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Image encryption is an effective way to protect image data. However, existing image encryption algorithms are still unable to strike a good balance between security and efficiency. To overcome the shortcomings of these algorithms, an image encryption algorithm based on plane-level image filtering and discrete logarithmic transformation (IEA-IF-DLT) is proposed. By utilizing the hash value more rationally, our proposed IEA-IF-DLT avoids the overhead caused by repeated generations of chaotic sequences and further improves the encryption efficiency through plane-level and three-dimensional (3D) encryption operations. Aiming at the problem that common modular addition and XOR operations are subject to differential attacks, IEA-IF-DLT additionally includes discrete logarithmic transformation to boost security. In IEA-IF-DLT, the plain image is first transformed into a 3D image, and then three rounds of plane-level permutation, plane-level pixel filtering, and 3D chaotic image superposition are performed. Next, after a discrete logarithmic transformation, a random pixel swapping is conducted to obtain the cipher image. To demonstrate the superiority of IEA-IF-DLT, we compared it with some state-of-the-art algorithms. The test and analysis results show that IEA-IF-DLT not only has better security performance, but also exhibits significant efficiency advantages.

A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum (qT) logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small-x logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consistently regulate them. We introduce a new pure rapidity regulator and a corresponding MS¯\\[ \\overline{\\mathrm{MS}} \\]-like scheme, which handles rapidity divergences while maintaining the homogeneity of the power expansion. We find that power-law rapidity divergences appear at subleading power, which give rise to derivatives of parton distribution functions. As a concrete example, we consider the qT spectrum for color-singlet production, for which we compute the complete qT2/Q2 suppressed power corrections at Oαs\\[ \\mathcal{O}\\left({\\alpha}_s\\right) \\], including both logarithmic and nonlogarithmic terms. Our results also represent an important first step towards carrying out a resummation of subleading-power rapidity logarithms.

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in SL(3,R)/SL(3,Z). We also provide a complementary convergence statement, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock’s from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.

Darmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of$p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results.