# Questions tagged [duality]

Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.

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### Do Poincaré duality algebras need to be defined over a field?

I asked the below question here on MSE, but after some time and a bounty offering I have not received an answer. A graded commutative, connected $\mathbb{k}$-algebra $A$ is called a Poincaré duality ...
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### Poincare duality-differential geometry

Let $M$ be a smooth and compact manifold with boundary $\partial M = X \times F$ on which the structure of a smooth locally trivial bundle $$\pi: \partial M \longrightarrow X$$ where the $X$ ...
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### What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
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### Derivation of dual for infinite linear program

I'm reading the section on Linear Programming in Barbu and Precupanu's Convexity and Optimization in Banach Spaces, and had a couple of questions concerning their derivation of the dual problem for ...
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### Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?

Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...
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### Proving the existence of a dual for an infinite linear program

I am concerned with proving the existence of the dual of an infinite linear program. In addition to the writings of Rockafellar, Luenberger, and Boyd & Vandenberghe on: subdifferentials, Legendre-...
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### Which topological spaces admit embeddings into Euclidean spaces

I'm interested in the dual question to: continuous images of open intervals, about surjections onto open intervals. Namely, if $X$ is a topological space, when can we guarantee that there exists a ...
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### Why are Serre functors always exact?

Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...
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### Convert the maximization problem with log objective function into exponential cone programming

Below is the problem definition and the detailed process I converted it into a formulation solvable by exponential cone solver. There are other constraints but for simplicity I omit them. However my ...
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### References on dualities on quantum field theory for mathematicians

Dualities on QFT–also called Quantum Field Theory dynamics–is a huge and fundamental research area. However, despite underpinning major mathematical breakthroughs such as the work of Kapustin and ...
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### I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module

I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...
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### If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am ...
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### How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?

$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\hom}{\mathcal{Hom}} \DeclareMathOperator{\ox}{\mathcal{O}_X}$Let $f:X \to Y$ be a proper morphism. In section 6.4. of Liu's book he introduces ...
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### What is the dual of generating Boolean subalgebra by subexpressions of a modal formula?

I am supposed to be answering this question rather than asking it but I really cannot figure out. There is a variation on Stone duality linking algebraic and (descriptive) Kripke semantics for (...
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### Divided power algebra is artinian as a module over the polynomial ring

I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow. In the paper Homological algebra on a complete intersection, with ...
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### Legendre transform on signed measure space

Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
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### Examples of isomorphic W* algebra with non-homeomorphic weak topology

Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
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### Reflection-invariant monomial ideals and Alexander duality

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...
### Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems. Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
This question was posted on MSE and got very little attention, so I'm also posting it here. Let $\mathcal{C}$ be a closed symmetric monoidal category and let \$PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...