Characterizing the Relationship Between Unitary Quantum Walks and Non-Homogeneous Random Walks

Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the two processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The first result establishes procedure for a stochastic matrix sequence to induce a random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. The second result establishes a similar procedure in the opposite direction. Given any random walk, a time-dependent quantum walk with the exact same vertex probability distribution is constructed. Interestingly, the matrices constructed by the first procedure allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial-time) sampling of quantum graph trajectories. Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.

A Detailed Study on Matrix Sequence of Generalized Narayana Numbers

In this paper, we introduce and investigate the generalized Narayana matrix sequence and we deal with, in detail, three special cases of this sequence which we call them Narayana, Narayana-Lucas and Narayana-Perrin matrix sequences. We present Binet’s formulas, generating functions, and the summation formulas for these sequences. We present the proofs to indicate how these sum formulas, in general, were discovered. Of course, all the listed sum formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there always exist interrelation between generalized Narayana, Narayana, Narayana-Lucas and Narayana-Perrin matrix sequences.

DNA Methylation, Deamination, and Translesion Synthesis Combine to Generate Footprint Mutations in Cancer Driver Genes in B-Cell Derived Lymphomas and Other Cancers

Cancer genomes harbor numerous genomic alterations and many cancers accumulate thousands of nucleotide sequence variations. A prominent fraction of these mutations arises as a consequence of the off-target activity of DNA/RNA editing cytosine deaminases followed by the replication/repair of edited sites by DNA polymerases (pol), as deduced from the analysis of the DNA sequence context of mutations in different tumor tissues. We have used the weight matrix (sequence profile) approach to analyze mutagenesis due to Activation Induced Deaminase (AID) and two error-prone DNA polymerases. Control experiments using shuffled weight matrices and somatic mutations in immunoglobulin genes confirmed the power of this method. Analysis of somatic mutations in various cancers suggested that AID and DNA polymerases η and θ contribute to mutagenesis in contexts that almost universally correlate with the context of mutations in A:T and G:C sites during the affinity maturation of immunoglobulin genes. Previously, we demonstrated that AID contributes to mutagenesis in (de)methylated genomic DNA in various cancers. Our current analysis of methylation data from malignant lymphomas suggests that driver genes are subject to different (de)methylation processes than non-driver genes and, in addition to AID, the activity of pols η and θ contributes to the establishment of methylation-dependent mutation profiles. This may reflect the functional importance of interplay between mutagenesis in cancer and (de)methylation processes in different groups of genes. The resulting changes in CpG methylation levels and chromatin modifications are likely to cause changes in the expression levels of driver genes that may affect cancer initiation and/or progression.

Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences

In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, the authors consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.

REAL TIME SEARCH OF AGRICULTURAL MACHINERY BASED ON MATRIX SEQUENCE SENSOR

Omni-directional vision sensor can provide information within the sensor range, and the directional angle of an object can be accurately obtained through omni-directional images. Based on this characteristic, an automatic navigation and positioning system for agricultural machinery is developed, and a three-dimensional positioning algorithm for agricultural wireless sensor networks based on cross particle swarm optimization is proposed. The method mainly includes three stages: convergence node selection, measurement distance correction and node location. Using the idea of crossover operation of genetic algorithm for reference, the diversity of particles is increased, and the influence of ranging error and the number of anchor nodes on positioning results is effectively improved. The location algorithm has the ability of global search. On the positioning node, the symmetric bidirectional ranging algorithm based on LFM (Linear frequency modulation) spread spectrum technology is used to calculate the distance between the positioning node and each beacon node, and the trilateral centroid positioning algorithm is used to calculate the coordinate position information of unknown nodes. Finally, the Kalman filter algorithm is used to superimpose the observed values of the target state to solve the influence of measurement noise on the positioning accuracy.

Global convergence of a modified Broyden family method for nonconvex functions

<p style='text-indent:20px;'>The Broyden family method is one of the most effective methods for solving unconstrained optimization problems. However, the study of the global convergence of the Broyden family method is not sufficient. In this paper, a new Broyden family method is proposed based on the BFGS formula of Yuan and Wei (Comput. Optim. Appl. 47: 237-255, 2010). The following approaches are used in the designed algorithm: (1) a modified Broyden family formula is given, (2) every matrix sequence <inline-formula><tex-math id="M1">\begin{document}$ \{B_k\} $\end{document}</tex-math></inline-formula> generated by the new algorithm possesses positive-definiteness, and (3) the global convergence of the new presented Broyden family algorithm with the Y-W-L inexact line search is obtained for general functions. Numerical performance shows that the modified Broyden family method is competitive with the classical Broyden family method.</p>

Matrix Representation of Bi-Periodic Jacobsthal Sequence

In this paper, we bring into light the matrix representation of bi-periodic Jacobsthal sequence, which we shall call the bi-periodic Jacobsthal matrix sequence. We dene it as with initial conditions J0 = I identity matrix, . We obtained the nth general term of this new matrix sequence. By studying the properties of this new matrix sequence, the well-known Cassini or Simpson's formula was obtained. We then proceed to find its generating function as well as the Binet formula. Some new properties and two summation formulas for this new generalized matrix sequence were also given.