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This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn. [ABSTRACT FROM AUTHOR]

We study optimal strategies in two-player stochastic games that are played on a finite graph, equipped with a general payoff function. The existence of optimal strategies that do not make use of memory and randomisation is a desirable property that vastly simplifies the algorithmic analysis of such games. Our main theorem gives a sufficient condition for the maximizer to possess such a simple optimal strategy. The condition is imposed on the payoff function, saying the payoff does not depend on any finite prefix (shift-invariant) and combining two trajectories does not give higher payoff than the payoff of the parts (submixing). The core technical property that enables the proof of the main theorem is that of the existence of ϵ -subgame-perfect strategies when the payoff function is shift-invariant. Furthermore, the same techniques can be used to prove a finite-memory transfer-type theorem: namely that for shift-invariant and submixing payoff functions, the existence of optimal finite-memory strategies in one-player games for the minimizer implies the existence of the same in two-player games. We show that numerous classical payoff functions are submixing and shift-invariant. [ABSTRACT FROM AUTHOR]

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Making machines learn like humans is the ultimate goal of artificial intelligence. Few-shot learning attempts to simulate the learning mechanism of humans, which is a task that can learn novel concepts from very few labeled samples. Due to the lack of sufficient labeled training data, existing methods often increase the risk of over-fitting or cause a considerable gap between clean and augmented data. To solve these problems, we present a novel compare network to perform robust few-shot learning in a meta-learned end-to-end manner. Specifically, we argue that it is desirable to learn a robust encoder that can draw inferences about other cases from one example. To this end, we improve the accuracy of few-shot learning by mining the internal mechanism of deep networks, which can leverage label information more effectively. By introducing shift-invariant blocks and a self-attention block in our architecture, all these components are seamlessly integrated into our framework, which can give feedback to each other without data augmentation. Furthermore, we provide ablative analyses of different blocks to help understand how each term contributes to performance. Extensive experiments demonstrate that our method can provide more robust results and outperform state-of-the-art few-shot learning methods.

Canonical polyadic decomposition (CPD) of multi-subject complex-valued fMRI data can be used to provide spatially and temporally shared components among groups with both magnitude and phase information. However, the CPD model is not well formulated due to the large subject variability in the spatial and temporal modalities, as well as the high noise level in complex-valued fMRI data. Considering that the shift-invariant CPD can model temporal variability across subjects, we propose to further impose a phase sparsity constraint on the shared spatial maps to denoise the complex-valued components and to model the inter-subject spatial variability as well. More precisely, subject-specific time delays are first estimated for the complex-valued shared time courses in the framework of real-valued shift-invariant CPD. Source phase sparsity is then imposed on the complex-valued shared spatial maps. A smoothed l0 norm is specifically used to reduce voxels with large phase values after phase de-ambiguity based on the small phase characteristic of BOLD-related voxels. The results from both the simulated and experimental fMRI data demonstrate improvements of the proposed method over three complex-valued algorithms, namely, tensor-based spatial ICA, shift-invariant CPD and CPD without spatiotemporal constraints. When comparing with a real-valued algorithm combining shift-invariant CPD and ICA, the proposed method detects 178.7% more contiguous task-related activations. [ABSTRACT FROM AUTHOR]

In this letter, we study a few properties of Complex Conjugate Pair Sums (CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI system whose impulse response is one period data of CCPS. For a given input $x(n)$ , we prove that the output of this system is equivalent to computing the first order derivative of $x(n)$. Further, with some constraints on the impulse response, the system output is also equivalent to the second order derivative. With this, we show that a fine edge detection in an image can be achieved using CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of projection for CCS is studied. Here the projection matrix has a circulant structure, which makes the computation of projections easier. Finally, we prove that CCS is shift-invariant and closed under the operation of circular cross-correlation. [ABSTRACT FROM AUTHOR]

In order to improve the accuracy of multi-focus image and infrared visible image fusion， based on the advantages of the good localization and shift invariance of the finite discrete shearlet transform, we propose a new image fusion algorithm，called FDST，which combines contrast with image gradient information correlation factor. Firstly, the FDST decomposes the registration images，and obtain the low frequency sub-band coefficients and high frequency sub-band coefficients of different scales and directions. The fusion principle o£ low frequency sub-band coefficients bases on the method o£ image gradient information correlation factor for weighting. The high frequency sub-band coefficients combine the high frequency and low frequency sub-band coefficients with contrast, and uses contrast as the criterion for choosing metric coefficients which is taken as the fusion rule. Finally，the low frequency information and high frequency information are reconstructed to obtain a fused image by the finite discrete shearlet inverse transform, and both subjective visual evaluation and objective performance assessments of the fusion results are implemented. The results indicate that the algorithm is superior to other fusion algorithms in subjective visual effects and objective evaluation. [ABSTRACT FROM AUTHOR]

We show that every closed shift-invariant subspace of L( K) is generated by the Λ-translates of a countable number of functions, where K is a local field of positive characteristic and Λ is an appropriate translation set. We use this result to provide a characterization of wavelets on such a field. [ABSTRACT FROM AUTHOR]