To Find a 'Weight' Function on ${(x, y)in mathbb{R}^2:, x,y geq 0}$ Measuring A-Priori-Given Properties
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I seek a map of ${(x, y)in mathbb{R}^2:, x,y geq 0}$ into ℝ, the value it yields being a 'weight according as' (x,y) has certain properties (and I'll call it a 'weight' for convenience) - in this instance (and letting $q$ denote either $x$ or $y$): greater as the size of both $x$ & $y$ are greater; greater the lesser the difference between $x$ & $y$, & 0 when $q=0$; symmetrical under exchange of $x$ & $y$; and insensitive to small changes in $q$ when $q$ is zero - that the gradient of the weight with respect to $q$ be zero when $q=0$. One I have devised is - essentially by translating the function $$rsin^22theta$$ into cartesian coordinates - is $$frac{4x^2y^2}{(x^2+y^2)^{3/2}} ;$$ or, if it is to be used for ordering only, $$frac{(x^2y^2)^2}{(x^2+y^2)^3} .$$ (The numerator is left in the form it is in to show that this is essentially a function of $x^2$ & $y^2$ - and because it would indeed be most economically computed by computing those quantities first & then the rest in terms of them.)
Although this is not a particularly complicated expression, I do wonder whether there might be some simpler expression that 'slicklier' explements the a-priori-specified properties; and wondering this is a particular instance of wondering whether in general there is a systematic method for, or theory of, finding the simplest & slickest weight explementing a-priori-specified properties ... as I have tended to find, through the various particular instances of this problem that I have grappled, that it is usually more difficult than at first expected.
real-analysis
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
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add a comment |
$begingroup$
I seek a map of ${(x, y)in mathbb{R}^2:, x,y geq 0}$ into ℝ, the value it yields being a 'weight according as' (x,y) has certain properties (and I'll call it a 'weight' for convenience) - in this instance (and letting $q$ denote either $x$ or $y$): greater as the size of both $x$ & $y$ are greater; greater the lesser the difference between $x$ & $y$, & 0 when $q=0$; symmetrical under exchange of $x$ & $y$; and insensitive to small changes in $q$ when $q$ is zero - that the gradient of the weight with respect to $q$ be zero when $q=0$. One I have devised is - essentially by translating the function $$rsin^22theta$$ into cartesian coordinates - is $$frac{4x^2y^2}{(x^2+y^2)^{3/2}} ;$$ or, if it is to be used for ordering only, $$frac{(x^2y^2)^2}{(x^2+y^2)^3} .$$ (The numerator is left in the form it is in to show that this is essentially a function of $x^2$ & $y^2$ - and because it would indeed be most economically computed by computing those quantities first & then the rest in terms of them.)
Although this is not a particularly complicated expression, I do wonder whether there might be some simpler expression that 'slicklier' explements the a-priori-specified properties; and wondering this is a particular instance of wondering whether in general there is a systematic method for, or theory of, finding the simplest & slickest weight explementing a-priori-specified properties ... as I have tended to find, through the various particular instances of this problem that I have grappled, that it is usually more difficult than at first expected.
real-analysis
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
$endgroup$
$begingroup$
@Eric Wofsey -- Ah yes ... thanks for that advice ... I always have had this bad habit of getting them confused that I still occasionally lapse into! I've fixed it now anyway. ¶ And another perennial lapsing I have is forgetting to '@' my comments! Apologies please, if you have had two lots of notification of this.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:07
$begingroup$
@Eric Wofsey -- I think I've misunderstood you: I first used the term 'measure' and then changed it to 'metric' very shortly before your comment arrived ... so when I read it, my mind just 'flipped' it into being a pointing-out of the incorrectitude of the word 'measure'. ¶ OK I'll review what I've put then I'll just say "map of X×Y into ℝ having certain properties", or something like that ... for the time-being.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:19
add a comment |
$begingroup$
I seek a map of ${(x, y)in mathbb{R}^2:, x,y geq 0}$ into ℝ, the value it yields being a 'weight according as' (x,y) has certain properties (and I'll call it a 'weight' for convenience) - in this instance (and letting $q$ denote either $x$ or $y$): greater as the size of both $x$ & $y$ are greater; greater the lesser the difference between $x$ & $y$, & 0 when $q=0$; symmetrical under exchange of $x$ & $y$; and insensitive to small changes in $q$ when $q$ is zero - that the gradient of the weight with respect to $q$ be zero when $q=0$. One I have devised is - essentially by translating the function $$rsin^22theta$$ into cartesian coordinates - is $$frac{4x^2y^2}{(x^2+y^2)^{3/2}} ;$$ or, if it is to be used for ordering only, $$frac{(x^2y^2)^2}{(x^2+y^2)^3} .$$ (The numerator is left in the form it is in to show that this is essentially a function of $x^2$ & $y^2$ - and because it would indeed be most economically computed by computing those quantities first & then the rest in terms of them.)
Although this is not a particularly complicated expression, I do wonder whether there might be some simpler expression that 'slicklier' explements the a-priori-specified properties; and wondering this is a particular instance of wondering whether in general there is a systematic method for, or theory of, finding the simplest & slickest weight explementing a-priori-specified properties ... as I have tended to find, through the various particular instances of this problem that I have grappled, that it is usually more difficult than at first expected.
real-analysis
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
$endgroup$
I seek a map of ${(x, y)in mathbb{R}^2:, x,y geq 0}$ into ℝ, the value it yields being a 'weight according as' (x,y) has certain properties (and I'll call it a 'weight' for convenience) - in this instance (and letting $q$ denote either $x$ or $y$): greater as the size of both $x$ & $y$ are greater; greater the lesser the difference between $x$ & $y$, & 0 when $q=0$; symmetrical under exchange of $x$ & $y$; and insensitive to small changes in $q$ when $q$ is zero - that the gradient of the weight with respect to $q$ be zero when $q=0$. One I have devised is - essentially by translating the function $$rsin^22theta$$ into cartesian coordinates - is $$frac{4x^2y^2}{(x^2+y^2)^{3/2}} ;$$ or, if it is to be used for ordering only, $$frac{(x^2y^2)^2}{(x^2+y^2)^3} .$$ (The numerator is left in the form it is in to show that this is essentially a function of $x^2$ & $y^2$ - and because it would indeed be most economically computed by computing those quantities first & then the rest in terms of them.)
Although this is not a particularly complicated expression, I do wonder whether there might be some simpler expression that 'slicklier' explements the a-priori-specified properties; and wondering this is a particular instance of wondering whether in general there is a systematic method for, or theory of, finding the simplest & slickest weight explementing a-priori-specified properties ... as I have tended to find, through the various particular instances of this problem that I have grappled, that it is usually more difficult than at first expected.
real-analysis
real-analysis
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
edited Dec 16 '18 at 6:30
AmbretteOrrisey
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
asked Dec 16 '18 at 2:41
AmbretteOrriseyAmbretteOrrisey
54210
54210
$begingroup$
@Eric Wofsey -- Ah yes ... thanks for that advice ... I always have had this bad habit of getting them confused that I still occasionally lapse into! I've fixed it now anyway. ¶ And another perennial lapsing I have is forgetting to '@' my comments! Apologies please, if you have had two lots of notification of this.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:07
$begingroup$
@Eric Wofsey -- I think I've misunderstood you: I first used the term 'measure' and then changed it to 'metric' very shortly before your comment arrived ... so when I read it, my mind just 'flipped' it into being a pointing-out of the incorrectitude of the word 'measure'. ¶ OK I'll review what I've put then I'll just say "map of X×Y into ℝ having certain properties", or something like that ... for the time-being.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:19
add a comment |
$begingroup$
@Eric Wofsey -- Ah yes ... thanks for that advice ... I always have had this bad habit of getting them confused that I still occasionally lapse into! I've fixed it now anyway. ¶ And another perennial lapsing I have is forgetting to '@' my comments! Apologies please, if you have had two lots of notification of this.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:07
$begingroup$
@Eric Wofsey -- I think I've misunderstood you: I first used the term 'measure' and then changed it to 'metric' very shortly before your comment arrived ... so when I read it, my mind just 'flipped' it into being a pointing-out of the incorrectitude of the word 'measure'. ¶ OK I'll review what I've put then I'll just say "map of X×Y into ℝ having certain properties", or something like that ... for the time-being.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:19
$begingroup$
@Eric Wofsey -- Ah yes ... thanks for that advice ... I always have had this bad habit of getting them confused that I still occasionally lapse into! I've fixed it now anyway. ¶ And another perennial lapsing I have is forgetting to '@' my comments! Apologies please, if you have had two lots of notification of this.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:07
$begingroup$
@Eric Wofsey -- Ah yes ... thanks for that advice ... I always have had this bad habit of getting them confused that I still occasionally lapse into! I've fixed it now anyway. ¶ And another perennial lapsing I have is forgetting to '@' my comments! Apologies please, if you have had two lots of notification of this.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:07
$begingroup$
@Eric Wofsey -- I think I've misunderstood you: I first used the term 'measure' and then changed it to 'metric' very shortly before your comment arrived ... so when I read it, my mind just 'flipped' it into being a pointing-out of the incorrectitude of the word 'measure'. ¶ OK I'll review what I've put then I'll just say "map of X×Y into ℝ having certain properties", or something like that ... for the time-being.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:19
$begingroup$
@Eric Wofsey -- I think I've misunderstood you: I first used the term 'measure' and then changed it to 'metric' very shortly before your comment arrived ... so when I read it, my mind just 'flipped' it into being a pointing-out of the incorrectitude of the word 'measure'. ¶ OK I'll review what I've put then I'll just say "map of X×Y into ℝ having certain properties", or something like that ... for the time-being.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:19
add a comment |
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$begingroup$
@Eric Wofsey -- Ah yes ... thanks for that advice ... I always have had this bad habit of getting them confused that I still occasionally lapse into! I've fixed it now anyway. ¶ And another perennial lapsing I have is forgetting to '@' my comments! Apologies please, if you have had two lots of notification of this.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:07
$begingroup$
@Eric Wofsey -- I think I've misunderstood you: I first used the term 'measure' and then changed it to 'metric' very shortly before your comment arrived ... so when I read it, my mind just 'flipped' it into being a pointing-out of the incorrectitude of the word 'measure'. ¶ OK I'll review what I've put then I'll just say "map of X×Y into ℝ having certain properties", or something like that ... for the time-being.
$endgroup$
– AmbretteOrrisey
Dec 16 '18 at 6:19