In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen ...uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff for the one-sided transposition shuffle at time n log n. We also study weighted generalisations of the one-sided transposition shuffle called biased one-sided transposition shuffles. We compute the full spectrum for every biased one-sided transposition shuffle, and prove the existence of a total variation cutoff for certain choices of weighted distribution. In particular, we recover the eigenvalues and well known mixing time of the classical random transposition shuffle. We study the hyperoctahedral group as an extension of the symmetric group, and formulate the one-sided transposition shuffle and random transposition shuffle as random walks on this new group. We determine the spectrum of each hyperoctahedral shuffle by developing a correspondence between their eigenvalues and standard Young bi-tableaux. We prove that the one-sided transposition shuffle on the hyperoctahedral group exhibits a cutoff at n log n, the same time as its symmetric group counterpart. We conjecture that this results extends to the biased one-sided transposition shuffles and the random transposition shuffle on the hyperoctahedral group.
This thesis develops the deformation theory of instantons on asymptotically conical G2-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the ...space of deformations to the kernel of a twisted Dirac operator on the G2-manifold and to the eigenvalues of a twisted Dirac operator on the nearly Kähler link. As an application, we use this framework to study the moduli spaces of known examples of G2-instantons living on the Bryant-Salamon manifolds and on R7. We develop two methods for determining eigenvalues of twisted Dirac operators on nearly Kähler 6- manifolds and apply this to calculate the virtual dimension of the moduli spaces that we study. In the case of the instanton of Günaydin-Nicolai, which lives on R7; we show how knowledge of the virtual dimension of the moduli space can be used to study uniqueness properties of this instanton.
The time-series forecasting makes a substantial contribution in timely decision-making. In this article, a recently developed eigenvalue decomposition of Hankel matrix (EVDHM) along with the ...autoregressive integrated moving average (ARIMA) is applied to develop a forecasting model for nonstationary time series. The Phillips-Perron test (PPT) is used to define the nonstationarity of time series. EVDHM is applied over a time series to decompose it into respective subcomponents and reduce the nonstationarity. ARIMA-based model is designed to forecast the future values for each subcomponent. The forecast values of each subcomponent are added to get the final output values. The optimized value of ARIMA parameters for each subcomponent is obtained using a genetic algorithm (GA) for minimum values of Akaike information criterion (AIC). Model performance is evaluated by estimating the future values of daily new cases of the recent pandemic disease COVID-19 for India, USA, and Brazil. The high efficacy of the proposed method is convinced with the results.
The effective Δmee2 in matter Denton, Peter B; Parke, Stephen J
Physical review. D,
11/2018, Volume:
98, Issue:
9
Journal Article
Peer reviewed
Open access
In this paper, we generalize the concept of an effective Δmee2 for νe/ν¯e disappearance experiments, which has been extensively used by the short baseline reactor experiments, to include the effects ...of propagation through matter for longer baseline νe/ν¯e disappearance experiments. This generalization is a trivial, linear combination of the neutrino mass squared eigenvalues in matter and thus is not a simple extension of the usually vacuum expression, although, as it must, it reduces to the correct expression in the vacuum limit. We also demonstrated that the effective Δmee2 in matter is very useful conceptually and numerically for understanding the form of the neutrino mass squared eigenstates in matter and hence for calculating the matter oscillation probabilities. Finally, we analytically estimate the precision of this two-flavor approach and numerically verify that it is precise at the subpercent level.
Abstract
By using the eigenvalue inclusion field theorem and the commutability of the Hadamard product, new upper bound estimations of
ρ
(
E
∘
F
) are given. The new estimations can be expressed only ...by the elements of two nonnegative matrices, which are operable and easy to calculate. Finally, an example is given to verify that the estimation formulas in this paper are more accurate than related conclusions.
Abstract
In this study, the lower bound expressions of the smallest eigenvalue
q
(
E
★
F
) are derived using the row sums of two M-matrices and the maximum of the main diagonal elements. The obtained ...lower bounds improve the existing related conclusions. Furthermore, the results of this paper are compared with those of related results by two specific examples to verify the accuracy of the results.
The Aα matrix of a graph G is defined by Nikiforov as Aα(G)=αD(G)+(1−α)A(G), where α∈0,1, A(G) and D(G) respectively denotes the adjacency matrix and the degree diagonal matrix of G. The eigenvalues ...of Aα(G) are called the Aα-eigenvalues of G. In this paper, we prove that, for a connected graph G of order n and with maximum degree Δ≥2 and for any α∈0,1), the multiplicity of an arbitrary Aα-eigenvalue λ of G, which is denoted as mα(G,λ), is bounded bymα(G,λ)≤(Δ−2)n+2Δ−1, the equality holds if and only if G and λ satisfy one of the following conditions:
(i) G=Kn with n≥3 and λ=αn−1;
(ii) G=Cn with even order, and λ∈{2α+2(1−α)cos2πjn:j=1,2,…,n−22};
(iii) G=Cn with odd order, and λ∈{2α+2(1−α)cos2πjn:j=1,2,…,n−12};
(iv) G=Kn2,n2 with n≥4, and λ=αn2.
Applying this result, we solve an open conjecture posed by Zhou et al. in 32 one year ago.
To simplify the analysis process of the multirelay magnetic coupling wireless power transfer system, a novel analysis method based on the quadratic eigenvalue problem (QEP) has been proposed. The ...second-order underlying equation of the system is given. The eigensolution represents the inherent properties of the entire system and provides important and useful information. The analytical solutions of currents induced in each coil are expressed in terms of the eigensolution of the corresponding QEP. Real parts and imaginary parts of the eigenvalues represent attenuation coefficients and system resonance frequencies, respectively. The effect of the load resistance on the eigenvalues is studied. It is found that the attenuation coefficient of an eigenvalue increases as the load resistance increases so that it can be ignored. The relationships between eigenvalues and some key frequencies are discussed, such as system resonance frequencies, zero-phase angular frequencies, and constant voltage/current frequencies. The results show that fixed zero-phase angular frequencies and constant current frequencies are equal to the imaginary parts of the eigenvalues with load resistance is equal to zero, and the constant voltage frequencies are equal to the imaginary parts of the complex eigenvalues in a strong damping stage. Finally, experimental results verify the theoretical analysis.
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