scholarly journals Local weak limits of Laplace eigenfunctions

2021 ◽  
Vol 3 (3) ◽  
pp. 481-515
Author(s):  
Maxime Ingremeau
Keyword(s):  
Weak Limits

scholarly journals Rigidity of Branching Microstructures in Shape Memory Alloys

2021 ◽  
Author(s):  
Theresa M. Simon
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory Alloys ◽  
Shape Memory ◽  
Differential Inclusion ◽  
Linearized Strain ◽  
Weak Limits ◽  
All Solutions ◽  
Rank One ◽  
Branched Structures ◽  
Habit Planes

AbstractWe analyze generic sequences for which the geometrically linear energy $$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{aligned}$$ E η ( u , χ ) : = η - 2 3 ∫ B 1 0 e ( u ) - ∑ i = 1 3 χ i e i 2 d x + η 1 3 ∑ i = 1 3 | D χ i | ( B 1 0 ) remains bounded in the limit $$\eta \rightarrow 0$$ η → 0 . Here $$ e(u) \,{:}{=}\,1/2(Du + Du^T)$$ e ( u ) : = 1 / 2 ( D u + D u T ) is the (linearized) strain of the displacement u, the strains $$e_i$$ e i correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and the partition into phases is given by $$\chi _i:{B_{1}\left( 0\right) } \rightarrow \{0,1\}$$ χ i : B 1 0 → { 0 , 1 } . In this regime it is known that in addition to simple laminates, branched structures are also possible, which if austenite was present would enable the alloy to form habit planes. In an ansatz-free manner we prove that the alignment of macroscopic interfaces between martensite twins is as predicted by well-known rank-one conditions. Our proof proceeds via the non-convex, non-discrete-valued differential inclusion $$\begin{aligned} e(u) \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, \end{aligned}$$ e ( u ) ∈ ⋃ 1 ≤ i ≠ j ≤ 3 conv { e i , e j } , satisfied by the weak limits of bounded energy sequences and of which we classify all solutions. In particular, there exist no convex integration solutions of the inclusion with complicated geometric structures.

Antinormally ordering of phase operators and the algebra of weak limits

1995 ◽  
Vol 113 (4-6) ◽  
pp. 427-432 ◽  
Author(s):  
John A. Vaccaro ◽  
Y. Ben-Aryeh
Keyword(s):  
Weak Limits ◽  
Phase Operators

scholarly journals Equivalent conditions for the tightness of a sequence of continuous Hilbert valued martingales

1987 ◽  
Vol 106 ◽  
pp. 113-119 ◽  
Author(s):  
Michel Métivier ◽  
Shintaro Nakao
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Sufficient Condition ◽  
Weak Limits ◽  
Equivalent Conditions

In D. Aldous gave a sufficient condition for the tightness of a sequence (Xn)n≥0 of right continuous (with left limits) processes taking their values in a separable complete metric space S. As already noted by Aldous this condition is far from being necessary when the processes Xn are not continuous. More precisely the Aldous-condition implies the left-quasi-continuity of all the weak limits of the sequence (Xn)n≥0.

A multivariate IFR class

1985 ◽  
Vol 22 (01) ◽  
pp. 197-204 ◽  
Author(s):  
Thomas H. Savits
Keyword(s):  
Exponential Distribution ◽  
Failure Rate ◽  
Random Vector ◽  
Increasing Failure Rate ◽  
Rate Distribution ◽  
Weak Limits ◽  
Multivariate Exponential Distribution ◽  
Multivariate Exponential

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

scholarly journals Regional assessment of forest effect on watershed hydrology: Slovakia as a case study  405

2013 ◽  
Vol 59 (No. 10) ◽  
pp. 405-415
Author(s):  
HlásnyT ◽  
SitkováZ ◽  
I. Barka
Keyword(s):  
Water Regime ◽  
Forest Cover ◽  
Stand Density ◽  
Watershed Hydrology ◽  
Natural Conditions ◽  
Tree Species Composition ◽  
Weak Limits ◽  
Spring Area ◽  
Regulatory Capacity

Recently, the importance of forest effect on watershed hydrology has been increasingly recognized due to an elevated threat of floods and expected alterations of water regime in watersheds induced by climate change. We assessed the trade-off between natural conditions of 61 basic watersheds in Slovakia and expected water-regulatory capacity of forest in these watersheds. A multi-criteria decision-making scheme was proposed to calculate a coefficient for each watershed indicating the need to regulate its water regime as given by natural conditions, and another coefficient indicating the magnitude of forest water-regulatory capacity given by forest structure and distribution. Factors indicating the forest water-regulatory capacity were extent of forest cover, forest fragmentation and distribution in watersheds relative to the spring area, forest stand density and vertical structure, and tree species composition. The results indicate that the present structure and distribution of forests in Slovakia has potential to moderately regulate the water regime at the scale of basic watersheds. We identified critical watersheds where natural conditions imply the unfavourable water regime and/or the forest water-regulatory capacity is weak. Limits of forest effect on watershed hydrology and caveats for interpreting the presented findings are discussed. 

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