Complete Invariance Research Articles

Invariants of totally real Lefschetz fibrations

In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants real Lefschetz chains. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of totally real Lefschetz fibrations. If however the fiber genus is 1, real Lefschetz chains are not sufficient to distinguish real Lefschetz fibrations. We show that by adding a certain binary decoration to real Lefschetz chains, we get a complete invariant.

The Polish Lie ring of vector fields on a smooth manifold is algebraically determined

Let L be a Polish (i.e., complete separable metrizable) Lie ring. L is said to be algebraically determined if, whenever R is a Polish Lie ring and φ:R→L is an algebraic isomorphism of Lie rings, then φ is a topological isomorphism. The purpose of this paper is to prove that the Lie ring of vector fields on a smooth manifold is an algebraically determined Polish Lie ring. A new fact about the ring of real numbers plays a crucial role in the proof of the general theorem. An application of the main theorem will be described to prove that certain algebraic objects are complete invariants for classifying smooth manifolds up to diffeomorphism.

ON THE ORIGIN OF STELLAR MASSES

It has been a longstanding problem to determine, as far as possible, the characteristic masses of stars in terms of fundamental constants; the almost complete invariance of this mass as a function of the star-forming environment suggests that this should be possible. Here I provide such a calculation. The typical stellar mass is set by the characteristic fragment mass in a star-forming cloud, which depends on the cloud's density and temperature structure. Except in the very early universe, the latter is determined mainly by the radiation released as matter falls onto seed protostars. The energy yield from this process is ultimately set by the properties of deuterium burning in protostellar cores, which determines the stars' radii. I show that it is possible to combine these considerations to compute a characteristic stellar mass almost entirely in terms of fundamental constants, with an extremely weak residual dependence on the interstellar pressure and metallicity. This result not only explains the invariance of stellar masses, it resolves a second mystery: why fragmentation of a cold, low-density interstellar cloud, a process with no obvious dependence on the properties of nuclear reactions, happens to select a stellar mass scale such that stellar cores can ignite hydrogen. Finally, the weak residual dependence on the interstellar pressure and metallicity may explain recent observational hints of a smaller characteristic mass in the high pressure, high metallicity cores of giant elliptical galaxies.

A Note on the Complete Rotational Invariance of Biradial Solutions to Semilinear Elliptic Equations

Abstract We investigate symmetry properties of solutions to equations of the form -Δu = + f(|x|, u) in ℝN for N ≥ 4, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact completely radial. A similar result holds for the semilinear Laplace-Beltrami equations on the sphere. Furthermore, we show that the condition on the Morse index is sharp. Finally we apply the result in order to estimate best constants of Sobolev type inequalities with different symmetry constraints.

A Cross-Validation of the Relational Entitlement and Proprietariness Scale

Relational entitlement and proprietariness refer to a set of beliefs supporting a view that a romantic partner is analogous to property and can be treated as such. This individual difference is expected to facilitate various forms of coercion and aggression in relationships. A means and covariance structures (MACS) approach within the framework of confirmatory factor analysis was conducted on two independent samples of undergraduate students (N A = 292, N B = 317) to cross-validate a previously developed six-factor structure of the Relational Entitlement and Proprietariness (REP) Scale. The analyses successfully validated the REP instrument, evidencing complete measurement invariance across the two samples. Additional tests were performed to explore potential sex differences in the scale. The first-order factor loading pattern of the hierarchical REP structure was largely invariant for the male and female samples. Similarly, the latent mean structure was equivalent for men and women with the exception of 13 observed measures. The findings of this study evidence that the REP measure is a factorially stable and reliable instrument to assess male and female relational entitlement and proprietariness behaviors.

Assessing Perceived Empathic and Social Self-Efficacy Across Countries.

The Perceived Empathic Self-Efficacy Scale (PESE) and the Perceived Social Self-Efficacy Scale (PSSE) were developed to assess, respectively, individuals' self-efficacy beliefs regarding both empathic responding to others' needs or feelings and managing interpersonal relationships. In this study of young adults, a unidimensional factorial structure of both scales was found in Italy, the United States, and Bolivia. Complete invariance at the metric level and partial invariance at the scalar level were found across gender and countries for both scales. The construct and incremental validity of both PESE and PSSE were further examined in a different sample of Italian young adults. Patterns of association of the PESE or PSSE with self-esteem, psychological well-being, and the use of adaptive and maladaptive coping strategies were found, often over and beyond their associations with empathy or extraversion, respectively.

Exploratory Structural Equation Modeling, Integrating CFA and EFA: Application to Students' Evaluations of University Teaching

This study is a methodological-substantive synergy, demonstrating the power and flexibility of exploratory structural equation modeling (ESEM) methods that integrate confirmatory and exploratory factor analyses (CFA and EFA), as applied to substantively important questions based on multidimentional students' evaluations of university teaching (SETs). For these data, there is a well established ESEM structure but typical CFA models do not fit the data and substantially inflate correlations among the nine SET factors (median rs = .34 for ESEM, .72 for CFA) in a way that undermines discriminant validity and usefulness as diagnostic feedback. A 13-model taxonomy of ESEM measurement invariance is proposed, showing complete invariance (factor loadings, factor correlations, item uniquenesses, item intercepts, latent means) over multiple groups based on the SETs collected in the first and second halves of a 13-year period. Fully latent ESEM growth models that unconfounded measurement error from communality showed almost no linear or quadratic effects over this 13-year period. Latent multiple indicators multiple causes models showed that relations with background variables (workload/difficulty, class size, prior subject interest, expected grades) were small in size and varied systematically for different ESEM SET factors, supporting their discriminant validity and a construct validity interpretation of the relations. A new approach to higher order ESEM was demonstrated, but was not fully appropriate for these data. Based on ESEM methodology, substantively important questions were addressed that could not be appropriately addressed with a traditional CFA approach.

On Definition of Skew Frames

Skew frames represent a common generalization of frames and orthomodular lattices. They could serve as Lindenbaum algebras of quantum intuitionistic logic as well as invariants of noncommutative C*-algebras. It is shown that lattices of open projections with skew (partial) operations are complete invariants of C*-algebras and that these operations are preserved by morphims of C*-algebras.

Conjugacy, orbit equivalence and classification of measure-preserving group actions

AbstractWe prove that if G is a countable discrete group with property (T) over an infinite subgroup H≤G which contains an infinite Abelian subgroup or is normal, then G has continuum-many orbit-inequivalent measure-preserving almost-everywhere-free ergodic actions on a standard Borel probability space. Further, we obtain that the measure-preserving almost-everywhere-free ergodic actions of such a G cannot be classified up to orbit equivalence by a reasonable assignment of countable structures as complete invariants. We also obtain a strengthening and a new proof of a non-classification result of Foreman and Weiss for conjugacy of measure-preserving ergodic almost-everywhere-free actions of discrete countable groups.

Glimm-Effros for coanalytic equivalence relations

AbstractAssuming every real has a sharp, we prove that for any equivalence relation either Borel reduces E0 or in a manner allows the assignment of bounded subsets of ω1 as complete invariants.

Measurement equivalence of osteoporosis-specific and general quality-of-life instruments in Aboriginal and non-Aboriginal women

To test the measurement equivalence (i.e., invariance) of osteoporosis-specific and general health-related quality-of-life (HRQOL) instruments in Canadian Aboriginal and non-Aboriginal women. A total of 258 Aboriginal and 181 non-Aboriginal women were recruited to the First Nations Bone Health Study from rural and urban sites in the province of Manitoba, Canada. The Medical Outcomes Study Short Form-36 (SF-36) and the mini-Osteoporosis Quality of Life Questionnaire (mini-OQLQ) were administered to study participants by trained interviewers. Confirmatory factor analysis techniques were adopted to test hypotheses about four forms of invariance for the two groups using likelihood ratio tests and other goodness-of-fit indices. For the mini-OQLQ instrument configural and metric invariance were satisfied, indicating that both groups have the same conceptualization of osteoporotic quality of life and the concepts have equivalent meaning. However, scalar and complete invariance were not satisfied for this instrument. The SF-36 exhibited complete invariance in the two groups. Measurement equivalence, which is required to conduct valid group comparisons, was not demonstrated for the disease-specific quality-of-life instrument but was supported for the general instrument. Ethnicity appears to influence responses about the effects of osteoporosis on quality of life.

Isomorphism of subshifts is a universal countable borel equivalence relation

We use the theory of Borel equivalence relations to analyze the equivalence relation of isomorphism among one-dimensional subshifts. We show that this equivalence relation is a universal countable Borel equivalence relation, so that it admits no definable complete invariants fundamentally simpler than the equivalence classes. We also see that the classification of higher dimensional subshifts up to isomorphism has the same complexity as for the one-dimensional case.

The complexity of classifying separable Banach spaces up to isomorphism

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.

Fold maps and immersions from the viewpoint of cobordism

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n + 1)-dimensional manifolds into an n-dimensional manifold Nn in terms of immersions with prescribed normal bundles. We compute that for Nn = Rn the cobordism group of simple fold maps is isomorphic to the direct sum of the (n - 1)-st stable homotopy group of spheres and the (n - 1)-st stable homotopy group of the space RP8. By using geometric invariants defined in the author�s earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group in terms of natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented bordism groups of simple fold maps.

Output feedback variable structure control for a class of uncertain switched systems

AbstractThis paper considers the output feedback variable structure control problem for a class of uncertain switched systems. A single sliding surface is constructed using only output information. Robust stability of the sliding motion restricted to the sliding surface is achieved and complete invariance to the matched and the mismatched uncertainties is guaranteed via a hysteresis switching law based on a reduced order observer. Output feedback variable structure controllers are given to guarantee that the redefined sliding surface is reached in finite time. A numerical example illustrates the effectiveness of the proposed design method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

The representation of object viewpoint in human visual cortex

Understanding the nature of object representations in the human brain is critical for understanding the neural basis of invariant object recognition. However, the degree to which object representations are sensitive to object viewpoint is unknown. Using fMRI we employed a parametric approach to examine the sensitivity to object view as a function of rotation (0°–180°), category (animal/vehicle) and fMRI-adaptation paradigm (short or long-lagged). For both categories and fMRI-adaptation paradigms, object-selective regions recovered from adaptation when a rotated view of an object was shown after adaptation to a specific view of that object, suggesting that representations are sensitive to object rotation. However, we found evidence for differential representations across categories and ventral stream regions. Rotation cross-adaptation was larger for animals than vehicles, suggesting higher sensitivity to vehicle than animal rotation, and was largest in the left fusiform/occipito-temporal sulcus (pFUS/OTS), suggesting that this region has low sensitivity to rotation. Moreover, right pFUS/OTS and FFA responded more strongly to front than back views of animals (without adaptation) and rotation cross-adaptation depended both on the level of rotation and the adapting view. This result suggests a prevalence of neurons that prefer frontal views of animals in fusiform regions. Using a computational model of view-tuned neurons, we demonstrate that differential neural view tuning widths and relative distributions of neural-tuned populations in fMRI voxels can explain the fMRI results. Overall, our findings underscore the utility of parametric approaches for studying the neural basis of object invariance and suggest that there is no complete invariance to object view in the human ventral stream.

Determining the range of real eigenvalues for the interval generalized eigenvalue problem

PurposeTo suggest a polynomial complexity method for determining the range of real eigenvalues in the case of the generalized eigenvalue problem when the elements of the matrices involved are independent intervals.Design/methodology/approachThe basic approach is to make use of approximate interval solutions as regards the right and left eigenvectors of the eigenproblem considered, the so‐called outer solutions, in order to determine the range.FindingsFirst, a new method for computing the outer solutions has been suggested. The main result of the paper, however, is the development of a simple method for determining the range of the real eigenvalues. Unlike the known general‐purpose methods that have exponential complexity, the present range determination method is much simpler as its complexity is only polynomial.Research limitations/implicationsThe method is applicable if certain sufficient conditions reported in the paper are satisfied (an incomplete quadratic system is to have a positive solution and the signs of the outer solutions should satisfy a complete or partial invariance).Practical implicationsThe method guarantees reliable numerical results when the original eigenproblems contain interval uncertainties as is, strictly speaking, most often the case in practice.Originality/valueTo the best of the author's knowledge, the present paper suggests, for the first time, a simple method of polynomial complexity for solving the problem considered which is inherently a NP‐hard problem (of exponential complexity).

Cross-cultural equivalence of the Beck Depression Inventory: A five-country analysis from the ODIN study

Background The Beck Depression Inventory (BDI) has demonstrated excellent psychometric properties and good performance as a screening measure in different contexts and languages. However, comparison of its structure across countries and languages remains understudied. Measurement invariance is a prerequisite for considering the BDI equivalent across versions, and for using it to make valid and interpretable comparisons of the severity of depression among different groups. Methods As part of a five-country (UK, Ireland, Spain, Norway, and Finland), two-stage epidemiological study of depressive disorder, 7934 persons were screened using the BDI. The item equivalence and measurement invariance of the BDI across the samples of the five countries was tested using Item Response Theory (IRT) and Multiple Indicators Multiple Causes (MIMIC) models. Results Overall results support the factorial validity of the BDI, with a unidimensional structure. Item 19 (weight loss) presented a clear misfit in the five countries. IRT models, as well as MIMIC models, suggest that complete measurement invariance cannot be assumed across the five countries. The Spanish sample accounted for the majority of the differences, with a moderate to low Differential Item Functioning for the other countries on the different items. Conclusions The BDI could be used cross-culturally in Europe, with particular cautions regarding the Spanish sample, and with the constraints and limits pointed out for the present results. The methodology used for the present work is suggested as a soundness approach for testing the cross-cultural validity of severity rating scales.

Local Lipschitz geometry of real weighted homogeneous surfaces

We compute Holder Complexes, i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces.

Applicability of action planning and coping planning to dental flossing among Norwegian adults: a confirmatory factor analysis approach

Using a prospective design and a representative sample of 25-yr-old Norwegians, this study hypothesized that action planning and coping planning will add to the prediction of flossing at 4 wk of follow-up over and above the effect of intention and previous flossing. This study tested the validity of a proposed 3-factor structure of the measurement model of intention, action planning, and coping planning and for its invariance across gender. A survey was conducted in three Norwegian counties, and 1,509 out of 8,000 randomly selected individuals completed questionnaires assessing the constructs of action planning and coping planning related to daily flossing. A random subsample of 500 participants was followed up at 4 wk with a telephone interview to assess flossing. Confirmatory factor analysis (CFA) confirmed the proposed 3-factor model after respecification. Although the chi-square test was statistically significant [chi(2) = 58.501, degrees of freedom (d.f.) = 17), complementary fit indices were satisfactory [goodness-of-fit index (GFI) = 0.99, root mean squared error of approximation (RMSEA) = 0.04]. Multigroup CFA provided evidence of complete invariance of the measurement model across gender. After controlling for previous flossing, intention (beta = 0.08) and action planning (beta = 0.11) emerged as independent predictors of subsequent flossing, accounting for 2.3% of its variance. Factorial validity of intention, action planning and coping planning, and the validity of action planning in predicting flossing prospectively, was confirmed by the present study.