471 research outputs found
Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames
In this paper, we settle a long-standing problem on the connectivity of
spaces of finite unit norm tight frames (FUNTFs), essentially affirming a
conjecture first appearing in [Dykema and Strawn, 2003]. Our central technique
involves continuous liftings of paths from the polytope of eigensteps to spaces
of FUNTFs. After demonstrating this connectivity result, we refine our analysis
to show that the set of nonsingular points on these spaces is also connected,
and we use this result to show that spaces of FUNTFs are irreducible in the
algebro-geometric sense, and also that generic FUNTFs are full spark.Comment: 33 pages, 4 figure
An operational definition of quark and gluon jets
While "quark" and "gluon" jets are often treated as separate, well-defined
objects in both theoretical and experimental contexts, no precise, practical,
and hadron-level definition of jet flavor presently exists. To remedy this
issue, we develop and advocate for a data-driven, operational definition of
quark and gluon jets that is readily applicable at colliders. Rather than
specifying a per-jet flavor label, we aggregately define quark and gluon jets
at the distribution level in terms of measured hadronic cross sections.
Intuitively, quark and gluon jets emerge as the two maximally separable
categories within two jet samples in data. Benefiting from recent work on
data-driven classifiers and topic modeling for jets, we show that the practical
tools needed to implement our definition already exist for experimental
applications. As an informative example, we demonstrate the power of our
operational definition using Z+jet and dijet samples, illustrating that pure
quark and gluon distributions and fractions can be successfully extracted in a
fully well-defined manner.Comment: 38 pages, 10 figures, 1 table; v2: updated to match JHEP versio
Irreducibility of the Gorenstein loci of Hilbert schemes via ray families
We analyse the Gorenstein locus of the Hilbert scheme of points on
i.e. the open subscheme parameterising zero-dimensional
Gorenstein subschemes of of degree . We give new sufficient
criteria for smoothability and smoothness of points of the Gorenstein locus. In
particular we prove that this locus is irreducible when and find its
components when . The proof is relatively self-contained and it does
not rely on a computer algebra system. As a by--product, we give equations of
the fourth secant variety to the -th Veronese reembedding of
for .Comment: v4: final. v2: expanded proof of Theorems A and B. 33 pages, comments
welcome
Classification of Differential Calculi on U_q(b+), Classical Limits, and Duality
We give a complete classification of bicovariant first order differential
calculi on the quantum enveloping algebra U_q(b+) which we view as the quantum
function algebra C_q(B+). Here, b+ is the Borel subalgebra of sl_2. We do the
same in the classical limit q->1 and obtain a one-to-one correspondence in the
finite dimensional case. It turns out that the classification is essentially
given by finite subsets of the positive integers. We proceed to investigate the
classical limit from the dual point of view, i.e. with ``function algebra''
U(b+) and ``enveloping algebra'' C(B+). In this case there are many more
differential calculi than coming from the q-deformed setting. As an
application, we give the natural intrinsic 4-dimensional calculus of
kappa-Minkowski space and the associated formal integral.Comment: 22 pages, LaTeX2e, uses AMS macro
Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials
The concept of cyclic tridiagonal pairs is introduced, and explicit examples
are given. For a fairly general class of cyclic tridiagonal pairs with
cyclicity N, we associate a pair of `divided polynomials'. The properties of
this pair generalize the ones of tridiagonal pairs of Racah type. The algebra
generated by the pair of divided polynomials is identified as a higher-order
generalization of the Onsager algebra. It can be viewed as a subalgebra of the
q-Onsager algebra for a proper specialization at q the primitive 2Nth root of
unity. Orthogonal polynomials beyond the Leonard duality are revisited in light
of this framework. In particular, certain second-order Dunkl shift operators
provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of
irreducibility is added; v3: version for Linear Algebra and its Applications,
one assumption added in Appendix about eq. (A.2
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