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Benedetto Silvestri

Publications and source records attributed to Benedetto Silvestri.

10 recordsLinked to original sources
arXiv2026

Operations on $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields

We extend to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the usual operations defined on $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$, then by employing the properties of the projective tensor product of locally convex spaces, we generalize to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$ the usual wedge product, insertion operator and exterior differential defined on $\mathrm{Alt}^{\bullet}(U,M)$ and establish their properties. Here $M$ is a smooth finite dimensional manifold, $U$ an open submanifold of $M$, $\mathcal{A}(U)$ the ring of smooth maps on $U$, $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth tensor fields of $M$ defined on $U$, $\mathrm{Alt}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth alternating tensor fields of $M$ defined on $U$, while $\mathscr{M}$ is a $\mathcal{A}(U)$-module. $\mathscr{M}$ depending by the operation might be either general, or one of the following function spaces: $\mathscr{L}_{c}^{1}(U,G,λ)$, $\mathscr{L}_{loc}^{1}(U,G,λ)$, $\mathcal{B}^{p}(U,G)$ with $G$ a Hausdorff locally convex space. This framenwork permits to construct the divergence of a $G$-valued $p$-times continuously differentiable vector field, to define the weak integral of $G$-valued compactly supported scalarly integrable maximal forms, to associate a $G$-valued measure with any locally integrable $G$-valued form, to establish a Stokes type theorem for $G$-valued compactly supported $p$-times continuously differentiable forms and to obtain a divergence type theorem for $G$-valued $p$-times continously differentiable vector fields.

math.FA↗
arXiv2026

Universal operatorial Noether current associated with a quasi-complete locally convex linear space valued Lagrangian defined on an open set of a product of normed spaces

In the context of locally convex vector valued integrable tensors on open sets of a $n$-dimensional manifold $M$ developped in our previous work \cite{sil09261621}, in this paper we construct a framework in which to address both a variational problem associated with a Lagrangian $\mathscr{L}$, and a conservation law on-shell associated with actions $ϕ$ and $ω$. $\mathscr{L}$ is a $C^{p}$-map valued in a quasi-complete Hausdorff lcs $\mathcal{Z}$ and defined on an open set $O$ of a normed space, $ϕ$ is a $C^{p}$-action parametrized by an open $A$ of a normed space $T$, acting over the space $\mathfrak{Z}^{O}$ of fields valued in a normed space $Y_{0}$ and defined on $W$ domain of a chart $ξ$ of $M$; $ω$ is a $C^{p}$-action on $W$ parametrized by $A$. Via a result in \cite{sil09261621}, we construct $\mathfrak{A}:\mathfrak{Z}^{O}\times A\to\mathcal{M}(W,\mathcal{Z})$ where $\mathfrak{A}(u,η)$ is the action functional associated with $\mathscr{L}$, $u$ and $η$. Here $\mathcal{M}(W,\mathcal{Z})=\mathcal{L}_{s}(\mathcal{H}(W),\mathcal{Z})$ is the lcs of measures on $W$ and valued in $\mathcal{Z}$. By letting $\mathcal{A}(u):A\to\mathcal{M}(W,\mathcal{Z})$ be such that $\mathcal{A}(u)(η)=\mathfrak{A}(u,η)$, we state that $\mathcal{A}(u)$ is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of $\mathcal{A}(u)$ on $0$ is $0$, then $\mathscr{N}[u]$ is conserved on-shell, and if $p=\infty$, then the divergence of $\mathscr{N}[u]$ equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here $\mathscr{N}[u]$ is a $\mathcal{L}_{b}(T,\mathcal{Z})$-valued $C^{(p-1)}$-vector field of $M$ on $U=ξ(W)$.

math.FA↗
arXiv2025

Construction of strong derivable maps via functional calculus of unbounded spectral operators in Banach spaces

We provide sufficient conditions for the existence of a strong derivable map and calculate its derivative by employing a result in our previous work on strong derivability of maps arising by functional calculus of an unbounded scalar type spectral operator $R$ in a Banach space and the generalization to complete locally convex spaces of a classical result valid in the Banach space context. We apply this result to obtain a sequence of integrals converging to an integral of a complete locally convex space extension of a map arising by functional calculus of $R$.

math.FA↗
arXiv2024

The 2-category of species of dynamical patterns

A new category $\mathfrak{dp}$, called of dynamical patterns addressing a primitive, nongeometrical concept of dynamics, is defined and employed to construct a $2-$category $2-\mathfrak{dp}$, where the irreducible plurality of species of context-depending dynamical patterns is organized. We propose a framework characterized by the following additional features. A collection of experimental settings is associated with any species, such that each one of them induces a collection of experimentally detectable trajectories. For any connector $T$, a morphism between species, any experimental setting $E$ of its target species there exists a set such that with each of its elements $s$ remains associated an experimental setting $T[E,s]$ of its source species, $T[\cdot,s]$ is called charge associated with $T$ and $s$. The vertical composition of connectors is contravariantly represented in terms of charge composition. The horizontal composition of connectors and $2-$cells of $2-\mathfrak{dp}$ is represented in terms of charge transfer. A collection of trajectories induced by $T[E,s]$ corresponds to a collection of trajectories induced by $E$ (equiformity principle). Context categories, species and connectors are organized respectively as $0,1$ and $2$ cells of $2-\mathfrak{dp}$ with factorizable functors via $\mathfrak{dp}$ as $1-$cells and as $2-$cells, arranged themself to form objects of categories, natural transformations between $1-$cells obtained as horizontal composition of natural transformations between the corresponding factors. We operate a nonreductionistic interpretation positing that the physical reality holds the structure of $2-\mathfrak{dp}$, where the fibered category $\mathfrak{Cnt}$ of connectors is the only empirically knowable part....

math-ph↗
arXiv2021

Stokes-type Integral Equalities for Scalarly Essentially Integrable Locally Convex Vector Valued Forms which are Functions of an Unbounded Spectral Operator

In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space $\langle B(G),σ(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of our previous articles. To obtain our equality we generalize the main result of that article, and employ the Stokes theorem for smooth locally convex vector valued forms established in a prodromic paper. Two facts are remarkable. Firstly the forms integrated involved in the equality are functions of a possibly unbounded scalar type spectral operator in $G$. Secondly these forms need not be smooth nor even continuously differentiable.

math.FA↗
arXiv2018

Natural transformations associated with a locally compact group and universality of the global Terrell law

Via the construction of a functor from $\mathsf{C}_{u}(H)$ to an auxiliary category we associate, with any triplet $(G,F,ρ)$, two natural transformations, $\mathfrak{m}_{\star}$ morphism of $\mathsf{Fct}(\mathsf{C}_{u}(H)^{op},\mathsf{Fct}(H,\mathsf{set}))$ and $\mathfrak{v}_{\natural}$ morphism of $\mathsf{Fct}(\mathsf{C}_{u}^{0}(H)^{op},\mathsf{Fct}(H,\mathsf{set}))$. $G$ and $F$ are locally compact groups, $ρ:F\to Aut(G)$ is a continuous morphism, $H$ is the external topological semidirect product of $G$ and $F$ relative to $ρ$, $\mathsf{C}_{u}^{0}(H)$ is a subcategory of $\mathsf{C}_{u}(H)$ a subcategory of the category of $C^{\ast}-$dynamical systems with symmetry group $H$ and equivariant morphisms. For $\mathfrak{A}$ in $\mathsf{C}_{u}^{0}(H)$ to assemble $\mathfrak{m}_{\star}^{\mathfrak{A}}$ we exploit the Connes characters generated by JLO cocycles $Φ$ on the unitization of certain $C^{\ast}-$crossed products relative to $\mathfrak{A}$, while to construct $\mathfrak{v}_{\natural}^{\mathfrak{A}}$ we exert the states of the $C^{\ast}-$algebra underlying $\mathfrak{A}$ associated in a convenient manner with the $0-$dimensional components of the $Φ$'s. Being $\mathsf{C}_{u}(H)$ the category of fissioning systems, we use $\mathfrak{m}_{\star}^{\mathfrak{A}}$ and $\mathfrak{v}_{\natural}^{\mathfrak{A}}$ to define the nucleon phases and the fragment states resulting next the fission processes of the fissioning system $\mathfrak{A}$ occur. We apply the naturality of $\mathfrak{m}_{\star}$ and $\mathfrak{v}_{\natural}$ to establish the universality of the global nucleon masses and the global Terrell law, stated as invariance of the light and heavy nucleon core masses and invariance of the prompt-neutron yield under controvariant action of $\mathsf{C}_{u}^{0}(H)$ and under action of $H$ over the field of fission processes.

math.OA↗
arXiv2018

Construction of a Natural Transformation from a Classical to a Quantum 0-Species

A natural transformation $\mathfrak{J}$ between functors valued in the category $\mathfrak{Chdv}_{0}$ is assembled. $\mathfrak{Chdv}_{0}$ is obtained by replacing both the categories $\mathrm{ptls}$ and $\mathrm{ptsa}$ with the category of topological linear spaces in the defining properties of the category $\mathfrak{Chdv}$ introduced in one of our previous papers. By letting a $\mathfrak{dp}$-valued functor be (classical) quantum whenever every its value is a dynamical pattern whose set map takes values in the set of (commutative) noncommutative topological unital $\ast-$algebras, and letting a (classical) quantum $0$-species be a $\mathfrak{Chdv}_{0}$-valued functor factorizing through the canonical functor from $\mathfrak{dp}$ to $\mathfrak{Chdv}_{0}$ into a (classical) quantum $\mathfrak{dp}$-valued functor, we have that the domain and codomain of $\mathfrak{J}$ are a classical and a quantum $0$-species respectively.

math.FA↗
arXiv2016

Use of bundles of locally convex spaces in problems of convergence of semigroups of operators defined on different Banach spaces. Applications to spectral stability problems

In this work we construct certain general bundles $<\mathfrak{M},ρ,X>$ and $<\mathfrak{B},η,X>$ of Hausdorff locally convex spaces associated with a given Banach bundle $<\mathfrak{E},π,X>$. Then we present conditions ensuring the existence of bounded sections $\mathcal{U}\in Γ^{x_{\infty}}(ρ)$ and $\mathcal{P}\in Γ^{x_{\infty}}(η)$ both continuous at a point $x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every $x\in X$.

math.FA↗
arXiv2012

Frechet differential of a power series in a Banach algebra

We present two new forms in which the Frechet differential of a power series in a Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T):A\mapsto [A,T]$.Then we apply the results to the analytic functional calculus in a complex Banach space.

math.FA↗