Tootfinder

Neuromorphic Online Clustering and Its Application to Spike Sorting
James E. Smith
https://arxiv.org/abs/2506.12555 https://arxiv.org…

Neuromorphic Online Clustering and Its Application to Spike Sorting
Active dendrites are the basis for biologically plausible neural networks possessing many desirable features of the biological brain including flexibility, dynamic adaptability, and energy efficiency. A formulation for active dendrites using the notational language of conventional machine learning is put forward as an alternative to a spiking neuron formulation. Based on this formulation, neuromorphic dendrites are developed as basic neural building blocks capable of dynamic online clustering. …

Chow cohomology and Lefschetz (1,1)-theorem
Ananyo Dan, Inder Kaur
https://arxiv.org/abs/2506.13220 https://arxiv.org/pdf/2506.13220

Chow cohomology and Lefschetz (1,1)-theorem
Any smooth, projective variety satisfies the Hodge conjecture in codimension one, known as the Lefschetz (1,1) theorem. Totaro formulated a version for singular varieties. He asked whether the natural Bloch-Gillet-Soulé cycle class map from the operational Chow group to the (1,1)-classes in the weight graded piece of the cohomology group is surjective? In this short article, we give a sufficient criterion for this to hold. In particular, we show that several singular varieties with at worst is…

On the validity of the Radon-Nikodym Theorem
Paolo Roselli, Michel Willem
https://arxiv.org/abs/2506.12023 https://arxiv.org/pdf/2506…

On the validity of the Radon-Nikodym Theorem
This paper presents a new general formulation of the Radon-Nikodym theorem in the setting of abstract measure theory. We introduce the notion of weak localizability for a measure and show that this property is both necessary and sufficient for the validity of a Radon-Nikodym-type representation under a natural compatibility relation between measures. The proof relies solely on elementary tools, such as Markov's inequality and the monotone convergence theorem. In addition to establishing the mai…

This https://arxiv.org/abs/2501.06931 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Discrete lossless convexification for pointing constraints
Discrete Lossless Convexification (DLCvx) formulates a convex relaxation for a specific class of discrete-time non-convex optimal control problems. It establishes sufficient conditions under which the solution of the relaxed problem satisfies the original non-convex constraints at specified time grid points. Furthermore, it provides an upper bound on the number of time grid points where these sufficient conditions may not hold, and thus the original constraints could be violated. This paper ext…

A panoramic view of exponential attractors
Radoslaw Czaja, Stefanie Sonner
https://arxiv.org/abs/2506.10388 https://arxiv.org/pdf/250…

A panoramic view of exponential attractors
We state necessary and sufficient conditions for the existence of $T$-discrete exponential attractors for semigroups in complete metric spaces. These conditions are formulated in terms of a covering condition for iterates of the absorbing set under the time evolution of the semigroup and imply the existence and finite-dimensionality of the global attractor. We then review, generalize and compare existing construction methods for exponential attractors and show that they all imply the covering c…

Mass inflation from rough initial data for the spherically symmetric Einstein-Maxwell-scalar field system with $\Lambda$
Flavio Rossetti
https://arxiv.org/abs/2506.08075

Mass inflation from rough initial data for the spherically symmetric Einstein-Maxwell-scalar field system with $Λ$
Recent rigorous results on black hole interiors clearly suggest that the strong cosmic censorship conjecture fails in its most fundamental, i.e. weak, formulation: violations are expected for a class of spherically symmetric charged black holes in the presence of a positive cosmological constant near extremality. These results require sufficiently regular solutions. Conversely, when non-smooth, finite-energy initial data are prescribed for linear waves propagating on a fixed black hole backgrou…

This https://arxiv.org/abs/2206.04896 has been replaced.
link: https://scholar.google.com/scholar?q=a

A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety
Ashby's law of requisite variety allows a comparison of systems with their environments, providing a necessary (but not sufficient) condition for system efficacy: a system must possess at least as much complexity as any set of environmental behaviors that require distinct responses from the system. However, to account for the dependence of a system's complexity on the level of detail -- or scale -- of its description, a multi-scale generalization of Ashby's law is needed. We define a class of c…

Kurzweil--Stieltjes integration on compact lines
Leandro Candido, Pedro L. Kaufmann
https://arxiv.org/abs/2506.07832 https://arxiv.or…

Kurzweil--Stieltjes integration on compact lines
We develop a version of the Kurzweil--Stieltjes integral on compact lines and establish its fundamental properties. For sufficiently regular integrators, we obtain convergence theorems and show that the presented integration process generalizes Lebesgue integration with respect to positive Radon measures. Additionally, we introduce a notion of derivation on compact lines which, when paired with the proposed integral, yields a formulation of the Fundamental Theorem of Calculus in this context.

Free Askey--Wilson functionals and geometric last passage percolation on a strip
Wlodek Bryc, Kamil Szpojankowski, Jacek Wesolowski
https://arxiv.org/abs/2506.01879

Free Askey--Wilson functionals and geometric last passage percolation on a strip
Barraquand, Corwin, and Yang arXiv:2306.05983 established that geometric last passage percolation (LPP) on a strip of $\mathbb{Z}^2$ has a unique stationary measure. Building on this, Barraquand arXiv:2409.08927 derived explicit contour integral formulas for the model's multipoint probability generating function. In this paper, we introduce free Askey--Wilson functionals and use them to extend these generating function formulas. Our framework yields explicit expressions valid over a broader ran…

This https://arxiv.org/abs/2505.19412 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

2d Cardy-Rabinovici model with the modified Villain lattice: Exact dualities and symmetries
The Cardy-Rabinovici model is a toy model of the lattice $U(1)$ gauge theories to study various oblique confinement states associated with the nonzero $θ$ angles. We reformulate the $2$d version of this model using the modified Villain lattice formalism, and we establish the exact $θ$ periodicity for the Witten effect and the strong-weak duality at the finite lattice spacings. We then study the phase structure of this model based on the duality, symmetry and anomaly, and the perturbative reno…