Homogeneity holds but additivity does not. I'm stuck

In summary, the conversation discusses finding a function f:R^2 -> R that satisfies the properties f(av) = a(f(v)) for all a in R and all v in R^2, but is not linear. An example function that satisfies these properties is f((x,y))=x^(1/3)*y^(2/3).
  • #1
steelphantom
159
0

Homework Statement


Give an example of a function f:R^2 -> R such that f(av) = a(f(v)) for all a in R and all v in R^2 but f is not linear.


Homework Equations


f(v + w) = f(v) + f(w) (Additivity)


The Attempt at a Solution



I really can't think of a function that will satisfy these properties. I know that for this to work, homogeneity must hold but additivity must not. I tried several functions, but none worked. Any hints? Thanks.
 
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  • #2
How about f((x,y))=x^(1/3)*y^(2/3)?
 
  • #3
Dick said:
How about f((x,y))=x^(1/3)*y^(2/3)?

Thanks! I see how you came up with that now. I tried f((x, y)) = x^2 + y^2 but the first property didn't hold since a changed. :rolleyes: Your function fixes that issue. Thanks again for the help.
 

Related to Homogeneity holds but additivity does not. I'm stuck

1. What does it mean for homogeneity to hold but additivity to not hold?

Homogeneity and additivity are two important properties that describe the relationship between variables in a scientific study. Homogeneity means that the relationship between two variables remains consistent across different levels or groups of a third variable. Additivity means that the relationship between two variables is the same regardless of the values of any other variables. So, if homogeneity holds but additivity does not, it means that the relationship between two variables is consistent across different groups, but the strength or direction of the relationship changes when other variables are added.

2. Why is it important to understand the difference between homogeneity and additivity?

Understanding the difference between homogeneity and additivity is crucial for accurately interpreting and analyzing scientific data. These properties can greatly impact the conclusions we draw from a study and can help identify potential confounding variables that may be influencing the relationship between variables. Additionally, knowing whether homogeneity and additivity hold can inform the appropriate statistical tests and methods to use in analyzing the data.

3. How can I determine if homogeneity holds while additivity does not?

To determine if homogeneity holds but additivity does not, you can visually inspect the relationship between the variables using scatter plots or by calculating correlation coefficients for different groups or levels of a third variable. Additionally, statistical tests such as ANOVA or regression can be used to assess the relationship between variables and identify any potential differences between groups or levels.

4. What are some possible reasons for homogeneity to hold but additivity to not hold?

There can be several reasons for homogeneity to hold but additivity to not hold. One possible explanation could be the presence of a confounding variable that is influencing the relationship between the variables. Another reason could be that the relationship between the variables is influenced by the level or group of the third variable, indicating an interaction effect. It is important to carefully examine the data and consider potential explanations for the observed pattern.

5. How can I address the issue of homogeneity holding but additivity not holding in my analysis?

If homogeneity holds but additivity does not, it is important to carefully consider the implications of this on your analysis and interpretation of the results. You may need to use different statistical methods or approaches, such as stratifying the data or using non-parametric tests, to accurately analyze the relationship between variables. It is also important to consider potential confounding variables or interaction effects and adjust for them in your analysis if possible.

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