# Micro Math Camp Definitions

Flashcards by Eric Andersen, updated more than 1 year ago
 Created by Eric Andersen about 5 years ago
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Flashcards on Micro Math Camp Definitions, created by Eric Andersen on 08/18/2015.

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 Question Answer Metric Space A pair $$\left(S, d \right)$$ where $$S$$ is a collection of points and $$d$$ is a distance function on $$S$$ that satisfies the properties of a metric. Cauchy-Schwarz Inequality For any vectors $$x, \, y$$ of an inner product space, $\left| \langle x, \, y \rangle \right|^2 \le \langle x, \, x \rangle \cdot \langle y, \, y \rangle.$ Equivalently, $\left| \langle x, \, y \rangle \right| \le \| x \| \cdot \| y \|.$ Discrete Metric The discrete metric is: $d \left(x, \, y \right) = \begin{cases} 1 \; \text{if } x \ne y, \\ 0 \; \text{if } x = y \end{cases}$ an $$\epsilon$$-ball of $$x$$ in $$S$$ $B_\epsilon \left( x \right) \equiv \left\{ y \in S \,|\, d \left( x,\,y \right) < \epsilon \right\}$ Open Set A set $$U \subseteq S$$ is open if $$\forall x \in U, \; \exists \epsilon > 0$$ such that $$B_\epsilon \left( x \right) \subseteq U$$ Metric Topology A metric topology of a metric space $$\left(S, \, d \right)$$ is the set of all open sets in $$S$$. That is: $\mathcal{T} \equiv \left\{ U \subseteq S \, | \, U \text{ is open} \right\}$ Power Set of $$X$$ The power set $$2^X$$ is the set of all subsets of $$X$$. Topological Space A set $$X$$ together with a set $$T \subseteq 2^X$$ that satisfies the three properties of a metric topology on S Closed Set A set $$U \subseteq S$$ is closed if $$S \setminus U$$ is open. Alternatively, a set $$C \subseteq S$$ is closed if $$\forall \left\{ x_n \right\}_{n=1}^{\infty} \subseteq C$$ for which $$\exists x \in S$$ such that $$x_n \rightarrow x, \, x \in C$$. Sequence A countably indexed collection of elements $$\left\{ x_n \right\}_{n=1}^{\infty} \subseteq S$$ Convergent Sequence A sequence converges to a limit point $$x \in S$$ if $$\forall \epsilon \in \mathbb{R}_+, \; \exists N_\epsilon$$ such that $$d \left(x_m, \, x \right) < \epsilon$$ for all $$m > N_\epsilon$$ Convergent Subsequence A sequence contains a convergent subsequence if there exists an index set $$N \subset \mathbb{N}$$ such that $$\left\{ x_n \right\}_{n \in \hat{N}} \rightarrow x$$ for some $$x \in S$$ Closure of a Set This is the smallest closed set containing $$C$$. $cl \left( C \right) = \bigcap \left\{ K \in 2^S \, | \, C \subseteq K; K \text{ closed.} \right\}$ Cauchy Sequence A sequence $$\left\{x_n\right\}_{n=1}^{\infty} \subseteq S$$ such that $$\forall \epsilon \in \mathbb{R}_+, \; \exists N_\epsilon \in \mathbb{N}$$ such that $$d \left(x_m,\,x_l\right) < \epsilon$$ for all $$m, l > N_\epsilon$$ Bounded Set A set $$U \subseteq S$$ is bounded if $$\exists M \in \mathbb{R}_+$$ such that $$\forall x,y \in U, \; d \left(x,\,y\right) \le M$$ Complete Set A set $$U \subseteq S$$ is complete if for every Cauchy sequence $$\left\{x_n\right\}_{n=1}^{\infty} \subseteq U, \; \exists x \in U$$ such that $$x_n \rightarrow x$$. Open Cover of a set $$K$$ A collection of sets $$\left\{ U_\alpha \right\}_{\alpha \in A}$$ such that $$U_\alpha \in \mathcal{T}$$ for all $$\alpha \in A$$ and $$K \subseteq \bigcup_{\alpha \in A} U_\alpha$$. Finite Subcover An open cover of $$K$$ admits a finite subcover if there exists a finite set $$A' \subseteq A$$ such that $$K \subseteq \bigcup_{\alpha \in A'} U_\alpha$$. Compact Set A subset $$K$$ of a metric space $$\left( S, \, d \right)$$ is compact if every open cover of $$K$$ has a finite subcover. Finite Intersection Property A collection $$\mathcal{A} \subseteq 2^S$$ has the finite intersection property if $$\bigcap \mathcal{A}' \ne \emptyset$$ for any finite, nonempty collection $$\mathcal{A}' \subseteq \mathcal{A}$$ Totally Bounded Metric Space A metric space $$S$$ is totally bounded if for any $$\epsilon > 0$$, there exists a finite collection of points $$x_1, \, \ldots, \, x_N$$ such that the open balls $$B_\epsilon \left( x_1 \right), \ldots, \, B_\epsilon \left(x_N\right)$$ cover $$S$$. Connected Metric Space A metric space $$S$$ is connected if there do not exist disjoint, nonempty open sets $$U$$ and $$U'$$ such that $$U \cup U' = S$$ Countably Infinite Set An infinite set $$S$$ is countably infinite if there exists a bijection $$f : S \rightarrow \mathbb{N}$$. Countable set A set is called countable if it is finite or countably infinite. Dense Set A set $$D$$ is dense in a metric space $$\left( S, \, d \right) \text{ if } cl \left( D \right) = S$$. Separable A metric space $$\left( S, \, d \right)$$ is separable if there exists a countable set $$D$$ that is dense in $$S$$. Continuity A function $$f : X \rightarrow Y$$ is continuous at $$x \in X \text{ if } \forall \left\{x_n\right\}_{n=1}^{\infty} \subseteq X$$ such that $$x_n \rightarrow x \in X$$, we have $$f \left( x_n \right) \rightarrow f \left( x \right)$$. Alternatively, if $$\forall \epsilon > 0, \, \exists \delta > 0$$ such that for all $$z \in X$$ such that $$d_X \left( x, \, z \right) < \delta, \, d_Y \left( f \left( x \right), \, f \left( z \right) \right) < \epsilon$$. Pointwise Convergence The sequence of functions $$\left\{f_n\right\}_{n=1}^{\infty}$$ converges pointwise to the function $$f \text{ if } \forall x \in X$$, the sequence $$f_n \left( x \right)$$ converges to $$f \left( x \right)$$. Uniform Convergence The sequence of functions $$\left\{f_n\right\}_{n=1}^{\infty}$$ converges pointwise to the function $$f \text{ if } \forall \epsilon > 0$$ there exists $$N_\epsilon$$ such that $$\forall x \in S \text{ and all } n \ge N_\epsilon, \, d \left( f_n \left(x \right), \, f \left( x \right) \right) < \epsilon$$. Convex Set Given a vector space $$\left( S, \, +, \, \cdot \right)$$, a set $$U \subseteq S$$ is convex if $$\forall x, \, y \in U \text{ and } \forall \alpha \in \left[0, \, 1 \right],$$ $\alpha x + \left(1-\alpha\right)y \in U$ Convex Hull of a Set $$co \left( U \right)$$ is the intersection of all convex sets containing $$U$$. Supremum Least Upper Bound Infinum Greatest Lower Bound $$i$$-th partial derivative $\lim_{h \rightarrow 0} \frac{ f \left( \tilde{x}_1, \ldots, \, \tilde{x}_i + h, \, \tilde{x}_{i+1}, \ldots, \, \tilde{x}_n \right) - f \left( \tilde{x}_1, \ldots, \, \tilde{x}_i, \, \tilde{x}_{i+1}, \ldots, \, \tilde{x}_n \right) }{h}$ Gradient $\nabla f \left( x \right) = \left[ \frac{ \partial f }{ \partial x_i} \right]$ is the $$N \times 1$$ vector of first partial derivatives of $$f \text{ at } x$$. Hessian Matrix of $$f \text{ at } x$$ matrix of well-defined partial and cross-partial derivatives Jacobian Matrix of $$f \text{ at } x$$ matrix of partial derivatives Differentiable Function If there is an $$M \times N$$ matrix $$Df\left(x\right)$$ such that for any sequence of vectors $$w \rightarrow 0$$ $\lim_{w \rightarrow 0} \frac{ \| f \left( x + w \right) - f \left( x \right) - Df\left(x\right)w \|}{\|w\|} = 0$ Taylor's Approximation $f\left(x\right) \approx f \left(x_0 \right) + \sum_{k=1}^{n} \frac{f^k \left(x_0 \right)}{k!} \left(x - x_0 \right)^k$ Concave Function A function $$f : A \rightarrow \mathbb{R}$$ defined on a convex set $$A \subset \mathbb{R}^N$$ is concave if and and only if $f \left( \alpha x' + \left(1-\alpha\right)x\right) \ge \alpha f\left(x'\right) + \left(1-\alpha\right)f\left(x\right)$ for all $$x, \, x' \in A$$ and all $$\alpha \in \left[0,\,1\right]$$ Convex Function A function $$f : A \rightarrow \mathbb{R}$$ defined on a convex set $$A \subset \mathbb{R}^N$$ is concave if and and only if $f \left( \alpha x' + \left(1-\alpha\right)x\right) \le \alpha f\left(x'\right) + \left(1-\alpha\right)f\left(x\right)$ for all $$x, \, x' \in A$$ and all $$\alpha \in \left[0,\,1\right]$$ Negative semidefinite matrix An $$N \times N$$ matrix $$A$$ is negative semidefinite if $$z'Az \le 0$$ for all $$z\in\mathbb{R}^{N}$$. A strict inequality implies negative definiteness. Positive semidefinite matrix An $$N \times N$$ matrix $$A$$ is positive semidefinite if $$z'Az \ge 0$$ for all $$z\in\mathbb{R}^{N}$$. A strict inequality implies positive definiteness. $$t$$-upper contour set The set of all points in $$A$$ with values greater than $$t$$. $C_t^+ \left(f\right) = \left\{x\in A \, | \, f\left(x\right) \ge t \right\}$ $$t$$-lower contour set The set of all points in $$A$$ with values less than $$t$$. $C_t^- \left(f\right) = \left\{x\in A \, | \, f\left(x\right) \le t \right\}$ Quasiconcave Function A function $$f: A \rightarrow \mathbb{R}$$ is quasiconcave if its upper contour sets $$C_t^+ \left( f \right)$$ are convex for any $$t\in\mathbb{R}$$. That is, $$\forall t\in\mathbb{R}$$ and $$\forall\, x,\,x'\in A$$: $\min \left\{ f\left(x\right),\, f\left(x'\right) \right\} \ge t \implies f\left(\alpha x + \left[1-\alpha\right]x'\right) \ge t.$ Equivalently, $$f$$ is quasiconcave if and only if $f \left(\alpha x + \left(1-\alpha\right)x^\prime\right) \ge \min \left\{ f\left(x\right),\, f\left(x'\right) \right\}$ for all $$x, \, x^\prime \in A$$ and $$\alpha \in \left[0, \, 1 \right]$$. Any concave function is quasiconcave. Any increasing function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is quasiconcave. Quasiconvex Function A function $$f: A \rightarrow \mathbb{R}$$ is quasiconvex if its lower contour sets $$C_t^- \left( f \right)$$ are convex for any $$t\in\mathbb{R}$$. That is, $$\forall t\in\mathbb{R}$$ and $$\forall\, x,\,x'\in A$$: $\max \left\{ f\left(x\right),\, f\left(x'\right) \right\} \le t \implies f\left(\alpha x + \left[1-\alpha\right]x'\right) \le t.$ Equivalently, $$f$$ is quasiconex if and only if $f \left(\alpha x + \left(1-\alpha\right)x^\prime\right) \le \max \left\{ f\left(x\right),\, f\left(x'\right) \right\}$ for all $$x, \, x^\prime \in A$$ and $$\alpha \in \left[0, \, 1 \right]$$. Local Maximizer/Minimizer of $$f$$ A local maximizer is a vector $$x^\ast \in A$$ where for a $$\delta > 0$$ $f\left(x^\ast\right) \ge f\left(x\right) \text{ for all } x \in B_\delta \left(x^\ast\right)$. A local minimizer is defined by reversing the inequality. Global Maximizer/Minimizer of $$f$$ A global maximizer is a vector $$x^\ast \in A$$ where for a $$\delta > 0$$ $f\left(x^\ast\right) \ge f\left(x\right) \text{ for all } x \in A$. A local minimizer is defined by reversing the inequality. Critical Points of $$f$$ All $$x \in A$$ such that $\nabla f \left( x \right) = 0$ Complementary Slackness requires $$\lambda_k^\ast$$ = 0 if the constraint $h_i \left(x\right) \le c_i$ does not bind Constraint Qualification An optimization problem satisfies constraint qualification if and only if the $$M \times N$$ matrix $G \left( x^\ast \right) = \left[ \begin{matrix} \frac{\partial g_1 \left( x^\ast \right)}{\partial x_1} & \ldots & \frac{\partial g_1 \left( x^\ast \right)}{\partial x_N} \\ \vdots & \ddots & \vdots \\ \frac{\partial g_M \left( x^\ast \right)}{\partial x_1} & \ldots & \frac{\partial g_M \left( x^\ast \right)}{\partial x_N} \\ \end{matrix} \right]$ satisfies $$\operatorname{rank} \left(G\right) = M$$ at an optimum $$x^\ast \in A$$. Correspondence Let $$X$$ and $$Y$$ be metric spaces. The correspondence $$\Gamma : X \rightrightarrows Y$$ is a function from $$X$$ to $$2^Y \setminus \emptyset$$. Upper Hemicontinuous Correspondence A correspondence $$\Gamma$$ is upper hemicontinuous at $$x \in X$$ if for every open set $$O \subseteq Y$$ with $$\Gamma \left( x \right) \subseteq O$$, there exists $$\delta > 0$$ such that $$\Gamma \left( B_\delta \left( x \right) \right) \subseteq O$$. $$\Gamma$$ is upper hemicontinuous if this holds for every $$x \in X$$. Compact-Valued Correspondence $$\Gamma$$ A correspondence $$\Gamma : X \rightrightarrows Y$$ is compact valued if $$\Gamma \left( x \right)$$ is compact in $$Y$$ for each $$x$$. Closed-Valued Correspondence $$\Gamma$$ A correspondence $$\Gamma : X \rightrightarrows Y$$ is closed valued if $$\Gamma \left( x \right)$$ is closed in $$Y$$ for each $$x$$. Convex-Valued Correspondence $$\Gamma$$ A correspondence $$\Gamma : X \rightrightarrows Y$$ is convex valued if $$\Gamma \left( x \right)$$ is convex in $$Y$$ for each $$x$$. Lower Hemicontinuous Correspondence A correspondence $$\Gamma : X \rightrightarrows Y$$ is lower hemicontinuous at $$x \in X$$ if for every open set $$O \subseteq Y$$ with $$\Gamma \left( x \right) \cap O \ne \emptyset$$, there exists a $$\delta > 0$$ such that $$\Gamma \left( x' \right) \cap O \ne \emptyset$$ for all $$x' \in B_\delta \left(x\right)$$. $$\Gamma$$ is lower hemicontinuous if this holds at each $$x \in X$$. Upper Inverse Image $\Gamma^{-1} \left( O \right) \equiv \left\{ x \in X \, | \, \Gamma \left(x\right) \subseteq O \right\}$ Lower Inverse Image $\Gamma_{-1} \left( O \right) \equiv \left\{ x \in X \, | \, \Gamma \left(x\right) \cap O \ne \emptyset \right\}$ Continuous Correspondence A correspondence $$\Gamma$$ is called continuous if it is both upper and lower hemicontinuous. Closed Graph Property A correspondence is closed at $$x \in X$$ if for all sequences $$x_n \rightarrow x \text{ and } y_n \rightarrow y$$, we have $$y \in \Gamma \left( x \right)$$ whenever $$y_n \in \Gamma \left( x_n \right)$$ for all $$n$$. If $$\Gamma$$ is closed at every $$x$$, then it has the closed graph property. Hyperplane in $$\mathbb{R}^{N}$$ A $$N-1$$ dimensional subset of $$\mathbb{R}^{N}$$ that can be described by a single linear equation: $H \left(p,\,r\right) = \left\{ x \in \mathbb{R}^N \, | \, p \cdot x = r \right\}$ where $$p$$ is a non-zero vector in $$\mathbb{R}^N$$ and $$r$$ is a real number. Separating Hyperplane Let $$A, B \subset \mathbb{R}^N, \, p \in \mathbb{R}^N \setminus \left\{ 0 \right\}$$. The hyperplane $$H \left( p, \, r \right)$$ separates $$A$$ and $$B$$ if $$p \cdot r \ge r \forall x \in A$$ and $$p \cdot y \le r \forall y \in B$$ Proper Separation of Sets The separation of sets $$A, \, B \subset \mathbb{R}^N \text{ by } H \left(p, \, r \right)$$ is proper if there exists $$x \in A \text{ and } y \in B \text{ such that } p \cdot x \ne p \cdot y$$ Strict Separation of Sets The hyperplane $$H \left(p, \, r \right)$$ strictly separates sets $$A, \, B \subset \mathbb{R}^N$$ if $p \cdot x > r \text{ for all} x \in A \text{ and } p \cdot y < r \text{ for all } y \in B$ Partially-Ordered Set (poset) A pair $$\left(X, \succsim \right)$$ where $$X$$ is a set and $$\succsim$$ is a partial order. Partial Order A binary relation $$\succsim \subseteq X \times X$$ that satisfies: (a) reflexivity: $$\forall x \in X, \, x \succsim x$$, (b) antisymmetry: $$\forall x, y \in X$$ if $$x \succsim y$$ and $$y \succsim x$$ then $$x = y$$, and (c) transitivity: $$\forall x, y, z \in X$$ if $$x \succsim y$$ and $$y \succsim z$$ then $$x \succsim z$$. Chain A poset is called a chain if the partial order $$\succsim$$ is complete on $$X$$. That is, for all $$x, y \in X$$ either $$x \succsim y$$ or $$y \succsim x$$. Bounds of a Poset Let $$E \subseteq X$$ be a subset of a poset. If we have $$x^\prime \succsim x$$ for all $$x \in E$$, then $$x^\prime$$ is an upper bound for E; similarly if $$x \succsim x^\prime$$ for all $$x \in E$$ then $$x^\prime$$ is a lower bound for $$E$$. Supremum of $$E$$ in $$X$$ Denoted $$\sup_X \left(E \right)$$, the supremum of $$E$$ in $$X$$ is an element $$x^\ast \in X$$ such that $$x^\ast$$ is an upper bound of $$E$$ and if $$x^\prime$$ is an upper bound of $$E$$, then $$x^\prime \succsim x^\ast$$. Infimum of $$E$$ in $$X$$ Denoted $$\inf_X \left(E \right)$$, the infimum of $$E$$ in $$X$$ is an element $$x_\ast \in X$$ such that $$x_\ast$$ is a lower bound of $$E$$ and if $$x'$$ is a lower bound of $$E$$, then $$x_\ast \succsim x'$$. Join of $$x \text{ and } y$$ For a two element subset $$\left\{x,\,y\right\} \subseteq X$$, the supremum of $$\left\{x,\,y\right\}$$ in $$X$$ is called the join and is denoted $$x \vee y$$. Meet of $$x \text{ and } y$$ For a two element subset $$\left\{x,\,y\right\} \subseteq X$$, the infimum of $$\left\{x,\,y\right\}$$ in $$X$$ is called the meet and is denoted $$x \wedge y$$. Lattice A poset $$\left(X,\,\succsim \right)$$ such that $$x \vee y$$ and $$x \wedge y$$ exist for every pair of elements $$x,y \in X$$. Sublattice Let $$\left(X,\,\succsim\right)$$ be a lattice. A set $$E \subseteq X$$ is called a sublattice of $$X$$ if $$\forall x, y \in E$$, we have $$\sup_X \left(\left\{x,\,y\right\}\right) \in E$$ and $$\inf_X \left( \left\{x,\,y\right\}\right) \in E$$. That is, $$E$$ contains the meet and join of all combinations of its elements. The collection of all nonempty sublattices for a lattice $$\left(X, \, \succsim \right)$$ is denoted $$\mathcal{L} \left( X \right)$$. Induced Set Ordering on $$\mathcal{L} \left( X \right)$$ Denoted $$\sqsubseteq$$ and defined by $$A \sqsubseteq B$$ if and only if for all $$x \in A \text{ and } y \in B$$, we have $$x \wedge y \in A$$ and $$x \vee y \in B$$. Complete Lattice A lattice $$\left(X, \, \succsim \right)$$ is called complete if $$\sup_X \left( E \right)$$ and $$\inf_X \left( E \right)$$ both exist for every nonempty subset $$E \subseteq X$$. Subcomplete Sublattice A sublattice $$E$$ of $$X$$ is subcomplete if $$\sup_X \left( F \right)$$ and $$\inf_X \left( F \right)$$ are both in $$E$$ for every nonempty set $$F \subseteq E$$. Increasing and Decreasing Monotone Functions Let $$\left(X,\,\succsim_X\right)$$ and $$\left( Y, \, \succsim_Y \right)$$ be posets. A function $$f \,: \, X \rightarrow Y$$ is increasing if $$x \succsim_X x' \implies f \left(x\right) \succsim_Y f\left(x'\right)$$ for all $$x, x' \in X$$. A function $$f \,: \, X \rightarrow Y$$ is decreasing if $$x \succsim_X x' \implies f \left(x'\right) \succsim_Y f \left( x \right)$$ for all $$x, x' \in X$$. The function is monotone if it is either increasing or decreasing. Increasing Differences Let $$X$$ and $$T$$ be posets and let $$S \subseteq X \times T$$. A function $$f \, : \, S \rightarrow \mathbb{R}$$ has increasing differences on $$S$$ if $$f \left(x, \, t'\right) - f\left(x, \, t \right)$$ is increasing in $$x$$ on the set $$\left\{ x \in X \, | \, \left( x, \, t' \right) \in S \text{ and } \left(x, \, t \right) \in S \right\}$$ for all $$t' \succ t$$. Supermodular Function Let $$X$$ be a lattice. A function $$f \, : \, X \rightarrow \mathbb{R}$$ is called supermodular iff $f \left( x \right) + f \left( x' \right) \le f \left( x \vee x' \right) + f \left( x \wedge x' \right)$ for all $$x, x' \in X$$. Quasisupermodular Function Let $$X$$ be a lattice and $$Y$$ be a poset. A function $$f : \, X \rightarrow Y$$ is quasisupermodular if $f \left( x \right) \succsim \left( \succ \right) f \left( x \wedge x' \right) \implies f \left( x \vee x' \right) \succsim \left( \succ \right) f \left( x' \right)$ for all $$x, \, x' \in X$$. Single Crossing Property Let $$X, \, T, \text{ and } Y$$ be posets and $$S \subseteq X \times T$$. A function $$f : \, S \rightarrow Y$$ satisfies the single crossing property on $$S$$ if $f \left( x', \, t \right) \succsim \left( \succ \right) f \left( x, \, t \right)$ $\implies$ $f \left( x', \, t' \right) \succsim \left( \succ \right) \, f \left( x, \, t' \right)$ for all $$x' \succ x$$ and $$t' \succ t$$ with $$\left\{x, \, x'\right\} \times \left\{t, \, t' \right\} \subseteq S$$.

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