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This study discusses the finite-time boundedness (FTB) of the state of linear systems with a finite-time H 2-gain performance constraint. With time-varying exogenous disturbances, the sufficient conditions for linear time-invariant and linear time-varying systems to satisfy both the FTB constraint and the finite-time H 2-gain performance constraint are developed. On the basis of the matrix inequality theory, a method for designing a controller by using state feedback is derived for each of the two classes of linear systems. These methods can effectively keep the state from reaching an unacceptable value and modulate the output performance within a specified finite-time interval. Two application examples of the terminal guidance scenario verify the effectiveness of the proposed methods.

Using the Lawson existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three-dimensional sphere. These surfaces contain the Clifford torus, the Lawson’s minimal surfaces, and seven new minimal surfaces with genera [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We will also discuss the relation between such surfaces and the maximal extendable group actions on subsurfaces of the three-dimensional sphere.

In the present study, an optimal laminar flow minichannel heat sink (MCHS) has been determined through three-dimensional simulations and the multi-objective optimization algorithm. The cross-sectional shape described by six variables is optimized by the multi-objective genetic algorithm (MOGA) and multi-objective particle swarm optimization (MOPSO). During the optimization, the thermal resistance θ and pumping power P are two conflicting objectives for evaluating the performances of the MCHS. After obtaining the non-inferior solutions, the technique for order preference by similarity to an ideal solution (TOPSIS) is applied as a decision-making method to determine the best compromise one. Results indicate that the TOPSIS can effectively reduce the P of the Pareto solutions without significantly increasing θ. Compared with the straight channel, the TOPSIS optimal solution could reduce θ by 7.47% or P by 31.54%. Meanwhile, the mechanism of performance improvement is analyzed by comparing the TOPSIS optimal solution and the straight channel with the same P. It is observed that the optimized channel shape changes the fluid distribution by increasing the heat transfer coefficient slightly and the heat transfer area by 12.22%, respectively.

Accurate forecasting of power load is of great significance to economic dispatch and safe operation of power grid. A short-term load forecasting model based on EEMD- LN-GRU is proposed according to the characteristics of uncertainty and nonlinearity of power load. In order to solve the problem of power load fluctuation and mode aliasing caused by empirical mode decomposition (EMD), the original load time series signal is decomposed into multiple intrinsic mode functions (IMF) and residual error component by ensemble empirical mode decomposition (EEMD). Each component signal is predicted by the gating recurrent unit (GRU) after layer normalization (LN). Finally, the components are predicted. The results are recombined, and then the peak valley value of the prediction sequence obtained by peak valley value correction strategy is modified to obtain the final load series. Taking the real load data of power plant of Slovak power company as an example, this method is compared with LSTM, GRU and other methods, and the results show that the proposed method has higher prediction accuracy for load forecasting.

Higher heating value (HHV) is one of the properties of biomass fuels which is essential in investigating their special characteristics and potentialities. In this paper, various techniques based on Gaussian process regression (GPR) were utilized to assess this value for biomass fuels, including several kernel functions, i.e., exponential, Matern, rational quadratic, and squared exponential. An extensive databank was collected from literature. The findings were compared, and the results indicated that Exponential-based model was more accurate, with the coefficient of regression ( R 2 ) of 0.961 and the mean relative error (% MRE) of 3.11 for total data. Compared to former models presented by previous researchers, the model proposed in this study showed a higher ability to predict output values. With various analyses, it can be concluded that the proposed method has a high rate of efficiency in assessing the HHV of various biomass.

This paper deals with the input–output finite-time stability of discrete time-varying linear systems in the presence of finite-time boundedness. The state boundedness and output stability of this s...

Purpose This study aims to find a novel solution to the calibration of three-axis magnetometers to suppress errors of sensors. The nature of the calibration process is parameter estimation and hence the purpose of the paper is to calculate the error parameters and eliminate sensor errors and obtain the true value of the pure magnetic field. Design/methodology/approach The paper puts forward a calibration method using an alternative iteration looping optimization (AILO) to estimate the parameters. The proposed method divided the parameters to be estimated into two parts: a portion less than one and the other greater than one. Parameters with different orders of magnitude are calculated respectively, which let one part to be a known quantity and the other part is derived by the known quantity; the derived quantity is used to calculate the known quantity again, and looping iteration multiple times until the iteration termination condition is satisfied. Findings The simulation and experimental results indicate that the calibration accuracy is improved at least by two orders by the proposed method compared to the two-step method and the linear decreasing weight particle swarm optimization (LDW-PSO) algorithm which proves the validity of the proposed method. Practical implications The proposed method can improve the calibration accuracy of total magnetic field, which provides a reference to the calibration of three-axis magnetometers. Originality/value A calibration method based on the AILO is proposed in this paper, which is used to improve the calibration accuracy of the three-axis magnetometer.

Suppose an orientation-preserving action of a finite group G on the closed surface \(\Sigma _g\) of genus \(g>1\) extends over the 3-torus \(T^3\) for some embedding \(\Sigma _g\subset T^3\). Then \(|G|\le 12(g-1)\), and this upper bound \(12(g-1)\) can be achieved for \(g=n^2+1, 3n^2+1, 2n^3+1, 4n^3+1, 8n^3+1, n\in {\mathbb {Z}}_+\). The surfaces in \(T^3\) realizing a maximal symmetry can be either unknotted or knotted. Similar problems in the non-orientable category are also discussed. The connection with minimal surfaces in \(T^3\) is addressed and the situation when the maximally symmetric surfaces above can be realized by minimal surfaces is identified.

Given a connected real Lie group and a contractible homogeneous proper $G$--space $X$ furnished with a $G$--invariant volume form, a real valued volume can be assigned to any representation $\rho\colon \pi_1(M)\to G$ for any oriented closed smooth manifold $M$ of the same dimension as $X$. Suppose that $G$ contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given $M$. From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed., Comment: 34 pages

We present a 3-manifold automorphism f : M → M with rank Fix ( f ⁎ ) = 2 rank π 1 ( M ) . Related facts are discussed.

We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+\cos \frac{\pi}{2g+1}+\cos \frac{2\pi}{2g+1})$ when $g\ge 7$, and for genus $g$ hyperbolic surface with order $4g$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+2\cos \frac{\pi}{2g})$ when $g\ge4$., Comment: 25 pages, 33 figures, accepted for publication in SCIENCE CHINA Mathematics

In this paper, we study the consensus problem of multi-agent systems with parametric uncertainties on directed graph communication topologies containing a spanning tree. The challenge lies in that the input–output property of a strictly proper system transfer function involving self and communication delays and the extremum of a delay-involved transfer function are both unclear. By establishing a new input–output property of the delay-involved strictly proper transfer function and applying a constructive approach in frequency domain, we obtain the extremum of the delay-involved transfer function, upon which, we establish a unified framework to resolve the consensus problem of multi-agent systems with parametric uncertainties and time delays, and the terminal convergence point value is achieved. Based upon Lyapunov stability theory and frequency domain input–output analysis, we demonstrate that the proposed unified consensus control framework ensures scaled weighted average consensus with the integral action of the synchronization signal vector. Simulation results are provided to demonstrate the effectiveness of the proposed consensus scheme.

Call a periodic map h on the closed orientable surface Σg extendable if h extends to a periodic map over the pair (S3,Σg) for possible embeddings e: Σg → S3. The authors determine the extendabilities for all periodical maps on Σ2. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair (S3,Σg). To do this the authors first list all periodic maps on Σ2, and indeed the authors exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be interesting. A by-product is that for each even g, the maximum order periodic map on Σg is extendable, which contrasts sharply with the situation in the orientation preserving category.

GDOP’s influence on observable degree is studied by analyzing a multi-antenna GPS/SINS attitude measuring system. Firstly, the mathematical model of the attitude measuring system, whose observations are single-differences, is provided; Secondly, it is proved that the model can be studied by PWCS theory, and GDOP’s influence on observable degree is explained intuitively. Then, the variance of observable degree is studied by simulation while GDOP is different, and the result is analysed. Simulation results show that, the observable degree becomes worse with the increasing GDOP value. Therefore, while designing such kind of Kalman filter, the influence to observable degree which made by GDOP should be considered adequately.

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-id$, or the mapping torus of a hyperbolic automorphism., Comment: 8 pages

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic genus $>1$ in $\mathbb{R}^3$. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces., Comment: 22 pages, 15 figures, accepted for publication in SCIENCE CHINA Mathematics

Due to the need of preparation of geomagnetic reference map, coons surface method is used to to generate reference map by processing the measured value of the total discrete magnetic field intensity for the selected area to the grid; Select four data grid effectiveness evaluation standard to evaluate three reference map respectively generated by Kriging method, minimum curvature method and coons surface method. Conclusion shows that in selected area, coons surface method can generate better mesh data index and grid data of high precision.

In this paper, it is shown that for any closed orientable $3$-manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable $3$-manifold with positive simplicial volume has virtually positive Seifert volume. The result reveals certain fundamental difference between the representation volume of the hyperbolic type and the Seifert type. The proof is based on developments and reactions of recent results on virtual domination and on virtual representation volumes of $3$-manifolds., 26 pages, exposition revised

In this paper, the theoretical and practical feasibility of estimating ego-motion in monocular navigation is under investigation. In particular, this paper concerns the observability issues of monocular navigation models. In order to recover the ego-motion of the monocular camera from 2D (two-dimension) image sequence, this paper proposes a novel recursive parametric model. This model differs from other traditional parametric models with the reduced state-dimension and the consistent observability of system state. In addition, the consistent observability of different models are compared by using time-varying observability analysis theory. At the end of this paper, two groups of experiments are implemented to validate the effectiveness of the proposed method.

The paper presents linear system Input–Output Finite-Time Stabilization (IO-FTS) method under Finite-Time Boundedness (FTB) constraint. A state feedback controller is designed, via Linear Matrix Inequalities (LMIs), to guarantee the system both IO-FTS and FTB. The proposed methods are applied to the guidance design of a class of terminal guidance systems to suppress disturbances with IO-FTS method and FTB constraints simultaneously satisfied. The simulation results illustrate the effectiveness of the proposed methods.

In this paper, we consider the consensus problem of multiple agents modeled by Euler–Lagrange (EL) equation, among which two classes of agents are addressed, i.e., some agents with exactly known parameters and the others with parametric uncertainties. We propose a distributed consensus protocol for the heterogeneous EL systems in which both time-delay and jointly connected topologies are taken into consideration. Based on graph theory, Lyapunov theory and Barbalat׳s lemma, the stability of the controller is proved. A distinctive feature of this work is to investigate the consensus problem of EL systems with heterogeneous dynamics, time-delay and jointly connected topologies in a unified theoretical framework. Simulation results are also provided to illustrate the effectiveness of the obtained results.

This paper studies the adaptive consensus problem of networked mechanical systems with time-varying delay and jointly-connected topologies. Two different consensus protocols are proposed. First, we present an adaptive consensus protocol for the connected switching topologies. Based on graph theory, Lyapunov stability theory and switching control theory, the stability of the proposed algorithm is demonstrated. Then we investigate the problem under the more general jointly-connected topologies, and with concurrent time-varying communication delay. The proposed consensus protocol consists of two parts: one is for the connected agents which contains the current states disagreement among them and the other is designed for the isolated agents which contains the states difference between the current and past. A distinctive feature of this work is to address the consensus control problem of mechanical systems with unknown parameters, time-varying delay and switching topologies in a unified theoretical framework. Numerical simulation is provided to demonstrate the effectiveness of the obtained results.

This paper presents finite-time control methods with H-infinity constraints for linear time-invariant (LTI) and time-varying (LTV) systems. The basic idea of the proposed approaches is to construct controllers for the LTI and LTV in such a way that a constant quadratic Lyapunov function and a time-varying quadratic Lyapunov function can be used to establish the finite-time stability and the H-infinity performance of the resulting closed-loop systems. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities (LMIs) and Differential Riccati Inequalities (DRIs) that are numerically feasible with commercially available software. Finally, the results are illustrated by application to the design of guidance law for a class of terminal guidance system.

Locating an inconspicuous object is critical in remote distance space-based measurement. An indirect location method is presented based on the locations of the salient regions, from which the location of the inconspicuous object is transferred indirectly. The location problem of inconspicuous object is considered via affine invariability, which existed in the remote measurement. Salient region detection and matching are carried out first to match the real-time images and template images, and then affine transform is established related to the two images via affine function. The ultimate aim of location is achieved by homography matrix, which is calculated by homologous feature points between the real time image and template image. At the end of this paper, experiment is performed to prove the validity of this method.

A metric direct (feature-less) online monocular SLAM algorithm is proposed, which, different from the traditional monocular SLAM algorithm, can obtain the absolute scale of the world. A calibrate object is used to calculate the relative pose of the first key frame and initialize the depth map, introducing the metric scale into the reconstruction meanwhile. Highly accurate pose estimation is achieved by direct image alignment, and the depth of image points is obtained by filtering over a large number of pixel-wise small-baseline stereo comparisons. With those, the environment is reconstructed in real-time as pose-graph of key frames with associated semi-dense depth maps. Also an explicitly scale-drift aware formulation is proposed to align the scale between key frames. The algorithm is tested online, and is proved to run in real-time on a CPU.

Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join of a complete graph of order $k$ and an empty graph of order $n-k$. In this paper, we show that the conjecture is true for trees of diameter at most four., Comment: 10 pages

We say that a knot $k_{1}$ in the $3$-sphere $1$-dominates another $k_{2}$ if there is a proper degree 1 map $E(k_{1})\to E(k_{2})$ between their exteriors, and write $k_{1}\ge k_{2}$. When $k_{1}\ge k_{2}$ but $k_{1}\ne k_{2}$ we write $k_{1}>k_{2}$. One expects in the latter eventuality that $k_{1}$ is more complicated. In this paper, we produce various sorts of evidence to support this philosophy.

This work studies distributed six degree-of-freedom spacecraft formation control with possible switching topology. By rearranging the combined translational and rotational dynamics into a unified Euler–Lagrange formulation, the developed controllers can be applied directly to maintain formation as well as desired relative attitudes. Based on a novel adaptive control architecture, distributed controllers are developed, which allow for parameter uncertainties and unknown external disturbances. By using the Lyapunov theory and a non-linear switching system theory, it is proved that arbitrary desired formation and relative attitudes among spacecrafts can be obtained with either fixed or switching communication topology. Simulations are provided that demonstrate the effectiveness of our theoretical results.

Let M be a compact orientable 3-manifold, and let F be a separating (resp. non-separating) incompressible surface in M which cuts M into two 3-manifolds M1 and M2 (resp. a manifold M1). Then M is called the surface sum (resp. self surface sum) of M1 and M2 (resp. M1) along F, denoted by M=M1∪FM2 (resp. M=M1∪F). In this paper, we will study how g(M) is related to χ(F), g(M1) and g(M2) when both M1 and M2 have high distance Heegaard splittings.

If there exists a diffeomorphism f on a closed, orientable n-manifold M such that the non-wandering set Ω(f) consists of finitely many orientable ( ±) attractors derived from expanding maps, then M is a rational homology sphere; moreover all those attractors are of topological dimension n−2. Expanding maps are expanding on (co)homologies.

For given closed orientable 3‐manifolds M and N let D.M;N/ be the set of mapping degrees from M to N . We address the problem: For which N is D.M;N/ finite for all M ? The answer is known for prime 3‐manifolds unless the target is a nontrivial graph manifold. We prove that for each closed nontrivial graph manifold N , D.M;N/ is finite for any graph manifold M . The proof uses a recently developed standard form of maps between graph manifolds and the estimation of the e PSL.2; R/‐volume for a certain class of graph manifolds.

For each composite number $n\ne 2^k$, there does not exist a single connected closed $(n+1)$-manifold such that any smooth, simply-connected, closed $n$-manifold can be topologically flat embedded into it. There is a single connected closed 5-manifold $W$ such that any simply-connected, 4-manifold $M$ can be topologically flat embedded into $W$ if $M$ is either closed and indefinite, or compact and with non-empty boundary., 21 pages, 3 figures

A quick proof of Bing's theorem indicated by the title is given. The proof also concludes Gumerov's result on covering degrees of solenoids., 4 pages

We construct the first examples of degree one maps between non-homeomorphic closed hyperbolic small 3-manifolds.

Let $OE_g$ (resp. $CE_g$ and $AE_g$) and resp. $OE^o_g$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces $\Sigma_g$ which extend over $(S^3, \Sigma_g)$ among all embeddings $\Sigma_g\to S^3$ and resp. unknotted embeddings $\Sigma_g\to S^3$. It is known that $OE^o_g\le 12(g-1)$, and we show that $12(g-1)$ is reached for an unknotted embedding $\Sigma_g \to S^3$ if and only if $g = 2$, 3, 4, 5, 6, 9, 11, 17, 25, 97, 121, 241, 601. Moreover $AE_g$ is $2g+2$; and $CE_g$ is $2g+2$ for even $g$, and $2g-2$ for odd $g$. Efforts are made to see intuitively how these maximal symmetries are embedded into the symmetries of the 3-sphere., Comment: 21 pages, 15 figures

International audience; In the original publication (http://​dx.​doi.​org/​10.​1007/​s00209-007-0113-8), the proof of Lemma 4 was incorrect and it is now corrected.

We say a knot $k$ in the 3-sphere $\mathbb S^3$ has {\it Property $IE$} if the infinite cyclic cover of the knot exterior embeds into $\mathbb S^3$. Clearly all fibred knots have Property $IE$. There are infinitely many non-fibred knots with Property $IE$ and infinitely many non-fibred knots without property $IE$. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property $IE$, then its Alexander polynomial $\Delta_k(t)$ must be either 1 or $2t^2-5t+2$, and we give two infinite families of non-fibred genus 1 knots with Property $IE$ and having $\Delta_k(t)=1$ and $2t^2-5t+2$ respectively. Hence among genus one non-fibred knots, no alternating knot has Property $IE$, and there is only one knot with Property $IE$ up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold., Comment: 24 pages, 9 figures

We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold M and any integer m> 0, a surgery on a link in M of at most 2m + 1 components will provide a closed, orientable, irreducible 3-manifold containing m disjoint, non-parallel, separating, incompressible surfaces of arbitrarily high genus. (2) There exists a 3-manifold M containing separating incompressible surfaces Sn of genus g(Sn) arbitrarily large, such that the amalgamation of minimal Heegaard splittings of two resulting 3-manifolds cutting along Sn can be stabilized g(Sn) − 3 times to a minimal Heegaard splitting of M .

We construct a class of simple, small knots in a handlebody of genus g , for any g >1.

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.

We study the question: which integers k can be realized as the degree of a map between two given closed (n-1)-connected 2n-manifolds? Mathematics Subject Classification (2000): 57R19, 55M25

Any 3-manifold 1-dominates at most finitely many 3-manifolds supporting S 3 S^3 geometry.

In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.

We determine the maximum order of cyclic group actions on the pair [Formula: see text] among all embeddings of closed surface [Formula: see text] into the 3-dimensional torus [Formula: see text] in the orientable category.

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact 3-manifolds orientable 3-manifold dominates only finitely many exterior of knots in $S^3$., 34 pages, 5 figures

We study the set ${\rm vol}\left(M,G\right)$ of volumes of all representations $\rho\co\pi_1M\to G$, where $M$ is a closed oriented $3$-manifold and $G$ is either ${\rm Iso}_+{\Hi}^3$ or ${\rm Iso}_e\t{\rm SL_2(\R)}$. By various methods, including relations between the volume of representations and the Chern--Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold $M$ with positive Gromov simplicial volume has a finite cover $\t M$ with ${\rm vol}(\t M,{\rm Iso}_+{\Hi}^3)\ne \{0\}$, and that any non-geometric 3-manifold $M$ containing at least one Seifert piece has a finite cover $\t M$ with ${\rm vol}(\t M,{\rm Iso}_e\t{\rm SL_2(\R)}) \ne \{0\}$. We also find 3-manifolds $M$ with positive simplicial volume but ${\rm vol}(M,{\rm Iso}_+{\Hi}^3)=\{0\}$, and non-trivial graph manifolds $M$ with ${\rm vol}(M,{\rm Iso}_e\t{\rm SL_2(\R)})=\{0\}$, proving that it is in general necessary to pass to some finite covering to guarantee that ${\rm vol}(M,G)\not=\{0\}$. Besides we determine ${\rm vol}\left(M, G \right)$ when $M$ supports the Seifert geometry., Comment: 43 pages. arXiv admin note: substantial text overlap with arXiv:1111.6153

For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of $\partial M$ or the volume of $M$ is bounded above. When the volume is bounded above, then area of $\partial M$ is bounded above and the length of closed geodesic on $\partial M$ is bounded below., Comment: 13 pages

An embedding of a space Y into the 3-sphere S3 is said to be strictly achiral if its image is pointwise fixed by an orientation reversing homeomorphism of S3. A space Y is said to be abstractly planar if it can be embedded into the 2-sphere S2. We first extend Kuratowski Theorem into a criterion for abstract planarity of polyhedra, then show that a polyhedron has a strictly achiral polyhedral embedding into S3 if and only if it is abstractly planar. Some related higher dimensional examples are also discussed.

A criterion is given for an immersed horizontal $\pi_1$ -injective surface in a graph manifold to be separable. Examples are constructed of such surfaces, which are not separable and do not satisfy the k-plane property, for any k. It is shown that the simple loop conjecture holds in graph manifolds and that any graph manifold with boundary has an immersed horizontal surface.