Row Sums Research Articles

Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers

In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.

On the little secondary Bruhat order

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More on the Bruhat order for $(0,1)$-matrices, Linear Algebra Appl., 421:219--232, 2007) suggested the study of the secondary Bruhat order on ${\cal A}(R,S)$ but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes ${\cal A}(R,S)$. Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of ${\cal A}(R,S)$.

Neighbors degree sum energy of graphs

In this paper, we introduce a new matrix for a graph G in which i th row sum and i th column sum are both equal to neighbors degree sum of i th vertex and define a new variant of graph energy called neighbors degree sum energy $$E_{N} (G)$$ of a graph G. The striking feature of this new matrix is that it is related with some well known degree based topological indices like Zagreb type indices, forgotten indices, etc. When $$E_N(G)$$ values of some molecules containing hetero atoms are correlated with their total $$\pi $$ –electron energy, we got a good correlation with the correlation coefficient $$r=0.982$$ . $$E_N(G)$$ values of some selected 25 polyaromatic hydrocarbons also showed excellent correlation with their $$\pi $$ –electron energy values with the correlation coefficient $$r=0.997$$ . Further, we computed neighbors degree sum energy of some standard classes of graphs and established some bounds and characterizations on largest eigenvalue of N(G) and neighbors degree sum energy of graphs.

Doubly (sub)stochastic operators on ℓp spaces

A square matrix is said to be doubly stochastic if its elements are non-negative and all row sums and column sums are equal one. An important tool in the study of majorization for infinite dimensional spaces ℓp(I) are doubly stochastic operators. These operators are a generalization of doubly stochastic matrices. In this paper, we compare some properties of doubly stochastic operators in finite and infinite dimensions. We will see that if D:ℓp(I)→ℓp(I) is a doubly stochastic operator then ‖D‖≤1. Moreover, the existence of such an operator with ‖D‖<1 is equivalent to 1<p<∞ and I is an infinite set. We discuss some other properties of doubly stochastic operators such as compactness and closedness and also, provide relevant applications of these operators in the existence of solutions for some infinite linear equations and functional equations.

Sign-restricted matrices of 0's, 1's, and −1's

We study sign-restricted matrices (SRMs), a class of rectangular (0,±1)-matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. We determine the maximum number of nonzeros in SRMs and characterize the possible row and column sum vectors. Moreover, a number of results on interchange operations are shown, both for SRMs and, more generally, for (0,±1)-matrices. The Bruhat order on ASMs can be extended to SRMs with the result a distributive lattice. Also, we study polytopes associated with SRMs and some relates decompositions.

A new DNA coding and hyperchaotic system based asymmetric image encryption algorithm.

In this paper, an asymmetric image encryption algorithm based on DNA coding and hyperchaotic system is designed. Unlike other image encryption schemes, for example, sharing of same private keys between sender and receiver, and fixed rules with simple operation, three stages are studied as follows to deal with these problems. Firstly, to eliminate the possible risk of key transmission and management, the initial values of the hyperchaotic system are generated for ahead by the RSA (Rivest-Shamir-Adleman) algorithm and the plain image, in which the sum of odd rows, even rows, odd columns, and even columns are computed respectively to extra the plain message from the plain image as input of RSA algorithm. Then, a mathematical map is established to transform all of them into initial values of the hyperchaotic system. Secondly, the pixel level permutation is performed to confuse the image according to the chaotic sequences generated. Finally, to solve the problem of fixed rules with simple operations in current DNA based image encryption algorithms, dynamical DNA encryption is designed to diffuse the permuted image. The process of DNA encryption includes DNA coding, DNA operation and DNA decoding. In particular, DNA rules are selected according to chaotic sequences dynamically, rather than fixed rules with simple operation. Theoretical analysis and numerical simulations show that the proposed algorithm is secure and reliable for image encryption.

Revisiting the approximated weight extraction methods in fuzzy analytic hierarchy process

There are simple approximated methods to extract the local weights from a pairwise comparison matrix which are row sums, inverse of column sums, arithmetic mean, and geometric mean. In this paper, first, we extend these methods to fuzzy analytic hierarchy process (FAHP) to extract the local weights as fuzzy numbers (FNs). Then, these weights are defuzzified using the center of gravity (COG) method. We also propose an approach to integrate different local weights obtained from different approximated methods to achieve a unified local weight. Moreover, this study proposes a novel and simple approach in which uncommon FNs are indirectly defuzzified based on COG method. This helps extend the multi-attribute decision-making methods to uncommon FNs. To illustrate the applicability of the proposed approaches, three numerical examples are given and their results are compared with some well-known FAHP methods in the literature.

Stability of ecosystems enhanced by species-interaction constraints.

Ecosystem stability is a central question both in theoretical and applied biology. Dynamical systems theory can be used to analyze how growth rates, carrying capacities, and patterns of species interactions affect the stability of an ecosystem. The response to increasing complexity has been extensively studied and the general conclusion is that there is a limit. While there is a complexity limit to stability at which global destabilisation occurs, the collapse rarely happens suddenly if a system is fully viable (no species is extinct). In fact, when complexity is successively increased, we find that the generic response is to go through multiple single-species extinctions before a global collapse. In this paper we demonstrate this finding via both numerical simulations and elaborations of theoretical predictions. We explore more biological interaction patterns, and, perhaps most importantly, we show that constrained interaction structures-a constant row sum in the interaction matrix-prevent extinctions from occurring. This makes an ecosystem more robust in terms of allowed complexity, but it also means singles-species extinctions do not precede or signal collapse-a drastically different behavior compared to the generic and commonly assumed case. We further argue that this constrained interaction structure-limiting the total interactions for each species-is biologically plausible.

A multi-regional generalized RAS updating technique

ABSTRACT We present an extension of the generalized RAS (GRAS) technique to a multi-regional (MR) or multi-national setting. The framework is applicable to updating/regionalizing/balancing any partitioned matrix that needs to conform to new row sums, column sums and additional non-overlapping aggregation constraints. The technique, which we refer to as MR-GRAS, also handles non-exhaustive constraints, in which case the missing values are endogenously generated in the updating process. We derive the closed-form solution of MR-GRAS, propose a simple iterative algorithm for its computation, and discuss the main analytical properties of the method as well as the normalization and interpretation of MR-GRAS multipliers. From a wide range of possible MR-GRAS applications, several updating frameworks of national and interregional supply and use tables are examined.

The [formula omitted]-Schröder paths and [formula omitted]-Schröder numbers

In this paper, we introduce a variant of the m-Schröder paths, i.e. of those lattice paths from (0,0) to (mn,0) with up steps U=(1,1), down steps D(m)=(1,1−m) and horizontal-down steps H(m)=(2,2−m), which never go below the x-axis. The number of m-Schröder paths of length mn is called the nth m-Schröder number, while the number of m-Schröder paths of length mn with no horizontal-down steps ending on the x-axis is called nth small m-Schröder number. We present two types of generalization of Schröder matrices. The entries are interpreted in terms of the number of partial m-Schröder paths. Then we show that these matrices are Riordan arrays, and we give some properties from this result. In particular, we obtain that the nth m-Schröder number is twice of the nth small m-Schröder number for n≥1, and we show that the A-sequence of the m-Schröder matrix is the sequence of the (m−1)-Schröder numbers. Finally, we study other two m-Schröder matrices whose row sums are the m-Schröder numbers and small m-Schröder numbers respectively.

On distance matrices of wheel graphs with an odd number of vertices

Let denote the wheel graph having n-vertices. If i and j are any two vertices of , define Let D be the matrix with entry equal to . The matrix D is called the distance matrix of . Suppose is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an matrix and a rank one matrix such that Here, is positive semidefinite, and all row sums are equal to zero.

Relative Heffter arrays and biembeddings

Relative Heffter arrays, denoted by H t ( m , n ; s , k ) , have been introduced as a generalization of the classical concept of Heffter array. A H t ( m , n ; s , k ) is an m × n partially filled array with elements in ℤ v , where v = 2 n k + t , whose rows contain s filled cells and whose columns contain k filled cells, such that the elements in every row and column sum to zero and, for every x ∈ ℤ v not belonging to the subgroup of order t , either x or − x appears in the array. In this paper we show how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph K (2 n k + t )/ t × t into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for t = k = 3, 5, 7, 9 and n ≡ 3 (mod 4) and for k = 3 with t = n , 2 n , any odd n .

On line sum optimization

We show that the column sum optimization problem, of finding a (0,1)-matrix with prescribed row sums which minimizes the sum of evaluations of given functions at its column sums, can be solved in polynomial time, either when all functions are the same or when all row sums are bounded by any constant. We conjecture that the more general line sum optimization problem, of finding a matrix minimizing the sum of given functions evaluated at its row sums and column sums, can also be solved in polynomial time.

An inverse formula for the distance matrix of a wheel graph with an even number of vertices

Let n≥4 be an even integer and Wn be the wheel graph with n vertices. The distance dij between any two distinct vertices i and j of Wn is the length of the shortest path connecting i and j. Let D be the n×n symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to dij. In this paper, we find a positive semidefinite matrix L˜ such that rank(L˜)=n−1, all row sums of L˜ equal to zero, and a rank one matrix wwT such thatD−1=−12L˜+4n−1wwT. An interlacing property between the eigenvalues of D and L˜ is also proved.

A method for computing the Perron root for primitive matrices

SummaryFollowing the Perron theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix A, there is a positive rank one matrix X such that B = A ∘ X, where ∘ denotes the Hadamard product of matrices, and such that the row (column) sums of matrix B are the same and equal to the Perron root. An iterative algorithm is presented to obtain matrix B without an explicit knowledge of X. The convergence rate of this algorithm is similar to that of the power method but it uses less computational load. A byproduct of the proposed algorithm is a new method for calculating the first eigenvector.

Phase transition in random contingency tables with non-uniform margins

For parameters $n,\delta ,B,$ and $C$, let $X=(X_{k\ell })$ be the random uniform contingency table whose first $\lfloor n^{\delta } \rfloor$ rows and columns have margin $\lfloor BCn \rfloor$ and the last $n$ rows and columns have margin $\lfloor Cn \rfloor$. For every $0<\delta <1$, we establish a sharp phase transition of the limiting distribution of each entry of $X$ at the critical value $B_{c}=1+\sqrt {1+1/C}$. In particular, for $1/2<\delta <1$, we show that the distribution of each entry converges to a geometric distribution in total variation distance whose mean depends sensitively on whether $B<B_{c}$ or $B>B_{c}$. Our main result shows that $\mathbb {E}[X_{11}]$ is uniformly bounded for $B<B_{c}$ but has sharp asymptotic $C(B-B_{c}) n^{1-\delta }$ for $B>B_{c}$. We also establish a strong law of large numbers for the row sums in top right and top left blocks.

English

For a given complex square matrix A with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of fc-regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds is verified by examples.

An Efficient Automatic Intrusion Detection in Cloud Using Optimized Fuzzy Inference System

Security incidents such as denial of service (DoS), scanning, malware code injection, viruses, worms, and password cracking are becoming common in a cloud environment that affects companies and may produce a financial loss if not detected in time. Such problems are handled by presenting an intrusion detection system (IDS) into the cloud. The existing cloud IDSs affect low detection accuracy, high false detection rate, and execution time. To overcome this problem, in this article, a gravitational search algorithm-based fuzzy inference system (GSA-FIS) is developed as intrusion detection. In this approach, fuzzy parameters are optimized using GSA. The proposed consist of two modules namely; possibilistic fuzzy c-means (PFCM) based clustering, training based on the GSA-FIS, and testing process. Initially, the incoming data is pre-processed and clustered with the help of PFCM. PFCM detects the noise of fuzzy c-means clustering (FCM), then conquers the coincident cluster problem of possibilistic fuzzy c-means (PCM) and eradicate the row sum constraints of fuzzy possibilistic c-means clustering (FPCM). After the clustering process, the clustered data is given to the optimized fuzzy inference system (OFIS). Here, normal and abnormal data are identified by the fuzzy score, while the training is done by the GSA through optimizing the entire fuzzy system. In this approach, four types of abnormal data are detected namely- probe, remote to local (R2L), user to root (U2R), and DoS. Simulation results show that the performance of the proposed GSA-FIS based IDS outperforms that of the different schemes in terms of precision, recall and F-measure.

Resolving issues of scaling for gramian-based input–output pairing methods

A key problem in process control is to decide which inputs should control which outputs. There are multiple ways to solve this problem, among them using gramian-based measures, which include the Hankel interaction index array, the participation matrix and the method. The gramian-based measures, however, have issues with input and output scaling. Generally, this is resolved by scaling all inputs and outputs to have equal range. However, we demonstrate how this can result in an incorrect pairing and examine alternative methods of scaling the gramian-based measures, using either row or column sums or by utilising the Sinkhorn-Knopp algorithm. To systematically analyse the benefits of the scaling schemes, a multiple-input multiple-output model generator is used to test the different schemes on a large number of systems. This assessment shows considerable benefits to be gained from the alternative scaling of the gramian-based measures, especially when using the Sinkhorn-Knopp algorithm.

Low-rank matrix completion and denoising under Poisson noise

Abstract This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyse several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares and a non-convex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed and the maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low-rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.