Abstract Consider a proper, isometric action by unimodular locally compact group $G$ on Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ $M$, and element $g \in G$, we define numerical index ${\operatorname {index}}_g(D)$, in terms of parametrix for trace associated to $g$. We prove equivariant Atiyah–Patodi–Singer theorem this index. ...

An algebraic construction for constant dimension subspace codes is called orbit code. It arises as the orbits under the action of a subgroup of the general linear group on subspaces in an ambient space. In particular orbit codes of a Singer subgroup of the general linear group has investigated recently. In this paper, we consider the normalizer of a Singer subgroup of the general linear group a...

Let E → X be an oriented surface bundle over a closed surface. Then the signature sign(E) is determined by the first Chern class of the flat vector bundle Γ associated to the monodromy homomorphism χ :π1(X)→ Sp2h(Z) of E, it is equal to −4〈c1(Γ ), [X]〉. The aim of this paper is to give an algebro-topological proof of this formula, i.e., one that does not use the Atiyah–Singer index theorem.  2...

Abstract: This study presents a novel direct-construction based fuzzy multiclass support vector classifier based on previous multi-class classification method by Crammer and Singer (2001). In our proposed method, the membership degree is computed by fuzzy c-means clustering, the optimal problem and its constraints of multiclass classification are reconstructed and its corresponding Lagrangian f...

Last time: Real 4-manifolds M with almost (many) complex structures but with no integrable almost complex structure, no complex structure. In understanding this situation , we observed the importance of the Riemann-Roch –Hirzebruch formula, and of the fact that it actually holds in the non-Kähler case(from the Atiyah-Singer Theorem). Namely for a compact complex manifold of any dimension and wi...

We study the action of Singer groups of projective geometries (and their subgroups) on (d − 1)-flats for arbitrary d. The possibilities which can occur are determined, and a formula for the number of orbits of each possible size is given. Motivated by an old problem of J.R. Isbell on the existence of certain permutation groups we pose the problem of determining, for given q and h, the maximum c...

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result [KR2, Th. 4.4] of the first paper to prove a residue formula ”à la Bott” for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of BismutGoette on the equivariant (Ray-Sing...

Using semi-classical analysis in ℝ n we present a quite general model for which the topological index formula of Atiyah–Singer predicts spectral flow with transition finite number eigenvalues between clusters (energy bands). This corresponds to physical phenomena that are well observed quantum energy levels small molecules [FZ00, FZ01], also geophysics oceanic or atmospheric equatorial waves [D...