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We exhibit a winning strategy for Synchronizer in the synchronization game on every synchronizing automaton in whose transition monoid the regular D-classes form subsemigroups, Comment: In Proceedings NCMA 2024, arXiv:2409.06120

We construct static and axially symmetric magnetically charged hairy black holes in the gravity-coupled Weinberg-Salam theory. Large black holes merge with the Reissner-Nordstr\"om (RN) family, while the small ones are extremal and support a hair in the form of a ring-shaped electroweak condensate carrying superconducting W-currents and up to $22\%$ of the total magnetic charge. The extremal solutions are asymptotically RN with a mass {\it below} the total charge, $M<|Q|$, due to the negative Zeeman energy of the condensate interacting with the black hole magnetic field. Therefore, they cannot decay into RN black holes. As their charge increases, they show a phase transition when the horizon symmetry changes from spherical to oblate. At this point they have the mass typical for planetary size black holes of which $\approx 11\%$ is stored in the hair. Being obtained within a well-tested theory, our solutions are expected to be physically relevant., Comment: 5+8 pages, 5+1 figures, matches the published version

The need for ever-faster information processing requires extremely small devices operating at frequencies approaching the terahertz and petahertz regimes. For the diagnostics of such devices, researchers need a spatiotemporal tool that surpasses the device under test in speed and spatial resolution and therefore, such a tool cannot be provided by electronics itself. Here we show how ultrafast electron beam probe with terahertz-compressed electron pulses can directly sense local electro-magnetic fields in electronic devices with femtosecond, micrometre and millivolt resolution under normal operation conditions. We analyse the dynamical response of a coplanar waveguide circuit and reveal the impulse response, signal reflections, attenuation and waveguide dispersion directly in the time domain. The demonstrated measurement bandwidth reaches 10 THz and the sensitivity to electric fields is tens of millivolts or -20 dBm. Femtosecond time resolution and the potential to directly integrate our technique into existing electron-beam inspection devices in semiconductor industry makes our femtosecond electron beam probe a promising tool for research and development of next-generation electronics at unprecedented speed and size.

For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka (The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61, 441-448 (2009)), gives a complete solution to the finite basis problem for semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a finite chain., Comment: 24 pages, 9 figures. Version 2 fixes a problem in the main construction from Section 5 and includes several minor improvements

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. It was conjectured that in each completely reachable automaton with $n$ states, every set of $k>0$ states is the image of a word of length at most $n(n-k)$. We confirm the conjecture for completely reachable automata with two input letters satisfying certain restrictions on the action of the letters., Comment: 14 pages, 7 figures

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis., Comment: 9 pages

The ability to perform attosecond-pump attosecond-probe spectroscopy (APAPS) is a longstanding goal in ultrafast science. While first pioneering experiments demonstrated the feasibility of APAPS, the low repetition rates (10-120 Hz) and the large footprints of existing setups have so far hindered the widespread exploitation of APAPS. Here we demonstrate two-color APAPS using a commercial laser system at 1 kHz, straightforward post-compression in a hollow-core fiber and a compact high-harmonic generation (HHG) setup. The latter enables the generation of intense extreme-ultraviolet (XUV) pulses by using an out-of-focus HHG geometry and by exploiting a transient blueshift of the driving laser in the HHG medium. Near-isolated attosecond pulses are generated, as demonstrated by one-color and two-color XUV-pump XUV-probe experiments. Our concept allows selective pumping and probing on extremely short timescales and permits investigations of fundamental processes that are not accessible by other pump-probe techniques., Comment: 8 pages, 5 figures

We exhibit new conditions under which a primitive automaton is synchronizing. In particular, we show that the primitivity of an automaton forces its synchronizability whenever the automaton has either a letter of defect 1 or a word of rank 2., Comment: 8 pages, 3 figures. In version 3, a few typos have been fixed, and two pictures have need added

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite semigroup., Comment: 12 pages. In version 2: the proofs of two key lemmas are expanded; two remarks are added; references are updated; a few typos and language errors are fixed

The branching fractions of $\tau$ lepton decays with the production of strange scalar mesons both in the ground and first radially excited states $\tau \to \nu_\tau [K^*_0(700), K^*_0(1430)]$ and $\tau \to \nu_\tau [K^*_0(700)\pi, K^*_0(1430)\pi, K^*_0(700) K, K^*_0 (700)\eta]$ are calculated in the extended $U(3) \times U(3)$ chiral quark NJL model. All mesons are considered as $q\bar{q}$ systems. The obtained results should be considered as predictions for future experiments., Comment: 9 pages, 5 figures

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite $\mathcal{R}$-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups., Comment: 37 pages, 5 figures. In version 3, the reference list is updated and a few typos are fixed

We construct the multi-charge generalizations for the electroweak magnetic monopole solution of Cho and Maison within a wide range of values of the magnetic charge. We use the same ansatz for the axially symmetric fields as the one previously employed to construct the electroweak sphalerons and compare the internal structure of monopoles with that of sphalerons. The monopoles have zero dipole moment but a finite quadrupole momentum that rapidly increases with growing magnetic charge. For large charges, the monopole configurations are strongly squashed and show inside a bubble of symmetric phase filled with a U(1) hypercharge field produced by a pointlike magnetic charge at the origin, strong enough to suppress all other fields and restore the full gauge symmetry. The bubble is surrounded by a large belt of broken phase containing a magnetically charged ring filled with a nonlinear W-condensate, squeezed between two superconducting rings of opposite electric currents. In the far field region there remains only the magnetic field supported by the total magnetic charge contained at the origin and in the magnetic ring. The axially symmetric monopoles are probably just a special case of more general monopole solutions not possessing any continuous symmetries. The Cho-Maison monopole is stable but the stability of its multi-charge generalizations is not yet confirmed. All electroweak monopoles have infinite energy due to the pointlike U(1) charge at the origin, but the energy is expected to become finite after taking gravity into account, which should provide a cutoff via creating an event horizon to shield the U(1) charge., Comment: 48 pages, many figures

Grammic monoids have recently been introduced by Christian Choffrut in terms of the action of the free monoid over a fixed ordered alphabet $X$ on the set of rows of Young tableaux filled with elements from $X$ via Schensted's insertion. For $X=\{a,b,c\}$ with $a

Let ${\bf Sr}(n, 1)$ denote the ai-semiring variety defined by the identity $x^n\approx x$, where $n>1$. We characterize all subdirectly irreducible members of a semisimple subvariety of ${\bf Sr}(n, 1)$. Based on this result, we prove that ${\bf Sr}(n, 1)$ is hereditarily finitely based (resp., hereditarily finitely generated) if and only if $n<4$ and that the lattice of subvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, we show that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms a variety and so we affirmatively answer the restricted Burnside problem for ${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main result obtained by Gajdo\v{s} and Ku\v{r}il (Semigroup Forum 80: 92--104, 2010)., Comment: 16 pages

Shaping electron beams with the cycles of light provides femtosecond and attosecond time resolution in electron microscopy and enables fundamental quantum-coherent measurements. However, efficient light-electron control requires a prolonged interaction between the two beams for cascaded transfer of photon energy and momentum to the freely propagating electrons. Here we report the use of traveling evanescent terahertz waves to achieve velocity matching and thereby high acceleration gradients both in space and in time. With experiment and simulations, we demonstrate attosecond streaking, temporal pulse compression, acceleration and spatial focusing of sub-relativistic electron pulses with a single evanescent-wave element under the control of selected terahertz delays and phases. Based on these results, we propose to use a symmetric arrangement with two evanescent terahertz waves to generate isolated attosecond electron pulses in a beam with realistic parameters. These results establish subluminal terahertz waves as a promising tool for ultrafast electron pulse control.

We observe that for each $n\ge 2$, the identities of the stylic monoid with $n$ generators coincide with the identities of $n$-generated monoids from other distinguished series of $\mathscr{J}$-trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids., Comment: 3 pages

For every group $G$, the set $\mathcal{P}(G)$ of its subsets forms a semiring under set-theoretical union $\cup$ and element-wise multiplication $\cdot$ and forms an involution semigroup under $\cdot$ and element-wise inversion ${}^{-1}$. We show that if the group $G$ is finite, non-Dedekind, and solvable, neither the semiring $(\mathcal{P}(G),\cup,\cdot)$ nor the involution semigroup $(\mathcal{P}(G),\cdot,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set., Comment: 18 pages, 1 figure

We characterize complete deterministic finite automata with two input letters in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word. The characterization leads to a polynomial-time algorithm for recognizing this class of automata., Comment: 13 pages, 4 figures. A conference version of this paper has been accepted for LATIN 2022. The present version incorporates many useful comments and suggestions by the anonymous reviewers of the conference version

The branching fractions of the decays $\tau \to a_{1} \pi \nu_\tau$, $\tau \to K_{1} \pi \nu_\tau$ and $\tau \to K_{1} K \nu_\tau$ are calculated in the framework of the extended $U(3)\times U(3)$ chiral Nambu--Jona-Lasinio model. There are no experimental data for these decays in the present time, thus the obtained results are considered as predictions. The comparison of these results to the theoretical results of other authors is carried out., Comment: 8 pages, 4 figures

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups., Comment: 14 pages, 1 figure

We apply a generalized field ansatz to describe the spherically symmetric sector of classical solutions of the electroweak theory. This sector contains Abelian magnetic monopoles labeled by their magnetic charge $n=\pm 1,\pm 2,\ldots$, the non-Abelian monopole for $n=\pm 2$ found previously by Cho and Maison (CM), and also the electric oscillating solutions. All magnetic monopoles have infinite energy. We analyze their perturbative stability and use the method of complex spacetime tetrad to separate variables and reduce the perturbation equations to multi-channel Schroedinger-type eigenvalue problems. The spectra of perturbations around the CM monopole do not contain negative modes hence this solution is stable. The $n=\pm 1$ Abelian monopole is also stable, but all monopoles with $|n|\geq 2$ are unstable with respect to perturbations with angular momentum $j=|n|/2-1$. The Abelian $|n|=2$ monopole is unstable only within the $j=0$ sector whereas the CM monopole also has $|n|=2$ and belongs to the same sector, hence it may be viewed as a stable remnant of the decay of the Abelian monopole. One may similarly conjecture that stable remnants exist also for monopoles with $|n|>2$, hence the CM monopole may be just the first member of a sequence of non-Abelian monopoles with higher magnetic charges. Only the CM monopole is spherically symmetric while all non-Abelian monopoles with $|n|>2$ are not rotationally invariant., Comment: 49 pages, 5 figures

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. We characterize such automata in terms of graphs and trees., Comment: 29 pages, 16 figures

We analyze possibilities to obtain a globally regular stationary generalization for the ultrastatic wormhole with a repulsive scalar field found by Bronnikov and by Ellis in 1973. The extreme simplicity of this static solution suggests that its stationary version could be obtainable analytically and should be globally regular, but no such generalization has been found. We analyze the problem and find that the difficulty originates in the vacuum theory, since the scalar field can be eliminated within the Eris-Gurses procedure. The problem then reduces to constructing the spinning generalization for the vacuum wormhole sourced by a thin ring of negative tension. Solving the vacuum Ernst equations determines the $g_{00}$, $g_{0\varphi}$ metric components and hence the AMD mass ${\rm M}$ and angular momentum ${\rm J}$, all of these being specified by the ring source. The scalar field can be included into consideration afterwords, but this only affects $g_{rr}$ and $g_{\vartheta\vartheta}$ without changing the rest. Within this approach, we analyze a number of exact stationary generalizations for the wormhole, but none of them are satisfactory. However, the perturbative expansion around the static vacuum background contains only bounded functions and presumably converges to an exact solution. Including the scalar field screens the singularity at the ring source and renders the geometry regular. This solution describes a globally regular spinning wormhole with two asymptotically flat regions. Even though the source itself is screened and not visible, the memory of it remains in $g_{00}$ and $g_{0\varphi}$ and accounts for the ${\rm M}\propto {\rm J}^2$ relation typical for a rotating extended source. Describing stationary spacetimes with an extended source is a complicated problem, which presumably explains the difficulty in finding the solution., Comment: 37 pages, references added, minor style improvements

MXenes are two-dimensional materials with a rich set of remarkable chemical and electromagnetic properties, the latter including saturable absorption and intense surface plasmon resonances. To fully harness the functionality of MXenes for applications in optics, electronics and sensing, it is important to understand the interaction of light with MXenes on atomic and femtosecond dimensions. Here, we use ultrafast electron diffraction and high-resolution electron microscopy to investigate the laser-induced structural dynamics of Ti3C2Tx nanosheets. We find an exceptionally fast lattice response with an electron-phonon coupling time of 230 femtoseconds. Repetitive femtosecond laser excitation transforms Ti3C2Tx through a structural transition into a metamaterial with deeply sub-wavelength nanoripples that are aligned with the laser polarization. By a further laser illumination, the material is reversibly photo-switchable between a flat and rippled morphology. The resulting nanostructured MXene metamaterial with directional nanoripples is expected to exhibit an anisotropic optical and electronic response as well as an enhanced chemical activity that can be switched on and off by light.

The 6-element Brandt monoid $B_2^1$ admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of $B_2^1$ as an inverse semigroup. As a consequence, we exhibit an easy proof that the semiring identities of $B_2^1$ are not finitely based., Comment: 6 pages

It was found recently that the anisotropies in the homogeneous Bianchi I cosmology considered within the context of a specific Horndeski theory are damped near the initial singularity instead of being amplified. In this work we extend the analysis of this phenomenon to cover the whole of the Horndeski family. We find that the phenomenon is absent in the K-essence and/or Kinetic Gravity Braiding theories, where the anisotropies grow as one approaches the singularity. The anisotropies are damped at early times only in more general Horndeski models whose Lagrangian includes terms quadratic and cubic in second derivatives of the scalar field. Such theories are often considered as being inconsistent with the observations because they predict a non-constant speed of gravitational waves. However, the predicted value of the speed at present can be close to the speed of light with any required precision, hence the theories actually agree with the present time observations. We consider two different examples of such theories, both characterized by a late self-acceleration and an early inflation driven by the non-minimal coupling. Their anisotropies show a maximum at intermediate times and approach zero at early and late times. The early inflationary stage exhibits an instability with respect to inhomogeneous perturbations, suggesting that the initial state of the universe should be inhomogeneous. However, more general Horndeski models may probably be stable., Comment: 21 pages, several figures

A binary relation on a finite set is called a Hall relation if it contains a permutation of the set. Under the usual relational product, Hall relations form a semigroup which is known to be a block-group, that is, a semigroup with at most one idempotent in each $\mathrsfs{R}$-class and each $\mathrsfs{L}$-class. Here we show that in a certain sense, the converse is true: every block-group divides a semigroup of Hall relations on a finite set., Comment: 8 pages; a typo in the formulation of Proposition 3 has been fixed

We study asymptotically flat black holes with massive graviton hair within the ghost-free bigravity theory. There have been contradictory statements in the literature about their existence -- such solutions were reported some time ago, but later a different group claimed the Schwarzschild solution to be the only asymptotically flat black hole in the theory. As a result, the controversy emerged. We have analyzed the issue ourselves and have been able to construct such solutions within a carefully designed numerical scheme. We find that for given parameter values there can be one or two asymptotically flat hairy black holes in addition to the Schwarzschild solution. We analyze their perturbative stability and find that they can be stable or unstable, depending on the parameter values. The masses of stable hairy black holes that would be physically relevant range form stellar values up to values typical for supermassive black holes. One of their two metrics is extremely close to Schwarzschild, while all their "hair" is hidden in the second metric that is not coupled to matter and not directly seen. If the massive bigravity theory indeed describes physics, the hair of such black holes should manifest themselves in violent processes like black hole collisions and should be visible in the structure of the signals detected by LIGO/VIRGO., Comment: 57 pages, many figures, matches the published version

We approach the task of computing a carefully synchronizing word of optimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experiments demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used. We compare our results with the ones obtained by the first author for exact synchronization, which is another version of synchronization studied in the literature, and draw some theoretical conclusions., Comment: 34 pages, 13 figures. This is an extended version of arXiv:1903.10549

It is shown that the endomorphism monoids of the category $2\mathfrak{Cob}$ of all $2$-cobordisms do not have finitely axiomatizable equational theories. The same holds for the {topological annular category} and various quotients of the latter, like the affine Temperley--Lieb category. Analogous results are obtained for versions with the algebraic signature enriched by involutions. Von-Neumann-regular extensions of these categories are also considered., Comment: 56 pages

We consider anisotropic cosmologies in a particular shift-symmetric Horndeski theory containing the $G^{\mu\nu}\partial_\mu\phi \partial_\nu\phi$ coupling, where $G^{\mu\nu}$ is the Einstein tensor. This theory admits stable in the future self-accelerating cosmologies whose tensor perturbations propagate with the velocity very close to the speed of light such that the theory agrees with the gravity wave observations. Surprisingly, we find that the anisotropies within the Bianchi I homogeneous spacetime model are screened at early time by the scalar charge, whereas at late times they are damped in the usual way. Therefore, contrary to what one would normally expect, the early state of the universe in the theory cannot be anisotropic and (locally) homogeneous in the absence of spatial curvature. The early universe cannot be isotropic either, because it should then be unstable with respect to inhomogeneous perturbations. As a result, the early universe should be inhomogeneous. At the same time, we find that in the spatially curved Bianchi IX case the anisotropies can be strong at early times even in the presence of a scalar charge., Comment: 18 pages, 2 figures, matches the published version

We study the metric-affine versions of scalar-tensor theories whose connection enters the action only algebraically. We show that the connection can be integrated out in this case, resulting in an equivalent metric theory. Specifically, we consider the metric-affine generalisations of the subset of the Horndeski theory whose action is linear in second derivatives of the scalar field. We determine the connection and find that it can describe a scalar-tensor Weyl geometry without a Riemannian frame. Still, as this connection enters the action algebraically, the theory admits the dynamically equivalent (pseudo)-Riemannian formulation in the form of an effective metric theory with an extra K-essence term. This may have interesting phenomenological applications., Comment: 8 pages, to appear in the thematic issue of Int. Journ. Mod. Phys. A - proceedings of the 10th Friedmann Seminar on Gravitation and Cosmology, St. Petersburg, Russia, June 23-29, 2019

In the present study we have demonstrated epitaxial stabilization of the metastable magnetically-hard $\varepsilon$-Fe$_2$O$_3$ phase on top of a thin MgO(111) buffer layer grown onto the GaN (0001) surface. The primary purpose to introduce a 4\,nm-thick buffer layer of MgO in between Fe$_2$O$_3$ and GaN was to stop thermal migration of Ga into the iron oxide layer. Though such migration and successive formation of the orthorhombic GaFeO$_3$ was supposed earlier to be a potential trigger of the nucleation of the isostructural $\varepsilon$-Fe$_2$O$_3$, the present work demonstrates that the growth of single crystalline uniform films of epsilon ferrite by pulsed laser deposition is possible even on the MgO capped GaN. The structural properties of the 60\,nm thick Fe$_2$O$_3$ layer on MgO / GaN were probed by electron and x-ray diffraction, both suggesting that the growth of $\varepsilon$-Fe$_2$O$_3$ is preceded by formation of a thin layer of $\gamma$-Fe$_2$O$_3$. The presence of the magnetically hard epsilon ferrite was independently confirmed by temperature dependent magnetometry measurements. The depth-resolved x-ray and polarized neutron reflectometry reveal that the 10\,nm iron oxide layer at the interface has a lower density and a higher magnetization than the main volume of the $\varepsilon$-Fe$_2$O$_3$ film. The density and magnetic moment depth profiles derived from fitting the reflectometry data are in a good agreement with the presence of the magnetically degraded $\gamma$-Fe$_2$O$_3$ transition layer between MgO and $\varepsilon$-Fe$_2$O$_3$. The natural occurrence of the interface between magnetoelectric $\varepsilon$- and spin caloritronic $\gamma$- iron oxide phases can enable further opportunities to design novel all-oxide-on-semiconductor devices., Comment: Accepted to Physical Review Materials, 7 pages, 5 figures

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap. 14, no.1 (1985), 38--42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit., Comment: 22 pages

We analyse what happens when the Horndeski Lagrangian is varied within the Palatini approach by considering the metric and connection as independent variables. Assuming the connection to be torsionless, there can be infinitely many metric-affine versions $L_{\rm P}$ of the original Lagrangian which differ from each other by terms proportional to the non-metricity tensor. After integrating out the connection, each $L_{\rm P}$ defines a metric theory, which can either belong to the original Horndeski family, or it can be of a more general DHOST type, or it shows the Ostrogradsky ghost. We analyse in detail the subclass of the theory for which the equations are linear in the connection and find that its metric-affine version is ghost-free. We present a detailed classifications of homogeneous and isotropic cosmologies in these theories. Taking into consideration other pieces of the Horndeski Lagrangian which are non-linear in the connection leads to more complex metric-affine theories which generically show the ghost. In some special cases the ghost can be removed by carefully adjusting the non-metricity contribution, but it is unclear if this is always possible. Therefore, the metric-affine generalisations of the Horndeski theory can be ghost-free, but not all of them are ghost-free, neither are they the only metric-affine theories for a gravity-coupled scalar field which can be ghost-free., Comment: 36 pages, 7 figures, many comments and references added; to appear in JCAP

We approach the task of computing a carefully synchronizing word of minimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experimental results demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used., Comment: 15 pages, 3 figures

A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of `related' normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad \cite{mem,cross} as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.

Up to isomorphism, there exist two non-isomorphic two-element monoids. We show that the identities of the free product of every pair of such monoids admit no finite basis., Comment: 5 pages