Browsing by Subject "frame"

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Access status: Restricted ,
Elementy teorii baz w przestrzeni Hilberta
(Data obrony: 2018-03-15) Fornal, Natalia
Wydział Matematyki Stosowanej
Access status: Open Access ,
Frames and factorization of graph Laplacians
(2015) Jørgensen, Palle E.T.; Tian, Feng
Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=(V,E)$, $V$ for vertices, and $E$ for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair ($\mathscr{H}_{E}$, $\Delta$) where $\mathscr{H}_{E}$ in an energy Hilbert space, and $\Delta\left(=\Delta_{c}\right)$ is the $c$-graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $\mathscr{H}_{E}$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.
Access status: Restricted ,
Potencjały krat
(Data obrony: 2020-07-14) Miłkowska, Marta
Wydział Matematyki Stosowanej
Access status: Open Access ,
Rezerwa wytrzymałości rygli systemu ramowego szkieletu żelbetowego
(2007) Kinasz, Roman
Thrust forces which arise on support of crossbars of a frame of a reinforced concrete skeleton of a building positively influence their work and as consequence-increase load-carrying ability. During carrying out of inspection of four-storeyed six flying frames the estimation of distribution of efforts thrust on support of reinforced concrete crossbars on all floors in view of features pivots connections of crossbars with columns is executed. Results of the executed researches have confirmed presence and influences of thrust efforts on support of reinforced concrete crossbars of a frame on increase in their carrying ability, that in the works many researchers theoretically expected. The essential increase in load-carrying ability of reinforced concrete crossbars of overlappings (from 40 up to 4%) depending on height of their accommodation in a frame system is received. Crossbars of a covering practically have no reserves on the second group of limiting conditions in connection with the big width of disclosing of normal cracks in them.
Access status: Open Access ,
Spectrum of J-frame operators
(Wydawnictwa AGH, 2018) Giribet, Juan Ignacio; Langer, Matthias; Leben, Leslie; Maestripieri, Alejandra; Martínez Pería, Francisco; Trunk, Carsten
A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H},[\cdot,\cdot ])$ which is compatible with the indefinite inner product $[\cdot,\cdot ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.