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A lower bound for the decoder error probability (P sub E (u)) of a linear maximum distance separable (MDS) code is derived by counting the dominant types of decoding words around code words. It is shown that the lower bound derived is similar in form, and close numerically, to the upper bound derived.
Thermodynamic models have been formulated to predict lower and upper bounds for the mass of pressurant gas required to pressurize a cryogenic tank and then expel liquid from the tank. Limiting conditions are based on either thermal equilibrium or zero energy exchange between the pressurant gas and initial tank contents. The models are independent of gravity level and allow specification of autogenous or non-condensible pressurants. Partial liquid fill levels may be specified for initial and final conditions. Model predictions are shown to successfully bound results from limited normal-gravity tests with condensable and non-condensable pressurant gases. Representative maximum collapse factor maps are presented for liquid hydrogen to show the effects of initial and final fill level on the range of pressurant gas requirements. Maximum collapse factors occur for partial expulsions with large final liquid fill fractions.
The dark dimension scenario, predicting one extra mesoscopic dimension in the micron range, has emerged by applying various swampland principles to the dark energy. In this note we find that realizing the grand unification of gauge forces is highly constraining in this context. Without actually constructing any grand unified theory (GUT) models, we argue that the mere assumption of grand unification of forces in this scenario, together with the experimental bounds on massive replicas of the Standard Model gauge bosons, predicts an upper bound for the GUT scale, 𝑀 GUT ≲ 10 16 GeV. Combined with the experimental bound on the proton lifetime, this predicts that the 𝑋 gauge boson mediating proton decay is a 5D solitonic string of Planckian tension stretched across a length scale 𝐿 ∼ (1–10 TeV) −1 ending on gauge branes of the same diameter ∼𝐿. This leads to a mass of 𝑀 𝑋 ∼ 10 15 –10 16 GeV. In particular assuming grand unification in the dark dimension scenario results in a tower of Kaluza-Klein excitations of Standard Model gauge bosons on the gauge branes in the 1–10 TeV range. This suggests that the diameter/separation 𝐿 of the gauge branes correlates with both the weak scale ∼1/𝐿 near a TeV and the GUT scale ∼𝑀$^2_5$𝐿 at 10 16 GeV.
A lower bound on the minimum free Euclidean distance of trellis-coded modulation (TCM) is derived that guarantees the existence of good TCM codes of any complexity. The bound is used to compare trellis codes combined with phase-shift keying, pulse amplitude modulation, and quadrature amplitude-shift keying modulation. This random coding bound is the first lower bound on the free distance of trellis codes, is tighter than any upper bound for large constraint lengths, and predicts the asymptotic performance of TCM when the complexity of the code becomes large. The bound can be used with any code rate and any modulation scheme and shows that the free distance increases linearly with the constraint length for large values of the constraint length.
We derive tight bounds on the cache misses for evaluation of explicit stencil operators on structured grids. Our lower bound is based on the isoperimetrical property of the discrete octahedron. Our upper bound is based on a good surface to volume ratio of a parallelepiped spanned by a reduced basis of the interference lattice of a grid. Measurements show that our algorithm typically reduces the number of cache misses by a factor of three, relative to a compiler optimized code. We show that stencil calculations on grids whose interference lattice have a short vector feature abnormally high numbers of cache misses. We call such grids unfavorable and suggest to avoid these in computations by appropriate padding. By direct measurements on a MIPS R10000 processor we show a good correlation between abnormally high numbers of cache misses and unfavorable three-dimensional grids.
The area-average strength of the open magnetic field in the polar coronal holes can be estimated from the radial component of the magnetic field measured by Ulysses in the solar wind, the fraction of the solar sphere covered by the polar coronal holes, and the fraction of the heliosphere filled by the fast solar wind from the polar coronal holes. For the present minimum phase of the solar cycle, the estimated strength is approximately 10 G. Using this strength for the ambient open field in the standard reconnection model for jets in coronal holes, we obtain for any given jet-front speed a lower bound on the initial temperature of the expanding jet-front plasma, and an upper bound on the ambient plasma density at the reconnection site. These two bounds indicate the following. For jet-front speeds of approximately 1000 km/s, (1) the reconnection site has to be in the low corona or upper transition region (n(e) is less than 10(exp 9) cm(exp -3)), not in the lower transition region or chromosphere, (2) the jet-front plasma is initially heated to T greater than approximately 10(exp 7) K, and (3) hence a compact X-ray flare is produced at the base of the jet. For jet-front speeds less than approximately 100 km/s, (1) the jet can be produced by reconnection in the lower transition region (approximately 10(exp 9) less than n(e) less than approximately 10(exp 10) cm(exp-3)) or upper chromosphere (approximately 10(exp 10) less than n(e) less than approximately 10(exp 12) cm-3), (2) the initial temperature of the jet-front plasma can be less than 10(exp 6) K, and (3) hence some EUV and H(alpha) jet-type macrospicules may be produced with no detectable X-ray emission.
In many realistic data filtering problems, the cross correlation of the state estimation error and the state forcing function is unknown due to the poor knowledge of the time history of the forcing function. In this paper, the conservative and minimal approximation to the cross correlation terms is presented. It requires only the knowledge of the estimation error covariance and the forcing function covariance, with the choice of an associated free parameter left to the user. If the estimation error covariance and/or the forcing function covariance are bounded from above but not known exactly, the cross correlation approximation using those upper bounds remains conservative. This cross correlation approximation leads to a conservative approximation to the estimation error covariance matrix differential equation between measurement times. The free parameter is determined as the analytic solution to an associated optimal control problem. The procedure is expanded to include discrete linear measurement incorporation.
An overview of current theories of the formation of our Solar System, with emphasis on giant planets, is presented. The most detailed models are based upon observations of planets and smaller bodies within our own Solar System and of young stars and their environments. Terrestrial planets are believed to grow via pairwise accretion until the spacing of planetary orbits becomes large enough that the configuration is stable for the age of the system. Giant planets begin their growth as do terrestrial planets, but they become massive enough that they are able to accumulate substantial amounts of gas before the protoplanetary disk dissipates. Larger disk mass allows for faster growth of solid planetary bodies. The ability of a solid planet to trap gas from the protoplanetary disk increases rapidly as its mass increases (because the depth of its gravitational potential well increases), but decreases as the planetesimal accretion rate is increased (as it becomes hotter). The net effect of increasing disk mass is that gas giant planets form more rapidly, but with larger core masses. Observations of circumstellar disks suggest an upper bound on the time available prior to dissipation of the gas, and planetary models place upper limits on core sizes. Together, these constraints suggest that Jupiter and Saturn formed in 1-10 million years, and the density of solids in the region of their formation was a few times as large as the lower bound provided by the traditional minimum mass nebula.
An overview of current theories of the formation of our Solar System, with emphasis on giant planets, is presented. The most detailed models are based upon observations of planets and smaller bodies within our own Solar System and of young stars and their environments. Terrestrial planets are believed to grow via pairwise accretion until the spacing of planetary orbits becomes large enough that the configuration is stable for the age of the system. Giant planets begin their growth as do terrestrial planets, but they become massive enough that they are able to accumulate substantial amounts of gas before the protoplanetary disk dissipates. Larger disk mass allows for faster growth of solid planetary bodies. The ability of a solid planet to trap gas from the protoplanetary disk increases rapidly as its mass increases (because the depth of its gravitational potential well increases), but decreases as the planetesimal accretion rate is increased (as it becomes hotter). The net effect of increasing disk mass is that gas giant planets form more rapidly, but with larger core masses. Observations of circumstellar disks suggest an upper bound on the time available prior to dissipation of the gas, and planetary models place upper limits on core sizes. Together, these constraints suggest that Jupiter and Saturn formed in 1 - 10 million years, and the density of solids in the region of their formation was a few times as large as the lower bound provided by the traditional minimum mass nebula.
Time domain stability robustness analysis and design for linear multivariable uncertain systems with bounded uncertainties is the central theme of the research. After reviewing the recently developed upper bounds on the linear elemental (structured), time varying perturbation of an asymptotically stable linear time invariant regulator, it is shown that it is possible to further improve these bounds by employing state transformations. Then introducing a quantitative measure called the stability robustness index, a state feedback conrol design algorithm is presented for a general linear regulator problem and then specialized to the case of modal systems as well as matched systems. The extension of the algorithm to stochastic systems with Kalman filter as the state estimator is presented. Finally an algorithm for robust dynamic compensator design is presented using Parameter Optimization (PO) procedure. Applications in a aircraft control and flexible structure control are presented along with a comparison with other existing methods.
Using Filipov's Theorem, it is shown that the conditions oif nonfinite escape of trajectories, reachability, and convexity of the dynamics over all admissible controls are needed for the existence of a time optimal solution for the robotic equation. With a lower bound for the finite-escape time established using a Liapunov approach, and an upper bound for the time to reach the target established using the exact linearization idea, a single inequality is found which is closely related to the coriolis and the centrifugal terms, the absence of which implies that the domain of existence of the optimal solution can be made arbitrarily large with a large torque constraint. As the work space is finite, this is essentially a global result in practical situations.
This note addresses the issue of 'conservatism' in the time domain stability robustness bounds obtained by the Liapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Liapunov approach with respect to the basis of the vector space in which the Liapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the 'optimal' transformation. Several examples are presented to illustrate the proposed analysis.
Upper bounds of eigenvalues of simply-supported rectangular plate using Rayleigh-Ritz method
A new approach is proposed to the evaluation of the lower and upper bounds of the global heat production rate within planetary interiors. The approach is based on the relationship between the internal energy change and the volume change of a planetary object. For illustration, the approach is applied to the moon. Using an average global surface heat flux of 18 erg/sq cm-sec, and assuming constancy of the lunar radius during the past 3.2 billion years. the lunar heat release within the past 3.2 billion years is estimated at (30 to 40) x 10 to the tenth erg/cu cm. This is equivalent to the present day uranium concentration of 35 to 50 ppb provided the radiogenic isotopes are of the same proportion as that given by Toksoz et al. (1978).
The problem of formation of the aerosol of space origin in Earth s atmosphere is examined. Meteoroids of the mass range of 10-18-10-8 g are considered as a source of its origin. The lower bound of the mass range is chosen according to the data presented in literature, the upper bound is determined in accordance with the theory of Whipple s micrometeorites. Basing on the classical equations of deceleration and heating for small meteor bodies we have determined the maximal temperatures of the particles, and altitudes at which they reach critically low velocities, which can be called as velocities of stopping . As a condition for the transformation of a space particle into an aerosol one we have used the condition of non-reaching melting temperature of the meteoroid. The simplified equation of deceleration without earth gravity and barometric formula for the atmosphere density are used. In the equation of heat balance the energy loss for heating is neglected. The analytical solution of the simplified equations is used for the analysis.
The calculation of currents induced by a transverse electric plane wave normally incident upon an infinite strip embedded in a grounded dielectric slab is used to infer a lower bound on the resonant frequency (or resonant-E-plane dimension) for rectangular microstrip antennas. An upper bound is provided by the frequency for which the E-plane dimension is a half-wavelength.
A mathematical model of the annoyance created at an airport by aircraft operations is developed. The model incorporates population distribution considerations around an airport and the annoyance caused by aircraft noise. The objective function of this model corresponds to seeking to minimize total population annoyance created by all aircraft operations in a 24-hour period. Several factors are included in this model as constraint bounded. Demand for flight services is incorporated by including lower bounds on the number of operations by type of aircraft, runway used and time period. Also upper bounds on the number of operations for each runway are included. The mathematical model as formulated is recognized as corresponding to a nonlinear integer mathematical programming problem.
Upper and lower bounds for ground-state second- order perturbation energy