Abstract: We investigate the relationship between the delooping level (dell) and the finitistic dimension of left and right serial quiver algebras. These 2-syzygy finite algebras have finite delooping level, and it can be calculated with an easy and finite algorithm. When the algebra is right serial, its right finitistic dimension is equal to its left delooping level. When the algebra is left serial, the above equality only holds under certain conditions. We provide examples to demonstrate this and include discussions on the sub-derived (sub-ddell) and derived delooping level (ddell). Both sub-ddell and ddell are improvements of the delooping level. We motivate their definitions and showcase how they can behave better than the delooping level in certain situations throughout the paper.

Abstract: The finitistic dimension conjecture is closely connected to the symmetry of the finitistic dimension. Recent work indicates that such connection extends to one of its upper bounds, the delooping level. In this paper, we show that the same holds for the derived delooping level, which is an improvement of the delooping level. This reduces the finitistic dimension conjecture to considering algebras whose opposite algebra has (derived) delooping level zero. We thereby demonstrate ways to utilize the new concept of derived delooping level to obtain new results and present additional work involving tensor product of algebras.

Abstract: In this paper, we develop new ideas regarding the finitistic dimension conjecture, or the findim conjecture for short. Specifically, we improve upon the delooping level by introducing three new invariants called the effective delooping level edell, the sub-derived delooping level sub-ddell, and the derived delooping level ddell. They are all better upper bounds for the opposite Findim. Precisely, we prove

Findim Λᵒᵖ = edell Λ ≤ ddell Λ (or sub-ddell Λ) ≤ dell Λ

and provide examples where the last inequality is strict (including the recent example from Kershaw–Rickard where dell Λ = ∞, but ddell Λ = 1 = Findim Λᵒᵖ). We further enhance the connection between the findim conjecture and tilting theory by showing finitely generated modules with finite derived delooping level form a torsion-free class ℱ. Therefore, studying the corresponding torsion pair (𝒯, ℱ) will shed more light on the little finitistic dimension. Lastly, we relate the delooping level to the φ-dimension φ dim, a popular upper bound for findim, and give another sufficient condition for the findim conjecture.

Abstract: The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights {ωxy}xy∈E, satisfying the condition that ω_xy=\bar{ω_yx} for every directed edge xy. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph.

In this paper we establish the spectrum of the magnetic Laplacian, as a set of real numbers with multiplicities, on the Sierpinski gasket graph (SG) where the magnetic fluxes equal α through the upright triangles, and β through the downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map U involving non-rational functions, which takes into account α, β, and the spectral parameter λ. In doing so we provide a quantitative answer to a question of Bellissard [Renormalization Group Analysis and Quasicrystals (1992)] on the relationship between the dynamical spectrum and the actual magnetic spectrum.

Our main theorems lead to two applications. In the case α=β, we demonstrate the approximation of the magnetic spectrum by the filled Julia set of U, the Sierpinski gasket counterpart to Hofstadter's butterfly. Meanwhile, in the case α,β∈{0,12}, we can compute the determinant of the magnetic Laplacian determinant and the corresponding asymptotic complexity.

Abstract: Making consistently profitable financial decisions in a continuously evolving and volatile stock market has always been a difficult task. Professionals from different disciplines have developed foundational theories to anticipate price movement and evaluate securities such as the famed Capital Asset Pricing Model (CAPM). In recent years, the role of artificial intelligence (AI) in asset pricing has been growing. We aim to further enhance AI's potential and utility by introducing a variant of the cross-entropy loss, called return-weighted cross-entropy (RWCE), that drives top growth while requiring only a limited amount of input information. Using only publicly accessible stock data (open/close/high/low, trading volume, sector information) and several technical indicators constructed from them, we propose an efficient daily trading system that detects top growth opportunities. Our best models achieve 61.73% annual return on daily rebalancing with an annualized Sharpe Ratio of 1.18 over 1340 testing days from 2019 to 2024, and 37.61% annual return with an annualized Sharpe Ratio of 0.97 over 1360 testing days from 2005 to 2010. The main drivers for success, especially independent of any domain knowledge, are the novel return-weighted loss function, the integration of categorical and continuous data, and the ML model architecture. We also demonstrate the superiority of our proposed RWCE loss via several performance metrics and statistical evidence.