Nonzero Elements Research Articles

Near-lossless Compression of Tc-99 m DMSA Scan Images Using Discrete Cosine Transformation.

The objective of this study was to optimize the threshold for discrete cosine transform (DCT) coefficients for near-lossless compression of Tc-99 m Dimercaptosuccinic acid (DMSA) scan images using discrete cosine transformation. Two nuclear medicine (NM) Physicians after reviewing several Tc-99 m DMSA scan images provided 242 Tc-99 m DMSA scan images that had scar. These Digital imaging and communication in medicine (DICOM) images were converted in the Portable Network Graphics (PNG) format. DCT was applied on these PNG images, which resulted in DCT coefficients corresponding to each pixel of the image. Four different thresholds equal to 5, 10, 15, and 20 were applied and then inverse discrete cosine transformation was applied to get the compressed Tc-99 m DMSA scan images. Compression factor was calculated as the ratio of the number of nonzero elements after thresholding DCT coefficients to the number of nonzero elements before thresholding DCT coefficients. Two NM physicians who had provided the input images visually compared the compressed images with its input image, and categorized the compressed images as either acceptable or unacceptable. The quality of compressed images was also assessed objectively using the following eight image quality metrics: perception-based image quality evaluator, structural similarity index measure (SSIM), multiSSIM, feature similarity indexing method, blur, global contrast factor, contrast per pixel, and brightness. Pairwise Wilcoxon signed-rank sum tests were applied to find the statistically significant difference between the value of image quality metrics of the compressed images obtained at different thresholds and the value of the image quality metrics of its input images at the level of significance = 0.05. At threshold 5, (1) all compressed images (242 out of 242 Tc-99 m DMSA scan images) were acceptable to both the NM Physicians, (2) Compressed image looks identical to its original image and no loss of clinical details was noticed in compressed images, (3) Up to 96.65% compression (average compression: 82.92%) was observed, and (4) Result of objective assessment supported the visual assessment. The quality of compressed images at thresholds 10, 15, and 20 was significantly better than that of input images at P < 0.0001. However, the number of unacceptable compressed images at thresholds 10, 15, and 20 was 6, 38, and 70, respectively. Up to 96.65%, near-losses compression of Tc-99 m DMSA images was found using DCT by thresholding DCT coefficients at a threshold value equal to 5.

Compressive frequency-difference direction-of-arrival estimation

Direction-of-arrival estimation is difficult for signals spatially undersampled by more than half the wavelength. Frequency-difference beamforming [Abadi, Song, and Dowling (2012). J. Acoust. Soc. Am. 132, 3018–3029] offers an alternative approach to avoid such spatial aliasing by using multifrequency signals and processing them at a lower frequency, the difference-frequency. As with the conventional beamforming method, lowering the processing frequency sacrifices spatial resolution due to a beam broadening. Thus, unconventional beamforming is detrimental to the ability to distinguish between closely spaced targets. To overcome spatial resolution deterioration, we propose a simple yet effective method by formulating the frequency-difference beamforming as a sparse signal reconstruction problem. Similar to compressive beamforming, the improvement (compressive frequency-difference beamforming) promotes sparse nonzero elements to obtain a sharp estimate of the spatial direction-of-arrival spectrum. Analysis of the resolution limit demonstrates that the proposed method outperforms the conventional frequency-difference beamforming in terms of separation if the signal-to-noise ratio exceeds 4 dB. Ocean data from the FAF06 experiment support the validity.

Sparse representation learning using [formula omitted] compressed sensing and rank-revealing QR factorization

Compressive sensing can be conceptualized for a classification problem in the context of representation learning, which is frequently applied for signal reconstruction via a few measurements. A novel compressive sensing-based approach is presented to substantially reduce the number of required samples for classification. This technique considers the pixels of image as sensors and it seeks to identify the most optimum sensors as decision variables in feature space. In this study, Spatial sensor locations are acquired through learning to identify the regions within an image where the most discriminative information for classification is embedded. ℓ1−2 minimization, as nonconvex but Lipschitz continuous, is solved to obtain the least non-zero elements of the full measurement vector to completely reconstruct the discriminative vector in feature space. Optimal sensors are localized from the training-set and subsequent test images are categorized based on learned sensors. The algorithm consists of three primary components: sparse minimization, feature space, and discrimination vector. ℓ1−2 minimization, rank-revealing QR factorization (RRQR) and SVM are considered for enhanced sparsity exploitation, feature space extraction and discrimination vector acquiring, respectively. The proposed method is evaluated on four different experiments and is compared against a state-of-art technique. The results demonstrate the superiority of proposed method to the compared method.

A proof of the polynomial conjecture for nilpotent Lie groups monomial representations

Let G = exp ⁡ ( g ) G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra g \mathfrak g , H = exp ⁡ ( h ) H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G G with Lie algebra h \mathfrak h , χ \chi a unitary character of H H and τ = ind H G χ \tau = \text {ind}_H^G \chi the monomial representation of G G induced by χ \chi . Let D τ ( G / H ) D_{\tau }(G/H) be the algebra of the G G -invariant differential operators on the fiber bundle over the base space G / H G/H associated to the data ( H , χ ) (H,\chi ) . We prove the polynomial conjecture due to Corwin-Greenleaf stating that if τ \tau is of finite multiplicities, the algebra D τ ( G / H ) D_{\tau }(G/H) is isomorphic to the algebra C [ Γ τ ] H {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H H -invariant polynomial functions on the affine subspace Γ τ = { ℓ ∈ g ∗ ; ℓ | h = − − 1 d χ } {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of g ∗ {\mathfrak g}^* . In this case, we show that any non-zero element of D τ ( G / H ) D_{\tau }(G/H) admits a fundamental tempered solution.

The existence spectrum for regular sparse anti-magic squares

Magic squares and anti-magic squares are the important research objects in combinatorics, and they are also well used in constructing graph labelings. Suppose that n and d are two positive integers with d<n. An n×n array A based on X={0,1,2,⋯,nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n,d), if every non-zero element of X occurs exactly once in A and the set of the row-sums, column-sums and two main diagonal-sums of A consists of 2n+2 consecutive integers. An SAMS(n,d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal, denoted by RSAMS(n,d). In this paper, the existence of an RSAMS(n,d) is investigated and it is proved that a necessary condition for the existence of an RSAMS(n,d) is also sufficient, i.e., there exists an RSAMS(n,d) if and only if 2≤d<n.

Alternating submodules for partition algebras, rook algebras, and rook-Brauer algebras

Letting n≥2k, the partition algebra CAk≥2(n) has two one-dimensional subrepresentations that correspond in a natural way to the alternating and trivial characters of the symmetric group Sk. In 2019, Benkart and Halverson introduced and proved evaluations in the two distinguished bases of CAk(n) for nonzero elements in the one-dimensional regular CAk(n)-submodule that corresponds to the Young symmetrizer ∑σ∈Skσ; in 2016, Xiao proved an explicit formula for the analogue of the sign representation for the rook monoid algebra. In this article, we lift Xiao's formula to a diagram basis evaluation in the partition algebra CAk(n). We prove that our diagram basis evaluation for this lifting, which we denote as Altk∈CAk(n), generates a one-dimensional module under the action of multiplication by arbitrary elements in CAk(n). Our explicit formula for Altk gives us a cancellation-free formula for the other one-dimensional regular CAk(n)-module, with regard to Benkart and Halverson's lifting of ∑σ∈Skσ. We then use a sign-reversing involution to evaluate our one-dimensional generators in the orbit basis, and we use our explicit formula for Altk to lift Young's N- and P-functions so as to allow set-partition tableaux as arguments, and we use this lifting to construct Young-type matrix units for CA2(n) and CA3(n).

On the Non-Zero Divisor Graphs of Some Finite Commutative Rings

The study of rings and graphs has been explored extensively by researchers. To gain a more effective understanding on the concepts of the rings and graphs, more researches on graphs of different types of rings are required. This manuscript provides a different study on the concepts of commutative rings and undirected graphs. The non-zero divisor graph, Γ(R) of a ring R is a simple undirected graph in which its set of vertices consists of all non-zero elements of R and two different vertices are joint by an edge if their product is not equal to zero. In this paper, the commutative rings are the ring of integers modulo n where n=8k and k≤3. The zero divisors are found first using the definition and then the non-zero divisor graphs are constructed. The manuscript explores some properties of non-zero divisor graph such as the chromatic number and the clique number. The result has shown that Γ(Z8k) is perfect.

Enhancing genomic mutation data storage optimization based on the compression of asymmetry of sparsity.

Background: With the rapid development of high-throughput sequencing technology and the explosive growth of genomic data, storing, transmitting and processing massive amounts of data has become a new challenge. How to achieve fast lossless compression and decompression according to the characteristics of the data to speed up data transmission and processing requires research on relevant compression algorithms. Methods: In this paper, a compression algorithm for sparse asymmetric gene mutations (CA_SAGM) based on the characteristics of sparse genomic mutation data was proposed. The data was first sorted on a row-first basis so that neighboring non-zero elements were as close as possible to each other. The data were then renumbered using the reverse Cuthill-Mckee sorting technique. Finally the data were compressed into sparse row format (CSR) and stored. We had analyzed and compared the results of the CA_SAGM, coordinate format (COO) and compressed sparse column format (CSC) algorithms for sparse asymmetric genomic data. Nine types of single-nucleotide variation (SNV) data and six types of copy number variation (CNV) data from the TCGA database were used as the subjects of this study. Compression and decompression time, compression and decompression rate, compression memory and compression ratio were used as evaluation metrics. The correlation between each metric and the basic characteristics of the original data was further investigated. Results: The experimental results showed that the COO method had the shortest compression time, the fastest compression rate and the largest compression ratio, and had the best compression performance. CSC compression performance was the worst, and CA_SAGM compression performance was between the two. When decompressing the data, CA_SAGM performed the best, with the shortest decompression time and the fastest decompression rate. COO decompression performance was the worst. With increasing sparsity, the COO, CSC and CA_SAGM algorithms all exhibited longer compression and decompression times, lower compression and decompression rates, larger compression memory and lower compression ratios. When the sparsity was large, the compression memory and compression ratio of the three algorithms showed no difference characteristics, but the rest of the indexes were still different. Conclusion: CA_SAGM was an efficient compression algorithm that combines compression and decompression performance for sparse genomic mutation data.

An intervention study of VR sports games on the perceptual and motor abilities of children with incomplete intellectual development

Abstract Childhood is a critical stage for the development of perceptual and motor abilities, and strengthening the training of children with incomplete mental development at this stage will affect the development of motor skills during their growth. In this paper, we take VR technology as a starting point to build a perceptual model and introduce a convolutional sparse representation algorithm. First, a sparse representation with few non-zero elements is found to optimize a function consisting of a data fidelity term and a sparse induced penalty function. Then, the sum of the convolution of the filter and the convolution sparse feature map, i.e., the convolution operation, is computed to generate the translation invariants. Then the convolutional sparse coding method is introduced to the traditional unsupervised problem by calculating the minimization objective function and solving it in an iterative manner alternatively. Finally, the constituents of the signal are analyzed and the discrete equivalence of the convolution is derived based on the Fourier transform to derive the intervening variables. The experimental results showed that the mean value of the post-test of motor ability compared with the pre-test increased by 4.6 through an eight-week VR sports game training intervention study test on different children with incomplete mental development. Therefore, it is of great theoretical and practical significance to understand the characteristics of perceptual and motor abilities of children with incomplete intellectual development and to develop corresponding programs for VR sports game training according to their characteristics.

Automated discovery of angle theorems

We consider geometry theorems whose premises and statement comprise a set of bisector conditions. Each premise and the statement can be represented as the rows of a “bisector matrix”: one with three non zero elements per row, one element with value -2 and the others with value 1. The existence of a theorem corresponds to rank deficiency in this matrix. Our method of theorem discovery starts with identification of rank deficient bisector matrices. Some such matrices can be represented as graphs whose vertices correspond to matrix rows and whose edges correspond to matrix columns. We show that if a bisector matrix which can be represented as a graph is rank deficient, then the graph is bicubic. We give an algorithm for finding the rank deficient matrices for a Hamiltonian bicubic graph, and report on the results for graphs with 6,8,10 and 12 vertices. We discuss a method of deriving rank deficient bisector matrices with more than 2 non-zero elements. We extend the work to matrices containing rows corresponding to angle trisectors.

Magic partially filled arrays on abelian groups

AbstractIn this paper we introduce a special class of partially filled arrays. A magic partially filled array on a subset of an abelian group is a partially filled array of size with entries in such that (i) every appears once in the array; (ii) each row contains filled cells and each column contains filled cells; (iii) there exist (not necessarily distinct) elements such that the sum of the elements in each row is and the sum of the elements in each column is . In particular, if , we have a zero‐sum magic partially filled array . Examples of these objects are magic rectangles, ‐magic rectangles, signed magic arrays, (integer or noninteger) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, that is, of an where . We also construct zero‐sum magic partially filled arrays when is the abelian group or the set of its nonzero elements.

Indeks Harmonik dan Indeks Gutman Graf Nilradikal pada Gelanggang Komutatif dengan Satuan

Graph theory is a topic that is still an important subject to discuss. This is because until now graph theory has many practical applications in various disciplines, for example in biology, computer science, economics, informatics engineering, linguistics, mathematics, health, and social sciences. This study discusses the Gutman index of nilradical graphs in the commutative ring with unity.A nilradical graph whose vertices are non-zero nilpotent elements, when the domain is a commutative ring with units, it forms a complete graph only if the commutative ring with units we use is limited to a positive integer modulo n (Z_n). Where n is the square of the prime number p which is less than equal to 3. It is known that the general pattern of harmonic indices and Gutman indices of nilradical graphs in the commutative ring with unity are H(N(Z_(n=p^2 ) ))=((p-2)^2+(p -2))/2(p-2) and Gut(N(Z_(n=p^2 ) ))=(1/2 ((p-2)^2+(p-2))) (p- 2)^2 respectively. In its application, this general form can be used as a numerical parameter of a graph in chemical graph theory, molecular topology, and mathematical chemistry.

On Ranks of Matrices over Noncommutative Domains

We consider matrices with entries in some domain, i.e., in a ring, not necessarily commutative, not containing non-trivial zero divisors. The concepts of the row rank and the column rank are discussed. (Coefficients of linear dependencies belong to the domain ; left coefficients are used for rows, right coefficients for columns.) Assuming that the domain satisfies the Ore conditions, i.e., the existence of non-zero left and right common multiples for arbitrary non-zero elements, it is proven that these row and column ranks are equal, which allows us to speak about the rank of a matrix without specifying which rank (row or column) is meant. In fact, the existence of non-zero left and right common multiples for arbitrary non-zero elements of is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and (-)difference operators, both ordinary and with partial derivatives and differences.

THE MATHEMATICAL ANALYSIS AND IMPROVEMENT OF MATRIX ALGORITHM FOR FEEDER FAULT LOCATION BASED ON FTU IN THE DISTRIBUTION NETWORK

The mathematical analysis and improvement of matrix algorithm based on FTU has been proposed to solve fault location problem in complex multi-source and multi fault distribution networks of power system. Based on the structural characteristics of the distribution system, a description matrix D is established by assuming the power sources and a positive direction and it derived from the correlation between the positive direction and different power lines. As a function of the fault current direction transmitted by the terminal power supply unit which is our feeder terminal unit (FTU), the fault information vector F is set up. By combining the description matrix D and searching for non-zero elements in the fault information vector, the fault location vector is discovered according to the fault location criteria. During the observation of the elements in the fault location vector, by analyzing we immediately found the location of the defect area. This algorithm can solve problems that cannot be completely solved by other algorithms, such as power terminal failures, ring network failures, and multi-source failures. This method does not require matrix multiplication or normalization. The widespread use of distributed generation (DG) has made the distribution network more complex, and it leads to the failure of the traditional matrix algorithm. Therefore, the matrix algorithm is further improved to adapt the complex characteristics of DG. Considering the accuracy, we avoid random search operations by filtering fault candidate scenarios based on fault confidence and Each algorithm run 100 times in a loop, and the average time taken for a single run is used as a measure of the computational efficiency of the algorithm. The matrix algorithm utilizes the information uploaded by FTU to SCADA to create network description matrix, fault information matrix, which are then used to obtain the fault judgment matrix for fault location and isolation and it proves that the algorithm judgment is effective.

Some New Class of Four-Dimensional Absolute Valued Algebras

An absolute valued algebra is a nonzero real algebra that is equipped with a multiplicative norm ||ab|| = ||a|| ||b||. We prove that if A is a four-dimensional absolute valued algebras containing a nonzero central element a which satisfies one of the following identities: (1) (a^2, a, a) = 0 or (a, a, a^2) = 0 (2) (a^2, a^2, a) = 0 or (a, a^2, a^2) = 0 (3) (a^2, a^2, a^2) = 0, with (a / a^2) ≠ 0 (4) (a^2, a, a^2) = 0. Then A contains a commutative sub-algebra of dimension two. Moreover, A is isomorphic to A_1, A_2, A_3, A_4, B_1 or B_2 in the first two cases and isomorphic to A_1, A_2, A_3, A_4, B_1, B_2, B_3 or B_4, in the last two cases.

Reversible image-hiding algorithm based on singular value sampling and compressive sensing

A reversible image-hiding algorithm based on a novel chaotic system is proposed using compressive sensing (CS) and singular value sampling (SVS) techniques. In the first stage, a novel mathematical model is constructed to extract the plain messages from the secret plain image, and the public-key Rivest–Shamir–Adleman (RSA) algorithm is adopted to encrypt these messages, obtaining the corresponding cipher messages. Then, another mathematical model of key transformation is constructed to transform above messages into the initial keys, which is used to produce a random key stream. In the second stage, the secret plain image is scrambled by a pre-encryption operation, and the corresponding singular values are obtained by singular value decomposition (SVD). Then, these values are partitioned, and zero blocks are identified and removed. Thereafter, the singular values of the blocks with non-zero elements are sampled by CS, with filled by zero elements again. In the third stage, high energy coefficients are removed and replaced by zero elements to obtain new sampling values. Then, the carrier image is processed by discrete wavelet transform (DWT). Next, new sampling values are embedded into the wavelet coefficients, and the inverse DWT is performed. Thus, a new carrier image containing the secrets is obtained. The advantages are: (1) A new chaotic system ImpTDCS is proposed to have a better behavior. (2) A novel model EMM is built to extract plain messages from secret image. (3) Multi-images can be embedded, which can hide more secret information each time. (4) SVD is operated followed by SVS on non-zero blocks, reducing transmission bandwidth. (5) High energy coefficients of SVS are removed before embedding operation, guaranteeing effectively the visual quality of carrier image containing secrets.

Positive lower density for prime divisors of generic linear recurrences

AbstractLet $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$ . Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.

On Absolute Valued Algebras Containing a Central Algebraic Element

Let be an absolute valued algebra containing a nonzero central algebraic element. Then is a pre-Hilbert algebra and is finite dimensional in the following cases: 1) A satisfies (x, x, x)=0. 2) A satisfies (x2, x2 , x2 )=0. 3) A satisfies (x, x2, x)=0. In these cases is isomorphic to or . It may be conjectured that every absolute valued algebra containing a nonzero central element is pre-Hilbert algebra.

On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.

Zonal graphs of small cycle rank

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. There is a connection between zonal labelings of connected bridgeless cubic plane graphs and the Four Color Theorem. Zonal labelings of cycles play a role in this connection. The cycle rank of a connected graph of order n and size m is m − n + 1 . Thus, cycles have cycle rank 1. All zonal connected graphs of cycle rank at most 2 are determined.