The current paper investigates a class of asymptotically linear Schrödinger equations with the nonlinear term f∈ C( ℝ,ℝ) s.t. lim _| t| →∞ f( t) /t=σ _0 , where σ _0 stands for the threshold of essential spectrum of Schrödinger operator. Clearly the Palais-Smale condition fails to hold in this case. Especially under the hypothesis ( V_2) , the lack of compactness occurs at the interaction between nonlinear term and continuum spectrum. For this reason, we introduce a bootstrap iteration approach for elliptic equation on ℝ^N . The iteration is self-contained and can be regarded as a generalization of Agmon-Douglis-Nirenberg theorem (see Agmon et al. (Comm Pure Appl Math 17:35—92, 1964), Agmon et al. (Comm Pure Appl Math 12:623—727, 1959)). The proof characterizes iteration steps independent of the choice of the parameter λ , which are indeed manipulated by intrinsic natures of potentials and nonlinear terms, and furthermore presents precise estimates for asymptotically linear functions or continuous nonlinear terms restricted on a bounded domain Ω with smooth boundary ∂Ω in ℝ ^N . Additionally, a comparison theorem for the spectrum of Schrödinger operator is also established in this paper. With above preparations, we can get a nontrivial solution without mountain pass geometry, and more importantly make an explicit description of nondegeneracy of solutions with monotonicity hypothesis.

We present a systematic analysis of the velocity structure functions (VSFs) of 167 molecular clouds with angular sizes greater than $\sim$176 arcmin$^2$ in three sectors of the Galactic mid-plane. We calculated the 1st- to 3rd-order VSFs and found that 60\% of the VSFs exhibit power-law distributions. The relative power-law exponents are consistent with predictions from intermittent turbulence models. Column density weighting reduces the proportion of power-law VSFs and steepens the VSF slopes, implying a reduction of turbulent energy in high-density regions. All clouds show small-scale intermittency, with slightly stronger intermittency in those molecular clouds showing none power-law VSFs. Negative VSF exponents that may indicate gravitational collapse are not observed in our sample. The scaling exponents of the observed VSFs do not correlate with the virial parameters of the molecular clouds. These two observations suggest that gravity-dominated scales in molecular clouds still need further investigation. Consistent VSF scaling exponents for the molecular clouds with significant power-law VSFs suggest large-scale external driving of turbulence in these molecular clouds. However, the driving mechanisms are likely not universal, as the power-law scaling coefficients in our results show relatively large scatter. The fact that nearly 40\% of the VSFs deviate to some extent from power-law distributions suggests that the influence of local environments on the internal turbulence of molecular clouds may not be negligible.

In this paper, we establish the existence of one solution for a Schrödinger equation with jumping nonlinearities: -Δ u+V(x)u=f(x,u), x∈ℝ^N, and u(x)→ 0, |x|→ +∞, where V is a potential function on which we make hypotheses, and in particular allow V which is unbounded below, and f(x,u)=au^-+bu^++g(x,u). No restriction on b is required, which implies that f(x,s)s^-1 may interfere with the essential spectrum of -Δ+V for s→ +∞. Using the truncation method and the Morse theory, we can compute critical groups of the corresponding functional at zero and infinity, then prove the existence of one negative solution.

In this paper we present the distribution of molecular gas in the Milky Way Galactic plane from l = [59.75, 74.75]^∘ and b = [-5.25, +5.25]^∘, using the MWISP ^12CO/^13CO/C^18O emission line data. The molecular gas in this region can be mainly attributed to the Local spur, Local arm, Perseus arm, and Outer arm. Statistics of the physical properties of the molecular gas in each arm, such as excitation temperature, optical depth, and column density, are presented. Using the DBSCAN algorithm, we identified 15 extremely distant molecular clouds with kinematic distances of 14.72-17.77 kpc and masses of 363-520 M_⊙, which we find could be part of the Outer Scutum-Centaurus (OSC) arm identified by and . It is also possible that, 12 of these 15 extremely distant molecular clouds constitute an independent structure between the Outer and the OSC arms or a spur. There exist two Gaussian components in the vertical distribution of the molecular gas in the Perseus spiral arm. These two Gaussian components correspond to two giant filaments parallel to the Galactic plane. We find an upward warping of the molecular gas in the Outer spiral arm with a displacement of around 270 pc with respect to the Galactic mid-plane.

In this work, we study the properties of molecular clouds in the second quadrant of the Milky Way Midplane, from l = 104.degrees 75 to l = 119.degrees 75, and b = -5.degrees 25 to b = 5.degrees 25, using the (CO)-C-12, (CO)-C-13, and (CO)-O-18 J = 1 - 0 emission line data from the Milky Way Imaging Scroll Painting project. We identify 857 and 300 clouds in the (CO)-C-12 and (CO)-C-13 spectral cubes, respectively, using the DENDROGRAM + SCIMES algorithms. The distances of the molecular clouds are estimated, and physical properties such as the mass, size, and surface densities of the clouds are tabulated. The molecular clouds in the Perseus Arm are about 30-50 times more massive, and 4-6 times larger than the clouds in the Local Arm. This result, however, is likely to be biased by distance selection effects. The surface densities of the clouds are enhanced in the Perseus Arm, with an average value of similar to 100 M (circle dot) pc(-2). Here. we select the 40 most extended (>0.35 arcdeg(2)) molecular clouds from the (CO)-C-12 catalog to build the H-2 column density probability distribution function (N-PDF). Some 78% of the N-PDFs of the selected molecular clouds are well fitted with log-normal functions with only small deviations at high densities, corresponding to star-forming regions with scales of similar to 1-5 pc in the Local Arm, and similar to 5-10 pc in the Perseus Arm. About 18% of the selected molecular clouds have power-law N-PDFs at high densities. In these molecular clouds, the majority of the regions fitted with the power law correspond to molecular clumps at sizes of similar to 1 pc, or filaments at widths of similar to 1 pc.

We identify 225 filaments from an H 2 column density map constructed using simultaneous 12 CO, 13 CO and C 18 O(J=1-0) observations carried out as a part of the Milky Way Imaging Scroll Painting(MWISP) project.We select 46 long filaments with lengths above 1.2 pc to analyze the filament column density profiles.We divide the selected filaments into 397 segments and calculate the column density profiles for each segment.The symmetries of the profiles are investigated.The proportion of intrinsically asymmetrical segments is 65.3%,and that of intrinsically symmetrical ones is 21.4%.The typical full width at half maximum(FWHM) of the intrinsically symmetrical filament segments is ～ 0.67 pc with the Plummer-like fitting,and ～ 0.50 pc with the Gaussian fitting,respectively.The median FWHMs derived from the second-moment method for intrinsically symmetrical and asymmetrical profiles are ～ 0.44 and 0.46 pc,respectively.Close association exists between the filamentary structures and the YSOs in the region.

Molecular Clouds in the Second Quadrant of the Milky Way Mid-plane from l=104. ◦75 to l=119. ◦75 and b=−5. ◦25 to b=5. ◦25 Yuehui Ma, 2 Hongchi Wang, 3 Chong Li, 2 Lianghao Lin, 3 Yan Sun, and Ji Yang 1Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210033, China 2University of Chinese Academy of Sciences, 19A Yuquan Road, Shijingshan District, Beijing 100049, China 3School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China

Let [Formula: see text] be a Riemannian manifold with finite volume and [Formula: see text] be a linear topological space. We consider the strongly indefinite superlinear problem [Formula: see text] where [Formula: see text] is a self-adjoint linear operator, [Formula: see text] is a real Hilbert space with the compact embedding [Formula: see text] if [Formula: see text] for some [Formula: see text], and [Formula: see text]. We obtain the existence of two solutions provided that [Formula: see text] and [Formula: see text] for a certain choice of [Formula: see text], [Formula: see text]. Moreover, we prove that, if [Formula: see text] and [Formula: see text] small enough, there exist prescribed number of nontrivial solutions. As applications, the corresponding results hold true for nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold.

The detection of clumps (cores) in molecular clouds is an important issue in sub-millimetre astronomy. However, the completeness of the identification and the accuracy of the returned parameters of the automated clump identification algorithms are still not clear. In this work, we test the performance and bias of the GaussClumps, ClumpFind, FellWalker, Reinhold, and Dendrograms algorithms in identifying simulated clumps. By designing the simulated clumps with various sizes, peak brightness, and crowdedness, we investigate the characteristics of the algorithms and their performance. In the aspect of detection completeness, FellWalker, Dendrograms, and GaussClumps are the first, second, and third best algorithms, respectively. The numbers of correct identifications of the six algorithms gradually increase as the size and signal-to-noise ratio (SNRs) of the simulated clumps increase and they decrease as the crowdedness increases. In the aspect of the accuracy of retrieved parameters, FellWalker and Dendrograms exhibit better performance than the other algorithms. The average deviations in clump parameters for all algorithms gradually increase as the size and SNR of clumps increase. Most of the algorithms except FellWalker exhibit significant deviation in extracting the total flux of clumps. Taken together, FellWalker, GaussClumps, and Dendrograms exhibit the best performance in detection completeness and extracting parameters. The deviation in virial parameter for the six algorithms is relatively low. When applying the six algorithms to the clump identification for the Rosette molecular cloud, ClumpFind1994, ClumpFind2006, GaussClumps, FellWalker, and Reinhold exhibit performance that is consistent with the results from the simulated test.

Using the PMO-13.7 m millimeter telescope at Delingha in China, we have conducted a large-scale simultaneous survey of(12)CO,(CO)-C-13, and C(18)OJ = 1 - 0 emission toward the sky region centered atl = 2097,b = -225 with a coverage of 40 x 45. The majority of the emission in the region comes from the clouds with velocities lying in the range from -3 to 55 km s(-1), at kinematic distances from 0.5 to 7.0 kpc. The molecular clouds in the region are concentrated into three velocity ranges. The molecular clouds associated with the 10 Hiiregions/candidates are identified and their physical properties are presented. Massive stars are found within Sh2-280, Sh2-282, Sh2-283, and BFS54, and we suggest them to be the candidate excitation sources of the Hiiregions. The distributions of the excitation temperature and line width with the projected distance from the center of Hiiregion/candidate suggest that the majority of the 10 Hiiregions/candidates and their associated molecular gas are three-dimensional structures, rather than two-dimensional structures.

We have conducted a large-field simultaneous survey of 12CO, 13CO and C18O J = 1 - 0 emission toward the Orion A giant molecular cloud (GMC) with a sky coverage of similar to 4.4 deg2 using the Purple Mountain Observatory (PMO)-13.7 m millimeter-wavelength telescope. We use the probability distribution function of the column density (N-PDF) to investigate the distribution of molecular hydrogen in the Orion A GMC. The H2 column density, derived from the 13CO emission, of the GMC is dominated by a log-normal distribution in the range from similar to 4 x 1021 to similar to 1.5 x 1023 cm-2 with excesses both at the low-density and high-density ends. The excess of the low-density end is possibly caused by an extended and low-temperature (similar to 10 K) component with velocities in the range of 5 - 8 km s-1. Compared with the northern sub-regions, the southern sub-regions of the Orion A GMC contain less gas with column density in NH2 > 1.25 x 1022 cm-2. The dispersions of the N-PDFs of the sub-regions are found to correlate with the evolutionary stages of the clouds across the Orion A GMC. The structure hierarchy of Orion A GMC is explored with the DENDROGRAM algorithm, and it is found that the GMC is composed of two branches. All structures except one in the tree have virial parameters less than 2, indicating self-gravity is important on the spatial scales from similar to 0.3 to similar to 4 pc. Although power-laws and departures from lognormal distributions are found at the high-density end of N-PDFs for active star-forming regions, the N-PDFs of structures in the Orion A GMC are predominantly lognormal on scales from R similar to 0.4 to 4 pc.

Data of $^{12}$CO/$^{13}$CO/C$^{18}$O $J=1\to0$ emission toward the Galactic plane region of $l=35^\circ$ to $45^\circ$ and $b= -5^\circ$ to $+5^\circ$ are available with the Milky Way Imaging Scroll Painting (MWISP) project. Using the data, we found a giant molecular filament (GMF) around $l\approx38\sim42^\circ$, $b\approx-3.5\sim0^\circ$, $V_{LSR} \approx 27 \sim 40$ km~s$^{-1}$, named the GMF MWISP G041-01. At a distance of 1.7 kpc, the GMF is about 160 pc long. With a median excitation temperature about 7.5 K and a median column density about $10^{21}$ cm$^{-2}$, this GMF is very cold and very diffuse compared to known GMFs. Using the morphology in the data cube, the GMF is divided into four components among which three show filamentary structure. Masses of the components are $ 10^3 \sim 10^4 M_\odot$, with a total mass for the whole filament being about $7\times10^4 M_\odot$ from the LTE method. $^{13}$CO cores inside each component are searched. Virial parameters are about 2.5 for these cores and have a power-law index of -0.34 against the mass. The mass fraction of dense cores traced by $^{13}$CO to the diffuse clouds traced by $^{12}$CO are about 7% for all components of the GMF. We found signatures of possible large scale filament-filament collision in the GMF.

We have conducted a large-field simultaneous survey of (CO)-C-12, (CO)-C-13, and (CO)-O-18 J = 1 - 0 emission toward the Cassiopeia A (Cas A) supernova remnant (SNR), which covers a sky area of 3 degrees.5 x 3 degrees.1. The Cas giant molecular cloud (GMC) mainly consists of three individual clouds with masses on the order of 10(4)-10(5) M-circle dot. The total mass derived from the (CO)-C-13 emission of the GMC is 2.1 x 10(5) M-circle dot and is 9.5 x 10(5) M-circle dot from the (CO)-C-12 emission. Two regions with broadened (6-7 km s(-1)) or asymmetric (CO)-C-12 line profiles are found in the vicinity (within a 10' x 10' region) of the Cas A SNR, indicating possible interactions between the SNR and the GMC. Using the GAUSSCLUMPS algorithm, 547 (CO)-C-13 clumps are identified in the GMC, 54% of which are supercritical (i.e., alpha(vir) < 2). The mass spectrum of the molecular clumps follows a power-law distribution with an exponent of -2.20. The pixel-by-pixel column density of the GMC can be fitted with a log-normal probability distribution function (N-PDF). The median column density of molecular hydrogen in the GMC is 1.6 x 10 (21) cm(-2) and half the mass of the GMC is contained in regions with H-2 column density lower than 3 x 10(21) cm(-2), which is well below the threshold of star formation. The distribution of the YSO candidates in the region shows no agglomeration.

In this paper, we mainly use the Galerkin approximation method and the iteration inequalities of the Maslov type index theory to study the properties of subharmonic solutions for the Hamiltonian systems ( ) =

Due to high gain, Avalanche Photodetector (APD) in Geiger mode can detect single-photon signal. In this paper, a method for assessing the reliability of silicon APDs in Geiger mode is proposed, including the rapid estimation of APD fault activation energy (Ea), the analysis of APD failure mechanisms and assessment of lifetime. The test time and temperature interval were optimized using Arrhenius formula according to the stress limit and failure activation energy. Compared to 980,000h, the expected lifetime of our samples, the test time is greatly shortened, proving the high efficiency of our method. The main cause of fail may be the increase of surface leakage current, which will be verified in subsequent tests. Failure rate increases with time in wear out phase, and strongly depends on the structure of the device. Therefore, it is of great significance to improve the device design.

Using the PMO-13.7 m millimeter telescope at Delingha in China, we have conducted a large-scale simultaneous survey of (CO)-C-12, (CO)-C-13, and (CO)-O-18 J = 1 - 0 emission toward the Rosette molecular cloud (RMC) region with a sky coverage of 3 degrees.5 x 2 degrees.5. The majority of the emission in the region comes from the RMC complex, with velocities lying in the range from -2 to 20.5 km s(-1). Beyond this velocity range, 73 molecular clumps are identified with kinematic distances from 2.4 to 11 kpc. Based on the spatial and velocity distribution, nine individual clouds, C1-C9, have been identified for the RMC complex. It appears that the C3 cloud is different from other clouds in the RMC complex in view of its characteristic velocity, excitation temperature, and velocity dispersion. Most of the young stellar clusters in the region are located in positions of both high column density and high excitation temperature. Seven new molecular filaments are discovered in the RMC complex. Evidence for cloud-cloud collision is found in the region of young stellar clusters REFL9 and PouF, showing that these young stellar clusters probably result from a cloud-cloud collision. The abundance ratios of (CO)-C-13 to (CO)-O-18 in the region have a mean value of 13.7, which is 2.5 times larger than the solar system value, showing that UV photons from the nearby OB clusters have a strong influence on the chemistry of clouds in the RMC complex.

This paper is concerned with multiplicity results for semilinear elliptic equations of the type \(-\Delta u+a(x)u=\left| u\right| ^{p-2}u+f\left( x,u\right) \) in \(\Omega \), \(u=0\) on \(\partial \Omega \). We obtain at least two nontrivial solutions for this type of equations with the case that 0 is not a local minimizer of the corresponding functional under suitable hypotheses. The method we used here is based on a linking structure relevant to the Nehari manifold.

In this paper we first study the solvability of the equation (0.1) {-Delta u = bu(+) + au(-) +t phi x is an element of Omega u = 0, x is an element of Omega by using Morse theory. If (a, b) is an element of Q(l)\Sigma, lambda(l) is simple and phi > 0, the connection between the numbers of the solutions and the location of (a, b) in Q(l) were established. Moreover, we give a complete expression for the behaviors of the reduced functional (J) over tilde ((a,b)) (v) as (a, b) locating in different regions divided by Schechter's curves. Since to find the critical points of (J) over tilde ((a,b)) (v) is equivalent to find the solutions of -Delta u = bu(+) + au(-), it is more convenient to deal with the problem which has been reduced to a finite dimensional space.

We present CO observations toward three large supernova remnants (SNRs) in the third Galactic quadrant using the Purple Mountain Observatory Delingha 13.7m radio telescope. The observations are part of the high-resolution CO survey of the Galactic plane between Galactic longitudes l=-10deg to 250deg and latitudes b=-5deg to 5d. CO emission was detected toward the three SNRs: G205.5+0.5 (Monoceros Nebula), G206.9+2.3 (PKS 0646+06), and G213.0-0.6. Both of SNRs G205.5+0.5 and G213.0-0.6 exhibit the morphological agreement (or spatial correspondences) between the remnant and the surrounding molecular clouds (MCs), as well as kinematic signatures of shock perturbation in the molecular gas. We confirm that the two SNRs are physically associated with their ambient MCs and the shock of SNRs is interacting with the dense, clumpy molecular gas. SNR G206.9+2.3, which is close to the northeastern edge of the Monoceros Nebula, displays the spatial coincidence with molecular partial shell structures at VLSR 15km/s. While no significant line broadening has been detected within or near the remnant, the strong morphological correspondence between the SNR and the molecular cavity implies that SNR G206.9+2.3 is probably associated with these CO gas and is evolving in the low-density environment. The physical features of individual SNRs, together with the relationship between SNRs and their nearby objects, are also discussed.

This paper contains the existence of four solutions of Schrödinger equations with jumping nonlinearities. The proof procedure is supported by a lot of new results. Initially, a consequence is rendered as a minimax principle on H1(RN), which allows us to achieve the feasibility verification of the (PS) condition. Furthermore, the constructions of minimal and maximal curves of Fučík spectrum in Ql (see the introduction for the definition of Ql) warrant an intensive investigation. That we encounter some thorny problems is largely due to the absence of compact embedding and the appearance of essential spectrum. Based on a nontrivial argument, we can compute critical groups of homogeneous functional at zero if (a,b) is free of Fučík spectrum and (a,b)∈Ql. This together with convexity and concavity offers a detailed description of the two curves by a series of sophisticated tricks. Additionally, we present a new version of Morse theory in view of the fact that classical version doesn't work directly for weak smooth functional on H1(RN). Finally, we prove a weak maximum principle for RN, which serves as a tool to get a critical point in positive and negative cone respectively and also compute critical groups of critical points of mountain pass type. With the help of above preparations, we attain the ultimate aim by Morse inequalities and various exact homology sequences.