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η-invariant
In mathematics, the eta invariant of a self-adjoint elliptic differential operator, elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using zeta function regularization. It was introduced by who used it to extend the Hirzebruch signature theorem to manifolds with boundary. The name comes from the fact that it is a generalization of the Dirichlet eta function. They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold. defined the signature defect of the boundary of a manifold as the eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at ''s''=0 or 1 of a Shimizu L-function. Definition The eta invariant of self-adjoint operator ''A'' is given by ''η''''A' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Differential Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real Method of characteristics, characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to Hyperbolic partial differential equation, hyperbolic and Parabolic partial differential equation, parabolic equations generally solve elliptic equations. Definitions Let L be a Differential operator, linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition Given a nonnegative integer ''m'', an order-m linear differential operator is a map P from a function space \mathcal_1 on \mathbb^n to another function space \mathcal_2 that can be written as: P = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta Function Regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization (physics), regularization or summability method that assigns finite values to Divergent series, divergent sums or products, and in particular can be used to define determinants and trace (linear algebra), traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to Condition number, ill-conditioned sums appearing in number theory. Definition There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series One method is to define its zeta regularized sum to be ζ''A''(−1) if this is defined, where the zeta function is defined for large Re(''s'') by : \zeta_A(s) = \frac+\frac +\cdots if this sum converges, and by analytic continuation elsewhere. In the case when ''a''''n'' = ''n'', the zeta function is the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hirzebruch Signature Theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combination of Pontryagin numbers called the L-genus. It was used in the proof of the Hirzebruch–Riemann–Roch theorem. Statement of the theorem The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series : = \sum_ = 1 + - +\cdots . The first two of the resulting L-polynomials are: * L_1 = \tfrac13 p_1 * L_2 = \tfrac1(7p_2 - p_1^2) (for further ''L''-polynomials see McTague, Carl (2014"Computing Hirzebruch L-Polynomials" or ). By taking for the p_i the Pontryagin classes p_i(M) of the tangent bundle of a 4''n'' dimensional smooth closed oriented manifold M one obtains the L-classes of M. Hirzebruch showed that the n-th L-class of M evaluated on the fundamental class of M, /m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Eta Function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdots. This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ''ζ''(''s'') — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ''ζ''*(''s''). The following relation holds: \eta(s) = \left(1-2^\right) \zeta(s) Both the Dirichlet eta function and the Riemann zeta function are special cases of polylogarithms. While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is entire and \eta( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature Defect
In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. introduced the signature defect for the cusp singularities of Hilbert modular surface In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular vari ...s. defined the signature defect of the boundary of a manifold as the eta invariant, the value as ''s'' = 0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at ''s'' = 0 or 1 of a Shimizu L-function. References * *{{Citation , last1=Hirzebruch , first1=Friedrich E. P. , title=Hilbert modular surfaces , doi=10.5169/seals-46292 , mr=0393045 , year=1973 , journal=L'Enseignement Mathé ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Modular Surface
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before. Definitions If ''R'' is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(''R'') acts on the product ''H''×''H'' of two copies of the upper half plane ''H''. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces: *The surface ''X'' is the quotient of ''H''×''H'' by SL2(''R''); it is not compact and usually has quotient singularities coming from point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shimizu L-function
In mathematics, the Shimizu ''L''-function, introduced by , is a Dirichlet series associated to a totally real algebraic number field. defined the signature defect of the boundary of a manifold as the eta invariant, the value as ''s''=0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular vari ... can be expressed in terms of the value at ''s''=0 or 1 of a Shimizu L-function. Definition Suppose that ''K'' is a totally real algebraic number field, ''M'' is a lattice in the field, and ''V'' is a subgroup of maximal rank of the group of totally positive units preserving the lattice. The Shimizu L-series is given by :L(M,V,s) = \sum_ \frac References * * * Zeta and L-functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
