Recent content by moh salem

  1. M

    Undergrad Is the Solution of a Differential Equation Always Unique?

    Thank you so much for all members.
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    Graduate Writing Elements in $L^p(X,Y,\mu)$ Mathematically

    "What is the definition of an equivalence class of functions with respect to the || ||p norm?"
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    Graduate When to Use Modulus of Convexity and Modulus of Smoothness in Calculations?

    Question: What do you mean the following concepts? and when to use? Let ##\delta _{X}:[0,2]\longrightarrow \lbrack 0,1]## }be the modulus of convexity of X, and defined by ##\delta _{X}(\varepsilon )=\inf \{1-\frac{1}{2}\left\Vert x-y\right\Vert :\left\Vert x\right\Vert \leq 1, \left\Vert...
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    Graduate Writing Elements in $L^p(X,Y,\mu)$ Mathematically

    \begin{equation*}Let\text{ } (X,\mathcal{A} ,\mu ) \text{ }be \text{ }a \text{ }complete\text{ } \sigma -finite\text{ } measure\text{ } space \\and \text{ }Y \text{ }be \text{ }a \text{ }separable\text{ } Banach\text{ } space\text{ } supplied \text{ }with \text{ }the \text{ }norm\text{ }...
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    Undergrad Is the Solution of a Differential Equation Always Unique?

    Is the solution of differential equation be unique always?
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    Graduate When to Use Modulus of Convexity and Modulus of Smoothness in Calculations?

    See the pdf file
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    Graduate Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

    Thank u Mr. mathman.
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    Graduate Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

    Yes, I mean \text{ } x_{0} = (0,0). but, if \text{ } x_{0} = (0,1). What is equal to N_{K}((0,1))?
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    Graduate Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

    \text{ }yes,\text{ } x_{0} = (0,0)
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    Graduate Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

    Let \text{ } H \text{ }be \text{ }a \text{ } Hilbert \text{ } space, \text{ }K \text{ }be \text{ }a \text{ }closed\text{ }convex\text{ }subset \text{ } of \text{ }H \text{ }and \text{ }x_{0}\in K. \text{ }Then \\N_{K}(x_{0}) =\{y\in K:\langle y,x-x_{0}\rangle \leq 0,\forall x\in K\} .\text{...
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    Graduate Measurability and integration of set-valued maps

    Thanks mathman.
  12. M

    Graduate Measurability and integration of set-valued maps

    [FONT=Times New Roman]Definition(set-valued map): Let X and Y be two nonempty sets and P(Y)={A:A⊆Y,A≠φ}. A set-valued map is a map F:X→P(Y) i.e. ∀x∈X, F(x)⊆Y for examples, (1) Let F:ℝ→P(ℝ) s.t. F(x)=]α,∞[,∀x∈X. Then F is a set-valued map. (2) Let F:ℝ→P(ℝ²) s.t. F(x)={(x,y):y=αx, α∈ℝ}.Then F is a...
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    Graduate Measurability and integration of set-valued maps

    I mean from the first paragraph is to: Let \text{ }F:X\longrightarrow P(Y) \text{ }be \text{ }a \text{ }set-valued \text{ }map \text{ }and \text{ } f:X\longrightarrow Y \text{ }be \text{ }a \text{ }measurable\\ \text{ }single-valued \text{ }map, \text{ }where \text{ } F(x)=\{f(x)\}.\\...
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    Graduate How Can Set-Valued Maps Be Integrated in Measurable Spaces?

    sorry, F : X→P(Y)\{φ} be a measurable, integrably bounded set-valued maps with closed images.
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    Graduate How Can Set-Valued Maps Be Integrated in Measurable Spaces?

    If \ \mu \ is\ nonatomic, \text{ }then \begin{equation*} \overline{\int_{X}Fd\mu }=\overline{co}\int_{X}F\text{ }d\mu =\int_{X}\overline{co}\text{ }F\text{ }d\mu , \end{equation*}Where, F :X\longrightarrow P(Y)\backslash \{\phi \}\ is \text{ } called\text{ }a\text{ } set-valued\text{ }...