512,995 research outputs found

    Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics

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    Let A be p × p positive definite matrix. A p-vector x such that Ax = x is called an eigenvector with the associated with eigenvalue . Equivalent characterizations are: (i) cos = 1, where is the angle between x and Ax. (ii) (x0Ax)−1 = xA−1x. (iii) cos = 1, where is the angle between A1/2x and A−1/2x. We ask the question what is x such that cos as defined in (i) is a minimum or the angle of separation between x and Ax is a maximum. Such a vector is called an anti-eigenvector and cos an anti-eigenvalue of A. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as min[(x0Ax)(x0A−1x)]−1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle between A1/2x and A−1/2x as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an r(< p) dimensional subspace S and its transform AS. Similar problems are considered for a general matrix A and its singular values leading to anti-singular values. Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented

    Semiclassical analysis of a complex quartic Hamiltonian

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    It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half \mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter \lambda.Comment: 4 page

    Bound States of Non-Hermitian Quantum Field Theories

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    The spectrum of the Hermitian Hamiltonian 12p2+12m2x2+gx4{1\over2}p^2+{1\over2}m^2x^2+gx^4 (g>0g>0), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian H=12p2+12m2x2gx4H={1\over2}p^2+{1 \over2}m^2x^2-gx^4, where the coupling constant gg is real and positive, is PT{\cal PT}-symmetric. As a consequence, the spectrum of HH is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When gg is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian PT{\cal PT}-symmetric gϕ4-g\phi^4 quantum field theory for all dimensions 0D<30\leq D<3 but is not present in the conventional Hermitian gϕ4g\phi^4 field theory.Comment: 14 pages, 3figure

    Phase diagram of quarter-filled band organic salts, [EDT-TTF-CONMe2]2X, X = AsF6 and Br

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    An investigation of the P/T phase diagram of the quarter-filled organic conductors, [EDT-TTF-CONMe2]2X, is reported on the basis of transport and NMR studies of two members, X=AsF6 and Br of the family. The strongly insulating character of these materials in the low pressure regime has been attributed to a remarkably stable charge ordered state confirmed by 13C NMR and the only existence of 1/4 Umklapp e-e scattering favoring a charge ordering instead of the 1D Mott localization seen in (TM)2X which are quarter-filled compounds with dimerization. A non magnetic insulating phase instead of the spin density wave state is stabilized in the deconfined regime of the phase diagram. This sequence of phases observed under pressure may be considered as a generic behavior for 1/4-filled conductors with correlations

    Muscle architecture and strength changes induced by different resistance training frequencies and detraining

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    International Journal of Exercise Science 15(4): 1661-1679, 2022. The purpose of the present study was to investigate muscle thickness and strength outcomes of the quadriceps femoris induced by different resistance training (RT) frequencies and detraining. In addition, muscle architecture (MA) parameters were also assessed. Twenty-seven healthy resistance-trained subjects (men, n = 17; women, n = 10; 20.8 ± 1.9 years; RT experience = 3.3 ± 1.6 years) volunteered to participate in this study. One leg of each subject was randomly allocated into the 2 sessions per week condition (2x) and the contralateral leg was then placed in the 4 sessions per week condition (4x). There were 16 RT sessions in 2x and 4x. After 4 weeks, 4x were divided into 2 other conditions: more 4 weeks with 2x(4x (+2x)) and detraining (4x (+Det)). Muscle thickness (MT), fascicle length (FL), pennation angle (PA) of the quadriceps muscles and one-repetition maximum for unilateral knee extension (1RMKE) were evaluated. A significant increase of 1RMKE in 2x, 4x, and 4x (+2x) and a decrease in 4x (+Det) was observed (all p \u3c 0.05). The MA showed similar results in most dependent variables for MT, FL and PA. Specifically 4x (+Det) condition demonstrated antagonistic results when compared to the 4x (+2x) in MT of rectus femoris (p = 0.001) and increased FL in vastus intermedius (p = 0.001)