Let K be an algebraic number field. With K, we associate the ring of finite adeles [Formula: see ... more Let K be an algebraic number field. With K, we associate the ring of finite adeles [Formula: see text]. Following a recent result of Weisbart on diffusions on finite rational adeles [Formula: see text], we define the Vladimirov operator [Formula: see text] on [Formula: see text] and define the Brownian motion on the group [Formula: see text]. We also consider the Schrödinger operator [Formula: see text] with a potential operator V given by a non-negative continuous function v on [Formula: see text]. We prove a version of the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text].
LetGbe a semi direct productG= ℝd⋊ ℝk. OnGwe consider a class of second order left-invariant diff... more LetGbe a semi direct productG= ℝd⋊ ℝk. OnGwe consider a class of second order left-invariant differential operators of the formLα=∑j=1de2λj(a)∂xj2+∑j=1k(∂aj2−2αj∂aj),${{\cal L}_\alpha } = \sum\limits_{j = 1}^d {\mkern 1mu} e{{\mkern 1mu} ^{2{\lambda _j}(a)}}\partial _{{x_j}}^2 + \sum\limits_{j = 1}^k {(\partial _{{a_j}}^2 - 2{\alpha _j}{\partial _{{a_j}}})},$wherea∈ ℝkandλ1,...,λd∈ (ℝk)*. It is known that bounded 𝓛α-harmonic functions onGare precisely the “Poisson integrals” ofL∞(ℝd) against the Poisson kernelναwhich is a smooth function on ℝd. We prove an upper bound ofναand its derivatives.
This paper deals with Gärdenfors’ theory of conceptual spaces. Let $${\mathcal {S}}$$ S be a conc... more This paper deals with Gärdenfors’ theory of conceptual spaces. Let $${\mathcal {S}}$$ S be a conceptual space consisting of 2-type fuzzy sets equipped with several kinds of metrics. Let a finite set of prototypes $$\tilde{P}_1,\ldots ,\tilde{P}_n\in \mathcal {S}$$ P ~ 1 , … , P ~ n ∈ S be given. Our main result is the construction of a classification algorithm. That is, given an element $${\tilde{A}}\in \mathcal {S},$$ A ~ ∈ S , our algorithm classifies it into the conceptual field determined by one of the given prototypes $$\tilde{P}_i.$$ P ~ i . The construction of our algorithm uses some physical analogies and the Newton potential plays a significant role here. Importantly, the resulting conceptual fields are not convex in the Euclidean sense, which we believe is a reasonable departure from the assumptions of Gardenfors’ original definition of the conceptual space. A partitioning algorithm of the space $$\mathcal {S}$$ S is also considered in the paper. In the application section...
In 2010 Durda, Caron, and Buchanan published a paper in INFOR: Information systems and Operationa... more In 2010 Durda, Caron, and Buchanan published a paper in INFOR: Information systems and Operational Research, entitled: An application of operational research to computational linguistics: Word ambiguity. In this article the authors developed "a new measure of word ambiguity (e.g., homonymy and polysemy) for use in psycholinguistic research". In our work we propose some modification of their algorithm.
Acta Universitatis Lodziensis. Folia Linguistica Rossica, 2018
Radziecki matematyk Andriej Nikołajewicz Kołmogorow (1903‒1987), autor ponad pół tysiąca prac nau... more Radziecki matematyk Andriej Nikołajewicz Kołmogorow (1903‒1987), autor ponad pół tysiąca prac naukowych, był jednym z najwybitniejszych matematyków XX wieku i w pewnym momencie być może nawet największym. W 1960 roku, Kołmogorow rozpoczął intensywne badania w dziedzinie językoznawstwa i teorii wiersza. Wydaje się, że niewielu polskich matematyków, którzy znają niektóre z rezultatów Kołmogorowa w tej czy innej dziedzinie matematyki, wie o jego pracach w dziedzinach nauk humanistycznych. Można podejrzewać, że wielu literaturoznawców i lingwistów również nie wie o istnieniu tych prac. Celem tego artykułu jest wyjaśnienie przyczyn, dla których Kołmogorow zajmował się humanistyką (historią, językoznawstwem, teorią wiersza).
Cognitive Studies | Études cognitives, Dec 3, 2017
This paper proposes a new classification algorithm for the partitioning of a conceptual space. Al... more This paper proposes a new classification algorithm for the partitioning of a conceptual space. All the algorithms which have been used until now have mostly been based on the theory of Voronoi diagrams. This paper proposes an approach based on potential theory, with the criteria for measuring similarities between objects in the conceptual space being based on the Newtonian potential function. The notion of a fuzzy prototype, which generalizes the previous definition of a prototype, is introduced. Furthermore, the necessary conditions that a natural concept must meet are discussed. Instead of convexity, as proposed by Gärdenfors, the notion of geodesically convex sets is used. Thus, if a concept corresponds to a set which is geodesically convex, it is a natural concept. This definition applies, for example, if the conceptual space is an Euclidean space. As a by-product of the construction of the algorithm, an extension of the conceptual space to d-dimensional Riemannian manifolds is obtained.
Abstract. In this note we give an improvement of the estimate of the Poisson kernels for second o... more Abstract. In this note we give an improvement of the estimate of the Poisson kernels for second order differential operators on homo-geneous manifolds of negative curvature obtained for the first time, using some probabilistic techniques, in [I] and then improved by the
propriate set S of generators of order not greater than 2 and a finite set of probability measure... more propriate set S of generators of order not greater than 2 and a finite set of probability measures {p,,..., pk) with their supports in S such that pl*...*pk = 2, where l ( g) = IGI- ' for every g EG. 2000 Mathematics Subject Classification: 60BI5, 43A05. Key words and phrases: Finite Coxcter groups, factorization,
We obtain estimates for derivatives of the Poisson kernels for the second order di#erential opera... more We obtain estimates for derivatives of the Poisson kernels for the second order di#erential operators on homogeneous manifolds of negative curvature both in the coercive and noncoercive case. 1.
Abstract. We consider the Green functions G for second-order coercive differen-tial operators on ... more Abstract. We consider the Green functions G for second-order coercive differen-tial operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R+. Estimates for derivatives of the Green functions G with respect to the N and A-variables are obtained. This paper completes a previous work of the author (see [12, 13]) where estimates for derivatives of the Green functions for the noncoercive operators has been obtained. Here we show how to use the previous methods and results from [12] in order to get analogous estimates for coercive operators. 1. Introduction. Let M be a connected, simply connected homogeneous manifolds of negative cur-vature. Such a manifold is a solvable Lie group S = NA, a semi-direct product of a nilpotent Lie group N and an Abelian group A = R+. Moreover, for an H belonging to the Lie algebra a of A, the real parts of the eigenvalues of AdexpH |n,
Abstract. We consider the Green functions for second-order left-invariant differential operators ... more Abstract. We consider the Green functions for second-order left-invariant differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R+. We obtain estimates for mixed derivatives of the Green functions both in the coercive and non-coercive case. The current paper completes the previous results obtained by the author in a series of papers [14, 15, 16, 19]. 1.
Abstract. We consider the Green functions for second order non-coercive differential operators on... more Abstract. We consider the Green functions for second order non-coercive differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R+. We obtain estimates for the mixed derivatives of the Green functions that complements a previous work by the same author [17]. 1.
In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker'... more In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker's classical invariance principle. We consider a fuzzy random walk ${S^*_n}=X^*_1+\cdots+X^*_n,$ where $\{X^*_i\}_1^{\infty}$ is a sequence of mutually independent and identically distributed $d$-dimensional fuzzy random variables whose $\alpha$-cuts are assumed to be compact and convex. Our reasoning and technique are based on the well known conjugacy correspondence between convex sets and support functions, which allows for the association of an appropriately normalized and interpolated time-continuous fuzzy random process with a real valued random process in the space of support functions. We show that each member of the associated family of dual sequences tends in distribution to a standard Brownian motion.
The main aim of this note is to find an explicit integral formula for the heat kernel of a certai... more The main aim of this note is to find an explicit integral formula for the heat kernel of a certain second-order left invariant differential operator on a solvable Lie group, being a semi-direct product RnR, by means of a skew-product formula for diffusions. An explicit formula of a different kind for the special case of the operator considered in this note (the operator without the drift term) was found recently by Calin, Chang and Li. As a corollary from our main result we get a simple formula for the return probability from which the asymptotic behaviour for small and large values of time follows easily.
The general problem of how to construct stochastic processes which are confined to stay in a pred... more The general problem of how to construct stochastic processes which are confined to stay in a predefined cone (in the one-dimensional but also multi-dimensional case also referred to as \emph{subordinators}) is of course known to be of great importance in the theory and a myriad of applications.\par But fuzzy stochastic processes are considered in this context for the first time in this paper:\par By first relating with each proper convex cone $C$ in $\R^{n}$ a certain cone of fuzzy vectors $C^*$ and subsequently using some specific Banach space techniques we have been able to produce as many pairs $(L^*_t, C^*)$ of fuzzy \L processes $L^*_t$ and cones $C^*$ of fuzzy vectors such that $L^*_t$ are $C^*$-$\,$subordinators.
We consider the Green functions G for second-order coercive dieren- tial operators on homogeneous... more We consider the Green functions G for second-order coercive dieren- tial operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R + . Estimates for derivatives of the Green functions G with respect to the N and A-variables are obtained. This paper completes a previous work of the author (see (12, 13)) where estimates for derivatives of the Green functions for the noncoercive operators has been obtained. Here we show how to use the previous methods and results from (12) in order to get analogous estimates for coercive operators.
Let W be a finite Coxeter group and let λW be the Haar measure on W, i.e., λW (w) = |W |−1 for ev... more Let W be a finite Coxeter group and let λW be the Haar measure on W, i.e., λW (w) = |W |−1 for every w ∈ W. We prove that there exist a symmetric set T ̸= W of generators of W consisting of elements of order not greater than 2 and a finite set of probability measures {μ1, . . . , μk} with their supports in T such that their convolution product μ1 ∗ . . . ∗ μk = λW . 2000 AMS Mathematics Subject Classification: Primary: 60B15; Secondary: 20F55.
Let Γ be a sub-semigroup of G = GL(d, R), d > 1. We assume that the action of Γ on R d is strongl... more Let Γ be a sub-semigroup of G = GL(d, R), d > 1. We assume that the action of Γ on R d is strongly irreducible and that Γ contains a proximal and expanding element. We describe contraction properties of the dynamics of Γ on R d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space P d−1. In the case where Γ is a sub-semigroup of GL(d, R)∩M (d, Z) and Γ has the above properties, we deduce that the Γ-orbits on T d = R d /Z d are finite or dense.
Let $S$ be a semi direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, no... more Let $S$ be a semi direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with $\R^k,$ $k>1.$ We consider a class of second order left-invariant differential operators on $S$ of the form $\mathcal L_\alpha=L^a+\Delta_\alpha,$ where $\alpha\in\R^k,$ and for each $a\in\R^k,$ $L^a$ is left-invariant second order differential operator on $N$ and $\Delta_\alpha=\Delta- ,$ where $\Delta$ is the usual Laplacian on $\R^k.$ Using some probabilistic techniques (e.g., skew-product formulas for diffusions on $S$ and $N$ respectively) we obtain an upper bound for the Poisson kernel for $\mathcal L_\alpha.$ We also give an upper estimate for the transition probabilities of the evolution on $N$ generated by $L^{\sigma(t)},$ where $\sigma$ is a continuous function from $[0,\infty)$ to $\R^k.$
Let K be an algebraic number field. With K, we associate the ring of finite adeles [Formula: see ... more Let K be an algebraic number field. With K, we associate the ring of finite adeles [Formula: see text]. Following a recent result of Weisbart on diffusions on finite rational adeles [Formula: see text], we define the Vladimirov operator [Formula: see text] on [Formula: see text] and define the Brownian motion on the group [Formula: see text]. We also consider the Schrödinger operator [Formula: see text] with a potential operator V given by a non-negative continuous function v on [Formula: see text]. We prove a version of the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text].
LetGbe a semi direct productG= ℝd⋊ ℝk. OnGwe consider a class of second order left-invariant diff... more LetGbe a semi direct productG= ℝd⋊ ℝk. OnGwe consider a class of second order left-invariant differential operators of the formLα=∑j=1de2λj(a)∂xj2+∑j=1k(∂aj2−2αj∂aj),${{\cal L}_\alpha } = \sum\limits_{j = 1}^d {\mkern 1mu} e{{\mkern 1mu} ^{2{\lambda _j}(a)}}\partial _{{x_j}}^2 + \sum\limits_{j = 1}^k {(\partial _{{a_j}}^2 - 2{\alpha _j}{\partial _{{a_j}}})},$wherea∈ ℝkandλ1,...,λd∈ (ℝk)*. It is known that bounded 𝓛α-harmonic functions onGare precisely the “Poisson integrals” ofL∞(ℝd) against the Poisson kernelναwhich is a smooth function on ℝd. We prove an upper bound ofναand its derivatives.
This paper deals with Gärdenfors’ theory of conceptual spaces. Let $${\mathcal {S}}$$ S be a conc... more This paper deals with Gärdenfors’ theory of conceptual spaces. Let $${\mathcal {S}}$$ S be a conceptual space consisting of 2-type fuzzy sets equipped with several kinds of metrics. Let a finite set of prototypes $$\tilde{P}_1,\ldots ,\tilde{P}_n\in \mathcal {S}$$ P ~ 1 , … , P ~ n ∈ S be given. Our main result is the construction of a classification algorithm. That is, given an element $${\tilde{A}}\in \mathcal {S},$$ A ~ ∈ S , our algorithm classifies it into the conceptual field determined by one of the given prototypes $$\tilde{P}_i.$$ P ~ i . The construction of our algorithm uses some physical analogies and the Newton potential plays a significant role here. Importantly, the resulting conceptual fields are not convex in the Euclidean sense, which we believe is a reasonable departure from the assumptions of Gardenfors’ original definition of the conceptual space. A partitioning algorithm of the space $$\mathcal {S}$$ S is also considered in the paper. In the application section...
In 2010 Durda, Caron, and Buchanan published a paper in INFOR: Information systems and Operationa... more In 2010 Durda, Caron, and Buchanan published a paper in INFOR: Information systems and Operational Research, entitled: An application of operational research to computational linguistics: Word ambiguity. In this article the authors developed "a new measure of word ambiguity (e.g., homonymy and polysemy) for use in psycholinguistic research". In our work we propose some modification of their algorithm.
Acta Universitatis Lodziensis. Folia Linguistica Rossica, 2018
Radziecki matematyk Andriej Nikołajewicz Kołmogorow (1903‒1987), autor ponad pół tysiąca prac nau... more Radziecki matematyk Andriej Nikołajewicz Kołmogorow (1903‒1987), autor ponad pół tysiąca prac naukowych, był jednym z najwybitniejszych matematyków XX wieku i w pewnym momencie być może nawet największym. W 1960 roku, Kołmogorow rozpoczął intensywne badania w dziedzinie językoznawstwa i teorii wiersza. Wydaje się, że niewielu polskich matematyków, którzy znają niektóre z rezultatów Kołmogorowa w tej czy innej dziedzinie matematyki, wie o jego pracach w dziedzinach nauk humanistycznych. Można podejrzewać, że wielu literaturoznawców i lingwistów również nie wie o istnieniu tych prac. Celem tego artykułu jest wyjaśnienie przyczyn, dla których Kołmogorow zajmował się humanistyką (historią, językoznawstwem, teorią wiersza).
Cognitive Studies | Études cognitives, Dec 3, 2017
This paper proposes a new classification algorithm for the partitioning of a conceptual space. Al... more This paper proposes a new classification algorithm for the partitioning of a conceptual space. All the algorithms which have been used until now have mostly been based on the theory of Voronoi diagrams. This paper proposes an approach based on potential theory, with the criteria for measuring similarities between objects in the conceptual space being based on the Newtonian potential function. The notion of a fuzzy prototype, which generalizes the previous definition of a prototype, is introduced. Furthermore, the necessary conditions that a natural concept must meet are discussed. Instead of convexity, as proposed by Gärdenfors, the notion of geodesically convex sets is used. Thus, if a concept corresponds to a set which is geodesically convex, it is a natural concept. This definition applies, for example, if the conceptual space is an Euclidean space. As a by-product of the construction of the algorithm, an extension of the conceptual space to d-dimensional Riemannian manifolds is obtained.
Abstract. In this note we give an improvement of the estimate of the Poisson kernels for second o... more Abstract. In this note we give an improvement of the estimate of the Poisson kernels for second order differential operators on homo-geneous manifolds of negative curvature obtained for the first time, using some probabilistic techniques, in [I] and then improved by the
propriate set S of generators of order not greater than 2 and a finite set of probability measure... more propriate set S of generators of order not greater than 2 and a finite set of probability measures {p,,..., pk) with their supports in S such that pl*...*pk = 2, where l ( g) = IGI- ' for every g EG. 2000 Mathematics Subject Classification: 60BI5, 43A05. Key words and phrases: Finite Coxcter groups, factorization,
We obtain estimates for derivatives of the Poisson kernels for the second order di#erential opera... more We obtain estimates for derivatives of the Poisson kernels for the second order di#erential operators on homogeneous manifolds of negative curvature both in the coercive and noncoercive case. 1.
Abstract. We consider the Green functions G for second-order coercive differen-tial operators on ... more Abstract. We consider the Green functions G for second-order coercive differen-tial operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R+. Estimates for derivatives of the Green functions G with respect to the N and A-variables are obtained. This paper completes a previous work of the author (see [12, 13]) where estimates for derivatives of the Green functions for the noncoercive operators has been obtained. Here we show how to use the previous methods and results from [12] in order to get analogous estimates for coercive operators. 1. Introduction. Let M be a connected, simply connected homogeneous manifolds of negative cur-vature. Such a manifold is a solvable Lie group S = NA, a semi-direct product of a nilpotent Lie group N and an Abelian group A = R+. Moreover, for an H belonging to the Lie algebra a of A, the real parts of the eigenvalues of AdexpH |n,
Abstract. We consider the Green functions for second-order left-invariant differential operators ... more Abstract. We consider the Green functions for second-order left-invariant differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R+. We obtain estimates for mixed derivatives of the Green functions both in the coercive and non-coercive case. The current paper completes the previous results obtained by the author in a series of papers [14, 15, 16, 19]. 1.
Abstract. We consider the Green functions for second order non-coercive differential operators on... more Abstract. We consider the Green functions for second order non-coercive differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R+. We obtain estimates for the mixed derivatives of the Green functions that complements a previous work by the same author [17]. 1.
In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker'... more In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker's classical invariance principle. We consider a fuzzy random walk ${S^*_n}=X^*_1+\cdots+X^*_n,$ where $\{X^*_i\}_1^{\infty}$ is a sequence of mutually independent and identically distributed $d$-dimensional fuzzy random variables whose $\alpha$-cuts are assumed to be compact and convex. Our reasoning and technique are based on the well known conjugacy correspondence between convex sets and support functions, which allows for the association of an appropriately normalized and interpolated time-continuous fuzzy random process with a real valued random process in the space of support functions. We show that each member of the associated family of dual sequences tends in distribution to a standard Brownian motion.
The main aim of this note is to find an explicit integral formula for the heat kernel of a certai... more The main aim of this note is to find an explicit integral formula for the heat kernel of a certain second-order left invariant differential operator on a solvable Lie group, being a semi-direct product RnR, by means of a skew-product formula for diffusions. An explicit formula of a different kind for the special case of the operator considered in this note (the operator without the drift term) was found recently by Calin, Chang and Li. As a corollary from our main result we get a simple formula for the return probability from which the asymptotic behaviour for small and large values of time follows easily.
The general problem of how to construct stochastic processes which are confined to stay in a pred... more The general problem of how to construct stochastic processes which are confined to stay in a predefined cone (in the one-dimensional but also multi-dimensional case also referred to as \emph{subordinators}) is of course known to be of great importance in the theory and a myriad of applications.\par But fuzzy stochastic processes are considered in this context for the first time in this paper:\par By first relating with each proper convex cone $C$ in $\R^{n}$ a certain cone of fuzzy vectors $C^*$ and subsequently using some specific Banach space techniques we have been able to produce as many pairs $(L^*_t, C^*)$ of fuzzy \L processes $L^*_t$ and cones $C^*$ of fuzzy vectors such that $L^*_t$ are $C^*$-$\,$subordinators.
We consider the Green functions G for second-order coercive dieren- tial operators on homogeneous... more We consider the Green functions G for second-order coercive dieren- tial operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A = R + . Estimates for derivatives of the Green functions G with respect to the N and A-variables are obtained. This paper completes a previous work of the author (see (12, 13)) where estimates for derivatives of the Green functions for the noncoercive operators has been obtained. Here we show how to use the previous methods and results from (12) in order to get analogous estimates for coercive operators.
Let W be a finite Coxeter group and let λW be the Haar measure on W, i.e., λW (w) = |W |−1 for ev... more Let W be a finite Coxeter group and let λW be the Haar measure on W, i.e., λW (w) = |W |−1 for every w ∈ W. We prove that there exist a symmetric set T ̸= W of generators of W consisting of elements of order not greater than 2 and a finite set of probability measures {μ1, . . . , μk} with their supports in T such that their convolution product μ1 ∗ . . . ∗ μk = λW . 2000 AMS Mathematics Subject Classification: Primary: 60B15; Secondary: 20F55.
Let Γ be a sub-semigroup of G = GL(d, R), d > 1. We assume that the action of Γ on R d is strongl... more Let Γ be a sub-semigroup of G = GL(d, R), d > 1. We assume that the action of Γ on R d is strongly irreducible and that Γ contains a proximal and expanding element. We describe contraction properties of the dynamics of Γ on R d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space P d−1. In the case where Γ is a sub-semigroup of GL(d, R)∩M (d, Z) and Γ has the above properties, we deduce that the Γ-orbits on T d = R d /Z d are finite or dense.
Let $S$ be a semi direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, no... more Let $S$ be a semi direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with $\R^k,$ $k>1.$ We consider a class of second order left-invariant differential operators on $S$ of the form $\mathcal L_\alpha=L^a+\Delta_\alpha,$ where $\alpha\in\R^k,$ and for each $a\in\R^k,$ $L^a$ is left-invariant second order differential operator on $N$ and $\Delta_\alpha=\Delta- ,$ where $\Delta$ is the usual Laplacian on $\R^k.$ Using some probabilistic techniques (e.g., skew-product formulas for diffusions on $S$ and $N$ respectively) we obtain an upper bound for the Poisson kernel for $\mathcal L_\alpha.$ We also give an upper estimate for the transition probabilities of the evolution on $N$ generated by $L^{\sigma(t)},$ where $\sigma$ is a continuous function from $[0,\infty)$ to $\R^k.$
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