Higher Dimensional Harmonic Volume Can be Computed as an Iterated Integral

1992 ◽  
Vol 35 (3) ◽  
pp. 328-340 ◽  
Author(s):  
William M. Faucette

AbstractIn this paper it is shown that the computation of higher dimensional harmonic volume, defined in [1], can be reduced to Harris' computation in the onedimensional case (See [3]), so that higher dimensional harmonic volume may be computed essentially as an iterated integral. We then use this formula to produce a specific smooth curve , namely a specific double cover of the Fermat quartic, so that the image of the second symmetric product of in its Jacobian via the Abel-Jacobi map is algebraically inequivalent to the image of under the group involution on the Jacobian.

Author(s):  
Shinobu Hosono ◽  
Bong H Lian ◽  
Shing-Tung Yau

Abstract We continue our study on the hypergeometric system $E(3,6)$ that describes period integrals of the double cover family of K3 surfaces. Near certain special boundary points in the moduli space of the K3 surfaces, we construct the local solutions and determine the so-called mirror maps expressing them in terms of genus 2 theta functions. These mirror maps are the K3 analogues of the elliptic $\lambda $-function. We find that there are two nonisomorphic definitions of the lambda functions corresponding to a flip in the moduli space. We also discuss mirror symmetry for the double cover K3 surfaces and their higher dimensional generalizations. A follow-up paper will describe more details of the latter.


Author(s):  
John Sheridan

Abstract The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.


2015 ◽  
Vol 353 (5) ◽  
pp. 445-448
Author(s):  
Indranil Biswas ◽  
Sanjay Kumar Singh

Author(s):  
Y. Q. Du ◽  
M. J. Pan ◽  
Q. Li ◽  
L. Li
Keyword(s):  

2018 ◽  
Author(s):  
Peter De Wolf ◽  
Zhuangqun Huang ◽  
Bede Pittenger

Abstract Methods are available to measure conductivity, charge, surface potential, carrier density, piezo-electric and other electrical properties with nanometer scale resolution. One of these methods, scanning microwave impedance microscopy (sMIM), has gained interest due to its capability to measure the full impedance (capacitance and resistive part) with high sensitivity and high spatial resolution. This paper introduces a novel data-cube approach that combines sMIM imaging and sMIM point spectroscopy, producing an integrated and complete 3D data set. This approach replaces the subjective approach of guessing locations of interest (for single point spectroscopy) with a big data approach resulting in higher dimensional data that can be sliced along any axis or plane and is conducive to principal component analysis or other machine learning approaches to data reduction. The data-cube approach is also applicable to other AFM-based electrical characterization modes.


2020 ◽  
Vol 9 (10) ◽  
pp. 8545-8557
Author(s):  
K. P. Singh ◽  
T. A. Singh ◽  
M. Daimary
Keyword(s):  

Author(s):  
Nicholas Mee

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.


Sign in / Sign up

Export Citation Format

Share Document