soranzo units.it, tironi units.it The problem of finding compact sequential spaces of arbitrary o... more soranzo units.it, tironi units.it The problem of finding compact sequential spaces of arbitrary order is presented and its dependence on the assumptions of the Theory of Sets is examined. In particular several constructions due to Baškirov, Kannan and Dow are compared. (Joint with Alessandro Soranzo.)
M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the oppo... more M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the opposite of the category Top of topological spaces is a quasivariety. J. Adámek and M.-C. Pedicchio proved that there exists a duality D between the category TopSys of topological systems (defined by S. Vickers) and the category Grids. In both papers a description of the full subcategory D(Top) of the category Grids is given. In this paper we describe internally all grids isomorphic to the objects of the full coreflective subcategory D(Loc) of the category Grids, i.e. we characterize internally all grids of the form D(C), where C is a localic topological system (here Loc is the category of locales regarded as a full subcategory of TopSys). Since, obviously, the category Frm of frames is equivalent to D(Loc), we can say that in this paper those grids which could be called frames are characterized internally. An internal characterization of all grids which correspond (in the above sense) to the frames having T 1 spectra and a generalization of the well-known fact that the spectrum of a locale is a sober space are obtained as well.
y Abstract. The notion of PBS-sublattice is introduced and, using it, a simplication of the resul... more y Abstract. The notion of PBS-sublattice is introduced and, using it, a simplication of the results of (6) and of some results of (5) is ob- tained. Two propositions concerning Wallman-type compactications are presented as well.
Abstract. A new cardinal invariant, the quasi-character, is introduced and some of its interestin... more Abstract. A new cardinal invariant, the quasi-character, is introduced and some of its interesting properties are studied, particula r ly in the class of chain-net or pseudo-radial spaces. Main results are that the quasi-character coincides with the tightness for pseudo-...
Following the recent establishment of an exact kinetic theory realized by the Master kinetic equa... more Following the recent establishment of an exact kinetic theory realized by the Master kinetic equation which describes the statistical behavior of the Boltzmann-Sinai Classical Dynamical System (CDS), in this paper the problem is posed of the construction of the related global existence and regularity theorems. For this purpose, based on the global prescription of the same CDS for arbitrary single and multiple collision events, first global existence is extablished for the N −body Liouville equation which is written in Lagrangian differential and integral forms. This permits to reach the proof of global existence both of generic N −body probability density functions (PDF) as well as of particular solutions which maximize the statistical Boltzmann-Shannon entropy and are factorized in terms of the corresponding 1−body PDF. The latter PDF is shown to be uniquely defined and to satisfy the Master kinetic equation globally in the extended 1−body phase space. Implications concerning the global validity of the asymptotic Boltzmann equation and Boltzmann H-theorem are discussed.
In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit... more In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. The statistical description is couched in terms of the Master kinetic equation, i.e., the kinetic equation which realizes the axiomatic "\textit{ab initio}" approach to the classical statistical mechanics of finite hard-sphere systems recently developed (Tessarotto \textit{et al.}, 2013-2017). The issue addressed here concerns the prescription of the BG-limit operator and specifically the non-commutative property of $\mathcal{L}_{BG}$ with the free-streaming operator which enters the same kinetic equation.\ It is shown\ that the form of the resulting limit equation remains in principle non-unique, its precise realization depending critically on the way the action of the same operator is prescribed. Implications for the global pre...
The concept of essential sequence and essential space is given within the class of pseudoradial s... more The concept of essential sequence and essential space is given within the class of pseudoradial spaces. This concept tries to clarify the behaviour of convergent (long) sequences. Various examples and implications are proved among classes of pseudoradial (and radial, semiradial, almost radial) essential spaces. It is shown that essentially radial spaces are the same as pseudoradial Whyburn spaces. A list of open problems is given.
Abstract We consider the notion of strongly pseudoradial space. Among other things, we examine it... more Abstract We consider the notion of strongly pseudoradial space. Among other things, we examine its relation with various similar notions, including the weak Whyburn property. Our investigation will suggest several interesting questions.
Recently it was proved, by Gerlits, Nagy and Szentmiklossy, that the space C p (X) of continuous ... more Recently it was proved, by Gerlits, Nagy and Szentmiklossy, that the space C p (X) of continuous real functions on X, with the topology of pointwise convergence, is radial if and only if it is Frechet and that there exists a space X for which C p (X) is pseudoradial but not Frechet, To find the precise border between the properties of being pseudoradial and Frechet for C p (X). we introduce the classes of u-pseudoradial and {/-almost radial spaces. If S = (f a) a<x is a Asequence, a function q>: X-> A is called an S-function if f a (x)-= fq,(X)(x) for every a ^ q>(x) and every x e X. (S,f) is said to be an wfA-sequence if it is an co-sequence or it is a /A-sequence (A > co) and has a continuous S-function. A space C p (X) is called w-almost radial if for any nonclosed A in it, there is an wtA-sequence (S f) such that S is a A-sequence in A and/e A-A. Various properties of w-pseudoradial and of w-almost radial spaces are proved. In particular, that, if £ is an ordinal number, then C p (f) is pseudoradial if and only if it is w-almost radial. This implies that there exist w-almost radial spaces C p (X) which are not Frechet.
In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological... more In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonoff-type topologies O on M is given. The continuous maps of a special form between spaces of the type (M,O) are described in an isomorphism theorem. The problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well. Some topological properties of the hyperspaces (M,O) with Tychonoff-type topologies O are briefly discussed.
In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological... more In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonoff-type topologies O on M is given. The continuous maps of a special form between spaces of the type (M,O) are described in an isomorphism theorem. The problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well. Some topological properties of the hyperspaces (M,O) with Tychonoff-type topologies O are briefly discussed.
soranzo units.it, tironi units.it The problem of finding compact sequential spaces of arbitrary o... more soranzo units.it, tironi units.it The problem of finding compact sequential spaces of arbitrary order is presented and its dependence on the assumptions of the Theory of Sets is examined. In particular several constructions due to Baškirov, Kannan and Dow are compared. (Joint with Alessandro Soranzo.)
M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the oppo... more M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the opposite of the category Top of topological spaces is a quasivariety. J. Adámek and M.-C. Pedicchio proved that there exists a duality D between the category TopSys of topological systems (defined by S. Vickers) and the category Grids. In both papers a description of the full subcategory D(Top) of the category Grids is given. In this paper we describe internally all grids isomorphic to the objects of the full coreflective subcategory D(Loc) of the category Grids, i.e. we characterize internally all grids of the form D(C), where C is a localic topological system (here Loc is the category of locales regarded as a full subcategory of TopSys). Since, obviously, the category Frm of frames is equivalent to D(Loc), we can say that in this paper those grids which could be called frames are characterized internally. An internal characterization of all grids which correspond (in the above sense) to the frames having T 1 spectra and a generalization of the well-known fact that the spectrum of a locale is a sober space are obtained as well.
y Abstract. The notion of PBS-sublattice is introduced and, using it, a simplication of the resul... more y Abstract. The notion of PBS-sublattice is introduced and, using it, a simplication of the results of (6) and of some results of (5) is ob- tained. Two propositions concerning Wallman-type compactications are presented as well.
Abstract. A new cardinal invariant, the quasi-character, is introduced and some of its interestin... more Abstract. A new cardinal invariant, the quasi-character, is introduced and some of its interesting properties are studied, particula r ly in the class of chain-net or pseudo-radial spaces. Main results are that the quasi-character coincides with the tightness for pseudo-...
Following the recent establishment of an exact kinetic theory realized by the Master kinetic equa... more Following the recent establishment of an exact kinetic theory realized by the Master kinetic equation which describes the statistical behavior of the Boltzmann-Sinai Classical Dynamical System (CDS), in this paper the problem is posed of the construction of the related global existence and regularity theorems. For this purpose, based on the global prescription of the same CDS for arbitrary single and multiple collision events, first global existence is extablished for the N −body Liouville equation which is written in Lagrangian differential and integral forms. This permits to reach the proof of global existence both of generic N −body probability density functions (PDF) as well as of particular solutions which maximize the statistical Boltzmann-Shannon entropy and are factorized in terms of the corresponding 1−body PDF. The latter PDF is shown to be uniquely defined and to satisfy the Master kinetic equation globally in the extended 1−body phase space. Implications concerning the global validity of the asymptotic Boltzmann equation and Boltzmann H-theorem are discussed.
In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit... more In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. The statistical description is couched in terms of the Master kinetic equation, i.e., the kinetic equation which realizes the axiomatic "\textit{ab initio}" approach to the classical statistical mechanics of finite hard-sphere systems recently developed (Tessarotto \textit{et al.}, 2013-2017). The issue addressed here concerns the prescription of the BG-limit operator and specifically the non-commutative property of $\mathcal{L}_{BG}$ with the free-streaming operator which enters the same kinetic equation.\ It is shown\ that the form of the resulting limit equation remains in principle non-unique, its precise realization depending critically on the way the action of the same operator is prescribed. Implications for the global pre...
The concept of essential sequence and essential space is given within the class of pseudoradial s... more The concept of essential sequence and essential space is given within the class of pseudoradial spaces. This concept tries to clarify the behaviour of convergent (long) sequences. Various examples and implications are proved among classes of pseudoradial (and radial, semiradial, almost radial) essential spaces. It is shown that essentially radial spaces are the same as pseudoradial Whyburn spaces. A list of open problems is given.
Abstract We consider the notion of strongly pseudoradial space. Among other things, we examine it... more Abstract We consider the notion of strongly pseudoradial space. Among other things, we examine its relation with various similar notions, including the weak Whyburn property. Our investigation will suggest several interesting questions.
Recently it was proved, by Gerlits, Nagy and Szentmiklossy, that the space C p (X) of continuous ... more Recently it was proved, by Gerlits, Nagy and Szentmiklossy, that the space C p (X) of continuous real functions on X, with the topology of pointwise convergence, is radial if and only if it is Frechet and that there exists a space X for which C p (X) is pseudoradial but not Frechet, To find the precise border between the properties of being pseudoradial and Frechet for C p (X). we introduce the classes of u-pseudoradial and {/-almost radial spaces. If S = (f a) a<x is a Asequence, a function q>: X-> A is called an S-function if f a (x)-= fq,(X)(x) for every a ^ q>(x) and every x e X. (S,f) is said to be an wfA-sequence if it is an co-sequence or it is a /A-sequence (A > co) and has a continuous S-function. A space C p (X) is called w-almost radial if for any nonclosed A in it, there is an wtA-sequence (S f) such that S is a A-sequence in A and/e A-A. Various properties of w-pseudoradial and of w-almost radial spaces are proved. In particular, that, if £ is an ordinal number, then C p (f) is pseudoradial if and only if it is w-almost radial. This implies that there exist w-almost radial spaces C p (X) which are not Frechet.
In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological... more In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonoff-type topologies O on M is given. The continuous maps of a special form between spaces of the type (M,O) are described in an isomorphism theorem. The problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well. Some topological properties of the hyperspaces (M,O) with Tychonoff-type topologies O are briefly discussed.
In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological... more In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonoff-type topologies O on M is given. The continuous maps of a special form between spaces of the type (M,O) are described in an isomorphism theorem. The problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well. Some topological properties of the hyperspaces (M,O) with Tychonoff-type topologies O are briefly discussed.
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Papers by Gino Tironi