Well-defined expression

Résumé

In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well defined (and thus not a function). The term well defined can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if f(x)=\frac{1}{x}, then even though f(0) is undefined does not mean that the function is not well defined – but simply that 0 is not in the domain of f. Example Let

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https://en.wikipedia.org/wiki/Well-defined_expression

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Properties of the shear stress peak radiated ahead of rapidly accelerating rupture fronts that mediate frictional slip

David Simon Kammer, Jean-François Molinari, Daniel Pino Muñoz, Mathilde Edmée Lucette Radiguet de la Bastaie

We study rapidly accelerating rupture fronts at the onset of frictional motion by performing high-temporal-resolution measurements of both the real contact area and the strain fields surrounding the propagating rupture tip. We observe large-amplitude and localized shear stress peaks that precede rupture fronts and propagate at the shear-wave speed. These localized stress waves, which retain a well-defined form, are initiated during the rapid rupture acceleration phase. They transport considerable energy and are capable of nucleating a secondary supershear rupture. The amplitude of these localized waves roughly scales with the dynamic stress drop and does not decrease as long as the rupture front driving it continues to propagate. Only upon rupture arrest does decay initiate, although the stress wave both continues to propagate and retains its characteristic form. These experimental results are qualitatively described by a self-similar model: a simplified analytical solution of a suddenly expanding shear crack. Quantitative agreement with experiment is provided by realistic finite-element simulations that demonstrate that the radiated stress waves are strongly focused in the direction of the rupture front propagation and describe both their amplitude growth and spatial scaling. Our results demonstrate the extensive applicability of brittle fracture theory to fundamental understanding of friction. Implications for earthquake dynamics are discussed.

National Academy of Sciences2016

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Achieving fundamental understanding of enantioselective heterogeneous synthesis is marred by the permanent presence of multitudinous arrangements of catalytically active sites in real catalysts. In this study, we address this issue by using structurally comparatively simple, well-defined, and chiral intermetallic PdGa{111} surfaces as catalytic substrates. We demonstrate the impact of chirality transfer and ensemble effect for the thermally activated azide-alkyne Huisgen cycloaddition between 3-(4-azidophenyl)propionic acid and 9-ethynylphenanthrene on these threefold symmetric intermetallic surfaces under ultrahigh vacuum conditions. Specifically, we encounter a dominating ensemble effect for this reaction as on the Pd-3-terminated PdGa{111} surfaces no stable heterocoupled structures are created, while on the Pd-1-terminated PdGa{111} surfaces, the cycloaddition proceeds regioselectively. Moreover, we observe chirality transfer from the substrate to the reaction products, as they are formed enantioselectively on the Pd-1-terminated PdGa{111} surfaces. Our results evidence a determinant ensemble effect and the immense potential of PdGa as asymmetric heterogeneous catalyst. Mechanistic insight into enantioselective reactions at intrinsically chiral surfaces can be challenging to obtain. Here the catalytic activity of Pd-1- and Pd-3-terminated PdGa{111} surfaces is shown to differ substantially, with Pd-1-terminated surfaces promoting on-surface azide- alkyne cycloadditions enantioand regioselectively.

NATURE RESEARCH2021

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Protecting Circuits From Computationally Bounded And Noisy Leakage

Sebastian Faust

Physical computational devices leak side-channel information that may, and often does, reveal secret internal states. We present a general transformation that compiles any circuit into a circuit with the same functionality but resilience against well-defined classes of leakage. Our construction requires a small, stateless, and computation-independent leak-proof component that draws random elements from a fixed distribution. In essence, we reduce the problem of shielding arbitrarily complex circuits to the problem of shielding a single, simple component. Our approach is based on modeling the adversary as a powerful observer that inspects the device via a limited measurement apparatus. We allow the apparatus to access all the bits of the computation (except those inside the leak-proof component), and the amount of leaked information to grow unbounded over time. However, we assume that the apparatus is limited in the amount of output bits per iteration and the ability to decode certain linear encodings. While our results apply in general to such leakage classes, in particular, we obtain security against (a) constant-depth circuits leakage, where the leakage function is computed by an AC(0) circuit (composed of NOT gates and unbounded fan-in AND and OR gates); (b) noisy leakage, where the leakage function reveals all the bits of the internal state of the circuit, but each bit is perturbed by independent binomial noise-i.e., flipped with some probability p. Namely, for some number p is an element of (0, 1/2], each bit of the computation is flipped with probability p, and remains unchanged with probability 1 - p.

Siam Publications2014

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