Countable Sequence Research Articles

Staircase patterns in Hirzebruch surfaces

The ellipsoidal capacity function of a symplectic four manifold X measures how much the form on X must be dilated in order for it to admit an embedded ellipsoid of eccentricity z . In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions, X is said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfaces H_{b} formed by blowing up the projective plane with weight b . We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parameters b that are blocked, i.e. have no staircase, and an uncountable number of other values of b that do admit staircases. The remaining b -values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long as b is not one of these special rational values, any staircase in H_{b} has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations on H_{b} is echoed in the structure of the set of blocked b -intervals.

On Bivariate Distributions with Singular Part

There are many families of bivariate distributions with given marginals. Most families, such as the Farlie–Gumbel–Morgenstern (FGM) and the Ali–Mikhail–Haq (AMH), are absolutely continuous, with an ordinary probability density. In contrast, there are few families with a singular part or a positive mass on a curve. We define a general condition useful to detect the singular part of a distribution. By continuous extension of the bivariate diagonal expansion, we define and study a wide family containing these singular distributions, obtain the probability density, and find the canonical correlations and functions. The set of canonical correlations is described by a continuous function rather than a countable sequence. An application to statistical inference is given.

A probabilistic approach of ultraviolet renormalization in the boundary Sine-Gordon model

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field X defined on a subset of \({\mathbb {R}}^d\) by the exponential of its cosine, namely \(\exp (\alpha \smallint \cos (\beta X))\). It has gathered significant attention due to its importance in quantum field theory and to its connection with the study of log-gases in statistical mechanics. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral \(\smallint \cos (\beta X)\) can be defined properly when \(\beta ^2<d\) using the standard Wick normalization of \(\cos (\beta X)\), a more involved renormalization procedure is needed when \(\beta ^2\in [d,2d)\). In particular it exhibits a countable sequence of phase transitions accumulating to the left of \(\beta =\sqrt{2d}\), each transition corresponding to the addition of an extra term in the renormalization scheme. The final threshold \(\beta =\sqrt{2d}\) corresponds to the Kosterlitz–Thouless (KT) phase transition of the \(\log \)-gas. In this paper, we present a novel probabilistic approach to renormalization of the two-dimensional boundary (or 1-dimensional) Sine-Gordon model up to the KT threshold \(\beta =\sqrt{2d}\). The purpose of this approach is to propose a simple and flexible method to treat this problem which, unlike the existing renormalization group techniques, does not rely on translation invariance for the covariance kernel of X or the reference measure along which \(\cos (\beta X)\) is integrated. To this purpose we establish by induction a general formula for the cumulants of a random variable defined on a filtered probability space expressed in terms of brackets of a family of martingales; to the best of our knowledge, the recursion formula is new and might have other applications. We apply this formula to study the cumulants of (approximations of) \(\smallint \cos (\beta X)\). To control all terms produced by the induction procedure, we prove a refinement of classical electrostatic inequalities, which allows us to bound the energy of configurations in terms of the Wasserstein distance between \(+\) and − charges.

The size of the class of countable sequences of ordinals

Assume Z F + A D + D C R \mathsf {ZF} + \mathsf {AD} + \mathsf {DC}_\mathbb {R} . There is no injection of &gt; ω 1 ω 1 {}^{&gt;\omega _{1}}{\omega _{1}} (the set of countable length sequences of countable ordinals) into ω O N {}^\omega \mathrm {ON} (the class of ω \omega length sequences of ordinals). There is no injection of [ ω 1 ] ω 1 [{\omega _{1}}]^{{\omega _{1}}} (the powerset of ω 1 {\omega _{1}} ) into &gt; ω 1 O N {}^{&gt;{\omega _{1}}}\mathrm {ON} (the class of countable length sequences of ordinals).

Spectral analysis of Hahn-Dirac system

In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green’s function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2ω,q((ω0, a); E).

Discrete frames for [formula omitted] arising from tiling systems on [formula omitted

A discrete frame for L2(Rd) is a countable sequence {ej}j∈J in L2(Rd) together with real constants 0<A≤B<∞ such thatA‖f‖22≤∑j∈J|〈f,ej〉|2≤B‖f‖22, for all f∈L2(Rd). We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable “tiling system”. We provide all relevant details for constructions in the case of Mn(R)⋊GLn(R), although the methods discussed here are general and could be adapted to some other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of n=2.

Gritty sets and functions

We define a subset of the unit interval to be gritty if it intersects every subinterval with positive measure but does not contain any subinterval. A countable sequence of pairwise disjoint gritty sets is constructed. These are then used to construct a function which is strictly increasing but has zero derivative on a set of positive measure and a function which is unbounded on a subset of positive measure of every subinterval yet has a finite integral.

Замкнутость бигармонической системы базисных потенциалов

The shift systems of the fundamental solution of biharmonic equation are considered. The shift sequence belongs to the complement of the bounded single-bond domain $Q$, so that all system functions satisfy the biharmonic equation in $Q$. The space of biharmonic functions $G_{2}(Q)$ is introduced as a closure in the $L_{2}(Q)$ norm of the linear shell of all kinds of shifts of the fundamental solution of biharmonic equation. The following problem is considered: what conditions the countable sequence of shifts must have so that the linear shell closure matches the entire $G _ {2} (Q) $ space, or, in other words, when is the considered system complete in $G _ {2} (Q)$? The problem for the harmonic case was considered by V.G. Lezhnev, he introduced a sufficient basis condition and proved the completeness and linear independence of shift systems of the fundamental solution of Laplace's equation.The work generalizes the basis condition to the biharmonic case and proves the closedness and linear independence of biharmonic system. The basis condition is related to the singularity condition of biharmonic functions. The proof relies on the property of continuity of potentials with a biharmonic nucleus and on the Rikier's boundary value problem solution.

Continuity and hypercyclicity of composition operators on algebras of symmetric analytic functions on Banach spaces

We consider some conditions for continuity and hypercyclicity of composition operators on algebras of symmetric analytic functions of bounded type on $$\ell _1, L_\infty [0,1)$$, and $$L_\infty [0,\infty )\,{\cap }\, L_1[0,\infty )$$. We establish hypercyclicity of some special composition operators, namely of compositions with translations on algebras of symmetric analytic functions and some other algebras generated by countable sequences of homogeneous polynomials.

Attractors of trees of maps and of sequences of maps between spaces with applications to subdivision

An extension of the Banach fixed-point theorem for a sequence of maps on a complete metric space (X, d) has been presented in a previous paper. It has been shown that backward trajectories of maps $$X\rightarrow X$$ converge under mild conditions and that they can generate new types of attractors such as scale-dependent fractals. Here we present two generalizations of this result and some potential applications. First, we study the structure of an infinite tree of maps $$X\rightarrow X$$ and discuss convergence to a unique “attractor” of the tree. We also consider “staircase” sequences of maps, that is, we consider a countable sequence of metric spaces $$\{(X_i,d_i)\}$$ and an associated countable sequence of maps $$\{T_i\}$$, $$T_i:X_{i}\rightarrow X_{i-1}$$. We examine conditions for the convergence of backward trajectories of the $$\{T_i\}$$ to a unique attractor. An example of such trees of maps are trees of function systems leading to the construction of fractals which are both scale-dependent and location dependent. The staircase structure facilitates linking all types of linear subdivision schemes to attractors of function systems.

The hydrodynamic limit of a randomized load balancing network

Randomized load balancing networks arise in a variety of applications, and allow for efficient sharing of resources, while being relatively easy to implement. We consider a network of parallel queues in which incoming jobs with independent and identically distributed service times are assigned to the shortest queue among a subset of $d$ queues chosen uniformly at random, and leave the network on completion of service. Prior work on dynamical properties of this model has focused on the case of exponential service distributions. In this work, we analyze the more realistic case of general service distributions. We first introduce a novel particle representation of the state of the network, and characterize the state dynamics via a countable sequence of interacting stochastic measure-valued evolution equations. Under mild assumptions, we show that the sequence of scaled state processes converges, as the number of servers goes to infinity, to a hydrodynamic limit that is characterized as the unique solution to a countable system of coupled deterministic measure-valued equations. As a simple corollary, we also establish a propagation of chaos result that shows that finite collections of queues are asymptotically independent. The general framework developed here is potentially useful for analyzing a larger class of models arising in diverse fields including biology and materials science.

Non-Stationary Fractal Interpolation

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k ∈ N where each F k maps H ( X ) → H ( X ) and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.

Liouville quantum gravity and the Brownian map I: the mathrm{QLE}(8/3,0) metric

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter gamma , and it has long been believed that when gamma = sqrt{8/3}, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work considers a growth process called quantum Loewner evolution (QLE) on a sqrt{8/3}-LQG surface {mathcal {S}} and defines d_{{mathcal {Q}}}(x,y) to be the amount of time it takes QLE to grow from x in {mathcal {S}} to y in {mathcal {S}}. We show that d_{{mathcal {Q}}}(x,y) is a.s. determined by the triple ({mathcal {S}},x,y) (which is far from clear from the definition of QLE) and that d_{{mathcal {Q}}} a.s. satisfies symmetry (i.e., d_{{mathcal {Q}}}(x,y) = d_{{mathcal {Q}}}(y,x)) for a.a. (x, y) pairs and the triangle inequality for a.a. triples. This implies that d_{{mathcal {Q}}} is a.s. a metric on any countable sequence sampled i.i.d. from the area measure on {mathcal {S}}. We establish several facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric a.s. extends uniquely and continuously to the entire sqrt{8/3}-LQG surface and that the resulting measure-endowed metric space is TBM.

The subcompleteness of diagonal Prikry forcing

Let $$D$$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $$D$$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $$D$$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $$D$$ is subcomplete above $$\mu $$, where $$\mu $$ is any regular cardinal below the first limit point of $$D$$.

LOCALIZED TRANSFUNCTIONS

A transfunction is a function which maps between sets of finite measures on measurable spaces. Push-forward operators form one important class of examples of transfunctions and are identified with their respective measurable functions. In this regard, transfunctions are a generalization of measurable functions between measurable spaces. Additionally, there are naturally arising transfunctions with nice properties which are not measurable functions. Transfunctions which are weakly $\sigma$-additive (commutable with addition over countable sequences of orthogonal measures) between second-countable metric spaces are of particular interest and are primarily developed in this paper. We study such transfunctions which are localized: sending source measures carried by small open sets to target measures also carried by small open sets. With the right settings and assumptions, we develop some theorems which characterize continuous functions and measurable functions, and show that the behavior of localized transfunctions can be approximated by measurable functions and by continuous functions, but only up to some error. We also characterize transfunctions that correspond to Markov operators. In our investigation of transfunctions we are motivated by several potential applications, including Monge-Kantorovich transportation problem or population dynamics that will be presented in some detail in this paper.

Almost fully closed mappings and quasi-$F$-compacta.

The notion of an almost fully closed mapping, which generalizes the definition of a fully closed map, is introduced. It is shown that the composition of almost fully closed mappings of compacta with the first axiom of countability is also almost fully closed, and so is the limit of a countable inverse transfinite sequence of the same mappings. A class of quasi- $F$ - compacta (which contains the class of Fedorchuk compacta ) is defined. For this class, some statements are proved that generalize and strengthen the known theorems on $F$ - compacta .

Metric topological groups: their metric approximation and metric ultraproducts

We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hyperlinear groups and weakly sofic groups.

Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes

We use a countable-support product of invariant Jensen's forcing notions to define a model of set theory in which the uniformization principle fails for some planar set all of whose vertical cross- sections are countable sets and, more specifically, Vitali classes. We also define a submodel of that model, in which there exists a countable sequence of Vitali classes whose union is not a countable set. Of course, the axiom of choice fails in this submodel.

The Triple Lattice PETs

ABSTRACTPolytope exchange transformations (PETs) are higher dimensional generalizations of interval exchange transformations (IETs) which have been well-studied for more than 40 years. A general method of constructing PETs based on multigraphs was described by R. Schwartz in 2013. In this paper, we describe a one-parameter family of multigraph PETs called the triple lattice PETs. We show that there exists a renormalization scheme for the triple lattice PETs on the parameter space (0, 1). We analyze the limit set Λφ with respect to the parameter . By renormalization, we show that Λφ is the nested intersection of a countable sequence of finite unions of isosceles trapezoids. The Hausdorff dimension of Λφ satisfies the inequality .

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.