In differential geometry, given a spin structure on an ${\displaystyle n}$-dimensional orientable Riemannian manifold${\displaystyle (M,g),}$ one defines the spinor bundle to be the complex vector bundle${\displaystyle {\pi }_{\mathbf{S}}:\mathbf{S}\to M}$ associated to the corresponding principal bundle${\displaystyle {\pi }_{\mathbf{P}}:\mathbf{P}\to M}$ of spin frames over ${\displaystyle M}$ and the spin representation of its structure group${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)}$ on the space of spinors${\displaystyle {\mathrm{\Delta }}_n.}$.

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A section of the spinor bundle${\displaystyle \mathbf{S}}$ is called a spinor field.

1. The Spin DJ controller was jointly developed for optimal integration with the award-winning djay software, and delivers Mac users of all levels the experience of mixing, scratching and playing their music like a turntable DJ. Djay 3 seamlessly integrates with the user’s iTunes library and transforms the Mac into a complete DJ setup, while.
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Formal definition[edit]

Let ${\displaystyle (\mathbf{P},F_{\mathbf{P}})}$ be a spin structure on a Riemannian manifold${\displaystyle (M,g),}$ that is, an equivariant lift of the oriented orthonormal frame bundle${\displaystyle {\mathrm{F}}_{SO}(M)\to M}$ with respect to the double covering ${\displaystyle \rho :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{S}\mathrm{O}(n)}$ of the special orthogonal group by the spin group.

The spinor bundle${\displaystyle \mathbf{S}}$ is defined ^[1] to be the complex vector bundle

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${\displaystyle \mathbf{S}=\mathbf{P}{\times }_{\kappa }{\mathrm{\Delta }}_n}$

associated to the spin structure${\displaystyle \mathbf{P}}$ via the spin representation${\displaystyle \kappa :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{U}({\mathrm{\Delta }}_n),}$ where ${\displaystyle \mathrm{U}(\mathbf{W})}$ denotes the group of unitary operators acting on a Hilbert space${\displaystyle \mathbf{W}.}$ It is worth noting that the spin representation ${\displaystyle \kappa }$ is a faithful and unitary representation of the group ${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)}$.^[2]

See also[edit]

Notes[edit]

1. ^Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN978-0-8218-2055-1 page 53
2. ^Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN978-0-8218-2055-1 pages 20 and 24

Further reading[edit]

• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN978-0-691-08542-5.
• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN978-0-8218-2055-1

Tokyo, Japan & Starnberg, Germany – November 10, 2009: Vestax ( www.vestax.com ), an industry leader in professional DJ equipment, and algoriddim ( www.algoriddim.net ), a leader in next-generation music software for the Mac, today announced the release of Spin, a brand new all-in-one bundle of digital DJ hardware and algoriddim’s djay 3 software. The Spin DJ controller was jointly developed for optimal integration with the award-winning djay software, and delivers Mac users of all levels the experience of mixing, scratching and playing their music like a turntable DJ. djay 3 seamlessly integrates with the user’s iTunes library and transforms the Mac into a complete DJ setup, while Vestax’s hardware replicates the control and physical sensation of mixing and scratching on actual records.

Spin will be available exclusively in Apple Stores in the USA and Canada starting today.

The bundling of Vestax’s DJ MIDI controller and algoriddim’s djay 3 provides a powerful solution for all DJs, from beginner to pro. The Spin DJ controller mimics the classic setup of two turntables and a mixer, and easily connects to any Mac via USB. The user can then control djay 3 using two high resolution touch-sensitive jog wheels for scratching and remixing like on an analog turntable. Using the crossfader and controls for speed, looping, cue points, effects, and EQ, the user can smoothly fade from one song to another and apply professional-grade effects. The Spin controller also includes a built-in audio interface with a microphone input and two stereo outputs for headphones and master.

“Introducing DJing and reaching out to a new generation – the elder to the younger, the never before to the professional and experienced – has been one of our goals since the beginning of the DJ Midi controller movement,” said Charles Akira Ono, Vice President of the Vestax Corporation. “With our know-how and team of engineers dedicated in pro audio, quality and design, we’ve partnered up with software company algoriddim to bring you the perfect mix. Call it the best mash-up for bringing in the New Year, as we assure you will experience DJing at a new level of fun.”

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“The Spin controller is a simple, yet powerful solution for all DJs, whether they’re just beginning to experiment with their music or they’ve been mixing for years,” said Karim Morsy, CEO of algoriddim. “We designed djay 3 to bridge the gap between everyday music enthusiasts and professional DJs, so we are thrilled to be able to offer our customers a hardware component that brings them that much closer to the real thing and beyond. It has been a real pleasure to closely work with Vestax on delivering Mac users everything they need to turn their Mac into a full-blown DJ system.”

Feature Highlights:

• Touch Sensor Jog Wheels: High resolution, touch-sensitive jog wheels enable the user to DJ like in analog mode when scratching or mixing;
• Multi Channel Audio System: Spin comes with high quality sound for monitoring, master output and a separate microphone input;
• Compact and Light: Completely travel friendly, the unit is perfect for Mac users on-the-go. A simple USB cable and djay 3 will have the user spinning wherever they please.

Spin is exclusively available in Apple Stores for $249.95, including djay 3 in the bundle. djay 3 is also available for a free 15-day trial or for purchase from www.djay-software.com, and is also available for purchase from Apple retail locations for $49.95 or €39.95 (+ VAT).

About Vestax
Found in 1977, in Tokyo, Japan by Mr. Hidesato Shiino, Vestax has lead the way in numerous fields in the pro audio and musical instruments industry. Vestax Corporation was incorporated with the intention to satisfy three core values. Values which, at the time of incorporation, the founders thought were not being satisfied by other manufacturers, design teams or creators within the music industry. These founders sought to create a company that would lead the way for culture, quality and innovation. A company that would satisfy the needs and expectations of consumers. Those three values then, still govern business activity at Vestax today and they remain as strongly entrenched as ever. To learn more about Vestax, please visit www.vestax.com, www.facebook.com/pages/Vestax/42602083750, and twitter.com/VestaxJapan.

About algoriddim
algoriddim is a Germany-based software company that specializes in next-generation Mac audio and multimedia applications. Since its inception, algoriddim has pursued sophisticated, intuitive and user-friendly software applications. Its mission is to develop software that is accessible to the average consumer while offering the quality and power of professional tools. algoriddim consistently seeks out innovative methods for accommodating the needs of its customers, and its flagship music software, djay, is now serving a broad international audience of music lovers, professional DJs and world-class musicians. To learn more about algoriddim, please visit www.algoriddim.net, www.facebook.com/djaysoftware and twitter.com/djaysoftware.

About Spin
A leader in the DJ industry of DJ MIDI Controllers, Vestax was unsatisfied with the current DJ products on the market outside of the common Pro Audio gear. With the combination of djay and Apple Stores, Vestax is proud to bring you the newest in hands-on music + creativity. Spin will allow you to experience djing at an entire new level. To learn more about Spin, please visit www.djay-software.com/spin.

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Contact
Frederik Seiffert
algoriddim
pr@algoriddim.net

Press Contact U.S.
Vanessa Boynton
Matter Communications
(401) 351-9503
algoriddim@matternow.com

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For Support please contact:
Vestax Customer Support
1-888-498-4788
spin@vestax.jp

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