THE HYDRODYNAMIC FLOW OF NEMATIC LIQUID CRYSTALS IN R\u3csup\u3e3\u3c/sup\u3e

This manuscript demonstrates the well-posedness (existence, uniqueness, and regularity of solutions) of the Cauchy problem for simplified equations of nematic liquid crystal hydrodynamic flow in three dimensions for initial data that is uniformly locally L3(R3) integrable (L3U(R3)). The equations examined are a simplified version of the equations derived by Ericksen and Leslie. Background on the continuum theory of nematic liquid crystals and their flow is provided as are explanations of the related mathematical literature for nematic liquid crystals and the Navier–Stokes equations

Geometric Cross-Modal Comparison of Heterogeneous Sensor Data

In this work, we address the problem of cross-modal comparison of aerial data streams. A variety of simulated automobile trajectories are sensed using two different modalities: full-motion video, and radio-frequency (RF) signals received by detectors at various locations. The information represented by the two modalities is compared using self-similarity matrices (SSMs) corresponding to time-ordered point clouds in feature spaces of each of these data sources; we note that these feature spaces can be of entirely different scale and dimensionality. Several metrics for comparing SSMs are explored, including a cutting-edge time-warping technique that can simultaneously handle local time warping and partial matches, while also controlling for the change in geometry between feature spaces of the two modalities. We note that this technique is quite general, and does not depend on the choice of modalities. In this particular setting, we demonstrate that the cross-modal distance between SSMs corresponding to the same trajectory type is smaller than the cross-modal distance between SSMs corresponding to distinct trajectory types, and we formalize this observation via precision-recall metrics in experiments. Finally, we comment on promising implications of these ideas for future integration into multiple-hypothesis tracking systems.Comment: 10 pages, 13 figures, Proceedings of IEEE Aeroconf 201

The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity

In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, Cap A , \operatorname {Cap}_{\mathcal {A}}, where A \mathcal {A} -capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the p p -Laplace equation and whose solutions in an open set are called A \mathcal {A} -harmonic. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: $$[ Cap A ⁡ ( λ E 1 + ( 1 − λ ) E 2 ) ] 1 ( n − p ) ≥ λ [ Cap A ⁡ ( E 1 ) ] 1 ( n − p ) + ( 1 − λ ) [ Cap A ⁡ ( E 2 ) ] 1 ( n − p ) \left [\operatorname {Cap}_\mathcal {A} ( \lambda E_1 + (1-\lambda ) E_2 )\right ]^{\frac {1}{(n-p)}} \geq \lambda \, \left [\operatorname {Cap}_\mathcal {A} ( E_1 )\right ]^{\frac {1}{(n-p)}} + (1-\lambda ) \left [\operatorname {Cap}_\mathcal {A} (E_2 )\right ]^{\frac {1}{(n-p)}}$$ when 1 &gt; p &gt; n , 0 &gt; λ &gt; 1 , 1&gt;p&gt;n, 0 &gt; \lambda &gt; 1, and E 1 , E 2 E_1, E_2 are convex compact sets with positive A \mathcal {A} -capacity. Moreover, if equality holds in the above inequality for some E 1 E_1 and E 2 , E_2, then under certain regularity and structural assumptions on A , \mathcal {A}, we show that these two sets are homothetic. In the second part of this article we study a Minkowski problem for a certain measure associated with a compact convex set E E with nonempty interior and its A \mathcal {A} -harmonic capacitary function in the complement of E E . If μ E \mu _E denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure μ \mu on S n − 1 \mathbb {S}^{n-1} , find necessary and sufficient conditions for which there exists E E as above with μ E = μ . \mu _E = \mu . We show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem for volume as well as in the work of Jerison in \cite{J} for electrostatic capacity. Using the Brunn-Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation when p ≠ n − 1 p\neq n- 1 and translation and dilation when p = n − 1 p = n-1 .</p

Well-Posedness of Nematic Liquid Crystal Flow in Luloc3(R3)L^3_{\hbox{uloc}}(\R^3)

In this paper, we establish the local well-posedness for the Cauchy problem of the simplified version of hydrodynamic flow of nematic liquid crystals (\ref{LLF}) in $\mathbb R^3$ for any initial data $(u_0,d_0)$ having small $L^3_{\hbox{uloc}}$-norm of $(u_0,\nabla d_0)$. Here $L^3_{\hbox{uloc}}(\mathbb R^3)$ is the space of uniformly locally $L^3$-integrable functions. For any initial data $(u_0, d_0)$ with small $\displaystyle |(u_0,\nabla d_0)|_{L^3(\mathbb R^3)}$, we show that there exists a unique, global solution to (\ref{LLF}) which is smooth for $t>0$ and has monotone deceasing $L^3$-energy for $t\ge 0$.Comment: 29 page

(1) GNGA FOR GENERAL REGIONS: SEMILINEAR ELLIPTIC PDE AND CROSSING EIGENVALUES

Abstract. We consider the semilinear elliptic PDE ∆u + f(λ, u) = 0 with the zero-Dirichlet boundary condition on a family of regions, namely stadions. Linear problems on such regions have been widely studied in the past. We seek to observe the corresponding phenomena in our nonlinear setting. Using the Gradient Newton Galerkin Algorithm (GNGA) of Neuberger and Swift, we document bifurcation, nodal structure, and symmetry of solutions. This paper provides the first published instance where the GNGA is applied to general regions. Our investigation involves both the dimension of the stadions and the value λ as parameters. We find that the so-called crossings and avoided crossings of eigenvalues as the dimension of the stadions vary influences the symmetry and variational structure of nonlinear solutions in a natural way. 1