Dimension | EPFL Graph Search

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one

THE CORONA THEOREM FOR THE DRURY-ARVESON HARDY SPACE AND OTHER HOLOMORPHIC BESOV-SOBOLEV SPACES ON THE UNIT BALL IN C-n

We prove that the multiplier algebra of the Drury-Arveson Hardy space H-n(2) on the unit ball in C-n has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space B-p(sigma) has the "baby corona property" for all sigma >= 0 and 1 < p < infinity. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

2011

Landmarking for Navigational Streaming of Stored High-Dimensional Media

Pascal Frossard, Yuan Yuan

Modern media data such as 360 degrees videos and light field (LF) images are typically captured in much higher dimensions than the observers' visual displays. To efficiently browse high-dimensional media, a navigational streaming model is considered: a client navigates the media space by dictating a navigation path to a server, who in response transmits the corresponding pre-encoded media data units (MDU) to the client one-by-one in sequence. Assuming that the MDU quality is pre-chosen and fixed, the problem resides in selecting and storing redundant representations of MDUs at the server in order to best trade off storage and transmission costs, while enabling adequate user's random access. We address this problem with a landmark-based MDU optimization framework. The media space is divided into neighborhoods, each containing one landmark (a chosen MDU). MDUs in a neighborhood use the associated landmark as a predictor for inter-coding. Thus, for any MDU transition within the same neighborhood, only one inter-coded MDU transmission is required when the landmark resides in the decoder buffer. It results in lower transmission cost and enables navigational random access. To optimize an MDU structure, we employ tree-structured vector quantizer (TSVQ) to first optimize landmark locations, then iteratively add P-MDUs as refinements using a fast branch-and-bound technique. Taking interactive LF images and viewport adaptive 360 degrees images as illustrative applications, and I-, P- and previously proposed merge frames to intra- and inter-code MDUs, we show experimentally that landmarked MDU structures can noticeably reduce the expected transmission cost compared with MDU structures without landmarks.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

The non-linear stochastic wave equation in high dimensions

Daniel Conus

The main topic of this thesis is the study of the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time. We are interested in questions of existence and uniqueness of solutions, as well as in properties of solutions, such as existence of high order moments and Hölder-continuity properties. The stochastic wave equation is formulated as an integral equation in which appear stochastic integrals with respect to martingale measures (in the sense of J.B. Walsh). Since, in dimensions greater than 3, the fundamental solution of the wave equation is neither a function nor a non-negative measure, but a general Schwartz distribution, we first develop an extension of the Dalang-Walsh stochastic integral that makes it possible to integrate a wide class of Schwartz distributions. This class contains the fundamental solution of the wave equation, under a hypothesis on the spectral measure of the noise that has already been used in the literature. With this extended stochastic integral, we establish existence of a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension. Uniqueness of the solution is established within a specific class of processes. In the case of a fine multiplicative noise, we obtain a series representation of the solution and estimates on the p-th moments of the solution (p ≥ 1). From this, we deduce Hölder-continuity of the solution under standard assumptions. The Hölder exponent that we obtain is optimal. For the case of general multiplicative noise, we construct a framework for working with appropriate iterated stochastic integrals and then derive a truncated Itô-Taylor expansion for the solution of the stochastic wave equation. The convergence of this expansion remains an open problem, so we conclude with some remarks that suggest an Itô-Taylor series expansion for the solution.

EPFL2008

The non-linear stochastic wave equation in high dimensions

Daniel Conus

EPFL2008