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If a reduced labelling of a graph with $n+1$ vertices is not equal to $n+1$, then it is equal to the reduced labelling of the same graph with $n$ vertices. |
If $f$ is a face of the $n$-cube, then either $f$ is contained in a hyperplane of the form $x_j = 0$ or $x_j = p$ for some $j \leq n$, or $f$ is contained in the hyperplane $x_n = p$. |
If $p$ is a prime number, $n$ is a positive integer, and $lab$ is a labeling of the $n$-simplices of the $p$-Kuhn triangulation of the $(n-1)$-sphere such that $lab$ is $0$ on the vertices and $1$ on the facets, then the number of $n$-simplices of the $p$-Kuhn triangulation of the $n$-sphere whose vertices are labeled ... |
A $k$-simplex of dimension $0$ is a single point. |
If $p$ is a positive integer and $n$ is a nonnegative integer, then the number of $n$-dimensional simplices in the $p$-simplex with a given labeling is odd. |
Suppose $n$ and $p$ are natural numbers, and $p > 0$. Suppose that for every $n$-tuple of natural numbers $x$ such that $x_i \leq p$ for all $i$, we have that $label(x)$ is either $0$ or $1$. Suppose that for every $n$-tuple of natural numbers $x$ such that $x_i \leq p$ for all $i$, we have that if $x_i = 0$, then $lab... |
Suppose $P$ and $Q$ are properties of sequences of real numbers. If $P$ is preserved by the function $f$ and $P$ implies that the sequence is in the unit interval, then there exists a function $l$ such that $l$ is bounded by $1$, $l$ is $0$ on the $0$s of $P$ and $Q$, $l$ is $1$ on the $1$s of $P$ and $Q$, $l$ is $0$ o... |
If $f$ is a continuous function from the unit cube to itself, then there exists a point $x$ in the unit cube such that $f(x) = x$. |
If $f$ is a continuous map from a compact convex set $S$ with nonempty interior to itself, then there exists a fixed point of $f$ in $S$. |
Suppose $f$ is a continuous function from the closed ball of radius $e$ centered at $a$ to itself. Then there exists $x$ in the closed ball of radius $e$ centered at $a$ such that $f(x) = x$. |
If $f$ is a continuous function from a compact convex set $S$ to itself, then $f$ has a fixed point. |
The boundary of a closed ball in a Euclidean space is not a retract of the ball. |
The sphere of radius $r$ is contractible if and only if $r \leq 0$. |
A sphere is connected if and only if its dimension is at least 2 or its radius is non-positive. |
A sphere is path-connected if and only if its dimension is at least 2 or its radius is non-positive. |
If $S$ is a bounded set in $\mathbb{R}^n$ and $T$ is a set such that $f: \overline{S} \to T$ is a continuous function with $f(S) \subseteq T$ and $f(x) = x$ for all $x \in T$, then $S \subseteq T$. |
If $S$ is a convex set with nonempty interior, $T$ is a convex set, and $T$ is contained in the affine hull of $S$, then $T - \{a\}$ is a deformation retract of $S - \{a\}$. |
If $S$ is a bounded convex set with a point $a$ in its relative interior, then the relative frontier of $S$ is a retract of the affine hull of $S$ minus the point $a$. |
If $S$ is a compact convex set and $a \in \operatorname{relint} S$, then the relative boundary of $S$ is a retract of the punctured affine hull of $S$. |
If $S$ is a bounded convex set with nonempty interior, $T$ is a convex set containing the boundary of $S$, and $T$ is contained in the affine hull of $S$, then the boundary of $S$ is homotopy equivalent to $T$ with a point removed. |
If $S$ is a convex bounded set with a point $a$ in its relative interior, then the relative frontier of $S$ is homotopy equivalent to the affine hull of $S$ with $a$ removed. |
If $S$ is a convex, bounded set in $\mathbb{R}^n$ with affine dimension not equal to $1$, then the relative frontier of $S$ is path connected. |
If $S$ is a convex, bounded set with affine dimension not equal to $1$, then the relative frontier of $S$ is connected. |
If $a$ is not in $S$, then the Borsuk map $f: S \to \mathbb{R}^n$ defined by $f(x) = \frac{x - a}{\|x - a\|}$ is continuous. |
The image of a set $s$ under the map $x \mapsto \frac{x - a}{\|x - a\|}$ is contained in the unit sphere if and only if $a \notin s$. |
If $a$ and $b$ are in the same path component of the complement of $s$, then the Borsuk maps from $a$ and $b$ are homotopic in $s$. |
If $s$ is a compact set in $\mathbb{R}^n$ and $c$ is a bounded component of the complement of $s$, then there is no continuous map from $s \cup c$ to the unit sphere that maps $s$ to the antipodal map of $s$ and $c$ to the unit sphere. |
If $f$ is a continuous function from a compact convex set $T$ to itself, and if $S$ is a set such that for all $x \in S$ and $y \in T$, we have $x + (y - f(y)) \in T$, then $f$ is surjective. |
If $f$ is a continuous function from the closed ball of radius $e$ centered at $a$ to itself, and $x$ is a point in the interior of the ball, then there exists a point $y$ in the ball such that $f(y) = x$. |
Suppose $f$ is a continuous function from an open set $S$ to a Euclidean space. Suppose $x$ is an interior point of $T \subseteq S$. Then $f(x)$ is an interior point of $f(T)$. |
Suppose $f$ is a continuous function from an open set $S$ to itself, and $f$ has a continuous inverse $g$. If $f$ is differentiable at $x \in S$, then $g$ is differentiable at $f(x)$. |
If $f$ is a continuous function from an open set $S$ to itself, and $f$ has an inverse $g$ that is also continuous, then $g$ is differentiable at $y$ if and only if $f$ is differentiable at $g(y)$. |
If $f$ is a differentiable function on an open set $S$ and $g$ is its inverse, then $g$ is differentiable at $f(x)$ for all $x \in S$. |
Suppose $f$ is a continuous function defined on a convex set $S$ and $f$ is holomorphic on the interior of $S$ except for a finite set of points $k$. Suppose $\gamma$ is a closed path in $S$ that does not pass through any of the points in $k$. Then the integral of $f(w)/(w-z)$ along $\gamma$ is $2\pi i$ times the windi... |
If $f$ is holomorphic on a convex set $S$ and $z$ is an interior point of $S$, then the Cauchy integral formula holds for $f$ and $z$. |
If $f$ is a holomorphic function on the open ball $B(z, r)$ and $w \in B(z, r)$, then the function $g(u) = f(u)/(u - w)$ has a contour integral around the circle $C(z, r)$ equal to $2\pi i f(w)$. |
If $f$ is holomorphic on a ball centered at $z$ with radius $r$, and $w$ is a point in the ball, then the integral of $f(u)/(u - w)$ around the circle centered at $z$ with radius $r$ is $2\pi i f(w)$. |
Suppose $f$ is a function whose $k$th derivative $f^{(k)}$ is continuous on the image of a path $\gamma$. Suppose that $\gamma$ is a path with bounded speed, and that the $k$th derivative of $f$ has a contour integral along $\gamma$. Then the $(k+1)$st derivative of $f$ has a contour integral along $\gamma$, and the $(... |
If $f$ is a continuous function on the image of a circle of radius $r$ centered at $z$, and if $\int_{\gamma} \frac{f(u)}{(u - w)^k} du$ exists for all $w \in B(z, r)$, then $\int_{\gamma} \frac{f(u)}{(u - w)^(k + 1)} du$ exists for all $w \in B(z, r)$, and the derivative of $\int_{\gamma} \frac{f(u)}{(u - w)^k} du$ wi... |
If $f$ is a holomorphic function on a ball $B(z,r)$ and $w \in B(z,r)$, then the function $g(u) = f(u)/(u-w)^2$ is integrable on the circle $C(z,r)$ and $f'(w) = \frac{1}{2\pi i} \int_{C(z,r)} g(u) du$. |
If $f$ is a function whose derivative exists and is continuous on an open set $S$, then the derivative is holomorphic on $S$. |
If $f$ is holomorphic on an open set $S$, then its derivative $f'$ is holomorphic on $S$. |
If $f$ is analytic on $S$, then its derivative $f'$ is analytic on $S$. |
If $f$ is holomorphic on an open set $S$, then all of its derivatives are holomorphic on $S$. |
If $f$ is analytic on $S$, then $f^{(n)}$ is analytic on $S$. |
If $f$ is holomorphic on an open set $S$ and $x \in S$, then the $n$th derivative of $f$ exists and is equal to the $(n+1)$st derivative of $f$ at $x$. |
If $f$ is holomorphic on an open set $S$ and $g$ is a path with image in $S$, then $f \circ g$ is a path. |
If $f$ is continuous on every triangle in $S$ and the integral of $f$ around every triangle in $S$ is zero, then $f$ is analytic on $S$. |
Suppose $f$ is a continuous function defined on an open set $S$ such that for every triangle $T$ contained in $S$, the integral of $f$ along the boundary of $T$ is zero. Then $f$ is analytic on $S$. |
If $f$ is continuous on an open set $S$ and if the integral of $f$ around every triangle in $S$ is zero, then $f$ is analytic on $S$. |
The $n$th derivative of the linear function $f(z) = cz$ is $f^{(n)}(z) = c$ if $n = 1$ and $0$ otherwise. |
The $n$th derivative of a constant function is zero for $n \geq 1$. |
The $n$th derivative of the identity function is $0$ for $n \geq 2$, $1$ for $n = 1$, and $z$ for $n = 0$. |
The $n$th derivative of the identity function is $z$ if $n = 0$, $1$ if $n = 1$, and $0$ otherwise. |
If $f$ is a function with a derivative of $1$ at $z$ and $f(z) = z$, then $f^n$ has a derivative of $1$ at $z$. |
If $f$ is holomorphic on an open set $S$, then the $n$th derivative of $-f$ is $-f^{(n)}$. |
If $f$ and $g$ are holomorphic functions on an open set $S$, then the $n$th derivative of $f + g$ is equal to the $n$th derivative of $f$ plus the $n$th derivative of $g$. |
If $f$ and $g$ are holomorphic functions on an open set $S$, then the $n$th derivative of $f - g$ is equal to the $n$th derivative of $f$ minus the $n$th derivative of $g$. |
The binomial coefficient $\binom{n+1}{k}$ is equal to $\binom{n}{k} + \binom{n}{k-1}$. |
If $f$ and $g$ are holomorphic functions on a set $S$, then the $n$th derivative of $f \cdot g$ is given by the formula $\sum_{i=0}^n \binom{n}{i} f^{(i)} g^{(n-i)}$. |
If $f$ and $g$ are holomorphic functions on an open set $S$ and $f(z) = g(z)$ for all $z \in S$, then the $i$th derivative of $f$ at $z$ is equal to the $i$th derivative of $g$ at $z$. |
If $f$ is holomorphic on an open set $T$, and $u$ is a complex number, then the $n$th derivative of $f(u \cdot z)$ is $u^n \cdot f^{(n)}(u \cdot z)$. |
If $f$ and $g$ are analytic at $z$, then the $n$th derivative of $f + g$ at $z$ is equal to the $n$th derivative of $f$ at $z$ plus the $n$th derivative of $g$ at $z$. |
If $f$ and $g$ are analytic at $z$, then the $n$th derivative of $f - g$ at $z$ is equal to the $n$th derivative of $f$ at $z$ minus the $n$th derivative of $g$ at $z$. |
If $f$ is analytic at $z$, then the $n$th derivative of $-f$ at $z$ is $-f^{(n)}(z)$. |
If $f$ and $g$ are analytic at $z$, then the $n$th derivative of $f \cdot g$ at $z$ is equal to the sum of all products of $i$th derivatives of $f$ and $(n-i)$th derivatives of $g$ for $i$ from $0$ to $n$. |
If $f$ is a continuous function on an open set $S$ and $f$ is holomorphic on $S - K$, where $K$ is a finite set, then $f$ is holomorphic on $S$. |
If $f$ is holomorphic on the complement of a finite set $K$ and $f$ has a limit at every point of $K$, then $f$ is holomorphic on $S$. |
If $f$ is a continuous function on a convex set $S$ and $f$ is holomorphic on the interior of $S$ except for a finite number of points, then the Cauchy integral formula holds for $f$ on $S$. |
If $f$ is a holomorphic function on a ball $B(z,r)$ and $w \in B(z,r)$, then the function $g(u) = f(u)/(u-w)^{k+1}$ has a contour integral along the circle of radius $r$ centered at $z$ equal to $(2\pi i/k!)f^{(k)}(w)$. |
If $f$ is a holomorphic function on a ball $B(z,r)$ and $w \in B(z,r)$, then the $k$th derivative of $f$ at $w$ is equal to the contour integral of $f(u)/(u-w)^{k+1}$ over the circle of radius $r$ centered at $z$. |
If $f$ is a continuous function on the closed ball of radius $r$ centered at $z$ and holomorphic on the open ball of radius $r$ centered at $z$, then the $k$th derivative of $f$ at $w$ is equal to the $k$th derivative of $f$ at $w$ divided by $(2 \pi i)^k$ times the contour integral of $f(u)/(u - w)^{k + 1}$ around the... |
If $f$ is a continuous function on the closed ball of radius $r$ centered at $z$ and $f$ is holomorphic on the open ball of radius $r$ centered at $z$, then the contour integral of $f(u)/(u - w)^2$ around the circle of radius $r$ centered at $z$ is equal to $2\pi i$ times the derivative of $f$ at $w$. |
If $f$ is holomorphic on a ball $B(z,r)$, then the power series $\sum_{n=0}^\infty \frac{f^{(n)}(z)}{n!}(w-z)^n$ converges to $f(w)$ for all $w \in B(z,r)$. |
If $f$ is a holomorphic function on the complex plane and $f(z) \to 0$ as $z \to \infty$, then $f(z) = 0$ for all $z$. |
If $f$ is a holomorphic function on $\mathbb{C}$ and $f$ converges to $l$ at infinity, then $f$ is constant. |
If $f$ is a holomorphic function on $\mathbb{C}$ and $f$ is unbounded, then $f$ has a zero. |
If $a_0 = 0$ or $a_i \neq 0$ for some $i \in \{1, \ldots, n\}$, then there exists a complex number $z$ such that $\sum_{i=0}^n a_i z^i = 0$. |
If a sequence of holomorphic functions converges uniformly on a ball to a function $g$, then $g$ is holomorphic. |
If a sequence of functions $f_n$ converges uniformly to a function $f$ on a ball $B(z,r)$, and each $f_n$ is differentiable at $z$, then $f$ is differentiable at $z$ and the derivative of $f$ at $z$ is the limit of the derivatives of $f_n$ at $z$. |
Suppose $f_n$ is a sequence of holomorphic functions on an open set $S$, and $f_n$ converges uniformly to $g$ on every compact subset of $S$. Then $g$ is holomorphic on $S$. |
If $f_n$ is a sequence of functions that are differentiable on an open set $S$, and if $f_n$ converges uniformly to a function $g$ on $S$, then $g$ is differentiable on $S$ and the derivative of $g$ is the limit of the derivatives of $f_n$. |
Suppose $f_n$ is a sequence of functions defined on an open set $S$ and $h$ is a summable sequence of nonnegative real numbers. Suppose that for each $n$, $f_n$ is differentiable on $S$ and that for each $x \in S$, the sequence $f_n(x)$ is dominated by $h$. Then there exists a function $g$ defined on $S$ such that for ... |
Suppose $f_n$ is a sequence of functions defined on an open set $S$ such that $f_n$ converges uniformly to a function $f$ on every compact subset of $S$. Suppose also that $f_n$ is differentiable on $S$ for each $n$, and that the sequence of derivatives $f_n'$ converges uniformly to a function $g$ on every compact subs... |
Suppose $f_n$ is a sequence of complex-valued functions defined on an open set $S$, and $f_n'$ is the derivative of $f_n$. Suppose that for each $n$, $f_n'$ is continuous on $S$. Suppose that for each $x \in S$, there exists a sequence $h_n$ of nonnegative real numbers such that $h_n$ converges to $0$ and $|f_n(y)| \le... |
Suppose $f_n$ is a sequence of complex-valued functions defined on an open set $S$, and each $f_n$ is differentiable. If for each $x \in S$, there exists a sequence $h_n$ of nonnegative real numbers such that $h_n$ converges to $0$ and $|f_n(y)| \leq h_n$ for all $y$ in some neighborhood of $x$, then the series $\sum_{... |
If the power series $\sum_{n=0}^\infty a_n r^n$ converges, then there exists a function $g$ and its derivative $g'$ such that for all $z$ with $|z| < r$, the power series $\sum_{n=0}^\infty a_n z^n$ converges to $g(z)$ and the power series $\sum_{n=0}^\infty n a_n z^{n-1}$ converges to $g'(z)$, and $g'(z)$ is the deriv... |
Suppose $a_n$ is a sequence of complex numbers such that $\sum_{n=0}^\infty a_n r^n$ converges for some $r > 0$. Then there exists a function $g$ and its derivative $g'$ such that for all $z$ in the ball of radius $r$ centered at $w$, the power series $\sum_{n=0}^\infty a_n (z - w)^n$ converges to $g(z)$ and the power ... |
If the power series $\sum_{n=0}^\infty a_n(w-z)^n$ converges to a function $f$ for all $w$ in a ball around $z$, then $f$ is holomorphic in that ball. |
A function is holomorphic on a ball if and only if its Taylor series converges to the function on the ball. |
If the power series $\sum_{n=0}^\infty a_n(w-z)^n$ converges to $f(w)$ for all $w$ in the ball of radius $r$ centered at $z$, then $f$ is analytic on the ball. |
A function $f$ is analytic on a ball $B(z,r)$ if and only if its Taylor series converges to $f$ on $B(z,r)$. |
If two holomorphic functions agree on the derivatives at a point, then they agree on a ball around that point. |
If $f$ is a holomorphic function on a ball centered at $z$ and all of its derivatives at $z$ are zero, then $f$ is zero on the ball. |
If $f$ is a holomorphic function on a connected open set $S$ and all of its derivatives vanish at some point $z \in S$, then $f$ vanishes on all of $S$. |
If two holomorphic functions agree on all derivatives at a point in a connected open set, then they are equal on the whole set. |
If $f$ is a holomorphic function on a connected open set $S$ and all of its derivatives vanish at some point $z \in S$, then $f$ is constant on $S$. |
Suppose $f$ is holomorphic on a set $S$ and $a$ is an interior point of $S$. Then the function $F$ defined by $F(z) = \frac{f(z) - f(a)}{z - a}$ if $z \neq a$ and $F(a) = f'(a)$ is holomorphic on $S$. |
Suppose $g$ is holomorphic on a set $S$ and $a$ is an interior point of $S$. Suppose that for all $z \in S - \{a\}$, we have $g(z) = (z - a)f(z)$. Then the function $f(z) - \frac{g(a)}{z - a}$ is holomorphic on $S$. |
If $f$ is holomorphic on an open set $S$, then the function $g$ defined by $g(z) = \frac{f(z) - f(a)}{z - a}$ if $z \neq a$ and $g(a) = f'(a)$ is holomorphic on $S$. |
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