Bounding length of Vectors

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I have problems to bound a potential.



Let $w_i$ be vectors for $i = 0,dots, n$ and $||w_i||$ the Euclidean norm of $w_i$.



$Phi = 4 sum_i=0^n ||w_i||^2$



$Phi' leq ||w_0 - w_1||^2 + ||w_n-1 - w_n||^2 + sum_i=1^n-1 ||w_i-1 + w_i+1||^2 $



I can compute the difference:



$Phi - Phi' geq sum_i=1^n-1 ||w_i-1 - w_i+1||^2 + ||w_0-w_1||^2 + ||w_n-1 - w_n||^2$



--> $Phi' leq Phi$



Does anyone see how to obtain a sharper bound?
E.g. $Phi' leq c cdot Phi$ for $c < 1$?




vectors euclidean-geometry upper-lower-bounds




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  • 1




    What is $Phi$ a function of? What do you mean by $Phi '$?.
    – Kavi Rama Murthy
    Aug 6 at 9:29










  • $Phi$ is the state of a system and $Phi'$ is the system state after one time step. So it's a relative change
    – Jannik
    Aug 6 at 9:31















up vote
0
down vote

favorite












I have problems to bound a potential.



Let $w_i$ be vectors for $i = 0,dots, n$ and $||w_i||$ the Euclidean norm of $w_i$.



$Phi = 4 sum_i=0^n ||w_i||^2$



$Phi' leq ||w_0 - w_1||^2 + ||w_n-1 - w_n||^2 + sum_i=1^n-1 ||w_i-1 + w_i+1||^2 $



I can compute the difference:



$Phi - Phi' geq sum_i=1^n-1 ||w_i-1 - w_i+1||^2 + ||w_0-w_1||^2 + ||w_n-1 - w_n||^2$



--> $Phi' leq Phi$



Does anyone see how to obtain a sharper bound?
E.g. $Phi' leq c cdot Phi$ for $c < 1$?




vectors euclidean-geometry upper-lower-bounds




share|cite|improve this question















  • 1




    What is $Phi$ a function of? What do you mean by $Phi '$?.
    – Kavi Rama Murthy
    Aug 6 at 9:29










  • $Phi$ is the state of a system and $Phi'$ is the system state after one time step. So it's a relative change
    – Jannik
    Aug 6 at 9:31













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have problems to bound a potential.



Let $w_i$ be vectors for $i = 0,dots, n$ and $||w_i||$ the Euclidean norm of $w_i$.



$Phi = 4 sum_i=0^n ||w_i||^2$



$Phi' leq ||w_0 - w_1||^2 + ||w_n-1 - w_n||^2 + sum_i=1^n-1 ||w_i-1 + w_i+1||^2 $



I can compute the difference:



$Phi - Phi' geq sum_i=1^n-1 ||w_i-1 - w_i+1||^2 + ||w_0-w_1||^2 + ||w_n-1 - w_n||^2$



--> $Phi' leq Phi$



Does anyone see how to obtain a sharper bound?
E.g. $Phi' leq c cdot Phi$ for $c < 1$?




vectors euclidean-geometry upper-lower-bounds




share|cite|improve this question











I have problems to bound a potential.



Let $w_i$ be vectors for $i = 0,dots, n$ and $||w_i||$ the Euclidean norm of $w_i$.



$Phi = 4 sum_i=0^n ||w_i||^2$



$Phi' leq ||w_0 - w_1||^2 + ||w_n-1 - w_n||^2 + sum_i=1^n-1 ||w_i-1 + w_i+1||^2 $



I can compute the difference:



$Phi - Phi' geq sum_i=1^n-1 ||w_i-1 - w_i+1||^2 + ||w_0-w_1||^2 + ||w_n-1 - w_n||^2$



--> $Phi' leq Phi$



Does anyone see how to obtain a sharper bound?
E.g. $Phi' leq c cdot Phi$ for $c < 1$?





vectors euclidean-geometry upper-lower-bounds





share|cite|improve this question










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asked Aug 6 at 9:05









Jannik

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  • 1




    What is $Phi$ a function of? What do you mean by $Phi '$?.
    – Kavi Rama Murthy
    Aug 6 at 9:29










  • $Phi$ is the state of a system and $Phi'$ is the system state after one time step. So it's a relative change
    – Jannik
    Aug 6 at 9:31













  • 1




    What is $Phi$ a function of? What do you mean by $Phi '$?.
    – Kavi Rama Murthy
    Aug 6 at 9:29










  • $Phi$ is the state of a system and $Phi'$ is the system state after one time step. So it's a relative change
    – Jannik
    Aug 6 at 9:31








1




1




What is $Phi$ a function of? What do you mean by $Phi '$?.
– Kavi Rama Murthy
Aug 6 at 9:29




What is $Phi$ a function of? What do you mean by $Phi '$?.
– Kavi Rama Murthy
Aug 6 at 9:29












$Phi$ is the state of a system and $Phi'$ is the system state after one time step. So it's a relative change
– Jannik
Aug 6 at 9:31





$Phi$ is the state of a system and $Phi'$ is the system state after one time step. So it's a relative change
– Jannik
Aug 6 at 9:31
















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