We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the translation invariance of associated systems of Diophantine equations to derive powerful congruence constraints on the underlying variables. There are application...

We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimension higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variationa...

In this paper, an algorithm is proposed for the classification of malignancy in mammogram. The types of malignancy can be classified as circumscribed, speculated, architectural distortion, miscellaneous, asymmetry and calcification. Chebyshev moments (CM), being discrete, are capable to combat the discritisation errors as compared to Zernike Moments (ZM), which are continuous moments. It is als...

In macaque inferotemporal cortex (IT), neurons have been found to respond selectively to complex shapes while showing broad tuning ("invariance") with respect to stimulus transformations such as translation and scale changes and a limited tuning to rotation in depth. Training monkeys with novel, paperclip-like objects, Logothetis et al. 9 could investigate whether these invariance properties ar...

The positional specificity of short-term visual memory for a variety of 3-D shapes was investigated in a series of 'same'/'different' discrimination experiments, with computer-rendered stimuli displayed either at the same or at different locations in the visual field. For animal-like shapes, we found complete translation invariance, regardless of the interstimulus similarity, and irrespective o...

An examination of the translation invariance of V0 under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.

Image copy detection is an important technology of copyright protection. This paper proposes efficient hashing method for image using 2D-2D (two-directional two-dimensional) PCA (Principal Component Analysis). The key the discovery translation invariance PCA. With property invariance, a novel model extracting rotation-invariant low-dimensional features designed by combining PCT (Polar Coordinat...

Nontrivial translation matrices occur for spin (A,B) ⊕ (C,D) with | A − C | = | B−D | = 1/2, necessarily associating a (C,D) field with a spin (A,B) field. Including translation matrices in covariant non-unitary Poincaré representations also introduces new gauge terms in the construction of massless particle fields from canonical unitary fields. In the usual procedure without spacetime translat...

A new version of the cluster expansion is proposed without breaking translation and rotation invariance. As an application this technique, we construct connected Schwinger functions regularized $$\phi ^4$$ theory in a continuous way.

This note defines the premium of a put option on the firm as a measure of insolvency risk. The put premium is not a coherent risk measure as defined by Artzner et al. (1999). It satisfies all the axioms for a coherent risk measure except one, the translation invariance axiom. However, it satisfies a weakened version of the translation invariance axiom that we label translation monotonicity. The...