component at mm and sub-mm wave ranges was estimated by using the equations given by \\citet{Cur90}.\nUsing the parameters of the stellar wind in \\citet{Cur90}, the flux densities of the free-free emission at mm and sub-mm wave ranges were estimated to be less than 1 mJy (dotted line in Fig~\\ref{fig12}).\nTherefore, the small index number is unlikely to be due to the contribution of the free-free component.\nThe power law fit assumes that the emission is optically thin and that the Rayleigh-Jeans approximation is valid.\nHowever, the observed index $\\alpha\\sim$2 is close to the value for the blackbody radiation, suggesting that the emission is optically thick.\nIn addition, a brightness temperature of 35 K suggests that the Rayleigh-Jeans approximation is not applicable in the mm and sub-mm wave ranges.\nTherefore, we applied an optically thick fit without Rayleigh-Jeans approximation using the formula,\n\\begin{equation}\nS_{\\nu}={\\Omega}_s B_{\\nu}[1-{\\rm exp}(-{\\tau}_{\\nu})],\n\\end{equation}\nWhere $S_{\\nu}$ is the flux density, ${\\Omega}_s$ is the source size, B$_{\\nu}$ is the Planck function, and ${\\tau}_{\\nu}$ is the dust optical depth, which is assumed to follow the power low, ${\\tau}_{\\nu} {\\propto} {\\nu}^{\\beta}$.\nAssuming that the source size is 0\\farcs37$\\times$0\\farcs26 and a dust temperature of 40 K, the SED fit provides $\\beta$ = 1.3 and ${\\tau}_{350 {\\rm GHz}}$ = 7.5 (dash-dotted line in Fig~\\ref{fig12}).\nThe average optical depth at 350 GHz for a disk of mass (gas+dust) $M_D$ and radius $R_D$ is given by \n\\begin{equation}\n<\\tau_{350 {\\rm GHz}}> = \\left( \\frac{0.5}{{\\rm cos} \\theta} \\right) \\left( \\frac{M_D}{0.1 M_{\\odot}} \\right) \\left( \\frac{R_D}{100 {\\rm AU}} \\right) ^{-2},\n\\end{equation}\nwhere $\\theta$ is the disk inclination angle to the line of sight \\citep{Jor07}.\nUsing this relation, the mass of the disk with ${\\tau}_{350 {\\rm GHz}}$ = 7.5, $R_D$ = 45 AU, and $\\theta$ = 69$^{\\circ}$ (assuming that the disk is perpendicular to the jet axis) is estimated to be 0.11 $M_{\\odot}$.\nThe mass derived here is approximately twice as large as the lower limit of 0.047 $M_{\\odot}$, which is derived under the assumption of optically thin emission.\n\n\\subsection{Stellar mass loss rate and its implication to the protostellar evolution}\n\nSince protostellar jet is considered to be closely linked to the mass accretion, the stellar mass-loss rate gives us a rough estimate of the mass accretion rate onto the star.\nTheoretical estimate for the ratio of mass outflow to mass accretion rate ($\\dot{M}_{\\rm out}\/\\dot{M}_{\\rm acc}$) is ${\\sim}$1\/3 for an X-wind type magneto-centrifugal wind \\citep[e.g.][]{Shu94}.\nIf the $\\dot{M}_{\\rm out}\/\\dot{M}_{\\rm acc}$ ratio is assumed to be $\\sim$0.3, the total mass-loss rate (blue + red) derived from the CO flux, 2.4$\\times$10$^{-6}$ $M_{\\odot}$ yr$^{-1}$, gives us the mass accretion rate of 8$\\times$ 10$^{-6}$ $M_{\\odot}$ yr$^{-1}$.\nIn spite of rather high accretion rate, the observed bolometric luminosity is only 7.5 $L_{\\odot}$, suggesting that the mass of the central star is still very small.\nIf most of the observed bolometric luminosity is released by means of accretion, the mass of the central star is calculated by using the relation, $M_*$ = $L_{\\rm acc}R_{*}\/G\\dot{M}_{\\rm acc}$, where $M_*$ is the mass of the central star, $L_{\\rm acc}$ is the accretion luminosity, and $\\dot{M}_{\\rm acc}$ is the mass accretion rate onto the protostar.\nThe radius of the protostar, $R_*$, is considered to be $\\sim$1 $R_{\\odot}$ in the earliest evolutionary stage with very low mass and $\\sim$3 $R_{\\odot}$ in the later stage \\citep[e.g.][]{Sta88}.\nThe mass of the central star is estimated to be 0.03--0.09 $M_{\\odot}$ for an accretion luminosity of 7.5 $L_{\\odot}$.\nWith a stellar mass of 0.03--0.09 $M_{\\odot}$, the Keplerian velocity at the surface of the protostar becomes $\\sim$80 km s$^{-1}$.\nIn this case, the jet velocity to Keplerian velocity ratio becomes $\\sim$2, which is reasonable if the jet is launched by magneto-centrifugal force \\citep{Shu94,Pud07}.\nIf a constant mass accretion rate of 8$\\times$ 10$^{-6}$ $M_{\\odot}$ yr$^{-1}$ is assumed, the age of the central star is estimated to be (4--12)$\\times$10$^3$ yr.\nThe timescale derived here is consistent to the kinematic age of the larger scale outflow of $\\sim$0.3 pc scale \\citep{Bac90}.\nHowever, morphology of the EHV jets implies that the mass accretion was variable.\nThe highly-colimated EHV jets terminate at $\\sim$20\\arcsec from the source, suggesting that L1448C(N) experienced lower activity phase in the past and enhanced its activity significantly in the last $\\sim$150 yr.\n\n\n\\subsection{Clumpy structure in the L1448C(N) jet}\n\nHigh resolution SiO and CO maps show that the EHV bullets B1 and R1 identified by \\citet{Bac90} consist of chains of knots.\nIf the BI-II and RI-II knots are included, the knots are aligned with almost equal intervals of $\\sim$2\\arcsec (500 AU).\nSimilar knotty structure with semi-regular intervals is also seen in the SiO and CO jets in HH211 \\citep{Hir06, Pal06, Lee07b} and HH212 \\citep{Cod07, Lee07a}.\nIn the case of HH211 and HH212, the knots seen in the SiO and CO have their counterparts in the near infrared H$_2$ emission except the innermost knots pairs that were highly obscured.\nIn the case of the L1448C(N) outflow, H$_2$ emission knots are seen only in the northern blueshifted side and not in the southern redshifted side \\citep{Dav94, Eis00}.\nThis is probably because the axis of the L1448C(N) jet is inclined from the plane of the sky and the near infrared emission in the southern part is obscured by the dense gas envelope traced by the NH$_3$ emission \\citep{Cur90}.\nIn the northern side, the morphology of the SiO jet coincides well with that of the H$_2$ jet \\citep{Eis00}, which also shows a kink between BI and BII components.\nTherefore it is likely that the knots in the L1448 jet are the internal bow shocks in the jet beam as in the cases of HH211\\citep{Hir06, Pal06, Lee07b} and HH212 \\citep{Cod07, Lee07a}.\nIn fact, some of the SiO knots are partially resolved in the transverse direction, and the RII-a knot shows an arc-shaped structure typical of a bow shock (Fig. 3f).\nThe SiO emission is weak at the positions of the BI-II and RI-II knots, suggesting that the shocks at these positions are rather weak as compared to the other knot positions.\nSince the jet is deflected at the positions of BI-II and RI-II knots, it is likely that the jet material there is impacting less dense material surrounding the jet beam.\n\n\n\\subsection{Jet bending}\n\nAs shown in Fig. 5, the blue part and the red part of the jet are misaligned by $\\sim$5$^{\\circ}$ and forming a C-shaped structure bending toward the west.\nSuch a C-shaped bending of the jet could be due to the Lorenz forces between the jet and interstellar magnetic field \\citep{Fen98}, the orbital motion of the jet source in a binary system \\citep{Fen98, Mas02}, or dynamical pressure from external medium \\citep{Fen98}. \nIn the case of the Lorenz forces, a C-shaped bending is expected if the poloidal current in the jet and counter jet flows in the same direction \\citep{Fen98}.\nHowever, this mechanism is difficult to account for the observed bending of the L1448C(N) jet, because typical interstellar magnetic field with several tens of microgauss is not strong enough to bend the jet beam with a density of $>$10$^6$ cm$^{-3}$.\n\nIf the C-shaped bending is produced by the orbital motion of a binary system, the orbital radius and orbital velocity can be estimated by using the analytical model of \\citet{Mas02}.\nHere the jet is assumed to be ejected at a velocity of $v_j$ from one of the binary protostars in a circular orbit of radius $r_0$ and orbital velocity $v_0$.\nThe $z$-axis is parallel to the orbital rotation axis, and $z$=0 is the orbital plane.\nAs shown in Fig. 3 of \\citet{Mas02}, the deflection angle $\\alpha$ of the jet beam near the source is approximated by the $x$=${\\kappa}z\/{\\rm cos}i$ line, where $\\kappa$=$v_0\/v_j$ and $i$ is the inclination angle of the orbital axis with respect to the plane of the sky.\nIn the case of the L1448C(N) jet, the deflection angle $\\alpha$ is estimated to be 2.5$^{\\circ}$, which corresponds to the half of the misalignment angle.\nUsing the jet velocity of $\\sim$160 km s$^{-1}$, the orbital velocity is calculated to be 6.5 km s$^{-1}$.\nSince the mass of the protostar with jet is only 0.03--0.09 $M_{\\odot}$, the total mass of the binary system is considered to be less than 0.18 $M_{\\odot}$.\nTherefore, the radius and period of the orbital morion are estimated to be smaller than 4.2 AU and 20 yr, respectively.\nHowever, such a short period orbital motion cannot account for the observed C-shaped pattern, because\nthe C-shaped bending is seen in the BI and RI parts of the jet with a length of $\\sim$2000 AU with a dynamical time scale of 47 yr.\nIn order to produce the C-shaped pattern with the orbital motion, the orbital period needs to be longer than twice of the dynamical time scale.\n \nIn the case of dynamical pressure of external medium, ambient gas with $n$(H$_2$)$<$10$^4$ cm$^{-3}$ cannot account for bending the jet with a density of $>$10$^6$ cm$^{-3}$, unless the protostar is moving with a velocity that is comparable to the jet velocity.\nOn the other hand, the dynamical pressure caused by the outflow from the nearby protostar, L1448N, cannot be ruled out.\nAs shown in the CO map of \\citet{Bac90}, the redshifted lobe of the L1448N outflow overlaps with the blue lobe of the L1448C(N) outflow.\nThe interaction between the two outflows from L1448C(N) and L1448N has been suggested by \\citet{Bac95}, because the large scale outflow from L1448C(N) shows a considerable bending at the place where the two outflows are overlapping.\nSince the redshifted emission from L1448N outflow reaches close to the position of L1448C(N) \\citep{Bac95}, it is possible that the jet from L1448C(N) is propagating under the influence of the L1448N outflow.\nIn this case, the dynamical pressure from L1448N outflow acts from north to south, and deflects the jet beams to the west if they were ejected to the the northwest and southeast directions. \n\n\\subsection{Deflection and wiggling of the jet}\n\nIn addition to the C-shaped bending, the jet is also deflected toward the east by $\\sim$40$^{\\circ}$ at the position of BI-II and toward the south by $\\sim$15$^{\\circ}$ at the RI-II position.\nSince both sides of the jet are deflected at almost same distance from the central star, the jet deflection is likely to be caused by some variability intrinsic to the driving source rather than by external perturbation.\nThe observed morphology is similar to the S-shaped point-reflection symmetric pattern that is expected if the disk is precessing or wobbling.\nAlthough the jet is not exactly the S-shape but asymmetric in deflection angle, this is probably because of the projection effect.\nIn a binary protostellar system with a disk misaligned with the orbital plane of the binary, the disk wobbles with a period approximately half of the binary orbital period and precesses with a period of $\\sim$20 orbital period \\citep{Bat00}.\nS-shaped point symmetry will be observed if the precession or wobbling time scale is longer than four times of the dynamical timescale.\nSince the jet deflection occurs at BI-II and RI-II the time scale of which is $\\sim$50 yr, the time scale of the precession or wobbling should be longer than $\\sim$200 yr.\nTherefore, if the deflection is due to the precession, the lower limit of the binary orbital period is $\\sim$10 yr.\nSince this orbital period of $\\sim$10 yr is comparable to the period of small scale wiggling shown in Fig. 6, 15--20 yr, this orbital motion can also explain the wiggling feature.\nIf the binary system consists of equal mass protostars with 0.03--0.09 $M_{\\odot}$, the orbital radius is estimated to be 2.4--4.2 AU.\nOn the other hand, if the jet deflection is due to the wobbling of the disk, the orbital period and the separation of the binary are estimated to be $\\sim$400 yr and 30 AU, respectively.\nSince the estimated separation of the binary is smaller than the size of the disk observed with the 350 GHz continuum emission, it is possible that the observed 90 AU scale disk harbors two sources separated by 60 AU.\nHowever, the binary with a separation of $>$60 AU cannot account for the small scale wiggling feature.\n\n\\subsection{Velocity variation of the jet}\n\nThe P-V diagram of the SiO shows that the velocity of the jet varies semi-periodically.\nThe velocity variation is more obvious in Fig.~\\ref{fig13}, which plots the velocity centroid of the SiO emission in the redshifted part of the jet as a function of the distance from L1448C(N).\nIt is shown that the typical amplitude of the variation in velocity centroid is $\\sim$7 km s$^{-1}$.\nSuch a velocity variation is expected if the jet is precessing, the jet is launched from an orbiting object, or the ejection velocity itself varies as a function of time \\citep[e.g.][]{Smi97}.\nThe period of the velocity variation estimated from the de-projected jet velocity and knot separation is $\\sim$15--20 yr.\nSince this time scale is much shorter than the precession time scale that is estimated to be 200 yr, it is unlikely that the velocity variation is caused by the precession of the jet.\nOn the other hand, the orbital motion of the driving source can account for the velocity variation; the binary system with an orbital period of $\\sim$15--20 yr, an orbital radius of 2.4--4.2 AU, a total mass of 0.06--0.18 $M_{\\odot}$ has an orbital velocity of $\\sim$4.7--6.2 km s$^{-1}$, which is comparable to the amplitude of the radial velocity variation.\nHowever, the orbital motion cannot explain the relation between the SiO intensity and velocity gradient.\nAs shown in the P-V map (Fig.~\\ref{fig7}), each SiO knot has its higher velocity in the upstream side and lower velocity in the downstream side.\nThe opposite velocity gradient is always seen in the faint emission between the knots.\nSuch a structure is more likely to be formed by means of periodic variation of the ejection velocity. \nIn such a case, the SiO knots are considered to be formed as the fast moving material plunge into the slow moving material in the downstream \\citep[e.g.][]{Sto93, Sut97}.\nThe periodic variation in the ejection velocity is probably due to the modulation of mass accretion by means of companion.\nIn such a case, the variation amplitude of the jet velocity corrected for the inclination is calculated to be $\\sim$20 km s$^{-1}$. \nThis velocity amplitude corresponds to the shock velocity, which is consistent to the velocity of C-type shocks that can account for the excitation conditions of the far infrared molecular lines \\citep{Nis99, Nis00}.\nSimilar velocity gradients in the knots with the faster part in the upstream side and the slower part in the downstream was also observed in the optical jet of HH111 \\citep{Rag02} and in the CO and SiO jets from IRAS 04166+2706 \\citep{San09}.\n\n\n\\subsection{Driving mechanism of the CO outflow}\n\nThe P-V diagram of the CO $J$=3--2 along the axis (Fig.~\\ref{fig7}) exhibits two kinematic components, i.e. the EHV jet with an almost constant velocity and the outflow shell with a parabolic velocity pattern.\nAlthough highly-collimated jet is clearly seen in both SiO and CO, the parabolic velocity pattern seen in the outflow shells is reproduced by the wind-driven model.\nTherefore, the observational results require a wide-opening angle wind and a collimated jet at the same time.\n\nOne possible mechanism to explain the observed jet+shell structure is the ^^ ^^ unified model'' proposed by \\citet{Sha06}, in which highly-collimated jet component is explained as an on-axis density enhancement of the {\\it X}-wind type of wide opening angle wind launched magnetocentrifugally.\nIn this model, the jet along the axis corresponds to the densest part of the primary wind, and the shell is mostly consisted of the swept-up ambient material.\nTheir numerical model successfully reproduced the structure of a dense and narrow jet surrounded by a conical shell. \nThe other models that can explain the two components structure are proposed by \\citet{Mac08} and \\citet{Ban06}.\nThe model proposed by \\citet{Mac08} predicts two distinct flows from the adiabatic core and the protostar.\nThe flow from the adiabatic core driven by the magnetocentrifugal mechanism has a low velocity and a wide opening angle, while the flow from the protostar, which is mainly driven by the magnetic pressure gradient force, has a high velocity and is well collimated.\nOn the other hand, a model proposed by \\citet{Ban06} predicts the structure with the jet powered by magnetocentrifugal force enclosed by the large-scale outflow driven by the magnetic pressure.\nAlthough these two models reproduce the jet+shell structure similar to the observational results, the velocities of the shells predicted in these model are rather small ($\\sim$5 km s$^{-1}$ for the model of \\citet{Mac08}) because of the shallow gravitational potential at the launching point.\nIn the case of the L1448C(N) outflow, the terminal velocity of the outflow shellreaches to ${\\Delta}V {\\sim} {\\pm}$70 km s$^{-1}$ without inclination correction, which is comparable to the velocity of the EHV jet.\nIn order to launch such a high velocity wind, the launching point of this wind should be close to the launching point of the EHV jet.\nTherefore, the models with two components launched from two different regions do not explain the jet+shell structure in the L1448C(N) outflow.\nIn the case of the {\\it X}-wind with density stratification \\citep{Sha06}, the rather high velocity in the shell component is naturally explained because the shell is driven by the high-velocity primary wind launched at the same region as the EHV jet.\n\n\n\\subsection{Origin of the SiO in the jet}\n\nIt is considered that the SiO molecules observed in jets and outflows are formed as a consequence of grain sputtering in a C-shock releasing Si-bearing material into the gas phase, followed by the reaction with O and OH \\citep{Sch97, Cas97, Gus08a}.\nThe multi-transition SiO lines from the L1448C(N) jet observed by \\citet{Nis07} have been successfully modeled by the steady-state C-shock model of \\citet{Gus08a} with a pre-shock density of $\\sim$10$^5$ cm$^{-3}$ and a shock velocity of 30--45 km s$^{-1}$.\nHowever, the conversion of Si into SiO is initially rather slow in their models, and the predicted SiO line emission predominantly arises from postshock gas $>$100 yr after the passage of shocks.\nSince most of the SiO knots in the L1448C(N) jet have dynamical timescale shorter than the SiO formation timescale, the steady-state C-type shock models of \\citet{Gus08a} does not account for the SiO in the knots close to the central star, especially in the innermost knot pair with extremely short time scale less than 10 yr.\nOne possible explanation is that the SiO molecules existed on the grain mantles and are released into the gas-phase by means of shocks as suggested by \\citet{Gus08b}.\n\nAnother possibility is the formation of SiO in high density primary jet \\citep{Sha06}.\n\\citet{Gla89, Gla91} studied the formation of molecules in protostellar winds, which are originally neutral atomic, and found that significant quantities of SiO can be formed quickly in the close vicinity ($<$0.1 AU) of the central star if the mass-loss rate is high ($>$10$^{-6}$ $M_{\\odot}$).\nSince the mass-loss rate of the L1448C(N) jet is high enough, this scenario of in situ formation also can be the origin of the SiO in the jet.\nThe chemical models of \\citet{Gla89, Gla91} also predict significant amount of CO synthesized in the winds; the CO abundance reaches an equilibrium value of 4$\\times$10$^{-4}$ under the conditions in which observable amount of SiO is formed.\nThe morphological and kinematical similarity of the CO and SiO jets supports the idea that both CO and SiO are formed in the protostellar wind.\n\n\n\\subsection{Properties of L1448C(S)}\n\nThe secondary source L1448C(S) is located at $\\sim$8\\farcs3 (2000 AU) southeast of L1448C(N).\nThe 350 GHz continuum flux of this source is $\\sim$60 mJy, which is five times smaller than the flux from L1448C(N).\nIf an optically thin condition is assumed, the mass of the circumstellar material surrounding L1448C(S) is estimated from the observed flux using the dust mass opacity of 1.75 cm$^2$ g$^{-1}$ \\citep{Oss94} and the formula given by \\citet{Jor07}.\nFor a dust temperature of $\\sim$40 K, the estimated mass in the optically thin limit is 8.6$\\times$10$^{-3}$ $M_{\\odot}$.\nSince L1448C(S) is associated with a molecular outflow, it is highly probable that this source is a protostar rather than a mere dust clump at the cavity wall as claimed by \\citet{Mau10}.\nThe NH$_3$ data of \\citet{Cur99} suggest that L1448C(S) is formed in the same dense core as L1448C(N).\nHowever, the SED of L1448C(S) is significantly different from that of L1448C(N).\nIn table 4, the broad-band spectra of L1448C(N) and L1448C(S) measured at different wavelengths are listed.\nIn the near-infrared, L1448C(S) is much dimmer than L1448C(N); only L1448C(N) appeared in the $K_s$ band image of \\citet{Tob07}.\nOn the contrary, in the mid-inrfrared at three IRAC bands, band 2 (4.5 $\\mu$m), band 3 (5.8 $\\mu$m), and band 4 (8.0 $\\mu$m), L1448C(S) becomes brighter than L1448C(N); especially in the bands 3 and 4, L1448C(S) is more than six times brighter than L1448C(N).\nIn the MIPS 24 $\\mu$m image, L1448C(S) is also seen clearly \\citep{Tob07, Reb07}.\nThe flux from L1448C(S) at 24 $\\mu$m looks similar to that from L1448C(N), although the accurate flux value of each source is not easy to measure because of the confusion.\nIn the sub-mm and mm wavebands, L1448C(S) is much weaker than L1448C(N).\nThe masses of the circumstellar material surrounding L1448C(S), $\\sim$0.01 $M_{\\odot}$, is approximately 10 times smaller than that of L1448C(N), $\\sim$0.1 $M_{\\odot}$.\nDue to the small amount of circumstellar material, the central star of L1448C(S) is likely to be less obscured in the mid-infrared as compared to that of L1448C(N) enshrouded by the thick cocoon.\nThe outflow activities in two sources are also different significantly.\nThe CO outflow from L1448C(S) is compact and substantially weaker than the L1448C(N) outflow.\nThe momentum flux of the L1448C(S) outflow is only $\\sim$10$^{-6}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$, which is two or three orders of magnitude smaller than that of the L1448C(N) outfow and is comparable to those of class I outflows studied by \\citet{Bon96}.\nIn addition, there is no EHV component nor SiO emission associated with the outflow from L1448C(S).\n\nThe small amount of circumstellar material of less than 0.01 $M_{\\odot}$ suggests that L1448C(S) may have accumulated most of its circumstellar mass.\nThe less energetic outflow also support the idea that L1448C(S) has a nature close to class I rather than class 0.\nThese results imply that two sources, L1448C(N) and L1448C(S) are formed in different epochs in the same dense core.\nAnother possibility is the effect of the L1448C(N) outflow.\nSince L1448C(S) is located at the same line of sight as the L1448C(N) outflow, it is possible that the outflowing gas has stripped away the dense gas surrounding L1448C(S).\nAlthough the three-dimensional geometries of the sources and outflows are not clear, the high rotational temperature of NH$_3$ observed at the position of L1448C(S) \\citep{Cur99} suggests the possibility that the gas around L1448C(S) is heated by the interaction with the jet from L1448C(N).\nIn such a case, the {\\it apparent} age of L1448C(S) is older than that of L1448C(N) simply because the amount of material left around L1448C(S) is smaller than that around L1448C(N).\nThe effect of outflow is also proposed to explain the difference of the apaprent evolutionary stage of protostellar pair L1448N(A) and L1448N(B), which is located at $\\sim$75\\arcsec northwest of L1448C \\citep{Oli06}.\n\nThe third scenario is the disintegration of an unstable multiple system as proposed by \\citet{Rei00}.\nSince nonhierarchical triple systems are unstable, they break up, ejecting the lightest member, while the remaining two members form a close binary system with a highly eccentric orbit.\nIn this scenario, the disks around escaping stars will be highly truncated; the typical disk size is expected to be around half of the distance between the stars in the close triple encounter.\nIf L1448C(S) is the escaping member, the small amount of its circumstellar material can be explained by means of disk truncation.\nThis scenario also implies that L1448C(N) is a close binary system.\nThe observed deflection, wiggling, and periodic velocity variation of the jet suggest the possibility that L1448C(N) is a close binary system with an orbital radius of $\\sim$2--4 AU.\nTherefore, it is possible that such a close binary system was formed by means of disintegration of a triple system.\nIn order to assess this scenario, kinematic information of L1448C(S) becomes important.\nAlthough previous NH$_3$ results of \\citet{Cur99} did not show peculiar motions in the dense core containing L1448C(N) and L1448C(S), a detailed study with higher angular resolution would be helpful.\n\n\n\n\n\\section{CONCLUSIONS}\nThe central region of the highly-colimated molecular outflow driven by L1448C was mapped in the SiO $J$=8--7, CO $J$=3--2, and 350 GHz continuum emission with the SMA at $\\sim$1 arcsecond resolution.\nOur main conclusions are the following:\n\\begin{enumerate}\n\\item The 350 GHz continuum emission was detected from two {\\it Spitzer} sources L1448C(N) and L1448C(S). \nThe continuum emission from L1448C(N) consists of an extended component and a compact component. \nThe compact component is elongated perpendicular to the outflow axis, and is likely to be a circumstellar disk with a size of $\\sim$90 AU. \nThe spectral index of this compact component derived from the data from 86 GHz to 350 GHz is ${\\alpha}{\\sim}$2, suggesting the possibility that the continuum emission is optically thick at 350 GHz.\nThe mass of the disk is estimated to be $\\sim$0.11 $M_{\\odot}$.\n\\item The continuum flux from L1448C(S) is $\\sim$60 mJy, which is $\\sim$10 times lower than the flux from L1448C(N), although L1448C(S) is brighter than L1448C(N) in the mid infrared wavebands.\nThe mass of the circumstellar material surrounding L1448C(S) is estimated to be 8.6$\\times$10$^{-3}$ $M_{\\odot}$.\n\\item A narrow jet from L1448C(N) along the outflow axis was observed in the SiO and the high-velocity CO. The width of the jet measured in the SiO images is $\\sim$200 AU FWHM on average. \nThe jet consists of a chain of emission knots with an inter-knot spacing of $\\sim$500 AU.\nIt is likely that the knots in the L1448 jet are the internal bow shocks in the jet beam.\n\\item The dynamical timescale of the innermost pair of knots, which are significant in the SiO but barely seen in the CO, is only $\\sim$10 yr. \nIt is likely that the SiO may have been formed quickly in the protostellar wind through the gas-phase reaction, or been formed on the dust grain and directly released into the gas phase by means of shocks.\n\\item The low velocity CO emission delineates two V-shaped shells with a common apex at L1448C(N). The kinematics of this shell component is reproduced by the model of wide opening angle wind. \nTherefore, the outflow from L1448C(N) consists of both highly-collimated jets and shells produced by wide-opening angle wind.\nThe observed jet+shell structure can be explained by the ^^ ^^ unified model'' proposed by \\citet{Sha06}, in which highly-collimated jet components are explained as an on-axis density enhancement of the {\\it X}-wind type of wide opening angle wind.\n\\item The Jet from L1448C(N) is extremely active with a momentum supply rate of $\\sim$5$\\times$10$^{-4}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$ and a mechanical luminosity of $\\sim$7 $L_{\\odot}$. \nThe mass accretion rate derived from the mass loss rate is $\\sim$10$^{-5}$ $M_{\\odot}$ yr$^{-1}$.\nSuch a high mass-accretion rate and a rather low bolometric luminosity of the central source, 7.5 $L_{\\odot}$, imply that the central protostar is still in the very early phase of its evolution with a mass of 0.03--0.09 $M_{\\odot}$ and a dynamical age of (4--12)$\\times$10$^3$ yr.\n\\item The blue part and the red part of the jet are misaligned by $\\sim$5$^{\\circ}$ and forming a C-shaped bending toward the west.\nThe possible origin of this bending is the dynamical pressure caused by the outflow from the nearby protostar, L1448N.\n\\item The jet is deflected toward the east in the blueshifted part and toward the south in the redshifted part. \nIn addition, the jet is wiggling with a period of $\\sim$15--20 yr.\nThe deflection and wiggling of the jet can be explained if the driving source is a member of the binary system with an orbital radius of 2--4 AU.\n\\item The jet shows a semi-periodic variation in radial velocity with an amplitude of $\\sim$7 km s$^{-1}$.\nEach SiO knot has its higher velocity in the upstream side and lower velocity in the downstream side. The opposite velocity gradient is seen in the faint emission between the knots.\nIt is likely that the ejection velocity varies periodically by means of modulation of mass accretion. \n\\item The bipolar outflow in the NE-SW direction centered at L1448C(S) was discovered in the CO $J$=3--2. \nThis provides a strong evidence that L1448C(S) is a protostar.\nThe momentum flux of this outflow is only $\\sim$10$^{-6}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$, which is two to three orders of magnitude smaller than that of L1448C(N) outflow, and is comparable to those of class I outflow.\n\\item L1448C(S) is surrounded by a rather small amount of circumstellar material of less than 0.01 $M_{\\odot}$ and is powering a less energetic outflow, suggesting that this source has a nature close to class I rather than class 0, even though this source is formed in the same dense core as L1448C(N).\nOne possible scenario to explain this dichotomy is the effect of the outflow from L1448C(N); significant amount of material in the envelope surrounding L1448C(S) might have been stripped off by the powerful outflow from L1448C(N).\nAnother possibility is the disintegration of an unstable multiple system, in which L1448C(S) is an escaping member with a truncated disk.\n\n\\end{enumerate}\n\n\\acknowledgments\n\nWe wish to thank all the SMA staff in Hawaii, Cambridge, and Taipei for their \nenthusiastic help during these observations. \nN. Hirano thanks M. Machida and S. Inutsuka for fruitful discussion.\nN. Hirano is supported by NSC grant 96-2112-M-001-023.\n\n \\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{Intro}\nGiven a (commutative) integral domain $D$ with fraction field $K$, we define $\\textnormal{Int}(D) := \\{f \\in K[X] \\mid f(D) \\subseteq D\\}$, which is the ring of integer-valued polynomials on $D$. Integer-valued polynomials and the properties of $\\textnormal{Int}(D)$ have been well studied; the book \\cite{CaCh} covers the major theory in this area and provides an extensive bibliography. In recent years, researchers have begun to study a generalization of $\\textnormal{Int}(D)$ to polynomials that act on a $D$-algebra rather than on $D$ itself \\cite{EFJ}, \\cite{EvrJoh}, \\cite{Fri1}, \\cite{Fri1Corr}, \\cite{Fri2}, \\cite{LopWer}, \\cite{Per1}, \\cite{PerDivDiff}, \\cite{PerFinite}, \\cite{PerWer}, \\cite{PerWer1}, \\cite{Wer}. For this generalization, we let $A$ be a torsion-free $D$-algebra such that $A \\cap K = D$, and let $B = K \\otimes_D A$, which is the extension of $A$ to a $K$-algebra. By identifying $K$ and $A$ with their images under the injections $k \\mapsto k \\otimes 1$ and $a \\mapsto 1 \\otimes a$, we can evaluate polynomials in $K[X]$ at elements of $A$. This allows us to define $\\textnormal{Int}_K(A) := \\{f \\in K[X] \\mid f(A) \\subseteq A\\}$, which is the ring of integer-valued polynomials on $A$ with coefficients in $K$. With notation as above, the condition $A \\cap K = D$ ensures that $D[X] \\subseteq \\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D)$.\n\n\\begin{Def}\\label{Nontrivial}\nWe say that $\\textnormal{Int}_K(A)$ is \\textit{nontrivial} if $\\textnormal{Int}_K(A) \\ne D[X]$.\n\\end{Def}\n\nThe goal of this paper is to determine when $\\textnormal{Int}_K(A)$ is nontrivial. Some results in this direction were proved by Frisch in \\cite[Lem. 4.1]{Fri2} and \\cite[Thm. 4.3]{Fri2}; these are restated below in Proposition \\ref{Fin gen nontrivial}. In the traditional case, necessary and sufficient conditions for $\\textnormal{Int}(D)$ to be nontrivial were given by Rush in \\cite{Rush}. Using Rush's criteria, we prove (Theorem \\ref{Nontriv fin gen criterion}) that when $D$ is any integral domain and $A$ is finitely generated as a $D$-module, $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial. Part of this work involves conditions under which we have $D[X] \\subseteq \\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A)$ for some $n$, where $M_n(D)$ is the algebra of $n \\times n$ matrices with entries in $D$. This led us to investigate whether having $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$ implies that $A \\cong M_n(D)$. While this is not true in general, the result does hold if $D$ is a Dedekind domain and $A$ can be embedded in $M_n(D)$ (Theorem \\ref{Uniqueness of M_n(D)}).\n\nIf we drop the assumption that $A$ is finitely generated as a $D$-module, determining whether $\\textnormal{Int}_K(A)$ is nontrivial becomes more complicated. However, when $D$ is Dedekind, we are able to give necessary and sufficient conditions for $\\textnormal{Int}_K(A)$ to be nontrivial (Theorem \\ref{criterion}). Our work on this topic also allows us to prove that if $D$ is Dedekind, then $\\textnormal{Int}_K(A)$ has Krull dimension 2 (Corollary \\ref{Krull dimension}). This generalizes another theorem of Frisch \\cite[Thm. 5.4]{Fri1} where it was assumed that $A$ was finitely generated as a $D$-module.\n\n\\section{Integral Algebras of Bounded Degree}\\label{Non-trivial}\nThroughout, $D$ denotes an integral domain with field of fractions $K$, and $A$ denotes a $D$-algebra. We will always assume that $A$ satisfies certain conditions, which we call our \\textit{standard assumptions}.\n\n\\begin{Def}\\label{Standard assumptions}\nWhen $A$ is a torsion-free $D$-algebra such that $A \\cap K = D$, we say that $A$ is a $D$-algebra with \\textit{standard assumptions}. When $A$ is finitely generated as a $D$-module, we say that $A$ is of \\textit{finite type}.\n\\end{Def}\n\nAs mentioned in the introduction, the condition that $A \\cap K = D$ implies that\n\\begin{equation*}\nD[X] \\subseteq \\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D)\n\\end{equation*}\nand it is natural to consider when $D[X] = \\textnormal{Int}_K(A)$ or $\\textnormal{Int}_K(A) = \\textnormal{Int}(D)$. This latter equality is investigated in \\cite{IntdecompII}, where the following theorem is proved. Unless stated otherwise, all isomorphisms are ring isomorphisms.\n\n\\begin{Thm}\\label{Int_K(A)=Int(D)}\n\\cite[Thms. 2.10, 3.10]{IntdecompII} Let $D$ be a Dedekind domain with finite residue rings. Let $A$ be a $D$-algebra of finite type with standard assumptions. For each maximal ideal $P$ of $D$, let $\\widehat{A}_P$ and $\\widehat{D}_P$ be the $P$-adic completions of $A$ and $D$, respectively. Then, the following are equivalent.\n\\begin{enumerate}[(1)]\n\\item $\\textnormal{Int}_K(A) = \\textnormal{Int}(D)$.\n\\item For each nonzero prime $P$ of $D$, there exists $t \\in \\mathbb{N}$ such that $A\/PA \\cong \\bigoplus_{i=1}^t D\/P$.\n\\item For each nonzero prime $P$ of $D$, there exists $t \\in \\mathbb{N}$ such that $\\widehat{A}_P \\cong \\bigoplus_{i=1}^t \\widehat{D}_P$.\n\\end{enumerate}\n\\end{Thm}\n\nIn this paper, we examine the containment $D[X] \\subseteq \\textnormal{Int}_K(A)$. In the traditional setting of integer-valued polynomials, the ring $\\textnormal{Int}(D)$ is said to be \\textit{trivial} if $\\textnormal{Int}(D) = D[X]$, and we adopt the same terminology for $\\textnormal{Int}_K(A)$. Clearly, for $\\textnormal{Int}_K(A)$ to be nontrivial it is necessary that $\\textnormal{Int}(D)$ be nontrivial, so we begin by reviewing the situation for $\\textnormal{Int}(D)$. Section I.3 of \\cite{CaCh} and a paper by Rush \\cite{Rush} give several results regarding the triviality or non-triviality of $\\textnormal{Int}(D)$. We will summarize these theorems after recalling several definitions.\n\n\\begin{Def}\nAn ideal $\\mathfrak{a}$ of $D$ is said to be the colon ideal or conductor ideal of $q \\in K$ if \n$$\\mathfrak{a} = (D :_D q) = \\{d \\in D \\mid dq \\in D\\}.$$\nFor a commutative ring $R$, we denote by $\\textnormal{nil}(R)$ the nilradical of $R$, which is the set of all nilpotent elements of $R$, or, equivalently, the intersection of all nonzero prime ideals of $R$. For $x \\in \\textnormal{nil}(R)$, we let $\\nu(x)$ equal the nilpotency of $x$, i.e., the smallest positive integer $n$ such that $x^n = 0$. If $I\\subseteq R$ is an ideal, let $V(I) = \\{P \\in \\text{Spec}(R) \\mid P \\supseteq I\\}$.\n\\end{Def}\n\nThe following proposition summarizes several sufficient and necessary conditions on $D$ in order for $\\textnormal{Int}(D)$ to be nontrivial.\n\\begin{Prop}\\label{IntD nontrivial}\\mbox{}\n\\begin{enumerate}[(1)]\n\\item \\cite[Cor. I.3.7]{CaCh} If $D$ is a domain with all residue fields infinite, then $\\textnormal{Int}(D)$ is trivial.\n\\item \\cite[Prop. I.3.10]{CaCh} Let $D$ be a domain. If there is a proper conductor ideal $\\mathfrak{a}$ of $D$ such that $D\/\\mathfrak{a}$ is finite, then $\\textnormal{Int}(D)$ is nontrivial.\n\\item \\cite[Thm. I.3.14]{CaCh} Let $D$ be a Noetherian domain. Then, $\\textnormal{Int}(D)$ is nontrivial if and only if there is a prime conductor ideal of $D$ with finite residue field.\n\\item \\cite[Cor. 1.7]{Rush} Let $D$ be an integral domain. Then, the following are equivalent:\n\\begin{enumerate}[(i)]\n\\item $\\textnormal{Int}(D)$ is nontrivial.\n\\item There exist $a,b\\in D$ with $b\\notin aD$ such that the two sets $\\{\\;|D\/P|\\; \\mid P \\in V((aD:b))\\}$ and $\\{\\nu(x) \\mid x\\in {\\rm nil}(D\/(aD:b))\\}$ are bounded.\n\\end{enumerate}\n\\end{enumerate}\n\\end{Prop}\n\nIf $A$ is finitely generated as a $D$-module, Frisch has shown that the analogs of the above conditions in Proposition \\ref{IntD nontrivial} hold for $\\textnormal{Int}_K(A)$:\n\n\\begin{Prop}\\label{Fin gen nontrivial}\nLet $D$ be a domain. Let $A$ be a $D$-algebra of finite type with standard assumptions.\n\\begin{enumerate}[(1)]\n\\item \\cite[Lem. 4.1]{Fri2} Assume there is a proper conductor ideal $\\mathfrak{a}$ of $D$ such that $D\/\\mathfrak{a}$ is finite. Then, $\\textnormal{Int}_K(A)$ is nontrivial.\n\\item \\cite[Thm. 4.3]{Fri2} Assume that $D$ is Noetherian. Then, $\\textnormal{Int}_K(A)$ is nontrivial if and only if there is a prime conductor ideal of $D$ with finite residue field.\n\\end{enumerate}\n\\end{Prop}\n\nIn particular, \\cite[Thm. 4.3]{Fri2} shows that for a Noetherian domain $D$ and a finitely generated algebra $A$, $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial. In Theorem \\ref{Nontriv fin gen criterion}, we will show that this holds even if $D$ is not Noetherian. Additionally, we can weaken our assumptions on $A$. Recall the following definition, which can be found in \\cite{Jac} or \\cite{Lam}, among other sources.\n\n\\begin{Def}\nLet $R$ be a commutative ring and $A$ an $R$-algebra. We say that $A$ is an \\emph{algebraic algebra} (over $R$) if every element of $A$ satisfies a polynomial equation with coefficients in $R$. We say that $A$ is an \\emph{algebraic algebra of bounded degree} if there exists $n\\in\\mathbb{N}$ such that the degree of the minimal polynomial equation of each of its elements is bounded by $n$. If we insist that each element of $A$ satisfy a monic polynomial with coefficients in $R$, then we say that $A$ is an \\emph{integral algebra} over $R$.\n\\end{Def}\n\nAlgebraic algebras are usually discussed over fields, in which case an algebraic algebra is also an integral algebra. Over a domain however, the two structures are not equivalent. For example, $A = \\mathbb{Z}[\\frac{1}{2}]$ is an algebraic algebra over $\\mathbb{Z}$ that is not an integral algebra. In this case, $A$ does not satisfy our standard assumption that $A \\cap \\mathbb{Q}$ should equal $\\mathbb{Z}$. However, if we instead take $A = \\mathbb{Z} \\oplus \\mathbb{Z}[\\frac{1}{2}]$ (so that $B = \\mathbb{Q} \\otimes_\\mathbb{Z} A \\cong \\mathbb{Q} \\oplus \\mathbb{Q}$, $D$ is the diagonal copy of $\\mathbb{Z}$ in $B$, and $K$ is the diagonal copy of $\\mathbb{Q}$ in $B$), then $A$ is an algebraic algebra over $D$, $A$ is not an integral algebra over $D$, and $A \\cap K = D$.\n\nNote also that if $A$ is finitely generated as a $D$-module, then $A$ is an integral algebra of bounded degree, with the bound given by the number of generators (see \\cite[Thm. 1, Chap. V]{BourbakiAlg} or \\cite[Prop. 2.4]{AtMc}). However, the converse does not hold. For instance, $A = D[X_1, X_2, \\ldots]\/(\\{X_i X_j \\mid i, j \\geq 1 \\})$ is not finitely generated, but if $f \\in A$ with constant term $d \\in D$, then $f$ satisfies the polynomial $(X-d)^2$. Thus, this $A$ is an integral algebra of bounded degree. \n\nFor our purposes, the importance of having a bounding degree $n$, is that it guarantees that $\\textnormal{Int}_K(A)$ contains $\\textnormal{Int}_K(M_n(D))$, where $M_n(D)$ denotes the algebra of $n \\times n$ matrices with entries in $D$.\n\n\\begin{Lem}\\label{Matrix containment}\nLet $D$ be a domain and let $A$ be a $D$-algebra with standard assumptions. Assume that $A$ is an integral $D$-algebra of bounded degree $n$. Then, $\\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A)$.\n\\end{Lem}\n\\begin{proof}\nLet $a \\in A$ and let $\\mu_a \\in D[X]$ be monic of degree $n$ such that $\\mu_a(a) = 0$. Let $f(x) = g(X)\/d \\in \\textnormal{Int}_K(M_n(D))$, where $g \\in D[X]$ and $d \\in D \\setminus \\{0\\}$. By \\cite[Lem. 3.4]{FriSep}, $g$ is divisible modulo $dD[X]$ by every monic polynomial in $D[X]$ of degree $n$; hence, $\\mu_a$ divides $g$ modulo $d$. It follows that $g(a) \\in dA$ and $f(a) \\in A$. Since $a$ was arbitrary, $f \\in \\textnormal{Int}_K(A)$.\n\\end{proof}\n\n\\begin{Rem}\nThe converse of Lemma \\ref{Matrix containment} does not hold, even in the case when $\\textnormal{Int}_K(M_n(D))$ is nontrivial, as Example \\ref{integralalgebraboundeddegree not necessary} below will show.\n\\end{Rem}\n\nThus, in the case of an integral algebra of bounded degree $n$, to prove that $\\textnormal{Int}_K(A)$ is nontrivial it suffices to show that $\\textnormal{Int}_K(M_n(D))$ is nontrivial. This task is more tractable, because the polynomials given in the next definition can be used to map $M_n(D)$ into $M_n(P)$, where $P$ is a maximal ideal of $D$ with a finite residue field.\n\n\\begin{Def}\\label{BCL polynomials}\nFor each prime power $q$ and each $n > 0$, let \n\\begin{equation*}\n\\phi_{q,n}(X) = (X^{q^n} - X)(X^{q^{n-1}}-X) \\cdots (X^q - X).\n\\end{equation*}\n\\end{Def}\n\n\\begin{Lem}\\label{BCL lemma}\n\\cite[Thm. 3]{BrawCarLev} Let $\\mathbb{F}_q$ be the finite field with $q$ elements. Then, $\\phi_{q,n}$ sends each matrix in $M_n(\\mathbb{F}_q)$ to the zero matrix. Consequently, if $P \\subset D$ is a maximal ideal of $D$ with residue field $D\/P \\cong \\mathbb{F}_q$, then $\\phi_{q, n}$ maps $M_n(D)$ into $M_n(P)$.\n\\end{Lem}\n\n\\begin{Prop}\\label{Nontriv matrix criterion}\nLet $D$ be a domain. If $\\textnormal{Int}(D)$ is nontrivial, then $\\textnormal{Int}_K(M_n(D))$ is nontrivial, for all $n \\geq 1$.\n\\end{Prop}\n\\begin{proof}\nLet $n \\geq 1$ be fixed. Since $\\textnormal{Int}(D)$ is nontrivial, by \\cite[Cor. 1.7]{Rush} there exist $a,b\\in D$ with $b\\notin aD$ such that $\\{\\;|D\/P|\\; \\mid P \\in V((aD:b))\\}$ and $\\{\\nu(x) \\mid x\\in \\textnormal{nil}(D\/(aD:b))\\}$ are bounded. Let $I = (aD:b)$. Note that, because the former condition holds, each prime ideal containing $I$ is maximal, so the nilradical of $D\/I$ is equal to the Jacobson radical of $D\/I$.\n\nLet $\\{q_1,\\ldots,q_s\\}=\\{\\;|D\/P|\\; \\mid P\\in V(I)\\}$. By Lemma \\ref{BCL lemma}, we have $\\phi_{q,n}(M_n(D))\\subseteq M_n(P)$ for each maximal ideal $P\\subset D$ whose residue field has cardinality $q$. Then\n\\begin{equation*}\ng(X)=\\prod_{i=1,\\ldots,s}\\phi_{q_i,n}(X)\n\\end{equation*}\nis a monic polynomial such that $g(M_n(D))\\subseteq \\prod_{i}M_n(P_i)\\subseteq M_n(J)$, where $J=\\sqrt{I}$. Considering everything modulo $I$, we have $\\overline{g}(M_n(D\/I))\\subseteq M_n(J\/I)$. \n\nNow, since $\\{\\nu(x) \\mid x\\in \\textnormal{nil}(D\/I)\\}$ is bounded, the nilpotency of every element in $J\/I$ is bounded by some positive integer $t$. It is a standard exercise that a matrix over a commutative ring with nilpotent entries is a nilpotent matrix. Moreover, it easily follows that the nilpotency of every matrix in $M_n(J\/I)$ is bounded by some $m\\in\\mathbb{N}$, depending only on $t$ and $n$. Hence, $\\overline{g}(X)^{m}$ maps every matrix $M_n(D\/I)$ to $0$, so that $g(X)^{m}$ maps $M_n(D)$ into $M_n(I)$. Finally, it is now easy to see that the polynomial $\\frac{b}{a}\\cdot g(X)^m$ is in $\\textnormal{Int}_K(M_n(D))$ but not in $D[X]$.\n\\end{proof}\n\nCombining Lemma \\ref{Matrix containment} with Proposition \\ref{Nontriv matrix criterion}, we obtain our desired theorem.\n\n\\begin{Thm}\\label{Nontriv fin gen criterion}\nLet $D$ be a domain and let $A$ be $D$-algebra with standard assumptions. Assume that $A$ is an integral $D$-algebra of bounded degree. Then, $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial. In particular, if $A$ is finitely generated as a $D$-module, then $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial.\n\\end{Thm}\n\nLemma \\ref{Matrix containment} shows that, for an integral algebra $A$ of bounded degree $n$, the following containments hold:\n\\begin{equation*}\nD[X] \\subseteq \\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D).\n\\end{equation*}\nWhile our focus has been on whether $\\textnormal{Int}_K(A)$ equals $D[X]$, for the remainder of this section we will consider the containment $\\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A)$. In particular, we will examine to what extent $\\textnormal{Int}_K(M_n(D))$ is unique among rings of integer-valued polynomials. That is, if $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$, then can we conclude that $A \\cong M_n(D)$? In general, the answer is no, as we show below in Example \\ref{Quaternion example}. However, in Theorem \\ref{Uniqueness of M_n(D)} we will prove that for $D$ Dedekind, if $A$ can be embedded in $M_n(D)$, then having $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$ implies that $A \\cong M_n(D)$.\n\nWe first recall the definition of a null ideal of an algebra.\n\n\\begin{Def}\\label{Null ideal}\nLet $R$ be a commutative ring and $A$ an $R$-algebra. The \\textit{null ideal} of $A$ with respect to $R$, denoted $N_R(A)$, is the set of polynomials in $R[X]$ that kill $A$. That is, $N_R(A) = \\{ f \\in R[X] \\mid f(A) = 0\\}$. In particular, $N_{D\/P}(A\/PA) = \\{f \\in (D\/P)[X] \\mid f(A\/PA) = 0\\}$ denotes the null ideal of $A\/PA$ with respect to $D\/P$.\n\\end{Def}\n\nThere is a close relationship between polynomials in rings of integer-valued polynomials and polynomials in null ideals.\n\n\\begin{Lem}\\label{Null ideal lemma}\nLet $D$ be a domain and let $A$ and $A'$ be $D$-algebras with standard assumptions.\n\\begin{enumerate}[(1)]\n\\item Let $g(X)\/d \\in K[X]$, where $g \\in D[X]$ and $d \\ne 0$. Then, $g(X)\/d \\in \\textnormal{Int}_K(A)$ if and only if the residue of $g$ (mod $d$) is in $N_{D\/dD}(A\/dA)$. \n\\item $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(A')$ if and only if $N_{D\/dD}(A\/dA) = N_{D\/dD}(A'\/dA')$ for all $d \\in D$.\n\\end{enumerate}\n\\end{Lem}\n\\begin{proof}\nNotice that $g \\in \\textnormal{Int}_K(A)$ if and only if $g(A) \\subseteq dA$ if and only if $g(A\/dA) = 0$ mod $d$. This proves (1), and (2) follows easily.\n\\end{proof}\n\n\\begin{Ex}\\label{Quaternion example}\nLet $D = \\mathbb{Z}_{(p)}$ be the localization of $\\mathbb{Z}$ at an odd prime $p$. Take $A$ to be the quaternion algebra $A = D \\oplus D\\mathbf{i} \\oplus D\\mathbf{j} \\oplus D\\mathbf{k}$, where $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ are the imaginary quaternion units satisfying $\\mathbf{i}^2 = \\mathbf{j}^2 = -1$ and $\\mathbf{i}\\mathbf{j} = \\mathbf{k} = -\\mathbf{j}\\mathbf{i}$. It is well known (cf.\\ \\cite[Exercise 3A]{Goodearl} or \\cite[Sec. 2.5]{DavSarVal}) that $A\/p^k A \\cong M_2(\\mathbb{Z}\/p^k \\mathbb{Z}) \\cong M_2(D\/p^k D)$ for all $k > 0$. By Lemma \\ref{Null ideal lemma}, $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}_\\mathbb{Q}(M_2(D))$. However, $A$ contains no nonzero nilpotent elements (and is in fact contained in the division ring $\\mathbb{Q} \\oplus \\mathbb{Q}\\mathbf{i} \\oplus \\mathbb{Q}\\mathbf{j} \\oplus \\mathbb{Q}\\mathbf{k}$) and so cannot be isomorphic to $M_2(D)$.\n\\end{Ex}\n\nThus, in general, $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(M_n(D))$ does not imply that $A \\cong M_n(D)$. However, as mentioned above, we do have such an isomorphism if $A$ can be embedded in $M_n(D)$. Proving this theorem involves some results of Racine \\cite{Racine}, \\cite{Racine2} about maximal subalgebras of matrix rings, which we now summarize.\n\n\\begin{Prop}\\label{Racine classification}\\mbox{}\n\\begin{enumerate}[(1)]\n\\item (\\cite[Thm. 1]{Racine}) Let $\\overline{A}$ be a maximal $\\mathbb{F}_q$-subalgebra of $M_n(\\mathbb{F}_q)$. Let $V$ be an $\\mathbb{F}_q$-vector space of dimension $n$, so that $M_n(\\mathbb{F}_q)\\cong{\\rm End}_{\\mathbb{F}_q}(V)$. Then, $\\overline{A}$ is one of the following two types.\n\\begin{itemize}\n\\item[(I)] The stabilizer of a proper nonzero subspace of $V$. That is, $\\overline{A} = S(W) = \\{\\varphi\\in {\\rm End}_{\\mathbb{F}_q}(V) \\mid \\varphi(W)\\subseteq W\\}$, where $W$ is a proper nonzero $\\mathbb{F}_q$-subspace of $V$.\n\n\\item[(II)] The centralizer of a minimal field extension of $\\mathbb{F}_q$. That is, $\\overline{A} = C_{{\\rm End}_{\\mathbb{F}_q}(V)}(\\mathbb{F}_{q^l})=\\{\\varphi\\in{\\rm End}_{\\mathbb{F}_q}(V) \\mid \\varphi x=x\\varphi, \\forall x\\in \\mathbb{F}_{q^l} \\}$, where $l\\in\\mathbb{Z}$ is a prime dividing $n$.\n\\end{itemize}\n\n\\item (\\cite[Theorem p.\\ 12]{Racine2}) Let $D$ be a Dedekind domain and let $A$ be a maximal $D$-subalgebra of $M_n(D)$. Then, there exists a maximal ideal $P$ of $D$ such that $A\/P A$ is a maximal subalgebra of $M_n(D\/P)$. \n\\end{enumerate}\n\\end{Prop}\n\nRacine's classification allows us to establish a partial uniqueness result for the null ideal of $M_n(\\mathbb{F}_q)$, and hence for $\\textnormal{Int}_K(M_n(D))$.\n\n\\begin{Lem}\\label{FqsubalgebrasMnFq}\nLet $\\overline{A}$ be an $\\mathbb{F}_q$-subalgebra of $M_n(\\mathbb{F}_q)$ such that $N_{\\mathbb{F}_q}(\\overline{A})=N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$. Then $\\overline{A} = M_n(\\mathbb{F}_q)$.\n\\end{Lem}\n\\begin{proof}\nSuppose the claim is not true, so that $\\overline A$ is contained in a maximal $\\mathbb{F}_q$-subalgebra of $M_n(\\mathbb{F}_q)$; hence, without loss of generality, we may assume that $\\overline A\\subsetneq M_n(\\mathbb{F}_q)$ is a maximal $\\mathbb{F}_q$-subalgebra. We will show that $N_{\\mathbb{F}_q}(\\overline{A})$ properly contains $N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$. Note that $N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q)) = (\\phi_{q,n}(X))$ by \\cite[Thm. 3]{BrawCarLev}, where $\\phi_{q,n}$ is the polynomial from Definition \\ref{BCL polynomials}.\n\nLet $V$ be an $\\mathbb{F}_q$-vector space of dimension $n$, so that $M_n(\\mathbb{F}_q)\\cong{\\rm End}_{\\mathbb{F}_q}(V)$. Assume first that $\\overline{A} = S(W)$ is of Type I as in Proposition \\ref{Racine classification}, and let $m = \\dim_{\\mathbb{F}_q}(W)$. Note that conjugating $\\overline{A}$ by an element of $GL(n, q)$ will change the matrices in $\\overline{A}$, but not the polynomials in the null ideal $N_{\\mathbb{F}_q}(\\overline{A})$. Moreover, up to conjugacy by an element in $GL(n, q)$, we may assume that $W$ has basis $e_1, e_2, \\ldots, e_m$, where $e_i$ is the standard basis vector with $1$ in the $i^\\text{th}$ component and 0 elsewhere. Under this basis, the matrices in $\\overline{A}$ are block matrices of the form \n$\\big(\\begin{smallmatrix}\nA_1 & A_2 \\\\ 0 & A_3\n\\end{smallmatrix}\\big)$ \nwhere $A_1$ is $m \\times m$ and $A_3$ is $(n-m) \\times (n-m)$. \n\nOne consequence of this representation is that every matrix in $S(W)$ has a reducible characteristic polynomial. As shown in the proof of \\cite[Thm. 3]{BrawCarLev}, $\\phi_{q, n}$ is the least common multiple of all monic polynomials in $\\mathbb{F}_q[X]$ of degree $n$. Hence, $\\phi_{q, n} \\in N_{\\mathbb{F}_q}(\\overline{A})$, because the characteristic polynomial of each matrix in $\\overline{A}$ divides $\\phi_{q,n}$. However, if $\\phi$ is the quotient of $\\phi_{q,n}$ by an irreducible polynomial in $\\mathbb{F}_q[X]$ of degree $n$, then $\\phi \\in N_{\\mathbb{F}_q}(\\overline{A})$, but $\\phi \\notin N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$. Thus, $N_{\\mathbb{F}_q}(\\overline{A})$ properly contains $N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$.\n\nNow, assume that $\\overline{A}$ is of Type II of Proposition \\ref{Racine classification}, so that $\\overline{A} = C_{{\\rm End}_{\\mathbb{F}_q}(V)}(\\mathbb{F}_{q^l})$ for some prime $l$ dividing $n$. Then, by \\cite[Thm. VIII.10]{McD}, we have $\\overline{A} \\cong M_{n\/l}(\\mathbb{F}_{q^l})$, and so \n\\begin{equation*}\nN_{\\mathbb{F}_q}(\\overline{A})=(\\phi_{q^l,n\/l}(X))\\supsetneq (\\phi_{q,n}(X)) = N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q)).\n\\end{equation*}\nAs before, the null ideal of $\\overline{A}$ strictly contains the null ideal of $M_n(\\mathbb{F}_q)$.\n\\end{proof}\n\n\\begin{Thm}\\label{Uniqueness of M_n(D)}\nLet $D$ be a Dedekind domain with finite residue fields. Let $A$ be a $D$-algebra of finite type with standard assumptions. Assume that $n \\geq 1$ is such that $A$ can be embedded in $M_n(D)$. Then, $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(M_n(D))$ if and only if $A \\cong M_n(D)$.\n\\end{Thm}\n\\begin{proof}\nClearly, $A \\cong M_n(D)$ implies that $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(M_n(D))$. So, assume that $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$. As we will prove shortly in Lemma \\ref{wellbehaviourlocalization}, $\\textnormal{Int}_K(A)$ (and likewise $\\textnormal{Int}_K(M_n(D))$) is well-behaved with respect to localization at primes of $D$: for each prime $P$ of $D$, we have $\\textnormal{Int}_K(A)_P = \\textnormal{Int}_K(A_P)$. Hence, $\\textnormal{Int}_K(M_n(D_P)) = \\textnormal{Int}_K(A_P)$ for each $P$. Since $D$ is Dedekind, $D_P$ is a discrete valuation ring, so there exists $\\pi \\in D_P$ such that $PD_P = \\pi D_P$. Moreover, we have $D_P\/\\pi D_P \\cong D\/P$ and $A_P \/ \\pi A_P \\cong A\/PA$, so that $N_{D_P\/\\pi D_P}(A_P \/ \\pi A_P) = N_{D\/P}(A\/PA)$ (and likewise for $M_n(D)$). By Lemma \\ref{Null ideal lemma} (2), we conclude that the null ideals $N_{D\/P}(M_n(D\/P))$ and $N_{D\/P}(A\/P A)$ are equal for all maximal ideals $P$ of $D$.\n\nNow, suppose by way of contradiction that the image of $A$ in $M_n(D)$ does not equal $M_n(D)$. As in Lemma \\ref{FqsubalgebrasMnFq}, we may assume that the image of $A$ in $M_n(D)$ is a maximal $D$-subalgebra of $M_n(D)$. By Proposition \\ref{Racine classification}, there exists a maximal ideal $P$ of $D$ such that $A\/P A$ is isomorphic to a maximal subalgebra of $M_n(D\/P)$. By Lemma \\ref{FqsubalgebrasMnFq}, the null ideals $N_{D\/P}(A\/P A)$ and $N_{D\/P}(M_n(D\/P))$ are not equal. This is a contradiction. Therefore, $A \\cong M_n(D)$.\n\\end{proof}\n\n\\section{General Case}\\label{General case section}\n\nWe return now to the study of when $\\textnormal{Int}_K(A)$ is nontrivial. Because of Theorem \\ref{Nontriv fin gen criterion}, $A$ being an integral $D$-algebra of bounded degree can be sufficient for $\\textnormal{Int}_K(A)$ to be nontrivial, but it is not necessary. There exist $D$-algebras $A$ that are neither finitely generated, nor algebraic over $D$ (let alone integral or of bounded degree), but for which $\\textnormal{Int}_K(A)$ is nontrivial, as the next example shows.\n\n\\begin{Ex}\\label{integralalgebraboundeddegree not necessary}\nLet $D = \\mathbb{Z}$ and let $A = \\prod_{i \\in \\mathbb{N}} \\mathbb{Z}$ be an infinite direct product of copies of $\\mathbb{Z}$. Then, the element $(1, 2, 3, \\ldots)$ cannot be killed by any polynomial in $\\mathbb{Z}[X]$, so $A$ is not algebraic over $\\mathbb{Z}$. However, since operations in $A$ are done component-wise, any polynomial in $\\textnormal{Int}(\\mathbb{Z})$ is also in $\\textnormal{Int}_\\mathbb{Q}(A)$. Hence, $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}(\\mathbb{Z})$, so in particular $\\textnormal{Int}_\\mathbb{Q}(A)$ is nontrivial. \n\\end{Ex}\n\nUltimately, the previous example works because for each prime $p$ there exists a polynomial that sends each element of $A\/pA$ to 0. More explicitly, each element of $\\prod_{i \\in \\mathbb{N}} \\mathbb{F}_p$ is killed by the polynomial $X^p-X$. This suggests that for $\\textnormal{Int}_K(A)$ to be nontrivial, it may be enough that there exists a finite index prime $P$ of $D$ with $A\/PA$ algebraic of bounded degree over $D\/P$ (since $D\/P$ is a field in this case, this is equivalent to having $A\/PA$ be integral of bounded degree over $D\/P$). We will prove below in Theorem \\ref{criterion} that if $D$ is a Dedekind domain, then this condition is necessary and sufficient for $\\textnormal{Int}_K(A)$ to be nontrivial.\n\nOur work will involve localizing $D$, $A$, and $\\textnormal{Int}_K(A)$ at $P$, and exploiting properties of $D_P$. In \\cite[Prop. 3.2]{Wer}, it is shown that if $D$ is Noetherian and $A$ is a free $D$-module of finite rank, then $\\textnormal{Int}_K(A)_P = \\textnormal{Int}_K(A_P)$ (in fact, \\cite[Prop. 3.2]{Wer} will hold if $A$ is merely finitely generated as a $D$-module). The next lemma shows that we can drop this finiteness assumption if $D$ is Dedekind.\n\n\\begin{Lem}\\label{wellbehaviourlocalization}\nLet $D$ be a Dedekind domain and $A$ a $D$-algebra with standard assumptions. Let $P$ be a prime ideal of $D$. Then $\\textnormal{Int}_K(A_P) = \\textnormal{Int}_K(A)_P$.\n\\end{Lem}\n\\begin{proof}\nThe containment $\\textnormal{Int}_K(A)_P \\subseteq \\textnormal{Int}_K(A_P)$ follows from the proof of \\cite[Prop. 3.2]{Wer}, which itself is an adaptation of a technique of Rush involving induction on the degrees of the relevant polynomials (see \\cite[Thm. I.2.1]{CaCh} or \\cite[Prop. 1.4]{Rush}). \n\nFor the other inclusion, let $f \\in \\textnormal{Int}_K(A_P)$ and write $f(X)=\\frac{g(X)}{d}$ for some $g \\in D[X]$ and $d\\in D \\setminus \\{0\\}$. Since $D$ is Dedekind, we may write $dD=P^aI$, where $a \\geq 0$ and $I$ is an ideal of $D$ coprime with $P$ (possibly equal to $D$ itself). If $a=0$ then $f \\in D_P[X] \\subseteq \\textnormal{Int}_K(A)_P$. If $a \\geq 1$, let $c\\in I \\setminus P$. We claim that $cf \\in\\textnormal{Int}_K(A)$, from which the statement follows since $c \\in D \\setminus P$. \n\nIf $Q \\subset D$ is a prime ideal different from $P$, then $cf \\in D_Q[X] \\subseteq \\textnormal{Int}_K(A_Q)$; that is, $cf(A_Q) \\subset A_Q$. Now, $f(A) \\subseteq f(A_P) \\subseteq A_P$ by assumption, so $cf(A) \\subset cA_P = A_P$, since $c \\notin P$. Since $A=\\bigcap_{Q\\in{\\rm Spec}(D)} A_Q$, it follows that $cf(A)\\subset A$, and we are done.\n\\end{proof}\n\n\nRecall (Definition \\ref{Null ideal}) that the null ideal of $A$ in $R$ is $N_R(A) = \\{ f \\in R[X] \\mid f(A) = 0\\}$.\n\n\\begin{Prop}\\label{equivalent conditions}\nLet $D$ be a Dedekind domain and $A$ a $D$-algebra with standard assumptions. Let $P$ be a prime ideal of $D$. Then, the following are equivalent.\n\\begin{enumerate}[(1)]\n\\item $N_{D\/P}(A\/PA) \\supsetneq (0)$.\n\\item $D_P[X] \\subsetneq \\textnormal{Int}_K(A_P)$.\n\\item $D\/P$ is finite and $A\/PA$ is a $D\/P$-algebraic algebra of bounded degree.\n\\end{enumerate}\n\\end{Prop}\n\\begin{proof}\n$(1) \\Rightarrow (2)$ Let $g\\in D[X]$ be a monic pullback of a nontrivial element $\\overline{g}\\in N_{D\/P}(A\/PA)$ and let $\\pi\\in P\\setminus P^2$. Then, $g(A_P)\\subseteq PA_P=\\pi A_P$, so $\\frac{g(X)}{\\pi}\\in \\textnormal{Int}_K(A_P)\\setminus D_P[X]$. \n\n$(2) \\Rightarrow (1)$ Let $f(X)=\\frac{g(X)}{d} \\in \\textnormal{Int}_K(A_P) \\setminus D_P[X]$, with $g \\in D[X] \\setminus P[X]$ and $d\\in P$. Let $v_P$ denote the canonical valuation on $D_P$. If $v_P(d)=e>1$ and $\\pi \\in P\\setminus P^2$, then $\\pi^{e-1} f(X)$ is still an element of $\\textnormal{Int}_K(A_P)$ which is not in $D_P[X]$. So, $g(A_P) \\subseteq \\frac{d}{\\pi^{e-1}} A_P \\subseteq \\pi A_P$. Hence, $\\overline{g}\\in(D_P\/PD_P)[X]\\cong(D\/P)[X]$ is a nontrivial element of $N_{D\/P}(A\/PA)$.\n\n$(1) \\Leftrightarrow (3)$ Note that\n\\begin{equation*}\nN_{D\/P}(A\/PA)=\\bigcap_{\\overline{a}\\in A\/PA}N_{D\/P}(\\overline{a})=\\bigcap_{\\overline{a}\\in A\/PA}(\\mu_{\\overline{a}}(X))\n\\end{equation*}\nwhere, for each $\\overline{a}\\in A\/PA$, $\\mu_{\\overline{a}}\\in (D\/P)[X]$ is the minimal polynomial of $\\overline{a}$ over the field $D\/P$.\n\nIf $N_{D\/P}(A\/PA)$ is nonzero, then it is equal to a principal ideal generated by a monic non-constant polynomial $\\overline{g}\\in (D\/P)[X]$. Since $N_{D\/P}(A\/PA)\\subseteq N_{D\/P}(D\/P)$, it follows that $D\/P$ is finite (if not, then $N_{D\/P}(D\/P) = (0)$, because the only polynomial which is identically zero on an infinite field is the zero polynomial). Moreover, each element $\\overline{a}\\in A\/PA$ is algebraic over $D\/P$ (otherwise the corresponding $N_{D\/P}(\\overline{a})$ is zero) and its degree over $D\/P$ is bounded by $\\deg(\\overline{g})$. \n\nConversely, assume $D\/P$ is finite and $A\/PA$ is a $D\/P$-algebraic algebra of bounded degree $n$. Then, there are finitely many polynomials over $D\/P$ of degree at most $n$, and the product of all such polynomials is a nontrivial element of $N_{D\/P}(A\/PA)$.\n\\end{proof}\n\nWe can now establish the promised criterion for $\\textnormal{Int}_K(A)$ to be nontrivial.\n\n\\begin{Thm}\\label{criterion}\nLet $D$ be a Dedekind domain and let $A$ be a $D$-algebra with standard assumptions. Then $\\textnormal{Int}_K(A)$ is nontrivial if and only if there exists a prime ideal $P$ of $D$ of finite index such that $A\/PA$ is a $D\/P$-algebraic algebra of bounded degree.\n\\end{Thm}\n\\begin{proof}\nClearly, $D[X]\\subsetneq \\textnormal{Int}_K(A)$ if and only if there exists a prime ideal $P\\subset D$ such that the two $D$-modules $D[X]$ and $\\textnormal{Int}_K(A)$ are not equal locally at $P$, that is, $D_P[X]\\subsetneq \\textnormal{Int}_K(A)_P$. Since $\\textnormal{Int}_K(A)_P=\\textnormal{Int}_K(A_P)$ by Lemma \\ref{wellbehaviourlocalization}, we can apply Proposition \\ref{equivalent conditions} and we are done.\n\\end{proof}\n\n\\begin{Ex}\\label{Nontriv examples}\nTheorem \\ref{criterion} applies to the following examples.\n\\begin{itemize}\n\\item[(1)] Let $D=\\mathbb{Z}$ and $A=\\overline{\\mathbb{Z}}$, the absolute integral closure of $\\mathbb{Z}$. Then, for each $n\\in\\mathbb{N}$, there exists $\\alpha\\in\\overline{\\mathbb{Z}}$ of degree $d>n$ such that $O_{\\mathbb{Q}(\\alpha)}=\\mathbb{Z}[\\alpha]$. It follows that for each prime $p\\in\\mathbb{Z}$, $\\overline{\\mathbb{Z}}\/p\\overline{\\mathbb{Z}}$ is an algebraic $\\mathbb{Z}\/p\\mathbb{Z}$-algebra of unbounded degree. Thus, $\\textnormal{Int}_{\\mathbb{Q}}(\\overline{\\mathbb{Z}})=\\mathbb{Z}[X]$.\n\n\\item[(2)] Let $D=\\mathbb{Z}_{(p)}$ and $A=\\mathbb{Z}_p$. Then, $\\mathbb{Z}_p\/p\\mathbb{Z}_p\\cong\\mathbb{Z}\/p\\mathbb{Z}$, so $\\mathbb{Z}_{(p)}[X]\\subsetneq \\textnormal{Int}_{\\mathbb{Q}}(\\mathbb{Z}_p)$.\n\n\\item[(3)] Let $D=\\mathbb{Z}$ and $A=\\widehat{\\mathbb{Z}}=\\prod_{p\\in\\mathbb{P}}\\mathbb{Z}_p$, the profinite completion of $\\mathbb{Z}$, where $\\mathbb{P}$ denotes the set of all prime numbers. For each prime $p\\in\\mathbb{Z}$, we have $p\\widehat{\\mathbb{Z}}=\\prod_{p'\\not=p}\\mathbb{Z}_{p'}\\times p\\mathbb{Z}_p$, so $\\widehat{\\mathbb{Z}}\/p\\widehat{\\mathbb{Z}}\\cong \\mathbb{Z}_p\/p\\mathbb{Z}_p\\cong\\mathbb{Z}\/p\\mathbb{Z}$. Thus, $\\mathbb{Z}[X]\\subsetneq\\textnormal{Int}_{\\mathbb{Q}}(\\widehat{\\mathbb{Z}})$.\n\\end{itemize}\n\\end{Ex}\n\nIf $\\widehat{A}$ is the $P$-adic completion of a $D$-algebra $A$, then we can say more about $\\textnormal{Int}_K(\\widehat{A})$. The following lemma also appears in \\cite{IntdecompII}. We include it in its entirety since the proof is quite short.\n\n\\begin{Lem}\\label{DVR lemma}\nLet $D$ be a discrete valuation ring (DVR) with maximal ideal $P = \\pi D$. Let $A$ be a $D$-algebra with standard assumptions, and let $\\widehat{A}$ be the $P$-adic completion of $A$. Then, $\\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A)$.\n\\end{Lem}\n\\begin{proof}\nThe containment $\\textnormal{Int}_K(\\widehat{A}) \\subseteq \\textnormal{Int}_K(A)$ is clear, since $A$ embeds in $\\widehat{A}$. Conversely, let $f \\in \\textnormal{Int}_K(A)$ and $\\alpha \\in \\widehat{A}$. Suppose $f(X) = g(X)\/\\pi^k$, where $g \\in D[X]$ and $k \\in \\mathbb{N}$. If $k = 0$, then the conclusion is clear, so assume that $k > 1$. \n\nVia the canonical projection $\\widehat{A} \\to A\/\\pi^k A$, we see that there exists $a\\in A$ such that $\\alpha \\equiv a \\pmod{\\pi^k \\widehat{A}}$. Since the coefficients of $g$ are central in $A$, we get $g(\\alpha)\\equiv g(a) \\pmod{\\pi^k \\widehat{A}}$. Thus, $f(\\alpha)=f(a)+\\lambda\/\\pi^k$, where $\\lambda\\in\\pi^k \\widehat{A}$, so that $f(\\alpha)\\in \\widehat{A}$. Hence, $f \\in \\textnormal{Int}_K(\\widehat{A})$ and $\\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A)$.\n\\end{proof} \n\nThus, in Example \\ref{Nontriv examples} (2), we have $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}(\\mathbb{Z}_{(p)})$. Moreover, in Example \\ref{Nontriv examples} (3) we have $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}(\\mathbb{Z})$ (see also \\cite{ChabPer} where the profinite completion of $\\mathbb{Z}$ was considered in order to study the polynomial overrings of $\\textnormal{Int}(\\mathbb{Z})$). A more general example, which results in proper containments among all of $D[X]$, $\\textnormal{Int}_K(A)$, and $\\textnormal{Int}(D)$, is the following.\n\n\\begin{Ex}\\label{DVR example}\nLet $D$ be a DVR with maximal ideal $P = \\pi D$ and finite residue field. Let $A$ be a $D$-algebra of finite type with standard assumptions and such that $\\textnormal{Int}_K(A) \\subsetneq \\textnormal{Int}(D)$. Let $\\widehat{A}$ be the $P$-adic completion of $A$. Then, $P$ satisfies the conditions of Theorem \\ref{criterion} with respect to $A$, so $D[X] \\subsetneq \\textnormal{Int}_K(A)$; and $\\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A)$ by Lemma \\ref{DVR lemma}. Thus,\n\\begin{equation*}\nD[X] \\subsetneq \\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A) \\subsetneq \\textnormal{Int}(D).\n\\end{equation*}\nIn general, $\\widehat{A}$ is not finitely generated as a $D$-module (this is the case, for instance, when $A$ is countable but $\\widehat{A}$ is uncountable). So, $\\widehat{A}$ can provide an example of a $D$-algebra that is not finitely generated and for which the integer-valued polynomial ring is properly contained between $D[X]$ and $\\textnormal{Int}(D)$.\n\\end{Ex}\n\n\\begin{Rem}\\label{Quaternion example again}\nLemma \\ref{DVR lemma} also gives us another approach to Example \\ref{Quaternion example}. With notation as in that example, we have $\\widehat{A} \\cong M_2(\\mathbb{Z}_p)$ (indeed, this follows from the fact that $A\/p^k A \\cong M_2(\\mathbb{Z}\/p^k \\mathbb{Z})$ for all $k > 0$). Thus, $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}_\\mathbb{Q}(M_2(\\mathbb{Z}_p)) = \\textnormal{Int}_\\mathbb{Q}(M_2(\\mathbb{Z}_{(p)}))$ even though $A \\not\\cong M_2(\\mathbb{Z}_{(p)})$.\n\\end{Rem}\n\nWe close this paper by using the conditions of Proposition \\ref{equivalent conditions} to prove that when $D$ is Dedekind, $\\textnormal{Int}_K(A)$ has Krull dimension 2. This result was shown by Frisch \\cite[Thm. 5.4]{Fri1} in the case where $A$ is of finite type. Our work does not require $A$ to be finitely generated, and somewhat surprisingly does not require a full classification of the prime ideals of $\\textnormal{Int}_K(A)$.\n\nRecall that a nonzero prime ideal $\\mathfrak{P}$ of $\\textnormal{Int}_K(A)$ is called unitary if $\\mathfrak{P} \\cap D \\ne (0)$, and is called non-unitary if $\\mathfrak{P} \\cap D = (0)$.\n\n\\begin{Thm}\\label{Height thm}\nLet $D$ be a Dedekind domain and let $A$ be a $D$-algebra with standard assumptions. Let $\\mathfrak{P}$ be a nonzero prime ideal of $\\textnormal{Int}_K(A)$.\n\\begin{enumerate}[(1)]\n\\item If $\\mathfrak{P}$ is non-unitary, then $\\mathfrak{P}$ has height 1.\n\\item If $\\mathfrak{P}$ is unitary, then let $P = \\mathfrak{P} \\cap D$.\n\\begin{itemize}\n\\item[(i)] If $P$ does not satisfy any of the conditions of Proposition \\ref{equivalent conditions}, then $\\mathfrak{P}$ has height 2.\n\n\\item[(ii)] If $P$ satisfies one of the conditions of Proposition \\ref{equivalent conditions}, then $\\mathfrak{P}$ is maximal \nand has height at most 2.\n\\end{itemize}\n\\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n(1) Following \\cite[Lem. 5.3]{Fri1}, the non-unitary prime ideals of $\\textnormal{Int}_K(A)$ are in one-to-one correspondence with the prime ideals of $K[X]$. Since $K[X]$ has dimension 1, the non-unitary primes of $\\textnormal{Int}_K(A)$ are all of height 1.\n\n(2) Let $P$ be a nonzero prime of $D$. Assume first that $P$ does not satisfy any of the conditions of Proposition \\ref{equivalent conditions}. Then, $D_P[X] = \\textnormal{Int}_K(A_P) = \\textnormal{Int}_K(A)_P$. It follows that the unitary primes of $\\textnormal{Int}_K(A)$ are in one-to-one correspondence with the primes of $D_P[X]$. Since $D$ is Dedekind, we know that $D_P[X]$ has dimension 2; hence, all the primes of $\\textnormal{Int}_K(A)$ under consideration have height 2.\n\nFor the remainder of the proof, assume that $P = \\mathfrak{P} \\cap D$ does satisfy the conditions of Proposition \\ref{equivalent conditions}. Since $\\mathfrak{P} \\cap D = P$, the prime ideal $\\mathfrak{P}$ survives in $\\textnormal{Int}_K(A)_P=\\textnormal{Int}_K(A_P)$ and clearly its extension $\\mathfrak{P}^e$ is still a prime unitary ideal (so, $\\mathfrak{P}^e\\cap D_P=PD_P$). It is sufficient to show that $\\mathfrak{P}^e$ is a maximal ideal, so we may work over the localizations. Thus, without loss of generality we will assume that $D$ is a DVR. In particular, this means that $P =\\pi D$, for some $\\pi\\in D$.\n\nLet $\\overline{g}\\in N_{D\/P}(A\/PA)$, $\\overline{g}\\not=0$, and let $g\\in D[X]$ be a pullback of $g(X)$. Then $g(A) \\subseteq PA=\\pi A$. Consequently, for each $f\\in\\textnormal{Int}_K(A)$ we have $(g\\circ f)(A)\\subset \\pi A$. Consider the ideal $\\mathfrak{A} = \\{F \\in \\textnormal{Int}_K(A) \\mid F(A) \\subseteq PA\\}$ of $\\textnormal{Int}_K(A)$. Because $P = \\pi D$ is principal, we have $\\mathfrak{A} = \\pi \\textnormal{Int}_K(A)$, which is contained in $\\mathfrak{P}$. Hence, for each $f \\in \\textnormal{Int}_K(A)$, $g \\circ f \\in \\mathfrak{P}$. \n\nNow, if we consider the $D\/P$-algebra $\\textnormal{Int}_K(A)\/\\mathfrak{P}$, we see that each element of $\\textnormal{Int}_K(A)\/\\mathfrak{P}$ is annihilated by $\\overline{g}(X)$. But $\\textnormal{Int}_K(A)\/\\mathfrak{P}$ is a domain, and for it to be annihilated by a nonzero polynomial, it must be finite. Thus, in fact $\\textnormal{Int}_K(A)\/\\mathfrak{P}$ is a finite field, and so $\\mathfrak{P}$ is maximal.\n\nFinally, to show that $\\mathfrak{P}$ has height at most 2, let $\\mathfrak{Q}$ be a prime of $\\textnormal{Int}_K(A)$ such that $(0) \\subsetneq \\mathfrak{Q} \\subseteq \\mathfrak{P}$. If $\\mathfrak{Q}$ is unitary, then we have $\\mathfrak{Q} \\cap D = P$, and by our work above $\\mathfrak{Q}$ is maximal, hence equal to $\\mathfrak{P}$. If $\\mathfrak{Q}$ is non-unitary, then it has height 1 by part (1) of the theorem. It follows that $\\mathfrak{P}$ has height at most 2.\n\\end{proof}\n\n\\begin{Cor}\\label{Krull dimension}\nLet $D$ be a Dedekind domain with quotient field $K$. Let $A$ be a $D$-algebra with standard assumptions. Then, $\\textnormal{Int}_K(A)$ has Krull dimension 2.\n\\end{Cor}\n\\begin{proof}\nIf $\\textnormal{Int}_K(A) = D[X]$, then its dimension equals that of $D[X]$, which is 2. So, assume that $\\textnormal{Int}_K(A)$ is nontrivial. By Theorem \\ref{criterion}, there exists a prime $P$ of $D$ that satisfies the conditions of Proposition \\ref{equivalent conditions}.\n\nLet $\\mathfrak{P} = \\{f \\in \\textnormal{Int}_K(A) \\mid f(0) \\in P \\}$. Since $\\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D)$, $\\mathfrak{P}$ is an ideal of $\\textnormal{Int}_K(A)$, and it is easily seen to be prime and unitary, with $\\mathfrak{P} \\cap D = P$. Moreover, it contains the non-unitary ideal $XK[X] \\cap \\textnormal{Int}_K(A)$. Hence, $\\mathfrak{P}$ has height at least 2, and so $\\dim(\\textnormal{Int}_K(A)) \\geq 2$. However, $\\dim(\\textnormal{Int}_K(A)) \\leq 2$ by Theorem \\ref{Height thm}, so we conclude that $\\dim(\\textnormal{Int}_K(A)) = 2$.\n\\end{proof}\n\n\\subsection*{Acknowledgments}\n\\noindent This research has been supported by the grant ``Assegni Senior'' of the University of Padova. The authors wish to thank the referee for several suggestions which improved the quality of the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n{S}{cene} text recognition is an essential process in computer vision tasks. Many practical applications such as traffic sign reading, product recognition, intelligent inspection, and image searching, benefit from the rich semantic information of scene text. With the development of scene text detection methods \\cite{gomez2017textproposals,khare2016blind,sun2015robust,zhu2016could}, scene character recognition has emerged at the forefront of this research topic and is regarded as an open and very challenging research problem \\cite{su2017accurate}.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\\begin{minipage}[c]{0.15\\textwidth}\n\\centering\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/regular-a.jpg}\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/regular-b.jpg}\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/regular-c.jpg}\n\\end{minipage}%\n}%\n\\subfigure[]{\n\\begin{minipage}[c]{0.15\\textwidth}\n\\centering\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-d.jpg}\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-e.jpg}\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-f.jpg}\n\\end{minipage}%\n}%\n\\subfigure[]{\n\\begin{minipage}[c]{0.15\\textwidth}\n\\centering\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-g.jpg}\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-h.jpg}\n \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-i.jpg}\n\\end{minipage}%\n}%\n\n\\caption{Examples of regular and irregular scene text. (a) Regular text. (b) Slanted and perspective text. (c) Curved text.}\n\\label{fig:1-scene-text}\n\\end{figure}\n\nNowadays, regular text recognition methods \\cite{bissacco2013photoocr,neumann2012real,shi2017end,su2017accurate,wang2012end} have achieved notable success. Moreover, methods based on convolutional neural networks \\cite{bissacco2013photoocr,jaderberg2016reading,wang2012end} have been broadly applied. Integrating recognition models with recurrent neural networks \\cite{he2016reading,shi2017end,shi2016robust} and attention mechanisms \\cite{cheng2017focusing,cheng2017arbitrarily,lee2016recursive,yang2017learning} yields better performance for these models.\n\nNevertheless, most current recognition models remain too unstable to handle multiple disturbances from the environment. Furthermore, the various shapes and distorted patterns of irregular text cause additional challenges in recognition. As illustrated in Fig. \\ref{fig:1-scene-text}, scene text with irregular shapes, such as perspective and curved text, is still very challenging to recognize.\n\nReading text is naturally regarded as a multi-classification task involving sequence-like objects \\cite{shi2017end}. Usually, the characters in one text are of the same size. However, characters in different scene texts can vary in size. Therefore, we propose the multi-object rectified attention network (MORAN), which can read rotated, scaled and stretched characters in different scene texts. The MORAN consists of a multi-object rectification network \\textbf{(MORN)} to rectify images and an attention-based sequence recognition network \\textbf{(ASRN)} to read the text. We separate the difficult recognition task into two parts. First, as one kind of spatial transformer, the MORN rectifies images that contain irregular text. As Fig. \\ref{fig:2-MORAN-system} shows, after the rectification by the MORN, the slanted text becomes more horizontal, tightly-bounded, and easier to read. Second, ASRN takes the rectified image as input and outputs the predicted word.\n\nThe training of the MORN is guided by the ASRN, which requires only text labels. Without any geometric-level or pixel-level supervision, the MORN is trained in a weak supervision way. To facilitate this manner of network training, we initialize a basic coordinate grid. Every pixel of an image has its own position coordinates. The MORN learns and generates an offset grid based on these coordinates and samples the pixel value accordingly to rectify the image. The rectified image is then obtained for the ASRN.\n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=8.5cm,height=3.2cm]{picture\/Moran-system.jpg}\n\n\\put(11,33){Input}\n\\put(10,28){Image}\n\\put(45,33){Rectified}\n\\put(48,28){Image}\n\\put(82,26){Result}\n\n\\put(79,17){JOHNNY}\n\\put(28,26){MORN}\n\\put(66,26){ASRN}\n\n\\put(29,7){Weak}\n\\put(25,3){Supervision}\n\n\\put(65,7){Text Label}\n\\put(64,3){Supervision}\n\n\\end{overpic}\n\n\\caption{Overview of the MORAN. The MORAN contains a MORN and an ASRN. The image is rectified by the MORN and given to the ASRN. The dashed lines show the direction of gradient propagation, indicating that the two sub-networks are jointly trained.}\n\\label{fig:2-MORAN-system}\n\\end{figure}\n\nWith respect to the ASRN, a decoder with an attention mechanism is more likely to predict the correct words because of the rectified images. However, Cheng et al. \\cite{cheng2017focusing} found that existing attention-based methods cannot obtain accurate alignments between feature areas and targets. Therefore, we propose a fractional pickup method to train the ASRN. By adopting several scales of stretch on different parts of the feature maps, the feature areas are changed randomly at every iteration in the training phase. Owing to training with fractional pickup, the ASRN is more robust to the variation of context. Experiments show that the ASRN can accurately focus on objects.\n\nIn addition, we designed a curriculum learning strategy for the training of the MORAN. Because the MORN and ASRN are mutually beneficial in terms of performance, we first fix one of them to more efficiently optimize the other. Finally, the MORN and ASRN are optimized in an end-to-end fashion to improve performance. In short, the contributions of our research are as follows:\n\n\\begin{itemize}\n\n\\item We propose the MORAN framework to recognize irregular scene text. The framework contains a multi-object rectification network (MORN) and an attention-based sequence recognition network (ASRN). The image rectified by the MORN is more readable for the ASRN.\n\n\\item Trained in a weak supervision way, the sub-network MORN is flexible. It is free of geometric constraints and can rectify images with complicated distortion.\n\n\\item We propose a fractional pickup method for the training of the attention-based decoder in the ASRN. To address noise perturbations, we expand the visual field of the MORAN, which further improves the sensitivity of the attention-based decoder.\n\n\\item We propose a curriculum learning strategy that enables the MORAN to learn efficiently. Owing to the training with this strategy, the MORAN outperforms state-of-the-art methods on several standard text recognition benchmarks, including the IIIT5K, SVT, ICDAR2003, ICDAR2013, ICDAR2015, SVT-Perspective, and CUTE80 datasets.\n\n\\end{itemize}\n\nThe rest of the paper is organized as follow. Section 2 reviews related work. Section 3 details the proposed method. Experimental results are given in Section 4, and the conclusions are presented in Section 5.\n\n\\section{Related Work}\n\\label{section:Related work}\nIn recent years, the recognition of scene text has greatly advanced because of the rapid development of neural networks \\cite{gu2017recent}. Zhu et al. \\cite{zhu2016scene} and Ye et al. \\cite{ye2015text} have provided an overview of the major advances in the field of scene text detection and recognition. Based on the sliding window method \\cite{wang2011end,wang2010word}, pattern features extracted by a neural network become dominant with respect to the hand crafted features, such as the connected components \\cite{neumann2012real}, strokelet generation \\cite{yao2014strokelets}, histogram of oriented gradients descriptors \\cite{dalal2005histograms,su2014accurate}, tree-structured models \\cite{shi2014end}, semi-markov conditional random fields \\cite{seok2015scene} and generative shape models \\cite{lou2016generative}. For instance, Bissacco \\cite{bissacco2013photoocr} applied a network with five hidden layers for character classification. Using convolutional neural networks (CNNs), Jaderberg et al. \\cite{Jaderberg2015Deep} and Yin et al. \\cite{yin2017scene} proposed respective methods for unconstrained recognition.\n\nWith the widespread application of recurrent neural networks (RNNs) \\cite{cho2014learning,hochreiter1997long}, CNN-based methods are combined with RNNs for better learning of context information. As a feature extractor, the CNN obtains the spatial features of images. Then, the RNN learns the context of features. Shi et al. \\cite{shi2017end} proposed an end-to-end trainable network with both CNNs and RNNs, named CRNN. Guided by the CTC loss \\cite{graves2006connectionist}, the CRNN-based network learns the conditional probability between predictions and sequential labels.\n\nFurthermore, attention mechanisms \\cite{bahdanau2014neural} focus on informative regions to achieve better performance. Lee et al. \\cite{lee2016recursive} proposed a recursive recurrent network with attention modeling for scene text recognition. Yang et al. \\cite{yang2017learning} addressed a two-dimensional attention mechanism. Cheng et al. \\cite{cheng2017focusing} used the focusing attention network (FAN) to correct shifts in attentional mechanisms and achieved more accurate position predictions.\n\nCompared with regular text recognition work, irregular text recognition is more difficult. One kind of irregular text recognition method is the bottom-up approach \\cite{cheng2017arbitrarily,yang2017learning}, which searches for the position of each character and then connects them. Another is the top-down approach \\cite{liu2016star,shi2016robust}. This type of approach matches the shape of the text, attempts to rectify it, and reduces the degree of recognition difficulty.\n\nIn the bottom-up manner, a two-dimensional attention mechanism for irregular text was proposed by Yang et al. \\cite{yang2017learning}. Based on the sliced Wasserstein distance \\cite{rabin2011wasserstein}, the attention alignment loss is adopted in the training phase, which enables the attention model to accurately extract the character features while ignoring the redundant background information. Cheng et al. \\cite{cheng2017arbitrarily} proposed an arbitrary-orientation text recognition network, which uses more direct information of the position to instruct the network to identify characters in special locations.\n\nIn the top-down manner, STAR-Net \\cite{liu2016star} used an affine transformation network that transforms the rotated and differently scaled text into more regular text. Meanwhile, a ResNet \\cite{he2016deep} is used to extract features and handle more complex background noise. RARE \\cite{shi2016robust} regresses the fiducial transformation points on sloped text and even curved text, thereby mapping the corresponding points onto standard positions of the new image. Using thin-plate-spline \\cite{bookstein1989principal} to back propagate the gradients, RARE is end-to-end optimized.\n\nOur proposed MORAN model uses the top-down approach. The fractional pickup training method is thus designed to improve the sensitivity of the MORAN to focus on characters. For the training of the MORAN, we propose a curriculum learning strategy for better convergence.\n\n\\section{Methodology}\n\nThe MORAN contains two parts. One is the MORN, which is trained in a weak supervision way to learn the offset of each part of the image. According to the predicted offsets, we apply sampling and obtain a rectified text image. The other one is ASRN, a CNN-LSTM framework followed by an attention decoder. The proposed fractional pickup further improves attention sensitivity. The curriculum learning strategy guides the MORAN to achieve state-of-the-art performance.\n\n\\subsection{Multi-Object Rectification Network}\n\\label{section:Multi-Object Rectification Network}\nCommon methods to rectify patterns such as the affine transformation network, are limited by certain geometric constraints. With respect to the affine transformation, it is limited to rotation, scaling, and translation. However, one image may have several kinds of deformations, and the distortion of scene text will thus be complicated. As shown in Fig. \\ref{fig:compare-stn}, the characters in the image become slanted after rectification by the affine transformation. The black edges introduce additional noise. Therefore, transformations with geometric constraints can not cover all complicated deformations.\n\nAnother method that is free of geometric constraints, is the deformable convolutional network \\cite{deformable2017}. Using deformable convolutional kernels, the feature extractor automatically selects informative features. We attempted to combine the recognition network with a deformable convolutional network. However, as a sequence-to-sequence problem, irregular text recognition is more challenging. The network sometimes failed to converge. The best accuracy rate on IIIT5K we achieved was only 78.1\\%, which is far behind the state-of-the-art result (91.2\\%).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5cm,height=5cm]{picture\/compare-stn.jpg}\n\\caption{Comparison of the MORN and affine transformation. The MORN is free of geometric constraints. The main direction of rectification predicted by the MORN for each character is indicated by a yellow arrow. The offset maps generated by the MORN are visualized as a heat map. The offset values on the boundary between red and blue are zero. The directions of rectification on both sides of the boundary are opposite and outward. The depth of the color represents the magnitude of the offset value. The gradual-change in color indicates the smoothness of the rectification.}\n\\label{fig:compare-stn}\n\\end{figure}\n\nBecause the recognition models remain inadequately strong to handle multiple disturbances from various shapes, we consider rectifying images to reduce the difficulty of the recognition. As demonstrated in Fig. \\ref{fig:3-MORAN-overview}, the MORN architecture rectifies the distorted image. The MORN predicts the offset of each part of the image without any geometric constraint. Based on the predicted offsets, the image is rectified and becomes easier to recognize.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=17cm]{picture\/Moran-overview.jpg}\n\\caption{Overall structure of MORAN. }\n\\label{fig:3-MORAN-overview}\n\\end{figure*}\n\nFurthermore, the MORN predicts the position offsets but not the categories of characters. The character details for classification are not necessary. We hence place a pooling layer before the convolutional layer to avoid noise and reduce the amount of calculation.\n\n\\begin{table}[h]\n\\centering\n\\caption{Architecture of the MORN}\n\\label{table:The architecture of MORN}\n\\begin{tabular*}{8.5cm}{c|p{3.5cm}<{\\centering}|c}\n\\hline\nType & Configurations & Size \\\\\n\\hline\nInput & - & 1$\\times$32$\\times$100 \\\\\n\\hline\nMaxPooling & k2, s2 & 1$\\times$16$\\times$50 \\\\\n\\hline\nConvolution & maps:64, k3, s1, p1 & 64$\\times$16$\\times$50 \\\\\n\\hline\nMaxPooling & k2, s2 & 64$\\times$8$\\times$25 \\\\\n\\hline\nConvolution & maps:128, k3, s1, p1 & 128$\\times$8$\\times$25 \\\\\n\\hline\nMaxPooling & k2, s2 & 128$\\times$4$\\times$12 \\\\\n\\hline\nConvolution & maps:64, k3, s1, p1 & 64$\\times$4$\\times$12 \\\\\n\\hline\nConvolution & maps:16, k3, s1, p1 & 16$\\times$4$\\times$12 \\\\\n\\hline\nConvolution & maps:2, k3, s1, p1 & 2$\\times$4$\\times$12 \\\\\n\\hline\nMaxPooling & k2, s1 & 2$\\times$3$\\times$11 \\\\\n\\hline\nTanh & - & 2$\\times$3$\\times$11 \\\\\n\\hline\nResize & - & 2$\\times$32$\\times$100 \\\\\n\\hline\n\\end{tabular*}\n\\begin{tablenotes}\n\\item Here k, s, p are kernel, stride and\npadding sizes, respectively. For example, $k3$ represents a $3\\times3$ kernel size.\n\\end{tablenotes}\n\\end{table}\n\nThe architecture of the MORN is given in Table\\ref{table:The architecture of MORN}. Each convolutional layer is followed by a batch normalization layer and a ReLU layer except for the last one. The MORN first divides the image into several parts and then predicts the offset of each part. With an input size of $32\\times100$, the MORN divides the image into $3\\times 11 = 33$ parts. All the offset values are activated by $Tanh(\\cdot)$, resulting in values within the range of $(-1, 1)$. The offset maps contain two channels, which denote the x-coordinate and y-coordinate respectively. Then, we apply bilinear interpolation to smoothly resize the offset maps to a size of $32\\times100$, which is the same size of the input image. After allocating the specific offset to each pixel, the transformation of the image is smooth. As demonstrated in Fig.\\ref{fig:compare-stn}, the color depth gradually changes on both sides of the boundary between the red and blue colors in the heat map, which evidences the smoothness of the rectification. There are no indented edges in the rectified image.\n\nMoreover, because every value in the offset maps represents the offset from the original position, we generate a basic grid from the input image to represent the original positions of the pixels. The basic grid is generated by normalizing the coordinate of each pixel to $[-1, 1]$. The coordinates of the top-left pixel are $(-1, -1)$, and those of the bottom-right one are $(1, 1)$. Pixels at the same positions on different channels have the same coordinates. Similar to the offset maps, the grid contains two channels, which represent the x-coordinate and y-coordinate, respectively. Then, the basic grid and the resized offset maps are summed as follows,\n\\begin{equation}\noffset_{(c,i,j)}^{'} = offset_{(c,i,j)}+basic_{(c,i,j)} , c = 1,2\n\\end{equation}\nwhere $(i,j)$ is the position of the $i$-th row and $j$-th column.\n\nBefore sampling, the x-coordinate and y-coordinate on the offset maps are normalized to $[0, W]$ and $[0, H]$, respectively. Here, $H\\times W$ is the size of the input image. The pixel value of $i$-th row and $j$-th column in rectified image $I'$ is,\n\\begin{equation}\nI'_{(i, j)} = I_{(i^{'}, j^{'})} \\label{sampling}\n\\end{equation}\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\ni^{'} = offset_{(1, i, j)}^{'} \\\\\nj^{'} = offset_{(2, i, j)}^{'}\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $I$ is the input image. Further, $i^{'}$ is obtained from the first channel of the offset maps, whereas $j^{'}$ is from the second channel. Both $i^{'}$ and $j^{'}$ are real values as opposed to integers so rectified image $I'$ is sampled from $I$ using bilinear interpolation.\n\nBecause Equation (\\ref{sampling}) is differentiable, the MORN can back-propagate the gradients. The MORN can be trained in a weak supervision way with images and associated text labels only, which means that it does not need pixel-level labeling information about the deformation of the text.\n\nAs Fig. \\ref{fig:4-ori-rectified-img} shows, the text in the input images is irregular. However, the text in the rectified images is more readable. Slanted or perspective texts become tightly bound after rectification. Furthermore, redundant noise is eliminated by the MORN for the curved texts. The background textures are removed in the rectified images of Fig. \\ref{fig:4-ori-rectified-img} (b).\n\n\\begin{figure}[h]\n\\centering\n\\subfigure[Perspective texts]{\n\\begin{minipage}[c]{0.45\\textwidth}\n\\centering\n \\includegraphics[width=7cm,height=4cm]{picture\/ori_img.jpg}\n\\end{minipage}%\n}%\n\n\\subfigure[curved texts]{\n\\begin{minipage}[c]{0.45\\textwidth}\n\\centering\n \\includegraphics[width=7cm,height=4cm]{picture\/ori_img_curve.jpg}\n\\end{minipage}%\n}%\n\n\n\\caption{Results of the MORN on challenging image text. The input images are shown on the left and the rectified images are shown on the right. The heat maps visualize offset maps as well as Fig. \\ref{fig:compare-stn}. (a) Slanted and perspective text. (b) Curved text, which is more challenging for recognition. Removed background textures are indicated by red circles.}\n\\label{fig:4-ori-rectified-img}\n\\end{figure}\n\nThe advantages of the MORN are manifold. 1) The rectified images are more readable owing to the regular shape of the text and the reduced noise. 2) The MORN is more flexible than the affine transformation. It is free of geometric constraints, which enables it to rectify images using complicated transformations. 3) The MORN is more flexible than methods using a specific number of regressing points. Existing method \\cite{shi2016robust} cannot capture the text shape in details if the width of the image is large. Thus the MORN has no limit with respect to the image size, especially the width of the input image. 4) The MORN does not require extra labelling information of character positions. Therefore, it can be trained in a weak supervision way by using existing training datasets.\n\n\\subsection{Attention-based Sequence Recognition Network}\n\\label{section:Attention-based Sequence Recognition Network}\nAs Fig. \\ref{fig:3-MORAN-overview} shows, the major structure of the ASRN is a CNN-BLSTM framework. We adopt a one-dimensional attention mechanism at the top of CRNN. The attention-based decoder, proposed by Bahdanau et al. \\cite{bahdanau2014neural}, is used to accurately align the target and label. It is based on an RNN and directly generates the target sequence $(y_{1},y_{2} ...,y_{N})$. The largest number of steps that the decoder generates is $T$. The decoder stops processing when it predicts an end-of-sequence token $``EOS\"$ \\cite{sutskever2014sequence}. At time step $t$, output $y_t$ is,\n\\begin{equation}\ny_{t} = Softmax(W_{out}s_{t}+b_{out})\n\\end{equation}\nwhere $s_{t}$ is the hidden state at time step $t$. We update $s_{t}$ using GRU \\cite{cho2014learning}. State $s_{t}$ is computed as:\n\\begin{equation}\ns_t = GRU(y_{prev}, g_{t}, s_{t-1})\n\\end{equation}\nwhere $y_{prev}$ denotes the embedding vectors of the previous output $y_{t-1}$ and $g_{t}$ represents the glimpse vectors, respectively calculated as,\n\\begin{equation}\ny_{prev} = Embedding(y_{t-1})\n\\end{equation}\n\\begin{equation}\ng_{t} = \\sum_{i=1}^L(\\alpha_{t,i} h_{i}) \\label{equ-g}\n\\end{equation}\nwhere $h_{i}$ denotes the sequential feature vectors and $L$ is the length of the feature maps. In addition, $\\alpha_{t,i}$ is the vector of attention weights as follows,\n\\begin{equation}\n\\alpha_{t,i} = {exp(e_{t,i})} \/ {\\sum_{j=1}^L(exp(e_{t,j}))} \\label{equ-alpha}\n\\end{equation}\n\\begin{equation}\ne_{t,i} = Tanh(W_{s}s_{t-1}+W_{h}h_{i}+b) \\label{equ-e}\n\\end{equation}\n\nHere, $W_{out}$, $b_{out}$, $W_{s}$, $W_{h}$ and $b$ are trainable parameters. Note that $y_{prev}$ is embedded from the ground truth of the last step in the training phase, whereas the ASRN only uses the predicted output of the last step as $y_{t-1}$ in the testing phase.\n\nThe decoder outputs the predicted word in an unconstrained manner in lexicon-free mode. If lexicons are available, we evaluate the probability distributions for all words and choose the word with the highest probability as the final result.\n\nThe architecture of the ASRN is given in Table\\ref{table:The architecture of ASRN}. Each convolutional layer is followed by a batch normalization layer and a ReLU layer.\n\\begin{table}[t]\n\\centering\n\\caption{Architecture of the ASRN}\n\\label{table:The architecture of ASRN}\n\\begin{tabular*}{8.5cm}{c|p{3.5cm}<{\\centering}|c}\n\\hline\nType & Configurations & Size \\\\\n\\hline\nInput & - & 1$\\times$32$\\times$100 \\\\\n\\hline\nConvolution & maps:64, k3, s1, p1 & 64$\\times$32$\\times$100 \\\\\n\\hline\nMaxPooling & k2, s2 & 64$\\times$16$\\times$50 \\\\\n\\hline\nConvolution & maps:128, k3, s1, p1 & 128$\\times$16$\\times$50 \\\\\n\\hline\nMaxPooling & k2, s2 & 128$\\times$8$\\times$25 \\\\\n\\hline\nConvolution & maps:256, k3, s1, p1 & 256$\\times$8$\\times$25 \\\\\n\\hline\nConvolution & maps:256, k3, s1, p1 & 256$\\times$8$\\times$25 \\\\\n\\hline\nMaxPooling & k2, s2x1, p0x1 & 256$\\times$4$\\times$26 \\\\\n\\hline\nConvolution & maps:512, k3, s1, p1 & 512$\\times$4$\\times$26 \\\\\n\\hline\nConvolution & maps:512, k3, s1, p1 & 512$\\times$4$\\times$26 \\\\\n\\hline\nMaxPooling & k2, s2x1, p0x1 & 512$\\times$2$\\times$27 \\\\\n\\hline\nConvolution & maps:512, k2, s1 & 512$\\times$1$\\times$26 \\\\\n\\hline\nBLSTM & hidden unit:256 & 256$\\times$1$\\times$26 \\\\\n\\hline\nBLSTM & hidden unit:256 & 256$\\times$1$\\times$26 \\\\\n\\hline\nGRU & hidden unit:256 & 256$\\times$1$\\times$26 \\\\\n\\hline\n\\end{tabular*}\n\\begin{tablenotes}\n\\item Here, k, s, p are kernel, stride and\npadding sizes, respectively. For example, $s2\\times1$ represents a $2\\times1$ stride size. ``BLSTM\" stands for bidirectional-LSTM. ``GRU\" is in attention-based decoder.\n\\end{tablenotes}\n\\end{table}\n\n\\subsection{Fractional Pickup}\n\nThe decoder in the ASRN learns the matching relationship between labels and target characters in images. It is a data-driven process. The ability to choose regions that are focus-worthy is enhanced by the feedback of correct alignment.\n\nHowever, scene text is surrounded by various types of noise. Often, the decoder is likely to be deceived into focusing on ambiguous background regions in practical applications. If the decoder generates an incorrect region of focus, the non-corresponding features are chosen, which can cause a failed prediction.\n\nSome challenging samples for recognition are presented in Fig. \\ref{fig:5-fractional-pickup}. In this figure, the images contain text with shadows and unclear boundaries between characters or complicated backgrounds. Moreover, the focus regions generated by the decoder are narrow, which increases the probability of drifting from the correct regions.\n\n\\begin{figure}[t]\n\\centering\n\\rule{8cm}{0.05em}\n\\\\\n-------------------------------------------------------------\n\\begin{overpic}[width=8cm,height=3cm]{picture\/fp_1.jpg}\n\n\\put(13,39){Without FP}\n\\put(67,39){With FP}\n\n\\put(20,-3){\\color{blue}{hot}\\color{red}{l}}\n\\put(72,-3){\\color{blue}{hotel}}\n\\end{overpic}\n\n-------------------------------------------------------------\n\n\\begin{overpic}[width=8cm,height=3cm]{picture\/fp_2.jpg}\n\\put(20,-3){\\color{blue}{a}\\color{red}{r}\\color{blue}{ge}\\color{red}{h}}\n\\put(72,-3){\\color{blue}{angels}}\n\\end{overpic}\n\n-------------------------------------------------------------\n\n\\begin{overpic}[width=8cm,height=3cm]{picture\/fp_3.jpg}\n\\put(20,-3){\\color{red}{e}\\color{blue}{aser}}\n\\put(72,-3){\\color{blue}{laser}}\n\\end{overpic}\n\n-------------------------------------------------------------\n\n\\caption{Difference in $\\alpha_{t}$ for training with and without fractional pickup. Here $\\alpha_{t}$ is visualized as a heat map. We delete the $\\alpha_{t}$ corresponding to ``EOS\".}\n\\label{fig:5-fractional-pickup}\n\\end{figure}\n\nWe propose a training method called fractional pickup that fractionally picks up the neighboring features in the training phase. An attention-based decoder trained by fractional pickup method can perceive adjacent characters. The wider field of attention contributes to the robustness of the MORAN.\n\nWe hence adopt fractional pickup at each time step of the decoder. In other words, a pair of attention weights are selected and modified at every time step. At time step $t$, $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$ are updated as,\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\alpha_{t,k}^{,} = \\beta \\alpha_{t,k}+(1-\\beta)\\alpha_{t,k+1} \\\\\n\\alpha_{t,k+1}^{,} = (1-\\beta) \\alpha_{t,k}+\\beta\\alpha_{t,k+1}\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere decimal $\\beta$ and integer $k$ are randomly generated as,\n\\begin{equation}\n\\beta = rand(0,1)\n\\end{equation}\n\\begin{equation}\nk = rand[1,T-1]\n\\end{equation}\nHere, T is the maximum number of steps of the decoder.\n\n\\textbf{Variation of Distribution}\nFractional pickup adds randomness to $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$ in the decoder. This means that, even for the same image, the distribution of $\\alpha_{t}$ changes every time step in the training phase. As noted in Equation (\\ref{equ-g}), the glimpse vectors $g_{t}$ grabs the sequential feature vectors $h_{i}$ according to the various distributions of $\\alpha_{t}$, which is equivalent to the changes in feature areas. The randomness of $\\beta$ and $k$ avoids over-fitting and contributes to the robustness of the decoder.\n\n\\textbf{Shortcut of Forward Propagation}\nSequential feature vector $h_{i}$ is the output of the last bidirectional-LSTM in the ASRN. As shown in Fig. \\ref{fig:short-cut}, for step $k+1$ in the bidirectional-LSTM, a shortcut connecting to step $k$ is created by fractional pickup. The shortcut retains some features of the previous step in the training phase, which is the interference to the forget gate in bidirectional-LSTM. Therefore, fractional pickup provides more information about the previous step and increases the robustness for the bidirectional-LSTM in the ASRN.\n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=4cm]{picture\/short-cut.jpg}\n\\put(-5,65){$h_{i}$}\n\\end{overpic}\n\\caption{Fractional pickup creates a shortcut of forward propagation. The shortcut is drawn as a red arrow.}\n\\label{fig:short-cut}\n\\end{figure}\n\n\\textbf{Broader Visual Field}\nTraining with fractional pickup disturbs the decoder through the local variation of $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$. Note that $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$ are neighbors. Without fractional pickup, the error term of sequence feature vector $h_{k}$ is,\n\\begin{equation}\n\\delta_{h_{k}} = \\delta_{g_{t}}\\alpha_{t,k}\n\\end{equation}\nwhere $\\delta_{g_{t}}$ is the error term of glimpse vector $g_{t}$. $\\delta_{h_{k}}$ is only relevant to $\\alpha_{t,k}$. However, with fractional pickup, the error item becomes,\n\\begin{equation}\n\\delta_{h_{k}} = \\delta_{g_{t}}(\\beta \\alpha_{t,k}+(1-\\beta)\\alpha_{t,k+1})\n\\end{equation}\nwhere $\\alpha_{t,k+1}$ is relevant to $h_{k+1}$, as noted in Equations (\\ref{equ-alpha}) and (\\ref{equ-e}), which means $\\delta_{h_{k}}$ is influenced by the neighbouring features. Owing to the disturbance, back-propagated gradients are able to dynamically optimize the decoder over a broader range of neighbouring regions.\n\nThe MORAN trained with fractional pickup method generates a smoother $\\alpha_{t}$ at each time step. Accordingly, it extracts features not only of the target characters, but also of the foreground and background context. As demonstrated in Fig. \\ref{fig:5-fractional-pickup}, the expanded visual field enables the MORAN to correctly predict target characters. To the best of our knowledge, this is the first attempt to adopt a shortcut in the training of the attention mechanism.\n\n\\subsection{Curriculum Training}\n\\label{section:curriculum-training}\n\nThe MORAN is end-to-end trainable with random initialization. However, end-to-end training consumes considerable time. We found that the MORN and ASRN can hinder each other during training. A MORN cannot be guided to rectify images when the input images have been correctly recognized by the high-performance ASRN. For the same reason, the ASRN will not gain robustness because the training samples have already been rectified by the MORN. The reasons above lead to inefficient training.\n\nTherefore, we propose a curriculum learning strategy to guide each sub-network in MORAN. The strategy is a three-step process. We first optimize the MORN and ASRN respectively and then join them together for further end-to-end training. The difficulty of training samples is gradually increased. The training set is denoted as $D = \\left \\{I_{i}, Y_{i} \\right \\}, i=1...N $. We minimize the negative log-likelihood of conditional probability of $D$ as follows:\n\n\\begin{equation}\nLoss = -\\sum_{i=1}^N{ \\sum_{t=1}^{\\left| Y_{i} \\right|}{\\log p(Y_{i,t} \\left| \\right. I_{i}; \\theta)} }\n\\end{equation}\nwhere $Y_{i,t}$ is the ground truth of the $t$-th character in $I_{i}$. $\\theta$ denotes the parameters of MORAN.\n\n\\textbf{First Stage for ASRN}\nWe first optimize the ASRN by using regular training samples. The dataset released by Gupta et al. \\cite{gupta2016synthetic} has tightly bounded annotations, which makes it possible to crop a text region with a tightly bounded box. The ASRN is first trained with these regular samples. Then, we simply crop every text using a minimum circumscribed horizontal rectangle to obtain irregular training samples. The commonly used datasets released by Jaderberg et al. \\cite{jaderberg2014synthetic} and Gupta et al. \\cite{gupta2016synthetic} offer abundant irregular training samples. We use them for the following training. Taking advantage of them, we optimize ASRN, which thus achieves higher accuracy.\n\n\\textbf{Second Stage for MORN}\nThe ASRN trained using regular training samples is chosen to guide the MORN training. This ASRN is not adequately robust for irregular text recognition so it is able to provide informative gradients for the MORN. We fix the parameters of this ASRN, and stack it after the MORN. If the transformation of the MORN does not reduce the difficulty of recognition, few meaningful gradients will be provided by the ASRN. The optimization of MORN would not progress. Only the correct transformation that decreases difficulty for recognition will give positive feedback to the MORN.\n\n\\textbf{Third Stage for End-to-end Optimization}\nAfter the MORN and ASRN are optimized individually, we connect them for joint training in an end-to-end fashion. Joint training enables MORAN to complete end-to-end optimization and outperform state-of-the-art methods.\n\n\\section{Experiments}\nIn this section we describe extensive experiments conducted on various benchmarks, including regular and irregular datasets. The performances of all the methods are measured by word accuracy.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Comparison of pooling layers in lexicon-free mode. ``No\", ``AP\" and ``MP\" respectively indicate no pooling layer, an average-pooling layer and a max-pooling layer at the top of the MORN. The kernel size is 2. ``s\" represents the stride. }\n\\label{table:comparison-of-pooling}\n\\begin{tabular}{|c| c | c | c | c | c | c | c | c | c}\n\\hline\n\\multirow{2}{*}{} & s & IIIT5K & SVT & IC03 & IC13 & SVT-P & CUTE80 & IC15 \\\\\n\\cline{2-9}\n\\hline\nNo & - & 85.7 & 87.9 & 92.9 & 91.5 & 75.8 & 65.9 & 59.4 \\\\\nAP & 2 & 89.2 & 87.4 & 94.8 & 91.1 & 75.9 & 71.1 & 64.6 \\\\\nAP & 1 & 89.3 & 87.9 & 94.7 & 91.6 & 75.9 & 72.9 & 64.9 \\\\\nMP & 2 & 90.4 & 88.2 & 94.5 & 91.8 & \\textbf{76.1} & 76.4 & 68.4 \\\\\nMP & 1 & \\textbf{91.2} & \\textbf{88.3} & \\textbf{95.0} & \\textbf{92.4} & \\textbf{76.1} & \\textbf{77.4} & \\textbf{68.8} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[t]\n\\centering\n\\caption{Performance of the MORAN. }\n\\label{table:Performance-of-MORAN}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multirow{1}{*}{Method} & IIIT5K & SVT & IC03 & IC13 & SVT-P & CUTE80 & IC15 \\\\\n\\cline{2-8}\n\\hline\nEnd-to-end training & 89.9 & 84.1 & 92.5 & 90.0 & 76.1 & 77.1 & 68.8 \\\\\n\\hline\nOnly ASRN & 84.2 & 82.2 & 91.0 & 90.1 & 71.0 & 64.6 & 65.6 \\\\\nMORAN without FP & 89.7 & 87.3 & 94.5 & 91.5 & 75.5 & 77.1 & 68.6 \\\\\nMORAN with FP & \\textbf{91.2} & \\textbf{88.3} & \\textbf{95.0} & \\textbf{92.4} & \\textbf{76.1} & \\textbf{77.4} & \\textbf{68.8} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Datasets}\n\n\\textbf{IIIT5K-Words (IIIT5K)} \\cite{mishra2012scene} contains 3000 cropped word images for testing. Every image has a 50-word lexicon and a 1000-word lexicon. The lexicon consists of a ground truth and some randomly picked words.\n\n\\textbf{Street View Text (SVT)} \\cite{wang2011end} was collected from the Google Street View, consisting of 647 word images. Many images are severely corrupted by noise and blur, or have very low resolutions. Each image is associated with a 50-word lexicon.\n\n\\textbf{ICDAR 2003 (IC03)} \\cite{lucas2003icdar} contains 251 scene images that are labeled with text bounding boxes. For fair comparison, we discarded images that contain non-alphanumeric characters or those have less than three characters, following Wang, Babenko, and Belongie \\cite{wang2011end}. The filtered dataset contains 867 cropped images. Lexicons comprise of a 50-word lexicon defined by Wang et al. \\cite{wang2011end} and a ``full lexicon\". The latter lexicon combines all lexicon words.\n\n\\textbf{ICDAR 2013 (IC13)} \\cite{karatzas2013icdar} inherits most of its samples from IC03. It contains 1015 cropped text images. No lexicon is associated with this dataset.\n\n\\textbf{SVT-Perspective (SVT-P)} \\cite{quy2013recognizing} contains 645 cropped images for testing. Images are selected from side-view angle snapshots in Google Street View. Therefore, most images are perspective distorted. Each image is associated with a 50-word lexicon and a full lexicon.\n\n\\textbf{CUTE80} \\cite{risnumawan2014robust} contains 80 high-resolution images taken in natural scenes. It was specifically collected for evaluating the performance of curved text recognition. It contains 288 cropped natural images for testing. No lexicon is associated with this dataset.\n\n\\textbf{ICDAR 2015 (IC15)} \\cite{karatzas2015icdar} contains 2077 cropped images including more than 200 irregular text. No lexicon is associated with this dataset.\n\n\\subsection{Implementation Details}\n\\textbf{Network: }Details about the MORN and the ASRN of MORAN are given in Table\\ref{table:The architecture of MORN} and Table\\ref{table:The architecture of ASRN} respectively. The number of hidden units of GRU in the decoder is $256$. The ASRN outputs 37 classes, including 26 letters, 10 digits and a symbol standing for $``EOS\"$.\n\n\\textbf{Training Model: }As stated in Section \\ref{section:curriculum-training}, the training of the MORAN is guided by a curriculum learning strategy. The training data consists of 8-million synthetic images released by Jaderberg et al. \\cite{jaderberg2014synthetic} and 6-million synthetic images released by Gupta et al. \\cite{gupta2016synthetic}. No extra data is used. We do not use any geometric-level or pixel-level labels in our experiments. Without any fine-tuning for each specific dataset, the model is trained using only synthetic text. With ADADELTA \\cite{zeiler2012adadelta} optimization method, we set learning rate to $1.0$ at the beginning and decreased it to $0.01$ in the third stage of the curriculum learning strategy. Following the similar settings in \\cite{liu2016star}, we found that a decreased learning rate contributes to better convergence. The batch size was set to 64. We trained the model for 600,000, 20,000 and 300,000 iterations respectively in three stages of the curriculum learning strategy. The training totally consumed 30 hours.\n\n\\textbf{Implementation: }We implemented our method under the framework of PyTorch \\cite{pytorch}. CUDA 8.0 and CuDNN v7 backends are used in our experiments so our model is GPU-accelerated. All the images are resized to $32\\times 100$. With an NVIDIA GTX-1080Ti GPU, the MORAN takes 10.4ms to recognize an image containing five characters in lexicon-free mode.\n\n\\subsection{Performance of the MORAN}\n\nWe used a max-pooling layer at the top of the MORN. To evaluate the effectiveness of this technique, a comparison of pooling layers with different configurations is shown in Table \\ref{table:comparison-of-pooling}. The accuracy is the highest when we use a max-pooling layer with a kernel size of 2 and stride of 1.\n\nBefore conducting a comparison with other methods, we list three results with a progressive combination of methods in Table \\ref{table:Performance-of-MORAN}. The MORAN trained in an end-to-end manner already achieves very promising performance. In curriculum learning, the first experiment is carried out using only an ASRN. Then, a MORN is added to the bottom of the above network to rectify the images. The last result is from the entire MORAN, including the MORN and ASRN trained with the fractional pickup method. The contribution of each part of our method is hence clearly demonstrated. For ICDAR OCR tasks, we report the total edit distance in Table \\ref{table:Performance-of-MORAN(TED)}.\n\n\\begin{table}[h]\n\\centering\n\\caption{Performance of the MORAN (total edit distance). }\n\\label{table:Performance-of-MORAN(TED)}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multirow{1}{*}{Method} & IC03 & IC13 & IC15 \\\\\n\\cline{2-4}\n\\hline\nEnd-to-end training & 29.1 & 57.7 & 368.8 \\\\\n\\hline\nOnly ASRN & 33.8 & 69.1 & 376.8 \\\\\nMORAN without FP & 22.7 & 45.3 & 345.2 \\\\\nMORAN with FP & \\textbf{19.8} & \\textbf{42.0} & \\textbf{334.0} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Comparisons with Rectification Methods}\n\\textbf{Affine Transformation}: The results using the affine transformation are provided by Liu et al. \\cite{liu2016star}. For fair comparison, we replace the ASRN by the R-Net proposed by Liu et al. \\cite{liu2016star}. A direct comparison of the results is shown in Table \\ref{table:comparison-stn}. As demonstrated in Fig.\\ref{fig:compare-stn} and described in Section \\ref{section:Multi-Object Rectification Network}, affine transformation is limited by the geometric constraints of rotation, scaling and translation. However, the distortion of scene text is complicated. The MORAN is more flexible than affine transformation. It is able to predict smooth rectification for images free of geometric constraints.\n\n\\begin{table}[h]\n\\centering\n\\caption{Comparison with STAR-Net. }\n\\label{table:comparison-stn}\n\\begin{tabular}{|p{2.2cm}<{\\centering}|p{0.9cm}<{\\centering}|p{0.6cm}<{\\centering}|p{0.6cm}<{\\centering}| p{0.6cm}<{\\centering}|p{1.1cm}<{\\centering}|}\n\\hline\n\\multirow{1}{*}{Method} & IIIT5K & SVT & IC03 & IC13 & SVT-P \\\\\n\\cline{2-6}\n\\hline\nLiu et al. \\cite{liu2016star} & 83.3 & 83.6 & 89.9 & \\textbf{89.1} & 73.5 \\\\\n\\hline\nOurs & \\textbf{87.5} & \\textbf{83.9} & \\textbf{92.5} & \\textbf{89.1} & \\textbf{74.6} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\textbf{RARE} \\cite{shi2016robust}: The results of RARE given by Shi et al. \\cite{shi2016robust} are in the Table \\ref{table:Results on general benchmarks} and Table \\ref{table:Results on irregular text}. We directly compare the network using exactly the same recognition network as that proposed in RARE. The results are shown in Table \\ref{table:comparison-rare}.\n\nThe MORAN has some benefits and drawbacks comparing with RARE. RARE using fiducial points can only capture the overall text shape of an input image, whereas the MORAN can rectify every character in an image. As shown in Fig. \\ref{fig:comparison-other}, all the characters in the image rectified by the MORAN are more normal in appearance than those of RARE. Furthermore, the MORAN without any fiducial points is theoretically able to rectify text of infinite length.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Comparison with RARE. }\n\\label{table:comparison-rare}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\multirow{1}{*}{Method} & IIIT5K & SVT & IC03 & IC13 & SVT-P & CUTE80 \\\\\n\\cline{2-7}\n\\hline\nShi et al. \\cite{shi2016robust} & 81.9 & 81.9 & 90.1 & 88.6 & 71.8 & 59.2 \\\\\n\\hline\nOurs & \\textbf{87.9} & \\textbf{83.9} & \\textbf{92.7} & \\textbf{90.0} & \\textbf{73.2} & \\textbf{72.6} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure}[h]\n\\centering\n\\begin{overpic}[width=8.5cm,height=5cm]{picture\/rare-a-wide.jpg}\n\n\\put(70,50){Predict:\\color{red}{stink}\\color{green}{er}}\n\\put(75,30){GT:denver}\n\\put(70,10){Predict:\\color{green}{denver}}\n\n\\end{overpic}\n\n\\caption{Comparison of the MORAN and RARE. All characters are cropped for further comparison. The recognition results are on the right. ``GT\" denotes the ground truth.}\n\n\\label{fig:comparison-other}\n\\end{figure}\n\nThe training of MORAN is more difficult than that of RARE. We thus designed a curriculum learning strategy to enable the stable convergence of the MORAN. In terms of RARE, although it is end-to-end optimized with special initialization, randomly initialized network may result in failure of convergence.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Results on general benchmarks. ``50\" and ``1k\" are lexicon sizes. ``Full\" indicates the combined lexicon of all images in the benchmarks. ``None\" means lexicon-free.}\n\\label{table:Results on general benchmarks}\n\\begin{tabular}{| c | c c c | c c | c c c | c |}\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{3}{c|}{IIIT5K} & \\multicolumn{2}{c|}{SVT} &\\multicolumn{3}{c|}{IC03} & IC13 \\\\\n\\cline{2-10}\n & 50 & 1k & None & 50 & None & 50 & Full & None & None \\\\\n\\hline\nAlmaz\\'{a}n et al \\cite{almazan2014word} & 91.2 & 82.1 & - & 89.2 & - & - & - & - & - \\\\\nYao et al. \\cite{yao2014strokelets} & 80.2 & 69.3 & - & 75.9 & - & 88.5 & 80.3 & - & - \\\\\nR.-Serrano et al. \\cite{rodriguez2015label} & 76.1 & 57.4 & - & 70.0 & - & - & - & - & - \\\\\nJaderberg et al. \\cite{jaderberg2014deep} & - & - & - & 86.1 & - & 96.2 & 91.5 & - & - \\\\\nSu and Lu \\cite{su2014accurate} & - & - & - & 83.0 & - & 92.0 & 82.0 & - & - \\\\\nGordo \\cite{gordo2015supervised} & 93.3 & 86.6 & - & 91.8 & - & - & - & - & - \\\\\nJaderberg et al. \\cite{Jaderberg2015Deep} & 95.5 & 89.6 & - & 93.2 & 71.7 & 97.8 & 97.0 & 89.6 & 81.8 \\\\\nJaderberg et al. \\cite{jaderberg2016reading} & 97.1 & 92.7 & - & 95.4 & 80.7* & \\textbf{98.7} & \\textbf{98.6} & 93.1* & 90.8* \\\\\nShi, Bai, and Yao \\cite{shi2017end} & 97.8 & 95.0 & 81.2 & \\textbf{97.5} & 82.7 & \\textbf{98.7} & 98.0 & 91.9 & 89.6 \\\\\nShi et al. \\cite{shi2016robust} & 96.2 & 93.8 & 81.9 & 95.5 & 81.9 & 98.3 & 96.2 & 90.1 & 88.6 \\\\\nLee and Osindero \\cite{lee2016recursive} & 96.8 & 94.4 & 78.4 & 96.3 & 80.7 & 97.9 & 97.0 & 88.7 & 90.0 \\\\\nLiu et al. \\cite{liu2016star} & 97.7 & 94.5 & 83.3 & 95.5 & 83.6 & 96.9 & 95.3 & 89.9 & 89.1 \\\\\nYang et al. \\cite{yang2017learning} & 97.8 & 96.1 & - & 95.2 & - & 97.7 & - & - & -\\\\\nYin et al. \\cite{yin2017scene} & 98.7 & 96.1 & 78.2 & 95.1 & 72.5 & 97.6 & 96.5 & 81.1 & 81.4 \\\\\nCheng et al. \\cite{cheng2017focusing} & 98.9 & 96.8 & 83.7 & 95.7 & 82.2 & 98.5 & 96.7 & 91.5 & 89.4 \\\\\nCheng et al. \\cite{cheng2017arbitrarily} & \\textbf{99.6} & \\textbf{98.1} & 87.0 & 96.0 & 82.8 & 98.5 & 97.1 & 91.5 & - \\\\\n\\hline\nOurs & 97.9 & 96.2 & \\textbf{91.2} & 96.6 & \\textbf{88.3} & \\textbf{98.7} & 97.8 & \\textbf{95.0} & \\textbf{92.4} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Results on General Benchmarks}\nThe MORAN was evaluated on general benchmarks in which most of the testing samples are regular text and a small part of them are irregular text. The MORAN was compared with 16 methods and the results are shown in Table \\ref{table:Results on general benchmarks}.\n\nIn Table \\ref{table:Results on general benchmarks}, the MORAN outperforms all current state-of-the-art methods in lexicon-free mode. As Jaderberg \\cite{jaderberg2016reading} treated each word as a category and the model cannot predict out-of-vocabulary words, we highlight these results by adding an asterisk. FAN \\cite{cheng2017focusing} trained with pixel-level supervision is also beyond the scope of consideration. We hence compare the MORAN with the baseline of FAN.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Results on irregular datasets. ``50\" is lexicon sizes. ``Full\" indicates the combined lexicon of all images in the benchmarks. ``None\" means lexicon-free.}\n\\label{table:Results on irregular text}\n\\begin{tabular}{|c|p{1.cm}<{\\centering} p{1.cm}<{\\centering} p{1.cm}<{\\centering} |p{1.5cm}<{\\centering}|p{1.5cm}<{\\centering}|}\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{3}{c|}{SVT-Perspective} & CUTE80 & IC15 \\\\\n\\cline{2-6}\n & 50 & Full & None & None & None \\\\\n\\hline\nABBYY et al. \\cite{wang2011end} & 40.5 & 26.1 & - & - & - \\\\\nMishra et al. \\cite{mishra2012scene} & 45.7 & 24.7 & - & - & - \\\\\nWang et al. \\cite{wang2012end} & 40.2 & 32.4 & - & - & - \\\\\nPhan et al. \\cite{quy2013recognizing} & 75.6 & 67.0 & - & - & - \\\\\nShi et al. \\cite{shi2016robust} & 91.2 & 77.4 & 71.8 & 59.2 & - \\\\\nYang et al. \\cite{yang2017learning} & 93.0 & 80.2 & 75.8 & 69.3 & - \\\\\nLiu et al. \\cite{liu2016star} & \\textbf{94.3} & 83.6 & 73.5 & - & - \\\\\nCheng et al. \\cite{cheng2017focusing} & 92.6 & 81.6 & 71.5 & 63.9 & 66.2 \\\\\nCheng et al. \\cite{cheng2017arbitrarily} & 94.0 & 83.7 & 73.0 & 76.8 & 68.2 \\\\\n\\hline\nOurs & \\textbf{94.3} & \\textbf{86.7} & \\textbf{76.1} & \\textbf{77.4} & \\textbf{68.8}\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\\subsection{Results on Irregular Text}\n\n\\begin{figure}[t]\n\\rule{8.25cm}{0.05em}\n\\\\\n\\\\\n\\\\\n\\centering\n\\begin{overpic}[width=8.5cm,height=8cm]{picture\/irregular-wide.jpg}\n\n\\put(5,100){Input Image}\n\\put(35,100){Rectified Images}\n\\put(67,101){Ground Truth}\n\\put(70,97){Prediction}\n\\put(0,94){-----------------------------------------------------------------}\n\n\\put(75,90){west}\n\\put(75,84){\\color{blue}{west}}\n\\put(70,77){---------------}\n\n\\put(75,71){united}\n\\put(75,64){\\color{blue}{united}}\n\\put(70,60){---------------}\n\n\n\\put(75,55){arsenal}\n\\put(75,50){\\color{blue}{arsenal}}\n\\put(70,45){---------------}\n\n\n\\put(75,40){football}\n\\put(75,35){\\color{blue}{football}}\n\\put(70,30){---------------}\n\n\n\\put(72,26){manchester}\n\\put(72,20){\\color{blue}{m}\\color{red}{essageid}}\n\\put(70,15){---------------}\n\n\n\\put(71,10){briogestone}\n\\put(71,4){\\color{red}{contracers}}\n\n\\put(0,-2){-----------------------------------------------------------------}\n\\end{overpic}\n\n\\caption{Effects of different curve angles of scene text. The first four rows are text with small curve angles and the last two rows are text with large curve angles. The MORAN can rectify irregular text with small curve angles.}\n\n\\label{fig:6-irregular-samples}\n\\end{figure}\n\nThe MORAN was also evaluated on irregular text datasets to reveal the contribution of the MORN. The results on SVT-Perspective, CUTE80 and IC15 are shown in Table \\ref{table:Results on irregular text}. The MORAN is still the best of all methods.\n\nFor the SVT-Perspective dataset, many samples are low-resolution and perspective. The result of the MORAN with 50-word lexicon is the same as that of the method of Liu et al. \\cite{liu2016star}. However, the MORAN outperforms all methods in the setting without any lexicon.\n\nIn addition to perspective text, the MORAN is able to recognize curved text. Some examples are demonstrated in Fig. \\ref{fig:6-irregular-samples}. The MORAN is able to rectify most curved text in CUTE80 and correctly recognize them. It is hence adequately robust to rectify text with small curve angle.\n\n\\subsection{Limitation of the MORAN}\n\nFor fair comparisons and good repeatability, we chose the widely used training datasets, which contain only horizontal synthetic text. Therefore, because of complicated background, the MORAN will fail when the curve angle is too large. Such cases are given in the last two rows of Fig. \\ref{fig:6-irregular-samples}. MORAN mistakenly regards the complicated background as foreground. However, such samples are rare in training datasets.\n\nFurthermore, with the existing training datasets and without any data augmentation, the MORAN focuses more on horizontal irregular text. Note that there are many vertical text in IC15. However, the MORAN is not designed for vertical text. Our method was proposed for the complicated deformation of text within a cropped horizontal rectangle.\n\nThe experiments above are all based on cropped text recognition. A MORAN without a text detector is not an end-to-end scene text recognition system. Actually, in more application scenarios, irregular and multi-oriented text are challenging both for detection and recognition, which have attracted great interest. For instance, Liu et al. \\cite{yuliang2017detecting} and Ch'ng et al. \\cite{CK2017} released complicated datasets. Sain et al. \\cite{sain2018multi} and He et al. \\cite{he2018multi} proposed methods to improve the performance of multi-oriented text detection. Therefore, scene text recognition still remains a challenging problem waiting for solutions.\n\n\\section{Conclusion}\nIn this paper, we presented a multi-object rectified attention network (MORAN) for scene text recognition. The proposed framework involves two stages: rectification and recognition. First, a multi-object rectification network, which is free of geometric constraints and flexible enough to handle complicated deformations, was proposed to transform an image containing irregular text into a more readable one. The rectified patterns decrease the difficulty of recognition. Then, an attention-based sequence recognition network was designed to recognize the rectified image and outputs the characters in sequence. Moreover, a fractional pickup method was proposed to expand the visual field of the attention-based decoder. The attention-based decoder thus obtains more context information and gains robustness. To efficiently train the network, we designed a curriculum learning strategy to respectively strengthen each sub-network. The proposed MORAN is trained in a weak-supervised way, which requires only images and the corresponding text labels. Experiments on both regular and irregular datasets, including IIIT5K, SVT, ICDAR2003, ICDAR2013, ICDAR2015, SVT-Perspective and CUTE80, demonstrate the outstanding performance of the MORAN.\n\nIn future, it is worth extending this method to deal with arbitrary-oriented text recognition, which is more challenging due to the wide variety of text and background. Moreover, the improvements in end-to-end text recognition performance come not just from the recognition model, but also from detection model. Therefore, finding a proper and effective way to combine the MORAN with a scene text detector is also a direction worth of study.\n\n\\section*{Acknowledgement}\nThis research was supported by the National Key R\\&D Program of China (Grant No.: 2016YFB1001405), GD-NSF (Grant No.: 2017A030312006), NSFC (Grant No.: 61472144, 61673182), GDSTP (Grant No.: 2015B010101004, 2015B010130003, 2017A030312006), GZSTP (Grant No.: 201607010227).\n\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Sample structures and low field transport data}\n\\begin{figure*}[h!]\n\\includegraphics[width=16cm] {FigS1.png}\n\\caption{(color online) (a) A schematic drawing of the sample structure. The samples studied here are from the same growth in Ref.\\cite{Oh_AM}, and their thicknesses are also characterized therein. A 20-QL In$_2$Se$_3$ \/15-QL (Sb$_0.65$In$_0.35$)$_2$Te$_3$ layer was first grown on a sapphire (Al$_2$O$_3$) substrate as a buffer layer to match the lattice constant of Sb$_2$Te$_3$. This buffer layer helps to reduce the defect density in the Sb$_2$Te$_3$ layer and allows tuning the carriers from p-type to n-type through titanium doping. Then, the Sb$_2$Te$_3$ layer is deposited and capped \\textit{in situ} with another 15-QL (Sb$_0.65$In$_0.35$)$_2$Te$_3$ layer to protect it from aging and maintain a symmetric environment between its top and bottom surfaces. (b) Carrier concentrations in the studied samples. The carrier densities in the samples are characterized with magneto-transport at low temperature with the Van der Pauw geometry. The carrier types are different in these two samples due to different amounts of Titanium dopants. From the sign of the slope in the $R_{Hall}$ vs $B$ plot, the 8-QL sample is determined to be p-doped, and the 10-QL sample is n-doped. The extracted carrier densities from the slope of the data give $n_e= 2.2 \\times 10^{11}$ cm$^{-2}$ and $n_h=3.1 \\times 10^{11}$ cm$^{-2}$ for the 10-QL and 8-QL samples, respectively. }\n\\end{figure*}\n\n\\newpage\n\\section{Additional data on the 8-QL sample}\n\\begin{figure*}[h!]\n\\includegraphics[width=16cm] {FigS2.png}\n\\caption{(color online) Normalized magneto-infrared spectra of the 8-QL sample for (a)(c) transmission and (b) reflection measurements outside and inside the substrate Reststrahlen band, respectively. All the modes are labeled with a Latin or Greek letter. The blue dash lines are guides to the eye showing the mode evolution in magnetic field. Compared to the 10-QL sample, the A mode is not observable here, probably due to the broad linewidth. All spectra are shifted for clarity.}\n\\end{figure*}\n\n\\section{Symmetrization and anti-symmetrization of the surface state dispersion}\nThe conduction surface state (CSS) and valence surface state (VSS) can be written generally as \\cite{eh_Theory1}:\n\\begin{align*}\n E_{CSS}=E_0+Ck^2+\\sqrt{A^2k^2+(M-Bk^2)^2},\\\\\n E_{VSS}=E_0+Ck^2-\\sqrt{A^2k^2+(M-Bk^2)^2}.\n\\end{align*}\nThe first term is the overall offset $E_0$, and the second term is the band asymmetry term described by $C$. The last term is the dominant component of the SS, which is described by the linear band parameter $A$, the Dirac mass $M$, and band inversion parameters $B$ and exhibits a electron-hole (e-h) symmetry. We can arrive at the e-h symmetry part $E_{sym}$ and e-h asymmetry part $E_{asy}$ by\n\\begin{align*}\n &E_{sym}=(E_{CSS}-E_{VSS})\/2=\\sqrt{A^2k^2+(M-Bk^2)^2},\\\\\n &E_{asy}=(E_{CSS}+E_{VSS})\/2=Ck^2+E_0.\n\\end{align*}\nFrom here, it is evident to see that the anti-symmetrized part leads to the band asymmetry term $Ck^2$ and an offset $E_0$, and the symmetrized part to the dominant dispersion also with an offset from the Dirac mass $M$. For clear comparisons between the dispersions of different thicknesses, we shift all the $E_{Sym}$ and $E_{Asy}$ to the same origin in Fig. 4(c-d) in the main text. \n\nFinally, we note that when the gap $M$ is sufficiently large, we can Taylor expand $E_{sym}$:\n\\begin{align*}\n E_{sym}=\\sqrt{A^2k^2+(M-Bk^2)^2}=M+\\frac{(A^2-2M B)k^2}{2M}+\\dots,\n\\end{align*}\n which explains the quadratic behavior in the ultrathin film limit.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRandom matrix theory (RMT)~\\cite{RMT} accurately describes eigenvalue\ncorrelations in complex systems. More precisely, if we consider a\nquantum system governed by a Hamiltonian $H$ whose classical\ncounterpart is chaotic, then the statistical properties of the\neigenvalue spectrum of $H$ can be modeled by an ensemble of matrices\nwith random entries (distributed according to some statistical weight)\nand with the same global symmetries as $H$. This description is\ninsensitive to the details of the interaction and predicts universal\nfeatures that are unveiled when different spectra are rescaled\n(unfolded) to the same mean density.\n\nAmong the many applications of RMT in mathematics and in physics, a\nparticularly interesting one is relevant for the description of the\nspectrum of the Dirac operator in quantum chromodynamics. QCD\nin the $\\varepsilon$-regime can be described by a chiral RMT with the\nsame chiral and flavor symmetries as QCD~\\cite{chRMT}. This approach\ncan also be extended to non-vanishing temperature and\/or chemical\npotential and has been confirmed in many numerical studies. In this\nformulation, the anti-Hermitian massless Dirac operator $D=\\gamma_\\mu\n\\left(\\partial_\\mu + i A_\\mu \\right)$ is described in terms of a\nmatrix with an off-diagonal block structure,\n\\begin{equation}\n \\label{dmatrix}\n D\\to \\left( \n \\begin{tabular}{cc}\n 0 & $i W$ \\\\\n $i W^\\dagger$ & 0 \\\\\n \\end{tabular}\n \\right) \\: , \n\\end{equation}\nwhere $W$ is a complex $(n+\\nu)\\times n$ matrix and $\\nu$ plays the\nrole of the topological charge.\n\nDepending on the color gauge group $G$ and on the fermion field\nrepresentation, the Dirac operator may also be invariant under some\ndiscrete antiunitary symmetries, leading to the following symmetry\nclasses~\\cite{Verbaarschot:1994qf}:\n\\begin{enumerate}\n\\item For $G=\\text{SU}(2)$ and fermions in the fundamental\n representation, the pseudo-real nature of the group generators\n allows us to recast the Dirac operator in a form with real matrix\n entries. The corresponding matrix ensemble is the chiral orthogonal\n ensemble (chOE) with Dyson index $\\beta_D=1$.\n\\item For $G=\\text{SU}(N_C)$ with $N_C \\ge 3$ and fermions in the\n fundamental representation, the Dirac operator generically has\n complex entries. The appropriate matrix ensemble is the chiral\n unitary ensemble (chUE) with Dyson index $\\beta_D=2$.\n\\item For gauge group $G=\\text{SU}(N_C)$ and fermions in the adjoint\n representation, the generators are antisymmetric matrices with\n imaginary entries, and the Dirac operator can be written as a matrix\n of real quaternions. The associated matrix ensemble is the chiral\n symplectic ensemble (chSE) with Dyson index $\\beta_D=4$.\n\\end{enumerate}\nThe behavior of the universal quantities depends on the symmetry\nclasses listed above. In particular, the probability density $P(s)$\nfor the spacing $s$ of adjacent unfolded levels can be computed\nexactly and is well approximated by the Wigner surmise,\n\\begin{align}\n \\label{eq:Wigner}\n P(s) = a\\,s^{\\beta_D}e^{-bs^2}\\quad\\text{with}\\quad\n a=2\\,\\frac{\\Gamma^{\\beta_D+1} \\left( \\beta_D\/2 +1 \\right)}\n {\\Gamma^{\\beta_D+2} \\left( (\\beta_D+1)\/2 \\right)} \\:,\\quad\n b= \\frac{\\Gamma^2\\left( \\beta_D\/2+1 \\right)}\n {\\Gamma^2 \\left( (\\beta_D +1)\/2 \\right)}\\:.\n\\end{align}\nFor quantum systems whose classical analog is integrable, $P(s)$ is\ngiven by the result for a Poisson process, $P(s)=e^{-s}$.\n\nThe massless staggered Dirac operator on a lattice in $d$ dimensions\nwith lattice spacing $a$,\n\\begin{equation}\n \\label{eq:DKS}\n (D_\\text{KS})_{x,y} = \\frac{1}{2a} \\sum_{\\mu=1}^{d}\n (-1)^{\\sum\\limits_{\\nu<\\mu}\\!\\! x_\\nu} \\left[ \\delta_{x+\\hat\\mu,y} \n U_\\mu^\\dagger(x) - \\delta_{x-\\hat\\mu,y} U_\\mu(x-\\hat\\mu) \\right] \\: , \n\\end{equation}\nwhich is widely used in numerical simulations, exhibits a peculiar\nfeature: For gauge group SU(2) and fundamental fermions, its\nantiunitary symmetry is that of the chSE \\cite{Halasz:1995vd} instead\nof the chOE symmetry of the continuum operator.\\footnote{A similar\n situation occurs for adjoint fermions: The staggered Dirac operator\n has chOE symmetry in this case, as opposed to the chSE symmetry of\n the continuum operator.} This discrepancy is due to the replacement\nof the $\\gamma$-matrices by the staggered phases in $D_\\text{KS}$.\nFig.~\\ref{fundsmallbetafig} confirms the expectation using $P(s)$ as\nan example.\n\n\\begin{figure}[-t]\n \\includegraphics[width=.33\\textwidth]{fundL10beta3}\n \\includegraphics[width=.33\\textwidth]{fundL14beta3}\n \\includegraphics[width=.33\\textwidth]{fundL16beta3}\n \\caption{The distribution of the unfolded level spacing $s$ obtained\n for different lattice sizes from the SU(2) staggered Dirac\n operator with fundamental fermions and gauge action parameter\n $\\beta=4\/g^2=3.0$ (histograms) is consistent with the chSE, which\n is not the symmetry of the continuum Dirac operator.}\n \\label{fundsmallbetafig}\n\\end{figure}\n\nA transition of the symmetry properties of the staggered Dirac\noperator from chSE to chOE is expected in the continuum limit. A\nfirst indication of such a transition has been reported in\nRef.~\\cite{Follana:2006zz}. We plan to study this transition in more\ndetail, but here we first consider a numerically cheaper case, namely\nthe free limit. This limit is approached by increasing $\\beta$ at\nfixed (or mildly varying) lattice size, i.e., the physical volume is\nshrinking to zero. From the RMT point of view, this limit is\ninteresting since it might result in a transition to Poisson behavior\nin, e.g., $P(s)$. We shall see that the situation is actually a bit\nmore complicated.\n\n\n\\section{The Dirac spectrum of vacuum configurations and the influence\n of Polyakov loops}\n\\label{sec:vac}\n\nIn this section we present a short theoretical interlude in\npreparation for our numerical results. Let us consider a particular\ngauge configuration. For reasons that will become clear below, we now\nconstruct a corresponding vacuum configuration (i.e., a configuration\nwith all plaquettes equal to unity) that is built from uniform links\nin an Abelian subgroup of SU(2) which reproduce the average traced\nPolyakov loops $P_\\mu$ (for all directions $\\mu$) of the configuration\nunder consideration. The eigenvalue spectrum of the staggered Dirac\noperator \\eqref{eq:DKS} can be computed analytically for such a vacuum\nconfiguration. If the lattice extent in the $\\mu$-direction is\n$L_\\mu$, we obtain\n\\begin{equation}\n \\label{clustersandpolyakovloops}\n \\lambda=\\pm i \\sqrt{\\sum_{\\mu=1}^d \\sin^2 \\left[ \\frac{2 \\pi}{L_\\mu}\n \\left( k_\\mu + c_\\mu + \\frac{\\arccos P_\\mu}{2 \\pi} \\right)\\right] }\n \\quad \\text{with }\\: k_\\mu\\in\\mathbb N\\:,\\; 0 \\le k_\\mu < \\frac{L_\\mu}2\\:,\n\\end{equation}\nwhere $c_\\mu = 0$ $\\left(c_\\mu= \\frac{1}{2}\\right)$ for (anti-)\nperiodic boundary conditions (b.c.s) of the Dirac operator in\ndirection $\\mu$. In the free limit, i.e., for $\\beta \\to \\infty$, the\nPolyakov loops take values in the center $\\mathbb Z_2$ of SU(2), i.e.,\n$P_\\mu=\\pm1$. In this case it is clear from\nEq.~\\eqref{clustersandpolyakovloops} that changing $P_\\mu$ from $+1$\nto $-1$, or vice versa, is equivalent to switching between periodic\nand antiperiodic b.c.s in that direction. In the following we always\nuse (anti-) periodic b.c.s for $\\mu=1$, 2, 3 ($\\mu=4$). Close to the\nfree limit, i.e., for large values of $\\beta$, the distribution of\n$P_\\mu$ is peaked at $\\pm1$. The eigenvalues predicted by\nEq.~\\eqref{clustersandpolyakovloops} are degenerate, see below.\n\n\n\\section{Numerical results for the eigenvalue spectrum close to the\n free limit}\n\n\\begin{figure}[-t]\n \\includegraphics[width=.33\\textwidth]{polyakov1}\n \\includegraphics[width=.33\\textwidth]{polyakov2}\n \\includegraphics[width=.33\\textwidth]{polyakov6}\n \\includegraphics[width=.33\\textwidth]{polyakov8}\n \\includegraphics[width=.33\\textwidth]{polyakov18}\n \\includegraphics[width=.33\\textwidth]{L6_polyakov5}\n \\caption{Separation of scales in the level spacings close to the\n free limit. The eigenvalues obtained for each configuration\n (black dots) arrange themselves in clusters of eight. These\n clusters are spread about the well-separated plateaux\n corresponding to the free case (dashed blue lines).\n Eq.~\\protect\\eqref{clustersandpolyakovloops} yields an accurate\n prediction for the location of each cluster (solid red\n lines). The different plateau structures are due to the different\n signs of $P_\\mu\\approx\\pm1$ in each configuration. The last plot\n confirms that the agreement between the data and\n Eq.~\\protect\\eqref{clustersandpolyakovloops} persists on larger\n lattices, for which the theoretical formula predicts more\n clusters.}\n \\label{threescalesfig}\n\\end{figure}\n\nWhen $\\beta$ is increased to very large values, we observe that the\neigenvalue spectrum of $D_\\text{KS}$ arranges itself as shown in\nFig.~\\ref{threescalesfig}. Only the eigenvalues with positive\nimaginary part are plotted, and an overall double (Kramers) degeneracy\n\\cite{Hands:1990wc} has been divided out from all of our results. The\ndashed blue lines, which will be called \\emph{plateaux} in the\nfollowing, correspond to the highly degenerate eigenvalues predicted\nby Eq.~\\eqref{clustersandpolyakovloops} in the free limit, i.e., for\n$P_\\mu=\\pm1$. The numerically obtained eigenvalues form\n\\emph{clusters} consisting of eight eigenvalues each, and these\nclusters are located close to the plateaux of the free limit. We\nobserve a clear separation of three energy scales (from largest to\nsmallest),\n\\begin{enumerate}\\itemsep-1mm\n\\item the spacings between the plateaux of the free limit,\n\\item the spacings between adjacent clusters (which, by definition, do\n not overlap), and\n\\item the spacings between adjacent eigenvalues within a cluster.\n\\end{enumerate}\n\nThe question now arises to what extent the locations of the clusters\nfor a particular configuration can be described by the levels obtained\nfrom Eq.~\\eqref{clustersandpolyakovloops} for the vacuum configuration\nconstructed as described in Sec.~\\ref{sec:vac}. The answer is given\nby the solid red lines, which correspond to the predictions of\nEq.~\\eqref{clustersandpolyakovloops} using the $P_\\mu$ computed from\nthe configuration under consideration. These lines are essentially\nhidden by the data points and thus give very good approximations to\nthe cluster locations. This statement holds for all configurations,\nsome of which are shown in Fig.~\\ref{threescalesfig}.\n\nWe can understand the observation that the clusters contain eight\nnearly degenerate eigenvalues. A careful analysis shows that the\neigenvalues predicted by Eq.~\\eqref{clustersandpolyakovloops} have a\nmultiplicity of $2^d$. For $d=4$, this predicts an eightfold\ndegeneracy in addition to Kramers' degeneracy. A small perturbation\nof the vacuum configuration lifts this eightfold degeneracy but not\nthe Kramers degeneracy, which is exact for $D_\\text{KS}$.\n\nWe also remark that in the continuum limit, in which the lattice\nspacing goes to zero at fixed physical volume, the eigenvalues of\n$D_\\text{KS}$ should arrange themselves in multiplets corresponding to\nthe taste degeneracy of staggered fermions. This effect was observed\nfor SU(3)~\\cite{quadruplets} (quadruplets) and\nSU(2)~\\cite{Follana:2006zz} (doublets) for improved versions of\n$D_\\text{KS}$, but it should not be confused with the effect we are\nstudying here.\n\nOur observations may be related to other recent work~\\cite{chisbconf}\ndiscussing the connection between the spectrum of the Dirac operator\n(which is relevant for chiral symmetry breaking) and the Polya\\-kov\nloop (which is an order parameter for confinement in the quenched\ntheory).\n\n\n\\section{Spectral fluctuations on different scales}\n\nWe now turn to a study of spectral correlations close to the free\nlimit, using again the nearest-neighbor spacing distribution $P(s)$ as\nan example. To construct $P(s)$, the average spectral density must be\nseparated from the spectral fluctuations by an unfolding procedure.\nBecause of the separation of scales observed above, a uniform\nunfolding of the entire spectral density is not sensible close to the\nfree limit. Rather, we should consider the spectral fluctuations\nseparately on the three scales we identified.\n\nFirst, we construct $P(s)$ for the level spacings within the clusters\nby unfolding the spectral density only within a given cluster and then\naveraging $P(s)$ over all clusters. Fig.~\\ref{still_chse} (left)\nshows that $P(s)$ within the clusters continues to agree with the chSE\neven for very large values of $\\beta$. This is consistent with the\ntheoretical expectation, since the perturbation that lifts the\ndegeneracy of the eigenvalues in each cluster has the same symmetries\nas the full $D_\\text{KS}$ operator, which are those of the chSE.\n\nSecond, to construct $P(s)$ for the spacings between clusters, we\ndefine a cluster by the average of its eight members and unfold the\ndensity of the clusters. Fig.~\\ref{still_chse} (right) shows that the\nresulting $P(s)$ differs from the chSE. It also differs from the\nPoisson distribution, but we believe that this is due to the small\nlattice size and to the fact that on a $10^4$ lattice the free\nstaggered operator has many ``accidental'' degeneracies. These\ndegeneracies can be removed by choosing a lattice with\n$L_\\mu=2\\ell_\\mu$, where the $\\ell_\\mu$ are four different prime\nnumbers. As an example, we generated quenched configurations close to\nthe free limit ($\\beta=10000$) for a $34\\times38\\times46\\times58$\nlattice, computed the averaged traced Polyakov loops $P_\\mu$ and used\nthese to calculate the ``cluster spectrum'' according to\nEq.~\\eqref{clustersandpolyakovloops}. The resulting $P(s)$ is shown\nin Fig.~\\ref{poissonlargelatticefig} (left) and now agrees with the\nPoisson distribution.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.33\\textwidth]{fundL10beta10000}\n \\hspace*{20mm}\n \\includegraphics[width=.33\\textwidth]{clusters_L10_beta_10000}\n \\caption{$P(s)$ for the eigenvalue spacings within the clusters\n (left) and for the spacings between clusters (right), both for\n $L^4=10^4$ and $\\beta=10000$.}\n \\label{still_chse}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.33\\textwidth]{incommensurable_sizes_clusters}\n \\hspace*{20mm}\n \\includegraphics[width=.33\\textwidth]{incommensurable_sizes_plateaux}\n \\caption{$P(s)$ for the ``cluster spectrum'' predicted by\n Eq.~\\protect\\eqref{clustersandpolyakovloops} for a single\n configuration on a $ 34 \\times 38 \\times 46 \\times 58 $ lattice at\n $\\beta=10000$ (left) and $P(s)$ for the free Dirac eigenvalues (or\n plateaux) on the same lattice (right). Both agree with the\n Poisson distribution.}\n \\label{poissonlargelatticefig}\n\\end{figure}\n\nThird, we consider $P(s)$ for the spacings between the free\neigenvalues (or plateaux), which are known analytically\n(Eq.~\\eqref{clustersandpolyakovloops} with $P_\\mu=\\pm1$). Again it is\nsensible to remove accidental degeneracies by choosing a ``prime\nlattice''. The result for a $34\\times38\\times46\\times58$ lattice,\nobtained after unfolding the free eigenvalues, is shown in\nFig.~\\ref{poissonlargelatticefig} (right) and agrees with Poisson as\nexpected \\cite{GoesToPoisson}.\n\nAlthough the two plots in Fig.~\\ref{poissonlargelatticefig} look very\nsimilar, it should be noted that they come from data at very different\nscales. The average spacing between the levels of the free spectrum\nis more than ten times larger than the average spacing between the\nlevels predicted by Eq.~(\\ref{clustersandpolyakovloops}).\n\n\n\\section{Summary and outlook}\n\nWe have investigated the spectrum of the staggered Dirac operator with\nSU(2) gauge fields close to the free limit. Three different energy\nscales emerge:\n\\begin{enumerate}\n\\item Overall plateau structure: The spectrum arranges itself in\n clusters of eight eigenvalues each, lying close to the plateaux\n predicted for the free Dirac operator. The plateau structure only\n depends on the lattice geometry (i.e., on the $L_\\mu$ and on the\n b.c.s) and on the signs of the average traced Polyakov loops in the\n different directions. (Note that the distribution of the traced\n Polyakov loops is peaked at $\\pm1$, corresponding to the center\n elements of SU(2).)\n\\item Plateau-breaking and cluster separation at an intermediate\n scale: At a finer scale, the spread of the clusters about the\n plateaux of the free limit is due to the deviations of the $P_\\mu$\n from $\\pm1$ and can be accurately modeled by\n Eq.~\\eqref{clustersandpolyakovloops}.\n\\item Eigenvalue splitting within the clusters: The system dynamics\n removes the degeneracy of the eight eigenvalues belonging to the\n same cluster.\n\\end{enumerate}\nIn the regime we have studied, these three scales are well separated\nand can be unambiguously disentangled from each other.\n\nThe nearest-neighbor spacing distribution $P(s)$ computed within the\nclusters shows a behavior compatible with the chSE, consistent with\nthe symmetries of the staggered Dirac operator. For large enough\n``prime lattices'', the spacing distributions between the clusters and\nbetween the plateaux tends to the Poisson distribution. In the near\nfuture, we will also present a study of the spectrum of the Dirac\noperator for adjoint fermions close to the free limit. Ultimately, of\ncourse, we would like to obtain a more detailed understanding of the\ncontinuum limit, in which a chSE to chOE (for SU(2) with fundamental\nfermions) or chOE to chSE (for adjoint fermions) transition is\nexpected.\n\n\\acknowledgments\n\nWe thank J.J.M.~Verbaarschot for helpful discussions and acknowledge\nsupport from DFG (FB, SK, TW) and from the Alexander von Humboldt\nFoundation (MP).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}