Folklore Theorem Research Articles

On a Theorem of Lafforgue

Abstract We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.

One-sided derivative of distance to a compact set

We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point γ(t) on a geodesic γ and a compact set K is a right-differentiable function of t. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles.

Vector-valued holomorphic functions in several variables

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

Spectral continuity for aperiodic quantum systems: Applications of a folklore theorem

This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson–Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay–Rudin–Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.

Analytical techniques for linear response formula of equilibrium states

In this paper, we expose some of the analytical framework used to prove the differentiability of equilibrium states probabilities with respect to the dynamics. We explore the relationship between Birkhoff metrics in cones and their corresponding anisotropic spaces and prove some useful folklore theorems. We apply that framework to revisit several examples recently studied in the research works about equilibrium states.

A note on effective descent for overconvergent isocrystals

In this short note we explain the proof that proper surjective and faithfully flat maps are morphisms of effective descent for overconvergent isocrystals. We then show how to deduce the folklore theorem that for an arbitrary variety over a perfect field of characteristic p, the Frobenius pull-back functor is an equivalence on the overconvergent category.

Gradient flow line near birth–death critical points

Near a birth–death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shift. The proof is based on the Whitney normal form, a Conley index construction, and an adiabatic limit analysis for an associated fast–slow differential equation.

Does $\ell _{p}$ -Minimization Outperform $\ell _{1}$ -Minimization?

In many application areas ranging from bioinformatics to imaging, we are faced with the following question: can we recover a sparse vector $x_{o} \in \mathbb {R}^{N}$ from its undersampled set of noisy observations $y \in \mathbb {R}^{n}$ , $y= A x_{o}+ w$ . The last decade has witnessed a surge of algorithms and theoretical results to address this question. One of the most popular schemes is the $\ell _{p}$ -regularized least squares given by the following formulation: $\hat x(\gamma ,p ) \in {\arg \min }_{x}~({1}/{2})\| {y - Ax} \|_{2}^{2} + \gamma {\| x \|_{p}^{p}}$ , where $p \in [{0, 1}]$ . Among these optimization problems, the case $p = 1$ , also known as LASSO, is the best accepted in practice, for the following two reasons. First, thanks to the extensive studies performed in the fields of high-dimensional statistics and compressed sensing, we have a clear picture of LASSO’s performance. Second, it is convex and efficient algorithms exist for finding its global minima. Unfortunately, neither of the above two properties hold for $0 \leq p . However, they are still appealing because of the following folklores in the high-dimensional statistics. First, $\hat x(\gamma , p )$ is closer to $x_{o}$ than $\hat {x}(\gamma ,1)$ . Second, if we employ iterative methods that aim to converge to a local minima of $ {\arg \min }_{x}~({1}/{2})\| {y - Ax} \|_{2}^{2} + \gamma {\| x \|_{p}^{p}}$ , then under good initialization, these algorithms converge to a solution that is still closer to $x_{o}$ than $\hat {x}(\gamma ,1)$ . In spite of the existence of plenty of empirical results that support these folklore theorems, the theoretical progress to establish them has been very limited. This paper aims to study the above-mentioned folklore theorems and establish their scope of validity. Starting with approximate message passing (AMP) algorithm as a heuristic method for solving $\ell _{p}$ -regularized least squares, we study the following questions. First, what is the impact of initialization on the performance of the algorithm? Second, when does the algorithm recover the sparse signal $x_{o}$ under a “good” initialization? Third, when does the algorithm converge to the sparse signal regardless of the initialization? Studying these questions will not only shed light on the second folklore theorem, but also lead us to the answer the first one, i.e., the performance of the global optima $\hat x(\gamma , p )$ . For that purpose, we employ the replica analysis 1 to show the connection between the solution of AMP and $\hat {x}(\gamma , p)$ in the asymptotic settings. This enables us to compare the accuracy of $\hat x(\gamma ,p )$ and $\hat x(\gamma ,1 )$ . In particular, we will present an accurate characterization of the phase transition and noise sensitivity of $\ell _{p}$ -regularized least squares for every $0 \leq p \leq 1$ . Our results in the noiseless setting confirm that $\ell _{p}$ -regularized least squares (if $\gamma $ is tuned optimally) exhibits the same phase transition for every $0 \leq p and this phase transition is much better than that of LASSO. Furthermore, we show that in the noisy setting, there is a major difference between the performance of $\ell _{p}$ -regularized least squares with different values of $p$ . For instance, we will show that for very small and very large measurement noises, $p = 0$ and $p = 1$ outperform the other values of $p$ , respectively. 1 Replica method is a widely accepted heuristic in statistical physics for analyzing large disordered systems.

Finsler geodesics, periodic Reeb orbits, and open books

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books.

Double branched covers of theta-curves

We prove a folklore theorem of Thurston, which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot [Formula: see text] is prime, if and only if lifting the third arc of the theta-curve to the double branched cover over [Formula: see text] produces a prime knot. We apply this result to Kinoshita’s theta-curve.

(0, 2) trialities

Motivated by the connection between 4-manifolds and 2d N=(0,2) theories, we study the dynamics of a fairly large class of 2d N=(0,2) gauge theories. We see that physics of such theories is very rich, much as the physics of 4d N=1 theories. We discover a new type of duality that is very reminiscent of the 4d Seiberg duality. Surprisingly, the new 2d duality is an operation of order three: it is IR equivalence of three different theories and, as such, is actually a triality. We also consider quiver theories and study their triality webs. Given a quiver graph, we find that supersymmetry is dynamically broken unless the ranks of the gauge groups and flavor groups satisfy stringent inequalities. In fact, for most of the graphs these inequalities have no solutions. This supports the folklore theorem that generic 2d N=(0,2) theories break supersymmetry dynamically.

Contact Anosov flows on hyperbolic 3–manifolds

Geodesic flows of Riemannian or Finsler manifolds have been the only known contact Anosov flows. We show that even in dimension 3 the world of contact Anosov flows is vastly larger via a surgery construction near an E ‐transverse Legendrian link that encompasses both the Handel‐Thurston and Goodman surgeries and that produces flows not topologically orbit equivalent to any algebraic flow. This includes examples on many hyperbolic 3‐manifolds, any of which have remarkable dynamical and geometric properties. To the latter end we include a proof of a folklore theorem from 3‐manifold topology: In the unit tangent bundle of a hyperbolic surface, the complement of a knot that projects to a filling geodesic is a hyperbolic 3‐manifold.

Folklore Theorems, Implicit Maps, and Indirect Inference

The delta method and continuous mapping theorem are among the most extensively used tools in asymptotic derivations in econometrics. Extensions of these methods are provided for sequences of functions that are commonly encountered in applications and where the usual methods sometimes fail. Important examples of failure arise in the use of simulation-based estimation methods such as indirect inference. The paper explores the application of these methods to the indirect inference estimator (IIE) in first order autoregressive estimation. The IIE uses a binding function that is sample size dependent. Its limit theory relies on a sequence-based delta method in the stationary case and a sequence-based implicit continuous mapping theorem in unit root and local to unity cases. The new limit theory shows that the IIE achieves much more than (partial) bias correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the autoregressive coefficient is in the locality of unity, reducing the bias and the variance of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in spite of the fact that the IIE is a continuously differentiable function of the MLE, the limit distribution of the IIE is not simply a scale multiple of the MLE, but depends implicitly on the full binding function mapping. The unit root case therefore represents an important example of the failure of the delta method and shows the need for an implicit mapping extension of the continuous mapping theorem.

NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

Hyperbolic sets with nonempty interior

In this paper we study hyperbolic sets with nonempty interior. Weprove the folklore theorem that every transitive hyperbolic setwith interior is Anosov. We also show that on a compact surfaceevery locally maximal hyperbolic set with nonempty interior isAnosov. Finally, we give examples of hyperbolic sets with nonemptyinterior for a non-Anosov diffeomorphism.

THEORETICAL PEARL: A simple proof of a folklore theorem about delimited control

We formalize and prove the folklore theorem that the static delimited-control operators shift and reset can be simulated in terms of the dynamic delimited-control operators control and prompt. The proof is based on small-step operational semantics.

A Simple Proof of a Folklore Theorem about Delimited Control

We formalize and prove the folklore theorem that the static delimited-control operators shift and reset can be simulated in terms of the dynamic delimited-control operators control and prompt. The proof is based on small-step operational semantics.

THEORETICAL PEARLS: A bargain for intersection types: a simple strong normalization proof

This pearl gives a discount proof of the folklore theorem that every strongly $\beta$-normalizing $\lambda$-term is typable with an intersection type. (We consider typings that do not use the empty intersection $\omega$ which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. This is a simplification over existing proofs that consider any longest reduction path. The choice of reduction strategy avoids the need for weakening or strengthening of type derivations. The proof becomes a bargain because it works for more intersection type systems, while being simpler than existing proofs.

A Simple Proof of a Folklore Theorem about Delimited Control

We formalize and prove the folklore theorem that the static delimited-control operators shift and reset can be simulated in terms of the dynamic delimited-control operators control and prompt. The proof is based on a small-step operational semantics that takes the form of an abstract machine.

Representations with scalarK-types and applications

We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about spherical representationas to representations with scalarK-types. We then apply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which are unipotent in the sense of Vogan.