arXiv research

The paper defines a path metric on a stable component of polynomial families.

This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable cohomotopy invariants to Seiberg-Witten invariants without restriction on Betti numbers.…

We introduce a new convex formulation for stable principal component pursuit (SPCP) to decompose noisy signals into low-rank and sparse representations. For numerical solutions of our SPCP formulation, we first develop a convex variational framework and then accelerate it with quasi-Newton methods. We show, via synthet…

Characterizes hyperbolic links with stable maps to the plane.

We solve principal component regression (PCR), up to a multiplicative accuracy $1+γ$, by reducing the problem to $\tilde{O}(γ^{-1})$ black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In…

Proposes ANOVA-TPNN for stable interpretation of complex functions.

We study stable constant mean curvature (CMC) hypersurfaces $Σ$ in slabs in a product space $M\times\r,$ where $M$ is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $Σ$ is not a cylinder then it is locally a vertical graph. Moreover, in case $M$ is $\h^n,\r^n$ or $…

Let M be a hyperbolizable, nontrivial compression body without toroidal boundary components. In this paper, we characterize which discrete and faithful representations of the fundamental group of M into PSL(2,C) are separable-stable. The set of separable-stable representations forms a domain of discontinuity for the ac…

We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work in all dimensions and without variational assumptions on the Allen-Cahn solution…

New method selects FMM components via variational Bayes.

We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $Σ$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify the $σ_{mod}$-regularity for convex projective structures and positive repr…

Study abelian cycles in Torelli group homology, proving new results in stable rational homology.

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic rea…

The multivariate version of the Mixed Tempered Stable is proposed. It is a generalization of the Normal Variance Mean Mixtures. Characteristics of this new distribution and its capacity in fitting tails and capturing dependence structure between components are investigated. We discuss a random number generating procedu…

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop…

The paper extends symplectic techniques to generalized complex geometry.

The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic sta…

Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.

Study identifies stable configurations of intertwined threads with repulsive interactions.

Adapts a short argument to derive a stability theorem for smooth maps.

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of …

In [20], Ros and Vergasta proved that an immersed orientable compact stable constant mean curvature surface $Σ$ with free boundary in a closed ball $B\subset\mathbb{R}^3$ must be a planar equator, a spherical cap or a surface of genus 1 with at most two boundary components. In this article, by using a modified Hersch t…

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are give…

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $σ(f)$ of its framed singularity link, the generalized normal degree $ν(f)$, and the algebraic number of cusps $κ(f)$ of any extensi…

Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.

Statistic dynamics of financial systems is investigated, basing on a model of randomly coupled equation system driven by stochastic Langevin force. It is found that in stable regime the noise power spectrum of the system is of 1/f^alpha form, with the exponent alpha=3/2 in case of Hermitian coupling matrices, or slight…

We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $Σ$ is rotationally symmetric in the following cases: (1) $n=2$, $\mathcal B$ is a slab and $Σ$ has genus zero, (2) $n\geq 2$, $\mathc…

Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple interacting losses. The behavior of gradient-based methods in games is not well under…

We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored…

Algorithm learns dynamics from past observations.

We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…

Study 2-loop part of Johnson cokernel using trace map.

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb{M}$ without boundary, \begin{equation*} -Δ_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when $Δ_g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb…

We present an algorithm, AROFAC2, which detects the (CP-)rank of a degree 3 tensor and calculates its factorization into rank-one components. We provide generative conditions for the algorithm to work and demonstrate on both synthetic and real world data that AROFAC2 is a potentially outperforming alternative to the go…

We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\# m\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ or $\# m (S^2 \times S^2)$. Given the relation between boundary Lefschetz fibrations and stable general…

FlowMM models stable crystal structures efficiently.

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group o…

This paper extends stable blanket theory to models with hidden variables and causal cycles.

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expa…

We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched over the disc. Our results addresses the problem to decide equiva…

SPPCSO addresses multicollinearity in high-dimensional data, improving model stability and predictive accuracy.

We introduce the \emph{metric spectrum}, which measures the exponential rate of approximation to an isolated invariant set of points starting in its stable set, and relate it to the Lyapunov spectrum. We determine the metric spectrum of each Morse component of the finest Morse decomposition of a linear induced flow on …

We develop a mean-field theory for multi-component ICA in high dimensions.

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…

New method estimates volatility for processes with jumps of unbounded variation.

Retraining stabilizes model influence on data.

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, γ). This is the space consisting of triples, (F_{g,n}, φ, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, φis a parameterization of the boundary, and f : F_{g,n} \to X is a contin…

We show that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if its (un)stable lamination is a projective limit of meridians. The proof is through 3-dimensional hyperbolic geometry, and involves an investigation of algebraic limits …