V∈ E_r={P∈R[X_1,X_2,..,X_d]∣degP≤r} If exp^{−V(x)} ∈ L^2(R^d) then V admits local min

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mona123

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V∈ E_r={P∈R[X_1,X_2,..,X_d]∣degP≤r} If exp^{−V(x)} ∈ L^2(R^d) then V admits local min

Can someone please help me to answer this question:

we consider V∈ [FONT=MathJax_Math-italic]E_[FONT=MathJax_Math-italic]r[/FONT][FONT=MathJax_Main]={[/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Main][[/FONT][FONT=MathJax_Math-italic]X_[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]X_[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main].[/FONT][FONT=MathJax_Main].[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]X_[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main]][/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]e[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math-italic]r[/FONT][/FONT]} If [FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Math-italic]p^{[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]V[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])}[/FONT][/FONT] ∈ [FONT=MathJax_Math-italic]L^[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]R^[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main])[/FONT][/FONT] then V admits a local minimum.
Thanks in advance
 
Last edited:
mona123 said:
Can someone please help me to answer this question:

we consider V∈ [FONT=MathJax_Math-italic]E_[FONT=MathJax_Math-italic]r[/FONT][FONT=MathJax_Main]={[/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Main][[/FONT][FONT=MathJax_Math-italic]X_[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]X_[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main].[/FONT][FONT=MathJax_Main].[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]X_[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main]][/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]e[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Math-italic]P[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math-italic]r[/FONT][/FONT]} If [FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Math-italic]p^{[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]V[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])}[/FONT][/FONT] ∈ [FONT=MathJax_Math-italic]L^[FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]R^[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main])[/FONT][/FONT] then V admits a local minimum.
Thanks in advance
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What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting
 
Hi Ishuda,
this is what i wrote but i don't know if that will help:
if we prove the following implication does it help to answer the initial question?: If [FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Math-italic]p[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]V[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math-italic]L[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main])[/FONT][/FONT] then the limit of [FONT=MathJax_Math-italic]V[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][/FONT] as [FONT=MathJax_Main]|[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]|[/FONT][/FONT]goes to $+\infty$ is $+\infty$[FONT=MathJax_Main]+[FONT=MathJax_Main]∞ is[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]∞[/FONT][/FONT]
 
mona123 said:
Hi Ishuda,
this is what i wrote but i don't know if that will help:
if we prove the following implication does it help to answer the initial question?: If [FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Math-italic]p[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]V[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math-italic]L[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]R[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main])[/FONT][/FONT] then the limit of [FONT=MathJax_Math-italic]V[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][/FONT] as [FONT=MathJax_Main]|[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]|[/FONT][/FONT]goes to $+\infty$ is $+\infty$[FONT=MathJax_Main]+[FONT=MathJax_Main]∞ is[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]∞[/FONT][/FONT]
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Since notations are sometimes different for different people, please restate the problem 'in English'.
 
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