Miracle-dz/UniFAR-FTUs huggingface dataset:
Hopf Algebras in Combinatorics

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some Hopf algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer Hopf algebra of permutations. Among the results surveyed are the Littlewood-Richardson rule and other symmetric function identities, Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, and Hazewinkel's polynomial freeness of QSym. The notes are written with a graduate student reader in mind, being mostly self-contained but requiring a good familiarity with multilinear algebra and -- for the representation-theory applications -- basic group representation theory. Comments are welcome, as there will be updates.
Normalizers inside amalgamated free product von Neumann algebras

Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group \Gamma_1 * \Gamma_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer N_M(A)'' of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M_1 *_B M_2. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in crossed products by \Gamma_1 * \Gamma_2.
Extending partial isometries of generalized metric spaces

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in\mathcal{K}$ there is some $B\in\mathcal{K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used to show that the isometry group of the Urysohn space over $\mathcal{R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.
The core dominance parameter and Fermi detection of extragalactic radio sources

By cross-correlating an archive sample of 542 extragalactic radio sources with the Fermi-LAT Third Source Catalog (3FGL), we have compiled a sample of 80 γ-ray sources and 462 non-Fermi sources with available core dominance parameter (RCD), and core and extended radio luminosity; all the parameters are directly measured or derived from available data in the literature. We found that RCD has significant correlations with radio core luminosity, γ-ray luminosity and γ-ray flux; the Fermi sources have on average higher RCD than non-Fermi sources. These results indicate that the Fermi sources should be more compact, and the beaming effect should play a crucial role in the detection of γ-ray emission. Moreover, our results also show Fermi sources have systematically larger radio flux than non-Fermi sources at fixed RCD, indicating larger intrinsic radio flux in Fermi sources. These results show a strong connection between radio and γ-ray flux for the present sample and indicate that the non-Fermi sources are likely due to the low beaming effect, and/or the low intrinsic γ-ray flux. This supports a scenario that has been published in the literature: a co-spatial origin of the activity for the radio and γ-ray emission, suggesting that the origin of the seed photons for the high-energy γ-ray emission is within the jet.
Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are the necessary conditions for the Chekhov-Eynard-Orantin topological recursion.
Kinematic Wave Models of Network Vehicular Traffic

The kinematic wave theory, originally proposed by (Lighthill and Whitham, 1955b; Richards, 1956), has been a good candidate for studying vehicular traffic. In this dissertation, we study kinematic wave models of network traffic, which are expected to be theoretically rigorous, numerically reliable, and computationally efficient. For traffic systems with inhomogeneous links, merges, diverges, or mixed-type vehicles, we study the kinematic waves in their Riemann solutions and develop numerical solution methods of the Godunov type and the supply-demand type. For a network traffic system, we propose a multi-commodity kinematic wave (MCKW) model and an implementation of it. The model observes First-In-First-Out principle in the order of a time interval and is numerically convergent. Further, we apply this simulation model to study equilibrium states and periodic waves in road networks. Finally, we summarize our work and discuss future research directions.
On the rigidity of lagrangian products

Motivated by work of the first author, this paper studies symplectic embedding problems of lagrangian products that are sufficiently symmetric. In general, lagrangian products arise naturally in the study of billiards. The main result of the paper is the rigidity of a large class of symplectic embedding problems of lagrangian products in any dimension. This is achieved by showing that the lagrangian products under consideration are symplectomorphic to toric domains, and by using the Gromov width and the cube capacity introduced by Gutt and Hutchings to obtain rigidity.
A method of state-selective transfer of atoms between microtraps based on the Franck–Condon principle

We present a method of transferring a cold atom between spatially separated microtraps by means of a Raman transition between the ground motional states of the two traps. The intermediate states for the Raman transition are the vibrational levels of a third microtrap, and we determine the experimental conditions for which the overlap of the wavefunctions leads to an efficient transfer. There is a close analogy with the Franck–Condon principle in the spectroscopy of molecules. The spin-dependent manipulation of neutral atoms in microtraps has important applications in quantum information processing. We also show that, starting with several atoms, precisely one atom can be transferred to the final potential well hence giving deterministic preparation of single atoms.
A new model for the Milky Way bar

Abstract We use Schwarzschild's orbit-superposition technique to construct self-consistent models of the Galactic bar. Using χ2 minimisation, we find that the best-fit Galactic bar model has a pattern speed Ωp=60 km s−1 kpc−1, disk mass Md=1.0×1011M⊙ and bar angle θbar=20° for an adopted bar mass Mbar=2×1010M⊙. The model can reproduce not only the three-dimensional and projected density distributions but also velocity and velocity dispersion data from the BRAVA survey. We also predict the proper motions in the range l=[−12°,12°], b=[−10°,10°], which appear to be higher than observations in the longitudinal direction. The model is stable within a timescale of 0.5 Gyr, but appears to deviate from steady-state on longer timescales. Our model can be further tested by future observations such as those from GAIA.
An algebraic Sato-Tate group and Sato-Tate conjecture

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to the Mumford-Tate group, but the group of components carries additional arithmetic information. We then check that in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-Tate group can also be described explicitly in terms of endomorphisms. In particular, we cover all abelian varieties (not necessarily absolutely simple) of dimension at most 3; this result figures prominently in the analysis of Sato-Tate groups for abelian surfaces given recently by Fite, Kedlaya, Rotger, and Sutherland.
Generalized multiplicative domains and quantum error correction

Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map, and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi's characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.
Well-Being Therapy and Lifestyle Intervention in Type 2 Diabetes: A Pilot Randomized Controlled Trial

ABSTRACT Objective This pilot randomized controlled trial evaluates the preliminary efficacy of a 4-month well-being therapy (WBT) and lifestyle intervention among adults with type 2 diabetes and overweight/obesity. Methods Fifty-eight patients were recruited from two outpatient clinics and randomized to receive the WBT-lifestyle intervention or the lifestyle intervention alone. Data were collected at baseline (T0), immediate postintervention (T1), 6-month follow-up (T2), and 12-month follow-up (T3). Primary efficacy outcomes included changes in weight, psychological distress, and well-being, whereas secondary efficacy outcomes included changes in lifestyle and physiological parameters. Results Compared with the lifestyle-alone intervention, the WBT-lifestyle intervention showed greater improvements in depression (p = .009, d = −0.6), hostility (p = .018, d = −0.6), and personal growth (p = .026, d = 0.5) at T1, in self-reported physical activity at T2 (p = .013, d = 0.7) and T3 (p = .040, d = 0.5), and in triglycerides (p = .019, d = −1.12) at T3. There were no differences between treatment groups in weight and other physiological parameters. Conclusions These findings suggest that WBT may be a valuable addition to lifestyle interventions for improving short-term psychological outcomes and promoting long-term healthy changes in physical activity, with a potential impact on physiological outcomes. Trial Registration:ClinicalTrials.gov identifier: NCT03609463.
On some determinant and matrix inequalities with a geometrical flavour

In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour. We first examine some inequalities which place work of Macbeath [13] in a more general setting and also relate to recent work of Gressman [8]. In particular, we establish optimisers for these determinant inequalities. We then use these inequalities to establish our main theorem which gives a geometric inequality of matrix type which improves and extends some inequalities of Christ in [5].
Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations

The low Mach number limit for the multi-dimensional full magnetohydrodynamic (MHD) equations, in which the effect of thermal conduction is taken into account, is rigorously justified within the framework of classical solutions with small density and temperature variations. Moreover, we show that for a sufficiently small Mach number, the compressible MHD equations admit a smooth solution on the time interval where the smooth solution of the incompressible MHD equations exists. In addition, the low Mach number limit for the ideal MHD equations with small entropy variation is also investigated. The convergence rates are obtained in both cases.
Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

An \textit{ideal} of $N$-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for $N$-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_N$ of all unitary equivalence classes of such $N$-tuples are established and certain ideals in $CDD_N$ are distinguished. It is proved that infinite operations in $CDD_N$ may be reconstructed from the direct sum operation of a pair. \textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense) uniqueness is established. The issue of classification of ideals in $CDD_N$ (up to isomorphism) is discussed. A model for $CDD_N$ is described and its concrete realization is presented. A new partial order of $N$-tuples of operators is introduced and its fundamental properties are established. Extremal importance of unitary disjointness of $N$-tuples and the way how it `tidies up' the structure of $CDD_N$ are emphasized.
Torsion, torsion length and finitely presented groups

Abstract We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some C′⁢(1/6){C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is C′⁢(1/6){C^{\prime}(1/6)}, and thus word-hyperbolic and virtually torsion-free.
From Fock's Transformation to de Sitter Space

As in Deformed Special Relativity, we showed recently that the Fock coordinate transformation can be derived from a new deformed Poisson brackets. This approach allowed us to establish the corresponding momentum transformation which keeps invariant the four dimensional contraction $p_{\mu} x^{\mu} $. From the resulting deformed algebra, we construct in this paper the corresponding first Casimir. After first quantization, we show by using the Klein-Gordon equation that the spacetime of the Fock transformation is the de Sitter one. As we will see, the invariant length representing the universe radius in the spacetime of Fock's transformation is exactly the radius of the embedded hypersurface representing the de Sitter spacetime.
On String Integrability: A Journey through the Two-Dimensional Hidden Symmetries in the AdS/CFT Dualities

One of the main topics in the modern String Theory are the AdS/CFT dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, that is, the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality. We review the fundamental concepts and properties of integrability in two-dimensional 𝜎-models and in the AdS/CFT context. The first part is focused on the AdS5/CFT4 duality, especially the classical and quantum integrability of the type IIB superstring on AdS5×S5 which is discussed in both pure spinor and Green-Schwarz formulations. The second part is dedicated to the AdS4/CFT3 duality with particular attention to the type IIA superstring on AdS4×ℂ𝑃3 and its integrability. This review is based on the author's PhD thesis discussed at Uppsala University the 21st September 2009.
On the Modular Completion of Certain Generating Functions

We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [27] on traces of singular moduli.
A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES

The author (Appl. Math. Comput. 218:860-865, 2011) introduced a new fractional integral operator given by, � I � a+f � (x) = � 1 � which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.
Dirac Operator on Complex Manifolds and Supersymmetric Quantum Mechanics

We explore a simple supersymmetric quantum mechanics (SQM) model describing the motion over complex manifolds in external gauge fields. The nilpotent supercharge Q of the model can be interpreted as a (twisted) exterior holomorphic derivative, such that the model realizes the twisted Dolbeault complex. The sum can be interpreted as the Dirac operator: the standard Dirac operator if the manifold is Kahler and the Dirac operator involving certain particular extra torsions for a generic complex manifold. Focusing on the Kahler case, we give new simple physical proofs of the two mathematical facts: (i) the equivalence of the twisted Dirac and twisted Dolbeault complexes and (ii) the Atiyah–Singer theorem.
Algebraic arctic curves in the domain-wall six-vertex model

The arctic curve, i.e. the spatial curve separating ordered (or ‘frozen’) and disordered (or ‘temperate) regions, of the six-vertex model with domain-wall boundary conditions is discussed for the root-of-unity vertex weights. In these cases, the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve. Some interesting examples are discussed in detail. The upper bound on the maximal degree of the equation in a generic root-of-unity case is obtained.
Combinatorial Lefschetz theorems beyond positivity

Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we introduce a version of the Kahler package beyond positivity, allowing us to prove the hard Lefschetz theorem for toric varieties (and beyond) even when the ample cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a non-degeneracy relation at torus-invariant subspaces, allowing us to state and prove a generalization of theorems of Hall and Laman in the setting of toric varieties and, more generally, the face rings of Hochster, Reisner and Stanley. This has several applications: - We fully characterize the possible face numbers of simplicial rational homology spheres, resolving the $g$-conjecture of McMullen in full generality and generalizing Stanley's earlier proof for simplicial polytopes. The same methods also verify a conjecture of Kuhnel: if $M$ is a triangulated closed $(d-1)$-manifold on $n$ vertices, then \[\binom{d+1}{j}\mathrm{b}_{j-1}(M)\ \le \ \binom{n-d+j-2}{j}\ \quad \text{for}\ 1\le j\le \frac{d}{2}. \] - We prove that for a simplicial complex that embeds into $\mathbb{R}^{2d}$, the number of $d$-dimensional simplices exceeds the number of $(d-1)$-dimensional simplices by a factor of at most $d+2$. This generalizes a result going back to Descartes and Euler, and resolves the Grunbaum-Kalai-Sarkaria conjecture. We obtain from this a generalization of the celebrated crossing lemma: For a map of a simplicial complex $\Delta$ into $\mathbb{R}^{2d}$, the number of pairwise intersections of $d$-simplices is at least \[\frac{f_d^{d+2}(\Delta)}{(d+3)^{d+2}f_{d-1}^{d+1}(\Delta)}\] provided $f_d(\Delta)> (d+3)f_{d-1}(\Delta)$.
On the mean order of connected induced subgraphs of block graphs

The average order of the connected induced subgraphs of a graph $G$ is called the mean connected induced subgraph (CIS) order of $G$. This is an extension of the mean subtree order of a tree, first studied by Jamison. In this article, we demonstrate that among all connected block graphs of order $n$, the path $P_n$ has minimum mean CIS order. This extends a result of Jamison from trees to connected block graphs, and supports the conjecture of Kroeker, Mol, and Oellermann that $P_n$ has minimum mean CIS order among all connected graphs of order $n$.
Entanglement spectra of q-deformed higher spin VBS states

We calculate the reduced density matrix of a block of integer spin-S’s in a q-deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of finite-size corrections on the entanglement spectrum and entropy in this generalized VBS model.
Recent lipoprotein(a) trials

Purpose of review Lipoprotein(a) (Lp(a)) is a genetically determined independent risk factor for cardiovascular disease and calcific aortic stenosis; thus, serum levels are minimally affected by conventional treatments for hypercholesterolemia and hypertriglyceridemia. New RNA therapies directly targeting Lp(a) have demonstrated efficacy in decreasing serum levels. Several recent trials have demonstrated efficacy and safety of these RNA therapeutics. Recent findings Single-stranded antisense oligonucleotides (ASOs) and small interfering RNA (siRNA) are two classes of RNA-targeted therapeutics that specifically target the LPA gene, which encodes for apolipoprotein(a), a dominant and rate-limiting component in the hepatic synthesis of Lp(a) particle. Pelacarsen (ASO), olpasiran (siRNA) and SLN360 (siRNA) are novel drugs that have demonstrated efficacy in lowering Lp(a) levels and excellent safety profiles. Summary Lp(a) is an independent risk factor for cardiovascular disease. RNA-directed therapies, pelacarsen, olpasiran and SLN360, have shown efficacy in dramatically lowering serum Lp(a) levels. Outcomes data will be the next frontier of Lp(a) trials.