many nice properties such as “the five conditions” in Connes [8], and conversely, it is proved that [8] any commutative spectral triple (A, H, D, γ) satisfying those five conditions is equivalent to a spectral triple consisting of the algebra of smooth functions on a Riemannian manifold M , the module of sections of a Clifford bundle over M and a **Dirac** type operator on it. Furthermore, if (A, H, D, γ) satisfies an additional important property – the Poincar´ e duality in **K**-**theory** – which means (A, H, D, γ) represents the fundamental class (i.e., a **K**-orientation) in **K** 0 (A), then it is

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by (2.10), where the **Dirac** operator is associated with a conformally flat metric and the twisting is carried out by an idempotent playing the role of a general vector bundle. In forthcoming work, we will present a simplification of the **local** **formula** and will elaborate on the relation between properties of the functions of the modular automorphism in the simplified form and stability of the **index** under perturbations that leave the **Dirac** operator in the same connected component of Fredholm **operators**. We end this article by considering the case of the canonical flat metric on T 2 θ , which corresponds to the trivial conformal factor **k** = 1 whose corresponding modular automorphism ∆ is the identity. Therefore, our **formula** for the **index**, in this case, reduces to a much simpler form, as one can replace the functions of the modular automorphism with the values of the functions at 1. This yields, for **k** = 1 [68]:

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operator: E 00 + E 00,∗ . We call it the generalized Dolbeault-**Dirac** operator and relate it to the **theory** of Clifford module and generalized **Dirac** **operators** in chapter 5. Using the heat kernel method for Clifford superconnection presented in [BGV91], we obtain the following **index** **formula** for the Euler characteristic of cohesive modules that extends classical the Hirzebruch-Riemann-Roch **formula**.

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The aim of this paper is to define a motive whose realizations afford modular forms (of arbitrary weight) on an indefinite division quaternion algebra. The idea of the construction, once again, is due to Jordan and Livn´e. However some remarks are in order. First, it is worth noting that although the realizations of the motive constructed in this paper are abstractly isomorphic to the D = disc(B)-new part of (two copies of) the realizations of the motive constructed in [Sc] via the Jacquet–Langlands correspondence, a “motivic Jacquet–Langlands correspondence” has not yet described that lifts this correspondence to the motivic setting. Therefore what we propose is the first construction –as a Chow motive– of D-new modular forms. Second, following their construction in this indefinite setting and working at the level of realizations gives the various incarnations of two copies of odd weight modular forms, rather than just one copy. It is not possible to canonically split them in a single copy: this is possible only including a splitting field for the quaternion algebra in the coefficients, but the resulting splitting depends on the choice of an identification of the base changed algebra with the split quaternion algebra. Indeed, we will construct a motive whose realizations afford two copies of odd weight modular forms. Finally, the idea of Jordan and Livn´e is to construct square roots of the Laplace **operators** after appropriately tensoring the source and the targets of ∆ n ; however the

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In this chapter, we will compute the **K** -**theory** of stunted **projective** space and their corresponding Adams’ operations. This will form a major component of the proof that R P n+ρn / R P n−1 is not coreducible. However, we will include every stunted **projective** space, for completeness, even if it is not used in the vector fields problem. We will generally use the notation R P n / R P m instead of R P n m+1 to emphasise that we only

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the polynomials and numbers are called the **twisted** q-Bernoulli polynomials and numbers, respectively. When **k** 1 and q 1, the polynomials and numbers are called the **twisted** Bernoulli polynomials and numbers, respectively. When **k** 1, q 1, and ζ 1, the polynomials and numbers are called the ordinary Bernoulli polynomials and numbers, respectively.

The uniqueness and existence theorems for the solutions of boundary value problems for systems of partial diﬀerential equations are suﬃciently well known. Such problems have remarkable applications in mathematical physics, the mechanics of deformable bodies, electromagnetism, relativistic quantum mechanics, and some of their natural generaliza- tions. Almost all such problems can be set in the context of Cliﬀord analysis (see [, ]). Cliﬀord analysis is centered around the concept of monogenic functions, i.e. null solu- tions of a ﬁrst order vector valued rotation invariant diﬀerential operator called the **Dirac** operator which factorizes the Laplace operator (see [, ]). As to the mathematical study of boundary value problems in Cliﬀord analysis, there are several diﬀerent approaches known in the literature. Without claiming completeness, we mention some of them. First of all, we have the approach originating with Bernstein, whose approach is to translate boundary value problems to the corresponding singular integral equations, then use the properties of the Fredholm operator to discuss the solvability of singular integral equations (see []). Another important approach is based on complex analysis. In this case, ﬁrst we use analytic function **theory** to solve these kinds of boundary value problems, then we use the results of boundary value problems to solve singular integral equations (see [, ]). The advantage of this method is that the explicit representation of solutions can be ob- tained, but in the higher dimensional space there still exist many obstacles to generalize this method. In this paper, we continue to use the method in [, ] to solve boundary value problems for the modiﬁed **Dirac** **operators**.

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In this section we compute the **K**–**theory** relevant to the non-compact BZDP model and show that it agrees exactly with the D-brane spectrum found using BCFT tech- niques. We do this in two different ways; first we use a long exact sequence similar to the one in [8], then we show that the result can be easily obtained by using the connection between Clifford Algebras and **K**–**theory**. The former method’s advan- tage is that it identifies which D-branes carry the same charges. This is particularily useful for torsion charged D-branes. The exact sequence method however, becomes quite cumbersome and it is sometimes difficult to disentangle the results.

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spectrum and norming constants was investigated in []. For the **Dirac** operator, the in- verse periodic and antiperiodic boundary value problems were given in [–]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a **Dirac** type-system were developed in [, ]. Uniqueness of the inverse problem for the **Dirac** operator with a dis- continuous coeﬃcient by the Weyl function was studied in [] and discontinuity condi- tions inside an interval were worked out in [, ]. The inverse problem for weighted **Dirac** equations was obtained in []. The reconstruction of the potential by the spectral function was given in []. For the **Dirac** operator with peculiarity, the inverse problem was found in []. Inverse nodal problems for the **Dirac** operator were examined in [, ]. In the case of potentials that belong entrywise to L p (, ), for some p ∈ [, ∞), the in-

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If K is a collineation realizing a projective deformation C of second order, ==3,4,..., and at the same time K realizes projective deformations of order i, 2, or then C is called weakly [r]

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of this system breaks down to the action of the special unitary group. The study of complex **Dirac** **operators** was initiated in 5–8; a systematic development of the associated function **theory**, including the invariance properties with respect to the underlying Lie groups and Lie algebras, is still in full progress see, e.g., 9–13.

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Again, a notion of two-sided H-monogenicity may be defined similarly. However, unless mentioned explicitly, we will only work with left H-monogenic matrix functions. This matrix approach has also been successfully applied in 17, 24 for the construction of a boundary values **theory** of h-monogenic functions.

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In § (2), we outline two fundamental reasons that call for a revision of Professor Einstein’s GTR. In § (3), we give an overview of the reading [5], and Professor Weyl’s [6] supposedly failed **theory** is brought back to life. In § (4), we demonstrate an important part of the unified **theory** to be developed, namely that the metric tensor is in-principle decomposable into a mathematical entity that has not ten free parameters, but four. In § (5), the spacetime upon with the present **theory** is build is laid down. In § (6), tensorial affinites are proposed in which even the desired spacetime is defined. Once the desired spacetime is defined, in § (7), the general field equations are written down and in § (8), it is shown that as a result of the tensorial nature of the affinities, it is possible to get rid of the non-linear terms of the resulting curvature tensor, thereby making the **theory** a linear **theory**. From this linear spacetime with tensorial affinities, in § (9), we write down the resulting field equation. Having laid down the **theory**, in § (10), we take stock of what has been achieved thus far. In § (11), we show how one can bring the gravitational force into the fold of the proposed unified **theory**. In § (12) we write down an appropriate geodesic equation that employs tensorial affinities. In § (13), we demonstrate that the proposed unified **theory** does contain Yang-Mills **Theory** and finally in § (14), (15) and (16), we give a general discussion, the conclusion drawn thereof and the recommendations for future works.

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In order to prove the main result, we also need a suitable Caccioppoli estimation (see Theorem .). Then by the technique of removable singularities, we can ﬁnd that solutions to an A-harmonic system satisfying a Lipschitz condition or in the case of a bounded mean oscillation can be extended to Cliﬀord valued solutions to the corresponding A-**Dirac** sys- tem.

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In the present paper, the semi-commutative differential oparators associated with the 1-dimensional **Dirac** operator are constructed. Using this results, the hierarchy of the mKdV (−) polynomials are expressed in terms of the KdV polyno- mials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (−) polynomials is constructed by the al- gebraic method.

The indeﬁnite inner product · , · **K** on a Krein space **K** gives rise to a classiﬁcation of elements of **K**. An element **k** ∈ **K** is called positive, negative, or neutral if **k**, **k** **K** > , **k**, **k** **K** < , or **k**, **k** **K** = respectively. A linear manifold or a subspace L in **K** is called indeﬁnite if it contains both positive and negative elements. We say that L is semi-deﬁnite if it is not indeﬁnite. A semi-deﬁnite subspace L is called non-negative (positive, uniformly positive) if x, x ≥ (x, x > , x, x ≥ δ x, (δ > )) for all x in L. A non-positive (nega- tive, uniformly negative) subspace is deﬁned in a similar way. We say that the subspace L is deﬁnite if x, x = if and only if x = .

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Case 1 . i 0 . In this case we may take u { 2,3} 1, and v 4, 5, 6 , the vertex w should be of distance 2 from v , hence should meet v and w u . If w meets v in one element we have 3/2(n − 6)(n − 7) choices for it and if it meets v in 2 elements again we have 3(n – 6) choices for it and the **formula** for Sz G 0 is obtained as above.

The complex algebra of physical space, CAPS, is the complex extension of. APS. It is also the hyperbolic extension of APS. Here we will introduce a[r]

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The energy vector in the **Dirac** **theory** has come when we would try to show the analogy between sign of helicity and the sign of energy, which we have then called sign of enginity. This energy vector need time vector whose components deserve physical senses.

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Let U and V be, respectively, an inﬁnite- and a ﬁnite-dimensional complex Banach algebras, and let U⊗ p V be their **projective** tensor product. We prove that (i) every compact Hermitian operator T 1 on U gives rise to a compact Hermitian operator T on U ⊗ p V having the properties that T 1 = T and sp(T 1 ) = sp(T ); (ii) if U and V are separable and U has Hermitian approximation property (HAP), then U ⊗ p V is also separable and has HAP; (iii) every compact analytic semigroup (CAS) on U induces the existence of a CAS on U⊗ p V having some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.

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