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In this paper, we discuss models of the common knowledge logic. The common knowledge logic is a multi-modal logic that includes the modal operators $\mathsf{K}_{i}$ ($i\in\mathcal{I}$, where $\mathcal{I}$ is a finite set of agents) and $\mathsf{C}$ in the language. The intended meanings of $\mathsf{K}_{i}\phi$ ($i\in\mathcal{I}$) and $\mathsf{C}\phi$ are ''the agent $i$ knows $\phi$'' ($i\in\mathcal{I}$) and ''$\phi$ is common knowledge among $\mathcal{I}$'', respectively. Semantically, this can be expressed as follows: $\mathsf{C}\phi$ is true if and only if all of $\phi$, $\mathsf{E}\phi$, $\mathsf{E}^{2}\phi$, $\mathsf{E}^{3}\phi,\ldots$ are true, where $\mathsf{E}\phi=\bigwedge_{i\in\mathcal{I}}\mathsf{K}_{i}\phi$. A Kripke frame that satisfies the condition is $\langle W,R_{\mathsf{K}_{i}} (i\in\mathcal{I}), R_{\mathsf{C}}\rangle$, where $R_{\mathsf{C}}$ is the reflexive and transitive closure of $R_{\mathsf{E}}=\bigcup_{i\in\mathcal{I}}R_{\mathsf{K}_{i}}$. We refer to such Kripke frames as CKL-frames. An algebra that satisfies the condition is a modal algebra with modal operators $\mathrm{K}_{i}$ ($i\in\mathcal{I}$) and $\mathrm{C}$, which satisfies that $\mathrm{C} x\leq\mathrm{E}\mathrm{C} x$ and $\mathrm{C} x$ is the greatest lower bound of the set $\{\mathrm{E}^{n} x\mid n\in\omega\}$, where $\mathrm{E} x=\bigwedge_{i\in\mathcal{I}} \mathrm{K}_{i} x$. We refer to such modal algebras as CKL-algebras. In this paper, we show that the class of CKL-frames is modally definable, whereas the class of CKL-algebras is not. That is, the class of CKL-algebras does not form a variety, and there exists a modal algebra in which the common knowledge logic is valid, but $\mathrm{C}x$ is not the greatest lower bound of the set $\{\mathrm{E}^{n} x\mid n\in\omega\}$.

In this paper, we give an extension of the J\'{o}nsson-Tarski representation theorem for both normal and non-normal modal algebras so that it preserves countably many infinitary meets and joins. To extend the J\'{o}nsson-Tarski representation to non-normal modal algebras we consider neighborhood frames instead of Kripke frames just as Do\v{s}en's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q-filters instead of prime filters. Then, we show that every predicate modal logic, whether it is normal or non-normal, has a model defined on a neighborhood frame with constant domains, and give completeness theorem for some predicate modal logics. We also show the same results for infinitary modal logics., Comment: The previous version of this paper is submitted by error

An experimental and computational investigation of the space-charge effects occurring in ultrafast photoelectron spectroscopy from the gas phase is presented. The target sample CF$_3$I is excited by ultrashort (100 fs) far-ultraviolet radiation pulses produced by a free-electron laser. The modification of the energy distribution of the photoelectrons, i.e. the shift and broadening of the spectral structures, is monitored as a function of the pulse intensity. The experimental results are compared with computational simulations which employ a Barnes-Hut algorithm to calculate the effect of individual Coulomb forces acting among the particles. In the presented model, a survey spectrum acquired at low radiation fluence is used to determine the initial energy distribution of the electrons after the photoemission event. The spectrum modified by the space-charge effects is then reproduced by $N$-body calculations that simulate the dynamics of the photoelectrons subject to the individual mutual Coulomb repulsion and to the attractive force of the positive ions. The employed numerical method accounts for the space-charge effects on the energy distribution and allows to reproduce the complete photoelectron spectrum and not just a specific photoemission structure. The simulations also provide information on the time evolution of the space-charge effects on the picosecond scale. Differences with the case of photoemission from solid samples are highlighted and discussed. The presented simulation procedure, although it omits the analysis of angular distribution, constitutes an effective simplified model that allows to predict and account for space-charge effects on the photoelectron energy spectrum in time-resolved photoemission experiments with high-intensity pulsed sources., Comment: 21 pages, 5 figures, 1 table

In this paper, we discuss a proof system $\mathsf{NGL}$ for the logic $\mathbf{GL}$ of provability, which is equipped with an $\omega$-rule. We show the three classes of transitive Kripke frames, the class which strongly validates the $\omega$-rule, the class which weakly validates the $\omega$-rule, and the class which is defined by the L\"{o}b formula, are mutually different, while all of them characterize $\mathbf{GL}$. This gives an example of a proof system $P$ and a class $C$ of Kripke frames such that $P$ is sound with respect to $C$ but the soundness cannot be proved by simple induction on the height of the derivations in $P$. We also show Kripke completeness of $\mathsf{NGL}$ in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations $\Box x\leq\Box\Box x$ and $\bigwedge_{n\in\omega}\Diamond^{n}1=0$ is not a variety., Comment: Previously, this version appeared as arXiv:2103.16857v4 which was submitted in error

We combine time resolved pump-probe Magneto-Optical Kerr Effect and Photoelectron Spectroscopy experiments supported by theoretical analysis to determine the relaxation dynamics of delocalized electrons in half-metallic ferromagnetic manganite $La_{1-x}Sr_{x}MnO_{3}$. We observe that the half-metallic character of $La_{1-x}Sr_{x}MnO_{3}$ determines the timescale of both the electronic phase transition and the quenching of magnetization, revealing a quantum isolation of the spin system in double exchange ferromagnets extending up to hundreds of picoseconds. We demonstrate the use of time-resolved hard X-ray photoelectron spectroscopy (TR-HAXPES) as a unique tool to single out the evolution of strongly correlated electronic states across a second-order phase transition in a complex material., Comment: Article 25 pages, 6 figures

We demonstrate ultrafast magnetization dynamics in a 5d transition metal using circularly-polarized x-ray free electron laser in the hard x-ray region. A decay time of light-induced demagnetization of L1${}_0$-FePt was determined to be $\tau_\textrm{Pt} = 0.6\ \textrm{ps}$ using time-resolved x-ray magnetic circular dichroism at the Pt L${}_3$ edge, whereas magneto-optical Kerr measurements indicated the decay time for total magnetization as $\tau_\textrm{total} = 0.1\ \textrm{ps}$. A transient magnetic state with the photo-modulated magnetic coupling between the 3d and 5d elements is firstly demonstrated., Comment: 6 pages,4 figures

In this paper, we give a duality theorem between the category of $\kappa$-additive complete atomic modal algebras and the category of $\kappa$-downward directed multi-relational Kripke frames, for any cardinal number $\kappa$. Multi-relational Kripke frames are not Kripke frames for multi-modal logic, but frames for monomodal logics in which the modal operator $\Diamond$ does not distribute over (possibly infinite) disjunction, in general. We first define homomorphisms of multi-relational Kripke frames, and then show the equivalence between the category of $\kappa$-downward directed multi-relational Kripke frames and the category $\kappa$-complete neighborhood frames, from which the duality theorem follows. We also present another direct proof of this duality based on the technique given by Minari.

In this paper, we introduce a proof system $\mathsf{NQGL}$ for a Kripke complete predicate extension of the logic $\mathbf{GL}$, that is, the logic of provability, which is defned by $\mathbf{K}$ and the L\"{o}b formula $\Box(\Box p\supset p)\supset\Box p$. $\mathsf{NQGL}$ is a modal extension of Gentzen's sequent calculus $\mathsf{LK}$. Although the propositional fragment of $\mathsf{NQGL}$ axiomatizes $\mathbf{GL}$, it does not have the L\"{o}b formula as its axiom. Instead, it has a non-compact rule, that is, a derivation rule with countably many premises. We show that $\mathsf{NQGL}$ enjoys cut admissibility and is complete with respect to the class of Kripke frames such that for each world, the supremum of the length of the paths from the world is finite.

Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi, and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic.