IA Scholar Query: Extensions of Iterative Congruences on Free Iterative Algebras.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 28 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Mathematical Components
https://scholar.archive.org/work/ahuebtxoqbcrbebz5rb2ulla4q
Mathematical Components is the name of a library of formalized mathematics for the Coq system. It covers a variety of topics, from the theory of basic data structures (e.g., numbers, lists, finite sets) to advanced results in various flavors of algebra. This library constitutes the infrastructure for the machine-checked proofs of the Four Color Theorem and of the Odd Order Theorem. The reason of existence of this book is to break down the barriers to entry. While there are several books around covering the usage of the Coq system and the theory it is based on, the Mathematical Components library is built in an unconventional way. As a consequence, this book provides a non-standard presentation of Coq, putting upfront the formalization choices and the proof style that are the pillars of the library. This books targets two classes of public. On the one hand, newcomers, even the more mathematically inclined ones, find a soft introduction to the programming language of Coq, Gallina, and the SSReflect proof language. On the other hand accustomed Coq users find a substantial account of the formalization style that made the Mathematical Components library possible.Assia Mahboubi, Enrico Tassiwork_ahuebtxoqbcrbebz5rb2ulla4qWed, 28 Sep 2022 00:00:00 GMTSpin/Pin-Structures and Real Enumerative Geometry
https://scholar.archive.org/work/3kcjwhzskba5fkeobbfmcq4njq
The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- and Pin-structures of Fukaya-Oh-Ohta-Ono and Solomon, establishes properties of these structures in both perspectives, and again shows them to be suitably equivalent. The second part uses the intrinsic perspective on (relative) Spin- and Pin-structures to detail constructions of orientations on the determinants of real Cauchy-Riemann operators and study their properties. The final part applies the results of the first two parts to the enumerative geometry of real curves and obtains an explicit comparison between the curve signs in the intrinsic definition of Welschinger and later Pin-structure dependent definitions. This comparison makes use of both the classical and instrinisc perspectives on Pin-structures and thus of the equivalence between them established in this monograph. The preface and the introductions to the three parts describe the present work in more detail.Xujia Chen, Aleksey Zingerwork_3kcjwhzskba5fkeobbfmcq4njqWed, 28 Sep 2022 00:00:00 GMTOn Completeness of Cost Metrics and Meta-Search Algorithms in -Calculus
https://scholar.archive.org/work/h4lp7jrk2fgldkq2bo5dj47qd4
In the paper we define three new complexity classes for Turing Machine undecidable problems inspired by the famous Cook/Levin's NP-complete complexity class for intractable problems. These are U-complete (Universal complete), D-complete (Diagonalization complete) and H-complete (Hypercomputation complete) classes. In the paper, in the spirit of Cook/Levin/Karp, we started the population process of these new classes assigning several undecidable problems to them. We justify that some super-Turing models of computation, i.e., models going beyond Turing machines, are tremendously expressive and they allow to accept arbitrary languages over a given alphabet including those undecidable ones. We prove also that one of such super-Turing models of computation - the $-Calculus, designed as a tool for automatic problem solving and automatic programming, has also such tremendous expressiveness. We investigate also completeness of cost metrics and meta-search algorithms in $-calculus.Eugene Eberbachwork_h4lp7jrk2fgldkq2bo5dj47qd4Mon, 26 Sep 2022 00:00:00 GMTOn the non-triviality of the torsion subgroup of the abelianized Johnson kernel
https://scholar.archive.org/work/c3ruc6mrwjd6vbhmnv7um57blq
The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the action of the mapping class group on the Malcev Lie algebra of the fundamental group of the surface. Using the same infinitesimal techniques, we also provide an alternative diagrammatic description of the rational abelianized Johnson kernel, and we include in the results the case of an oriented surface with one boundary component.Quentin Faes, Gwenael Massuyeauwork_c3ruc6mrwjd6vbhmnv7um57blqMon, 26 Sep 2022 00:00:00 GMTFactorization theorems and canonical representations for generating functions of special sums
https://scholar.archive.org/work/ir3pbnzjyvfi7i3rq7xveqwv4q
This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section of the thesis explores the topic of how to best rigorously define a so-termed "canonically best" matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type.Maxie Dion Schmidtwork_ir3pbnzjyvfi7i3rq7xveqwv4qSun, 25 Sep 2022 00:00:00 GMTA topological zero-one law and elementary equivalence of finitely generated groups
https://scholar.archive.org/work/ljrgwkhtazbnlem7ly27rkzkq4
Let 𝒢 denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces 𝒮⊆𝒢 satisfying the following zero-one law: for any sentence σ in the infinitary logic ℒ_ω_1, ω, the set of all models of σ in 𝒮 is either meager or comeager. In particular, we show that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we obtain that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. Our paper has a substantial expository component. We give streamlined proofs of some known results and survey ideas from topology, logic, and geometric group theory relevant to our work. We also discuss some open problems.D. Osinwork_ljrgwkhtazbnlem7ly27rkzkq4Sat, 24 Sep 2022 00:00:00 GMTExceptional sequences in semidistributive lattices and the poset topology of wide subcategories
https://scholar.archive.org/work/hhpustqm65gfbmjqv6nck7pkpi
Let Λ be a finite-dimensional algebra over a field K. We describe how Buan and Marsh's τ-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated Λ-modules. When Λ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with a bijection between their completely join-irreducible and completely meet-irreducible elements, known as rowmotion or simply the "κ-map". Generalizing known results for finite semidistributive lattices, we show that the κ-map determines exactly when a set of completely join-irreducible elements forms a "canonical join representation". A consequence is that the corresponding "canonical join complex" is a flag simplicial complex, as has been shown for finite semidistributive lattices and lattices of torsion classes of finite-dimensional algebras. Finally, in the case of lattices of torsion classes of finite-dimensional algebras, we demonstrate how Jasso's τ-tilting reduction can be encoded using the κ-map. We use this to define κ^d-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to τ-exceptional sequences in the algebra setting.Emily Barnard, Eric J. Hansonwork_hhpustqm65gfbmjqv6nck7pkpiFri, 23 Sep 2022 00:00:00 GMTCryptoSolve: Towards a Tool for the Symbolic Analysis of Cryptographic Algorithms
https://scholar.archive.org/work/v2buw5fkbjc5tj4zogzzvxcdmm
Recently, interest has been emerging in the application of symbolic techniques to the specification and analysis of cryptosystems. These techniques, when accompanied by suitable proofs of soundness/completeness, can be used both to identify insecure cryptosystems and prove sound ones secure. But although a number of such symbolic algorithms have been developed and implemented, they remain scattered throughout the literature. In this paper, we present a tool, CryptoSolve, which provides a common basis for specification and implementation of these algorithms, CryptoSolve includes libraries that provide the term algebras used to express symbolic cryptographic systems, as well as implementations of useful algorithms, such as unification and variant generation. In its current initial iteration, it features several algorithms for the generation and analysis of cryptographic modes of operation, which allow one to use block ciphers to encrypt messages more than one block long. The goal of our work is to continue expanding the tool in order to consider additional cryptosystems and security questions, as well as extend the symbolic libraries to increase their applicability.Dalton Chichesterwork_v2buw5fkbjc5tj4zogzzvxcdmmWed, 21 Sep 2022 00:00:00 GMTSmoothness of components of the Emerton-Gee stack for GL_2
https://scholar.archive.org/work/3j4ebjwvd5dnxjz672siqvzyvi
Let K be a finite unramified extension of ℚ_p, where p>2. [CEGS19] and [EG22] construct a moduli stack of two dimensional mod p representations of the absolute Galois group of K. We show that most irreducible components of this stack (including several non-generic components) are isomorphic to quotients of smooth affine schemes. We also use this quotient presentation to compute global sections on these components.Anthony Guzman, Kalyani Kansal, Iason Kountouridis, Ben Savoie, Xiyuan Wangwork_3j4ebjwvd5dnxjz672siqvzyviTue, 20 Sep 2022 00:00:00 GMTInductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice
https://scholar.archive.org/work/q2m4mq7huvacbphug5esfxbsve
Here we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, i.e. Martin-L\"of-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-L\"of's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-L\"of's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.Maria Emilia Maietti, Samuele Maschio, Michael Rathjenwork_q2m4mq7huvacbphug5esfxbsveTue, 20 Sep 2022 00:00:00 GMTAlgebraic Presentations of Dependent Type Theories
https://scholar.archive.org/work/rsnnmooesjhlbcxmqedquui53a
C-systems were defined by Cartmell as models of generalized algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky's construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky. We construct this equivalence as the restriction of an equivalence between more general structures, called CE-systems and E-systems, respectively. To this end, we identify C-systems and B-systems as "stratified" CE-systems and E-systems, respectively; that is, systems whose contexts are built iteratively via context extension, starting from the empty context.Benedikt Ahrens and Jacopo Emmenegger and Paige Randall North and Egbert Rijkework_rsnnmooesjhlbcxmqedquui53aTue, 20 Sep 2022 00:00:00 GMTA categorical framework for congruence of applicative bisimilarity in higher-order languages
https://scholar.archive.org/work/ucnkh74hybc67hla5zop3w2zd4
Applicative bisimilarity is a coinductive characterisation of observational equivalence in call-by-name lambda-calculus, introduced by Abramsky (1990). Howe (1996) gave a direct proof that it is a congruence, and generalised the result to all languages complying with a suitable format. We propose a categorical framework for specifying operational semantics, in which we prove that (an abstract analogue of) applicative bisimilarity is automatically a congruence. Example instances include standard applicative bisimilarity in call-by-name, call-by-value, and call-by-name non-deterministic λ-calculus, and more generally all languages complying with a variant of Howe's format.Tom Hirschowitz, Ambroise Lafontwork_ucnkh74hybc67hla5zop3w2zd4Tue, 20 Sep 2022 00:00:00 GMTA Solution of the 4th Clay Millennium Problem about the Navier-Stokes Equations
https://scholar.archive.org/work/2gvd54husfdv3dooe4cb2gxssy
In this paper it is solved the 4 th Clay Millennium problem about the Navier-Stokes equations, in the direction of regularity (no blow-up). This is proved for the Navier-Stokes equations for the non-periodic formulation and without external forcing (homogeneous case).The proof is based on discovering a new invariant as a 2D surface density of (rotatory) momentum, derived from the well-known Helmholtz-Kelvin-Stokes velocity circulation invariant. This invariant is indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity..It is proved that not only there is no Blow-up in finite time but not even at the time T=+∞.Konstantinos E. Kyritsis, University Of Ioanninawork_2gvd54husfdv3dooe4cb2gxssyMon, 19 Sep 2022 00:00:00 GMTSynthesizing Nested Relational Queries from Implicit Specifications
https://scholar.archive.org/work/j5j65p5q5zawtl5fr6rroyuaya
Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I⃗) is a logical specification involving the source data I⃗ and the interface data O. It is a valid definition of O in terms of I⃗, if any two models of the specification agreeing on I⃗ agree on O. In contrast, an explicit definition is a query that produces O from I⃗. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous effective implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be effectively converted to explicit definitions in the nested relational calculus NRC. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.Michael Benedikt, Pierre Pradic, Christoph Wernhardwork_j5j65p5q5zawtl5fr6rroyuayaSat, 17 Sep 2022 00:00:00 GMTA System of Interaction and Structure III: The Complexity of BV and Pomset Logic
https://scholar.archive.org/work/ndvjbeywrnderitb6xx3qj4mru
Pomset logic and BV are both logics that extend multiplicative linear logic (with Mix) with a third connective that is self-dual and non-commutative. Whereas pomset logic originates from the study of coherence spaces and proof nets, BV originates from the study of series-parallel orders, cographs, and proof systems. Both logics enjoy a cut-admissibility result, but for neither logic can this be done in the sequent calculus. Provability in pomset logic can be checked via a proof net correctness criterion and in BV via a deep inference proof system. It has long been conjectured that these two logics are the same. In this paper we show that this conjecture is false. We also investigate the complexity of the two logics, exhibiting a huge gap between the two. Whereas provability in BV is NP-complete, provability in pomset logic is Σ_2^p-complete. We also make some observations with respect to possible sequent systems for the two logics.Lê Thành Dũng Nguyên, Lutz Straßburgerwork_ndvjbeywrnderitb6xx3qj4mruFri, 16 Sep 2022 00:00:00 GMTSigned permutohedra, delta-matroids, and beyond
https://scholar.archive.org/work/2yzqr44rpvgeheytlc7pyjivwi
We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.Christopher Eur, Alex Fink, Matt Larson, Hunter Spinkwork_2yzqr44rpvgeheytlc7pyjivwiWed, 14 Sep 2022 00:00:00 GMTThe stable cohomology of the moduli space of curves with level structures
https://scholar.archive.org/work/qidkhko6xjastigsrtqmhwx5wy
We prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves: a polynomial algebra in the Miller-Morita-Mumford classes.Andrew Putmanwork_qidkhko6xjastigsrtqmhwx5wyTue, 13 Sep 2022 00:00:00 GMTThe Lubin-Tate Theory of Configuration Spaces: I
https://scholar.archive.org/work/zcyncnnmvbenfbnhvgmdgb6jdi
We construct a spectral sequence converging to the Morava E-theory of unordered configuration spaces and identify its E^2-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the E-theory of the weight p summands of iterated loop spaces of spheres (parametrising the weight p operations on 𝔼_n-algebras), as well as the E-theory of the configuration spaces of p points on a punctured surface. We read off the corresponding Morava K-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the 𝔽_p-homology of the space of unordered configurations of p particles on a punctured surface.Lukas Brantner, Jeremy Hahn, Ben Knudsenwork_zcyncnnmvbenfbnhvgmdgb6jdiTue, 13 Sep 2022 00:00:00 GMTSmall PSL(2, 𝔽) representations of Seifert fiber space groups
https://scholar.archive.org/work/lg5ebs7nynetzoqvpff27e5g7m
Let M be a Seifert fiber space with non-abelian fundamental group and admitting a triangulation with t tetrahedra. We show that there is a non-abelian PSL(2, 𝔽) quotient where |𝔽| < c(2^20t3^120t) for an absolute constant c>0 and use this to show that the lens space recognition problem lies in coNP for Seifert fiber space input. We end with a discussion of our results in the context of distinguishing lens spaces from other 3–manifolds more generally.Neil R Hoffman, Kathleen L Petersenwork_lg5ebs7nynetzoqvpff27e5g7mMon, 12 Sep 2022 00:00:00 GMTSimple Models and Biased Forecasts
https://scholar.archive.org/work/7v4nptklbbhbveilxm44tb2i4u
This paper proposes a framework in which agents are constrained to use simple time-series models to forecast economic variables and characterizes the resulting biases. It considers agents who can only entertain state-space models with no more than d states, where d measures the agents' cognitive abilities. When the true data-generating process does not have a d-state representation, agents end up with misspecified models and biased forecasts. Under some assumptions, agents attend to the most persistent observables at the expense of less persistent ones. This bias anchors agents' forward-looking decisions to persistent state variables and increases comovement among those decisions. The paper proceeds to study the implications of the theory in the context of new-Keynesian, real business cycle, and Diamond--Mortensen--Pissarides models. In each case, constraining agents to use simple models brings the outcomes more in line with stylized facts.Pooya Molaviwork_7v4nptklbbhbveilxm44tb2i4uMon, 12 Sep 2022 00:00:00 GMT