$\langle$reception date$\rangle$ $\langle$acception date$\rangle$
$\langle$publication date$\rangle$

key word${ISM:clouds}$ — key word${X-rays:binaries}$ — key
word${infrared:ISM}$ — key word${ultraviolet:ISM}$

# Near-Ultraviolet Radiation toward Molecular Cloud N4 in W 50/SS 433:
Evidence for Direct Interaction of the Jet with Molecular Cloud

Hiroaki Yamamoto ∗ Tatsumi Ishikawa and Tsutomu  T. Takeuchi Graduate School
of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602,
Japan<EMAIL_ADDRESS>
###### Abstract

We compared the molecular clouds in the western part of SS 433 with near-
ultraviolet radiation data obtained from GALEX. Near-ultraviolet radiation is
prominently confirmed toward only N4, while no near-ultraviolet radiation is
detected toward N1, N2, and N3. The radiative region of near-ultraviolet
radiation is nearly the same as the CO-emitting region in N4, and does not
extend beyond the jet seen in X-ray radiation. Near-ultraviolet radiation
cannot be explained solely by broadband continuous radiation and may originate
from line emissions. The intensity of near-ultraviolet radiation exhibits an
anti-correlation with that of 13CO($J$=3–2) emission. This anti-correlation,
along with strong far-infrared radiation in the region with weaker near-
ultraviolet radiation intensity compared to its surroundings, suggests that
near-ultraviolet radiation originates from behind the molecular cloud, heating
up the interstellar dust in N4. Subsequently, the dust in N4 reradiates in the
far-infrared band. In the same region, a high peak $T_{\rm MB}$ ratio of
12CO($J$=3–2)/12CO($J$=1–0) of $\sim$0.9, and a high kinetic temperature of
$T_{\rm k}$ $\sim$56 K in the molecular cloud indicate that CO molecules are
highly excited, and the molecular cloud is heated through photoelectric
heating. This heating results from electrons released due to the photoelectric
effect caused by the phenomenon where interstellar dust absorbs near-
ultraviolet radiation. In terms of the timescale of near-ultraviolet radiation
originating from line emissions, near-ultraviolet radiation towards N4 cannot
be explained by the shock of the blast wave from a supernova that created W
50. These findings also suggest that N4 directly interacts with the jet from
SS 433. As a result of this direct interaction, near-ultraviolet radiation is
emitted from an interacting layer between the jet and N4.

##  1 Introduction

The microquasar SS 433 is well known as an X-ray binary at nearly the center
of the supernova remnant W 50. SS 433 emanates a powerful jet with a velocity
of $\sim$0.26$c$ (Margon, 1984). Radio continuum radiation at 1.4 GHz has been
identified by Dubner et al. (1998), and the shell of this radiation is
elongated in the east–west direction due to bursts from the powerful jet.

The distance to SS 433 has been estimated in several ways (e.g., Vermeulen et
al. (1993); Dubner et al. (1998); Dehnen & Binney (1998); Blundell & Bowler
(2004); Lockman et al. (2007); Gaia Collaboration (2018); Gaia Collaboration
(2021)). The most reliable estimation of the distance is based on a deep-
integration radio image of SS 433 on the arcsecond scale, considering the
speed and precession period of the jet (Blundell & Bowler (2004); Lockman et
al. (2007)). The estimation places the distance at $\sim$5.5 kpc.

X-ray observations have revealed the spatial distribution and properties of
the jet emanating from SS 433 (e.g., Safi-Harb & Ögelman (1997); Kotani
(1998); Moldowan et al. (2005); Brinkmann et al. (2007)). The X-ray jet
extends by $\sim$0.7 degrees to west and $\sim$1 degree to east, corresponding
to $\sim$70 pc and $\sim$100 pc at a distance of 5.5 kpc, respectively. High
energy gamma-ray radiation in the range of a few hundred GeV to 100 TeV has
been detected by Fermi-LAT, HAWC (High Altitude Water Cherenkov Observatory),
and more recently LHAASO (Large High Altitude AirShower Observatory) toward
the W 50/SS 433 system (e.g., Abeysekara et al. (2018); Fang et al. (2020);
Cao et al. (2023)). The origin of such GeV-TeV gamma-ray radiation is
considered to be inverse Compton scattering.

Yamamoto et al. (2022) revealed an association of mid-far infrared radiation
with molecular cloud N4 (see next paragraph for details of N4), indicating
that possibly the interstellar dust in N4 heats up by the interaction of the
jet from SS 433.

CO observations toward the W 50/SS 433 system have been conducted by several
authors (e.g., Huang et al. (1983); Yamamoto et al. (2008); Su et al. (2018);
Liu et al. (2020); Yamamoto et al. (2022); Sakemi et al. (2023)). Yamamoto et
al. (2008) identified the molecular clouds associated with the W 50/SS 433
system. Four molecular clouds exist in the western part of SS 433, and are
lebeled N1–N4. Yamamoto et al. (2022) revealed the detailed physical
properties of N4, which is located nearest to SS 433 and along same line of
sight as X-ray jet. The kinetic temperature, $T_{\rm k}$, of the molecular
cloud is estimated to be $\sim$56 K at maximum in the western part of N4 from
12CO($J$=1–0, 3–2) and 13CO($J$=3–2). Several observational pieces of evidence
indicate that N4 is possibly interacting with the jet from SS 433.

Toward N4, GeV-TeV gamma-ray, X-ray, mid-far infrared, and radio radiation are
detected. In this paper, we compare the N4 with ultraviolet radiation, which
has never been confirmed in the W 50/SS 433 system, and reveal the detailed
nature of N4.

##  2 Observations

###  2.1 Nobeyama 45m telescope

Observations were conducted on the molecular clouds detected by NANTEN
observations conducted by Yamamoto et al. (2008) using the Nobeyama 45 m
(NRO45m) telescope. The front-end system utilized for the observations of the
12CO($J$=1–0) emission, with a rest frequency of 115.27120 GHz, was BEARS
(25-BEam Array Receiver System) (Sunada et al., 2000). The beam size of the
telescope at 115 GHz was $\sim$15$\arcsec$. The backend used was Auto
Correlators, with 32 MHz bandwidth mode employed for 25 beams (Sorai et al.,
2000). The velocity resolution was $\sim$0.1 km s-1. The observations were
conducted in the On-The-Fly mapping mode, involving scans along directions of
the Galactic longitude and latitude (Mangum et al. (2000); Sawada et al.
(2008a)). Orthogonal two scans were merged using the basket weaving method
(Emerson & Graeve, 1988). The observing grid for the OTF mapping mode was
5$\arcsec$, and the grid size of output data matched that of the JCMT archival
data at 7$\arcsec$.2. The data was smoothed to $\sim$20$\arcsec$ to reduce the
r.m.s. noise levels. The final r.m.s. noise level of the data was $\sim$1.4 K
ch-1. In this paper, we exclusively utilize the dataset pertaining to the N4
cloud in SS 433 as presented in Yamamoto et al. (2022).

###  2.2 JCMT archival data

We utilized archival datasets obtained with the JCMT15m telescope. The data
used in this study corresponds to Proposal ID M18BP027. In our analysis, we
exclusively employed datasets of 12CO, and 13CO($J$=3–2) emissions. The
frequency resolution of the data was 61 kHz, corresponding to the velocity
resolution of $\sim$0.05 km s-1. For this paper, we further smoothed the
velocity resolution to 0.1 km s-1. The spatial grid of the data was 7.2 for
the beam size of $\sim$15\. The data was smoothed to 20$\arcsec$ to facilinate
comparison with 12CO($J$=1–0). The main beam efficiency, $\eta_{\rm MB}$ of
0.72 was adopted to convert the intensity scale $T_{\rm a}$ in the archival
data to the main beam temperature, $T_{\rm MB}$, scale (Buckle et al., 2009).
The final r.m.s. noise level for the data of 12CO, and 13CO($J$=3–2) was
$\sim$1.6, and $\sim$1.9 K ch-1, respectively.

###  2.3 GALEX archival data

Image data at UV bands are provided by GALEX (Galaxy Evolution Explorer) and
made public. GALEX is a UV astronomical satellite launched in 2003 by NASA
(Morrissey et al., 2007). GALEX has far-UV (FUV) and near-UV (NUV) bands and
is performing an all-sky survey as well as deep surveys in some sky areas. In
this analysis, archival datasets from All-Sky Imaging Survey (AIS) GR6/GR7
(e.g., Bianchi et al. (2014)) are utilized. However, in the W 50/SS 433
region, only the NUV detector was operational, as the FUV detecter did not
function initially. The system angular resolution in the NUV band is
5$\arcsec$.6 at full width at half maximum. When comparing the NUV datasets
with CO datasets, the data was smoothed to 20$\arcsec$. The sensitivity of the
datasets is $m_{\rm AB}$ $\sim$21 magnitude before smoothing.

##  3 Availability of GALEX Data

Figure 1 shows datasets of GALEX in the field where the present analyzed
region is included. Figure 1(a) represents an artifact flag map. The details
are described in the caption of the figure. In Figure 1(a), four distinct
linear structures are seen, traversing the center of the field: two extending
from the upper right to lower left and two from the upper left to lower right
in yellow. These are identified as number 17, indicating ’edge + brteddge’
artifacts. Additionally, a prominent linear structure in green, stretching
from the upper-right to lower-left across the center of the field in Figure
1(a), is marked as number 136, indicating ’varmask + varpix’. These artifact
structures are expected to be removed in Figure 1(b). However, the linear
structure around ($\alpha$, $\delta$)(J2000) $\sim$ (19h9m56s, 5D24’0”) in
Figure 1(b) can not be conclusively identified as either real radiation or a
residual error of the artifact. This structure lies outside the radio shell of
the W50/SS433 system (see Figure 2) and beyond the scope of this paper. There
is an overlap of artifact and radiation near ($\alpha$, $\delta$)(J2000)
$\sim$ (19h10m20s, 5D7’16”) in the northeastern part of the NUV radiation
area. The extent of accurate artifact removal is uncertain. However, for the
purposes of this study, the NUV radiation around ($\alpha$, $\delta$)(J2000)
$\sim$ (19h10m20s, 5D7’16”) in Figure 1(b) is considered authentic.

##  4 Results

Figure 2 shows three composite color image, which consists of red: NUV
radiation, blue: 0.1–2.4 keV X-ray, and green: 1.4 GHz radio continuum
superposed on an integrated intensity map of 12CO($J$=1–0) emission. NUV
radiation is prominently confirmed toward N4, while no NUV radiation is
observed toward N1, and N2 which are situated at the leading edge of the radio
continuum shell. Liu et al. (2020) mentioned that N3 is situated in the radio
continuum shell, and at the leading edge of the X-ray jet. However, no NUV
radiation is also seen toward N3. As mentioned in Yamamoto et al. (2022), N4
is situated within the radio continuum shell and along the same line of sight
as the X-ray jet emanating from SS 433. The distribution of near-ultraviolet
radiation in N4 is also confirmed to the same line of sight as the X-ray jet
as well, and does not extend beyond that of X-ray radiation.

Notably, strong and extended NUV radiation is concentrated around ($\alpha$,
$\delta$)(J2000) $\sim$ (19h11m5s, 5D16’30”). This extended source of NUV
radiation could originate from HD 179124, characterized as a B9V spectral type
star, positioned at an estimated distance of $\sim$380 pc from the Sun as
inferred from its parallax. Therefore, this radiation source is unrelated to
the W 50/SS 433 system. While many small, point-like sources are also
confirmed toward the W 50/SS 433 system, they are not associated with the W
50/SS 433 system.

The upper-left image in Figure 3 provides an intensity distribution of NUV
radiation in N4 superposed on the distribution of peak $T_{\rm MB}$ of
12CO($J$=1–0) emission, obtained through Gaussian fitting and depicted in
contours. The bright NUV radiating region is split into two parts, west and
east. The area between these two bright NUV regions corresponds to a local
peak in 12CO($J$=1–0) emission, which we will discuss in more detail in the
next section. The intensity distribution of the region with bright NUV
radiation exhibits a patchy nature, with intensity fluctuations less than a
factor of 2. The spectra shown in Figure 3 represent the 12CO($J$=1–0) spectra
for each region, labeled with characters in the upper-left image. Additional
details regarding these spectra can be found in the caption of Figure 3. The
detection of weak CO emission around the molecular cloud, as defined in
Yamamoto et al. (2022), suggests that the extent of the molecular cloud is
more extensive than what was previously reported in Yamamoto et al. (2022),
and it closely matches the extent of NUV radiation. Consequently, this implies
a connection between NUV radiation and the molecular cloud. Further details
regarding the relationship between NUV radiation and the molecular cloud will
be discussed in the next section.

##  5 Discussion and Summary

First, we discuss the properties of NUV radiation. The radio continuum
radiation at 1.4 GHz toward N4 is weak and does not exhibit any morphology
related to the shape of the N4 cloud. Yamamoto et al. (2022) have identified
mid/far-infrared radiation toward N4, tracing thermal dust radiation with
temperature below 100 K, and PAH emission. This thermal dust radiation has
been detected only in the infrared to radio wavelength, indicating that
infrared radiation presented in Yamamoto et al. (2022) is not related to
current NUV radiation. If radiation from much hotter sources, such as stars
with temperatures of thousands of K or more, were present, extended radiation
would also be detectable in the near-infrared and visible wavelengths.
However, such extended radiation in the near-infrared and visible wavelengths
has not been detected in 2MASS and optical datasets. Cheung (2002) reported
that the broadband spectrum at radio, optical, and UV wavelengths at knot D in
3C279 originates from synchrotron radiation. However, in the present case, no
counterpart of NUV radiation at other wavelengths is detected, indicating that
the origin of NUV radiation is not synchrotron radiation. These results
suggest that NUV radiation in N4 is not a part of broadband continuous
radiation. Another possibility for the NUV radiation is emission lines.
Numerical simulation in Lahmann et al. (2020) and Lehmann et al. (2022)
suggests the presence of UV radiation in the shocked layer. In the GALEX NUV
band, ranging from 1750 $\rm{\AA}$ to 2800 $\rm{\AA}$, strong line emissions
such as CIII] 1909 $\rm{\AA}$ and MgII 2800 $\rm{\AA}$ are observed. While
these specific line emissions in NUV bands are not included in Lahmann et al.
(2020) and Lehmann et al. (2022), emissions such as Ly$\alpha$, Ly$\beta$, and
H$\alpha$ emissions are reported. Simulations show H$\alpha$ emission in the
shocked layer, but towards N4, H$\alpha$ emission has not been detected
(details of H$\alpha$ emission are described in the latter half of this
section). CIII] 1909 $\rm{\AA}$ and MgII 2800 $\rm{\AA}$ are often detected in
distant galaxies and/or QSOs (e.g., Steidel & Sargent (1991); Bunker et al.
(2023)). If the origin of NUV radiation toward N4 is emission lines, CIII] and
MgII become candidates for the NUV radiation. Sonneborn et al. (1997)
monitored the intensity of several line emissions including CIII] emission at
UV/NUV band after outburst in SN1987A. They found that the intensity of all of
observed line emissions peaked at $\sim$400 days after the outburst, and
disappeared at $\sim$2000 days. This indicates that the radiation timescale of
line emissions in the UN/NUV bands is several years after a strong shock.
Present NUV radiation seen in N4 suggests that N4 is now under strong shock
condition. If the age of W 50 is $\sim$105 years (Panferov, 2017), the blast
wave of the supernova that formed W 50 has already reached the boundary of the
cavity created by the massive star before it becomes a compact star, and line
emissions in the UV/NUV bands due to the effect of the supernova have already
disappeared. This suggests that the supernova is not the origin of NUV
radiation toward N4.

Figure 4 shows a detailed comparison of the spatial distribution of NUV
radiation with (a) integrated intensity of 13CO($J$=3–2) emission, (b) peak
$T_{\rm MB}$ ratio of 12CO($J$=3–2)/12CO($J$=1–0) emission, and (c) AKARI N160
(Doi et al., 2015). In Figure 4(a), the intensity of NUV decreases at the
intense 13CO($J$=3–2) emitting area around ($\alpha$, $\delta$)(J2000) $\sim$
(19h10m10.5s, 5D4’42”). The intensity of the extended 13CO($J$=3–2) emitting
area also shows an anti-correlation with that of NUV radiation. In Figure 5,
the peak $T_{\rm MB}$ ratio higher than 0.6 extends over N4, indicating that
the CO molecules in N4 are relatively excited compared to the typical
molecular clouds in the Galactic Plane (e.g., Sawada et al. (2008b)).
Particularly, the peak $T_{\rm MB}$ ratio in Figure 4(b) and Figure 5 is as
high as $\sim$0.9 where the intensity of NUV radiation decreases. The region
with a $T_{\rm k}$ of $\sim$56 K estimated in Yamamoto et al. (2022)
corresponds to the region where the intensity of NUV decreases. In Figure
4(c), far-infrared radiation peaks in the region where NUV radiation decreases
at ($\alpha$, $\delta$)(J2000) $\sim$ (19h10m10s, 5D4’44”). These results
suggest that NUV radiation is absorbed by interstellar dust in N4 and the
heated dust reradiates in the far-infrared wavelength. If this is true, NUV
radiation is radiated from behind the molecular cloud. Molecular cloud is
heated up by photoelectric heating through electrons released due to the
photoelectric effect caused by the phenomenon where interstellar dust absorbs
NUV radiation. As mentioned in section 4, while N3 is also situated within the
radio continuum shell (Liu et al., 2020), no NUV radiation has been detected.
$T_{\rm k}$ of N3 has been estimated to be $\sim$30 K, which is higher than
that of typical molecular clouds in the Galactic plane. They conclude that the
heating source for N3 is cosmic-rays, not X-rays, as the heating rate from the
X-ray radiation is three orders of magnitude lower than the cooling time
scale. In contrast, the heating source for N4 is identified as an X-ray jet.
This difference in heating sources may influence the physical phenomena that
produce NUV radiation. The details of this will be discussed in a forthcoming
paper.

Figure 6 shows a schematic view of the current condition of N4. The results
discussed in this section, along with the findings in Yamamoto et al. (2022),
suggest that N4 is currently interacting directly with the jet from SS 433.
Atoms of carbon and/or magnesium etc. are ionized and excited in the
interacting layer between the jet and N4. Then, those emissions are detected
as the NUV radiation with GALEX. Figure 7 presents a three-color composite
image, consisting of red: H$\alpha$ emission by IPHAS (Barentsen et al.,
2014), green: an intensity map of 12CO($J$=1–0) emission integrating from 45
to 60 km s-1, and blue: a map identical to the green one but integrating from
25 to 35 km s-1. The molecular cloud depicted in blue is identified as a giant
molecular filament situated in front of the W 50 /SS 433 system, with an
estimated distance of $\sim$1.7 kpc (Lin et al., 2020). This foreground
component overlaps with two-thirds of N4 at the southeastern part. The $N$(H2)
of the foreground component is estimated to be a few times 1021 cm-2,
corresponding to $A_{\rm v}$ of 2 to 5. Considering $A_{\rm v}$ in both the
foreground component and N4, as well as the number of stars identified in the
H$\alpha$ image, both inside and outside N4, H$\alpha$ emission does not seem
to be significantly dimmed by the molecular clouds. Therefore, it is inferred
that in N4, H$\alpha$ emission is not being radiated. This may result by the
moderate effective interacting velocity between the jet and N4 because
alfven’s critical ionization velocity of HII and CIII are $\sim$50.9 km s-1
and $\sim$19.7 km s-1, respectively. This velocity range of the shock is
similar to the velocity simulated in Lahmann et al. (2020). In the case of
Cygnus loop (Danforth et al., 2001), emissions of ionized atoms of oxygen,
carbon, nitrogen etc., provide insights into the detailed shock conditions in
the interacting layer between the cavity wall and the blast wave of the
supernova. Thus, to gain a better understanding of the details of the
interacting layer between the jet and N4, spectroscopic observations in the
UV/NUV bands are essential. The interacting layer of the jet and N4 is an
important site of shock-cloud interaction. The region of shock-cloud
interaction is the place where high-energy gamma-ray radiation is produced as
a result of the collision between cosmic-ray protons accelerated by diffusive
shock acceleration (Fermi, 1949) and the ISM. Strong and long-term shock-cloud
interaction continues while the jet emanates from SS 433, and the timescale of
shock-cloud interaction in the present case is longer than that in the case of
supernova remnants. This implies that the energy of the cosmic-ray protons may
increase beyond 1 PeV, and N4 may be a candidate for a PeVatron.

Figure 1: (a) An artifact flag map in the field where present region is
included. The number represents the summation of artifact effects occurred in
ecah pixel shown in table 1. (b) GALEX data image of the field.  Figure 2:
Three composite color image of red: NUV radiation taken with GALEX, green: 1.4
GHz radio continuum radiation taken with VLA, and blue: 0.1–2.4 keV radiation
taken with ROSAT superposed on integrated intensity map of 12CO($J$=1–0)
emission taken with NRO45m telescope in contours. The contour levels of
12CO($J$=1–0) emission are illustrated from 9.26 K km s-1 every 11.12 K km
s-1. A part of the red circle shows the edge of a field of view of GALEX
observation. N1 to N4 are labels of molecular clouds defined in Yamamoto et
al. (2008).  Figure 3: Upper left: intensity distribution of NUV radiation in
image superposed on peak $T_{\rm MB}$ distribution of 12CO($J$=1–0) emission
in contours. The contour levels are illustrated from 2.8 K every 5.6 K.
Circles labeled A to H indicate the positions of spectra in 12CO($J$=1–0)
represented in the figure. The Spectra shown in the figure were generated by
smoothing data within each circle, having a one-arcmin diameter, while the
resolution of data used in this analysis is 20$\arcsec$. In each spectral
panel, the lower horizontal dashed line represents the 0 K level, the middle
horizontal dashed line signifies the r.s.m. levels of the datasets before
smoothing data within each circle, and the upper horizontal dashed line
corresponds to the lowest contour level of the peak $T_{\rm MB}$ distribution
in the upper-left of the figure.  Figure 4: Comparison of NUV with (a)
integrated intensity of 13CO($J$=3–2) emission, (b) peak $T_{\rm MB}$ ratio of
12CO($J$=3–2)/12CO($J$=1–0) emission, and (c) Akari N160 (160$\mu$m). Contour
levels are (a) from 2 K km s-1 every 1.5 K km s-1, (b) from 0.3 every 0.1, and
(c) from 330 MJy sr-1 every 30 MJy sr-1, respectively. The dotted contours in
(a) represent the intensity decreases from the outside to the inside.  Figure
5: Peak $T_{\rm MB}$ ratio of 12CO($J$=3–2)/12CO($J$=1–0) emission. Contour
levels are from 0.3 every 0.1.  Figure 6: Upper: Schematic view of nature of
the interaction of the jet from SS 433 and N4. Colors of red, green, and blue
represent NUV radiation, molecular cloud traced in 12CO($J$=1–0), and the jet
from SS 433, NUV radiation, respectively. Observers are at downward in the
figure. Lower: Three composite color map of observational datasets. What
colors of red, green, and blue indicate is the same as colors in upper figure.
Contours represent integrated intensity of 13CO($J$=3–2) emission. Contour
levels are the same as those in figure 3(a). Horizontal dashed lines represent
edge of radiating area of NUV, molecular cloud, and the region where intensity
of NUV decreases compared to surroundings in the figure.  Figure 7: Three
composite color image of red: H$\alpha$ image by IPHAS (Barentsen et al.,
2014), Geen: an intensity of 12CO($J$=1–0) emission integrating 45 to 60 km
s-1, and blue: same as green but integrating 25 to 35 km s-1. CO intensities
are also presented in Contours. Contour levels are the same as those in figure
1. The lowest level of the contours are shown by thick ones and othres are
shown by thin ones. The dotted contours in blue shows that intensities
decrease from outside to inside.

Table 1: Details of Artifact Flags of Galex data Number | description
---|---
1 | (edge) Detector bevel edge reflection.
2 | (window) Detector window reflection.
4 | (dichroic) Dichroic reflection.
8 | (varpix) Variable pixel based on time slices.
16 | (brtedge) Bright star near field edge.
32 | Detector rim(annulus) proximity ( 0.6 deg fld ctr).
64 | (dimask) dichroic reflection artifact mask flag.
128 | (varmask) Masked pixel determined by varpix.
256 | (hotmask) Detector hot spots.
512 | (yaghost) Possible ghost image from YA slope.

This work has been supported by the Japan Society for the Promotion of Science
(JSPS) Grants-in-Aid for Scientific Research (19K03912, 21H01128). This work
has also been supported in part by the Collaboration Funding of the Institute
of Statistical Mathematics “New Perspective of the Cosmology Pioneered by the
Fusion of Data Science and Physics”. This research is based on observations
made with the GALEX mission, obtained from the MAST data archive at the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 526555.
The James Clerk Maxwell Telescope is operated by the East Asian Observatory on
behalf of The National Astronomical Observatory of Japan; Academia Sinica
Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science
Institute; Center for Astronomical Mega-Science (as well as the National Key
R&D Program of China with No. 2017YFA0402700). Additional funding support is
provided by the Science and Technology Facilities Council of the United
Kingdom and participating universities and organizations in the United Kingdom
and Canada. The Nobeyama 45 m radio telescope is operated by Nobeyama Radio
Observatory, a branch of National Astronomical Observatory of Japan. This
research is based on observations with AKARI, a JAXA project with the
participation of ESA. This paper makes use of data obtained as part of the INT
Photometric H$\alpha$ Survey of the Northern Galactic Plane (IPHAS,
www.iphas.org) carried out at the Isaac Newton Telescope (INT). The INT is
operated on the island of La Palma by the Isaac Newton Group in the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de
Canarias. All IPHAS data are processed by the Cambridge Astronomical Survey
Unit, at the Institute of Astronomy in Cambridge. The bandmerged DR2 catalogue
was assembled at the Centre for Astrophysics Research, University of
Hertfordshire, supported by STFC grant ST/J001333/1.

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$n\to\infty$. Since $\mathcal{H}_{K}(\Omega)$ is a Radon-Riesz space, we have
$\lVert f^{0}-f_{n}^{\lambda_{n}}\rVert\to 0$ when $n\to\infty$. This shows
that

$\lim_{n\to\infty}\sup_{x\in\Omega}\left\lvert\partial^{\vartheta}f^{0}(x)-\partial^{\vartheta}f_{n}^{\lambda_{n}}(x)\right\rvert\to
0,\quad\text{for any }\left\lvert\vartheta\right\rvert\leq 2.$

In Examples 6.1, 6.2, and 6.3, we use the reproducing kernels to solve the
approximate solutions directly by the representer theorems. Now we look at a
popular example of neural networks. Under the universal approximation of
neural networks, we will use the sigmoid functions to compute the approximate
solutions approximately by the approximation theorems.

######  Example 6.4.

Let $\Omega:=[-1,1]^{d}$. It is obvious that $\mathrm{L}_{1}(\Omega)$ is a
predual space of $\mathrm{L}_{\infty}(\Omega)$. Let $\mathcal{U}_{m}$ be the
collection of all multi-layer neural networks with $m$ coefficients, that is,

$f(x)=\sigma\left(\mathsf{W}_{l}\sigma\left(\cdots\sigma\left(\mathsf{W}_{2}\sigma\left(\mathsf{W}_{1}x+\boldsymbol{b}_{1}\right)+\boldsymbol{b}_{2}\right)\cdots\right)+\boldsymbol{b}_{l}\right),\quad\text{for
}x\in\Omega,$

where $\sigma$ is a continuous sigmoid function,
$\mathsf{W}_{1},\mathsf{W}_{2},\ldots,\mathsf{W}_{l}$ are weight matrixes, and
$\boldsymbol{b}_{1},\boldsymbol{b}_{2},\ldots,\boldsymbol{b}_{l}$ are bias
vectors as in [1, Section 6.4]. Thus
$\mathcal{U}_{m}\subseteq\mathrm{L}_{\infty}(\Omega)$ and there exists a
surjection $\Gamma_{m}$ from $\mathbb{R}^{m}$ onto $\mathcal{U}_{m}$. It is
easy to check that $\mathcal{U}_{m}$ is weakly* closed if the coefficients are
bounded. Since $\left\\{\mathcal{U}_{m}:m\in\mathbb{N}\right\\}$ satisfies the
universal approximation in $\mathrm{C}(\Omega)$ as in [4, 13],
$\left\\{\mathcal{U}_{m}:m\in\mathbb{N}\right\\}$ also satisfies the universal
approximation in $\mathrm{L}_{\infty}(\Omega)$.

Moreover $\langle
f,\xi^{\ast}\rangle=\int_{\Omega}f(x)\xi^{\ast}(x)\mathrm{d}x$ for
$f\in\mathrm{L}_{\infty}(\Omega)$ and $\xi^{\ast}\in\mathrm{L}_{1}(\Omega)$.
Thus the input data of standard sigmoid networks or convolutional neural
networks can be equivalently transferred to the generalized input data
$\boldsymbol{\xi}_{n}^{\ast}\in\left(\mathrm{L}_{1}(\Omega)\right)^{N_{n}}$,
for example of convolution operators. As in Lemma 2.2, we can show that
$\mathfrak{F}_{\mathcal{D}}$ is relatively compact for many kinds of the
classical data, for example of digital images. The multi-loss function
$\mathbb{L}_{n}$ is represented by the classical loss function $L$ as in
Equation (2.9). It is simple to choose the local Lipschitz continuous loss
functions to construct the algorithms of neural networks. Thus Lemma 2.3 (b)
assures that $\mathcal{L}$ is uniformly local Lipschitz continuous. This shows
that Condition (II) holds true.

For the neural networks, $R$ is usually unknown or unformulated, but $R_{n}$
is computed by $\boldsymbol{\xi}_{n}$ and $\mathbb{L}_{n}$. In the convergence
theorems, we only need to check the pointwise convergence for $R$. In the
numerical sense, Condition (I) can be seen to hold true if the convergence of
$R_{n}(f)$ exists for different test function
$f\in\mathrm{L}_{\infty}(\Omega)$.

Therefore, the approximation theorems and convergence theorems hold true for
many kinds of neural networks. By the theory of the regularized learning, we
will study the composite algorithms of neural networks and support vector
machines to solve partial differential equations adaptively in our next work.

Finally, we look at a simple example of an ill-posed problem in the Euclidean
space $\mathbb{R}^{2}$ equipped with the 2-norm.

######  Example 6.5.

Since $\mathbb{R}^{2}$ is a finite-dimensional Hilbert space, the weak*
topology of $\mathbb{R}^{2}$ is equal to the norm topology of $\mathbb{R}^{2}$
and $\langle f,\xi^{\ast}\rangle=f\cdot\xi^{\ast}$ for
$f,\xi^{\ast}\in\mathbb{R}^{2}$. Let

$\mathsf{A}:=\begin{pmatrix}1&0\\\ 0&0\\\
0&0\end{pmatrix},\quad\mathsf{A}_{n}:=\begin{pmatrix}1&0\\\ 0&1/n\\\
0&0\end{pmatrix}\text{ for
}n\in\mathbb{N},\quad\boldsymbol{b}:=\begin{pmatrix}1\\\ 1\\\ 1\end{pmatrix}.$

Now we discuss the linear equation $\mathsf{A}\boldsymbol{c}=\boldsymbol{b}$.
Let the expected risk function
$R(f):=\lVert\mathsf{A}f-\boldsymbol{b}\rVert^{2}_{2}$ for
$f\in\mathbb{R}^{2}$. Thus $\mathcal{S}^{0}\left(\mathbb{R}^{2}\right)$ is the
collection of all least-squared solutions of
$\mathsf{A}\boldsymbol{c}=\boldsymbol{b}$ and the best-approximate solution of
$\mathsf{A}\boldsymbol{c}=\boldsymbol{b}$ is the minimum-norm element of
$\mathcal{S}^{0}\left(\mathbb{R}^{2}\right)$ as in [7, Definition 2.1].

We look at the special generalized data

$\boldsymbol{\xi}_{n}:=\left(\xi_{n1}^{\ast},\xi_{n2}^{\ast},\xi_{n3}^{\ast}\right),\quad\boldsymbol{y}_{n}:=\left(1,1,1\right),\quad\text{for
all }n\in\mathbb{N}.$

where

$\xi_{n1}^{\ast}:=\begin{pmatrix}1\\\
0\end{pmatrix},~{}\xi_{n2}^{\ast}:=\begin{pmatrix}0\\\ 1/n\end{pmatrix},\text{
and }\xi_{n3}^{\ast}:=\begin{pmatrix}0\\\ 0\end{pmatrix}.$

It is easy to check that $\mathfrak{F}_{\mathcal{D}}$ is relatively compact in
$\mathbb{R}^{2}$. Let the multi-loss function

$\mathbb{L}_{n}\left(\boldsymbol{\xi}^{\ast},\boldsymbol{y},\boldsymbol{t}\right):=\frac{1}{3}\sum_{k=1}^{3}L\left(\xi^{\ast}_{k},y_{k},t_{k}\right),\quad\text{for
}\boldsymbol{\xi}^{\ast}\in\mathbb{R}^{3\times
2},~{}\boldsymbol{y}\in\mathbb{R}^{3},~{}\boldsymbol{t}\in\mathbb{R}^{3},$

where $n\in\mathbb{N}$ and

$L\left(\xi^{\ast},y,t\right):=\left\\{\begin{array}[]{cl}3\left\lvert
t-y\right\rvert^{2},&\text{if }y\in\left\\{1\right\\},\\\
0,&\text{otherwise},\end{array}\right.$

for $\xi^{\ast}\in\mathbb{R}^{2},~{}y\in\mathbb{R},~{}t\in\mathbb{R}$. Since
$L$ is a local Lipschitz continuous loss function, Lemma 2.3 (b) assures that
$\mathcal{L}$ is uniformly local Lipschitz continuous. This shows that
Condition (II) holds true.

Moreover, $R_{n}(f)=\lVert\mathsf{A}_{n}f-\boldsymbol{b}\rVert^{2}_{2}$ for
$f\in\mathbb{R}^{2}$. Thus $R_{n}$ converges pointwise to $R$ when
$n\to\infty$. This shows that Condition (I) holds true. [7, Theorem 2.5]
assures that the best-approximate solution of
$\mathsf{A}\boldsymbol{c}=\boldsymbol{b}$ is

$f^{0}:=\mathsf{A}^{\dagger}\boldsymbol{b}=\begin{pmatrix}1\\\
0\end{pmatrix},$

and the best-approximate solution of
$\mathsf{A}_{n}\boldsymbol{c}=\boldsymbol{b}$ is

$f_{n}:=\mathsf{A}_{n}^{\dagger}\boldsymbol{b}=\begin{pmatrix}1\\\
1+n\end{pmatrix},$

where $\mathsf{A}^{\dagger}$ and $\mathsf{A}_{n}^{\dagger}$ are the pseudo
inverses of $\mathsf{A}$ and $\mathsf{A}_{n}$, respectively. Thus $f_{n}$ is
also a minimizer of $R_{n}$ over $\mathbb{R}^{2}$. But $f_{n}$ is not
convergent when $n\to\infty$. This shows that $f_{n}$ is not a well-posed
approximate solution of $\mathsf{A}\boldsymbol{c}=b$ for any $n\in\mathbb{N}$.

Let $\Phi(r):=r^{2}$ for $r\in[0,\infty)$. Thus [7, Theorem 5.1] assures that
there exists the unique approximate solution
$f_{n}^{\lambda}\in\mathcal{S}_{n}^{\lambda}\left(\mathbb{R}^{2}\right)$ which
is written as

$f_{n}^{\lambda}=\left(\mathsf{A}_{n}^{T}\mathsf{A}_{n}+\lambda\mathsf{I}\right)^{-1}\mathsf{A}_{n}^{T}\boldsymbol{b}=\begin{pmatrix}1\\\
n/(1+\lambda n^{2})\end{pmatrix}.$

where $\mathsf{I}$ is an identity matrix. By Theorem 4.6, $f_{n}^{\lambda}$
can become a well-posed approximate solution for a $n\in\mathbb{N}$ and a
$\lambda>0$. For example, if we choose $\lambda_{n}:=1/\sqrt{n}$, then
$f_{n}^{\lambda_{n}}\to f^{0}$ when $n\to\infty$. However, if we choose
$\lambda_{n}:=1/n^{2}$, then $f_{n}^{\lambda_{n}}$ is not convergent when
$n\to\infty$. This shows that the approximate solution may not be well-posed
for any pair $n,\lambda$ even when $n\to\infty$ and $\lambda\to 0$.

######  Remark 6.2.

In Example 6.5, it is easily seen that the empirical risk functions are not
equicontinuous while they are equicontinuous on all bounded subset. Even
though we may not solve the ill-posed problems directly by the empirical
risks, the regularized learning provides another way to find the well-posed
approximate solutions. Example 6.5 also shows that the choice of $\lambda$
affects the convergence of the approximate solutions of the ill-posed
problems. As in Theorem 4.8, we will try to discuss the open problem to choose
the adaptive $\lambda$ to the special learning problems by the weak* topology.

## Final Remark

In this article, the theory of Banach spaces is the foundation of the
regularized learning to analyze the generalized data. By the techniques of
weak* topology, we complete the proof of the representer theorems,
approximation theorems, and convergence theorems of the regularized learning.
Specially, the strong convergence condition of the original problems can be
replaced to the weak convergence condition of the original problems and the
additional checkable condition of the generalized data for the proof of the
weak* convergence of the approximate solutions to the exact solutions. The
intention of the regularized learning is to find an approximate solution which
is computable by the machine. The work of the regularized learning provides
another road to study the algorithms of machine learning including:

  * •

the interpretability in approximation theory,

  * •

the nonconvexity and nonsmoothness in optimization theory,

  * •

the generalization and overfitting in regularization theory.

In our philosophical idea, the learning algorithms are locally interpretable
by the generalized data and the exact solutions of the original problems are
approximated globally by the learning algorithms. Based on the theorems of the
regularized learning, the existence and convergence of the approximate
solutions can be guaranteed by the nonconvex and nonsmooth loss functions and
thus the approximate solutions will be computed by the nonconvex and nonsmooth
optimization approaches. Moreover, the regularization terms are used to avoid
the overfitting such that the weak* compactness is motivated in the proof of
the weak* convergence of the approximate solutions. It is well known that the
data-driven methods are usually used to analyze the black-box models and the
model-driven methods are usually used to analyze the white-box models. For a
special example, the regularized learning gives a new method to construct the
composite algorithms to combine the black-box and white-box models in one
system. Our original idea is inspired by the eastern philosophy such as the
golden mean and the Tai Chi diagram. In our current researches, we discuss the
composite algorithms of support vector machines, neural networks, and decision
trees for the big data analysis in education and medicine by the theorems of
the regularized learning.

## Acknowledgments

The author would like to express his gratitude to Professor Liren Huang at the
South China Normal University. Huang was the master advisor of Ye at the South
China Normal University. When Ye was back to work at Guangzhou, Huang had
already retired for a long time. Huang still gave a great help of this article
for its valuable revisions and significant improvements. The author is
supported by the National Natural Science Foundation of China #12071157,
#12026602, the Guangdong Basic and Applied Basic Research Foundation
#2019A1515011995, and the Foundation of Department of Education of Guangdong
Province #2020ZDZX3004.

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