In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex...

a star Bipolar coordinates, a two-dimensional orthogonal coordinate system Bipolar set, a derivative of a polar set Bipolar theorem, a theorem in convex...

{\displaystyle f^{**}\leq f} . This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality (via the perturbation...

_{c\in C}\langle c,y\rangle .} Banach–Alaoglu theorem – Theorem in functional analysis Bipolar theorem – Theorem in convex analysis Polar cone – Concepts in...

Banach–Alaoglu theorem, setting τ {\displaystyle \tau } to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem. Let M {\displaystyle...

and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of...

cone C in X, the polar cone is equivalent to the polar set for C. Bipolar theorem Polar set Schaefer & Wolff 1999, pp. 215–222. Boyd, Stephen P.; Vandenberghe...

Cone (disambiguation) Cone (geometry) Cone (topology) Farkas' lemma Bipolar theorem Ordered vector space Boyd, Stephen; Vandenberghe, Lieven (2004-03-08)...

-closure of the convex balanced hull of A . {\displaystyle A.} The bipolar theorem: The bipolar of A , {\displaystyle A,} denoted by A ∘ ∘ , {\displaystyle A^{\circ...

that are compact (and thus complete) but not closed. The bipolar theorem states that the bipolar (that is, the polar of the polar) of a subset of a locally...

Bipolar transistors must be properly biased to operate correctly. In circuits made with individual devices (discrete circuits), biasing networks consisting...

only if S is an orthonormal basis; i.e., maximal orthonormal set. bipolar bipolar theorem. bounded A bounded operator is a linear operator between Banach...

the open neighborhoods of the origin. This fact together with the bipolar theorem means that via the operation of taking the polar of a subset, the collection...

∘ = H ∘ ∘ = H {\displaystyle U^{\circ }=H^{\circ \circ }=H} by the bipolar theorem, it follows that a continuous linear functional f {\displaystyle f}...

more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal...

biodegrade in biologically active environments Bipolar I disorder Boolean prime ideal theorem, a mathematical theorem Brachial plexus injury, an injury to the...

Mohr–Mascheroni theorem Poncelet–Steiner theorem Descartes' theorem Dinostratus' theorem Dividing a circle into areas Equal incircles theorem Five circles theorem Gauss...

vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy...

Atlas, Homma, and Marks. "An Artificial Neural Network for Spatio-Temporal Bipolar Patterns: Application to Phoneme Classification" (PDF). Neural Information...

acyclic orientation with a single source and a single sink is called a bipolar orientation. A transitive orientation is an orientation such that the resulting...

No. 35 on the Top Independent Albums chart. Townsend was diagnosed with bipolar disorder around 1998, a condition that was unknowingly exacerbated by his...

communication, the field of information theory (in particular, the Shannon–Hartley theorem) motivates the need for studying the signal-to-noise ratio (SNR). In a...

non-concentric circles can be aligned with the circles of a system of bipolar coordinates. In that case, the radical axis is simply the y {\displaystyle...

family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek...

load. Modern semiconductor switches such as MOSFETs or insulated-gate bipolar transistors (IGBTs) are well suited components for high-efficiency controllers...

remaining is the source. This is a common-source FET circuit. The analogous bipolar junction transistor circuit may be viewed as a transconductance amplifier...

vacuum tube triodes. The same analysis applies to modern devices such as bipolar junction and field-effect transistors. Consider a circuit of an ideal inverting...

transistor. A common type of phototransistor, the bipolar phototransistor, is in essence a bipolar transistor encased in a transparent case so that light...

sources are useful, for example, in modeling the behavior of amplifiers. A bipolar junction transistor can be modeled as a dependent current source whose...

possible circles that pass through both foci. These circles form the basis of bipolar coordinates. The circles of Apollonius of a triangle are three circles...