We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a ...universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.
In the application of WE43, it is found that Y2O3 inclusion formed in the process of casting seriously reduced the mechanical properties of the products. The reduction of the mechanical properties is ...even more distinct when it comes to the application in the thin walled complex-precision castings. In order to decrease the Y2O3 inclusions, Gd element was used to replace part of the Y element in Mg-Y-RE series alloys. The effect of Y content (Mg-xY-1Gd-2Nd-0.5Zn-0.5Zr) and Y/Gd ratio (Mg-xY-(5-x)Gd-2Nd-0.5Zn-0.5Zr) on the microstructure and mechanical properties of Mg–Y–RE alloys were investigated in this paper. With decreasing Y content, the grain size of the alloys increased, both ultimate tensile strength (UTS) and yield strength (YS) of alloys decreased monotonically. Replacing part of Y content with Gd and keeping the total rare earth content unchanged, the low Y content Mg-2Y-3Gd-2Nd-0.5Zn-0.5Zr alloy showed the same mechanical properties as the high Y content Mg-4Y-1Gd-2Nd-0.5Zn-0.5Zr alloy at both room and elevated temperatures. After solution treatment at 525 °C for 8 h and aging treatment at 225 °C for 10 h, the UTS, YS and elongation (ε) of Mg-2Y-3Gd-2Nd-0.5Zn-0.5Zr alloy reached 281.7 MPa, 198 MPa and 11.1% at room temperature, and 216.7 MPa, 171.6 MPa and 16.1% at 250 °C. The new low Y content Mg-2Y-2Nd-3Gd-0.5Zn-0.5Zr alloy is expected to replace the high Y content WE43 alloys, which can be used in the complex thin-walled parts of aviation products.
We study the discrete Schrödinger operators H_{\lambda ,\alpha ,\theta } on \ell ^2(\mathbb{Z}^{d+1}) with surface potential of the form V(n,x)=\lambda \delta (x)\tan \pi (\alpha \cdot n+\theta ), ...and H_{\lambda ,\alpha ,\theta }^{+} on \ell ^2(\mathbb{Z}^{d}\times \mathbb{Z}_+) with the boundary condition \psi _{(n,-1)}=\lambda \tan \pi (\alpha \cdot n+\theta )\psi _{(n,0)} , where \alpha \in \mathbb{R}^d is rationally independent. We show that the spectra of H_{\lambda ,\alpha ,\theta } and H_{\lambda ,\alpha ,\theta }^{+} are (-\infty ,\infty ) for all parameters. We can also determine the absolutely continuous spectra and Hausdorff dimension of the spectral measures if d=1.
Microstructures, tensile properties and high cycle fatigue behavior of sand-cast Mg–Gd–Y–Zr alloys with different Gd and Zr contents have been investigated. With decreasing Gd content from 9 to ...11wt%, the amount of eutectic phase in sand-cast Mg–xGd–3Y–0.5Zr alloys and of the β′ precipitates in peak-aged (T6) conditions decreased, and increasing Zr content from 0.3 to 0.5wt% led to a decrease in the grain size. The variation of Gd content had a slight influence on the tensile properties, while Zr content had an obvious influence on the tensile properties particularly the elongation. Gd and Zr affected the fatigue behavior in different ways, i.e. the increase of Zr content improved the fatigue strength while the increase of Gd content extends the fatigue life under relatively high stress.
In this study, the compositional dependence of the age hardening response and high temperature tensile properties of the Mg–xGd–3Y–0.5Zr (x=3, 6, 10, and 12wt%) alloys are investigated. The amount of ...cuboid-shaped phases and β′ precipitates increased significantly with increasing the Gd content. The Mg–10Gd–3Y–0.5Zr alloy exhibited the maximum ultimate tensile strength at room temperature, while at higher temperatures the Mg–12Gd–3Y–0.5Zr alloy exhibited the maximum yield strength and ultimate tensile strength. The yield strength and ultimate tensile strength of the Mg–12Gd–3Y–0.5Zr alloy increased with the test temperature and showed its maximum at 150°C, and then decreased as the temperature increased further. The Mg–12Gd–3Y–0.5Zr alloy maintained a high ultimate tensile strength of more than 300MPa up to 250°C. The superior high temperature tensile strength of the tested alloy is mainly associated with solution strengthening and precipitation hardening of the cuboid-shaped phases and β′ precipitates in Mg matrix. Especially, β′ precipitates can hinder the dislocation movement at high temperature.
In this work, solution and low-temperature (50–125 °C) aging treatment is utilized to develop a novel Mg-Li alloy with an excellent combination of strength and ductility. After optimizing the ...solution time, effects of aging treatment on microstructure and mechanical properties of the Mg-8Li-3Al-2Zn-0.5Y alloy were investigated. The highest strength is obtained after solution heat treatment at 350 °C for 4 h, accompanied with relatively low ductility. The following low-temperature aging treatment is used to balance the strength and ductility. With the increase of aging temperature, the ductility of the alloy was improved, accompanied by continued decrease in strength. The formation and aggregation of Al-Li phase during aging treatment is the major factor that influences the mechanical properties of the aged alloys. The reduced solid solution strengthening of Al and the aggregation of Al-Li phase are detrimental to strength. The evolution of Al-Li phase during aging treatment at different temperature is analyzed in detail. Through promoting the precipitation and preventing the aggregation of Al-Li phase by optimizing aging temperature, well-designed low-temperature aging treatment (75 °C × 4 h) results in favorable microstructural evolution and optimum combination of strength (yield strength = 216 MPa, ultimate tensile strength = 276 MPa) and ductility (elongation = 11.1%).
•The reduced solid solution strengthening of Al and the aggregation of Al-Li phase are detrimental to strength.•The evolution of Al-Li phase during aging treatment at different temperature is analyzed in detail.•Well-designed aging treatment results in optimum combination of strength and ductility.
This study was undertaken to investigate the influence of Er addition on microstructure and mechanical properties of as-cast Mg-10Li-5Zn alloy. With addition of Er, the grain size of as-cast alloys ...is greatly reduced, meanwhile, Mg-Zn-Er phase forms and increases gradually and Mg-Li-Zn phase decreases. The Mg-Zn-Er phase is mainly W-phase. The size of W-phase becomes larger with the increase of Er content from 0.5 wt% till 3.5 wt% and W-phase is mainly distributed from in the grain interiors to along the grain boundaries as well. The size, quantity and distribution of W-phase is an important factor that influences the mechanical properties of Mg-10Li-5Zn-xEr alloys. Overmuch coarse W-phase in the grain boundaries is detrimental to strength. The Mg-10Li-5Zn-0.5Er alloy exhibits an optimum combination of tensile properties with the ultimate tensile strength, yield strength and elongation of 198 MPa, 223 MPa and 14.7%. The optimal Er addition in as-cast Mg-10Li-5Zn alloy is ~0.5 wt%.
The association between playing Ma-jong, chess and cards (PMCC) and shoulder pain among elderly adults in China remains unclear. The present study aimed to examine the association based on a ...nationally representative large-scale survey.
The study used two waves of data from China Health and Retirement Longitudinal Study (CHARLS) in 2015 and 2018. All subjects met the inclusion criteria were classified based on the results of the CHARLS wave questionnaire. Logistic regression models were conducted to analyze the cross-sectional and longitudinal association between PMCC and shoulder pain. Odds ratios (OR) were reported to indicate the strength of the association.
In total, 8125 participants aged over 60 years (48.3 % male; mean age 67.9 ± 6.4 years) were enrolled in a cross-sectional association study in 2015, and further 6861 elderly adults were followed up in 2018. The overall prevalence of shoulder pain in the cross-sectional population was 15.6 %. In the non-PMCC and PMCC groups, the incidence of shoulder pain was 16.6 % and 10.7 %, respectively (P < 0.001). In the fully adjusted model, the results showed a significant association between PMCC and shoulder pain (OR: 0.798, 95 % CI: 0.662–0.963, P = 0.018). In the longitudinal analysis from 2015 to 2018, the incidence of new-occurrence shoulder pain in non-PMCC and PMCC groups was 22.1 %, and 18.5 %, respectively (P = 0.004). In the fully adjusted model, compared with the non-PMCC group, the risk of the occurrence of shoulder pain for individuals with PMCC was lower (OR: 0.832, 95 % CI: 0.709–0.975, P = 0.023).
PMCC is associated with lower incidence of shoulder pain, which provides evidence for PMCC as a potential protective factor in the occurrence of shoulder pain.
•Individuals with PMCC are associated with lower shoulder pain incidence.•Individuals with PMCC are associated with lower new-occurrence shoulder pain incidence.•PMCC is associated with lower incidence of shoulder pain, which provides evidence for PMCC as a potential protective factor in the occurrence of shoulder pain.
Let
H
0
be a discrete periodic Schrödinger operator on
ℓ
2
(
Z
d
)
:
H
0
=
-
Δ
+
V
,
where
Δ
is the discrete Laplacian and
V
:
Z
d
→
C
is periodic. We prove that for any
d
≥
3
, the Fermi variety at ...every energy level is irreducible (modulo periodicity). For
d
=
2
, we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for
d
=
2
and a constant potential
V
, the Fermi variety at
V
-level has exactly two irreducible components (modulo periodicity). We also prove that the Bloch variety is irreducible (modulo periodicity) for any
d
≥
2
. As applications, we prove that when
V
is a real-valued periodic function, the level set of any extrema of any spectral band functions, spectral band edges in particular, has dimension at most
d
-
2
for any
d
≥
3
, and finite cardinality for
d
=
2
. We also show that
H
=
-
Δ
+
V
+
v
does not have any embedded eigenvalues provided that
v
decays super-exponentially
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