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Added by
J. M. J. van Leeuwen
Publication date
2013
fields
Physics
and research's language is
English
Authors
J. M. J. van Leeuwen
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The conditions are investigated under which a row of increasing dominoes is able to keep tumbling over. The analysis is restricted to the simplest case of frictionless dominoes that only can topple not slide. The model is scale invariant, i.e. dominoes and distance grow in size at a fixed rate, while keeping the aspect ratios of the dominoes constant. The maximal growth rate for which a domino effect exist is determined as a function of the mutual separation.
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The primary emphasis of this study has been to explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. Restricting our consideration to the genoise, one of the basic cakes of classic French cuisine, we have obtained a semi-empirical formula for its baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The formula, which is based on the Diffusion equation, has three adjustable parameters whose values are estimated from data obtained by baking genoises in cylindrical pans of various diameters. The resulting formula for the baking time exhibits the scaling behavior typical of diffusion processes, i.e. the baking time is proportional to the (characteristic length scale)^2 of the cake. It also takes account of evaporation of moisture at the top surface of the cake, which appears to be a dominant factor affecting the baking time of a cake. In solving this problem we have obtained solutions of the Diffusion equation which are interpreted naturally and straightforwardly in the context of heat transfer; however, when interpreted in the context of the Schrodinger equation, they are somewhat peculiar. The solutions describe a system whose mass assumes different values in two different regions of space. Furthermore, the solutions exhibit characteristics similar to the evanescent modes associated with light waves propagating in a wave guide. When we consider the Schrodinger equation as a non-relativistic limit of the Klein-Gordon equation so that it includes a mass term, these are no longer solutions.
The physics of a row of toppling dominoes is discussed. In particular the forces between the falling dominoes are analyzed and with this knowledge, the effect of friction has been incorporated. A set of limiting situations is discussed in detail, such as the limit of thin dominoes, which allows a full and explicit analytical solution. The propagation speed of the domino effect is calculated for various spatial separations. Also a formula is given, which gives explicitly the main dependence of the speed as function of the domino width, height and interspacing.
Quantum metrology exploits entangled states of particles to improve sensing precision beyond the limit achievable with uncorrelated particles. All previous methods required detection noise levels below this standard quantum limit to realize the benefits of the intrinsic sensitivity provided by these states. Remarkably, a recent proposal has shown that, in principle, such low-noise detection is not a necessary requirement. Here, we experimentally demonstrate a widely applicable method for entanglement-enhanced measurements without low-noise detection. Using an intermediate magnification step, we perform squeezed state metrology 8 dB below the standard quantum limit with a detection system that has a noise floor 10 dB above the standard quantum limit. Beyond its conceptual significance, this method eases implementation complexity and is expected to find application in next generation quantum sensors.
The modular design of planar phased arrays arranged on orthogonal polygon-shaped apertures is addressed and a new method is proposed to synthesize domino-tiled arrays fitting multiple, generally conflicting, requirements. Starting from an analytic procedure to check the domino-tileability of the aperture, two multi-objective optimization techniques are derived to efficiently and effectively deal with small and medium/large arrays depending on the values of the bounds for the cardinality of the solution space of the admissible clustered solutions. A set of representative numerical examples is reported to assess the effectiveness of the proposed synthesis approach also through full-wave simulations when considering non-ideal models for the radiating elements of the array.
Our work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions $F_lambda,G_lambda$ are defined as certain partition functions of the six vertex model, with variables corresponding to row rapidities, and the labeling signatures $lambda=(lambda_1ge ldotsge lambda_N)in mathbb{Z}^N$ encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomials. Cauchy type summation identities for $F_lambda,G_lambda$ and their skew counterparts follow from the Yang-Baxter equation. Using algebraic Bethe Ansatz, we obtain a double alternant type formula for $F_lambda$ and a Sergeev-Pragacz type formula for $G_lambda$. In the spirit of the theory of Schur processes, we define probability measures on sequences of signatures with probability weights proportional to products of our symmetric functions. We show that these measures can be viewed as determinantal point processes, and we express their correlation kernels in a double contour integral form. We present two proofs: The first is a direct computation of Eynard-Mehta type, and the second uses non-standard, inhomogeneo
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