arXiv Math @arxiv_math@qoto.org

We find a Floer theoretic approach to obtain the transpose polynomial $W^T$ of an invertible curve singularity $W$. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical $A_\infty$-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of $W^T$. We find Lagrangians in the Milnor fiber of $W$ that are mirror to the indecomposable matrix factorizations of $W^T$ when $W^T$ is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror. There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian $L$ and corresponding disc potential function $W_L$, we first determine the quotient $X$ by the maximal symmetry group for the Milnor fiber. Second, we define a class $Γ$ of symplectic cohomology of $X$ based on the monodromy of the singularity $W$. Another disc counting function, $g$, is defined by the closed-open image of $Γ$ on $L$. We demonstrate that restricting to the hypersurface $g = 0$ transforms the disc potential function $W_L$ into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class $Γ$.

The theory of motivic superpolynomials is developed, including its extension to algebraic links, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis and DAHA recurrences for iterated torus links. The DAHA construction is provided. Applications are considered to affine Springer fibers and compactified Jacobians of type $A_n$ in the most general case (for arbitrary characteristic polynomials), rho-invariants of algebraic knots, motivic formulas for the DAHA-Jones polynomials, including the case of $A_1$. The key theme is the conjectural coincidence of motivic superpolynomials with the DAHA ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. The 2nd connection conjecture links the superpolynomials to the $L$-functions. DAHA theory is adjusted to the definition and main properties of superpolynomials. The motivic tools are systematically exposed, a certain unification of Schubert Calculus with the theory of plane curve singularities.

In this work, we study the composition operators on the little Lipschitz space ${\mathcal L}_0$ of a rooted tree $T$ defined as the subspace of the Lipschitz space ${\mathcal L}$, introduced in Colonna and Easley [4], that consists of the complex-valued functions $f$ on $T$ such that $$ \lim_{|v|\to\infty}|f(v)-f(v^-)|=0, $$ where $v^-$ is the vertex adjacent to the vertex $v$ in the path from the root to $v$ and $|v|$ denotes the number of edges from the root to $v$. Specifically, we give several sufficient conditions on a Lipschitz self-map $φ$ on $T$ for which the composition operator $C_φ$ maps ${\mathcal L}_0$ into itself, and estimate its operator norm. In addition we study the spectrum of $C_φ$ and the hypercyclicity of a constant multiple of $C_φ$.

A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$.

I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies.

Let $M, N$ be free modules over a Noetherian commutative ring $R$ and let $F$ be a field whose cardinality does not exceed the continuum. We prove the following: 1) The Injective Continuum Function Hypothesis, ICF, is equivalent to assertion that [ any two F-vector spaces are isomorphic iff their duals are so.] 2) If the dual of $M$ is a projective $R$-module and rank$_R(M)$ is infinite then the ring $R$ is Artinian. 3) If $R$ is Artinian and card$(R)$ does not exceed the continuum then the dual of $M$ is free. 4) If $M, N$ have isomorphic duals then they are themselves isomorphic (over $R$), when rank$_R(M)$ is not an $ω$-measurable cardinal and $R$ is a non-Artinian ring that is either Hilbert or countable. 5) If $R$ is a non-local domain then $R$ is a half-slender ring. We also prove that if the powersets of two given sets have equal cardinalities then there is a bijection from the one powerset to the other that preserves the symmetric difference of sets.

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

Let $\mathcal{D}$ be the whole space $\mathbb{R}^d$ or a sphere of radius $R>0$ in $\mathbb{R}^d$ with $1\leq d\leq3$. Consider the following cubic-quintic energy functional \begin{gather*} \begin{aligned} E_{N}(φ)=\frac{1}{2}\int_{\mathcal{D}}|\nabla φ|^2\,dx-\frac{N}{4}\int_{ \mathcal{D}}|φ|^{4}\,dx+\frac{N^{2}}{6}\int_{\mathcal{D}}|φ|^6\,dx \end{aligned} \end{gather*} minimized in $\left\lbrace φ\in H^1_0(\mathcal{D}): \int_{\mathcal{D}}|φ|^2\,dx=1\right\rbrace$. Firstly, we prove the limiting behavior of the ground state $φ_N$ as $N\to+\infty$, which corresponds to the \emph{Thomas-Fermi limit}. The limit profile is given by the Thomas-Fermi minimizer $u^{TF}=\sqrt{{3\cdot\mathbb{1}_Ω}/{4}}(x)$ with $Ω=B(0,\sqrt[d]{4/(3ω_d)})$. Moreover, we obtain a sharp vanishing rate for $φ_N$ that $\|φ_{N}\|_{L^\infty(\mathcal{D})}\sim N^{-{1}/{2}}$ as $N\to+\infty$. Finally, we prove that $\left\|N^{{1}/{2}}φ_{N}\left(N^{{1}/{d}}x\right)-u^{TF}(x)\right\|_{L^\infty\left(Ω\right)}\lesssim N^{-1/(2d)}$ as $N\to+\infty$.

Forward-backwards stochastic differential equations (FBSDEs) are central in optimal control, game theory, economics, and mathematical finance. Unfortunately, the available FBSDE solvers operate on \textit{individual} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs as these solvers must be re-run several times. \textit{Neural operators} (NOs) offer an alternative approach for \textit{simultaneously solving} large families of FBSDEs by directly approximating the solution operator mapping \textit{inputs:} terminal conditions and dynamics of the backwards process to \textit{outputs:} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. We confirm that ``small'' NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms, to any prescribed error $\varepsilon>0$ using a depth of $\mathcal{O}(\log(1/\varepsilon))$, a width of $\mathcal{O}(1)$, and a sub-linear rank; i.e. $\mathcal{O}(1/\varepsilon^r)$ for some $r<1$. This result is rooted in our second main contribution, which shows that convolutional NOs of similar depth, width, and rank can approximate the solution operator to a broad class of Elliptic PDEs. A key insight here is that the convolutional layers of our NO can efficiently encode the Green's function associated to the Elliptic PDEs linked to our FBSDEs. A byproduct of our analysis is the first theoretical justification for the benefit of lifting channels in NOs: they exponentially decelerate the growth rate of the NO's rank.

An overview of the authors' ideas about the process of completing a $p$-adically normed space in the setting of vertex operator algebras. We focus in particular on the $p$-adic Heisenberg VOA and its connections with $p$-adic modular forms.

In this paper, we establish Newton-Maclaurin type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton-Maclaurin inequality to a broader class of functions.

Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums $\sum_{i=1}^dX_i$ have the same distribution $p$ is a convex polytope $\mathcal{P}(p)$ and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes $\mathcal{P}(p), p\in \mathcal{D}_d,$ is a continuous function $l(p)$ over $\mathcal{D}_d$ and it is the density of a finite measure $μ_s$ on $\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $μ_s$ normalized over the simplex $\mathcal{D}$ belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on $\mathcal{D}$ and that when $d$ increases it converges to the mode.

Bulk-edge correspondence is a wide-ranging principle that applies to topological matter, as well as a precise result established in a large and growing number of cases. According to the principle, the distinctive topological properties of matter, thought of as extending indefinitely in space, are equivalently reflected in the excitations running along its boundary, when one is present. Indices encode those properties, and their values, when differing, are witness to a violation of that correspondence. We address such violations, as they are encountered in a hydrodynamic context. The model concerns a shallow layer of fluid in a rotating frame and provides a local description of waves propagating either across the oceans or along a coastline; it becomes topological when suitably modified at short distances. The edge index is sensitive to boundary conditions, as exemplified in earlier work, hence exhibiting a violation. Here we present classification of all (local, self-adjoint) boundary conditions and a parameterization of their manifold. They come in four families, distinguished in part by the degree of their underlying differential operators. Essentially, that degree counts the degrees of freedom of the hydrodynamic field that are constrained at the boundary by way of their normal derivatives. Generally, both the correspondence and its violation are typical. Within families though, the maximally possible amount of violation is increasing with its degree. Several indices of interest are charted for all boundary conditions. A single spectral mechanism for the onset of violations is furthermore identified. The role of a symmetry is investigated.

We prove that every ergodic transformation is Shannon orbit equivalent to a weak mixing transformation. The proof is based on the techniques introduced by Fieldsteel and Friedman to show that there is a mixing transformation for a given ergodic transformation $T$ which is, for all $a\geq1$, weak-$a$-equivalent to $T$ and, for all $b\in(0,1)$, strong-$b$-equivalent to $T$. In particular, we will adapt the construction of Fieldsteel and Friedman by which they permute the columns of each Rokhlin tower in a sequence of rapidly growing Rokhlin towers so that the corresponding cocycles converge to an orbit equivalence cocycle of $T$ such that the resulting transformation and orbit equivalence have the desired properties. In addition to this, we will demonstrate a flexible method for obtaining actions of $\mathbb{Z}^{2} $ which are Shannon orbit equivalent to a given ergodic transformation.

We propose a new approach to proving the uniqueness of solutions to a certain class of mean field games of controls. In this class, the equilibrium is determined by an aggregate quantity $Q(t)$, e.g. the market price or production, which then determines optimal trajectories for agents. Our approach consists in analyzing the relationship between $Q(t)$ and corresponding optimal trajectories to find conditions under which there is at most one equilibrium. We show that our conditions do not match those prescribed by the Lasry-Lions monotonicity condition, nor even displacement monotonicity, but they do apply to economic models that have been proposed in the literature.

We characterize two classical types of conformality of a holomorphic self-map of the unit disk at a boundary point - existence of a finite angular derivative in the sense of Carathéodory and the weaker property of angle preservation - in terms of the non-tangential asymptotic behaviour of the hyperbolic distortion of the map. These characterizations are given purely with reference to the intrinsic metric geometry of the unit disk. In particular, we relate the classical Julia-Wolff-Carathéodory theorem with the case of equality in the Schwarz-Pick lemma at the boundary. We also provide an operator-theoretic characterization of the existence of a finite angular derivative based on Hilbert space methods. As an application we study the backward dynamics of discrete dynamical systems induced by holomorphic self-maps, and characterize the regularity of the associated pre-models in terms of a Blaschke-type condition involving the hyperbolic distortion along regular backward orbits.

We investigate the differential smoothness of a certain family of skew Poincaré-Birkhoff-Witt extensions.

We prove that there is a normed barrelled space with dimension $\mathrm{non}(\mathcal M)$, which denotes the smallest cardinality of a non-meager subset of $\mathbb R$. Consequently, it is consistent with $\mathsf{ZFC}$ that there is a normed barrelled space with dimension $<\!\mathfrak{c}$. This answers a question of Sánchez Ruiz and Saxon. This is a special case of a more general theorem: for every infinite cardinal $κ$, every Banach space with density character $κ$ contains a barrelled subspace with dimension $\mathrm{cf}[κ]^ω\cdot \mathrm{non}(\mathcal M)$. We also prove that if the dual of a Banach space does not contain $c_0$ or $\ell^p$ for any $p \geq 1$, then that spaces does not have a barrelled subspaces with dimension $<\!\mathrm{cov}(\mathcal N)$, which denotes the smallest cardinality of a collection of Lebesgue null sets covering $\mathbb R$. In particular, it is consistent with $\mathsf{ZFC}$ that no classical Banach spaces contain barrelled subspaces with dimension $\mathfrak{b}$. This partly answers another question of Sánchez Ruiz and Saxon.

This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group $G$. The theory extends the $RO(G)$-graded cohomology of a $G$-space $B$ to a cohomology graded by $RO(ΠB)$, the representations of the equivariant fundamental groupoid of $B$. This paper is meant to serve as a guide to this theory and contains some new computations. We explain the key ingredients for defining parametrized cellular cohomology when $G$ is a finite group, with particular attention to the case of the cyclic group $G=C_2$. We compute some examples and observe that $RO(ΠB)$ is not always free. When $G$ is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with $G = C_2$ and $G=C_4$.

In this article, we present a method to construct $e$-power $b$-happy numbers of any height. Using this method, we construct a tree that encodes these happy numbers, their heights, and their ancestry--relation to other happy numbers. For fixed power $e$ and base $b$, we consider happy numbers with at most $k$ digits and we give a formula for the cardinality of the preimage of a single iteration of the happy function. We show that these happy numbers arise naturally as children of a given vertex in the tree. We conclude by applying this technique to $e$-power $b$-unhappy numbers of a given height.