Set Of Singular Values Research Articles

Spectral norm and energy of a digraph with respect to a VDB topological index

The set of singular values of a digraph with respect to a vertex-degree based topological index is the set of all singular values of its general adjacency matrix. The spectral norm is the largest singular value and the energy the sum of the singular values. In this paper we characterize the digraphs which have exactly one singular value different from zero and the digraphs for which all singular values are equal. As a consequence, we deduce sharp upper and lower bounds for the spectral norm and energy of digraphs. In addition to being a natural generalization, proving the results in the general setting of digraphs allows us to deduce new results on graph energy.

Experimental Analysis on Performance of Speech Utterance recognition using AI Models

In Automated speech recognition of the system performance is crucial and important to satisfy multiple requirements of HMI and, more recently, even in IoT-related applications as well. Concurrently, there has been an increase in demand for detecting strong critical features derived from speech utterances. This paper presents a performance of the developed machine learning algorithms with respect to audio digit speech recognition and classification. The prepared dataset contains a free range of words (from 1 to 10) from speakers of different age groups. The Audacity software used for preprocessing the audio files that includes removal of noise included in the signal and trimming the silence on either side of the word utterance. audio signal sampled at fs = 48kHz.We have developed four AI Models to recognise the word utterances. Audio signals are processed separately and derived two unique feature sets that includes statistical features set and singular values by performing SVD related to word utterances. The cepstral values for each utterance are obtained from state-of-the-art MFCC. Variance-covariance matrix is calculated from the generated MFCC matrix. The diagonal values which form the variance are recorded and denoted as feature set-1 for the word utterance and inputted to the machine learning algorithms. Performance matrices of the developed models are recorded. To keep the computational bottleneck associated with the use of feature sets to minimum, dimensionality reduction is carried out by applying singular value decomposition to the extracted MFCC matrix. The derived set of singular values ​​considered as feature set-2 is used to train and test the developed AI models with a ratio of 70:30. We presented and discussed the performance and results produced by MLP, KNN, SVM, Random Forest algorithms. In comparison, MLP and Random Forest were found to show excellent performance on both feature sets with 100% training accuracy and 99% test accuracy.

The Eremenko–Lyubich constant

Eremenko and Lyubich proved that an entire function whose set of singular values is bounded is expanding at points where its image has large modulus. These expansion properties have been at the centre of the subsequent study of this class of functions, now called the Eremenko-Lyubich class. We improve the estimate of Eremenko and Lyubich, and show that the new estimate is asymptotically optimal. As a corollary, we obtain an elementary proof that functions in the Eremenko-Lyubich class have lower order at least $1/2$.

Lateral Motion Control Invariance Helicopter on the Roll Angle. Analytical Synthesis

For the linearized fourth-order model of the isolated lateral motion of a single-rotor helicopter as a MIMO system containing two inputs, the control is analytically synthesized, which ensures the invariance of the roll angle motion in the presence of disturbances in the control channels, as well as the required placement of the poles of the closed-loop system, given any specific values from the area of their stability. The approach to the synthesis of invariant control consists in finding a matrix of feedback coefficients of a linear system that satisfies the invariance conditions, which are a system of power matrix equations of a certain design. The synthesis is based on the application of theor ems based on the use of the regularization condition of the matrix equation and the invariance conditions under disturbances in the control channels, as well as theorems that make it possible to place the poles of the MIMO system using the original decomposition of the control object. Regularization of a matrix equation is understood as a solution to the problem of providing a given set of singular values for an inverted symmetric square matrix. The invariance of the MIMO system is considered with respect to unmeasured disturbances inthe control channels. The use of such an approach to the synthesis of invariant control made it possible to obtain an analytical solution that is versatile and can be applied in various flight modes of single-rotor helicopters with different dynamic properties. The results of the numerical synthesis of the lateral motion of a singlerotor helicopter using the obtained laws of invariant control, which confirm the reliability of the analytical expressions, areshown.

Performance Analysis and Validation of Modified Singular Spectrum Analysis based on Simulation Torrential Rainfall Data

A popular method for time series analysis to extract the components of noise and trend from the time series data is called the singular spectrum analysis (SSA). However, the drawback of SSA is its problem in determining the appropriate window length, L for certain data set in confirming patent separation of the components of trend and noise. Another issue that crops up when using SSA is that, over time, the sum of day-to-day rainfall becomes nearly comparable. In this case, disjoints sets of singular values and distinctive series components could essentially be intermixed, resulting in poor separability between trend and noise components. The introduction of modified SSA is to mitigate the problems efficiently. The performance of modified SSA is measured by using w-correlation and RMSE based on simulated data. These results show that the parameter L = T/5 was suitable to use in short time series rainfall data. It can be proved by the plot of the extracted trend for modified SSA that appears to conform to the original data configuration for time series rainfall however there is the omission of components of noise predominantly for L = T/5 in detecting the uncharacteristically heavy downpour which could potentially initiate the occurrence of torrential rainfall. In addition, the result shows that average RMSE for reconstructed time series components of modified SSA is much smaller than SSA for each L

Entire functions with prescribed singular values

We introduce a new class of entire functions [Formula: see text] which consists of all [Formula: see text] for which there exists a sequence [Formula: see text] and a sequence [Formula: see text] satisfying [Formula: see text] for all [Formula: see text]. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set [Formula: see text] containing the origin and at least one more point is the set of singular values of some locally univalent function in [Formula: see text], hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set [Formula: see text] is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class [Formula: see text] contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

Singular values and non-repelling cycles for entire transcendental maps

Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$ has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.

On Connected Preimages of Simply-Connected Domains Under Entire Functions

Let f be a transcendental entire function, and let {U,V subset mathbb{C}} be disjoint simply-connected domains. Must one of {f^{-1}(U)} and {f^{-1}(V)} be disconnected? In 1970, Baker implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains. (A domain {Usubset mathbb{C}} is completely invariant under f if {f^{-1}(U)=U}.) It was recently observed by Julien Duval that Baker's argument, which has also been used in later generalisations and extensions of Baker's result, contains a flaw. We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, for the function {f(z)= e^z+z}, there is a collection of infinitely many pairwise disjoint simply-connected domains, each with connected preimage. We also answer a long-standing question of Eremenko by giving an example of a transcendental meromorphic function, with infinitely many poles, which has the same property. Furthermore, we show that there exists a function f with the above properties such that additionally the set of singular values S(f) is bounded; in other words, f belongs to the Eremenko–Lyubich class. On the other hand, if S(f) is finite (or if certain additional hypotheses are imposed), many of the original results do hold. For the convenience of the research community, we also include a description of the error in Baker's proof, and a summary of other papers that are affected.

Fatou components and singularities of meromorphic functions

AbstractWe prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $. We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

Image splicing forgery detection based on low-dimensional singular value decomposition of discrete cosine transform coefficients

Digital image forgery has significantly increased due to the rapid development of several tools of image manipulation. Based on the manipulation used to produce a tampered image, image forgery techniques can be characterized into three types: copy–move forgery, image splicing, and image retouching. Image splicing is achieved by adding regions from one image into another. This technique changes the content of the target image and causes variations in image features which are used to detect the forgery regions. In this study, an image splicing forgery detection method based on low-dimensional singular value decomposition of discrete cosine transform (DCT) coefficients has been presented. The suspicious input image is divided into multi-size blocks, and each block is transformed into 2D DCT. The DCT coefficients are calculated correspondingly to each block. The features from DCT are extracted using SVD algorithm. The roughness measure is calculated for the set of singular values obtained. Lastly, four types of statistical features—mean, variance, third-order moment skewness, and fourth-order moment kurtosis—are extracted from SVD features and are then arranged in a feature vector. Feature reduction has been applied by kernel principal component analysis. Finally, support vector machine is used to distinguish between the authenticated and spliced images. The proposed method was evaluated against three standard image datasets CASIA v1, DVMM v1, and DVMM v2. The proposed method shows an average detection accuracy of 97.15, 99.30, and 96.97 for DVMM v1, CASIA v1, and DVMM v2, respectively. These results outperform several current image splicing detection methods.

Selecting appropriate singular values of transmission matrix to improve precision of incident wavefront retrieval

A method of selecting appropriate singular values of the transmission matrix to improve the precision of incident wavefront retrieval in focusing light through scattering media is proposed. The optimal singular values selected by this method can reduce the degree of ill-conditionedness of the transmission matrix effectively, which indicates that the incident wavefront retrieved from the optimal set of singular values is more accurate than the incident wavefront retrieved from other sets of singular values. The validity of this method is verified by numerical simulation and actual measurements of the incident wavefront of coherent light through ground glass.

Conformal measures for meromorphic maps

In this paper we study the relation between the existence of a conformal measure on the Julia set $J(f)$ of a transcendental meromorphic map $f$ and the existence of zero of the topological pressure function $t \mapsto P(f, t)$ for the map $f$. In particular, we show that if $f$ is hyperbolic and there exists a $t$-conformal measure which is not totally supported on the set of escaping points, then $P(f, t) = 0$. On the other hand, for a wide class of maps $f$, including arbitrary maps with at most finitely many poles and finite set of singular values and hyperbolic maps with at most finitely many poles and bounded set of singular values, if $P(f, t) = 0$, we construct a $t$-conformal measure on $J(f)$. This partially answers a question of R.D. Mauldin.

Reconstruction and basis function construction of electromagnetic interference source signals based on Toeplitz‐based singular value decomposition

In this study, the authors propose a novel method, namely Toeplitz-based singular value decomposition (or TL-SVD for short) for the reconstruction and basis function construction of electromagnetic interference (EMI) source signals. Given a specific EMI source signal, they first construct a Toeplitz type data matrix. By applying singular value decomposition (SVD) to the constructed matrix, they obtain a set of singular values, which are further divided into two parts, corresponding to the clear and noisy components of input signals, respectively. The de-noised signal can then be reconstructed by reserving relatively larger singular values and abandoning smaller ones. Finally, by utilising the compositions of certain vectors resulting from the previous SVD step, the basis function can be constructed. To evaluate the performance of the proposed method, they conduct extensive experiments on both the synthetic data and real EMI signals, by comparing with several state-of-the-art signal reconstruction methods, such as discrete wavelet transform, EEMD, sparse representation based on K-SVD and OMP. Experimental results demonstrate that the proposed method can outperform comparison approaches.

On Innermost Circles of the Sets of Singular Values for Generic Deformations of Isolated Singularities

We will show that for each k≠1, there exists an isolated singularity of a real analytic map from \(\mathbb {R}^{4}\) to \(\mathbb {R}^{2}\) which admits a real analytic deformation such that the set of singular values of the deformed map has a simple, innermost component with k outward cusps and no inward cusps. Conversely, such a singularity does not exist if k=1.

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

We study the escaping set of functions in the class $\mathcal B^*$, that is, holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of $\mathbb C^*$. For functions in the class $\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the class $\mathcal B^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary $e\in\{0,\infty\}^\mathbb N$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to $\{0,\infty\}$ according to $e$ under iteration by $f$.

Hyperbolic entire functions and the Eremenko\u2013Lyubich class: Class mathcal {B} or not class mathcal {B}?

Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the Eremenko–Lyubich classmathcal {B} of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class mathcal {B}, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain.

Performance measure of switched control systems based on covariance tensor approach

In this paper, controller performance measure is considered for switched systems. The covariance tensor-based method is proposed for the controller performance measure for this systems evolution with the discrete-valued switching dynamics and local models for continuous dynamics. We define a measurement tensor to construct the measured process outputs data, and employ the data processing technique of higher-order singular value decomposition. Applying the higher-order singular value decomposition, we can obtain the sets of singular values from the measurement tensor, which are used jointly to evaluate the controller performance of the overall switched systems significantly. We develop the covariance tensor-based performance assessment method for the multivariate switched control systems with characteristic information being mined from the measurement tensor and derive the calculation approach base on the sets of singular values from the measurement tensor. It is shown that by applying higher-order singular value decomposition for measured outputs tensor data, the performance assessment results can exactly reflect the controller performance under the overall dynamical process. Finally, two cases study of the numerical simulation examples and a typical industrial process system well demonstrate the effectiveness of the proposed method.

Time Hierarchies and Model Reduction in Canonical Non-linear Models.

The time-scale hierarchies of a very general class of models in differential equations is analyzed. Classical methods for model reduction and time-scale analysis have been adapted to this formalism and a complementary method is proposed. A unified theoretical treatment shows how the structure of the system can be much better understood by inspection of two sets of singular values: one related to the stoichiometric structure of the system and another to its kinetics. The methods are exemplified first through a toy model, then a large synthetic network and finally with numeric simulations of three classical benchmark models of real biological systems.

A note on repelling periodic points for meromorphic functions with a bounded set of singular values

Let f be a meromorphic function with a bounded set of singular values and for which infinity is a logarithmic singularity. Then we show that f has infinitely many repelling periodic points for any minimal period n ≥ 1 , using a much simpler argument than the corresponding results for arbitrary entire transcendental functions.

Capacity of electromagnetic communication modes in a noise-limited optical system

We present capacity bounds of an optical system that communicates using electromagnetic waves between a transmitter and a receiver. The bounds are investigated in conjunction with a rigorous theory of degrees of freedom (DOF) in the presence of noise. By taking into account the different signal-to-noise ratio (SNR) levels, an optimal number of DOF that provides the maximum capacity is defined. We find that for moderate noise levels, the DOF estimate of the number of active modes is approximately equal to the optimum number of channels obtained by a more rigorous water-filling procedure. On the other hand, for very low- or high-SNR regions, the maximum capacity can be obtained using less or more channels compared to the number of communicating modes given by the DOF theory. In general, the capacity is shown to increase with increasing size of the transmitting and receiving volumes, whereas it decreases with an increase in the separation between volumes. Under the practical channel constraints of noise and finite available power, the capacity upper bound can be estimated by the well-known iterative water-filling solution to determine the optimal power allocation into the subchannels corresponding to the set of singular values when channel state information is known at the transmitter.