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Efficient Algorithms for Finite -Algebras
No full textFor a finite -algebra , i.e., for a -algebra whichis a finitely generated -module, we assume that is explicitlygiven by a system of -module generators , its relation module, and the structure constants of the multiplication in . Inthis setting we develop and analyze efficient algorithms for computingessential information about . First we provide polynomial time algorithmsfor solving linear systems of equations over and for basic ideal-theoreticoperations in . Then we develop ZPP (zero-error probabilitic polynomialtime) algorithms to compute the nilradical and the maximal ideals of0-dimensional affine algebras with or. The task of finding the associated primes of a finite-algebra is reduced to these cases and solved in ZPPIF (ZPPplus one integer factorization). With the same complexity, we calculate theconnected components of the set of minimal associated primes {\rmminPrimes}(R) and then the primitive idempotents of . Finally, we provethat knowing an explicit representation of is polynomial time equivalent toknowing a strong Gr\"obner basis of an ideal such that .Comment: Published in journal of Groups, Complexity, Cryptolog
Partial-dual polynomial as a framed weight system
No full textRecently, Chmutov proved that the partial-dual polynomial considered as afunction on chord diagrams satisfies the four-term relations. In this paper, weshow that this function on framed chord diagrams also satisfies the four-termrelations, i.e., is a framed weight system.Comment: 10pages, 11figure
On a version of the Problem B of Mahler involving derivatives
No full textIn 1902, Paul St\"ackel constructed an analytic function in aneighborhood of the origin, which was transcendental, and with the propertythat both and its inverse, as well as its derivatives, assumed algebraicvalues at all algebraic points in this neighborhood. Inspired by this result,Mahler in 1976 questioned the existence of an transcendental entire function such that and are subsets of This problem was solved by Marques andMoreira in 2017. As St\"acklel's result involved derivatives, it is natural toquestion whether we have an analogous result for transcendental entirefunctions involving derivatives. In this article, we show that there are anuncountable amount of such functions
Topics in elliptic problems: from semilinear equations to shape optimization
No full textIn this paper, which corresponds to an updated version of the author'sHabilitation lecture in Mathematics, we do an overview of several topics inelliptic problems. We review some old and new results regarding the Lane-Emdenequation, both under Dirichlet and Neumann boundary conditions, then focus onsign-changing solutions for Lane-Emden systems. We also survey some resultsregarding fully nontrivial solutions to gradient elliptic systems with mixedcooperative and competitive interactions. We conclude by exhibiting results onoptimal partition problems, with cost functions either related to Dirichleteigenvalues or to the Yamabe equation. Several open problems are referred alongthe text.Comment: Review article focused on the author's own work(expanded version of his Habilitation lecture).Draws heavily from: arXiv:2305.02870,arXiv:2211.04839,arXiv:2209.02113, arXiv:2109.14753,arXiv:2106.03661,arXiv:2106.00579,arXiv:1908.11090,arXiv:1807.03082, arXiv:1706.08391, arXiv:1701.05005, arXiv:1508.01783,arXiv:1412.4336,arXiv:1409.5693,arXiv:1405.5549,arXiv:1403.6313,arXiv:1307.3981,arXiv:1201.520
DeepSec: Deciding Equivalence Properties for Security Protocols -- Improved theory and practice
No full textAutomated verification has become an essential part in the securityevaluation of cryptographic protocols. In this context privacy-type propertiesare often modelled by indistinguishability statements, expressed as behaviouralequivalences in a process calculus. In this paper we contribute both to thetheory and practice of this verification problem. We establish new complexityresults for static equivalence, trace equivalence and labelled bisimilarity andprovide a decision procedure for these equivalences in the case of a boundednumber of protocol sessions. Our procedure is the first to decide traceequivalence and labelled bisimilarity exactly for a large variety ofcryptographic primitives -- those that can be represented by a subtermconvergent destructor rewrite system. We also implemented the procedure in anew tool, DeepSec. We showed through extensive experiments that it issignificantly more efficient than other similar tools, while at the same timeraises the scope of the protocols that can be analysed.Comment: 136 page
The independence of central banks: a reductio ad absurdum
No full textThis paper testifies to the fact that the proclaimed independence of central banks, as conceived by its founders, is nothing more than a chimera. We demonstrate that the hypothesis ‘inflation is a purely monetary phenomenon’ does not substantiate the case for independence. Further, the portrayal of the conservative central banker, the imaginary principal-agent contract, the alleged financial autonomy, along with the ban on budgetary financing, amount to flawed logic in arguing for the independence of the central bank. We also highlight that the idea of independence is not convincing due to the absence of well-defined outlines in its operational toolbox and the system of rules it relies upon
Expressiveness of SHACL Features and Extensions for Full Equality and Disjointness Tests
No full textSHACL is a W3C-proposed schema language for expressing structural constraintson RDF graphs. Recent work on formalizing this language has revealed a strikingrelationship to description logics. SHACL expressions can use three fundamentalfeatures that are not so common in description logics. These features areequality tests; disjointness tests; and closure constraints. Moreover, SHACL ispeculiar in allowing only a restricted form of expressions (so-called targets)on the left-hand side of inclusion constraints. The goal of this paper is to obtain a clear picture of the impact andexpressiveness of these features and restrictions. We show that each of thefour features is primitive: using the feature, one can express boolean queriesthat are not expressible without using the feature. We also show that therestriction that SHACL imposes on allowed targets is inessential, as long asclosure constraints are not used. In addition, we show that enriching SHACL with "full" versions of equalitytests, or disjointness tests, results in a strictly more powerful language
On certain partial sums involving squares of Hecke eigenvalues
No full textLet be the th Fourier coefficient of a cuspidal Hecke eigenform of even integral weight and trivial character that is a normalized new form for some level . We show that the partial sumsare not integral for
Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation
No full textIn this paper we study the algebraic properties of a new integrabledifferential-difference equation. This equation can be seen as a deformation ofthe modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidtequation in its continuous limit. Using its Lax representation we explicitlyconstruct a recursion operator for this equation and prove that it is aNijenhuis operator. Moreover, we present the bi-Hamiltonian structures for thisnew equation.Comment: 11 page
Robust non-computability of dynamical systems and computability of robust dynamical systems
No full textIn this paper, we examine the relationship between the stability of thedynamical system and the computability of its basins ofattraction. We present a computable system thatpossesses a computable and stable equilibrium point, yet whose basin ofattraction is robustly non-computable in a neighborhood of in the sensethat both the equilibrium point and the non-computability of its associatedbasin of attraction persist when is slightly perturbed. This indicates thatlocal stability near a stable equilibrium point alone is insufficient toguarantee the computability of its basin of attraction. However, we alsodemonstrate that the basins of attraction associated with a structurally stable- globally stable (robust) - planar system defined on a compact set arecomputable. Our findings suggest that the global stability of a system and thecompactness of the domain play a pivotal role in determining the computabilityof its basins of attraction.Comment: arXiv admin note: text overlap with arXiv:2109.1508