Tootfinder

On Rational Recursion for Holonomic Sequences
Bertrand Teguia Tabuguia, James Worrell
https://arxiv.org/abs/2404.19136 https://arxiv.…

On Rational Recursion for Holonomic Sequences
It was recently conjectured that every component of a discrete rational dynamical system is a solution to an algebraic difference equation that is linear in its highest-shift term (a quasi-linear equation). Holonomic sequences are trivially seen as solutions to such dynamical systems. We prove that the conjecture holds for holonomic sequences and propose two algorithms for converting holonomic recurrence equations into such quasi-linear equations. The two algorithms differ in their efficiency a…

Program Synthesis in Saturation
Petra Hozzov\'a, Laura Kov\'acs, Chase Norman, Andrei Voronkov
https://arxiv.org/abs/2402.18962 https://

Program Synthesis in Saturation
We present an automated reasoning framework for synthesizing recursion-free programs using saturation-based theorem proving. Given a functional specification encoded as a first-order logical formula, we use a first-order theorem prover to both establish validity of this formula and discover program fragments satisfying the specification. As a result, when deriving a proof of program correctness, we also synthesize a program that is correct with respect to the given specification. We describe pr…

This https://arxiv.org/abs/2404.04515 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csPL_…

Predictable Verification using Intrinsic Definitions
We propose a novel mechanism of defining data structures using intrinsic definitions that avoids recursion and instead utilizes monadic maps satisfying local conditions. We show that intrinsic definitions are a powerful mechanism that can capture a variety of data structures naturally. We show that they also enable a predictable verification methodology that allows engineers to write ghost code to update monadic maps and perform verification using reduction to decidable logics. We evaluate our …

This https://arxiv.org/abs/2402.13828 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csNE_…

Origami: (un)folding the abstraction of recursion schemes for program synthesis
Program synthesis with Genetic Programming searches for a correct program that satisfies the input specification, which is usually provided as input-output examples. One particular challenge is how to effectively handle loops and recursion avoiding programs that never terminate. A helpful abstraction that can alleviate this problem is the employment of Recursion Schemes that generalize the combination of data production and consumption. Recursion Schemes are very powerful as they allow the cons…

Kontsevich's Formula for Rational Curves from Classical and Quantum Perspectives
Greg Weiler
https://arxiv.org/abs/2403.19663 https://

Kontsevich's Formula for Rational Curves from Classical and Quantum Perspectives
Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more direct and combinatoric, the second more abstract. In order to achieve this, we introduce several moduli spaces, such as the Deligne-Mumford-Knudsen spaces and the Kontsevich spaces, and exploit their properties. In particular, the boundary structure of these …

This https://arxiv.org/abs/2312.13344 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Improving the five-point bootstrap
We present a new algorithm for the numerical evaluation of five-point conformal blocks in $d$-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the $\langleσσεσσ\rangle$ correlator in the 3d critical Is…

More than 20 years after I first learned to program (in Lisp!), I'm still so 🤩 about recursion, especially tree recursion. Pure magic 💫

Solvable Initial Value Problems Ruled by Discontinuous Ordinary Differential Equations
Olivier Bournez, Riccardo Gozzi
https://arxiv.org/abs/2405.00165 htt…

Solvable Initial Value Problems Ruled by Discontinuous Ordinary Differential Equations
We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call \emph{solvable intitial value problems} and we prove that for this class of problems the unique solution can always be obtained analytically via transfinite recursion. We present several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the…

The Mumford dynamical system and the Gelfand-Dikii recursion
Polina Baron
https://arxiv.org/abs/2402.16975 https://arxiv.org/pdf/2402…

The Mumford dynamical system and the Gelfand-Dikii recursion
In his paper "The Mumford Dynamical System and Hyperelliptic Kleinian Functions" (see arXiv:2402.09218), Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the (P,Q)-recursion introduced in his paper. In the present paper, we further develop the theory of (P,Q)-recursion and describe its connections to the Korteweg-de Vries hierarchy, the Lenard operator, and the Gelfand-Dikii recursion.

Normally I like to boost books that aren’t well known. But I just finished Recursion by Blake Crouch and it’s damn good. #bookstodon
https://www.goodreads.com/book/show/42046112-recursion

This https://arxiv.org/abs/2307.02180 has been replaced.
link: https://scholar.google.com/scholar?q=a

Runtime Repeated Recursion Unfolding in CHR: A Just-In-Time Online Program Optimization Strategy That Can Achieve Super-Linear Speedup
We introduce a just-in-time runtime program transformation strategy based on repeated recursion unfolding. Our online program optimization generates several versions of a recursion differentiated by the minimal number of recursive steps covered. The base case of the recursion is ignored in our technique. Our method is introduced here on the basis of single linear direct recursive rules. When a recursive call is encountered at runtime, first an unfolder creates specializations of the associate…

This https://arxiv.org/abs/2108.06995 has been replaced.
link: https://scholar.google.com/scholar?q=a

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function
We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices $Ω$ in our previous work. Furthermore, we observ…

This https://arxiv.org/abs/2312.05159 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Minimax Performance Limits for Multiple-Model Estimation
This article concerns the performance limits of strictly causal state estimation for linear systems with fixed, but uncertain, parameters belonging to a finite set. In particular, we provide upper and lower bounds on the smallest achievable gain from disturbances to the point-wise estimation error. The bounds rely on forward and backward Riccati recursions -- one forward recursion for each feasible model and one backward recursion for each pair of feasible models. We give simple examples where …

On the discrete analogues of Appell function $F_4$
Ravi Dwivedi, Dwivedi Sahai
https://arxiv.org/abs/2402.18931 https://arxiv.org/pdf…

On the discrete analogues of Appell function $F_4$
In this paper, we study the Appell function $F_4$ from discrete point of view. In particular, we obtain regions of convergence, difference-differential equations, finite and infinite summation formulas and a list of recursion relations satisfied by the discrete analogues of Appell function $F_4$.

Origami: (un)folding the abstraction of recursion schemes for program synthesis
Matheus Campos Fernandes, Fabricio Olivetti de Franca, Emilio Francesquini
https://arxiv.org/abs/2402.13828 https://arxiv.org/pdf/2402.13828
arXiv:2402.13828v1 Announce Type: new
Abstract: Program synthesis with Genetic Programming searches for a correct program that satisfies the input specification, which is usually provided as input-output examples. One particular challenge is how to effectively handle loops and recursion avoiding programs that never terminate. A helpful abstraction that can alleviate this problem is the employment of Recursion Schemes that generalize the combination of data production and consumption. Recursion Schemes are very powerful as they allow the construction of programs that can summarize data, create sequences, and perform advanced calculations. The main advantage of writing a program using Recursion Schemes is that the programs are composed of well defined templates with only a few parts that need to be synthesized. In this paper we make an initial study of the benefits of using program synthesis with fold and unfold templates, and outline some preliminary experimental results. To highlight the advantages and disadvantages of this approach, we manually solved the entire GPSB benchmark using recursion schemes, highlighting the parts that should be evolved compared to alternative implementations. We noticed that, once the choice of which recursion scheme is made, the synthesis process can be simplified as each of the missing parts of the template are reduced to simpler functions, which are further constrained by their own input and output types.

This https://arxiv.org/abs/2402.13206 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Two Formulas for the Number of Lines on Complex Projective Hypersurfaces
Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Libgober, and van der Waerden-Zagier. Schubert calculus via the splitting principle leads to an explicit general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combi…

Origami: (un)folding the abstraction of recursion schemes for program synthesis
Matheus Campos Fernandes, Fabricio Olivetti de Franca, Emilio Francesquini
https://arxiv.org/abs/2402.13828 https://arxiv.org/pdf/2402.13828
arXiv:2402.13828v1 Announce Type: new
Abstract: Program synthesis with Genetic Programming searches for a correct program that satisfies the input specification, which is usually provided as input-output examples. One particular challenge is how to effectively handle loops and recursion avoiding programs that never terminate. A helpful abstraction that can alleviate this problem is the employment of Recursion Schemes that generalize the combination of data production and consumption. Recursion Schemes are very powerful as they allow the construction of programs that can summarize data, create sequences, and perform advanced calculations. The main advantage of writing a program using Recursion Schemes is that the programs are composed of well defined templates with only a few parts that need to be synthesized. In this paper we make an initial study of the benefits of using program synthesis with fold and unfold templates, and outline some preliminary experimental results. To highlight the advantages and disadvantages of this approach, we manually solved the entire GPSB benchmark using recursion schemes, highlighting the parts that should be evolved compared to alternative implementations. We noticed that, once the choice of which recursion scheme is made, the synthesis process can be simplified as each of the missing parts of the template are reduced to simpler functions, which are further constrained by their own input and output types.

This https://arxiv.org/abs/2305.00424 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Policy Iteration Reinforcement Learning Method for Continuous-time Mean-Field Linear-Quadratic Optimal Problem
This paper employs a policy iteration reinforcement learning (RL) method to study continuous-time linear quadratic mean-field control problems in the infinite horizon. The drift and diffusion terms in the dynamics involve the state as well as the control. We investigate the stability and convergence of the RL algorithm using a Lyapunov Recursion. Instead of solving a pair of coupled Riccati equations, the RL technique focuses on strengthening an auxiliary function and the cost functional as the…

On a relation between $\lambda$-full well-ordered sets and weakly compact cardinals
Gabriele Gull\`a
https://arxiv.org/abs/2403.15904 https://

On a relation between $λ$-full well-ordered sets and weakly compact cardinals
We prove, via transfinite recursion, the existence, inside any linearly ordered set of appropriate regular cardinality $λ$, of a particular kind of well-ordered subsets characterized by the property of $λ$-fullness. Let $H$ be a set of regular cardinals: by using our results about well-ordered $λ$-full sets we show that if $\inf H$ is a weakly compact cardinal, then, for every LOTS $X$, $H$-compactness is equivalent to the nonexistence of gaps of types in $H$.

Construction of a new (3 1)-dimensional KdV equation and its closed-form solutions with solitary wave behaviour and conserved vectors
Nardjess Benoudina, Chaudry Massood Khalique, Ji Lin
https://arxiv.org/abs/2404.17156

Construction of a new (3 + 1)-dimensional KdV equation and its closed-form solutions with solitary wave behaviour and conserved vectors
This paper discusses the construction of a new $(3+1)$-dimensional Korteweg-de Vries (KdV) equation. By employing the KdV's recursion operator, we extract two equations, and with elemental computation steps, the obtained result is $ 3u_{xyt}+3u_{xzt}-(u_{t}-6uu_{x}+u_{xxx})_{yz}-2\left(u_{x}\partial_{x}^{-1}u_{y}\right)_{xz}-2\left(u_{x}\partial_{x}^{-1}u_{z}\right)_{xy}=0 $. We then transform the new equation to a simpler one to avoid the appearance of the integral in the equation. Thereafter,…

This https://arxiv.org/abs/2402.13206 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Two Formulas for the Number of Lines on Complex Projective Hypersurfaces
Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Libgober, and van der Waerden-Zagier. Schubert calculus via the splitting principle leads to an explicit general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combi…

Recursion formulas, concentration polytope, and sharp affine isoperimetric inequalities for the volume decomposition functionals
Yu-de Liu, Qiang Sun, Ge Xiong
https://arxiv.org/abs/2404.09152

Recursion formulas, concentration polytope, and sharp affine isoperimetric inequalities for the volume decomposition functionals
New sharp affine isoperimetric inequalities for volume decomposition functionals $X_{2}$ and $X_{3}$ in $\mathbb{R}^n$ are established. To attack these extremal problems, we find the recursion formulas of volume decomposition functionals, then introduce concentration polytope and characterize its geometric structure, especially its vertices and $1$-dimensional faces. The concentration polytope intuitively reinterprets the discrete logarithmic Minkowski problem.

WDVV-based recursion for open Gromov-Witten invariants
Roi Blumberg, Sara B. Tukachinsky
https://arxiv.org/abs/2402.10542 https://arx…

WDVV-based recursion for open Gromov-Witten invariants
We give a computability result for open Gromov-Witten invariants based on open WDVV equations. This is analogous to the result of Kontsevich-Manin for closed Gromov-Witten invariants. For greater generality, we base the argument on a formal object, the Frobenius superpotential, that generalizes several different definitions of open Gromov-Witten invariants. As an application, we show that open Gromov-Witten invariants for some products of projective spaces vanish.

"Like a Nesting Doll": Analyzing Recursion Analogies Generated by CS Students using Large Language Models
Seth Bernstein, Paul Denny, Juho Leinonen, Lauren Kan, Arto Hellas, Matt Littlefield Sami Sarsa, Stephen MacNeil
https://arxiv.org/abs/2403.09409

"Like a Nesting Doll": Analyzing Recursion Analogies Generated by CS Students using Large Language Models
Grasping complex computing concepts often poses a challenge for students who struggle to anchor these new ideas to familiar experiences and understandings. To help with this, a good analogy can bridge the gap between unfamiliar concepts and familiar ones, providing an engaging way to aid understanding. However, creating effective educational analogies is difficult even for experienced instructors. We investigate to what extent large language models (LLMs), specifically ChatGPT, can provide acce…

This https://arxiv.org/abs/2312.14063 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csDB_…

Polynomial Time Convergence of the Iterative Evaluation of Datalogo Programs
Datalogo is an extension of Datalog that allows for aggregation and recursion over an arbitrary commutative semiring. Like Datalog, Datalogo programs can be evaluated via the natural iterative algorithm until a fixed point is reached. However unlike Datalog, the natural iterative evaluation of some Datalogo programs over some semirings may not converge. It is known that the commutative semirings for which the iterative evaluation of Datalogo programs is guaranteed to converge are exactly those …

The Intro to CS exam my daughter is preparing has questions about the space complexity of recursive functions in C. I have no idea if they are supposed to include the stack and whether tail recursion was ever mentioned.

This https://arxiv.org/abs/2106.04339 has been replaced.
link: https://scholar.google.com/scholar?q=a

Quantization of classical spectral curves via topological recursion
We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair rep…

What do you mean, ancient computers with no stack? My first BASIC interpreter didn't have a stack, and it felt really modern! Well, back then. So I passed arrays around and an index. Worked "fine" even for recursion.
But then I wanted to port my recursive generate-a-christmas-tree algorithm to my little cousin's BASIC interpreter, as a Christmas present, and it wouldn't let me pass the array to the subroutine either. That's when things got a bit difficult ...<…

The ancient world before computers had stacks or heaps - The Old New Thing
A lot of computing got done even before we had stacks and heaps.

This https://arxiv.org/abs/2404.13175 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Open-Closed String Duality, Branes, and Topological Recursion
We consider matrix models exhibiting open-closed string duality in two-dimensional string theories with various amounts of supersymmetry. In particular, a relationship between matrix models in the $β= 2$ Wigner-Dyson class and models in the $(1 + 2Γ, 2)$ Altland-Zirnbauer class relates the perturbative solutions of the two systems' string equations. Point-like operator insertions in the closed string theory are mapped to the topological expansion of the free energy in the open string theory. …

Turns out I'm not done.
Damnit. It was supposed to be just a January project.
I'm adding another app along side the Mood-log and Monolith-Face for my Pinetime running WaspOS.
The plan is tagging the time of every meal or snack or drink or toke, tracking when you take your meds, or do your exercise, or whatever. User-configured categories.
The screen divides into four so my experimental hierarchy is:
* Food / Drink / Meds / Fun
* Food
* Meal
* Snack
* Sweets
* Cheese
* Drink
* Water
* Sugar
* Caffine
* Alcohol
* Medic
* Allergy
* Pain
* Vitamin
* ??
* Other
* Cigarette
* MJ
* ???
* ???
When the app is triggered it shows the time of last-milestone for each.
The ??? are empty for now. Tempted to put like LSD and MDMA in there but I've done them so infrequently it'd be not really worth tracking.
One day in and I'm liking knowing when I last fetched a glass of water or ate a snack with confidence. Not even importing into the daily graph yet.
What should it be called? "Level Up" seemed like it might encourage alcoholism given "tracking the number of drinks you actually drink" was part of the plan.
Working title is "milestone" mostly just for the alliteration of "Mood" and "Monolith" and "Milestone".
If you had a 2x2 menu of buttons to mark time-stamps that you could program into your watch, with as much recursion into sub-menus as you needed, how would you set yours up?

Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations
Ewa Damek, Sebastian Mentemeier
https://arxiv.org/abs/2403.13868

Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations
In recent works on the theory of machine learning, it has been observed that heavy tail properties of Stochastic Gradient Descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, Gürbüzbalaban et al. (arXiv:2006.04740) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_k=A_k X_{k-1}+B_k$, for independent and identically distributed pairs $(A_k, B_k)$, whe…

This https://arxiv.org/abs/2306.16881 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

Complexity results for modal logic with recursion via translations and tableaux
This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the $μ$-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the $μ$-calc…

This https://arxiv.org/abs/2010.10751 has been replaced.
link: https://scholar.google.com/scholar?q=a

Sample-path large deviations for a class of heavy-tailed Markov additive processes
For a class of additive processes driven by the affine recursion $X_{n+1} = A_n X_n + B_n$, we develop a sample-path large deviations principle in the $M_1'$ topology on $D [0,1]$. We allow $B_n$ to have both signs and focus on the case where Kesten's condition holds on $A_1$, leading to heavy-tailed distributions. The most likely paths in our large deviations results are step functions with both positive and negative jumps.

Auxiliary Calculations for Graphene-Based Quantum Hall Arrays Using Partially Recursive Star-Mesh Transformations
Dominick S. Scaletta, Albert F. Rigosi
https://arxiv.org/abs/2402.14520

Auxiliary Calculations for Graphene-Based Quantum Hall Arrays Using Partially Recursive Star-Mesh Transformations
A previous mathematical approach adopted for optimizing the number of total device elements required for obtaining high effective quantized resistances in graphene-based quantum Hall array devices (QHARS) has been further explored with partial recursion patterns. Designs would assume the use of epitaxial graphene elements, whose quantized Hall resistance at the ν=2 plateau (R_H \approx 12906.4 \Ohm) becomes the building block for larger effective, quantized resistances. Auxiliary calculations …

This https://arxiv.org/abs/2110.14427 has been replaced.
link: https://scholar.google.com/scholar?q=a

The ODE Method for Asymptotic Statistics in Stochastic Approximation and Reinforcement Learning
The paper concerns the stochastic approximation recursion, \[ θ_{n+1}= θ_n + α_{n + 1} f(θ_n, Φ_{n+1}) \,,\quad n\ge 0, \] where the {\em estimates} $θ_n\in\Re^d$ and $ \{ Φ_n \}$ is a Markov chain on a general state space. In addition to standard Lipschitz assumptions and conditions on the vanishing step-size sequence, it is assumed that the associated \textit{mean flow} $ \tfrac{d}{dt} \vartheta_t = \bar{f}(\vartheta_t)$, is globally asymptotically stable with stationary point denote…

Calculation of Poincar\'e recurrence time and irreversibility in Statistical Mechanics
Dominique Levesque, Nicolas Sourlas
https://arxiv.org/abs/2402.12910

Calculation of Poincaré recurrence time and irreversibility in Statistical Mechanics
One of the important questions in statistical mechanics is how irreversibility (time's arrow) occurs when Newton equations of motion are time reversal invariant. Boltzmann's argument is that the observed irreversibility is due to the very large number of particles. One objection to Boltzmann was based on Poincaré's recursion theorem: a classical hamiltonian confined system returns after some time, called today Poincaré recurrence time (PRT), close to its initial configuration. Boltzmann's rep…

Van Hove singularities in the density of states of a chaotic dynamical system
Bryn Davies
https://arxiv.org/abs/2404.12073 https://ar…

Van Hove singularities in the density of states of a chaotic dynamical system
The statistics of a chaotic recursion relation can be predicted by constructing an associated sequence of periodic elliptic operators. For such operators, the density of states is well understood, can be computed straightforwardly and explicit formulas can often be derived. The example studied here is a non-linear recursion relation which can be related to a sequence of periodic operators generated by a Fibonacci tiling rule. This link is used to derive an explicit formula for the limiting dist…

Two-component integrable extension of general heavenly equation
Wojciech Kry\'nski, Artur Sergyeyev
https://arxiv.org/abs/2402.10317 https://

Two-component integrable extension of general heavenly equation
We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate that if the metrics in question are hyper-para-Kähler, then our system reduces to the general heavenly equation. We also present an infinite hierarchy of nonlocal symmetries and a recursion operator for the system under study.

This https://arxiv.org/abs/2401.17211 has been replaced.
initial toot: https://mastoxiv.page/@arX…

Quantum dynamics in one and two dimensions via recursion method
We report an implementation of the recursion method that addresses quantum many-body dynamics in the nonperturbative regime. The method essentially amounts to constructing a Lanczos basis in the space of operators and solving coupled Heisenberg equations in this basis. The reported implementation has two key ingredients: a computer-algebraic routine for symbolic calculation of nested commutators and a procedure to extrapolate the sequence of Lanczos coefficients according to the universal opera…

This https://arxiv.org/abs/2403.13216 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

The Schwarzschild Black Hole from Perturbation Theory to all Orders
Applying the quantum field theoretic perturbiner approach to Einstein gravity, we compute the metric of a Schwarzschild black hole order by order in perturbation theory. Using recursion, this calculation can be carried out in de Donder gauge to all orders in Newton's constant. The result is a geometric series which is convergent outside a disk of finite radius, and it agrees within its region of convergence with the known de Donder gauge metric of a Schwarzschild black hole. It thus provides a …

Semidirect Product Decompositions for Periodic Regular Languages
Yusuke Inoue, Kenji Hashimoto, Hiroyuki Seki
https://arxiv.org/abs/2403.05088 https://

Semidirect Product Decompositions for Periodic Regular Languages
The definition of period in finite-state Markov chains can be extended to regular languages by considering the transitions of DFAs accepting them. For example, the language $(ΣΣ)^*$ has period two because the length of a recursion (cycle) in its DFA must be even. This paper shows that the period of a regular language appears as a cyclic group within its syntactic monoid. Specifically, we show that a regular language has period $P$ if and only if its syntactic monoid is isomorphic to a submono…

Recurrence formulae for spectral determinants
J. Cunha, P. Freitas
https://arxiv.org/abs/2404.12114 https://arxiv.org/pdf/2404.12114<…

Recurrence formulae for spectral determinants
We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and ea…

This https://arxiv.org/abs/2305.07321 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Searching problems above arithmetical transfinite recursion
We investigate some Weihrauch problems between $\mathsf{ATR}_2$ and $\mathsf{C}_{ω^ω}$ . We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to $\mathsf{ATR}_2$. Furthermore, we introduce the $ω$-model reflection $\mathsf{ATR}_2^{\mathrm{rfn}}$ of $\mathsf{ATR} $ and show that it is an upper bound for problems provable from the axiomatic system $\mathrm{ATR}_0$ which are of the form $\fora…

Auxiliary Calculations for Graphene-Based Quantum Hall Arrays Using Partially Recursive Star-Mesh Transformations
Dominick S. Scaletta, Albert F. Rigosi
https://arxiv.org/abs/2402.14520

Auxiliary Calculations for Graphene-Based Quantum Hall Arrays Using Partially Recursive Star-Mesh Transformations
A previous mathematical approach adopted for optimizing the number of total device elements required for obtaining high effective quantized resistances in graphene-based quantum Hall array devices (QHARS) has been further explored with partial recursion patterns. Designs would assume the use of epitaxial graphene elements, whose quantized Hall resistance at the ν=2 plateau (R_H \approx 12906.4 \Ohm) becomes the building block for larger effective, quantized resistances. Auxiliary calculations …

This https://arxiv.org/abs/2402.10542 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

WDVV-based recursion for open Gromov-Witten invariants
We give a computability result for open Gromov-Witten invariants based on open WDVV equations. This is analogous to the result of Kontsevich-Manin for closed Gromov-Witten invariants. For greater generality, we base the argument on a formal object, the Frobenius superpotential, that generalizes several different definitions of open Gromov-Witten invariants. As an application, we prove vanishing properties for open Gromov-Witten invariants on products of projective spaces.

This https://arxiv.org/abs/2203.16523 has been replaced.
link: https://scholar.google.com/scholar?q=a

Shifted Witten classes and topological recursion
The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold using the Givental--Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Usin…

This https://arxiv.org/abs/2305.00626 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On two-term hypergeometric recursions with free lower parameters
Let $F(n,k)$ be a hypergeometric function that may be expressed so that $n$ appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $\frac{1}{(n)_{k}}$. Only in exceptional cases is $F(n, k)$ such that Zeilberger's algorithm produces a two-term recursion for $\sum_{k = 0}^{\infty} F(n, k)$ obtained via the telescoping of the right-hand side of a difference equation of the form $p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k)$ for fixed $r \in \ma…

Almost-Sure Termination by Guarded Refinement
Simon Oddershede Gregersen, Alejandro Aguirre, Philipp G. Haselwarter, Joseph Tassarotti, Lars Birkedal
https://arxiv.org/abs/2404.08494

Almost-Sure Termination by Guarded Refinement
Almost-sure termination is an important correctness property for probabilistic programs, and a number of program logics have been developed for establishing it. However, these logics have mostly been developed for first-order programs written in languages with specific syntactic patterns for looping. In this paper, we consider almost-sure termination for higher-order probabilistic programs with general references. This combination of features allows for recursion and looping to be encoded throu…

This https://arxiv.org/abs/2308.01101 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond
We introduce and study invariant differential operators acting on the space $\mathcal{H}(Ω)$ of holomorphic functions on the complement ${Ω=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the "complexified unit circle" $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w =1\}$. We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for invest…

This https://arxiv.org/abs/2402.12910 has been replaced.
initial toot: https://mastoxiv.page/@…

Calculation of Poincaré recurrence time and irreversibility in Statistical Mechanics
One of the important questions in statistical mechanics is how irreversibility (time's arrow) occurs when Newton equations of motion are time reversal invariant. One objection to irreversibility is based on Poincaré's recursion theorem: a classical hamiltonian confined system returns after some time, so-called Poincaré recurrence time (PRT), close to its initial configuration. Boltzmann's reply was that for a $N \sim 10^{23} $ macroscopic number of particles, PRT is very large and exceeds the…

The Mumford Dynamical System and Hyperelliptic Kleinian Functions
Victor Buchstaber
https://arxiv.org/abs/2402.09218 https://arxiv.or…

The Mumford Dynamical System and Hyperelliptic Kleinian Functions
We establish differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the $(P,Q)$-recursion, which defines a sequence of functions $P_1,P_2,\ldots$ given the first function of this sequence $P_1$ and a sequence of parameters $h_1,h_2,\ldots$. The general solution of the $(P,Q)$-recursion is shown to give a solution for the parametric graded Korteweg--de Vries hierarchy. We prove that all solutions of the Mumford dynamical $g$-system are determ…

Inversion and Integral Identities in dCFTs
Georgios Sakkas
https://arxiv.org/abs/2403.05243 https://arxiv.org/pdf/2403.05243

Inversion and Integral Identities in dCFTs
This work derives an application from the identities of arXiv:hep-th/0602028 in order to invert four point functions in defect conformal field theories. For this, a recursion relation is established and the O(N) model with a line defect is considered as a testing ground of this application. Specifically, the CFT data are calculated from inversion of tilt and displacement four point functions. The recursion relation enables efficient computation of hypergeometrics at order $ε$ in the $ε$-expan…

Predictable Verification using Intrinsic Definitions
Adithya Murali, Cody Rivera, P. Madhusudan
https://arxiv.org/abs/2404.04515 https://

Predictable Verification using Intrinsic Definitions
We propose a novel mechanism of defining data structures using intrinsic definitions that avoids recursion and instead utilizes monadic maps satisfying local conditions. We show that intrinsic definitions are a powerful mechanism that can capture a variety of data structures naturally. We show that they also enable a predictable verification methodology that allows engineers to write ghost code to update monadic maps and perform verification using reduction to decidable logics. We evaluate our …

This https://arxiv.org/abs/2203.16523 has been replaced.
link: https://scholar.google.com/scholar?q=a

Shifted Witten classes and topological recursion
The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold using the Givental--Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Usin…

A New Algorithm With Lower Complexity for Bilevel Optimization
Haimei Huo, Zhixun Su
https://arxiv.org/abs/2404.11377 https://arxiv.o…

A New Algorithm With Lower Complexity for Bilevel Optimization
Many stochastic algorithms have been proposed to solve the bilevel optimization problem, where the lower level function is strongly convex and the upper level value function is nonconvex. In particular, exising Hessian inverse-free algorithms that utilize momentum recursion or variance reduction technqiues can reach an $ε$-stationary point with a complexity of $\tilde{O}(ε^{-1.5})$ under usual smoothness conditions. However, $\tilde{O}(ε^{-1.5})$ is a complexity higher than $O(ε^{-1.5})$. H…

Magnetic properties of low-angle twisted bilayer graphene at three-quarters filling
Kevin J. U. Vidarte, Caio Lewenkopf
https://arxiv.org/abs/2404.08177 ht…

Magnetic properties of low-angle twisted bilayer graphene at three-quarters filling
We present a theoretical investigation of the magnetic properties exhibited by twisted bilayer graphene systems with small twist angles, where the appearance of flat minibands strongly enhances electron-electron interaction effects. We consider a tight-binding Hamiltonian combined with a Hubbard mean-field interaction term and employ a real-space recursion technique to self-consistently calculate the system's local density of states. The $O({\cal N})$ efficiency of the recursion method makes it…

This https://arxiv.org/abs/2104.14316 has been replaced.
link: https://scholar.google.com/scholar?q=a

Separation of PSPACE and EXP
This article shows that PSPACE not equal EXP. A simple but novel proof technique has been used to separate these two classes. Whether an arbitrary Turing machine accepts an input when the running time is limited has been computed in this paper. Then, the limit goes to infinity. Thus, methods of the recursion theory can be applied to problems of computational complexity theory without violating the relativization barrier.

This https://arxiv.org/abs/2312.02572 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csDB_…

Efficient Enumeration of Recursive Plans in Transformation-based Query Optimizers
Query optimizers built on the transformation-based Volcano/Cascades framework are used in many database systems. Transformations proposed earlier on the logical query dag (LQDAG) data structure, which is key in such a framework, focus only on recursion-free queries. In this paper, we propose the recursive logical query dag (RLQDAG) which extends the LQDAG with the ability to capture and transform recursive queries, leveraging recent developments in recursive relational algebra. Specifically, th…

Banded totally positive matrices and normality for mixed multiple orthogonal polynomials
Am\'ilcar Branquinho, Ana Foulqui\'e-Moreno, Manuel Ma\~nas
https://arxiv.org/abs/2404.13965

Banded totally positive matrices and normality for mixed multiple orthogonal polynomials
This paper serves as an introduction to banded totally positive matrices, exploring various characterizations and associated properties. A significant result within is the demonstration that the collection of such matrices forms a semigroup, notably including a subset permitting positive bidiagonal factorization. Moreover, the paper applies this concept to investigate step line normality concerning the degrees of associated recursion polynomials. It presents a spectral Favard theorem, ensuring …

This https://arxiv.org/abs/2307.09064 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csPL_…

Newtonian Program Analysis of Probabilistic Programs
Due to their quantitative nature, probabilistic programs pose non-trivial challenges for designing compositional and efficient program analyses. Many analyses for probabilistic programs rely on iterative approximation. This article presents an interprocedural dataflow-analysis framework, called NPA-PMA, for designing and implementing (partially) non-iterative program analyses of probabilistic programs with unstructured control-flow, nondeterminism, and general recursion. NPA-PMA is based on New…

This https://arxiv.org/abs/2104.14316 has been replaced.
link: https://scholar.google.com/scholar?q=a

Separation of PSPACE and EXP
This article shows that PSPACE not equal EXP. A simple but novel proof technique has been used to separate these two classes. Whether an arbitrary Turing machine accepts an input when the running time is limited has been computed in this paper. Then, the limit goes to infinity. Thus, methods of the recursion theory can be applied to problems of computational complexity theory without violating the relativization barrier.

This https://arxiv.org/abs/2308.05705 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
We study four point planar loop amplitudes at an arbitrary point in the Coulomb branch of $\mathcal{N}=4$ super-Yang-Mills theory. We study two particle unitary cuts up to four loop order. We explicitly verify that bubble and triangle graphs do not contribute at one loop level and show that the results hold at higher loop level as well. We also write down an all loop recursion relation for two particle reducible graphs for four point amplitudes.

On the irreducibility and convergence of a class of nonsmooth nonlinear state-space models on manifolds
Armand Gissler, Alain Durmus, Anne Auger
https://arxiv.org/abs/2402.06447 <…

On the irreducibility and convergence of a class of nonsmooth nonlinear state-space models on manifolds
In this paper, we analyze a large class of general nonlinear state-space models on a state-space X, defined by the recursion $ϕ_{k+1} = F(ϕ_k,α(ϕ_k,U_{k+1}))$, $k \in\bN$, where $F,α$ are some functions and $\{U_{k+1}\}_{k\in\bN}$ is a sequence of i.i.d. random variables. More precisely, we extend conditions under which this class of Markov chains is irreducible, aperiodic and satisfies important continuity properties, relaxing two key assumptions from prior works. First, the state-space X…

This https://arxiv.org/abs/2402.13206 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Two Formulas for the Number of Lines on Complex Projective Hypersurfaces
Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Libgober, and van der Waerden-Zagier. Schubert calculus via the splitting principle leads to an explicit general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combi…

This https://arxiv.org/abs/2312.02572 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csDB_…

Efficient Enumeration of Recursive Plans in Transformation-based Query Optimizers
Query optimizers built on the transformation-based Volcano/Cascades framework are used in many database systems. Transformations proposed earlier on the logical query dag (LQDAG) data structure, which is key in such a framework, focus only on recursion-free queries. In this paper, we propose the recursive logical query dag (RLQDAG) which extends the LQDAG with the ability to capture and transform recursive queries, leveraging recent developments in recursive relational algebra. Specifically, th…

This https://arxiv.org/abs/2308.05705 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
We study four point planar loop amplitudes at an arbitrary point in the Coulomb branch of $\mathcal{N}=4$ super-Yang-Mills theory. We study two particle unitary cuts up to four loop order. We explicitly verify that bubble and triangle graphs do not contribute at one loop level and show that the results hold at higher loop level as well. We also write down an all loop recursion relation for two particle reducible graphs for four point amplitudes.

Magnetic Phase Diagram and Skyrmions of the Hubbard Model on the Beta-Mn Type Lattice
Yoshiro Kakehashi
https://arxiv.org/abs/2403.07443 https://

Magnetic Phase Diagram and Skyrmions of the Hubbard Model on the Beta-Mn Type Lattice
Magnetic phase diagram for the Hubbard model on the Beta-Mn type lattice has been calculated as a function of the Coulomb interaction energy parameter U and the electron number per atom n by using the generalized Hartree-Fock approximation combined with the recursion method for electronic-structure calculations. The ferromagnetic state, the ferrimagnetic state, and the helimagnetic state as well as the 3Q multiple spin density waves (3QMSDW) states have been obtained on the U-n plane. Their det…

This https://arxiv.org/abs/2402.13206 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Two Formulas for the Number of Lines on Complex Projective Hypersurfaces
Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Libgober, and van der Waerden-Zagier. Schubert calculus via the splitting principle leads to an explicit general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combi…

Revisiting log-periodic oscillations
Jean-Marc Luck
https://arxiv.org/abs/2403.00432 https://arxiv.org/pdf/2403.00432

Revisiting log-periodic oscillations
This work is inspired by a recent study of a two-dimensional stochastic fragmentation model. We show that the configurational entropy of this model exhibits log-periodic oscillations as a function of the sample size, by exploiting an exact recursion relation for the numbers of its jammed configurations. This is seemingly the first statistical-mechanical model where log-periodic oscillations affect the size dependence of an extensive quantity. We then propose and investigate in great depth a one…

This https://arxiv.org/abs/2402.00124 has been replaced.
link: https://scholar.google.com/scholar?q=a

Gravitational Bound Waveforms from Amplitudes
With the aim of computing bound waveforms from scattering amplitudes, we explore gravitational two-body dynamics using the Schwinger-Dyson equations and Bethe-Salpeter recursion. We show that the tree-level scattering waveform admits a natural analytic continuation, in rapidity, to the bound waveform, which we confirm from an independent calculation, in the Post-Newtonian expansion, of the time-domain multipoles at large eccentricity. We demonstrate consistency of this scattering-to-bound map w…

This https://arxiv.org/abs/2402.13206 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Two Formulas for the Number of Lines on Complex Projective Hypersurfaces
Two formulas for the classical number $C_n$ of lines on a generic hypersurface of degree $2n-3$ in $\mathbb{CP}^n$ are obtained which differ from the formulas by Libgober, and van der Waerden-Zagier. Schubert calculus via the splitting principle leads to an explicit general closed-form formula in terms of the Catalan numbers and elementary symmetric polynomials. This in turn yields $C_n$ as a linear difference recursion relation of unbounded order. Thus, for the sequence of certain linear combi…

This https://arxiv.org/abs/2312.13344 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Improving the five-point bootstrap
We present a new algorithm for the numerical evaluation of five-point conformal blocks in $d$-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the $\langleσσεσσ\rangle$ correlator in the 3d critical Is…