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We present analog clocks fitted to the Mars solar day. These clocks use the standard Earth-based second of the International System of Units (SI) as their operational unit of time, unlike current practice for Mars timekeeping. We discuss the importance of preserving the SI second. On this basis, we identify the two analog clocks most suitable for public use by a future Mars population. These are a 20-hour clock with a hand motion similar to that of the standard Earth clock, and a 24-hour clock with a novel "Martian" hand motion which strikes the hour when all 3 hands converge onto that hour mark on the dial. Both clocks have Earth-day equivalents to assist learning. We also present a 24-hour "SpaceClock", similar to the Martian clock but with no favored reference plane, hence equally readable from any viewing orientation.

We investigate an optomechanical system as a model of an autonomous mechanical pendulum clock in the quantum regime, whose operation relies only on incoherent (thermal) resources. The escapement of the clock, the mechanism that translates oscillatory motion into ticks, is provided by an emitter in the optical cavity and the operation of the clock relies on the existence of a limit cycle. Since the clock is based on an oscillatory degree of freedom, it can overcome the thermodynamic uncertainty relation and is thus more accurate than clocks that rely only on stochastic transitions. Furthermore, by increasing the amount of emitters in the cavity, the clock approaches the behavior expected for a macroscopic pendulum clock, where fluctuations become irrelevant while the clock dynamics becomes completely irreversible. This allows for investigating the quantum-to-classical transition of pendulum clocks.

Optical clocks have improved their frequency stability and estimated accuracy by more than two orders of magnitude over the best caesium microwave clocks that realise the SI second. Accordingly, an optical redefinition of the second has been widely discussed, prompting a need for the consistency of optical clocks to be verified worldwide. While satellite frequency links are sufficient to compare microwave clocks, a suitable method for comparing high-performance optical clocks over intercontinental distances is missing. Furthermore, remote comparisons over frequency links face fractional uncertainties of a few $10^{-18}$ due to imprecise knowledge of each clock's relativistic redshift, which stems from uncertainty in the geopotential determined at each distant location. Here, we report a landmark campaign towards the era of optical clocks, where, for the first time, state-of-the-art transportable optical clocks from Japan and Europe are brought together to demonstrate international comparisons that require neither a high-performance frequency link nor information on the geopotential difference between remote sites. Conversely, the reproducibility of the clocks after being transporte

A driven linear oscillator and a feedback mechanism are two necessary elements of any classical periodic clock. Here, we introduce a novel, fully quantum clock using a driven oscillator in the quantum regime and coherent quantum feedback. We show that if we treat the model semiclassically, this system supports limit cycles, or self-sustained oscillations, as needed for a periodic clock. We then analyse the noise of the system quantum mechanically and prove that the accuracy of this clock is higher compared to the clock implemented with the classical measurement feedback. We experimentally implement the model using two superconducting cavities with incorporated Josephson junctions and microwave circulators for the realisation of the quantum feedback. We confirm the appearance of the limit cycle and study the clock accuracy both in frequency and time domains. Under specific conditions of noisy driving, we observe that the clock oscillations are more coherent than the drive, pointing towards the implementation of a quantum autonomous clock.

Joint position and clock estimation is crucial in many wireless sensor network applications, especially in distance-based estimation with time-of-arrival (TOA) measurement. In this work, we consider a TOA-based ultra-wideband (UWB) sensor network, propose a novel clock rigidity theory and investigate the relation between the network graph properties and the feasibility of clock estimation with TOA timestamp measurements. It is shown that a clock framework can be uniquely determined up to a translation of clock offset and a scaling of all clock parameters if and only if it is infinitesimally clock rigid. We further prove that a clock framework is infinitesimally clock rigid if its underlying graph is generically bearing rigid in 2-dimensional space with at least one redundant edge. Combined with distance rigidity, clock rigidity provides a graphical approach for analyzing the joint position and clock problem. It is shown that a position-clock framework can be uniquely determined up to some trivial variations corresponding to both position and clock if and only if it is infinitesimally joint rigid. Simulation results are presented to demonstrate the clock estimation and joint positio

Clock synchronization is critically important in positioning, navigation and timing systems. While its performance has been intensively studied in a wide range of disciplines, much less is known for the fundamental thermodynamics of clock synchronization, what limits the precision and how to optimize the energy cost for clock synchronization. Here, we report the first experimental investigation of two stochastic clocks synchronization, unveiling the thermodynamic relation between the entropy cost and clock synchronization in an open cavity optomechanical system. Two autonomous clocks are synchronized spontaneously by engineering the controllable photon-mediated dissipative optomechanical coupling and the disparate decay rates of hybrid modes. The measured dependence of the degree of synchronization on entropy cost exhibits an unexpected non-monotonic characteristic, indicating that the perfect clock synchronization does not cost the maximum entropy and there exists an optimum. The investigation of transient dynamics of clock synchronization exposes a trade-off between energy and time consumption. Our results reveal the fundamental relation between clock synchronization and thermody

The bloom clock is a space-efficient, probabilistic data structure designed to determine the partial order of events in highly distributed systems. The bloom clock, like the vector clock, can autonomously detect causality violations by comparing its logical timestamps. Unlike the vector clock, the space complexity of the bloom clock does not depend on the number of nodes in a system. Instead it depends on a set of chosen parameters that determine its confidence interval, i.e. false positive rate. To reduce the space complexity from which the vector clock suffers, the bloom clock uses a 'moving window' in which the partial order of events can be inferred with high confidence. If two clocks are not comparable, the bloom clock can always deduce it, i.e. false negatives are not possible. If two clocks are comparable, the bloom clock can calculate the confidence of that statement, i.e. it can compute the false positive rate between comparable pairs of clocks. By choosing an acceptable threshold for the false positive rate, the bloom clock can properly compare the order of its timestamps, with that of other nodes in a highly accurate and space efficient way.

Gauge invariant local observables describing primordial scalar quantum fluctuations in Inflationary Cosmology are identified as elements of a type $II$ de Sitter crossed product algebra. This algebra is defined, after adding a reference frame clock, as the algebra of clock dressed local operators. Clock dressing sets, in the weak gravity limit, the Schrodinger equation for gauge invariant quantum fluctuations. Instead of using a slow roll inflaton potential to define the clock Hamiltonian and the clock state we suggest a natural double de Sitter clock making the whole algebra associated with the planar patch a type $II$ factor. The corresponding clock states are EPR squeezed states. Using this clock to define the needed clock dressing leads to concrete model independent predictions of the inflationary parameters. Some speculative remarks on potential ways to define a type $I$ upgrading are briefly discussed.

A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. We prove that the category of regular clock maps is a small-orthogonality class in the category of clock maps. Hence it is locally presentable. The geometric realization of any precubical set or transverse set gives rise to a regular clock map. Finally, we prove that for the underlying directed space of a regular clock map, the canonical quotient from directed paths to traces is always a homotopy equivalence.

We discuss a systematic way in which a relational dynamics can be established relative to periodic clocks both in the classical and quantum theories, emphasising the parallels between them. We show that: (1) classical and quantum relational observables that encode the value of a quantity relative to a periodic clock are only invariant along the gauge orbits generated by the Hamiltonian constraint if the quantity itself is periodic, and otherwise the observables are only transiently invariant per clock cycle (this implies, in particular, that counting winding numbers does not lead to invariant observables relative to the periodic clock); (2) the quantum relational observables can be obtained from a partial group averaging procedure over a single clock cycle; (3) there is an equivalence ('trinity') between the quantum theories based on the quantum relational observables of the clock-neutral picture of Dirac quantisation, the relational Schrödinger picture of the Page-Wootters formalism, and the relational Heisenberg picture that follows from quantum deparametrisation, all three taken relative to periodic clocks (implying that the dynamics in all three is necessarily periodic); (4) in

Clock-dependent probabilistic timed automata extend classical timed automata with discrete probabilistic choice, where the probabilities are allowed to depend on the exact values of the clocks. Previous work has shown that the quantitative reachability problem for clock-dependent probabilistic timed automata with at least three clocks is undecidable. In this paper, we consider the subclass of clock-dependent probabilistic timed automata that have one clock, that have clock dependencies described by affine functions, and that satisfy an initialisation condition requiring that, at some point between taking edges with non-trivial clock dependencies, the clock must have an integer value. We present an approach for solving in polynomial time quantitative and qualitative reachability problems of such one-clock initialised clock-dependent probabilistic timed automata. Our results are obtained by a transformation to interval Markov decision processes.

We present an elastic buffer design that enables all-digital clock recovery implementation with free-running receiver clock featuring negative and positive clock frequency offsets. Error-free real-time data transmission is demonstrated from -400 ppm to +400 ppm.

In this paper, we generalize the \textit{Clock Theorem} of Formal Knot Theory to knotoids in $S^2$. The clock theorem implies that clock states of a knotoid diagram form a lattice under transpositions. These states form the basis of many invariants of knotoids and linkoids including the Alexander polynomial, Mock Alexander polynomial and the Jones polynomial.

Being able to measure time, whether directly or indirectly, is a significant advantage for an organism. It allows for the timely reaction to regular or predicted events, reducing the pressure for fast processing of sensory input. Thus, clocks are ubiquitous in biology. In the present paper, we consider minimal abstract pure clocks in different configurations and investigate their characteristic dynamics. We are especially interested in optimally time-resolving clocks. Among these, we find fundamentally diametral clock characteristics, such as oscillatory behavior for purely local time measurement or decay-based clocks measuring time periods of a scale global to the problem. We include also sets of independent clocks ("clock bags"), sequential cascades of clocks and composite clocks with controlled dependency. Clock cascades show a "condensation effect" and the composite clock shows various regimes of markedly different dynamics.

Causal consistency is in an intermediate consistency model that can be achieved together with high availability and high performance requirements even in presence of network partitions. There are several proposals in the literature for causally consistent data stores. Thanks to the use of single scalar physical clocks, GentleRain has a throughput higher than other proposals such as COPS or Orbe. However, both of its correctness and performance relay on monotonic synchronized physical clocks. Specifically, if physical clocks go backward its correctness is violated. In addition, GentleRain is sensitive on the clock synchronization, and clock skew may slow write operations in GenlteRain. In this paper, we want to solve this issue in GenlteRain by using Hybrid Logical Clock (HLC) instead of physical clocks. Using HLC, GentleRain protocl is not sensitive on the clock skew anymore. In addition, even if clocks go backward, the correctness of the system is not violated. Furthermore, by HLC, we timestamp versions with a clock very close to the physical clocks. Thus, we can take causally consistency snapshot of the system at any give physical time. We call GentleRain protocol with HLCs Gentl

We survey the role of stable clocks in general relativity. Clock comparisons have provided important tests of the Einstein Equivalence Principle, which underlies metric gravity. These include tests of the isotropy of clock comparisons (verification of local Lorentz invariance) and tests of the homogeneity of clock comparisons (verification of local position invariance). Comparisons of atomic clocks with gravitational clocks test the Strong Equivalence Principle by bounding cosmological variations in Newton's constant. Stable clocks also play a role in the search for gravitational radiation: comparision of atomic clocks with the binary pulsar's orbital clock has verified gravitational-wave damping, and phase-sensitive detection of waves from inspiralling compact binaries using laser interferometric gravitational observatories will facilitate extraction of useful source information from the data. Stable clocks together with general relativity have found important practical applications in navigational systems such as GPS.

We present a derivation of the structure and dynamics of a ticking clock by showing that for finite systems a single natural principle serves to distinguish what we understand as ticking clocks from time-keeping systems in general. As a result we recover the bipartite structure of such a clock: that the information about ticks is a classical degree of freedom. We describe the most general form of the dynamics of such a clock, and discuss the additional simplifications to go from a general ticking clock to models encountered in literature. The resultant framework encompasses various recent research results despite their apparent differences. Finally, we introduce the information theory of ticking clocks, distinguishing their abstract information content and the actually accessible information.

In general relativity, the picture of spacetime assigns an ideal clock to each worldline. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of clocks along nearby worldlines. However, if time is defined operationally, as a pointer position of a physical clock that obeys the principles of general relativity and quantum mechanics, such a picture is at most a convenient fiction. Specifically, we show that the general relativistic mass-energy equivalence implies gravitational interaction between the clocks, while the quantum mechanical superposition of energy eigenstates leads to a non-fixed metric background. Based only on the assumption that both principles hold in this situation, we show that the clocks necessarily get entangled through time dilation effect, which eventually leads to a loss of coherence of a single clock. Hence, the time as measured by a single clock is not well-defined. However, the general relativistic notion of time is recovered in the classical limit of clocks.

A fundamental description of time can be consistent not only with the usual monotonic behavior but also with a periodic physical clock variable, coupled to the degrees of freedom of a system evolving in time. Generically, one would in fact expect some kind of oscillating motion of a system that is dynamical and interacts with its surroundings, as required for a fundamental clock that can be noticed by any other system. Unitary evolution does not require a monotonic clock variable and can be achieved more generally by formally unwinding the periodic clock movement, keeping track not only of the value of the clock variable but also of the number of cycles it has gone through at any moment. As a result, the clock is generically in a quantum state with a superposition of different clock cycles, a key feature that distinguishes oscillating clocks from monotonic time. Because the clock and an evolving system have a common conserved energy, the clock is in different cycles for different energy eigenstates of the system state. Coherence could therefore be lost faster than observed, for instance if a system that would be harmonic in isolation is made anharmonic by interactions with a fundam