265 research outputs found
Mean eigenvalues for simple, simply connected, compact Lie groups
We determine for each of the simple, simply connected, compact and complex
Lie groups SU(n), Spin and that particular region inside the unit
disk in the complex plane which is filled by their mean eigenvalues. We give
analytical parameterizations for the boundary curves of these so-called trace
figures. The area enclosed by a trace figure turns out to be a rational
multiple of in each case. We calculate also the length of the boundary
curve and determine the radius of the largest circle that is contained in a
trace figure. The discrete center of the corresponding compact complex Lie
group shows up prominently in the form of cusp points of the trace figure
placed symmetrically on the unit circle. For the exceptional Lie groups ,
and with trivial center we determine the (negative) lower bound on
their mean eigenvalues lying within the real interval . We find the
rational boundary values -2/7, -3/13 and -1/31 for , and ,
respectively.Comment: 12 pages, 8 figure
Infinitesimals without Logic
We introduce the ring of Fermat reals, an extension of the real field
containing nilpotent infinitesimals. The construction takes inspiration from
Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual
infinitesimals without any need of a background in mathematical logic. In
particular, on the contrary with respect to SIA, which admits models only in
intuitionistic logic, the theory of Fermat reals is consistent with classical
logic. We face the problem to decide if the product of powers of nilpotent
infinitesimals is zero or not, the identity principle for polynomials, the
definition and properties of the total order relation. The construction is
highly constructive, and every Fermat real admits a clear and order preserving
geometrical representation. Using nilpotent infinitesimals, every smooth
functions becomes a polynomial because in Taylor's formulas the rest is now
zero. Finally, we present several applications to informal classical
calculations used in Physics: now all these calculations become rigorous and,
at the same time, formally equal to the informal ones. In particular, an
interesting rigorous deduction of the wave equation is given, that clarifies
how to formalize the approximations tied with Hook's law using this language of
nilpotent infinitesimals.Comment: The first part of the preprint is taken directly form arXiv:0907.1872
The second part is new and contains a list of example
On all possible static spherically symmetric EYM solitons and black holes
We prove local existence and uniqueness of static spherically symmetric
solutions of the Einstein-Yang-Mills equations for any action of the rotation
group (or SU(2)) by automorphisms of a principal bundle over space-time whose
structure group is a compact semisimple Lie group G. These actions are
characterized by a vector in the Cartan subalgebra of g and are called regular
if the vector lies in the interior of a Weyl chamber. In the irregular cases
(the majority for larger gauge groups) the boundary value problem that results
for possible asymptotically flat soliton or black hole solutions is more
complicated than in the previously discussed regular cases. In particular,
there is no longer a gauge choice possible in general so that the Yang-Mills
potential can be given by just real-valued functions. We prove the local
existence of regular solutions near the singularities of the system at the
center, the black hole horizon, and at infinity, establish the parameters that
characterize these local solutions, and discuss the set of possible actions and
the numerical methods necessary to search for global solutions. That some
special global solutions exist is easily derived from the fact that su(2) is a
subalgebra of any compact semisimple Lie algebra. But the set of less trivial
global solutions remains to be explored.Comment: 26 pages, 2 figures, LaTeX, misprints corrected, 1 reference adde
Exploiting symmetries in SDP-relaxations for polynomial optimization
In this paper we study various approaches for exploiting symmetries in
polynomial optimization problems within the framework of semi definite
programming relaxations. Our special focus is on constrained problems
especially when the symmetric group is acting on the variables. In particular,
we investigate the concept of block decomposition within the framework of
constrained polynomial optimization problems, show how the degree principle for
the symmetric group can be computationally exploited and also propose some
methods to efficiently compute in the geometric quotient.Comment: (v3) Minor revision. To appear in Math. of Operations Researc
Antibodies for prevention and treatment of diseases caused by Clostridium difficile
The present invention relates to an antibody having specificity for an immunogenic determinant consisting of the pentasaccharide repeating unit of the Clostridium difficile glycopolymer PS-I: α-L-Rhap-(1→3)-β-D-Glcp-(1→4)-[α-L-Rhap-(1→3)]-α-D-Glcp-(1→2)-α-D-Glcp or a fragment thereof. Said antibody is able to prevent and treat diseases caused by C. difficile. The present invention further pertains to a method of treating or preventing a disease caused by the pathogen Clostridium difficile, which comprises administering to a subject said antibody or a vaccine composition comprising said antibody
A collaborative and evolving response to the needs of frontline workers, patients and families during the COVID-19 pandemic at Tygerberg Hospital, Western Cape Province, South Africa
The global devastation caused by the COVID-19 pandemic and its mental health impact is undeniable. The physical and psychological consequences are wide-ranging – affecting patients fighting the disease, frontline workers in the trenches with them, healthcare staff deployed in high-care settings, and families disconnected from their loved ones in their darkest hours. Within 6 weeks of the COVID-19 outbreak in South Africa, the Department of Psychiatry at Stellenbosch University established the TBH/SU COVID Resiliency Clinic to provide psychological support to frontline workers at Tygerberg Hospital. Identified barriers in healthcare workers accessing mental healthcare resulted in moving towards an on-site visibility to try to remove some of these barriers. This greater on-site presence enabled networking and building of relationships with frontline staff that over time highlighted other frontline needs, such as providing psychosocial and spiritual support to patients and their families. We share challenges, lessons learned and recommendations from two initiatives: the TBH/SU COVID-19 Resiliency Clinic, and an embedded COVID Care Team (CCT). We describe the establishment, roll-out and progress of the Clinic and the subsequent CCT
Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups
In this paper, we construct a Lagrangian submanifold of the moduli space
associated to the fundamental group of a punctured Riemann surface (the space
of representations of this fundamental group into a compact connected Lie
group). This Lagrangian submanifold is obtained as the fixed-point set of an
anti-symplectic involution defined on the moduli space. The notion of
decomposable representation provides a geometric interpretation of this
Lagrangian submanifold
The Bloch-Okounkov correlation functions of classical type
Bloch and Okounkov introduced an n-point correlation function on the infinite
wedge space and found an elegant closed formula in terms of theta functions.
This function has connections to Gromov-Witten theory, Hilbert schemes,
symmetric groups, etc, and it can also be interpreted as correlation functions
on integrable gl_\infty-modules of level one. Such gl_\infty-correlation
functions at higher levels were then calculated by Cheng and Wang. In this
paper, generalizing the type A results, we formulate and determine the n-point
correlation functions in the sense of Bloch-Okounkov on integrable modules over
classical Lie subalgebras of gl_\infty of type B,C,D at arbitrary levels. As
byproducts, we obtain new q-dimension formulas for integrable modules of type
B,C,D and some fermionic type q-identities.Comment: v2, very minor changes, Latex, 41 pages, to appear in Commun. Math.
Phy
Geometric K-Homology of Flat D-Branes
We use the Baum-Douglas construction of K-homology to explicitly describe
various aspects of D-branes in Type II superstring theory in the absence of
background supergravity form fields. We rigorously derive various stability
criteria for states of D-branes and show how standard bound state constructions
are naturally realized directly in terms of topological K-cycles. We formulate
the mechanism of flux stabilization in terms of the K-homology of non-trivial
fibre bundles. Along the way we derive a number of new mathematical results in
topological K-homology of independent interest.Comment: 45 pages; v2: References added; v3: Some substantial revision and
corrections, main results unchanged but presentation improved, references
added; to be published in Communications in Mathematical Physic
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