ON THE MINIMAL COSET COVERINGS OF THE SET OF SINGULAR AND OF THE SET OF NONSINGULAR MATRICES

2018 ◽  
Vol 52 (1 (245)) ◽  
pp. 8-11
Author(s):  
A.V. Minasyan

It is determined minimum number of cosets over linear subspaces in $ \mathbb{F}_q $ necessary to cover following two sets of $ A (n \mathclose{\times} n) $ matrices. For one of the set of matrices $ \det{A} = 0 $ and for the other set $ \det{A} \neq 0 $. It is proved that for singular matrices this number is equal to $ 1 \mathclose{+} q \mathclose{+} q^2 \mathclose{+} \ldots \mathclose{+} q^{n-1} $ and for the nonsingular matrices it is equal to $ (q^n \mathclose{-} 1)(q^n \mathclose{-} q)(q^n \mathclose{-} q^2) \cdots (q^n \mathclose{-} q^{n-1}) / q^{\large{\binom{n}{2}}} $.

Author(s):  
Hibiki Ono ◽  
Yoshifumi Manabe

Abstract This paper proposes new card-based cryptographic protocols to calculate logic functions with the minimum number of cards using private operations under the semi-honest model. Though various card-based cryptographic protocols were shown, the minimum number of cards used in the protocol has not been achieved yet for many problems. Operations executed by a player where the other players cannot see are called private operations. Private operations have been introduced in some protocols to solve a particular problem or to input private values. However, the effectiveness of introducing private operations to the calculation of general logic functions has not been considered. This paper introduces three new private operations: private random bisection cuts, private reverse cuts, and private reveals. With these three new operations, we show that all of AND, XOR, and copy protocols are achieved with the minimum number of cards by simple three-round protocols. This paper then shows a protocol to calculate any logical functions using these private operations. Next, we consider protocols with malicious players.


1983 ◽  
Vol 26 (3) ◽  
pp. 297-306 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.


Author(s):  
James Wiegold ◽  
H. Lausch

AbstractThe growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.


2000 ◽  
Vol 78 (2) ◽  
pp. 99-112
Author(s):  
W E Couch ◽  
M Surovy ◽  
R J Torrence

Motions of finite Toda lattices are known to be associated with linear wave equations whose general solutions can be expressed in terms of progressing waves, and this association is known to generalize to finite non-Abelian Toda lattices of n x n matrices and systems of n coupled linear wave equations. We present a nontrivial family of non-Abelian Toda lattice motions that can be specialized to ones that are not finite, but not infinitely extendible either, as they contain nonvanishing but singular matrices of rank (n – s). In these cases we give a natural continuation of the lattice dynamics by means of nonsingular matrices of dimension (n – s) x (n – s), and describe how to find s progressing wave solutions of the associated system of n coupled linear wave equations.PACS No.: 5.45-a


10.37236/9564 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Dhruv Rohatgi ◽  
John C. Urschel ◽  
Jake Wellens

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.


2018 ◽  
Vol XIX (1) ◽  
pp. 58-64
Author(s):  
Vasiliu Paul

A system is a set of elements that can be found in one of the following states: operating state and fault. Any system has two stable states: functioning and defect, which is why, in the theory of reliability, it is called a bivalent system. A subset of defective elements is called the system cut if all the other elements of the system are in operation and the system is defective. The width of a bivalent system is equal to the minimum number of elements the system cuts have. In this paper is presented an algorithm for automatic determination of the dual system width to a bivalent system, a Matlab script that implements the algorithm, a case study and subsequent directions of development.


2021 ◽  
Vol 15 (1) ◽  
pp. 235-248
Author(s):  
Mayank R. Kapadia ◽  
Chirag N. Paunwala

Introduction: Content Based Image Retrieval (CBIR) system is an innovative technology to retrieve images from various media types. One of the CBIR applications is Content Based Medical Image Retrieval (CBMIR). The image retrieval system retrieves the most similar images from the historical cases, and such systems can only support the physician's decision to diagnose a disease. To extract the useful features from the query image for linking similar types of images is the major challenge in the CBIR domain. The Convolution Neural Network (CNN) can overcome the drawbacks of traditional algorithms, dependent on the low-level feature extraction technique. Objective: The objective of the study is to develop a CNN model with a minimum number of convolution layers and to get the maximum possible accuracy for the CBMIR system. The minimum number of convolution layers reduces the number of mathematical operations and the time for the model's training. It also reduces the number of training parameters, like weights and bias. Thus, it reduces the memory requirement for the model storage. This work mainly focused on developing an optimized CNN model for the CBMIR system. Such systems can only support the physicians' decision to diagnose a disease from the images and retrieve the relevant cases to help the doctor decide the precise treatment. Methods: The deep learning-based model is proposed in this paper. The experiment is done with several convolution layers and various optimizers to get the maximum accuracy with a minimum number of convolution layers. Thus, the ten-layer CNN model is developed from scratch and used to derive the training and testing images' features and classify the test image. Once the image class is identified, the most relevant images are determined based on the Euclidean distance between the query features and database features of the identified class. Based on this distance, the most relevant images are displayed from the respective class of images. The general dataset CIFAR10, which has 60,000 images of 10 different classes, and the medical dataset IRMA, which has 2508 images of 9 various classes, have been used to analyze the proposed method. The proposed model is also applied for the medical x-ray image dataset of chest disease and compared with the other pre-trained models. Results: The accuracy and the average precision rate are the measurement parameters utilized to compare the proposed model with different machine learning techniques. The accuracy of the proposed model for the CIFAR10 dataset is 93.9%, which is better than the state-of-the-art methods. After the success for the general dataset, the model is also tested for the medical dataset. For the x-ray images of the IRMA dataset, it is 86.53%, which is better than the different pre-trained model results. The model is also tested for the other x-ray dataset, which is utilized to identify chest-related disease. The average precision rate for such a dataset is 97.25%. Also, the proposed model fulfills the major challenge of the semantic gap. The semantic gap of the proposed model for the chest disease dataset is 2.75%, and for the IRMA dataset, it is 13.47%. Also, only ten convolution layers are utilized in the proposed model, which is very small in number compared to the other pre-trained models. Conclusion: The proposed technique shows remarkable improvement in performance metrics over CNN-based state-of-the-art methods. It also offers a significant improvement in performance metrics over different pre-trained models for the two different medical x-ray image datasets.


2011 ◽  
Vol 30 (2) ◽  
pp. 101 ◽  
Author(s):  
Guillaume Noyel ◽  
Jesús Angulo ◽  
Dominique Jeulin

Computing an array of all pairs of geodesic distances between the pixels of an image is time consuming. In the sequel, we introduce new methods exploiting the redundancy of geodesic propagations and compare them to an existing one. We show that our method in which the source point of geodesic propagations is chosen according to its minimum number of distances to the other points, improves the previous method up to 32 % and the naive method up to 50 % in terms of reduction of the number of operations.


2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.


1972 ◽  
Vol 78 (1) ◽  
pp. 37-38 ◽  
Author(s):  
M. Khosh-Khui ◽  
M. Niknejad

SUMMARYTwo chickpea pure lines, one being short and wide in stature and the other tall and narrow, were crossed. Parents, F1 and F2 generations were grown in the field to study the inheritance of plant dimensions, as well as their correlation with seed yield. Heritability, in broad sense, for plant height and width was 36 and 20% respectively. Estimated minimum number of genes for plant height was two and for plant width was three pairs of genes. Plant height showed correlation values of 0·43 and -0·67 with yield of seed and 1000 seed weight respectively. There were no significant correlations between plant width and the same two yield components.


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