• Kyungpook Mathematical Journal
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• 제55권1호
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• pp.157-168
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• 2015

In this paper, we prove that every f-biharmonic map from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature,satisfying some condition, is f-harmonic. Also we present some properties for the f-biharmonicity of submanifolds of $\mathbb{S}^n$, and we give the classification of f-biharmonic curves in 3-dimensional sphere.

• 대한수학회지
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• 제42권3호
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• pp.543-553
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• 2005

In this paper, we prove the existence of nonconstant bounded harmonic maps on a Cartan-Hadamard manifold of pinched negative curvature by solving the asymptotic Dirichlet problem. To be precise, given any continuous data f on the boundary at infinity with image within a ball in the normal range, we prove that there exists a unique harmonic map from the manifold into the ball with boundary value f.

• 대한수학회보
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• 제29권2호
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• pp.233-236
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• 1992

Let f:M.rarw.N be a smooth map between Rioemannian manifolds M and N. If f maps geodesics of M to geodesics of N, f is called totally geodesic. As is well known, totally geodesic maps are harmonic and the image f(M) of a totally geodesic map f:M.rarw. N is an immersed totally geodesic submanifold of N (cf. .cint. 6.3 of [W]). We are interested in the following question: When is a harmonic map f:M .rarw. N with rank .leq. 1 everywhere on M totally geodesic\ulcorner In other words, when is the image of a harmonic map f:M .rarw. N with rank .leq. 1 everywhere on M geodesics of N\ulcorner In this note, we give some sufficient conditions on curvatures of M. It is interesting that no curvature assumptions on target manifolds are necessary in Theorems 1 and 2. Some properties of totally geodesic maps are also given in Theorem 3. We think our Theorem 3 is somewhat unusual in view of the following classical theorem of Eells and Sampson (see pp.124 of [ES]).

• 대한수학회지
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• 제49권5호
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• pp.977-991
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• 2012

Let (M,F) and (M',F') be two foliated Riemannian manifolds with M compact. If the transversal Ricci curvature of F is nonnegative and the transversal sectional curvature of F' is nonpositive, then any transversally harmonic map ${\phi}:(M,F){\rightarrow}(M^{\prime},F^{\prime})$ is transversally totally geodesic. In addition, if the transversal Ricci curvature is positive at some point, then ${\phi}$ is transversally constant.

• 대한수학회보
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• 제32권2호
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• pp.211-214
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• 1995

Let M and N be compact Riemannian manifolds and $f : M \to N$ be a smooth map. Following J. Eells, f is exponentially harmonic if it represents a critical point of the exponential energy integral $$ E(f) = \int_{M} exp(\left\$\mid$ df \right\$\mid$^2) dM $$ where $(\left\ df $\mid$\right\$\mid$^2$ is the energy density defined as $\sum_{i=1}^{m} \left\$\mid$ df(e_i) \right\$\mid$^2$, m = dimM, for orthonormal frame $e_i$ of M. The Euler- Lagrange equation of the exponential energy functional E can be written $$ exp(\left\$\mid$ df \right\$\mid$^2)(\tau(f) + df(\nabla\left\$\mid$ df \right\$\mid$^2)) = 0 $$ where $\tau(f)$ is the tension field along f. Hence, if the energy density is constant, every harmonic map is exponentially harmonic and vice versa.

• 대한수학회보
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• 제54권6호
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• pp.2091-2106
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• 2017

The authors investigate f-biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature, and derive that if $\int_{M}f^p{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$, $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}du{\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, the authors also get that if u satisfies some integral conditions, then it is minimal. These results give an affirmative partial answer to conjecture 4 (generalized Chen's conjecture for f-biharmonic submanifolds).

• 대한수학회지
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• 제61권1호
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• pp.183-205
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• 2024

In this paper, for an m-dimensional (m ≥ 5) complete non-compact submanifold M immersed in an n-dimensional (n ≥ 6) simply connected Riemannian manifold N with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of M, the norm of its weighted mean curvature vector |H_f| and the weighted real-valued function f, we can obtain: • several one-end theorems for M; • two Liouville theorems for harmonic maps from M to complete Riemannian manifolds with nonpositive sectional curvature.

• Coupled systems mechanics
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• 제8권4호
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• pp.351-375
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• 2019

The present work investigates the use of a rigid rocking block as a tool to reduce vibrations in a frame structure. The study is based on a simplified model composed by a 2-DOF linear system, meant to represent a general M-DOF frame structure, coupled with a rocking rigid block through a linear visco-elastic device, which connects only the lower part of the 2-DOF system. The possibility to restrain the block directly to the ground, by means of a second visco-elastic device, is investigated as well. The dynamic response of the model under an harmonic base excitation is then analysed in order to evaluate the effectiveness of the coupling in reducing the displacements and the drift of the 2-DOF system. The nonlinear equations of motion of the coupled assemblage 2-DOF-block are obtained by a Lagrangian approach and then numerically integrated considering some reference mechanical and geometrical quantities as variable parameters. It follows an extensive parametric analysis, whose results are summarized through behaviour maps, which portray the ratio between the maximum displacements and drifts of the system, with and without the coupling with the rigid block, for several combinations of system's parameters. When the ratio of the displacements is less than unity, the coupling is considered effective. Results show that the presence of the rocking rigid block improves the dynamics of the system in large ranges of the characterizing parameters.

• Coupled systems mechanics
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• 제9권6호
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• pp.539-562
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• 2020

In this paper the linear elastic coupling between a 2 degree of freedom shear-type frame system and a rigid block is analytically and experimentally investigated. As demonstrated by some of the authors in previous papers, it is possible to choose a coupling system able to guarantee advantages, whatever the mechanical characteristics of the frame. The main purpose of the investigation is to validate the analytical model. The nonlinear equations of motion of the coupled system are obtained by a Lagrangian approach and successively numerically integrated under harmonic and seismic excitation. The results, in terms of gain graphs, maps and spectra, represent the ratio between the maximum displacements or drifts of the coupled and uncoupled systems as a function of the system's parameters. Numerical investigations show the effectiveness of the nonlinear coupling for a large set of parameters. Thus experimental tests are carried out to verify the analytical results. An electro-dynamic long-stroke shaker sinusoidally and seismically forces a shear-type 2 d.o.f frame coupled with a rigid aluminium block. The experimental investigations confirm the effectiveness of the coupling as predicted by the analytical model.

• Journal of Power Electronics
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• 제18권6호
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• pp.1729-1750
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• 2018

Voltage source inverters (VSIs) are widely used to drive induction motors in industry applications. The quality of output waveforms depends on the switching sequences used in pulse width modulation (PWM). In this work, all existing optimal space vector pulse width modulation (SVPWM) switching strategies are studied. The performance of existing SVPWM switching strategies is optimized to realize a tradeoff between quality of output waveforms and switching losses. This study generalizes the existing optimal switching sequences for total harmonic distortions (THDs) and switching losses for different modulation indexes and reference angles with a parameter called quality factor. This factor provides a common platform in which the THDs and switching losses of different SVPWM techniques can be compared. The optimal spatial distribution of each sequence is derived on the basis of the quality factor to minimize harmonic current distortions and switching losses in a sector; the result is the minimum ripple loss SVPWM (MRSLPWM). By employing the sequences from optimized switching maps, the proposed method can simultaneously reduce THDs and switching losses. Two hybrid SVPWM techniques are proposed to reduce line current distortions and switching losses in motor drives. The proposed hybrid SVPWM strategies are MRSLPWM 30 and MRSLPWM 90. With a low-cost PIC microcontroller (PIC18F452), the proposed hybrid SVPWM techniques and the quality of output waveforms are experimentally validated on a 2 kVA VSI based on a three-phase two-level insulated gate bipolar transistor.