The Quantum A∞A_{\infty}-Relations on the Elliptic Curve

We define and prove the existence of the Quantum $A_{\infty}$-relations on the Fukaya category of the elliptic curve, using the notion of the Feynman transform of a modular operad, as defined by Getzler and Kapranov. Following Barannikov, these relations may be viewed as defining a solution to the quantum master equation of Batalin-Vilkovisky geometry.Comment: 43 page

Applications of Graph Integration to Function Comparison and Malware Classification

We classify .NET files as either benign or malicious by examining directed graphs derived from the set of functions comprising the given file. Each graph is viewed probabilistically as a Markov chain where each node represents a code block of the corresponding function, and by computing the PageRank vector (Perron vector with transport), a probability measure can be defined over the nodes of the given graph. Each graph is vectorized by computing Lebesgue antiderivatives of hand-engineered functions defined on the vertex set of the given graph against the PageRank measure. Files are subsequently vectorized by aggregating the set of vectors corresponding to the set of graphs resulting from decompiling the given file. The result is a fast, intuitive, and easy-to-compute glass-box vectorization scheme, which can be leveraged for training a standalone classifier or to augment an existing feature space. We refer to this vectorization technique as PageRank Measure Integration Vectorization (PMIV). We demonstrate the efficacy of PMIV by training a vanilla random forest on 2.5 million samples of decompiled .NET, evenly split between benign and malicious, from our in-house corpus and compare this model to a baseline model which leverages a text-only feature space. The median time needed for decompilation and scoring was 24ms

Invariant properties for finding distance in space of elasticity tensors

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor, hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor that is endowed with a particular symmetry and is closest to the given elasticity tensor

On deformation-gradient tensors as two-point tensors in curvilinear coordinates

We derive a general expression for the deformation-gradient tensor by invoking the standard definition of a gradient of a vector field in curvilinear coordinates. This expression shows the connection between the standard definition of a gradient of a vector field and the deformation gradient tensor in continuum mechanics. We illustrate its application in the context of problems discussed by Ogden [1997]

Proof of validity of first-order travel estimates

In the seminal paper by Dahlen et al. the authors formulate an important expression as a first-order estimate of traveltime delay. The authors left out a term which would at first glance seem nontrivial, on the basis that their intention was to derive the Fr\'echet derivative linking the observed delay to the model perturbation (Nolet 2009, pers. comm.). Here we show that the derivation by Dahlen et al. results in a first-order estimate even without anticipating a Fr\'echet derivative, but instead remaining deductively in their Taylor-series formulation. Although a mathematical technicality, this strengthens the result of Dahlen et al. by showing that it is intrinsically valid, requiring no external justification. We show also that ignoring the aforementioned term is not valid in general and needs to be supported by careful argument.Comment: 6 pages, 1 figur

On commutativity of Backus and Gazis averages

We show that the Backus (1962) equivalent-medium average, which is an average over a spatial variable, and the Gazis et al. (1963) effective-medium average, which is an average over a symmetry group, do not commute, in general. They commute in special cases, which we exemplify

On closest isotropic tensors and their norms

An anisotropic elasticity tensor can be approximated by the closest tensor belonging to a higher symmetry class. The closeness of tensors depends on the choice of a criterion. We compare the closest isotropic tensors obtained using four approaches: the Frobenius 36-component norm, the Frobenius 21-component norm, the operator norm and the L2 slowness-curve fit. We find that the isotropic tensors are similar to each other within the range of expected measurement errors

On the Backus average of layers with randomly oriented elasticity tensors

As shown by Backus (1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of a randomly oriented anisotropic elasticity tensor, which-one might expect-would result in an isotropic medium. However, we show-by means of a fundamental symmetry of the Backus average-that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, a relationship between the Backus and Gazis et al. (1963) averages

On Christoffel roots for nondetached slowness surfaces

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the $qP$ slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the $qP$ and $qSV$ slowness surfaces are not smooth

On modelling bicycle power-meter measurements: Part II. Relations between rates of change of model quantities

Power-meter measurements are used to study a model that accounts for the use of power by a cyclist. The focus is on relations between rates of change of model quantities, such as power and speed, both in the context of partial derivatives, where other quantities are constant, and Lagrange multipliers, where other quantities vary to maintain the imposed constraints