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Second-Order Parameterizations for the Complexity Theory of Integrable Functions
Aras Bacho, Martin Ziegler
https://arxiv.org/abs/2506.11210 https://

Second-Order Parameterizations for the Complexity Theory of Integrable Functions
We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of second-order parameterized complexity theory for spaces of continuous functions. Specifically we prove the mutual linear equivalence of three natural parameterizations of the space $\Lrm{p}$ of $p$-integrable complex functions on the real unit interval: (binary) $\Lrm{p}$-modulus, rate of convergence of Fourier series, and rate of approximation by s…

The Complexity of the Set of Validities of a Theory
Denis Hirschfeldt, Henry Towsner, Scott Weinstein
https://arxiv.org/abs/2506.08901 https://

The Complexity of the Set of Validities of a Theory
We study the collection of first-order logical schemata all of whose instances are theorems of a given theory $T$; we call these the validities of $T$ ($\mathsf{V}(T)$). It is easy to see that if $T$ is a decidable theory, then $\mathsf{V}(T)$ is distinct from the set of valid formulas of first-order logic as customarily understood. We provide a complete model-theoretic characterization of the complexity, in the sense of Turing degree, of $\mathsf{V}(T)$ for decidable theories $T$, and answer a…

From Theory to Practice: Advancing Multi-Robot Path Planning Algorithms and Applications
Teng Guo
https://arxiv.org/abs/2506.09914 https://

From Theory to Practice: Advancing Multi-Robot Path Planning Algorithms and Applications
The labeled MRPP (Multi-Robot Path Planning) problem involves routing robots from start to goal configurations efficiently while avoiding collisions. Despite progress in solution quality and runtime, its complexity and industrial relevance continue to drive research. This dissertation introduces scalable MRPP methods with provable guarantees and practical heuristics. First, we study dense MRPP on 2D grids, relevant to warehouse and parcel systems. We propose the Rubik Table method, achieving …

Sampling Theory for Super-Resolution with Implicit Neural Representations
Mahrokh Najaf, Gregory Ongie
https://arxiv.org/abs/2506.09949 https://

Sampling Theory for Super-Resolution with Implicit Neural Representations
Implicit neural representations (INRs) have emerged as a powerful tool for solving inverse problems in computer vision and computational imaging. INRs represent images as continuous domain functions realized by a neural network taking spatial coordinates as inputs. However, unlike traditional pixel representations, little is known about the sample complexity of estimating images using INRs in the context of linear inverse problems. Towards this end, we study the sampling requirements for recove…

New Trends in Kinetic Theory Towards the Complexity of Living Systems
Nicola Bellomo, Diletta Burini, Jie Liao
https://arxiv.org/abs/2506.08752 https://

New Trends in Kinetic Theory Towards the Complexity of Living Systems
The development of a mathematics for living systems is one of the most challenging prospects of this century. The search began with the pioneering contribution of Ilia Prigogine, who developed methods from statistical physics to describe the dynamics of vehicular traffic. This visionary seminal research contribution has given rise to a great deal of research activity, which began at the end of the last century and has been further developed in this century by several authors who have developed …

This https://arxiv.org/abs/2311.18804 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Holographic complexity of the Klebanov-Strassler background
We study the complexity of the gravity dual to the confining $SU(N)\times SU(N+M)$ Klebanov-Strassler gauge theory, which is an important test case for holographic complexity in higher-dimensional and nonconformal gauge/gravity dualities. We emphasize the dependence of the complexity on parameters of the gauge theory, finding a common behavior with confinement scale for several complexity functionals. We also analyze how the complexity diverges with the UV cut off, which is more complicated tha…

This https://arxiv.org/abs/2408.05762 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csDM_…

The complexity of computing the period and the exponent of a digraph
The period of a strongly connected digraph is the greatest common divisor of the lengths of all its cycles. The period of a digraph is the least common multiple of the periods of its strongly connected components. These notions play an important role in the theory of Markov chains and the analysis of powers of nonnegative matrices. While the time complexity of computing the period is well-understood, little is known about its space complexity. We show that the problem of computing the period of…

Towards Bridging Formal Methods and Human Interpretability
Abhijit Paul, Proma Chowdhury, Kazi Sakib
https://arxiv.org/abs/2506.09759 https://

Towards Bridging Formal Methods and Human Interpretability
Labeled Transition Systems (LTS) are integral to model checking and design repair tools. System engineers frequently examine LTS designs during model checking or design repair to debug, identify inconsistencies, and validate system behavior. Despite LTS's significance, no prior research has examined human comprehension of these designs. To address this, we draw on traditional software engineering and graph theory, identifying 7 key metrics: cyclomatic complexity, state space size, average branc…

This https://arxiv.org/abs/1908.09164 has been replaced.
link: https://scholar.google.com/scholar?q=a

A homological approach to chromatic complexity of algebraic K-theory
The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebrai…

This https://arxiv.org/abs/2401.16983 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Classification complexity of chaotic systems
In this paper, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1$ and $\infty$ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty$, while in dimension $1$ we get some partial results. More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e. is Borel bireducib…

Study of Complexity Factor and Stability of Dynamical Systems in $f(G)$ Gravity
M. Zeeshan Gul, W. Ahmad, M. M. M. Nasir, Faisal Javed, Orhan Donmez, Bander Almutairi
https://arxiv.org/abs/2506.03559

Study of Complexity Factor and Stability of Dynamical Systems in $f(G)$ Gravity
In this paper, we evaluate the complexity of the non-static cylindrical geometry with anisotropic matter configuration in the framework of modified Gauss-Bonnet theory. In this perspective, we calculate modified field equations, the C energy formula, and the mass function that helps to understand the astrophysical structures in this modified gravity. Furthermore, we use the Weyl tensor and obtain different structure scalars by orthogonally splitting the Riemann tensor. One of these scalars, $YT…

Stark-Coleman Invariants and Quantum Lower Bounds: An Integrated Framework for Real Quadratic Fields
Ruopengyu Xu, Chenglian Liu
https://arxiv.org/abs/2506.07640

Stark-Coleman Invariants and Quantum Lower Bounds: An Integrated Framework for Real Quadratic Fields
Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. While Stark's conjecture establishes theoretical connections between special units and class group structures, explicit constructions have remained elusive, and precise quantum complexity bounds for class group computations are lacking. Here we establish an integrated framework defining Stark-Coleman invariants $κ_p(K) = \log_p \left( \frac{\varepsilon_…

Resonance Complexity Theory and the Architecture of Consciousness: A Field-Theoretic Model of Resonant Interference and Emergent Awareness
Michael Arnold Bruna
https://arxiv.org/abs/2505.20580

Resonance Complexity Theory and the Architecture of Consciousness: A Field-Theoretic Model of Resonant Interference and Emergent Awareness
This paper introduces Resonance Complexity Theory (RCT), which proposes that consciousness emerges from stable interference patterns of oscillatory neural activity. These patterns, shaped by recursive feedback and constructive interference, must exceed critical thresholds in complexity, coherence, gain, and fractal dimensionality to give rise to conscious experience. The resulting spatiotemporal attractors encode subjective awareness as dynamic resonance structures distributed across the neural…

The Complexity of Correlated Equilibria in Generalized Games
Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Gabriele Farina
https://arxiv.org/abs/2506.01899

The Complexity of Correlated Equilibria in Generalized Games
Correlated equilibria -- and their generalization $Φ$-equilibria -- are a fundamental object of study in game theory, offering a more tractable alternative to Nash equilibria in multi-player settings. While computational aspects of equilibrium computation are well-understood in some settings, fundamental questions are still open in generalized games, that is, games in which the set of strategies allowed to each player depends on the other players' strategies. These classes of games model funda…

A Conceptual Introduction to Hetero-functional Graph Theory for Systems-of-Systems
Amro M. Farid, Amirreza Hosseini, John C. Little
https://arxiv.org/abs/2505.24046

A Conceptual Introduction to Hetero-functional Graph Theory for Systems-of-Systems
A defining feature of twenty first century engineering challenges is their inherent complexity, demanding the convergence of knowledge across diverse disciplines. Establishing consistent methodological foundations for engineering systems remains a challenge -- one that both systems engineering and network science have sought to address. Model-based systems engineering (MBSE) has recently emerged as a practical, interdisciplinary approach for developing complex systems from concept through imple…

Mapping correlations and coherence: adjacency-based approach to data visualization and regularity discovery
Guang-Xing Li
https://arxiv.org/abs/2506.05758 …

Mapping correlations and coherence: adjacency-based approach to data visualization and regularity discovery
The development of science has been transforming man's view towards nature for centuries. Observing structures and patterns in an effective approach to discover regularities from data is a key step toward theory-building. With increasingly complex data being obtained, revealing regularities systematically has become a challenge. Correlation is a most commonly-used and effective approach to describe regularities in data, yet for complex patterns, spatial inhomogeneity and complexity can often un…

Information-Optimal Sensing and Control in High-Intensity Laser Experiments
A. D\"opp, C. Eberle, J. Esslinger, S. Howard, F. Irshad, J. Schroeder, N. Weisse, S. Karsch
https://arxiv.org/abs/2506.04946

Information-Optimal Sensing and Control in High-Intensity Laser Experiments
High-intensity laser systems present unique measurement and optimization challenges due to their high complexity, low repetition rates, and shot-to-shot variations. We discuss recent developments towards a unified framework based on information theory and Bayesian inference that addresses these challenges. Starting from fundamental constraints on the physical field structure, we recently demonstrated how to capture complete spatio-temporal information about individual petawatt laser pulses. Bui…

Learning DNF through Generalized Fourier Representations
Mohsen Heidari, Roni Khardon
https://arxiv.org/abs/2506.01075 https://arxiv.…

Learning DNF through Generalized Fourier Representations
The Fourier representation for the uniform distribution over the Boolean cube has found numerous applications in algorithms and complexity analysis. Notably, in learning theory, learnability of Disjunctive Normal Form (DNF) under uniform as well as product distributions has been established through such representations. This paper makes five main contributions. First, it introduces a generalized Fourier expansion that can be used with any distribution $D$ through the representation of the distr…

VC-dimension of generalized progressions in some nonabelian groups
Gabriel Conant, Aycin Iplikci Arodirik, Tora Ozawa, David Zeng
https://arxiv.org/abs/2505.21789

VC-dimension of generalized progressions in some nonabelian groups
We analyze generalized progressions in some nonabelian groups using a measure of complexity called VC-dimension, which was originally introduced in statistical learning theory by Vapnik and Chervonenkis. Here by a "generalized progression" in a group $G$, we mean a finite subset of $G$ built from a fixed set of generators in analogy to a (multidimensional) arithmetic progression of integers. These sets play an important role in additive combinatorics and, in particular, the study of approximate…

A parallel algorithm for the computation of the Jones polynomial
Kasturi Barkataki, Eleni Panagiotou
https://arxiv.org/abs/2505.23101 https://

A parallel algorithm for the computation of the Jones polynomial
Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data that become available through experiments or Artificial Intelligence. In this context, the efficient computation of topological invariants and other metrics of entanglement becomes an urgent issue. The computation of common measures of topological complexity, su…

This https://arxiv.org/abs/2211.07055 has been replaced.
link: https://scholar.google.com/scholar?q=a

Geometric complexity theory for product-plus-power
According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result and establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree. We connect this new formulation to the orbit closure proble…