Orthogonality Condition
Orthogonality Condition - Related Videos
01:26
Orthogonal Trajectories
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2026
Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
10:32
Orthogonal Protein Purification Facilitated by a Small Bispecific Affinity Tag
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A novel and highly efficient two-step affinity chromatography protocol has been developed and is described in detail. The method is based on a small purification tag with two inherent affinities and is applicable to a wide range of target proteins with different properties.
08:43
Calibration Procedures for Orthogonal Superposition Rheology
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Cited by 4 •
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We present a detailed calibration protocol for a commercial orthogonal superposition rheology technique using Newtonian fluids including end-effect correction factor determination methods and recommendations for best practices to reduce experimental error.
06:46
Fear Conditioning
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2023
Fear Conditioning is a type of learning in which an association is established between a negative unpleasant event and a harmless stimulus. This leads to a fear of the harmless stimulus. This process is largely mediated by the amygdala, which is a brain region involved in emotions and stress reactions. Fear conditioning can be utilized in several ways to understand different aspects of learning and memory. This video presents an overview of the principles behind fear conditioning, discusses the...
12:57
Resonance Fluorescence of an InGaAs Quantum Dot in a Planar Cavity Using Orthogonal Excitation and Detection
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Cited by 5 •
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Resonant excitation of a single self-assembled quantum dot can be achieved using an excitation mode orthogonal to the fluorescence collection mode. We demonstrate a method using the waveguide and Fabry-Perot modes of a planar microcavity surrounding the quantum dots. The method allows complete freedom in the detection polarization.
FAQs
The inner product is not merely a calculation detail: it determines what “orthogonal” means in a given setting. For vectors, the relevant operation is typically a dot product, whereas functions may require an integral over a specified domain. Changing the inner product or domain can therefore change whether the orthogonality condition holds, which matters when selecting coordinates or expansions.
An orthogonal component contributes no inner-product interaction with the component against which it is measured. This separation makes coordinate representations and projections easier to organize, and it supports least-squares solutions by distinguishing the represented part from remaining components. In practice, choosing orthogonal elements can reduce the coupling among terms in a mathematical representation.
Orthogonal bases allow components to be treated separately through the inner product, rather than handled as mutually entangled terms. This structure simplifies coordinate representations and supports expansions in orthogonal bases. The same principle extends beyond finite-dimensional vectors to function expansions, where orthogonality helps organize contributions across a chosen domain.
First identify the elements and the inner product appropriate to the problem. Compute the dot product for vectors or the corresponding integral for functions over the specified domain, then determine whether the result vanishes. The calculation should use the same inner-product convention intended for the coordinates, projection, or expansion being analyzed.
In Fourier series and spectral methods, orthogonality separates contributions associated with different elements of an expansion. That separation lets a function or solution be analyzed through distinct components rather than one undifferentiated expression. For differential equations, the condition helps organize the analysis of solutions by exploiting the structure of the selected orthogonal functions or basis.
For functions, the integral used in the inner product is tied to a specified domain. The domain is therefore part of the test, not a cosmetic detail: changing it can change the value of the integral and consequently the conclusion about orthogonality. Stating the domain makes function expansions and differential-equation analyses mathematically interpretable.