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The existing watermarking algorithms invariant wavelet domain are weak at resisting geometric attacks and have small embedding capacities. In this paper, a robust double-encrypted watermarking algorithm based on the fractional Fourier transform (FRFT) and discrete cosine transform (DCT) in invariant wavelet domain is proposed. A hybrid domain is obtained by applying the redistributed invariant wavelet transform (RIDWT) and cosine transform to the enlarged host image. The low-frequency and high-frequency regions of the domain have advantages against different attacks; thus, the two components are both selected for watermark embedding. To solve the false positive problem, the enlarged watermark is double-encrypted by the Arnold transform and the FRFT before a singular value decomposition is applied to it, which enhances the security of the algorithm due to the generation of more keys. Multiparameter particle swarm optimization (MP-PSO) is used to obtain the optimal embedding factors for achieving the balance between invisibility and robustness. The simulation results and comparative experiments show that the proposed algorithm exhibits high robustness under the premise of satisfying security, reliability and invisibility, especially for geometric attacks such as rotation, cropping and translation.

Objective To assess different bridging anticoagulation therapies early after mechanical heart valve replacement (MHVR) in Chinese patients. Methods We performed a prospective, single-center, observational cohort study of 305 patients who underwent elective MHVR with different bridging anticoagulation regimens. Patients enrolled in the study were divided into three bridging therapy groups: the unfractionated heparin (UFH) group (n = 109), the low-molecular-weight heparin (LMWH) group (n = 97), and the UFH with sequential LMWH (UFH-LMWH) group (n = 99). All patients were followed for 4 weeks. Results Two patients experienced thromboembolic stroke events in the UFH group. The LMWH group was associated with an increase in the incidence of bleeding events compared with the UFH group (10.3% VS 2.8%; P = 0.03). With a comparison of LMWH and UFH group in secondary endpoints, the statistical test for significance indicated a trend of reduced ICU length of stay (P = 0.08), postoperative length of stay (P = 0.08) and time of achieving target INR (P = 0.06). The creatinine level (odds ratio = 1.03; 95% confidence interval = 1.01 to 1.05; P = 0.02) and hypertension (odds ratio = 3.72; 95% confidence interval = 1.35 to 10.28; P = 0.01) were risk factors for bleeding events. Conclusion For Chinese patients, the LMWH bridging anticoagulation presents the increased the incidence of bleeding events, but enables patients to benefit from achieving an early anticoagulation effect. Close follow-up and personalized management are required in patients with thromboembolic and bleeding risk factors. Trial registration Chinese Clinical Trial Registry ChiCTR1800019841. Registered 2 December 2018 retrospectively.

The offset linear canonical transform (OLCT) plays an important role in many fields of optics and signal processing. In this paper, we address the problem of signal filtering and reconstruction in the OLCT domain based on new convolution theorems. Firstly, we propose new convolution and product theorems for the OLCT, which state that a modified ordinary convolution in the time domain is equivalent to simple multiplication operations for the OLCT and the Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and is easy to implement in designing filters. The classical convolution theorem in the FT domain is shown to be a special case of our derived results. Then, a practical multichannel sampling expansion for band-limited signal with the OLCT is introduced. This sampling expansion constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. By designing OLCT filters, we can obtain derivative sampling and second-order derivative interpolation. Furthermore, potential applications of the multichannel sampling are discussed. Last, based on the new convolution structure, we investigate and discuss several applications, including swept-frequency filter analysis, image denosing and image encryption. Some illustrations and simulations are presented to verify the validity and effectiveness of the proposed method.

The affine Fourier transform (AFT) plays an important role in many fields of optics and signal processing. The Gabor transform (GT) is a kind of linear time–frequency representation (TFR). Compared with many bilinear TFRs, the GT does not have the cross-term problem. In this study, the authors propose a time–frequency analysis method based on the AFT and GT. First, they obtain an affine relation between the AFT and the modified GT (MGT). Since the MGT is closely related to the AFT, they can use it as an assistant tool for signal processing in the AFT domain. Moreover, many useful relations between the AFT and the MGT are derived, such as recovery relation, projection relation and power integration relation. Then, they demonstrate that the AFT also has the affine relation with other TFRs, such as the Gabor–Wigner transform and the general class of quadratic distribution. Last, using the new time–frequency analysis method associated with the AFT and MGT, they present the filter design for multiple component chirp signal separation. Moreover, the simulation results illustrate the effectiveness of the proposed method.

The objective of generalized sampling expansion (GSE) is the reconstruction of an unknown, continuously defined function $f\left(t\right)$ from samples of the responses from $M$ linear time-invariant (LTI) systems that are each sampled using the $1/M$ th Nyquist rate. In this paper, we investigate the GSE for lowpass and bandpass signals with multiple sampling rates in the fractional Fourier transform (FRFT) domain. First, we propose an improvement of Papoulis’ GSE, which has multiple sampling rates in the FRFT domain. Based on the proposed GSE, we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates. In addition, the Papoulis GSE and the previous GSE associated with FRFT are shown to be special instances of our results. Second, we address the problem of the GSE of fractional bandpass signals. A new GSE for fractional bandpass signals with equal sampling rates is derived. We show that the restriction of an even number of channels in the GSE for fractional bandpass signals is unnecessary, and perfect signal reconstruction is possible for any arbitrary number of channels. Further, we develop the GSE for a fractional bandpass signal with multiple sampling rates. Lastly, we discuss the application of the proposed method in the context of single-image super-resolution reconstruction based on GSE. Illustrations and simulations are presented to verify the validity and effectiveness of the proposed results.

In this paper, two new Lagrange neural networks are proposed for compressed sensing. The first one is for the standard recovery of sparse signals. The second one is for the recovery of sparse signals with noise. By drawing into delay, we also derive two delayed Lagrange neural networks. Both the two networks are based on the l0 norm approximation. They can be implemented by circuits.

Quadratic representations are very useful in the study of Euclidean Jordan algebras and complementarity problems. In this paper, we provide some characterizations of the complementarity properties for the quadratic representation P a . For example, P a has the E0-property; P a is monotone iff $${\pm a \in {\mathcal K}}$$ . In addition, the algebra and cone automorphism invariance of some E-properties are studied. By use of the quadratic representations, the Jordan quad E-property is proved to keep cone automorphism invariant in simple Jordan algebras. The pseudomonotone property is shown to be cone automorphism invariant.

In this paper, neural network approach is addressed for signal reconstruction under the frame of compressed sensing. By introducing implicit variables, we convert the basis pursuit denoising model into a quadratic programming problem. Based on a class of generalized Fischer–Burmeister complementarity functions, we establish a neural network method for the signal reconstruction of compressed sensing. A projection neural network is also presented to recover the original signals. These two neural networks can be implemented using integrated circuits and two block diagrams of the neural networks are presented. Based on our proposed method, some potential applications of the compressed sensing are discussed.

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. In this paper, first, we present much briefer and more direct derivations of sampling relation for time-limited signal in LCT domain. Moreover, based on the sampling relation expansion in the LCT domain, the equivalent of the classical Fourier series (FS) for the LCT is derived directly. Then, the discrete-time linear canonical transform (DTLCT) is proposed through the dual extension of linear canonical series (LCS), which is the generalization of discrete-time Fourier transform (DTFT).

The Gibbs phenomenon describes the behavior of the partial sums of a Fourier series (FS) in a neighborhood of a simple discontinuity of the function expanded. In this paper, we investigate the Gibbs phenomenon for linear canonical series (LCS) in linear canonical transform (LCT) domain. We show that a similar phenomenon is observed for the LCS of a function with jump discontinuities. First, the convergence and Gibbs phenomenon of the LCS are discussed for the real parts of the LCS. It is showed that if a function is continuous, then there is no Gibbs phenomenon. Moreover, we proved that the uniform convergence of the LCS for a non-periodic analysis function in the smooth region. Then, we present a theorem that the Gibbs constant that arise for the LCS appears to be the same to that occurring in standard FS.

BACKGROUND Cardiac masses are rare, but lead to high risk of stroke and death. Because of the different treatment methods, it is significant for clinicians to differentiate the nature of masses. Cardiac magnetic resonance (CMR) imaging has high intrinsic soft-tissue contrast and high spatial and temporal resolution and can provide evidence for differential diagnosis of cardiac masses. However, there is no evidence-based conclusion as to its accuracy. Therefore, the purpose of our study is to perform a systematic review on this issue and provide useful information for clinical diagnosis and treatment. METHODS We will perform a systematic search in EMBASE, Cochrane Library, PubMed and Web of Science for diagnostic studies using CMR to detect cardiac masses from inception to October, 2019. Two authors will independently screen titles and abstracts for relevance, review full texts for inclusion and conduct detail data extraction. The methodological quality will be assessed using the QUADAS-2 tool. If pooling is possible, we will use bivariate model for diagnostic meta-analysis to estimate summary sensitivity, specificity, positive likelihood ratio, negative likelihood ratio, and diagnostic odds ratio of CMR, as well as different sequences of CMR. Estimates of sensitivity and specificity from each study will be plotted in summary receive operating curve space and forest plots will be constructed for visual examination of variation in test accuracy. If enough studies are available, we will conduct sensitivity analysis and subgroup analysis. RESULTS The results of this systematic review and meta-analysis will be published in a peer-reviewed journal. CONCLUSION To our knowledge, this will be the first systematic review on the accuracy of CMR in the differential diagnosis of cardiac masses. This study will provide evidence and data to form a comprehensive understanding of the clinical value of CMR for cardiac masses patients. ETHICS AND DISSEMINATION Ethics approval and patient consent are not required, as this study is a systematic review. PROSPERO REGISTRATION NUMBER CRD42019137800.

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. In this paper, a new kind of wave packet transform (WPT) associated with the LCT is proposed, this new WPT (WPTL) is defined based on the ideal of the LCT and the WPT. Some properties and physical meaning of the WPTL are investigated. In particular, we show a version of the resolution of the identity of WPTL. Moreover, the relationship between the WPTL and the Wigner distribution (WD) is derived. At last, we introduce the concept of the fractional wavepacketgram, which is defined as the modulus square of the WPTL. It is proved that the fractional wavepacketgram is a member of the Cohen class time–frequency distribution where the kernel is a scale dependent ambiguity function.

In this paper, a concept of regulation functions is proposed, and some related properties and examples are explored. Based on this regulation function and some smoothing complementarity functions, we present a family of smoothing Newton methods to solve the symmetric cone complementarity problem. This algorithm allows a unified convergence analysis for some smoothing Newton methods. We show that the resulting Newton equation is well-defined and solvable, and provides a theory of global convergence.

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of real-time measurement of the spectrum of a signal in the FRFT domain. In this paper, we propose two approaches for real-time measurement of the FRFT of a signal based on modulation and bandpass filtering systems. The relation is established between the linear frequency modulation (LFM or chirp) spectrum and the FRFT of its envelope. In addition, two applications for spectrum measurement are presented in the FRFT domain. The LFM signal can be bandlimited in the Fourier transform (FT) domain through spectrum measurement associated with bandpass filtering method. The results can also be useful in the problems related to swept-frequency filter for measurement in the FRFT domain.

The wavelet transform (WT) and linear canonical transform (LCT) have been shown to be powerful tool for optics and signal processing. In this paper, firstly, we introduce a novel time-frequency transformation tool coined the generalized wavelet transform (GWT), based on the idea of the LCT and WT. Then, we derive some fundamental results of this transform, including its basis properties, inner product theorem and convolution theorem, inverse formula and admissibility condition. Further, we also discuss the time-fractional-frequency resolution of the GWT. The GWT is capable of representing signals in the time-fractional-frequency plane. Last, some potential applications of the GWT are also presented to show the advantage of the theory. The GWT can circumvent the limitations of the WT and the LCT.

The Gabor expansion (GE), which maps the time domain signal into the joint time and frequency domain, has been recognized as very useful for signal processing. However, sinusoidal analysis used in the traditional GE is not appropriate for a compact representation for chirp-type signals. In this paper, a generalized Gabor expansion (GGE) is proposed in order to rectify the limitations of the GE, the proposed expansion not only inherits the advantage of GE, but also has the capability of signal representations in the linear canonical transform (LCT) domain which is similar to the LCT. Basis functions of the proposed expansion are obtained via LCT basis. Compared with the traditional GE, the GGE can offer signal representations on a general, non-rectangular time–frequency plane tiling. Besides, the completeness and biorthogonality conditions of the GGE are derived.

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. The classical multichannel sampling theorem for common bandlimited signals has been extended differently to bandlimited signals associated with LCT. However, a practical issue associated with the reconstruction of the original bandpass signal from multichannel samples in LCT domain still remains unresolved. The purpose of this paper is to introduce a practical multichannel sampling theorem for bandpass signals in LCT domain. The sampling expansion which is constructed by the ordinary convolution in the time domain can reduce the effect of spectral leakage and is easy to implement. The classical multichannel sampling theorem and the well-known sampling theorems for the LCT are shown to be special cases of it. Some potential applications of the multichannel sampling are also presented to show the advantage of the theory.

In this paper, we introduce some concepts of E-properties for nonlinear transformations defined on Euclidean Jordan algebras, such as E-property, quad E-property, uniform Jordan E-property, E 0 -property. And then we study the implications among these E-properties. In particular, the quad E-property is equivalent to the E-property. The K -copositive property implies the E 0 -property. The uniform Jordan E-property is proved to have the E 0 and R 0 -properties. Based on the E-properties, we get some sufficient conditions to guarantee the solution existence for the nonlinear symmetric cone complementarity problem.

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates new sampling relations in the LCT domain. Firstly, the relationship between linear canonical series (LCS) and LCT is introduced. The LCS expansion coefficients are the sampled values of LCT. Then, based on the conventional Fourier series and LCS, two new sampling relations in the LCT domain are presented, where the signal in the time domain is reconstructed from the samples of its LCT directly. The first theorem considers signals band-limited in some LCT domain, and the second deals with signals band-limited in the conventional Fourier transform domain.

In this letter, we first propose new nearly tridiagonal commuting matrices of discrete Fourier transform (DFT) matrix and generalized DFT (GDFT) matrix. Then, using the block diagonalizations technique of circular-centrosymmetric and centrosymmetric matrices, we derive novel tridiagonal commuting matrices for each discrete cosine transform (DCT) and discrete sine transform (DST) matrices of the types I, IV, V, and VIII from the commuting matrices of the DFT and GDFT based on the relationships in matrix forms among DFT, GDFT, and various types of DCT and DST. Moreover, the novel tridiagonal commuting matrices of various types of DCT and DST do not have multiple eigenvalues. Last, with these novel commuting matrices, we can easily determine an orthonormal set of Hermite-like eigenvectors for each of their corresponding DCT or DST matrix.

Linear canonical transform (LCT) form a three-parameter family of intergral transforms with wide application in optics. In this paper, we investigate the linear canonical Wigner distribution (LCWD) which is based on the LCT and the classical Wigner distribution (WD). Firstly, the definition of LCWD is discussed. Moreover, the transformation law for the LCWD through a first-order optical system is derived. This new phase-space distribution provides analysis of signals in both space and LCT domains simultaneously. Then, the main properties of LCWD are investigated in detail. Finally, the application of the LCWD is presented. The LCWD is found to be the appropriate phase-space distribution function for light-beam characterization in first-order optical system. Moreover, the moment matrix formalism for beam characterization is studied.

Fractional Fourier transform (FRFT) plays an important role in many field of optics and signal processing. The recently developed concept of two side fractional quaternion Fourier transform (FRQFT) based on quaternion algebra and FRFT has been found useful for signal and image processing. It is a generalization of the quaternion Fourier transform (QFT). Many properties of the FRQFT are already known, but an extension of the QFT's Plancherel theorem is still missing. The purpose of this paper is to introduce extensions of this theorem. Firstly, we establish scalar value Plancherel theorem for the FRQFT. Then, the concept of right side fractional quaternion Fourier transform (FRQFTr) is proposed. Furthermore, we establish quaternion value Plancherel theorem for the FRQFTr. The Plancherel theorems in QFT domain are shown to be special cases of the achieved results.

Recently, Gowda and Sznajder [Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)] have introduced and studied automorphism invariance of some P-properties for linear transformations. This paper deals with this automorphism invariance of some other complementarity properties, such as \(\hbox {E}_0,\,\hbox {P}_0\), S, Z-properties. Particularly, we answer Gowda and Sznajder in positive that order P-property is algebra automorphism invariant in simple Jordan algebras. By replacing transposition with the invertibility in the concept of automorphism invariance, we propose a notion of similarity automorphism invariance. Most complementarity properties of linear transformations are also shown to be similarity invariant under algebra automorphisms and cone automorphisms.

In underwater acoustic communication systems, the channel equalization community has recently given much attention to decision feedback equalization (DFE). It is because that the DFE offers intersymbol interference (ISI) cancellation with reduced noise enhancement. However, its key algorithm such as constant modulus algorithm (CMA) has moderate convergence rate and steady-state mean square error (MSE), which is not sufficient for the receive system of communication. So a new cost function is defined and then a novel DFE based on such cost function is proposed. The efficiency of the proposed DFE is proved by computer simulations.

In the digital transmission system, an efficient fractionally-spaced equalizer (FSE) for blind equalization based on modified constant modulus algorithm (MCMA) and phase locked loop (PLL) is proposed. Comparing with conventional FSE based on famous CMA, the proposed algorithm has not only lower steady-state mean square error and faster convergence rate but also the ability to recover carrier phase rotation. The efficiency of the method is proved by computer simulation.

This paper addresses the problem of multidimensional signal reconstruction from generalized samples in fractional Fourier domain including the deterministic case and the stochastic case. The generalized sampling expansion is investigated for the case where the fractional bandlimited input depends on N real variable, i.e., f ( t ) = f ( t 1 , ⋯ , t N ) and is used as a common input to a parallel bank of m independent N dimensional linear fractional Fourier filters Hα,k(u), k = 1 , ⋯ , m . For the deterministic input, the input is assumed to have its N dimensional fractional Fourier transform bandlimited to the frequency rang |ui|≤Ωi, for i = 1 , ⋯ , N . If m, the number of fractional Fourier filters, is written as a product of positive integers in the form m = m 1 m 2 ⋯ m N , and if the fractional bandlimited input f(t) is processed by fractional Fourier filter Hα,k(u)resulting m outputs gk(t), then f(t) can be reconstructed in terms of the samples gk(nT), each output being sampled at the identical rates of Ω 1 csc α / m 1 π , Ω 2 csc α / m 2 π , ⋯ , Ω N csc α / m N π samples/second in t 1 , ⋯ , t N respectively. This contrasts with the rates of Ω 1 csc α / π , Ω 2 csc α / π , ⋯ , Ω N csc α / π in t 1 , ⋯ , t N needed for reconstruction of the unfiltered input f(t). Input sampling expansions in terms of samples of the output filters are given for both deterministic and stochastic inputs, the generalized sampling expansion for random input having the same form as for the deterministic case but interpreted in the mean-square sense. Our formulation and results are general and include derivative sampling and periodic nonuniform sampling in the fractional Fourier domain for multidimensional signals as special case. Finally, the potensional application of the multidimensional generalized sampling is presented to show the advantage of the theory. Especially, the application of multidimensional generalized sampling in the context of the image scaling about image super-resolution is also discussed. The simulation results of image scaling are also presented.

The well-known Lyapunov’s theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a ∈ V, we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for La and show that La has the R0-property if and only if a is invertible. Finally, we provide La with some characterizations of the E0-property and the nondegeneracy property.

Objective Peripheral arterial disease (PAD) is a common clinical manifestation of the systemic atherosclerotic process, and the ankle-brachial index (ABI) is an ideal tool to diagnose PAD. Currently, there have been few long-term follow-up studies focused on the associations of the ABI with all-cause mortality and cardiovascular disease (CVD) mortality in Chinese MetS patients. The aim of this study was to evaluate the usefulness of ABI to predict the prognosis of CVD in hospitalized Chinese patients with metabolic syndrome (MetS). Methods Participants from multi-center departments were followed up from November 2004 to January 2011. The study sample actually comprised 1,266 valid participants whose age was ≥35 years. Patients were separated into four groups, with an ABI ≤0.4, 0.41-0.7, 0.71-0.9 and 0.91-1.4. An ABI ≤0.9 was defined as PAD, and subjects with an ABI >1.4 were excluded because of the false negative rate. Factors related to all-cause and cardiovascular mortality were observed by Cox models and the log rank test. Potential confounding variables with values of p

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. In this paper, we address the problem of signal reconstruction from multichannel samples in the LCT domain based on a new convolution theorem. Firstly, a new convolution structure is proposed for the LCT, which states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for LCT and Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and easy to implement in the designing of filters. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, a practical multichannel sampling expansion for band limited signal with the LCT is introduced. This sampling expansion which is constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. Last, the potential application of the multichannel sampling is presented to show the advantage of the theory. Especially, the application of multichannel sampling in the context of the image superresolution is also discussed. The simulation results of superresolution are also presented.

By applying hybrid functions of general block-pulse functions and Legendre polynomials, linear Volterra integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of linear Volterra integrodifferential systems are derived. Using the results we obtain the optimal control and state as well as the optimal value of the objective functional. The numerical examples illustrate that the algorithms are valid.

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. This paper addresses the problem of signal reconstruction from multichannel and periodic nonuniform samples in the LCT domain. Firstly, the multichannel sampling theorem (MST) for band-limited signals with the LCT is proposed based on multichannel system equations, which is the generalization of the well-known sampling theorem for the LCT. We consider the problem of reconstructing the signal from its samples which are acquired using a multichannel sampling scheme. For this purpose, we propose two alternatives. The first scheme is based on the conventional Fourier series and inverse LCT operation. The second is based on the conventional Fourier series and inverse Fourier transform (FT) operation. Moreover, the classical Papoulis MST in FT domain is shown to be special case of the achieved results. Since the periodic nonuniformly sampled signal in the LCT has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived MST and the specific space-shifting property of the LCT. Last, the potential applications of the MST are presented to show the advantage of the theory.

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates the sampling of bandlimited signals in LCT domain. First, we propose the linear canonical series (LCS) based on the LCT, which is a generalized pattern of Fourier series. Moreover, the LCS inherits all the nice properties from the LCT. Especially, the Parseval’s relation is presented for the LCS, which is used to derive the sampling theorem of LCT. Then, utilizing the generalized form of Parseval’s relation for the complex LCS, we obtain the sampling expansion for bandlimited signals in LCT domain. The advantage of this reconstruction method is that the sampling expansion can be deduced directly not based on the Shannon theorem.

As a generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, but the correlation theorem, similar to the version of the Fourier transform (FT), is still to be determined. In this paper, firstly, we introduce a new convolution structure for the LCT, which is expressed by a one dimensional integral and easy to implement in filter design. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, based on the new convolution structure, the correlation theorem is derived, which is also a one dimensional integral expression. Last, as an application, utilizing the new convolution theorem, we investigate the sampling theorem for the band limited signal in the LCT domain. In particular, the formulas of uniform sampling and low pass reconstruction are obtained.

In this paper, we first investigate a two-parametric class of smoothing functions which contains the penalized smoothing Fischer–Burmeister function and the penalized smoothing CHKS function as special cases. Then we present a smoothing Newton method for the nonlinear complementarity problem based on the class of smoothing functions. Issues such as line search rule, boundedness of the level set, global and quadratic convergence are studied. In particular, we give a line search rule containing the common used Armijo-type line search rule as a special case. Also without requiring strict complementarity assumption at the P 0 -NCP solution or the nonemptyness and boundedness of the solution set, the proposed algorithm is proved to be globally convergent. Preliminary numerical results show the efficiency of the algorithm and provide efficient domains of the two parameters for the complementarity problems.

There recently has been much interest in smoothing Newton method for solving nonlinear complementarity problems. We extend such method to symmetric cone complementarity problems (SCCP). In this paper, we first investigate a one-parametric class of smoothing functions in the context of symmetric cones, which contains the Fischer–Burmeister smoothing function and the CHKS smoothing function as special cases. Then we propose a smoothing Newton method for the SCCP based on the one-parametric class of smoothing functions. For the proposed method, besides the classical step length, we provide a new step length and the global convergence is obtained. Finally, preliminary numerical results are reported, which show the effectiveness of the two step lengthes in the algorithm and provide efficient domains of the parameter for the complementarity problems.

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Frechet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.

Let Tn+1(R) be the algebra of all upper triangular n+1 by n+1 matrices over a 2-torsionfree commutative ring R with identity. In this paper, we give a complete description of the Jordan automorphisms of Tn+1(R), proving that every Jordan automorphism of Tn+1(R) can be written in a unique way as a product of a graph automorphism, an inner automorphism and a diagonal automorphism for n ≥ 1.

The aim of the generalized sampling expansion (GSE) is the reconstruction of an unknown continuously defined function f(t), from the samples of the responses of M linear time invariant (LTI) systems, each sampled by the 1/M th Nyquist rate. In this letter, we investigate the GSE in the fractional Fourier transform (FRFT) domain. Firstly, the GSE for fractional bandlimited signals with FRFT is proposed based on new linear fractional systems, which is the generalization of classical generalized Papoulis sampling expansion. Then, by designing fractional Fourier filters, we obtain reconstruction method for sampling from the signal and its derivative based on the derived GSE and the property of FRFT. Last, the potential application of the GSE is presented to show the advantage of the theory.

A popular approach to solving the complementarity problem is to reformulate it as an equivalent equation system via a complementarity function. In this paper, we propose a new class of functions, which contains the penalized natural residual function and the penalized Fischer–Burmeister function for symmetric cone complementarity problems. We show that this class of functions is indeed a class of complementarity functions. We finally prove that the merit function of this new class of complementarity functions is coercive.

The linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, however, the convolution theorems don't have the elegance and simplicity comparable to that of the Fourier transform (FT), which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. The purpose of this letter is to introduce a new convolution structure for the LCT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters. Some of well-known results about the convolution theorem in FT domain, fractional Fourier transform (FRFT) domain are shown to be special cases of our achieved results.

In this paper, integrodifferential systems are converted into a system of algebraic equations by hybrid of general block-pulse functions and the second Chebyshev polynomials. The approximate solutions of integrodifferential systems are derived. The numerical examples illustrate that the algorithm is valid.