What is it called when each element of one vector is greater than each element of another?












2












$begingroup$


I'm currently writing a term paper and while writing some proofs I've used an operator I don't know the name of.



Let us assume we have $x_1, x_2 in mathcal{R}^n$. Now lets define an operator '$succeq$' such that $x_1 succeq x_2$ implies that $x_1(i) geq x_2(i),,forall,, i in {1, .. n}$. Essentially, the operator implies that each element of one vector is greater than each element of the other. My question: is there a specific term for this operator? I strongly feel as though I've come across papers that have used this operator in proofs but I'm having a hard time finding them.



So far I've come across the concept of 'majorization' (http://mathworld.wolfram.com/Majorization.html) that seems to come close to what I want. Any help/information would be appreciated.






vector-spaces operator-theory operations-research







share|cite|improve this question









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    2












    $begingroup$


    I'm currently writing a term paper and while writing some proofs I've used an operator I don't know the name of.



    Let us assume we have $x_1, x_2 in mathcal{R}^n$. Now lets define an operator '$succeq$' such that $x_1 succeq x_2$ implies that $x_1(i) geq x_2(i),,forall,, i in {1, .. n}$. Essentially, the operator implies that each element of one vector is greater than each element of the other. My question: is there a specific term for this operator? I strongly feel as though I've come across papers that have used this operator in proofs but I'm having a hard time finding them.



    So far I've come across the concept of 'majorization' (http://mathworld.wolfram.com/Majorization.html) that seems to come close to what I want. Any help/information would be appreciated.






    vector-spaces operator-theory operations-research







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I'm currently writing a term paper and while writing some proofs I've used an operator I don't know the name of.



      Let us assume we have $x_1, x_2 in mathcal{R}^n$. Now lets define an operator '$succeq$' such that $x_1 succeq x_2$ implies that $x_1(i) geq x_2(i),,forall,, i in {1, .. n}$. Essentially, the operator implies that each element of one vector is greater than each element of the other. My question: is there a specific term for this operator? I strongly feel as though I've come across papers that have used this operator in proofs but I'm having a hard time finding them.



      So far I've come across the concept of 'majorization' (http://mathworld.wolfram.com/Majorization.html) that seems to come close to what I want. Any help/information would be appreciated.






      vector-spaces operator-theory operations-research







      share|cite|improve this question









      $endgroup$




      I'm currently writing a term paper and while writing some proofs I've used an operator I don't know the name of.



      Let us assume we have $x_1, x_2 in mathcal{R}^n$. Now lets define an operator '$succeq$' such that $x_1 succeq x_2$ implies that $x_1(i) geq x_2(i),,forall,, i in {1, .. n}$. Essentially, the operator implies that each element of one vector is greater than each element of the other. My question: is there a specific term for this operator? I strongly feel as though I've come across papers that have used this operator in proofs but I'm having a hard time finding them.



      So far I've come across the concept of 'majorization' (http://mathworld.wolfram.com/Majorization.html) that seems to come close to what I want. Any help/information would be appreciated.






      vector-spaces operator-theory operations-research




      vector-spaces operator-theory operations-research






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 2 at 21:34









      dar_devildar_devil

      236




      236






















          2 Answers
          2






          active

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          $begingroup$

          I don't really like a description of these as "vectors": I'd rather call them functions $mathbb Nto mathbb R$, or possibly with a domain consisting of some segment of $mathbb N$.



          Anyhow, in that context, I thought I had seen this called "the dominance order" on functions, which is a partial order you get when you say $fleq g$ if $f(x)leq g(x)$ for all $x$.



          For example here or here.



          However, while searching, I see that people use this term for lots of other partial orders that seem unrelated. Still, it seems like a pretty sensible name. You could say that one function(/vector) dominates another if it is greater at each point.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
            $endgroup$
            – dar_devil
            Jan 2 at 21:57










          • $begingroup$
            Actually, each function from the natural numbers to the reals is a vector.
            $endgroup$
            – Michael Hoppe
            Jan 2 at 22:00










          • $begingroup$
            @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
            $endgroup$
            – rschwieb
            Jan 2 at 22:58










          • $begingroup$
            @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
            $endgroup$
            – rschwieb
            Jan 3 at 1:18










          • $begingroup$
            @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
            $endgroup$
            – dar_devil
            Jan 7 at 20:01



















          1












          $begingroup$

          I've encountered the notation $x_1 succeq x_2$ in the book Convex Optimization by Boyd and Vandenberghe. If $K$ is a cone, then they define the "generalized inequality" $x succeq_K y$ to mean that $x - y in K$. In your case, $K$ is the nonnegative orthant.






          share|cite|improve this answer









          $endgroup$













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            2 Answers
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            2 Answers
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            $begingroup$

            I don't really like a description of these as "vectors": I'd rather call them functions $mathbb Nto mathbb R$, or possibly with a domain consisting of some segment of $mathbb N$.



            Anyhow, in that context, I thought I had seen this called "the dominance order" on functions, which is a partial order you get when you say $fleq g$ if $f(x)leq g(x)$ for all $x$.



            For example here or here.



            However, while searching, I see that people use this term for lots of other partial orders that seem unrelated. Still, it seems like a pretty sensible name. You could say that one function(/vector) dominates another if it is greater at each point.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
              $endgroup$
              – dar_devil
              Jan 2 at 21:57










            • $begingroup$
              Actually, each function from the natural numbers to the reals is a vector.
              $endgroup$
              – Michael Hoppe
              Jan 2 at 22:00










            • $begingroup$
              @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
              $endgroup$
              – rschwieb
              Jan 2 at 22:58










            • $begingroup$
              @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
              $endgroup$
              – rschwieb
              Jan 3 at 1:18










            • $begingroup$
              @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
              $endgroup$
              – dar_devil
              Jan 7 at 20:01
















            3












            $begingroup$

            I don't really like a description of these as "vectors": I'd rather call them functions $mathbb Nto mathbb R$, or possibly with a domain consisting of some segment of $mathbb N$.



            Anyhow, in that context, I thought I had seen this called "the dominance order" on functions, which is a partial order you get when you say $fleq g$ if $f(x)leq g(x)$ for all $x$.



            For example here or here.



            However, while searching, I see that people use this term for lots of other partial orders that seem unrelated. Still, it seems like a pretty sensible name. You could say that one function(/vector) dominates another if it is greater at each point.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
              $endgroup$
              – dar_devil
              Jan 2 at 21:57










            • $begingroup$
              Actually, each function from the natural numbers to the reals is a vector.
              $endgroup$
              – Michael Hoppe
              Jan 2 at 22:00










            • $begingroup$
              @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
              $endgroup$
              – rschwieb
              Jan 2 at 22:58










            • $begingroup$
              @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
              $endgroup$
              – rschwieb
              Jan 3 at 1:18










            • $begingroup$
              @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
              $endgroup$
              – dar_devil
              Jan 7 at 20:01














            3












            3








            3





            $begingroup$

            I don't really like a description of these as "vectors": I'd rather call them functions $mathbb Nto mathbb R$, or possibly with a domain consisting of some segment of $mathbb N$.



            Anyhow, in that context, I thought I had seen this called "the dominance order" on functions, which is a partial order you get when you say $fleq g$ if $f(x)leq g(x)$ for all $x$.



            For example here or here.



            However, while searching, I see that people use this term for lots of other partial orders that seem unrelated. Still, it seems like a pretty sensible name. You could say that one function(/vector) dominates another if it is greater at each point.






            share|cite|improve this answer









            $endgroup$



            I don't really like a description of these as "vectors": I'd rather call them functions $mathbb Nto mathbb R$, or possibly with a domain consisting of some segment of $mathbb N$.



            Anyhow, in that context, I thought I had seen this called "the dominance order" on functions, which is a partial order you get when you say $fleq g$ if $f(x)leq g(x)$ for all $x$.



            For example here or here.



            However, while searching, I see that people use this term for lots of other partial orders that seem unrelated. Still, it seems like a pretty sensible name. You could say that one function(/vector) dominates another if it is greater at each point.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Jan 2 at 21:48









            rschwiebrschwieb

            107k12102251




            107k12102251












            • $begingroup$
              Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
              $endgroup$
              – dar_devil
              Jan 2 at 21:57










            • $begingroup$
              Actually, each function from the natural numbers to the reals is a vector.
              $endgroup$
              – Michael Hoppe
              Jan 2 at 22:00










            • $begingroup$
              @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
              $endgroup$
              – rschwieb
              Jan 2 at 22:58










            • $begingroup$
              @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
              $endgroup$
              – rschwieb
              Jan 3 at 1:18










            • $begingroup$
              @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
              $endgroup$
              – dar_devil
              Jan 7 at 20:01


















            • $begingroup$
              Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
              $endgroup$
              – dar_devil
              Jan 2 at 21:57










            • $begingroup$
              Actually, each function from the natural numbers to the reals is a vector.
              $endgroup$
              – Michael Hoppe
              Jan 2 at 22:00










            • $begingroup$
              @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
              $endgroup$
              – rschwieb
              Jan 2 at 22:58










            • $begingroup$
              @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
              $endgroup$
              – rschwieb
              Jan 3 at 1:18










            • $begingroup$
              @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
              $endgroup$
              – dar_devil
              Jan 7 at 20:01
















            $begingroup$
            Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
            $endgroup$
            – dar_devil
            Jan 2 at 21:57




            $begingroup$
            Thank you for your answer! Out of curiosity, may I ask why you don't like a description of those as 'vectors'? I want to make sure I avoid inaccuracies or faux pax.
            $endgroup$
            – dar_devil
            Jan 2 at 21:57












            $begingroup$
            Actually, each function from the natural numbers to the reals is a vector.
            $endgroup$
            – Michael Hoppe
            Jan 2 at 22:00




            $begingroup$
            Actually, each function from the natural numbers to the reals is a vector.
            $endgroup$
            – Michael Hoppe
            Jan 2 at 22:00












            $begingroup$
            @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
            $endgroup$
            – rschwieb
            Jan 2 at 22:58




            $begingroup$
            @MichaelHoppe Nobody’s disputing that. I just said I didn’t like that description (in this case). Some folks tend to cling to tightly to vectors as “lists of numbers” and I prefer the function aspect of it.
            $endgroup$
            – rschwieb
            Jan 2 at 22:58












            $begingroup$
            @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
            $endgroup$
            – rschwieb
            Jan 3 at 1:18




            $begingroup$
            @dar_devil Just a preference in this context. Are you talking in the context of a vector space after all?
            $endgroup$
            – rschwieb
            Jan 3 at 1:18












            $begingroup$
            @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
            $endgroup$
            – dar_devil
            Jan 7 at 20:01




            $begingroup$
            @rschwieb That makes sense. Yes, I am talking in the context of vector spaces. I do think of vectors as "lists of numbers" rather than from the function aspect. I'm in a field that is more engineering than math and maybe that's why.
            $endgroup$
            – dar_devil
            Jan 7 at 20:01











            1












            $begingroup$

            I've encountered the notation $x_1 succeq x_2$ in the book Convex Optimization by Boyd and Vandenberghe. If $K$ is a cone, then they define the "generalized inequality" $x succeq_K y$ to mean that $x - y in K$. In your case, $K$ is the nonnegative orthant.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              I've encountered the notation $x_1 succeq x_2$ in the book Convex Optimization by Boyd and Vandenberghe. If $K$ is a cone, then they define the "generalized inequality" $x succeq_K y$ to mean that $x - y in K$. In your case, $K$ is the nonnegative orthant.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                I've encountered the notation $x_1 succeq x_2$ in the book Convex Optimization by Boyd and Vandenberghe. If $K$ is a cone, then they define the "generalized inequality" $x succeq_K y$ to mean that $x - y in K$. In your case, $K$ is the nonnegative orthant.






                share|cite|improve this answer









                $endgroup$



                I've encountered the notation $x_1 succeq x_2$ in the book Convex Optimization by Boyd and Vandenberghe. If $K$ is a cone, then they define the "generalized inequality" $x succeq_K y$ to mean that $x - y in K$. In your case, $K$ is the nonnegative orthant.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 2 at 21:56









                littleOlittleO

                29.9k646110




                29.9k646110






























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