On the Dynamics of Induced Maps on the Space of Probability Measures

For the generic continuous map and for the generic homeomorphism of the Cantor space, we study the dynamics of the induced map on the space of probability measures, with emphasis on the notions of Li-Yorke chaos, topological entropy, equicontinuity, chain continuity, chain mixing, shadowing and recurrence. We also establish some results concerning induced maps that hold on arbitrary compact metric spaces.Comment: 23 page

A note on the continuity of multilinear mappings in topological modules

In the present note, we obtain a criterion for the equicontinuity of families of multilinear mappings between topological modules. We also give an example which shows that the hypothesis imposed on the neighborhoods of zero is essential for the validity of our theorem

Time Spent in Home Care Tasks Related to Ownership and Uses of Home Care Equipment

The data for this research were taken from Utah\u27s contribution to the regional research project An Interstate Comparison of Urban/Rural Families\u27 Time Use. Data were collected between May 1977 and August 1978 from 210 two-parent/ two-child families. This thesis research studied the relationship between ownership and use of nine selected household appliances and time spent on the related housekeeping tasks for 208 of the families studied. Statistical analysis was done using t-tests for comparisons of time spent on the related task by owners and non-owners of each appliance. Analysis of variance was used to compare time spent on combined activities with ownership of differing numbers of appliances . The relationship between frequency of use and time spent on tasks was measured us ing the Pearson Product Moment correlation. The hypotheses tested were: 1. Ownership of home care equipment is not related to the amount of time spent in home care tasks. 2. Reported use of home care equipment is not related to the amount of time spent in home care tasks. Hypothesis Number 1 was accepted for all relationships tested with the exception of the dishwasher and time spent in dishwashing and the sewing machine and time spent in construction of clothing and household linens. The results indicated that the homemakers who owned a dishwasher spent less time in dishwashing than did non-owners. This was not true of the spouses, who spent very little time in dishwashing under either circumstance. The homemakers who owned a sewing machine spent considerably more time in construction of clothing and household linens than non-owners. When families were grouped by the number of appliances owned , no statistically significant relationships were found to exist between the number of appliances owned and the total time spent in home care tasks. Generally, those who owned many or few of the appliances spent more time in home care activities than did owners of four or five of the appliances. Hypothesis Number 2 was rejected for the relationships between dishwasher use and spouse time spent in dishwashing, sewing machine use and homemaker time spent in construction of clothing and household linens and use of power yard equipment and time spent in maintenance of home, yard, car and pets. The number of times the dishwasher was reported to have been used was related to the amount of time spent in dishwashing by spouses although the time was very limited. The lime spent in clothing and household linen construction increased with the number of times the sewing machine was used. This relationship would have been expected. Those who used their power yard equipment more often spent more time in maintenance of home, yard, car and pets. This was true for both the borne makers and the spouses. The assumed relationship between appliance ownership and use and time spent on home care activities was not found to exist for most appliances. The time savings potential of appliances had not been realized. The time spent on most tasks did not differ significantly between owners and non-owners, or by the reported number of times used

Pilot to Program: The full integration of PebblePad eportfolio learning into a Bachelor of Nursing course

ePortfolio e-learning has moved student learning from the didactic model to a student-led focus. The integration of PebblePad eportfolio learning has allowed students to take ownership of their learning in units where PebblePad is integrated, and what is evident is that students want ownership of all their learning and achievements throughout their nurse education

Expansivity and Shadowing in Linear Dynamics

In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators

Going paperless in student nurse clinical work integrated learning

The development of student nurses’ knowledge and skills in the Murdoch University (MU) undergraduate Bachelor of Nursing (BN) course is assessed against the Nursing and Midwifery Board of Australia [NMBA] (2016) Registered Nurse standards for practice, using the NMBA framework for assessing standards for practice (NMBA, 2015)..

A generalized Grobman-Hartman theorem

We prove that any generalized hyperbolic operator on any Banach space is structurally stable. As a consequence, we obtain a generalization of the classical Grobman-Hartman theorem.Comment: 9 page

On the existence of polynomials with chaotic behaviour

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold. © 2013 Nilson C. Bernardes Jr. and Alfredo Peris.The present work was done while the first author was visiting the Departament de Matematica Aplicada at Universitat Politecnica de Valencia (Spain). The first author is very grateful for the hospitality. The first author was supported in part by CAPES: Bolsista, Project no. BEX 4012/11-9. The second author was supported in part by MEC and FEDER, Project MTM2010-14909, and by GVA, Projects PROMETEO/2008/101 and PROMETEOII/2013/013.Bernardes, NC.; Peris Manguillot, A. (2013). On the existence of polynomials with chaotic behaviour. Journal of Function Spaces and Applications. 2013(320961). https://doi.org/10.1155/2013/320961S2013320961Bayart, F., & Matheron, E. (2009). Dynamics of Linear Operators. doi:10.1017/cbo9780511581113Grosse-Erdmann, K.-G., & Peris Manguillot, A. (2011). Linear Chaos. Universitext. doi:10.1007/978-1-4471-2170-1Rolewicz, S. (1969). On orbits of elements. Studia Mathematica, 32(1), 17-22. doi:10.4064/sm-32-1-17-22Herzog, G. (1992). On linear operators having supercyclic vectors. Studia Mathematica, 103(3), 295-298. doi:10.4064/sm-103-3-295-298Ansari, S. I. (1997). Existence of Hypercyclic Operators on Topological Vector Spaces. Journal of Functional Analysis, 148(2), 384-390. doi:10.1006/jfan.1996.3093Bernal-González, L. (1999). Proceedings of the American Mathematical Society, 127(04), 1003-1011. doi:10.1090/s0002-9939-99-04657-2Bonet, J., & Peris, A. (1998). Hypercyclic Operators on Non-normable Fréchet Spaces. Journal of Functional Analysis, 159(2), 587-595. doi:10.1006/jfan.1998.3315Bonet, J., Martínez-Giménez, F., & Peris, A. (2001). A Banach Space which Admits No Chaotic Operator. Bulletin of the London Mathematical Society, 33(2), 196-198. doi:10.1112/blms/33.2.196Shkarin, S. (2008). On the spectrum of frequently hypercyclic operators. Proceedings of the American Mathematical Society, 137(01), 123-134. doi:10.1090/s0002-9939-08-09655-xDe la Rosa, M., Frerick, L., Grivaux, S., & Peris, A. (2011). Frequent hypercyclicity, chaos, and unconditional Schauder decompositions. Israel Journal of Mathematics, 190(1), 389-399. doi:10.1007/s11856-011-0210-6Bernardes, N. C. (1998). ON ORBITS OF POLYNOMIAL MAPS IN BANACH SPACES. Quaestiones Mathematicae, 21(3-4), 311-318. doi:10.1080/16073606.1998.9632049Bernardes Jr., N. C. (1998). Proceedings of the American Mathematical Society, 126(10), 3037-3045. doi:10.1090/s0002-9939-98-04483-9Dineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. doi:10.1007/978-1-4471-0869-6Peris, A. (2001). Proceedings of the American Mathematical Society, 129(12), 3759-3761. doi:10.1090/s0002-9939-01-06274-8ARON, R. M., & MIRALLES, A. (2008). CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS. Glasgow Mathematical Journal, 50(2), 319-323. doi:10.1017/s0017089508004229Peris, A. (2003). Chaotic polynomials on Banach spaces. Journal of Mathematical Analysis and Applications, 287(2), 487-493. doi:10.1016/s0022-247x(03)00547-xMARTÍNEZ-GIMÉNEZ, F., & PERIS, A. (2010). CHAOTIC POLYNOMIALS ON SEQUENCE AND FUNCTION SPACES. International Journal of Bifurcation and Chaos, 20(09), 2861-2867. doi:10.1142/s0218127410027416Martínez-Giménez, F., & Peris, A. (2009). Existence of hypercyclic polynomials on complex Fréchet spaces. Topology and its Applications, 156(18), 3007-3010. doi:10.1016/j.topol.2009.02.010Bès, J., & Peris, A. (2007). Disjointness in hypercyclicity. Journal of Mathematical Analysis and Applications, 336(1), 297-315. doi:10.1016/j.jmaa.2007.02.043Bernardes, N. C., Bonilla, A., Müller, V., & Peris, A. (2013). Distributional chaos for linear operators. Journal of Functional Analysis, 265(9), 2143-2163. doi:10.1016/j.jfa.2013.06.019Le�n-Saavedra, F., & M�ller, V. (2004). Rotations of Hypercyclic and Supercyclic Operators. Integral Equations and Operator Theory, 50(3), 385-391. doi:10.1007/s00020-003-1299-8Grosse-Erdmann, K.-G., & Peris, A. (2010). Weakly mixing operators on topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 104(2), 413-426. doi:10.5052/racsam.2010.25Li, T.-Y., & Yorke, J. A. (1975). Period Three Implies Chaos. The American Mathematical Monthly, 82(10), 985. doi:10.2307/2318254Schweizer, B., & Smital, J. (1994). Measures of Chaos and a Spectral Decomposition of Dynamical Systems on the Interval. Transactions of the American Mathematical Society, 344(2), 737. doi:10.2307/2154504Bermúdez, T., Bonilla, A., Martínez-Giménez, F., & Peris, A. (2011). Li–Yorke and distributionally chaotic operators. Journal of Mathematical Analysis and Applications, 373(1), 83-93. doi:10.1016/j.jmaa.2010.06.011Hou, B., Cui, P., & Cao, Y. (2010). Chaos for Cowen-Douglas operators. Proceedings of the American Mathematical Society, 138(03), 929-929. doi:10.1090/s0002-9939-09-10046-1Martínez-Giménez, F., Oprocha, P., & Peris, A. (2009). Distributional chaos for backward shifts. Journal of Mathematical Analysis and Applications, 351(2), 607-615. doi:10.1016/j.jmaa.2008.10.049Schenke, A., & Shkarin, S. (2013). Hypercyclic operators on countably dimensional spaces. Journal of Mathematical Analysis and Applications, 401(1), 209-217. doi:10.1016/j.jmaa.2012.11.013BONET, J., FRERICK, L., PERIS, A., & WENGENROTH, J. (2005). TRANSITIVE AND HYPERCYCLIC OPERATORS ON LOCALLY CONVEX SPACES. Bulletin of the London Mathematical Society, 37(02), 254-264. doi:10.1112/s0024609304003698Shkarin, S. (2012). Hypercyclic operators on topological vector spaces. Journal of the London Mathematical Society, 86(1), 195-213. doi:10.1112/jlms/jdr08

Graph Theoretic Structure of Maps of the Cantor Space

In this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of the space of homeomorphisms and the space of self-maps of the Cantor space. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the comeager conjugacy class of the space of homeomorphisms of the Cantor space. The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is a comeager subset of homeomorphism space of the Cantor space each element of which has topological entropy zero. Our analogous investigation in the space of continuous self-maps of the Cantor space yields a surprising result: there is a comeager subset of the space of self-maps of the Cantor space such that any two elements of this set are conjugate to each other by an homeomorphism. Our description of this class also yields many old and new results concerning dynamics of a comeager subset of the space of continuous self-maps of the Cantor space.Comment: 26 page