Fan Theorem Research Articles

The Fan Theorem, its strong negation, and the determinacy of games

The Fan Theorem, its strong negation, and the determinacy of games

A minimax approach to duality for linear distributional sensitivity testing

We consider the dual formulation of the problem of finding the maximum of E ν [ f ( X ) ] , where ν is allowed to vary over all the probability measures on a Polish space X for which d c ( μ , ν ) ≤ r , with d c an optimal transport distance, f a real-valued function on X satisfying some regularity, μ a ‘baseline’ measure and r ≥ 0 . Whereas some derivations of the dual rely on Fenchel duality, applied on a vector space of functions in duality with a vector space of measures, we impose compactness on X to allow the use of the minimax theorem of Ky Fan, which does not require vector space structure.

The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map

Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

Coding of real‐valued continuous functions under WKL$\mathsf {WKL}$

AbstractIn the context of constructive reverse mathematics, we show that weak Kőnig's lemma () implies that every pointwise continuous function is induced by a code in the sense of reverse mathematics. This, combined with the fact that implies the Fan theorem, shows that implies the uniform continuity theorem: every pointwise continuous function has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier‐free axiom of choice.

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.

The application of a universal separating vector lemma to optimal sampled-data control problems with nonsmooth Mayer cost function

<p style='text-indent:20px;'>In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called <i>universal separating vector lemma</i> is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.</p>

Instance reducibility and Weihrauch degrees

We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in reverse constructive mathematics (formal Church's thesis, Brouwer's Continuity principle and Fan theorem, Excluded middle, Limited principle, Function choice, Markov's principle, etc.). We show that the instance degrees form a frame, i.e., a complete lattice in which finite infima distribute over set-indexed suprema. They turn out to be equivalent to the frame of upper sets of truth values, ordered by the reverse Smyth partial order. We study the overall structure of the lattice: the subobject classifier embeds into the lattice in two different ways, one monotone and the other antimonotone, and the $\lnot\lnot$-dense degrees coincide with those that are reducible to the degree of Excluded middle. We give an explicit formulation of instance degrees in a relative realizability topos, and call these extended Weihrauch degrees, because in Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees correspond precisely to Weihrauch degrees. The extended degrees improve the structure of Weihrauch degrees by equipping them with computable infima and suprema, an implication, the ability to control access to parameters and computation of results, and by generally widening the scope of Weihrauch reducibility.

Variational Quantum Singular Value Decomposition

Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.

König's lemma, weak König's lemma, and the decidable fan theorem

AbstractWe provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan theorem is not equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation.

Some improvements on the Ky Fan theorem for tensors

The improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

Decidable fan theorem and uniform continuity theorem with continuous moduli

AbstractThe uniform continuity theorem states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, is strictly stronger than the decidable fan theorem , but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that is equivalent to a similar principle for real‐valued functions on the Cantor space . These results extend Berger's [2] characterisation of for integer‐valued functions on and unify some characterisations of in terms of functions having continuous moduli.

A MARRIAGE OF BROUWER’S INTUITIONISM AND HILBERT’S FINITISM I: ARITHMETIC

AbstractWe investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics (i.e., which part is equiconsistent with $\textbf {PRA}$ or consistent provably in $\textbf {PRA}$ ) already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: (i) fan theorem for decidable fans but arbitrary bars; (ii) continuity principle and the axiom of choice both for arbitrary formulae; and (iii) $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO (except the choice along functions); but that limited principle of omniscience LPO makes the situation completely classical.

On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators.

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg-Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

On the existence and stability of solutions to stochastic equilibrium problems

In this paper we consider stochastic equilibrium problems involving parameter of probability measures. Employing the fixed point theorem of Knaster, Kuratowski, Mazurkiewicz, and Fan (KKM-Fan), conditions for the existence of solutions to such problems are established. We then propose new metric concepts on the underlying stochastic spaces and study some properties corresponding to these metrics. Afterwards, we study sufficient conditions for the solution mappings of such problems, that are closed, upper (lower) semicontinuous and continuous with respect to the mentioned metrics. Finally, the special cases of stochastic optimization problems are taken into account as the applications.

Characterising Brouwer’s continuity by bar recursion on moduli of continuity

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space $${{\mathbb {N}}}^{{\mathbb {N}}}$$ to the natural numbers $${\mathbb {N}}$$ which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar induction. These principles state that a certain kind of continuous function from $${{\mathbb {N}}}^{{\mathbb {N}}}$$ to $${\mathbb {N}}$$ admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space $${{\{ 0,1 \}}}^{{\mathbb {N}}}$$ using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types ( $${\mathsf {E}}\text {-}\mathsf {HA}^\omega $$ ), and those for the Baire space hold over $${\mathsf {E}}\text {-}\mathsf {HA}^\omega $$ extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction.

Constructive Decision via Redundancy-Free Proof-Search

We give a constructive account of Kripke–Curry’s method which was used to establish the decidability of implicational relevance logic (\(\mathbf{R}_{{\rightarrow }}\)). To sustain our approach, we mechanize this method in axiom-free Coq, abstracting away from the specific features of \(\mathbf{R}_{{\rightarrow }}\) to keep only the essential ingredients of the technique. In particular we show how to replace Kripke/Dickson’s lemma by a constructive form of Ramsey’s theorem based on the notion of almost full relation. We also explain how to replace König’s lemma with an inductive form of Brouwer’s Fan theorem. We instantiate our abstract proof to get a constructive decision procedure for \(\mathbf{R}_{{\rightarrow }}\) and discuss potential applications to other logical decidability problems.

Optimal control of systems governed by fractional Laplacian in the minimax framework

ABSTRACT We consider some optimal control problem governed by a class of differential equations with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes are proved. The proof of the main result relies on the variational structure of the problem. To show that differential equations with the Dirichlet fractional Laplacian have a weak solution we employ the Ky Fan Theorem.

SEPARATING THE FAN THEOREM AND ITS WEAKENINGS II

AbstractVarieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Earlier work showed all of these implications to be strict. Here we reprove one of the strictness results, using very different arguments. The technique used is a mixture of realizability, forcing in the guise of Heyting-valued models, and Kripke models.

Equivalence of bar induction and bar recursion for continuous functions with continuous moduli

We compare Brouwer's bar theorem and Spector's bar recursion for the lowest type in the context of constructive reverse mathematics. To this end, we reformulate bar recursion as a logical principle stating the existence of a bar recursor for every function which serves as the stopping condition of bar recursion. We then show that the decidable bar induction is equivalent to the existence of a bar recursor for every continuous function from NN to N with a continuous modulus. We also introduce fan recursion, the bar recursion for binary trees, and show that the decidable fan theorem is equivalent to the existence of a fan recursor for every continuous function from {0,1}N to N with a continuous modulus. The equivalence for bar induction holds over the extensional version of intuitionistic arithmetic in all finite types augmented with the characteristic principles of Gödel's Dialectica interpretation. On the other hand, we show the equivalence for fan theorem without using such extra principles.

Free choice sequences: A temporal interpretation compatible with acceptance of classical mathematics

This paper sketches a way of supplementing classical mathematics with a motivation for a Brouwerian theory of free choice sequences. The idea is that time is unending, i.e. that one can never come to an end of it, but also indeterminate, so that in a branching time model only one branch represents the ‘actual’ one. The branching can be random or subject to various restrictions imposed by the creating subject. The fact that the underlying mathematics is classical makes such perhaps delicate issues as the fan theorem no longer problematic. On this model, only intuitionistic logic applies to the Brouwerian free choice sequences, and there it applies not because of any skepticism about classical mathematics, but because there is no ‘end of time’ from the standpoint of which everything about the sequences can be decided.