Miracle-dz/UniFAR-FTUs huggingface dataset:
On the appearance of Eisenstein series through degeneration

Let G be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = G \ H be the associated finite volume hyperbolic Riemann surface. If ? is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If ? is hyperbolic, then, following ideas due to Kudla�Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If ? G corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.
Continued fraction enhancement of Bayesian computing

The agéd number theoretic concept of continued fractions can enhance certain Bayesian computations. The crux of this claim is due to continued fraction representations of numerically challenging special function ratios that arise in Bayesian computing. Continued fraction approximation via Lentz's Algorithm often leads to efficient and stable computation of such quantities. Copyright © 2012 John Wiley & Sons, Ltd.
Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline-Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.
A thorough look at the (in)stability of piston-theoretic beams

We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms provided through the flow effects. Three different configurations are considered: a clamped panel, a hinged panel, and a flag (a cantilever clamped at the leading edge, free at the trailing edge). After providing the functional framework for the dynamics, recent results on well-posedness and long-time behavior of the associated dynamical system for solutions are presented. Having provided this theoretical context, in-depth numerical stability analyses are provided, focusing both at the onset of flow-induced instability (flutter), and qualitative properties of the post-flutter dynamics across configurations. Modal approximations are utilized, as well as finite difference schemes.
THE CHOLESKY UPDATE FOR UNCONSTRAINED OPTIMIZATION

It is well known that the unconstrained Optimizatio n often arises in economies, finance, trade, law, meteorology, medicine, biology, chemist ry, engineering, physics, education, history, sociology, psychology, and so on. The classical Unc onstrained Optimization is based on the Updating of Hessian matrix and computed of its inve rse which make the solution is very expensive. In this work we will updating the LU fac tors of the Hessian matrix so we don’t need to compute the inverse of Hessian matrix, so called th e Cholesky Update for unconstrained optimization. We introduce the convergent of the up date and report our findings on several standard problems, and make a comparison on its per formance with the well-accepted BFGS update.
A stable partitioned FSI algorithm for incompressible flow and elastic solids

A stable partitioned algorithm for coupling incompressible flows with compressible elastic solids is described. This added-mass partitioned (AMP) scheme requires no sub-iterations, can be made fully second- or higher-order accurate, and remains stable even in the presence of strong added-mass effects. The approach extends the scheme of Banks et al. [1,2] for compressible flow, and uses Robin (mixed) boundary conditions with the fluid and solid solvers at the interface. The AMP Robin conditions are derived from a local characteristic decomposition in the solid at the interface. Two forms of the Robin conditions are derived depending on whether the fluid equations are advanced with a fractional-step method or not. A normal mode analysis for a discretization of an FSI model problem is performed to show that the new AMP algorithm is stable for any ratio of the solid and fluid densities, including the case of very light solids when the added-mass effects are large. In contrast, it is shown that a traditional partitioned algorithm involving a Dirichlet-Neumann coupling for the same FSI problem is formally unconditionally unstable for any ratio of densities. Exact traveling wave solutions are derived for three FSI model problems of increasing complexity, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy solids.
A Dynamical Mechanism for the Big Bang and Non-Regularizability for $w=1$

We consider a contracting universe and its transition to expansion through the big bang singularity with a time varying equation of state $w$, where $w$ approaches $1$ as the universe contracts to the big bang. We show that this singularity is non-regularizable. That is, there is no unique extension of the physical quantities after the transition, but rather infinitely many. This is entirely different from the case of $w > 1$ studied in \cite{Xue:2014}, where $w$ approaches a constant value $w_c > 1$ as the universe contracts. In that case a continuous transition through the big bang to yield a unique extension was possible only for a discrete set of $w_c$ satisfying coprime conditions. We also show that there exists another time variable, $N$, at the big bang singularity itself, at $t=0$, where $w$, varies as a function of $N$. This defines an {\em extended big bang state}. Within it, $H$ is infinity. In the extended state, $w$ varies from a universe dominated by the cosmological constant to $1$. After $w$ reaches $1$ then the big bang occurs and time $t >0$ resumes. This gives a dynamical mechanism for the big bang that is mathematically complete as a function of $N$. Dynamical systems methods are used with classical modeling.
Comparative non-invasive T1w/T2w myelin mapping of the night monkey: implications for cortical adaptation to a nocturnal niche environment

Night monkeys (Aotus) are the only genus of monkeys within the Simian lineage that successfully occupy the nocturnal niche environment. Their behavior is supported by their sensory organs’ unique morphological features; however, little is known about their evolutionary adaptations in sensory regions of the cerebral cortex. Here, we investigate this question by exploring the cortical organization of night monkeys using high-resolution in-vivo brain MRI and comparative cortical-surface T1w/T2w myeloarchitectonic mapping. Our results show that the night monkey cerebral cortex has a similar organization of cortical myelin distribution compared to the diurnal macaque and marmoset monkeys. T1w/T2w myelin and its gradient allowed us to parcellate high myelin areas, middle temporal complex (MT + MST) and auditory cortex, and low myelin area, Brodmann area 7 (BA7) in three species, despite the difference in gyri�cations. Relative to the total cortical surface area, those of MT + MST and the auditory cortex are signi�cantly larger in night monkeys than diurnal monkeys. In contrast, the relative area of BA7 is expanded in proportion to brain size across species. We propose that the selective expansion of sensory areas dedicated to visual motion and auditory processing in night monkeys may re�ect cortical adaptations to the nocturnal niche environment.
On truncated Weyl modules

Abstract We study structural properties of truncated Weyl modules. A truncated Weyl module is a local Weyl module for , where is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced . We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov–Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari–Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that related to Demazure flags and chains of inclusions of truncated Weyl modules.
Modified equations for variational integrators applied to Lagrangians linear in velocities

Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.
Analytic solutions for the approximated Kantorovich mass transfer problems by $p$-Laplacian approach

This manuscript discusses the approximation of a global maximizer of the Kantorovich mass transfer problem through the approach of $p$-Laplacian equation. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Kantorovich problem will be demonstrated.
Realization of level one representations of $U\sb q(\hat{\mathfrak {g}})$ at a root of unity

Using vertex operators, we construct explicitly Lusztig's $\mathbb Z[q, q^{-1}]$-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the q-dimension is still given by the Weyl-Kac character formula. As a consequence we also answer the corresponding question of realizing the affine Kac-Moody Lie algebras of simply laced type at level one in finite characteristic.
Automatic Speaker Recognition Based on Pitch Contours

The results of a study aimed at finding the importance of pitch for automatic speaker recognition are presented. Pitch contours were obtained for 60 utterances, each approximately 2‐sec in duration, of 10 female speakers. A data‐reduction procedure based on the Karhunen‐Loeve representation was found effective in representing the pitch information in each contour in a 20‐dimensional space. The data were divided into two portions; one part was used to design the speaker recognition system, while the other part was used to test the effectiveness of the design. The 20‐dimensional vectors representing the pitch contours of the design set were linearly transformed so that the ratio of interspeaker to intraspeaker variance in the transformed space was maximum. A reference utterance was formed for each speaker by averaging the transformed vectors of that speaker. The test utterance was assigned to the speaker corresponding to the reference utterance with the smallest Euclidean distance in the transformed space. The percentage of correct identifications using the above procedure was found to be 97%. The recognition rate increased to 98% with the addition of duration as an independent measure.
A uniform stability principle for dual lattices

We prove a highly uniform stability or “almost-near” theorem for dual lattices of lattices L ⊆ Rn. More precisely, we show that, for a vector x from the linear span of a lattice L ⊆ Rn, subject to λ1(L) ≥ λ > 0, to be ε-close to some vector from the dual lattice L of L, it is enough that the inner products ux are δ-close (with δ < 1/3) to some integers for all vectors u ∈ L satisfying ‖u‖ ≤ r, where r > 0 depends on n, λ, δ and ε, only. This generalizes an analogous result proved for integral lattices in [15]. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis. Informally, a property of objects of certain kind is “stable” if objects “almost satisfying” this property are already “close” to objects having the property. For that reason results establishing such a stability are frequently referred as a “almostnear” principles or theorems. Making precise the vague notions “almost satisfying” and “close” various rigorous notions of stability can be obtained. The study of stability of functional equations originates from a question about the stability of additive functions R → R and, more generally, of homomorphisms G→ H between metrizable topological groups, asked by Ulam, cf. [18], [21], [22]. Since that time Ulam’s type stability, modified in various ways, was studied for various (systems of) functional equations — see, e.g., Rassias [19], Székelyhidi [20]. A systematic and general approach to this topic in the realm of compact Hausdorff topological spaces, using nonstandard analysis was developed by Anderson [1]. The study of stability of the homomorphy property with respect to the compact-open topology was commenced by the second of the present authors [23], [24], [25]. The survey article by Boualem and Brouzet [4] reflects some recent development. In the present paper we will prove the stability theorem for dual lattices stated in the abstract, as well as some closely related results. Typically, such a stability result would be formulated in a weaker form, namely that every vector x from the linear span of L, behaving almost like a vector from the dual lattice L of L in the sense that all its inner products u x = u1x1 + . + unxn with vectors u from a “sufficiently big” subset of L are “sufficiently close” to some integer, is already “arbitrarily close” to a vector y ∈ L. Below is the precise formulation arising from this account. In it span(L) denotes the linear subspace of R generated by L, |a| Z = minc∈Z |a− c| = min ( a − ⌊a⌋, ⌈a⌉ − a ) denotes the distance of the real number a from the set of all integers Z, and ‖x‖ = √xx is the euclidean norm, induced by the usual inner (scalar) product x y on R. 2010 Mathematics Subject Classification. Primary 11H06; Secondary 11H31, 11H60, 03H05.
Counterexamples to ferromagnetic ordering of energy levels

The Heisenberg ferromagnet has symmetry group SU(2). The property known as ferromagnetic ordering of energy levels (FOEL) states that the minimum energy eigenvalue among eigenvectors with total spin s is monotone decreasing as a function of s. While this property holds for certain graphs such as open chains, in this note we demonstrate some counterexamples. We consider the spin 1/2 model on rings of length 2n for n = 2, 3, …, 8, and show that the minimum energy among all spin singlets is less than or equal to the minimum energy among all spin triplets, which violates FOEL. This also shows some counterexamples to the “Aldous ordering” for the symmetric exclusion process. We also review some of the literature related to these examples.
Kalimullin Pair and Semicomputability in $\alpha$-Computability Theory

We generalize some results on semicomputability by Jockusch \cite{jockusch1968semirecursive} to the setting of $\alpha$-Computability Theory. We define an $\alpha$-Kalimullin pair and show that it is definable in the $\alpha$-enumeration degrees $\mathcal{D}_{\alpha e}$ if the projectum of $\alpha$ is $\alpha^*=\omega$ or if $\alpha$ is an infinite regular cardinal. Finally using this work on $\alpha$-semicomputability and $\alpha$-Kalimullin pairs we conclude that every nontrivial total $\alpha$-enumeration degree is a join of a maximal $\alpha$-Kalimullin pair if $\alpha$ is an infinite regular cardinal.
Vassiliev invariants for pretzel knots

We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on g + 1 parameters n1,…,ng+1. These invariants are symmetric polynomials in n1,…,ng+1 whose degree coincide with their order. We also discuss their topological and integer-valued properties.
Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory

We further explore the implications of our framework in [arXiv:1301.1977, arXiv:1309.4775], and physically derive, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent, (i) a 5d AGT correspondence for any compact Lie group, (ii) a 5d and 6d AGT correspondence on ALE space of type ADE, and (iii) identities between the ordinary, q-deformed and elliptic affine W-algebras associated with the 4d, 5d and 6d AGT correspondence, respectively, which also define a quantum geometric Langlands duality and its higher analogs formulated by Feigin-Frenkel-Reshetikhin in [3,4]. As an offshoot, we are led to the sought-after connection between the gauge-theoretic realization of the geometric Langlands correspondence by Kapustin-Witten [5,6] and its algebraic CFT formulation by Beilinson-Drinfeld [7], where one can also understand Wilson and 't Hooft-Hecke line operators in 4d gauge theory as monodromy loop operators in 2d CFT, for example. In turn, this will allow us to argue that the higher 5d/6d analog of the geometric Langlands correspondence for simply-laced Lie (Kac-Moody) groups G (G_k), ought to relate the quantization of circle (elliptic)-valued G Hitchin systems to circle/elliptic-valued ^LG (^LG_k)-bundles over a complex curve on one hand, and the transfer matrices of a G (G_k)-type XXZ/XYZ spin chain on the other, where ^LG is the Langlands dual of G. Incidentally, the latter relation also serves as an M-theoretic realization of Nekrasov-Pestun-Shatashvili's recent result in [8], which relates the moduli space of 5d/6d supersymmetric G (G_k)-quiver SU(K_i) gauge theories to the representation theory of quantum/elliptic affine (toroidal) G-algebras.
DEGREE CONES AND MONOMIAL BASES OF LIE ALGEBRAS AND QUANTUM GROUPS

Abstract We provide ℕ-filtrations on the negative part Uq ( $\mathfrak{n}$ −) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$ , such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$ −. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$ -module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$ .
Robust fault detection filter using linear matrix inequalities’ approach for shaft crack diagnosis

Detecting cracks in rotating shafts is a challenging problem when using vibration-based diagnostics. This is due to the fact that a localized crack has a minimal influence on the global vibration response of the system. To increase sensitivity and reliability, the vibration response needs to be coupled with additional sources of information such as a mathematical model of the machine. Modern control theory techniques offer system-level mathematical models for both control and diagnostics. Focusing on the latter, a new and promising approach involves the use of unknown input observers. Such observers can be designed to employ robust fault detection filters (RFDFs) for isolating fault signatures while reducing the influence of real-world disturbances and noise. For the present study, a modified design procedure coupled with robust fault detection is utilized for shaft crack detection. The filter is designed using the linear matrix inequalities (LMI) technique. The LMI approach is applied to obtain the solution of the mixed H−/H∞ optimization problem, which arises during the synthesis of the RFDF. By reformulating the LMI conditions, the proposed RFDF design procedure is simplified and thus requires less iteration steps to find the optimal solution. A new feature of the present approach involves the application of the rigid finite element method for the formulation of the mathematical model of the rotor and the shaft crack. The numerical and experimental results confirm the advantages of the designed robust fault detection filter and its ability to detect shaft cracks. The filter is minimally sensitive to measurement noise while allowing for the identification of shallow cracks (2% or 5% deep). The cracks are manifested through the observance of very subtle vibration response changes. The results also confirm the effectiveness and accuracy of the rigid finite element modeling concerning the cracked rotor.
Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations [Formula: see text] of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] for symmetric bipartitions [Formula: see text] are infinite dimensional for all parameters [Formula: see text]. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text] arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text].
Classifying homogeneous cellular ordinal balleans up to coarse equivalence

For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $cof(X)=cof(Y)$ and $cov^\sharp(X)=cov^\sharp(Y)$ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of the cardinals $cof(X)$ and $cov^\sharp(X)$. For every limit ordinal $\gamma$ we shall define a ballean $2^{<\gamma}$ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality $cf(\gamma)$ plays a role analogous to the role of the Cantor cube $2^{\kappa}$ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to $2^{<\gamma}$. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube $2^\omega$.