Gorenstein coresolving categories

Let 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8 Enochs, E. E., Jenda, O. M. G. (1995). Gorenstein injective and projective modules. Math. Z. 220:611–633.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein FP-injective modules [20 Mao, L. X., Ding, N. Q. (2008). Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7:491–506.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein AC-injective modules [3 Bravo, D., Gillespie, J. (2016). Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44:2213–2233.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢?_𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢?_𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.     

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pⁿ-basis which generalizes the pⁿ-basis defined in [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar]], and the concept of an pⁿ-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pⁿ-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar], 3 Furuya , M. , Niitsuma , H. (2006). Regular local rings essentially of finite type over fields of prime characteristic. J. Algebra 306:703–711.[Crossref], [Web of Science ®] , [Google Scholar]]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9 Orbanz , U. ( 1973 ). Höhere Derivationen und Regularität . J. Reine Angew. Mathematik 262/263 : 194 – 204 .[Crossref] , [Google Scholar]]).     

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 ( 5 ): 43 – 49 [CROSSREF] [CSA] [Crossref] , [Google Scholar]) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999 Bogomolny , E. , Gerland , U. , Schmit , C. ( 1999 ). Models of intermediate spectral statistics . Phys. Rev. E 59 : 1315 – 1318 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004 Burq , N. , Zworski , M. ( 2004 ). Geometric control in the presence of a black box . JAMS 17 : 443 – 471 [CROSSREF] [CSA] [Web of Science ®] , [Google Scholar]).     

Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in ? ^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996 Namah , G. ( 1996 ). Asymptotic solution of a Hamilton–Jacobi equation . Asymptotic Anal. 12 ( 4 ): 355 – 370 . [CSA] [Web of Science ®] , [Google Scholar]), Namah and Roquejoffre (1999 Namah , G. , Roquejoffre , J.-M . ( 1999 ). Remarks on the long-time behavior of the solutions of Hamilton–Jacobi equations . Comm. PDE 24 ( 5–6 ): 883 – 893 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Roquejoffre (1998 Roquejoffre , J.-M . ( 1998 ). Comportement asymptotique des solutions d’équations de Hamilton–Jacobi monodimensionnelles . C. R. Acad. Sci. Paris Sér. I Math. 326 ( 2 ): 185 – 189 . [CSA] [Crossref] , [Google Scholar]), Fathi (1998 Fathi , A. ( 1998 ). Sur la convergence du semi-groupe de Lax–Oleinik semigroup . C. R. Acad. Sci. Paris Sér. I Math. 327 ( 3 ): 267 – 270 . [CSA] [Crossref] , [Google Scholar]), Barles and Souganidis (2000 Barles , G. , Souganidis , P. E. ( 2000 ). On the large time behavior of solutions of Hamilton–Jacobi equaitons . SIAM J. Math. Anal. 31 ( 4 ): 925 – 939 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Barles , G. , Souganidis , P. E. ( 2001 ). Space-time periodic solutions and long-time behavior of solutions to quasi-periodic parabolic equations . SIAM J. Math. Anal. 32 ( 6 ): 1311 – 1323 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator.     

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005 Bodnar , M. , Velazquez , J. J. L. ( 2005 ). Derivation of macroscopic equations for individual cell-based models: a formal approach . Math. Methods Appl. Sci. 28 ( 15 ): 1757 – 1779 .[Crossref], [Web of Science ®] , [Google Scholar]; Holm and Putkaradze, 2005 Holm , D. D. , Putkaradze , V. ( 2005 ). Aggregation of finite size particles with variable mobility . Phys. Rev. Lett. 95 : 226106 . [Google Scholar]; Mogilner and Edelstein-Keshet, 1999 Mogilner , A. , Edelstein-Keshet , L. ( 1999 ). A non-local model for a swarm . J. Math. Biol. 38 ( 6 ): 534 – 570 .[Crossref], [Web of Science ®] , [Google Scholar]; Morale et al., 2005 Morale , D. , Capasso , V. , Oelschläger , K. ( 2005 ). An interacting particle system modelling aggregation behavior: from individuals to populations . J. Math. Biol. 50 ( 1 ): 49 – 66 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Topaz and Bertozzi, 2004 Topaz , C. M. , Bertozzi , A. L. ( 2004 ). Swarming patterns in a two-dimensional kinematic model for biological groups . SIAM J. Appl. Math. 65 ( 1 ): 152 – 174 (electronic) .[Crossref], [Web of Science ®] , [Google Scholar]; Topaz et al., 2006 Topaz , C. M. , Bertozzi , A. L. , Lewis , M. A. ( 2006 ). A nonlocal continuum model for biological aggregation . Bull. Math. Biol. 68 ( 7 ): 1601 – 1623 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

Propagation of Singularities for Schrödinger Equations on the Euclidean Space with a Scattering Metric

Given a scattering metric on the Euclidean space. We consider the Schrödinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the “homogeneous wavefront set”. We also prove that the notion of the homogeneous wavefront set is essentially equivalent to that of the quadratic scattering wavefront set introduced by Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). One of the main results in Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) follows on the Euclidean space with a weaker, almost optimal condition on the potential.     

Approximations for Solutions of Lévy-Type Stochastic Differential Equations

Abstract

The problem of the construction of strong approximations with a given order of convergence for jump-diffusion equations is studied. General approximation schemes are constructed for Lévy-type stochastic differential equation. In particular, the article generalizes the results from [2 Gardoń , A. 2004 . The order of approximations for solutions of Ito-type stochastic differential equations with jumps . Stoch. Anal. Appl. 22 ( 3 ): 679 – 699 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 5 Kloeden , P.E. , and Platen , E. 1995 . Numerical Solutions of Stochastic Differential Equations . Springer-Verlag , Berlin . [Google Scholar]]. The Euler and the Milstein schemes are shown for finite and infinite Lévy measure.     

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT

In his recent work, [1] Simson, D. 2000. An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II. J. Algebra, 227: 670–705. [Crossref], [Web of Science ®] , [Google Scholar] and [2] Simson, D. 2001. On Small Right Pure Semisimple Rings and the Structure of their Auslander-Reiten Quiver. Communic. in Algebra, 29 in press[Web of Science ®] , [Google Scholar], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.     

ℤ3-Twisted Representations of Lattice Vertex Operator Algebras

Abstract

Let L be a positive definite even lattice and let g ∈ Aut L be a fixed point free automorphism of order 3. We determine the twisted Zhu's algebra A _? (V _L ) for the lattice vertex operator algebra V _L , where ? is an automorphism of V _L induced from g. As a result, we show that the set of all irreducible ?-twisted modules for V _L (up to isomorphism) are exactly those constructed by Dong and Lepowsky (1996 Dong, C. and Lepowsky, J. 1996. The algebraic structure of relative twisted vertex operators. J. Pure and Applied Algebra, 110: 259–295. [Crossref], [Web of Science ®] , [Google Scholar]) and Lepowsky (1985 Lepowsky, J. 1985. Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA, 82: 8295–8299. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

Optimal Exponential Utility in a Jump Bond Market

We consider the optimal exponential utility in a bond market with jumps basing on a model similar to Björk et al. [4 Björk , T. , Kabanov , Y. , and Runggaldier , W. 1997 . Bond market structure in the presence of marked point processes . Math. Finance 7 : 211 – 223 .[Crossref], [Web of Science ®] , [Google Scholar]], which is arbitrage free. Similar to the normalized integral with respect to the cylindrical martingale first introduced in Mikulevicius and Rozovskii [13 Mikulevicius , R. , and Rozovskii , B.L. 1998 . Normalized stochastic integrals in topological vector spaces . In : Séminaire de probabilités XXXII (Lecture Notes in Mathematics) . Springer , Berlin , pp. 137 – 165 . [Google Scholar]], we introduce the (𝕄, Q ⁰)-normalized martingale and local (𝕄, Q ⁰)-normalized martingale. For a given maturity T ₀ ∈ [0, T*], we describe the minimal entropy martingale (MEM) based on [T ₀, T*] by a backward semimartingale equation (BSE) w.r.t. the (𝕄, Q ⁰)-normalized martingale. Then we give an explicit form of the optimal approximate wealth to the optimal exp-utility problem by making use of the solution of the BSE. Finally, we describe the dynamics of the exp utility indifference valuation of a bounded contingent claim H ∈ L ^∞(? _{T 0} ) by another BSE under the minimal entropy martingale measure in the incomplete market.     

Extension of Goulden–Jackson cluster method on pattern occurrences in random sequences and comparison with Régnier–Szpankowski method

The Goulden–Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1 Goulden, I.P. and Jackson, D.M. 1979. An inversion theorem for cluster decompositions of sequences with distinguished subsequences. Journal of London Mathematical Society, Second Series, 20: 567–576. [Crossref], [Web of Science ®] , [Google Scholar]]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In this paper, we elaborate on one of the several extensions in [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. The extended result is then compared with the method of Régnier–Szpankowski [3 Régnier, M. and Szpankowski, W. 1997. On the approximate pattern occurrences in a text. Proceedings of the compression and complexity of sequences 1997, : 253–264.  [Google Scholar]], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier–Szpankowski method can be simplified to the extended Goulden–Jackson method.     

Self-Dual Hopf Quivers

ABSTRACT

We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004 van Oystaeyen , F. , Zhang , P. ( 2004 ). Quiver Hopf algebras . J. Algebra 280 ( 2 ): 577 – 589 . [CSA] [CROSSREF]  [Google Scholar]). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000 Andruskiewitsch , N. , Schneider , H.-J . ( 2000 ). Finite quantum groups and Cartan matrices . Adv. Math. 154 : 1 – 45 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.     

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

In this paper we pursue the work initiated in [6 Bahuaud , E. ( 2009 ). Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics . Pacific J. Math. 239 : 231 – 249 .[Crossref], [Web of Science ®] , [Google Scholar], 7 Bahuaud , E. , Gicquaud , R. ( 2011 ). Conformal compactification of asymptotically locally hyperbolic metrics . J. Geom. Anal. 21 : 1085 – 1118 .[Crossref], [Web of Science ®] , [Google Scholar]]: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to ?1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor.     

Determining an Unbounded Potential from Cauchy Data in Admissible Geometries

In [4 Dos Santos Ferreira , D. , Kenig , C.E. , Salo , M. , Uhlmann , G. ( 2009 ). Limiting Carleman weights and anisotropic inverse problems . Invent. Math. 178 : 119 – 171 .[Crossref], [Web of Science ®] , [Google Scholar]] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L ^n/2. In the process, we derive L ^p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8 Jerison , D. , Kenig , C.E. ( 1985 ). Unique continuation and absence of positive eigenvalues for Schrödinger operators . Ann. Math. 121 : 463 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]] and Kenig et al. [9 Kenig , C.E. , Ruiz , A. , Sogge , C.D. ( 1987 ). Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators . Duke Math. J. 55 : 329 – 347 .[Crossref], [Web of Science ®] , [Google Scholar]].     

2 × 2 Permanental Ideals of Hypermatrices

We analyze the structure of ideals generated by some classes of 2 × 2 permanents of hypermatrices, generalizing [9 Laubenbacher , R. C. , Swanson , I. ( 2000 ). Permanental ideals . J. Symbolic Comput. 30 : 195 – 205 .[Crossref], [Web of Science ®] , [Google Scholar]] on 2 × 2 permanental ideals of generic matrices. We compare the obtained structure to that of the corresponding determinantal ideals in [11 Swanson , I. , Taylor , A. ( 2013 ). Minimal primes of ideals arising from conditional independence statements . J. Algebra 392 : 299 – 314 .[Crossref], [Web of Science ®] , [Google Scholar]]: as expected, the permanental ideals have many more (minimal) components. In the last two sections, we examine a few related classes of permanental ideals.     

Elementary matrix reduction over J-stable rings

A commutative ring R is J-stable provided that R∕aR has stable range 1 for all a?J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3 Gillman, L., Henriksen, M. (1956). Some remarks about elementary divisor rings. Trans. Amer. Math. Soc. 82:362–365.[Crossref] , [Google Scholar], Theorem 8], [4 Larsen, M., Lewis, W., Shores, T. (1974). Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187:231–248.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 4.1], [7 McGovern, W. W. (2008). Bézout rings with almost stable range 1. J. Pure Appl. Algebra 212:340–348.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.7], [8 Moore, M. E. (1975). A strongly complement property of Dedekind domain. Czechoslovak Math. J. 25(100):282–283. [Google Scholar], Theorem], [9 Moore, M., Steger, A. (1971). Some results on completability in commutative rings. Pacific J. Math. 37:453–460.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.1], [14 Zabavsky, B. V. (1996). Generalized adequate rings. Ukrainian Math. J. 48:614–617.[Crossref] , [Google Scholar], Theorem 1] and [18 Zabavsky, B. V., Komarnyts’kyi, M. Y. (2010). Cohn-type theorem for adequacy and elementary divisor rings. J. Math. Sci. 167:107–111.[Crossref] , [Google Scholar], Theorem 7].     

Sharp Bounds on the Number of Resonances for Conformally Compact Manifolds with Constant Negative Curvature Near Infinity

Abstract

The purpose of this article is to prove a sharp bound on the number of resonances for the Laplacian on conformally compact manifolds with constant negative curvature near infinity, thus improving the polynomial bound of Guillopé and Zworki (Guillopé, L., Zworski, M. ([1995b] Guillopé, L. and Zworski, M. 1995b. Upper bounds on the number of resonances for noncompact Riemann surfaces. J. Funct. Anal., 129: 364–389. [Crossref], [Web of Science ®] , [Google Scholar]). Polynomial bound on the number of resonances for some complete spaces of constant negative curvature near infinity. Asympt. Anal. 11:1–22).     

On weakly clean rings

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2 Anderson, D. D., Camillo, V. P. (2002). Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30(7):3327–3336.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and were extensively investigated by Ahn and Anderson [1 Ahn, M.-S., Anderson, D. D. (2006). Weakly clean rings and almost clean rings. Rocky Mountain J. Math. 36:783–798.[Crossref], [Web of Science ®] , [Google Scholar]], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.