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Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.
The author derives an expression for one side of the local relative trace formula, at the level of Lie algebras, by combining methods of Arthur and Harish-Chandra with the structure theory for reductive symmetric spaces.
Probability distributions and the rules of inclusion-exclusion apply to derive general results about graph theory and number theory. In order to apply the rules of inclusion-exclusion in the setting some of these problems require, it is necessary to use specific equations and make several assumptions about integration and measure of the probability distributions.
Using the Weil--Gel'fand--Zak transform of Faddeev's quantum dilogarithm, we construct a compact state integral for positively shaped triangulations of oriented pseudo three-manifolds. The integral is labelled by a circle-valued first cohomology class of the vertex complement (a ``sector'') and takes boundary values in a line bundle over compatible boundary cocycles. We prove absolute convergence, smoothness, invariance under shape gauge transformations and under compatible shaped $2$--$3$ moves, branching independence, and a single-valued global meromorphic continuation in all shape and boundary parameters for each fixed quantum parameter. Gluing is expressed by a conditional Haar integral which retains the final cohomology class; the ordinary trace of integrated cut amplitudes computes an average of the sectors over a torus of classes determined by the cut, and is a sewing formula for every cut of a closed oriented three-manifold, but not for arbitrary pseudo-manifolds. For closed admissible triangulations, the original Teichmüller partition function is the normalized average of the sectors; this yields its global meromorphic continuation. A translation law relates sectors differing by twice an extending class, with phases given by linking pairings; for closed three-manifolds the sectors are supported in a distinguished coset of twice the torsion group and are determined by a single amplitude, which a nonvanishing twisted quadratic Gauss sum recovers from the original partition function.
We study positive-data learning of context-free languages whose nonempty factors are typed by a fixed morphism $s:Σ^+\to S$ into a finite semigroup. A polynomial-time set-driven operator reconstructs each substitutable target from finite positive witnesses; a separate conservative learner identifies nonempty targets in Gold's sense. The encoded characteristic-data size is polynomial in grammar size and a typed shortest-yield parameter. Ordinary grammar thickness suffices when the typing is determined by a fixed prefix-suffix window and a finite-group value; for linear targets, linear grammar size alone suffices. Conversely, a fixed two-element semilattice typing yields finite targets with grammars of size and ordinary thickness both $O(n)$ that require exponentially long characteristic samples. The lower bound applies to any set-driven operator admitting characteristic samples for the fixed-typing class. Retyping closure, examples beyond fixed windows, and a Dyck-language obstruction delineate the framework's scope.
Every regular N-gon defines a canonical family of regular polygons which are conforming to the bounds of the 'star polygons' determined by N. These star polygons are formed from truncated extended edges of the N-gon and the intersection points ('star' points) determine the parameters of the family. The First Family Theorem (FFT) shows how each star[k] point defines a matching S[k] 'tile' and the GeneralizedFFT shows how every S[k] can generate their own families with an S[k+1] on the left and (if space permits) an S[k'] on the right where k' = N/2-k for N even. If such an S[k]-S[k'] pair exists it will be called a 'dual' pair and share an edge. When N is even the S[1] tile and N will form a dual pair since N = S[N/2-1]. When the outer-billiards map Tau is introduced in Section 4 the singularity set W will generate two manifolds +W and -W depending on whether N is clockwise or ccw so the relationship between +W and -W depends on how N is related to its mirror image T(N) where T is a horizontal reflection. But N is 'chiral' so no rotation by multiples of 2Pi/N can overlay T(N) on N. This means that +W and -W are distinct and W is their 'union'. But any mapping of +W and -W to W cannot be smooth because their polarities are distinct and we assert that the S[k] dual tiles are the 'resonances' of this mapping. Indeed every dual S[k]-S[k'] pair will be split between +W and -W regions so their shared edge defines the boundaries of invariant regions. The Invariance Conjecture of Appendix II says that that every dual S[k] pair with at least k or k' co-prime with N will survive in W and the number of such pairs is the algebraic complexity C(N) = EulerPhi(N)/2 so every regular N-gon will have C(N) distinct dual pairs and hence C(N)-1 invariant regions. The boundaries of these regions are separatrices which act as eigenstates in a matching Hamiltonian composed of C(N) twist maps.
We recover some data of a module $M$ from the Ext-family $\{Ext^i_R(M,R)\}$. In this regard, we investigate homological subsets of $Spec(R)$, defined by the help of Ext-family. For example, we present a situation for which $\bigcup_{i=0}^\infty Ass(Ext^i_R(M,R))$ is infinite, thereby answering a question of Vasconcelos. We extend Grothendieck's calculation of $dim(Ext^g_R(M,R))$ and present some new situations to calculate $dim(Ext^\ast_R(M,R))$. Also, we compute support and the set of all associated prime ideals of the Ext-family in a serial of nontrivial cases.
Fix integers $m\ge2$ and $k\ge1$. For every sufficiently large fixed odd exponent $M$, choose $k$ independent elements uniformly from the sphere of radius $\ell$ in the free Burnside group $B_m(M)$. We prove that the quotient by their normal closure is infinite with probability at least $1-e^{-c\ell}$. More precisely, the quotient map has no nontrivial kernel element of length at most $κ\ell$, for constants $c,κ>0$. We also consider the analogue of the Gromov positive density model for the Burnside variety.
Artificial Intelligence (AI) started out with the ambition to reproduce the human mind, but as soon as the sheer scale of that ambition became apparent, it retreated into either studying specialized intelligent behaviours or proposing overarching architectural concepts for interfacing specialized intelligent components---conceived of as agents acting through organizational structures. The agent-based modelling paradigm proved to have interesting applications beyond the original domain, supporting the understanding, simulation, and the assessment of behaviour of socio-economic and legal structures on an aggregate level. Looking at contemporary AI, whereas text completion provides a successful example of specialized behaviour, surprisingly little of the rich foundation of multi-agent systems (MAS) and normative systems research is currently reflected in Agentic AI systems. Addressing this gap, the chapter examines cross-cutting conceptualisations, modelling problems, and design challenges shared by large-scale distributed AI and institutional systems. Three main contributions can be highlighted. First, by tracing the historical trajectory of AI, the chapter identifies an \textit{ecological} perspective on intelligence, examining both the value and limits of the \textit{compression-as-intelligence} hypothesis. Second, it shows how moving between internal cognitive mechanisms and external manifestations of agency supports a functional separation of concerns for both individual and collective agency. Third, it argues that \textit{epistemological}, \textit{normative}, and \textit{value pluralisms} are necessary conditions for sustaining the ecological character of intelligence. Furthermore, we indicate how AI techniques can support institutions by modelling social roles, simulating normative interactions, detecting and diagnosing failures, and supporting procedures of review and revision.
We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to 5, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from 4-dimensional topology.
The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x+1$ problem. The analysis rests on the dynamics of an iterative map in binary form, defined on the dyadic rationals of $[1/2,\,1)$. We estimate how the length of the binary expansion of the iterated numbers changes, and we regard the iterative map as a perturbation of a circle map without periodic points, with an explicit bound for the error between both maps. A condition on the periods of periodic orbits is given; it is the classical argument of Crandall, and it recovers the lower bound of Eliahou for the length of nontrivial cycles. This version corrects several errors of the first one; they are listed in the Addendum.
Auctions allocate scarce goods, from car licenses to online advertising, to buyers whose utilities are often not linear in money, as with budgets or risk aversion. We ask how a seller should sell to such buyers and what learning their preferences is worth. We propose demand-based auctions, which elicit demand curves and are dominant-strategy incentive compatible for any non-linear utilities. The revenue-maximizing one takes familiar second-price forms and is max-min optimal when the seller observes only demand curves. Under standard assumptions, learning risk attitudes beyond demand curves adds relatively little revenue, as does learning budgets unless they correlate with values.
We prove that Fekete configurations associated with a positive Hermitian line bundle over a compact Kähler manifold are asymptotically exact norming sets for random holomorphic sections. For every fixed $a>0$, the ratio of the supremum norm of a degree-$k$ section to its maximum on the degree-$\lceil ak\rceil$ Fekete configuration converges to one in probability. This holds for any background Kähler form: Fekete equidistribution provides enough sampling points near the maximum of the limiting pointwise variance. We obtain polynomial upper bounds for the tail of the ratio and almost-sure norm comparisons. In contrast, when the limiting variance is nonconstant, we construct separated arrays of cardinality comparable to the section-space dimension for which the ratio converges to a constant strictly greater than one. General separated arrays still satisfy probabilistic norming inequalities with degree-independent constants, and every such array is asymptotically exact if and only if the volume form is proportional to $c_1(L,h)^n$. We also prove a discrete $L^2$ norm comparison with exponentially small exceptional probability. The norm comparisons and convergence statements hold in both the Gaussian and spherical ensembles.
We prove the Ax-Schanuel property of the derivatives of mixed period mappings. We also prove the jet space reformulation of this result. The proofs use the Ax-Schanuel result for principal bundles with flat connections obtained by Blázquez-Sanz, Casale, Freitag, and Nagloo.
India's reservation policy entitles designated groups to fixed fractions of faculty positions. Positions are indivisible, arrive over time across departments, and appointments are irreversible. We show that no rule can keep cumulative reserved counts nondecreasing while rounding entitlements to adjacent integers at both the department and university levels; university-level compliance forces department-level errors that grow without bound. This impossibility explains why both solutions practiced in India provoked litigation and protests. A random-roster solution preserves irreversibility and department-level compliance, matches entitlements in expectation, and concentrates university-level errors. Dropping cumulative accounting restores exact rounding at both levels.
We give an explicit construction of the cluster scattering diagram for any acyclic exchange matrix of affine type. We show that the corresponding cluster scattering fan coincides both with the mutation fan and with a fan constructed in the almost-positive roots model.
When individuals make simultaneous decisions across multiple ordered dimensions, standard multivariate ordered choice models impose narrow bracketing, with each dimension decided as if the others did not exist. We develop a general rectangular structure model capturing broad bracketing while nesting the standard model. The framework introduces two layers of dependence, one through the threshold structure associated with decision rules and the other through the correlation of latent utilities. We provide microfoundations, prove identification, and discuss estimation. We showcase the model in applications to parental investment in children's education, cryptocurrency familiarity and expectations, and health insurance, where the latter separates moral hazard from adverse selection.
We characterize the hieroglyphs that embed in the torus, where a hieroglyph is a one-vertex graph with a rotation system. We prove equivalent criteria in terms of four forbidden hieroglyphs, the structure of the loop graph, reductions to three terminal forms, and the rank of the interlacement matrix. The proof uses the intersection form on the first homology of a disc with bands over $\mathbb{F}_2$ and the equality between its rank and twice the genus. We describe the effect of reductions and the role of boundary components. Counting the cycles of the boundary permutation yields a linear-time recognition algorithm for an indexed representation of the input.
In this paper, we study the multifractal formalism for Lyapunov exponents of typical one-step cocycles. We establish a variational relation between the Legendre transform of topological pressure of the generalized singular value function and measure-theoretic entropies. As a consequence, we show that the restricted variational principle of Lyapunov exponents holds for typical one-step cocycles. Furthermore, we prove a variational principle formula for the generalized singular value function.