# Questions tagged [duality-theorems]

For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.

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We have primal, minimize $z = 3u_1 + 0.5u_2$ subject to $$u_1 - 2u_2 \leq 4 \\ u_1 + u_2 \leq 2 \\ u_1, u_2 \geq 0$$ I found the dual \text{max: } z' = 4v_1 + v_2 \\ \text{subject to: } \\ v_1 + ... • 21 0 votes 0 answers 249 views ### proof of duality in projective geometry I am looking for a proof for duality principle in projective geometry. There is an axiomatic and then a homogeneous coordinates development of projective geometry where dual of lines to points and ... • 1,191 0 votes 1 answer 55 views ### Dot Products and Linear Progamming Let v_{1}, \ldots, v_{m} and w be vectors in \mathbb{R}^{n}. Suppose that whenever d \in \mathbb{R}^{n} satisfies v_{i} \cdot d \leq 0 for i = 1, \ldots, m, it also satisfies w \cdot d \... 7 votes 1 answer 218 views ### Yoneda's lemma: group morphisms give Hopf-algebra morphisms Let k be a commutative ring. Let \text{Alg} be the category of commutative k-algebras and \text{CHopf} the category of commutative Hopf-algebras. Let us also write [\text{Alg}, \text{Grp}] ... 6 votes 1 answer 264 views ### Learning roadmap and prerequisites for Isbell duality I'm looking for a roadmap to learning about Isbell duality. I know a reasonable amount about several of the "specific" dualities (Gelfand duality, AffSch - CRing, frames - locales, etc), ... • 177 1 vote 0 answers 37 views ### Dual of \{\nabla h, \ h \in W^{1,s}(\Omega)\} in (L^s(\Omega))^3. Let \Omega be a bounded regular domain of \mathbb{R}^d. Define the following space : X:=\{\nabla h, \ h \in W^{1,s}(\Omega)\} \subset (L^s(\Omega))^3. Can I say that the dual space of X is the ... • 2,078 6 votes 0 answers 127 views ### Hodge star operator and "Serre duality" I am familiar with the Hodge star operator or Hodge duality in the theory of finite-dimensional differentiable manifolds, which gives an isomorphism \star:\Omega^{i}(M)\longrightarrow\Omega^{n-i}(M) ... • 1,482 2 votes 0 answers 53 views ### How to formulate principle of duality in projective geometry in terms of category theory? In projective geometry, the principle of duality states that any theorem that holds for an incidence structure (P, L, I), where P are the points, L are the lines and I \subseteq P \times L is ... • 446 0 votes 1 answer 529 views ### What happens to an equality constraint in the primal when you translate to the dual? \begin{align} \text{maximize} &\quad x_1 +3x_2 & & \\ \text{subject to} &\quad x_1 - x_2 = 2 \\ & -2x_1 +3x_2 \ge 5 \\ & \quad\quad\quad ... • 1,553 1 vote 1 answer 29 views ### Rewrite \min _{z \in \mathbb{R}^{n}}\|z\|_{1} s.t. B z=d as an equivalent standard LP problem. I want to rewrite the following problem \min _{z \in \mathbb{R}^{n}}\|z\|_{1} s.t. B z=d \ where\ \boldsymbol{B} \in \mathbb{R}^{\boldsymbol{m} \times \boldsymbol{n}} and \ \boldsymbol{d} \in \...
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In "Categories and Sheaves" by Kashiwara and Schapira on the page 101, the definition of the dual pair in tensor category is given and in the next theorem it is shown that if $(X,Y)$ is a ...