If $x \in [0,1]$ has ternary expansion $(a_n)$, i.e. $x = 0.a_1a_2..$ with $a_n =0,1$ or $2$, define $N$ as the first index $n$ for which $a_n = 1$, and set $N = \infty$ if none of the $a_n$ are $1$ (when $x \in $ Cantor Set). Now, set $b_n = \frac{a_n}{2}$ for $n < […]

In the function topic of “function combinations” or “function algebra”, for the basic arithmetic operations of the following: 1) f + g 2) f – g 3) f * g 4) f / g To find the domain of these, one needs to find the domains of each f, g and find the intersection to […]

The full question actually is : If $A\subseteq \mathbb{N}$ and $f: \mathbb{N} \rightarrow \mathbb{N}$ is a function. Prove that if $f$ is a total recursive function and $A\subseteq \mathbb{N}$ recursive, then $f^{-1}(A)$ is recursive. I guess that the part that $A\subseteq \mathbb{N}$ is recursive is helping us to understand that this statement is true. However, […]

I posted the question before show the quadratic function is quasi-concave. But I can not understand the only answer, and the author delated his account already. I rephrase my question and show my second attempt here. I have a quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$, with $x_i$ nonnegative and $A \in[0,1)$. And w.l.o.g. we […]

The function that I am having trouble with is: $$ { f }_{ n }(x)=\begin{cases} 1\quad ,\quad x \in\{ 1,\frac { 1 }{ 2 } ,…,\frac { 1 }{ n } \} \\ 0\quad ,\quad otherwise \end{cases} $$ and in particular I am trying to show that the each $f_{n}$ is continuous at x = […]

For example, let’s say a term $A(x,y)$, a function of two random variables $x$ and $y$, is the argument of an expectation over $y$. The resulting term is no longer a function of $y$. Is there a mathematical symbol that explicitly says this? I know there are workarounds–in this arbitrary case, bar notation and dropping […]

Suppose f(x) = g(x) for all real x, and both f and g are sufficiently nice (perhaps we might limit them to be polynomials or analytic functions). Can we always manipulate one (with algebraic transformations) into the other? Is the same true for functions with multiple arguments? To say more precisely what I mean let […]

Consider these two diagrams of sets and functions (with $f$ and $f’$ invertible): \begin{array}{ccccccccc} A & \overset{f}{\longrightarrow} & B && &B & \overset{f^{-1}}{\longrightarrow} & A\\ u\downarrow& & v\downarrow & &; & v\downarrow & & u\downarrow\\ A’ & \overset{f’}{\longrightarrow} & B’ & && B’ & \overset{f’^{-1}}{\longrightarrow} & A’ \end{array} Can I say that the first commutes […]

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] – 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $$$$ I really have no clue as to how to go about this as I’ve never encountered such questions before. I would be truly grateful if somebody would please show me […]

I am looking for a function $f(x) \in [0,1]$ when $x \in (-\infty, +\infty)$. $f(x)$ increases very fast when $x$ is small starting from $-\infty$, and then very slow and eventually approach $1$ when $x$ is infinity.

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