34,694 research outputs found
Rank-based linkage I: triplet comparisons and oriented simplicial complexes
Rank-based linkage is a new tool for summarizing a collection of objects
according to their relationships. These objects are not mapped to vectors, and
``similarity'' between objects need be neither numerical nor symmetrical. All
an object needs to do is rank nearby objects by similarity to itself, using a
Comparator which is transitive, but need not be consistent with any metric on
the whole set. Call this a ranking system on . Rank-based linkage is applied
to the -nearest neighbor digraph derived from a ranking system. Computations
occur on a 2-dimensional abstract oriented simplicial complex whose faces are
among the points, edges, and triangles of the line graph of the undirected
-nearest neighbor graph on . In steps it builds an
edge-weighted linkage graph where
is called the in-sway between objects and . Take to be
the links whose in-sway is at least , and partition into components of
the graph , for varying . Rank-based linkage is a
functor from a category of out-ordered digraphs to a category of partitioned
sets, with the practical consequence that augmenting the set of objects in a
rank-respectful way gives a fresh clustering which does not ``rip apart`` the
previous one. The same holds for single linkage clustering in the metric space
context, but not for typical optimization-based methods. Open combinatorial
problems are presented in the last section.Comment: 37 pages, 12 figure
Fourier Coefficients of Weight Zero Mixed False Modular Forms
In this paper we employ the Circle Method to give exact formulae for Fourier
coefficients of an infinite family of weight zero mixed false modular forms
using and extending the techniques of Bringmann and Nazaroglu as well as
Rademacher. To do so we additionally provide a bound on a Kloosterman sum of
modulus .Comment: 53 pages, 3 figures, 1 table; Comments welcome
The Metaverse: Survey, Trends, Novel Pipeline Ecosystem & Future Directions
The Metaverse offers a second world beyond reality, where boundaries are
non-existent, and possibilities are endless through engagement and immersive
experiences using the virtual reality (VR) technology. Many disciplines can
benefit from the advancement of the Metaverse when accurately developed,
including the fields of technology, gaming, education, art, and culture.
Nevertheless, developing the Metaverse environment to its full potential is an
ambiguous task that needs proper guidance and directions. Existing surveys on
the Metaverse focus only on a specific aspect and discipline of the Metaverse
and lack a holistic view of the entire process. To this end, a more holistic,
multi-disciplinary, in-depth, and academic and industry-oriented review is
required to provide a thorough study of the Metaverse development pipeline. To
address these issues, we present in this survey a novel multi-layered pipeline
ecosystem composed of (1) the Metaverse computing, networking, communications
and hardware infrastructure, (2) environment digitization, and (3) user
interactions. For every layer, we discuss the components that detail the steps
of its development. Also, for each of these components, we examine the impact
of a set of enabling technologies and empowering domains (e.g., Artificial
Intelligence, Security & Privacy, Blockchain, Business, Ethics, and Social) on
its advancement. In addition, we explain the importance of these technologies
to support decentralization, interoperability, user experiences, interactions,
and monetization. Our presented study highlights the existing challenges for
each component, followed by research directions and potential solutions. To the
best of our knowledge, this survey is the most comprehensive and allows users,
scholars, and entrepreneurs to get an in-depth understanding of the Metaverse
ecosystem to find their opportunities and potentials for contribution
Dimension-8 SMEFT Analysis of Minimal Scalar Field Extensions of the Standard Model
We analyze the constraints obtainable from present data using the Standard
Model Effective Field Theory (SMEFT) on extensions of the Standard Model with
additional electroweak singlet or triplet scalar fields. We compare results
obtained using only contributions that are linear in dimension-6 operator
coefficients with those obtained including terms quadratic in these
coefficients as well as contributions that are linear in dimension-8 operator
coefficients. We also implement theoretical constraints arising from the
stability of the electroweak vacuum and perturbative unitarity. Analyzing the
models at the dimension-8 level constrains scalar couplings that are not
bounded at the dimension-6 level. The strongest experimental constraints on the
singlet model are provided by Higgs coupling measurements, whereas electroweak
precision observables provide the strongest constraints on the triplet model.
In the singlet model the present di-Higgs constraints already play a
significant role. We find that the current constraints on model parameters are
already competitive with those anticipated from future di- and tri-Higgs
measurements. We compare our results with calculations in the full model,
exhibiting the improvements when higher-order SMEFT terms are included. We also
identify regions in parameter space where the SMEFT approximation appears to
break down. We find that the combination of current constraints with the
theoretical bounds still admits regions where the SMEFT approach is not valid,
particularly for lower scalar boson masses.Comment: 66 Pages, 14 Figures, 4 Table
Can you hear your location on a manifold?
We introduce a variation on Kac's question, "Can one hear the shape of a
drum?" Instead of trying to identify a compact manifold and its metric via its
Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a
point on the manifold, up to symmetry, from its pointwise counting function
where here and form an orthonormal
basis for . This problem has a physical interpretation. You are placed
at an arbitrary location in a familiar room with your eyes closed. Can you
identify your location in the room by clapping your hands once and listening to
the resulting echos and reverberations?
The main result of this paper provides an affirmative answer to this question
for a generic class of metrics. We also probe the problem with a variety of
simple examples, highlighting along the way helpful geometric invariants that
can be pulled out of the pointwise counting function .Comment: 26 pages, 1 figur
Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma
A partition of is called a
-secluded partition if, for every ,
the ball intersects at most
members of . A goal in designing such secluded partitions is to
minimize while making as large as possible. This partition
problem has connections to a diverse range of topics, including deterministic
rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type
results.
In this work, we establish near-optimal relationships between and
. We show that, for any bounded measure partitions and for any
, it must be that . Thus, when is
restricted to , it follows that . This bound is tight up to log factors, as it is
known that there exist secluded partitions with and
. We also provide new constructions of secluded
partitions that work for a broad spectrum of and
parameters. Specifically, we prove that, for any
, there is a secluded partition with
and
. These new partitions are optimal up to
factors for various choices of and . Based
on the lower bound result, we establish a new neighborhood version of Sperner's
lemma over hypercubes, which is of independent interest. In addition, we prove
a no-free-lunch theorem about the limitations of rounding schemes in the
context of pseudodeterministic/replicable algorithms
Bayesian networks for disease diagnosis: What are they, who has used them and how?
A Bayesian network (BN) is a probabilistic graph based on Bayes' theorem,
used to show dependencies or cause-and-effect relationships between variables.
They are widely applied in diagnostic processes since they allow the
incorporation of medical knowledge to the model while expressing uncertainty in
terms of probability. This systematic review presents the state of the art in
the applications of BNs in medicine in general and in the diagnosis and
prognosis of diseases in particular. Indexed articles from the last 40 years
were included. The studies generally used the typical measures of diagnostic
and prognostic accuracy: sensitivity, specificity, accuracy, precision, and the
area under the ROC curve. Overall, we found that disease diagnosis and
prognosis based on BNs can be successfully used to model complex medical
problems that require reasoning under conditions of uncertainty.Comment: 22 pages, 5 figures, 1 table, Student PhD first pape
The -Number of Binomial Edge Ideals
The invariant -number was introduced very recently in the study
of Reed-Muller-type codes. Jaramillo and Villarreal (J Combin. Theory Ser. A
177:105310, 2021) initiated the study of the -number of edge
ideals. Inspired by their work, we take the initiation to study the
-number of binomial edge ideals in this paper. We discuss some
properties and bounds of the -number of binomial edge ideals. We
explicitly find the -number of binomial edge ideals locally at the
associated prime corresponding to the cutset . We show that the
-number of Knutson binomial edge ideals is less than or equal to
the -number of their initial ideals. Also, we classify all binomial
edge ideals whose -number is . Moreover, we try to relate the
-number with the Castelnuvo-Mumford regularity of binomial edge
ideals and give a conjecture in this direction
Eve, Adam and the Preferential Attachment Tree
We consider the problem of finding the initial vertex (Adam) in a
Barab\'asi--Albert tree process at large times.
More precisely, given , one wants to output a subset of vertices of so that the
initial vertex belongs to with probability at
least when is large. It has been shown by Bubeck, Devroye
& Lugosi, refined later by Banerjee & Huang, that one needs to output at least
and at most vertices to
succeed. We prove that the exponent in the lower bound is sharp and the key
idea is that Adam is either a ``large degree" vertex or is a neighbor of a
``large degree" vertex (Eve).Comment: 11 pages, comments are welcome
The Low-Scale Seesaw Solution to the MW and (g − 2) Anomalies
The recent CDF-II measurement of the W-boson mass shows a strong tension with the corresponding Standard Model prediction. Once active neutrino masses are explained in the context of the Low-Scale Seesaw mechanisms, this tension can be resolved. We investigate the possibility of explaining the longstanding muon anomalous magnetic moment anomaly within the same frameworks. We present a simplified extension of the Standard Model, accounting only for the second lepton generation, that describes a massive active neutrino and provides a combined solution to these anomalies. The model is renormalisable and introduces in the spectrum, beyond the sterile species of the Low-Scale Seesaw mechanism, only one pair of exotic vector-like leptons, doublets under the electroweak symmetry. We moreover discuss the extension of this model to the realistic three-family caseA.d.G. and L.M. acknowledge partial financial support by the Spanish Research Agency (Agencia Estatal de Investigación)
through the grant IFT Centro de Excelencia Severo Ochoa No CEX2020-
001007-S and by the grant PID2019-108892RB-I00 funded by MCIN/AEI/
10.13039/501100011033, by the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant
agreement No 860881-HIDDeN. The research of S.P. has received partial financial support by the Polish Science Centre (NCN), grant DEC2016/23/G/ST2/0430
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