Tootfinder

Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus
Hao Shen, Rongchan Zhu, Xiangchan Zhu
https://arxiv.org/abs/2402.19137

Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus
This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on $\mathbb{R}^+\times \mathbb{T}^2$ within the framework of paracontrolled calculus \cite{GIP15}. The model is given by the equation: \begin{equation*} (\partial_t-Δ) u=F(u)η \end{equation*} where $η\in C^{-1-κ}$ with $1/6>κ>0$, and $F\in C_b^2(\mathbb{R})$. Assume that $η\in C^{-1-κ}$ and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows fro…

This https://arxiv.org/abs/2309.15058 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Monoidal Structures in Orthogonal Calculus
Orthogonal Calculus, first developed by Weiss in 1991, provides a calculus of functors for functors from real inner product spaces to spaces. Many of the functors to which Orthogonal Calculus has been applied since carry an additional lax symmetric monoidal structure which has so far been ignored. For instance, the functor $V \mapsto \text{BO}(V)$ admits maps $$\text{BO}(V) \times \text{BO}(W) \to \text{BO}(V \oplus W)$$ which determine a lax symmetric monoidal structure. Our first main resul…

Rewriting and Inductive Reasoning
M\'arton Hajdu, Laura Kov\'acs, Michael Rawson
https://arxiv.org/abs/2402.19199 https://arx…

Rewriting and Inductive Reasoning
Rewriting techniques based on reduction orderings generate "just enough" consequences to retain first-order completeness. This is ideal for superposition-based first-order theorem proving, but for at least one approach to inductive reasoning we show that we are missing crucial consequences. We therefore extend the superposition calculus with rewriting-based techniques to generate sufficient consequences for automating induction in saturation. When applying our work within the unit-equational fr…

When Lawvere meets Peirce: an equational presentation of boolean hyperdoctrines
Filippo Bonchi, Alessandro Di Giorgio, Davide Trotta
https://arxiv.org/abs/2404.18795

When Lawvere meets Peirce: an equational presentation of boolean hyperdoctrines
Fo-bicategories are a categorification of Peirce's calculus of relations. Notably, their laws provide a proof system for first-order logic that is both purely equational and complete. This paper illustrates a correspondence between fo-bicategories and Lawvere's hyperdoctrines. To streamline our proof, we introduce peircean bicategories, which offer a more succinct characterization of fo-bicategories.

This https://arxiv.org/abs/2312.12071 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the de Rham homomorphism for $L_π$-cohomologies
We study the procedure of regularization in the context of the Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry. It provides us with the machinery to handle the de Rham homomorphism for $L_π$-cohomologies. In this respect, we obtain the condition resolving the question of triviality of the kernel for de Rham homomorphism. In particular, we specify the non-trivial cohomology classes explicitly for a sequence of parameters $π= \langle p_0,\,p_1,\dots,\,…

The Peculiar Destiny of Sentiment de Monsieur Leibnitz (May 1705 -- March 1706)
Sandra Bella (AHP-PReST)
https://arxiv.org/abs/2403.20052 https://

The Peculiar Destiny of Sentiment de Monsieur Leibnitz (May 1705 -- March 1706)
During the querelle des infiniment petits, Leibniz wrote several texts to justify using Differential calculus among Parisian savants. However, only three were published. Among these publications, ''Sentiment de Monsieur Leibnitz'' had a peculiar destiny. Although we are aware of the manuscript (Gotha FB A 448--449, Bl. 41--42), it is only recently that we have identified a copy of its impression in the British Library catalogue. This copy was printed in 1706 together with writings by other math…

This https://arxiv.org/abs/2404.16122 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

Generalized Optimization Modulo Theories
Optimization Modulo Theories (OMT) has emerged as an important extension of the highly successful Satisfiability Modulo Theories (SMT) paradigm. The OMT problem requires solving an SMT problem with the restriction that the solution must be optimal with respect to a given objective function. We introduce a generalization of the OMT problem where, in particular, objective functions can range over partially ordered sets. We provide a formalization of and an abstract calculus for the generalized OM…

Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case
Antoine Gloria, Siguang Qi
https://arxiv.org/abs/2402.19182

Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case
The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits…

The algebraic entropies of the Leavitt path algebra and the graph algebras agree
Wolfgang Bock, Crist\'obal Gil Canto, Dolores Mart\'in Barquero, C\'andido Mart\'in Gonz\'alez, Iv\'an Ruiz Campos, Alfilgen Sebandal
https://arxiv.org/abs/2402.18585

The algebraic entropies of the Leavitt path algebra and the graph algebras agree
In this note we prove that the algebras $L_K(E)$ and $KE$ have the same entropy. Entropy is always referred to the standard filtrations in the corresponding kind of algebra. The main argument leans on (1) the holomorphic functional calculus; (2) the relation of entropy with suitable norm of the adjacency matrix; and (3) the Cohn path algebras which yield suitable bounds for the algebraic entropies.

The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only
Delfim F. M. Torres
https://arxiv.org/abs/2404.14458

The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only
Through duality it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or right operators. The emergence of these novel fractional integration by parts formulas inspires the introduction of a new calculus of variations, where only one type of fractional derivative (left or right) is present. This applies to both the problem formulatio…

This https://arxiv.org/abs/2402.02905 has been replaced.
link: https://scholar.google.com/scholar?q=a

Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus
We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincaré principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The…

Extending Network Calculus To Deal With Partially Negative And Decreasing Service Curves
Anja Hamscher, Vlad-Cristian Constantin, Jens B. Schmitt
https://arxiv.org/abs/2403.18042 …

Extending Network Calculus To Deal With Partially Negative And Decreasing Service Curves
Network Calculus (NC) is a versatile analytical methodology to efficiently compute performance bounds in networked systems. The arrival and service curve abstractions allow to model diverse and heterogeneous distributed systems. The operations to compute residual service curves and to concatenate sequences of systems enable an efficient and accurate calculation of per-flow timing guarantees. Yet, in some scenarios involving multiple concurrent flows at a system, the central notion of so-called …

“He focused on issues like infrastructure & downplayed the importance of reproductive rights. ‘It's certainly an issue that needs to be dealt with,’ he told Politico, ‘but not our top issue. I don't think that this is the issue that wins or loses the race.’ That calculus turned out to be wrong, & Lands just gave Democrats good reason to believe Powell will be only the 1st of many Republicans to pay the price at the ballot box this year.”

Alabama Democrat flips GOP seat after campaigning against threats to IVF
Democrat Marilyn Lands defeated Republican Teddy Powell on Tuesday in a nationally watched special election for a GOP-held seat in the Alabama House of Representatives. While Republicans will ...

A Bionic Natural Language Parser Equivalent to a Pushdown Automaton
Zhenghao Wei, Kehua Lin, Jianlin Feng
https://arxiv.org/abs/2404.17343 https://<…

A Bionic Natural Language Parser Equivalent to a Pushdown Automaton
Assembly Calculus (AC), proposed by Papadimitriou et al., aims to reproduce advanced cognitive functions through simulating neural activities, with several applications based on AC having been developed, including a natural language parser proposed by Mitropolsky et al. However, this parser lacks the ability to handle Kleene closures, preventing it from parsing all regular languages and rendering it weaker than Finite Automata (FA). In this paper, we propose a new bionic natural language parser…

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/2310.05661 has been replaced.
link: https://scholar.google.com/scholar?q=a

The calculus of names -- The legacy of Jan Łukasiewicz
With his research on Aristotelian syllogistic, Jan Łukasiewicz (1929; 1934; 1939; 1951/57) initiated a branch of logic called the calculus of names. This section deals with axiomatic systems that analyze various fragments of the logic of names, i.e., one that study various forms of names and functors acting on them, as well as logical relationships between sentences in which these names and functors occur. In this work, we want not only to present the genesis of the calculus of names and its f…

MAG$\pi$!: The Role of Replication in Typing Failure-Prone Communication
Matthew Alan Le Brun, Ornela Dardha
https://arxiv.org/abs/2404.16213 https://

MAG$π$!: The Role of Replication in Typing Failure-Prone Communication
MAG$π$ is a Multiparty, Asynchronous and Generalised $π$-calculus that introduces timeouts into session types as a means of reasoning about failure-prone communication. Its type system guarantees that all possible message-loss is handled by timeout branches. In this work, we argue that the previous is unnecessarily strict. We present MAG$π$!, an extension serving as the first introduction of replication into Multiparty Session Types (MPST). Replication is a standard $π$-calculus construct u…

"As Europe shifts to clean energy, does it still need so much US LNG?"
#Europe #US #US #America

As Europe shifts to clean energy, does it still need so much US LNG?
Gas exporters point to Europe's energy demand to justify building more LNG terminals, but Europe's clean energy transition could change that calculus.

The Flower Calculus
Pablo DonatoPARTOUT
https://arxiv.org/abs/2402.15174 https://arxiv.org/pdf/2402.15174

The Flower Calculus
We introduce the flower calculus, a deep inference proof system for intuitionistic first-order logic inspired by Peirce's existential graphs. It works as a rewriting system over inductive objects called "flowers", that enjoy both a graphical interpretation as topological diagrams, and a textual presentation as nested sequents akin to coherent formulas. Importantly, the calculus dispenses completely with the traditional notion of symbolic connective, operating solely on nested flowers containing…

This https://arxiv.org/abs/2403.12394 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Calculus of variations on hypergraphs
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

A Graphical Calculus for Stable Curvature Invariants
Gregor Weingart
https://arxiv.org/abs/2404.16355 https://arxiv.org/pdf/2404.1635…

A Graphical Calculus for Stable Curvature Invariants
In this article we develop a graphical calculus for stable invariants of Riemannian manifolds akin to the graphical calculus for Rozansky-Witten invariants for hyperkähler manifolds; based on interpreting trivalent graphs with colored edges as stably invariant polynomials on the space of algebraic curvature tensors. In this graphical calculus we describe explicitly the Pfaffian polynomials central to the Theorem of Chern-Gauß-Bonnet and the normalized moment polynomials calculating the moment…

This https://arxiv.org/abs/1911.12944 has been replaced.
link: https://scholar.google.com/scholar?q=a

Pricing and hedging short-maturity Asian options in local volatility models
This paper discusses the short-maturity behavior of Asian option prices and hedging portfolios. We consider the risk-neutral valuation and the delta value of the Asian option having a Hölder continuous payoff function in a local volatility model. The main idea of this analysis is that the local volatility model can be approximated by a Gaussian process at short maturity $T.$ By combining this approximation argument with Malliavin calculus, we conclude that the short-maturity behaviors of Asian…

Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions
Eva-Maria Hekkelman, Edward McDonald, Teun D. H. van Nuland
https://arxiv.org/abs/2404.16338

Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions
We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regu…

This https://arxiv.org/abs/2304.12954 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Cofibrantly generated model structures for functor calculus
Model structures for many different kinds of functor calculus can be obtained by applying a theorem of Bousfield to a suitable category of functors. In this paper, we give a general criterion for when model categories obtained via this approach are cofibrantly generated. Our examples recover the homotopy functor and $n$-excisive model structures of Biedermann and Röndigs, with different proofs, but also include a model structure for the discrete functor calculus of Bauer, Johnson, and McCarthy.

This https://arxiv.org/abs/2402.02905 has been replaced.
link: https://scholar.google.com/scholar?q=a

Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus
We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincaré principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The…

This https://arxiv.org/abs/2312.11639 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_grqc_…

Cosmology in Lorentzian Regge calculus: causality violations, massless scalar field and discrete dynamics
We develop a model of spatially flat, homogeneous and isotropic cosmology in Lorentzian Regge calculus, employing 4-dimensional Lorentzian frusta as building blocks. By examining the causal structure of the discrete spacetimes obtained by gluing such 4-frusta in spatial and temporal direction, we find causality violations if the sub-cells connecting spatial slices are spacelike. A Wick rotation to the Euclidean theory can be defined globally by a complexification of the variables and an analyti…

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/2002.10175 has been replaced.
link: https://scholar.google.com/scholar?q=a

Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids
We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.

The Essense of Useful Evaluation Through Quantitative Types (Extended Version)
Pablo Barenbaum, Delia Kesner, Mariana Milicich
https://arxiv.org/abs/2404.18874

The Essense of Useful Evaluation Through Quantitative Types (Extended Version)
Several evaluation notions for lambda calculus qualify as reasonable cost models according to Slot and van Emde Boas' Invariance Thesis. A notable result achieved by Accattoli and Dal Lago is that leftmost-outermost reduction is reasonable, where the term representation uses sharing and the steps are useful. These results, initially studied in call-by-name, have also been extended to call-by-value. However, the existing formulations of usefulness lack inductive structure, making it challenging …

Multi-controlled Phase Gate Synthesis with ZX-calculus applied to Neutral Atom Hardware
Korbinian Staudacher, Ludwig Schmid, Johannes Zeiher, Robert Wille, Dieter Kranzlm\"uller
https://arxiv.org/abs/2403.10864

Multi-controlled Phase Gate Synthesis with ZX-calculus applied to Neutral Atom Hardware
Quantum circuit synthesis describes the process of converting arbitrary unitary operations into a gate sequence of a fixed universal gate set, usually defined by the operations native to a given hardware platform. Most current synthesis algorithms are designed to synthesize towards a set of single qubit rotations and an additional entangling two qubit gate, such as CX, CZ, or the Molmer Sorensen gate. However, with the emergence of neutral atom based hardware and their native support for gates …

This https://arxiv.org/abs/2306.13405 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion
This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fraction…

The frustration for many of us with Biden comes from a place of knowing that he could do something to alter Netenyahu's calculus. He could withhold military assistance until Netenyahu stops committing the genocide, or come out publicly against what he's doing. Instead, if reports are to be believed, Biden has stuck to privately criticizing Netenyahu, *which isn't working*. This is appeasement all over again, but because the victims are brown, everyone looks the other way.
<…

This https://arxiv.org/abs/2312.15070 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Consistent, multidimensional differential histogramming and summary statistics with YODA 2
Histogramming is often taken for granted, but the power and compactness of partially aggregated, multidimensional summary statistics, and their fundamental connection to differential and integral calculus make them formidable statistical objects, especially when very large data volumes are involved. But expressing these concepts robustly and efficiently in high-dimensional parameter spaces and for large data samples is a highly non-trivial challenge -- doubly so if the resulting library is to r…

Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable
Sasank Mouli
https://arxiv.org/abs/2403.03933 https://a…

Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable
For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over $\{+1,-1\}$ variables requires size $2^{Ω(n)}$. This shows that Polynomial Calculus sizes over the $\{0,1\}$ and $\{+1,-1\}$ bases are incomparable (since Tseitin tautologies show a separation in the other direction) and answers an open …

It was π-day, so I participated in a Pie-Your-Professor to raise funds for charity. Before-and-after. I'm next to the legendary, brilliant Thomas Goodwillie [Calculus of Functors, etc.].
(Non-Americans must rue that April has only 30 days, so their approximation is much worse.)

Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions
Eva-Maria Hekkelman, Edward McDonald, Teun D. H. van Nuland
https://arxiv.org/abs/2404.16338

Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions
We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regu…

This https://arxiv.org/abs/2310.13944 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Game semantics for lattice-based modal μ-calculus
In this paper, we generalize modal $μ$-calculus to the non-distributive (lattice-based) modal $μ$-calculus and formalize some scenarios regarding categorization using it. We also provide a game semantics for the developed logic. The proof of adequacy of this game semantics proceeds by generalizing the unfolding games on the power-set algebras to the arbitrary lattices and showing that these games can be used to determine the least and the greatest fixed points of a monotone operator on a latt…

Stochastic Processes and Mean Square Calculus on Fractal Curves
Alireza Khalili Golmankhaneh, Kerri Welch, Cristina Serpa, Ivanka Stamova
https://arxiv.org/abs/2403.09648

Stochastic Processes and Mean Square Calculus on Fractal Curves
In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square $F^α$-derivative, and fractal mean square integral. The mean square…

Limit formulas for the trace of the functional calculus of quantum channels for $SU(2)$
Robin van Haastrecht
https://arxiv.org/abs/2402.06054 https://

Limit formulas for the trace of the functional calculus of quantum channels for $SU(2)$
Lieb and Solovej \cite{liebsolBloch} studied traces of quantum channels, defined by the leading component in the decomposition of the tensor product of two irreducible representations of $SU(2)$, to establish a Wehrl-type inequality for integrals of convex functions of matrix coefficients. It is proved that the integral is the limit of the trace of the functional calculus of quantum channels. In this paper, we introduce new quantum channels for all the components in the tensor product and gener…

Network Calculus Characterization of Congestion Control for Time-Varying Traffic
Harvinder Lehal, Natchanon Luangsomboon, J\"org Liebeherr
https://arxiv.org/abs/2403.15303

Network Calculus Characterization of Congestion Control for Time-Varying Traffic
Models for the dynamics of congestion control generally involve systems of coupled differential equations. Universally, these models assume that traffic sources saturate the maximum transmissions allowed by the congestion control method. This is not suitable for studying congestion control of intermittent but bursty traffic sources. In this paper, we present a characterization of congestion control for arbitrary time-varying traffic that applies to rate-based as well as window-based congestion …

This https://arxiv.org/abs/2403.10560 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csIT_…

Holographic Phase Retrieval via Wirtinger Flow: Cartesian Form with Auxiliary Amplitude
We propose a new gradient method for holography, where a phase-only hologram is parameterized by not only the phase but also amplitude. The key idea of our approach is the formulation of a phase-only hologram using an auxiliary amplitude. We optimize the parameters using the so-called Wirtinger flow algorithm in the Cartesian domain, which is a gradient method defined on the basis of the Wirtinger calculus. At the early stage of optimization, each element of the hologram exists inside a complex…

This https://arxiv.org/abs/2312.03537 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Analysis on the cone of discrete Radon measures
We study analysis on the cone of discrete Radon measures over a locally compact Polish space $X$. We discuss probability measures on the cone and the corresponding correlation measures and correlation functions on the sub-cone of finite discrete Radon measures over $X$. For this, we consider on the cone an analogue of the harmonic analysis on the configuration space developed in [12]. We also study elements of the difference calculus on the cone: we introduce discrete birth-and-death gradients …

Product Calculus and Stokes Theorem
M. G. Naber
https://arxiv.org/abs/2403.08784 https://arxiv.org/pdf/2403.08784

Product Calculus and Stokes Theorem
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar multiplication and vector addition a set of vector spaces can be constructed analogous to what is found in exterior calculus. A product differential is defined which allows for the product derivative version of closed and exact forms. The product differential also allow…

Statistical validation of a deep learning algorithm for dental anomaly detection in intraoral radiographs using paired data
Pieter Van Leemput, Johannes Keustermans, Wouter Mollemans
https://arxiv.org/abs/2402.14022

Statistical validation of a deep learning algorithm for dental anomaly detection in intraoral radiographs using paired data
This article describes the clinical validation study setup, statistical analysis and results for a deep learning algorithm which detects dental anomalies in intraoral radiographic images, more specifically caries, apical lesions, root canal treatment defects, marginal defects at crown restorations, periodontal bone loss and calculus. The study compares the detection performance of dentists using the deep learning algorithm to the prior performance of these dentists evaluating the images without…

From Rewrite Rules to Axioms in the $\lambda$$\Pi$-Calculus Modulo Theory
Valentin BlotDEDUCTEAM, LMF, ENS Paris Saclay, Gilles DowekDEDUCTEAM, LMF, ENS Paris Saclay, Thomas Traversi\'eDEDUCTEAM, LMF, ENS Paris Saclay, MICS, Th\'eo WinterhalterDEDUCTEAM, LMF, ENS Paris Saclay
https://arxiv.org/abs/2402.09024

From Rewrite Rules to Axioms in the $λ$$Π$-Calculus Modulo Theory
The $λ$$Π$-calculus modulo theory is an extension of simply typed $λ$-calculus with dependent types and user-defined rewrite rules. We show that it is possible to replace the rewrite rules of a theory of the $λ$$Π$-calculus modulo theory by equational axioms, when this theory features the notions of proposition and proof, while maintaining the same expressiveness. To do so, we introduce in the target theory a heterogeneous equality, and we build a translation that replaces each use of the …

This https://arxiv.org/abs/2310.13944 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Game semantics for lattice-based modal μ-calculus
In this paper, we generalize modal $μ$-calculus to the non-distributive (lattice-based) modal $μ$-calculus and formalize some scenarios regarding categorization using it. We also provide a game semantics for the developed logic. The proof of adequacy of this game semantics proceeds by generalizing the unfolding games on the power-set algebras to the arbitrary lattices and showing that these games can be used to determine the least and the greatest fixed points of a monotone operator on a latt…

A couple of weeks ago when I began this PreCalc course I wasn't even confident that I could do the work. Two weeks in and things are starting to make sense. To my surprise I'm even enjoying it a little bit, to the point that I'm considering taking a full calculus course as an elective.

The Goodwillie calculus of polyhedral products
Guy Boyde, Niall Taggart
https://arxiv.org/abs/2402.07774 https://arxiv.org/pdf/2402.0…

The Goodwillie calculus of polyhedral products
We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. We do this by analysing the behaviour on calculus of the Denham-Suciu fibre sequence, the Iriye-Kishimoto decomposition of the polyhedral product constructed from a collection of pairs of cones and their bases, and the Hilton-Milnor decomposition. As a corollary we show that the Goodwillie calculus of these polyhedral products converges int…

The frustration for many of us with Biden comes from a place of knowing that he could do something to alter Netenyahu's calculus. He could withhold military assistance until Netenyahu stops committing the genocide, or come out publicly against what he's doing. Instead, if reports are to be believed, Biden has stuck to privately criticizing Netenyahu, *which isn't working*. This is appeasement all over again, but because the victims are brown, everyone looks the other way.
<…

(Towards a) Statistical Probabilistic Lazy Lambda Calculus
Radha Jagadeesan
https://arxiv.org/abs/2403.15570 https://arxiv.org/pdf/24…

(Towards a) Statistical Probabilistic Lazy Lambda Calculus
We study the desiderata on a model for statistical probabilistic programming languages. We argue that they can be met by a combination of traditional tools, namely open bisimulation and probabilistic simulation.

This https://arxiv.org/abs/2111.13384 has been replaced.
link: https://scholar.google.com/scholar?q=a

EOLANG and $φ$-calculus
Object-oriented programming (OOP) is one of the most popular paradigms used for building software systems. However, despite its industrial and academic popularity, OOP is still missing a formal apparatus similar to $λ$-calculus, which functional programming is based on. There were a number of attempts to formalize OOP, but none of them managed to cover all the features available in modern OO programming languages, such as C++ or Java. We have made yet another attempt and created $φ$-calculus.…

Network Calculus Bounds for Time-Sensitive Networks: A Revisit
Yuming Jiang
https://arxiv.org/abs/2403.13656 https://arxiv.org/pdf/24…

Network Calculus Bounds for Time-Sensitive Networks: A Revisit
Network calculus (NC), particularly its min-plus branch, has been extensively utilized to construct service models and compute delay bounds for time-sensitive networks (TSNs). This paper provides a revisit to the fundamental results. In particular, counterexamples to the most basic min-plus service models, which have been proposed for TSNs and used for computing delay bounds, indicate that the packetization effect has often been overlooked. To address, the max-plus branch of NC is also consider…

This https://arxiv.org/abs/2211.14525 has been replaced.
link: https://scholar.google.com/scholar?q=a

Calculus rules for proximal ε-subdifferentials and inexact proximity operators for weakly convex functions
We investigate inexact proximity operators for weakly convex functions. To this aim, we derive sum rules for proximal ε-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas. This allows us to investigate inexact proximity operators for weakly convex functions in terms of proximal ε-subdifferentials.

This https://arxiv.org/abs/2401.12721 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Wasserstein Diffusion on Multidimensional Spaces
Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^β$ on ${\mathcal P}(M)$, heuristically given as $$d\mathbb{P}^β(μ)=\frac{1}{Z} e^{-β\, \text{Ent}(μ| m)}\ d\mathbb{P}^*(μ);$$ (ii) associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\mathcal E…

(Towards a) Statistical Probabilistic Lazy Lambda Calculus
Radha Jagadeesan
https://arxiv.org/abs/2403.15570 https://arxiv.org/pdf/24…

(Towards a) Statistical Probabilistic Lazy Lambda Calculus
We study the desiderata on a model for statistical probabilistic programming languages. We argue that they can be met by a combination of traditional tools, namely open bisimulation and probabilistic simulation.

Stochastic Processes and Mean Square Calculus on Fractal Curves
Alireza Khalili Golmankhaneh, Kerri Welch, Cristina Serpa, Ivanka Stamova
https://arxiv.org/abs/2403.09648

Stochastic Processes and Mean Square Calculus on Fractal Curves
In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square $F^α$-derivative, and fractal mean square integral. The mean square…

This https://arxiv.org/abs/2307.15734 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Collinear fragmentation at NNLL: generating functionals, groomed correlators and angularities
Jet calculus offers a unique mathematical technique to bridge the area of QCD resummation with Monte Carlo parton showers. With the ultimate goal of constructing next-to-next-to-leading logarithmic (NNLL) parton showers we study, using the language of generating functionals, the collinear fragmentation of final-state partons. In particular, we focus on the definition and calculation of the Sudakov form factor, which physically describes the no-emission probability in an ordered branching proces…

This https://arxiv.org/abs/2403.12394 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Calculus of variations on hypergraphs
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

Lax Additivity
Merlin Christ, Tobias Dyckerhoff, Tashi Walde
https://arxiv.org/abs/2402.12251 https://arxiv.org/pdf/2402.12251…

Lax Additivity
We introduce notions of lax semiadditive and lax additive $(\infty,2)$-categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete $(\infty,2)$-categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and map…

This https://arxiv.org/abs/2204.07839 has been replaced.
link: https://scholar.google.com/scholar?q=a

Dependence Logics in Temporal Settings
Many forms of dependence manifest themselves over time, with behavior of variables in dynamical systems as a paradigmatic example. This paper studies temporal dependence in dynamical systems from a logical perspective, by enriching a minimal modal base logic of static functional dependencies. We first introduce a logic for dynamical systems featuring temporalized variables, provide a complete axiomatic proof calculus, and show that its satisfiability problem is decidable. Then, to capture expli…

Network Calculus Bounds for Time-Sensitive Networks: A Revisit
Yuming Jiang
https://arxiv.org/abs/2403.13656 https://arxiv.org/pdf/24…

Network Calculus Bounds for Time-Sensitive Networks: A Revisit
Network calculus (NC), particularly its min-plus branch, has been extensively utilized to construct service models and compute delay bounds for time-sensitive networks (TSNs). This paper provides a revisit to the fundamental results. In particular, counterexamples to the most basic min-plus service models, which have been proposed for TSNs and used for computing delay bounds, indicate that the packetization effect has often been overlooked. To address, the max-plus branch of NC is also consider…

This https://arxiv.org/abs/2204.07839 has been replaced.
link: https://scholar.google.com/scholar?q=a

Dependence Logics in Temporal Settings
Many forms of dependence manifest themselves over time, with behavior of variables in dynamical systems as a paradigmatic example. This paper studies temporal dependence in dynamical systems from a logical perspective, by enriching a minimal modal base logic of static functional dependencies. We first introduce a logic for dynamical systems featuring temporalized variables, provide a complete axiomatic proof calculus, and show that its satisfiability problem is decidable. Then, to capture expli…

This https://arxiv.org/abs/2305.06964 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Koszul self duality of manifolds
We show that Koszul duality for operads in $(\mathrm{Top},\times)$ can be expressed via generalized Thom complexes. As an application, we prove the Koszul self duality of the little disk modules $E_M$. We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.

This https://arxiv.org/abs/2310.02845 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

Note on a Translation from First-Order Logic into the Calculus of Relations Preserving Validity and Finite Validity
In this note, we give a linear-size translation from formulas of first-order logic into equations of the calculus of relations preserving validity and finite validity. Our translation also gives a linear-size conservative reduction from formulas of first-order logic into formulas of the three-variable fragment of first-order logic.

Rational approximation of operator semigroups via the $\mathcal B$-calculus
Alexander Gomilko, Yuri Tomilov
https://arxiv.org/abs/2403.14411 https://

Rational approximation of operator semigroups via the $\mathcal B$-calculus
We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the $\mathcal B$- functional calculus developed recently. O…

Simplifying explicit subtyping coercions in a polymorphic calculus with effects
Filip Koprivec, Matija Pretnar
https://arxiv.org/abs/2404.04218 https://

Simplifying explicit subtyping coercions in a polymorphic calculus with effects
Algebraic effect handlers are becoming increasingly popular way of structuring and reasoning about effectful computations, and their performance is often a concern. One of the proposed approaches towards efficient compilation is tracking effect information through explicit subtyping coercions. However, in the presence of polymorphism, these coercions are compiled to additional arguments of compiled functions, incurring significant overhead. In this paper, we present a polymorphic effectful ca…

This https://arxiv.org/abs/2209.14019 has been replaced.
link: https://scholar.google.com/scholar?q=a

Inertial Quasi-Newton Methods for Monotone Inclusion: Efficient Resolvent Calculus and Primal-Dual Methods
We introduce an inertial quasi-Newton Forward-Backward Splitting Algorithm to solve a class of monotone inclusion problems. While the inertial step is computationally cheap, in general, the bottleneck is the evaluation of the resolvent operator. A change of the metric makes its computation hard even for (otherwise in the standard metric) simple operators. In order to fully exploit the advantage of adapting the metric, we develop a new efficient resolvent calculus for a low-rank perturbed standa…

Lax Additivity
Merlin Christ, Tobias Dyckerhoff, Tashi Walde
https://arxiv.org/abs/2402.12251 https://arxiv.org/pdf/2402.12251…

Lax Additivity
We introduce notions of lax semiadditive and lax additive $(\infty,2)$-categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete $(\infty,2)$-categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and map…

Web bases in degree two from hourglass plabic graphs
Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, Jessica Striker, Joshua P. Swanson
https://arxiv.org/abs/2402.13978

Web bases in degree two from hourglass plabic graphs
Webs give a diagrammatic calculus for spaces of $U_q(\mathfrak{sl}_r)$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $U_q(\mathfrak{sl}_4)$-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary $U_q(\mathfrak{sl}_r)$. Here, we show that Fraser's basis agrees with that predicted by the hourglass plabic graph framework and gi…

Calculus of variations on hypergraphs
Mengqiu Shao, Yulu Tian, Liang Zhao
https://arxiv.org/abs/2403.12394 https://arxiv.org/pdf/2403…

Calculus of variations on hypergraphs
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

Admissibility of the Structural Rules in the Sequent Calculus with Equality
Franco Parlamento, Flavio Previale
https://arxiv.org/abs/2403.06887 https://

Admissibility of the Structural Rules in the Sequent Calculus with Equality
On the ground of a general theorem concerning the admissibility of the structural rules in sequent calculi with additional atomic rules, we develop a proof theoretic analysis for several extensions of the ${\bf G3[mic]}$ sequent calculi with rules for equality, including the one originally proposed by H.Wang. In the classical case we relate our results with the semantic tableau method for first order logic with equality. In particular we establish that, for languages without function symbols, i…

Profinite trees, through monads and the lambda-calculus
Vincent Moreau
https://arxiv.org/abs/2402.13086 https://arxiv.org/pdf/2402.13…

Profinite trees, through monads and the lambda-calculus
In its simplest form, the theory of regular languages is the study of sets of finite words recognized by finite monoids. The finiteness condition on monoids gives rise to a topological space whose points, called profinite words, encode the limiting behavior of words with respect to finite monoids. Yet, some aspects of the theory of regular languages are not particular to monoids and can be described in a general setting. On the one hand, Bojańczyk has shown how to use monads to generalize the …

This https://arxiv.org/abs/2011.08808 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lax monoidal adjunctions, two-variable fibrations and the calculus of mates
We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As a sample application we obtain an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint functors between symmetric monoidal …

This https://arxiv.org/abs/2308.05004 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Differentiability in infinite dimension and the Malliavin calculus
In this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [20]) and L. Gross (see [41]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus.

This https://arxiv.org/abs/2101.05754 has been replaced.
link: https://scholar.google.com/scholar?q=a

A Strong Bisimulation for a Classical Term Calculus
When translating a term calculus into a graphical formalism many inessential details are abstracted away. In the case of $λ$-calculus translated to proof-nets, these inessential details are captured by a notion of equivalence on $λ$-terms known as $\simeq_σ$-equivalence, in both the intuitionistic (due to Regnier) and classical (due to Laurent) cases. The purpose of this paper is to uncover a strong bisimulation behind $\simeq_σ$-equivalence, as formulated by Laurent for Parigot's $λμ$-ca…

This https://arxiv.org/abs/2401.07486 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A derivation of first variation formulas from the strain-displacement relations in thin shell theory
In this paper, we derive the first variation formulas for surfaces in 3-dimensional Euclidean space by using the ``strain-displacement relations'' known in thin shell theory. For applications to architectural surface design, we focus on the objective function which has linear Weingarten surfaces as stationary points. This article aims to provide an elementary and cross-disciplinary exposition for applications without any tensor calculus, and thus, we do not give any new mathematical results.

This https://arxiv.org/abs/2209.14019 has been replaced.
link: https://scholar.google.com/scholar?q=a

Inertial Quasi-Newton Methods for Monotone Inclusion: Efficient Resolvent Calculus and Primal-Dual Methods
We introduce an inertial quasi-Newton Forward-Backward Splitting Algorithm to solve a class of monotone inclusion problems. While the inertial step is computationally cheap, in general, the bottleneck is the evaluation of the resolvent operator. A change of the metric makes its computation hard even for (otherwise in the standard metric) simple operators. In order to fully exploit the advantage of adapting the metric, we develop a new efficient resolvent calculus for a low-rank perturbed standa…

Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$
Nikos Katzourakis, Roger Moser
https://arxiv.org/abs/2403.12625

Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$
We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(Ω)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is a positive matrix, and $Ω\Subset \mathbb R^n$. Higher order problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an establis…

This https://arxiv.org/abs/2011.08808 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lax monoidal adjunctions, two-variable fibrations and the calculus of mates
We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As a sample application we obtain an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint functors between symmetric monoidal …

A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq
Reynald Affeldt, Zachary Stone
https://arxiv.org/abs/2403.18229 https://

A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq
Formalization of real analysis offers a chance to rebuild traditional proofs of important theorems as unambiguous theories that can be interactively explored. This paper provides a comprehensive overview of the Lebesgue Differentiation Theorem formalized in the Coq proof assistant, from which the first Fundamental Theorem of Calculus (FTC) for the Lebesgue integral is obtained as a corollary. Proving the first FTC in this way has the advantage of decomposing into loosely-coupled theories of mod…

A Context for Manifold Calculus
Kensuke Arakawa
https://arxiv.org/abs/2403.03321 https://arxiv.org/pdf/2403.03321

A Context for Manifold Calculus
We develop Weiss's manifold calculus in the setting of $\infty$-categories, where we allow the target $\infty$-category to be any $\infty$-category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary $\infty$-categories with small limits.

This https://arxiv.org/abs/2311.18757 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Analytic Besov functional calculus for several commuting operators
This paper investigates when analytic Besov functions of $n$ variables act on the generators of $n$ commuting $C_0$-semigroups on a Banach space. The theory for $n=1$ has already been published, and the present paper uses a different approach to that case as well as extending to the cases when $n\ge2$. It also clarifies some spectral mapping properties and provides some operator norm estimates.

A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq
Reynald Affeldt, Zachary Stone
https://arxiv.org/abs/2403.18229 https://

A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq
Formalization of real analysis offers a chance to rebuild traditional proofs of important theorems as unambiguous theories that can be interactively explored. This paper provides a comprehensive overview of the Lebesgue Differentiation Theorem formalized in the Coq proof assistant, from which the first Fundamental Theorem of Calculus (FTC) for the Lebesgue integral is obtained as a corollary. Proving the first FTC in this way has the advantage of decomposing into loosely-coupled theories of mod…

Mirroring Call-by-Need, or Values Acting Silly
Beniamino Accattoli, Adrienne Lancelot
https://arxiv.org/abs/2402.12078 https://arxiv.…

Mirroring Call-by-Need, or Values Acting Silly
Call-by-need evaluation for the lambda-calculus can be seen as merging the best of call-by-name and call-by-value, namely the wise erasing behaviour of the former and the wise duplicating behaviour of the latter. To better understand how duplication and erasure can be combined, we design a degenerated calculus, dubbed call-by-silly, that is symmetric to call-by-need in that it merges the worst of call-by-name and call-by-value, namely silly duplications by-name and silly erasures by-value. We…

$L^p$-bounds in Safarov pseudo-differential calculus on manifolds with bounded geometry
Santiago G\'omez Cobos, Michael Ruzhansky
https://arxiv.org/abs/2403.13920

$L^p$-bounds in Safarov pseudo-differential calculus on manifolds with bounded geometry
Given a smooth complete Riemannian manifold with bounded geometry $(M,g)$ and a linear connection $\nabla$ on it (not necessarily a metric one), we prove the $L^p$-boundedness of operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: $ρ>1/3$ and $\nabla$ symmetric; and $\nabla$ flat with any values of $ρ$ and $δ$. More…

Uniform Substitution for Differential Refinement Logic
Enguerrand Prebet, Andr\'e Platzer
https://arxiv.org/abs/2404.16734 https://

Uniform Substitution for Differential Refinement Logic
This paper introduces a uniform substitution calculus for differential refinement logic dRL. The logic dRL extends the differential dynamic logic dL such that one can simultaneously reason about properties of and relations between hybrid systems. Refinements is useful e.g. for simplifying proofs by relating a concrete hybrid system to an abstract one from which the property can be proved more easily. Uniform substitution is the key to parsimonious prover microkernels. It enables the verbatim us…

This https://arxiv.org/abs/2308.05004 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Differentiability in infinite dimension and the Malliavin calculus
In this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [20]) and L. Gross (see [41]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus.

This https://arxiv.org/abs/2210.13886 has been replaced.
link: https://scholar.google.com/scholar?q=a

Properties and Characterisations of Cofree Cartesian Differential Categories
Cartesian differential categories come equipped with a differential operator which formalises the total derivative from multivariable calculus. Cofree Cartesian differential categories always exist over a specified base category, where the general construction is based on Faa di Bruno's formula. A natural question to ask is, when given an arbitrary Cartesian differential category, how can one check if it is cofree without knowing the base category? In this paper, we provide characterisations of…

This https://arxiv.org/abs/2302.05954 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

SCL(FOL) Revisited
This paper presents an up-to-date and refined version of the SCL calculus for first-order logic without equality. The refinement mainly consists of the following two parts: First, we incorporate a stronger notion of regularity into SCL(FOL). Our regularity definition is adapted from the SCL(T) calculus. This adapted definition guarantees non-redundant clause learning during a run of SCL. However, in contrast to the original presentation, it does not require exhaustive propagation. Second, we in…

Generalized Optimization Modulo Theories
Nestan Tsiskaridze, Clark Barrett, Cesare Tinelli
https://arxiv.org/abs/2404.16122 https://a…

Generalized Optimization Modulo Theories
Optimization Modulo Theories (OMT) has emerged as an important extension of the highly successful Satisfiability Modulo Theories (SMT) paradigm. The OMT problem requires solving an SMT problem with the restriction that the solution must be optimal with respect to a given objective function. We introduce a generalization of the OMT problem where, in particular, objective functions can range over partially ordered sets. We provide a formalization of and an abstract calculus for the generalized OM…

Spanning Matrices via Satisfiability Solving
Clemens Eisenhofer, Michael Rawson, Laura Kov\'acs
https://arxiv.org/abs/2402.10610 https://

Spanning Matrices via Satisfiability Solving
We propose a new encoding of the first-order connection method as a Boolean satisfiability problem. The encoding eschews tree-like presentations of the connection method in favour of matrices, as we show that tree-like calculi have a number of drawbacks in the context of satisfiability solving. The matrix setting permits numerous global refinements of the basic connection calculus. We also show that a suitably-refined calculus is a decision procedure for the Bernays-Schönfinkel class.

Embedding Differential Dynamic Logic in PVS
J. Tanner Slagel (NASA), Mariano Moscato (NIA), Lauren White (NASA), C\'esar A. Mu\~noz (NASA), Swee Balachandran (NIA), Aaron Dutle (NASA)
https://arxiv.org/abs/2404.15214

Embedding Differential Dynamic Logic in PVS
Differential dynamic logic (dL) is a formal framework for specifying and reasoning about hybrid systems, i.e., dynamical systems that exhibit both continuous and discrete behaviors. These kinds of systems arise in many safety- and mission-critical applications. This paper presents a formalization of dL in the Prototype Verification System (PVS) that includes the semantics of hybrid programs and dL's proof calculus. The formalization embeds dL into the PVS logic, resulting in a version of dL wh…

This https://arxiv.org/abs/2210.07085 has been replaced.
link: https://scholar.google.com/scholar?q=a

Towards Uniform Certification in QBF
We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments. This approach leads…

On the complexity of normalization for the planar $\lambda$-calculus
Anupam Das, Damiano Mazza, L\^e Th\`anh D\~ung Nguy\^en, Noam Zeilberger
https://arxiv.org/abs/2404.05276

On the complexity of normalization for the planar $λ$-calculus
We sketch a tentative proof of P-completeness for the $β$-convertibility problem on untyped planar (a.k.a. ordered or non-commutative) $λ$-terms.

Superposition with Delayed Unification
Ahmed Bhayat, Johannes Schoisswohl, Michael Rawson
https://arxiv.org/abs/2403.04775 https://ar…

Superposition with Delayed Unification
Classically, in saturation-based proof systems, unification has been considered atomic. However, it is also possible to move unification to the calculus level, turning the steps of the unification algorithm into inferences. For calculi that rely on unification procedures returning large or even infinite sets of unifiers, integrating unification into the calculus is an attractive method of dovetailing unification and inference. This applies, for example, to AC-superposition and higher-order supe…

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…

Quantitative Weakest Hyper Pre: Unifying Correctness and Incorrectness Hyperproperties via Predicate Transformers
Linpeng Zhang, Noam Zilberstein, Benjamin Lucien Kaminski, Alexandra Silva
https://arxiv.org/abs/2404.05097

Quantitative Weakest Hyper Pre: Unifying Correctness and Incorrectness Hyperproperties via Predicate Transformers
We present a novel \emph{weakest pre calculus} for \emph{reasoning about quantitative hyperproperties} over \emph{nondeterministic and probabilistic} programs. Whereas existing calculi allow reasoning about the expected value that a quantity assumes after program termination from a \emph{single initial state}, we do so for \emph{initial sets of states} or \emph{initial probability distributions}. We thus (i)~obtain a weakest pre calculus for hyper Hoare logic and (ii)~enable reasoning about so-…

This https://arxiv.org/abs/2212.11055 has been replaced.
link: https://scholar.google.com/scholar?q=a

Coalgebraic Satisfiability Checking for Arithmetic $μ$-Calculi
The coalgebraic $μ$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic $μ$-calculus includes an exponential-time upper bound on satisfiability checking, which however relies on the availability of tableau rules for the next-step modalities that are sufficiently well-behaved in a formally defined sense; in particular, rule ma…

This https://arxiv.org/abs/2309.04624 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

A General Probabilistic Framework in IMALL: A Concrete Categorical Perspective
We consider the linear lambda-calculus extended with the sup type constructor, which provides an additive conjunction along with a non-deterministic destructor. The sup type constructor has been introduced in the context of quantum computing. In this paper, we study this type constructor within a simple linear logic categorical model, employing the category of semimodules over a commutative semiring. We demonstrate that the non-deterministic destructor, either weighted or not, finds a suitable …

This https://arxiv.org/abs/2305.12601 has been replaced.
link: https://scholar.google.com/scholar?q=a

Simply typed convertibility is TOWER-complete even for safe lambda-terms
We consider the following decision problem: given two simply typed $λ$-terms, are they $β$-convertible? Equivalently, do they have the same normal form? It is famously non-elementary, but the precise complexity - namely TOWER-complete - is lesser known. One goal of this short paper is to popularize this fact. Our original contribution is to show that the problem stays TOWER-complete when the two input terms belong to Blum and Ong's safe $λ$-calculus, a fragment of the simply typed $λ$-cal…