Hermite and Gegenbauer polynomials in superspace using Clifford analysis

The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore a lot of basic properties, such as orthogonality relations, differential equations and recursion formulae are proven. Finally, an interesting physical application of the super Clifford-Hermite polynomials is discussed, thus giving an interpretation to the super-dimension.Comment: 18 pages, accepted for publication in J. Phys.

Spherical harmonics and integration in superspace

In this paper the classical theory of spherical harmonics in R^m is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.Comment: 22 pages, accepted for publication in J. Phys.

Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)

The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1|2),Spin(m)), in the context of the Dirac operator. This connects our results with the Clifford-Fourier transform studied in previous work.Comment: Second version, changes in title, introduction and section

Introductory clifford analysis

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications

Treatment of Parkinson’s Disease:Early, Late, and Combined

Medical therapy in de novo Parkinson’s disease typically starts with a dopamine agonist or levodopa in combination with a decarboxylase inhibitor or if symptoms are still very mild with a MAO-B inhibitor. When patients do not (or no longer) respond satisfactorily to these initial therapies, different drugs can be initiated or combined (i.e., “add-on” treatments). These add-on therapies not only comprise oral agents but also intra-jejunal and intra-cutaneous treatments and functional neurosurgical procedures. This chapter starts with the treatment of de novo Parkinson’s disease whereafter indications and expected effects of the different “add-on” therapies will be described. The “add-on” therapies will be described in a hierarchical way and treatment algorithms will be provided based on prevailing symptoms including non-motor symptoms. The symptoms that will be discussed are: (1) bradykinesia and “wearing-OFF, " (2) tremor at rest, (3) dyskinesia, (4) gait and postural symptoms including freezing of gait, and (5) important non-motor symptoms. Finally, a comprehensive add-on treatment algorithm will be provided that takes into account non-motor symptoms that may limit the efficacy and tolerability of the different add-on therapies.</p

Orthosymplectically invariant functions in superspace

The notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced. This leads to dimensional reduction theorems for differentiation and integration in superspace. These spherically symmetric functions can be used to solve orthosymplectically invariant Schroedinger equations in superspace, such as the (an)harmonic oscillator or the Kepler problem. Finally the obtained machinery is used to prove the Funk-Hecke theorem and Bochner's relations in superspace.Comment: J. Math. Phy

Reliability of the Multidimensional Pain Inventory and stability of the MPI classification system in chronic back pain

Contains fulltext : 109346.pdf (publisher's version ) (Open Access)ABSTRACT: BACKGROUND: This cross validation study examined the reliability of the Multidimensional Pain Inventory (MPI) and the stability of the Multidimensional Pain Inventory Classification System of the empirically derived subgroup classification obtained by cluster analysis in chronic musculoskeletal pain. Reliability of the German Multidimensional Pain Inventory was only examined once in the past in a small sample. Previous international studies mainly involving fibromyalgia patients showed that retest resulted in 33-38% of patients being assigned to a different Multidimensional Pain Inventory subgroup classification. METHODS: Participants were 204 persons with chronic musculoskeletal pain (82% chronic non-specific back pain). Subgroup classification was conducted by cluster analysis at 4 weeks before entry (=test) and at entry into the pain management program (=retest) using Multidimensional Pain Inventory scale scores. No therapeutic interventions in this period were conducted. Reliability was quantified by intraclass correlation coefficients (ICC) and stability by kappa coefficients (kappa). RESULTS: Reliability of the Multidimensional Pain Inventory scales was least with ICC = 0.57 for the scale life control and further ranged from ICC = 0.72 (negative mood) to 0.87 (solicitous responses) in the other scales. At retest, 82% of the patients in the Multidimensional Pain Inventory cluster interpersonally distressed (kappa = 0.69), 80% of the adaptive copers (kappa = 0.58), and 75% of the dysfunctional patients (kappa = 0.70) did not change classification. In total, 22% of the patients changed Multidimensional Pain Inventory cluster group, mainly into the adaptive copers subgroup. CONCLUSION: Test-retest reliability of the German Multidimensional Pain Inventory was moderate to good and comparable to other language versions. Multidimensional Pain Inventory subgroup classification is substantially stable in chronic back pain patients when compared to other diagnostic groups and other examiner-based subgroup Classification Systems. The MPI Classification System can be recommended for reliable and stable specification of subgroups in observational and interventional studies in patients with chronic musculoskeletal pain

On the Efetov-Wegner terms by diagonalizing a Hermitian supermatrix

The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable to find certain structures in random matrix theory. However it still poses serious problems since up to now the calculation of all Rothstein contributions known as Efetov-Wegner terms in physics was quite cumbersome. We derive the supermatrix Bessel function with all Efetov-Wegner terms for an arbitrary rotation invariant probability density function. As applications we consider representations of generating functions for Hermitian random matrices with and without an external field as integrals over eigenvalues of Hermitian supermatrices. All results are obtained with all Efetov-Wegner terms which were unknown before in such an explicit and compact representation.Comment: 23 pages, PACS: 02.30.Cj, 02.30.Fn, 02.30.Px, 05.30.Ch, 05.30.-d, 05.45.M

The gait and balance of patients with diabetes can be improved: a randomised controlled trial

Aims/hypothesis: Gait characteristics and balance are altered in diabetic patients. Little is known about possible treatment strategies. This study evaluates the effect of a specific training programme on gait and balance of diabetic patients. Methods: This was a randomised controlled trial (n = 71) with an intervention (n = 35) and control group (n = 36). The intervention consisted of physiotherapeutic group training including gait and balance exercises with function-orientated strengthening (twice weekly over 12weeks). Controls received no treatment. Individuals were allocated to the groups in a central office. Gait, balance, fear of falls, muscle strength and joint mobility were measured at baseline, after intervention and at 6-month follow-up. Results: The trial is closed to recruitment and follow-up. After training, the intervention group increased habitual walking speed by 0.149m/s (p < 0.001) compared with the control group. Patients in the intervention group also significantly improved their balance (time to walk over a beam, balance index recorded on Biodex balance system), their performance-oriented mobility, their degree of concern about falling, their hip and ankle plantar flexor strength, and their hip flexion mobility compared with the control group. After 6months, all these variables remained significant except for the Biodex sway index and ankle plantar flexor strength. Two patients developed pain in their Achilles tendon: the progression for two related exercises was slowed down. Conclusions/interpretation: Specific training can improve gait speed, balance, muscle strength and joint mobility in diabetic patients. Further studies are needed to explore the influence of these improvements on the number of reported falls, patients' physical activity levels and quality of life. Trial registration:: ClinicalTrials.gov NCT00637546 Funding:: This work was supported by the Swiss National Foundation (SNF): PBSKP-123446/