Miracle-dz/UniFAR-FTUs huggingface dataset:
Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In the case of minimizing harmonic maps we go further, and prove that the singular set $S(f)$, which is well known to satisfy $\dim S(f)\leq n-3$, is in fact $n-3$-rectifiable with uniformly {\it finite} $n-3$-measure. An effective version of this allows us to prove that $|\nabla f|$ has estimates in $L^3_{weak}$, an estimate which is sharp as $|\nabla f|$ may not live in $L^3$. The above results are in fact just applications of a new class of estimates we prove on the {\it quantitative} stratifications $S^k_{\epsilon,r}(f)$ and $S^k_{\epsilon}(f)\equiv S^k_{\epsilon,0}(f)$. Roughly, $S^k_{\epsilon}\subseteq M$ is the collection of points $x\in S^k_\epsilon$ for which no ball $B_r(x)$ is $\epsilon$-close to being $k+1$-symmetric. We show that $S^k_\epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate $Vol(B_r\,S_\epsilon^k)\leq C r^{n-k}$. The proofs require a new $L^2$-subspace approximation theorem for stationary harmonic maps, as well as new $W^{1,p}$-Reifenberg and rectifiable-Reifenberg type theorems. These results are generalizations of the classical Reifenberg, and give checkable criteria to determine when a set is $k$-rectifiable with uniform measure estimates. The new Reifenberg type theorems may be of some independent interest.
On Transiso Graph

In this note, we define a new graph $\Gamma_d(G)$ on a finite group $G$, where $d$ is a divisor of $|G|$. The vertices of $\Gamma_d(G)$ are the subgroups of $G$ of order $d$ and two subgroups $H_1$ and $H_2$ of $G$ are said to be adjacent if there exists $S_i \in \mathcal{T}(G,H_i)$ $(i=1,2)$ such that $S_1 \cong S_2$. We shall discuss the completeness of $\Gamma_d(G)$ for various groups like finite abelian groups, dihedral groups and some finite $p$-groups.
On zero-divisors of semimodules and semialgebras

Abstract We prove McCoy’s property for the zero-divisors of polynomials in semirings, investigate the zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule M⁢[G]{M[G]} has very few zero-divisors if and only if the S-semimodule M does so. The concept of Auslander semimodules is introduced as well. Then we introduce Ohm–Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. Finally, we investigate the set of zero-divisors of McCoy semialgebras. We also introduce strong Krull primes for semirings and investigate their extension in semialgebras.
Speech perception, memory, and cognition: Implications for automatic speech recognition systems

Recent experiments suggest that people form perceptual representations for speech as a result of experience with a specific language. These representations are argued to take the form of perceptual maps stored in memory that specify distances between speech stimuli. This point will be illustrated with data from cross‐language experiments on American and Japanese adults using the segments /r/ and /l/. The results show that the underlying perceptual space encompassing /r/ and /l/ varies greatly in American and Japanese listeners—their perceptual maps are dramatically different. Cross‐cultural developmental studies show that language‐specific speech representations are present early in life and differ in infants reared in different linguistic environments. Further work suggests that the perceptual maps resulting from speech experience not only influence auditory speech perception, but influence auditory‐visual speech perception in adults and speech production in infants. Implications for automatic speech rec.
Multiple polylogarithms in weight 4

We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,. ,n_d}(z_1,.,z_d), in depth d>1 and weight \sum_i n_i=4 in terms of the classical tetralogarithm Li_4. In the process, we prove a statement conjectured by Goncharov which can be rephrased as writing the sum of iterated integrals I_{3,1}(V(x,y),z), where V(x,y) denotes a formal version of the five term relation for the dilogarithm, in terms of Li_4-terms (we need 122 such).
A step towards the cluster positivity conjecture

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.
Self-dual, self-Petrie-dual and Möbius regular maps on linear fractional groups

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality, self-Petrie-duality and Mobius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5,5). The final section includes an enumeration of the $PSL(2,q)$ maps for $q\le81$ and a list of all the $PSL(2,q)$ maps which have any of these special properties for $q\le49$.
Homotopical intersection theory, III: Multirelative intersection problems

This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the assumption that it can be deformed off of any proper subcollection in a homotopy coherent way. In a certain range of dimensions, ours is a complete obstruction to finding the desired deformation. We apply this machinery to embedding problems and to the study of linking phenomena.
A sufficient condition for the compactness of the $\overline{\partial}$-Neumann operator on high level forms

By establishing a unified estimate of the twisted Kohn-Morrey-H\"{o}rmander estimate and the $q$-pseudoconvex Ahn-Zampieri estimate, we discuss variants of Property $(P_q)$ of D. Catlin and Property $(\widetilde{P_q})$ of J. McNeal on the boundary of a smooth pseudoconvex domain in $\mathbb{C}^n$ for certain high levels of $L^2$-integrable forms. These variant conditions on the one side, imply compactness of the $\overline{\partial}$-Neumann operator on the associated domain, on the other side, are different from the classical Property $(P_q)$ of D. Catlin and Property $(\widetilde{P_q})$ of J. McNeal. As an application of our result, we show that if the Hausdorff $(2n-2)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\overline{\partial}$-Neumann operator $N_{n-1}$ is compact on $(0,n-1)$-level $L^2$-integrable forms. This result generalizes Boas and Sibony's results on $(0,1)$-level forms.
A single parameter Hermite–Padé series representation for Apéry’s constant

Inspired by the results of Rhin and Viola (2001), the purpose of this work is to elaborate on a series representation for $\zeta \left( 3\right)$ which only depends on one single integer parameter. This is accomplished by deducing a Hermite-Pad\'e approximation problem using ideas of Sorokin (1998). As a consequence we get a new recurrence relation for the approximation of $\zeta(3)$ as well as a corresponding new continued fraction expansion for $\zeta(3)$, which do no reproduce Ap\'ery's phenomenon, i.e., though the approaches are different, they lead to the same sequence of diophantine approximations to $\zeta \left( 3\right) $. Finally, the convergence rates of several series representations of $\zeta(3)$ are compared.
Quotients of primes in an algebraic number ring

It has been established on many occasions that the set of quotients of prime numbers is dense in the set of positive real numbers. More recently, it has been proved that the set of quotients of primes in the Gaussian integers is dense in the complex plane. In this article, we not only extend this result to any imaginary quadratic number ring, but also prove that the set of quotients of primes in any real quadratic number ring is dense in the set of real numbers. To conclude, we show how to extend these results to an arbitrary algebraic number ring.
Scoring Rules for Continuous Probability Distributions

Personal, or subjective, probabilities are used as inputs to many inferential and decision-making models, and various procedures have been developed for the elicitation of such probabilities. Included among these elicitation procedures are scoring rules, which involve the computation of a score based on the assessor's stated probabilities and on the event that actually occurs. The development of scoring rules has, in general, been restricted to the elicitation of discrete probability distributions. In this paper, families of scoring rules for the elicitation of continuous probability distributions are developed and discussed.
Fourier multiplier theorems on Besov spaces under type and cotype conditions

In this paper we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents $p$ and $q$, which depend on the type $p$ and cotype $q$ of the underlying Banach spaces. In a previous paper we considered $L^p$-$L^q$-multiplier theorems. In the current paper we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the $L^p$-$L^q$-setting as well. We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties then the boundedness of the multiplier operator extrapolates to other values of $p$ and $q$ for which $\frac1p - \frac1q$ remains constant.
Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles

We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of algebraic d-critical loci introduced by Joyce in arXiv:1304.4508. This resolves a long-standing question in the categorification of Lagrangian intersection numbers and it may have important consequences in symplectic geometry and topological field theory. Our second main result proves that (oriented) complex Lagrangian intersections in complex symplectic manifolds for any pair of Lagrangian submanifolds in the complex analytic topology carry the structure of (oriented) analytic d-critical loci in the sense of arXiv:1304.4508.
Counterexamples on spectra of sign patterns

An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is called spectrally arbitrary if any monic real polynomial of degree $n$ can be realized as a characteristic polynomial of a matrix obtained by replacing the non-zero elements of $S$ by numbers of the corresponding signs. A sign pattern $S$ is said to be a superpattern of those matrices that can be obtained from $S$ by replacing some of the non-zero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known "Nilpotent Jacobian" method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern $S$ and its superpattern $S'$ such that $S$ is spectrally arbitrary but $S'$ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche.
Generation of Inertia–Gravity Waves in a Simulated Life Cycle of Baroclinic Instability

Abstract The excitation and propagation of inertia–gravity waves (IGWs) generated by an unstable baroclinic wave was examined with a high-resolution 3D nonlinear numerical model. IGWs arose spontaneously as the tropospheric jetstream was distorted by baroclinic instability and strong parcel accelerations took place, primarily in the jetstream exit region of the upper troposphere. Subsequent propagation of IGWs occurred in regions of strong windspeed-in the tropospheric and stratospheric jets, and in a cutoff low formed during the baroclinic lifecycle. IGWs on the flanks of these jets were rotated inward by differential advection and subsequently absorbed by the model's hyperdiffusion. Although absorption of IGWs at the sidewalls of the jet is an artifact of the model, IGW propagation was for the most pan confined to regions with an intrinsic period shorter than the local inertial period. Only a few IGWs were able to penetrate the middle stratosphere, due to weak winds or an unfavorable alignment of waveve.
Rigid body collisions with friction

An energetically consistent theory is presented for dynamics of partly elastic collisions between somewhat rough rigid bodies with friction that opposes slip. This theory is based on separately accounting for frictional and non-frictional sources of dissipation. Alternative theories derived from Newton’s impact law or Poisson’s impact hypothesis are shown to be valid only for central (collinear) or non-frictional collisions; generally the latter theories yield erroneous energy dissipation if small initial slip stops during collision between eccentric bodies. Collision processes are complex when small slip is stopped by friction; then either the direction of slip reverses or contact points roll without slip. An inconsistent theory based on Newton’s impact law can yield erroneous energy increases when slip stops during collision; the consistent theory always dissipates energy. The impact law that specifies a simple proportionality between normal components of contact velocity for incidence and rebound is not applicable in any range of incident velocities with small slip if the collision is non-collinear with friction. In Percussion the force or Impetus whereby one body is moved may cause another body against which it strikes to be put in motion, and withal lose some of its strength or swiftness. (J. Wallis, 1668)
Asymptotic Expansions for Median Estimate of a Parameter

We obtain asymptotic expansions for the distribution of a median (of empirical distribution) estimate of a parameter in additive noise with symmetric density. For Laplacian (i.e., two-sided exponential) density this estimate coincides with the maximum likelihood estimate. As a corollary we obtain asymptotic expansions for moments of these estimates. Numerical comparisons with exact data show that the use of asymptotic expansions significantly increases the accuracy of statistical inferences even for relatively small sample sizes.
ESSENTIAL MERIDIONAL SURFACES FOR TUNNEL NUMBER ONE KNOTS

We show that for each pair of positive integers g and n, there are tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exte- riors contain n disjoint, non-parallel, closed incompressible surfaces, each of genus n.
Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space

This paper is a close follow-up of Kaltenbacher and Tomba 2013 and Jin 2012, where Newton-Landweber iterations have been shown to converge either (unconditionally) without rates or (under an additional regularity assumption) with rates. The choice of the parameters in the method were different in each of these two cases. We now found a unified and more general strategy for choosing these parameters that enables both convergence and convergence rates. Moreover, as opposed to the previous one, this choice yields strong convergence as the noise level tends to zero, also in the case of no additional regularity. Additionally, the resulting method appears to be more efficient than the one from Kaltenbacher and Tomba 2013, as our numerical tests show.
Fourier Series on Spheres

Abstract Spectral methods for the numerical solution of problems in spherical (and spheroidal) geometries are discussed. A new class of expansions based on special Fourier series are shown to lead to efficient and accurate simulations. A detailed exposition is given of the properties of surface harmonic series and transform methods and their relation to the new Fourier series on spheres. With resolution 1/N in both longitude and latitude, spectral methods using surface harmonics require 0(N) arithmetic operations per retained mode per time step while those based on Fourier series on spheres require only 0(logN) operations per retained mode per time step.
Endoscopic classification of very cuspidal representations of quasi-split unitary groups

Let $\mathbf{G}$ be an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The goal of this paper is to describe the supercuspidal representations within certain L-packets of $\mathbf{G}$, which are classified by Arthur and Mok using the theory of endoscopy. The description is given in terms of the cuspidal types constructed by Bushnell-Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy certain regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder. To achieve our goal, we first interpret the question as to study the reducibilities of some parabolically induced representations, using a theory of M{\oe}glin and Shahidi; we then apply a relation, given by Blondel, between these reducibilities and the structures of some Hecke algebras, where the latter can be computed using a Theorem of Lusztig. We can interpret our final result as explicitly describing the local Langlands correspondence for $\mathbf{G}$.
The Beta- flatness Condition in CR Spheres Multiplicity Results

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riemann spheres under Beta-flatness condition. To give a lower bound for the number of solutions, we use Bahri methods based on the theory of critical points at infinity and a Poincare-Hopf type formula.
Ultrasensitive mode-localized mass sensor with electrically tunable parametric sensitivity

We use the phenomena of mode localization and vibration confinement in pairs of weakly coupled, nearly identical microelectromechanical (MEMS) resonators as an ultrasensitive technique of detecting added mass on the resonator. The variations in the eigenstates for induced mass additions are studied and compared with corresponding resonant frequency shifts in pairs of MEMS resonators that are coupled electrostatically. We demonstrate that the relative shifts in the eigenstates can be over three orders of magnitude greater than those in resonant frequency for the same addition of mass. We also investigate the effects of voltage controlled electrical spring tuning on the parametric sensitivity of such sensors and demonstrate sensitivities tunable by over 400%.
Invariance principles for tree-valued Cannings chains

We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under the conditions of M\"ohle and Sagitov, this convergence holds for all tree-valued Fleming-Viot processes under consideration in the dust-free case, and for the Fleming-Viot processes with values in the space of distance matrix distributions in the case with dust. Convergence to Fleming-Viot processes with values in the space of marked metric measure spaces in the case with dust is ensured by an additional assumption on the probability that a randomly sampled individual belongs to a non-singleton family.
Comparing fat graph models of moduli space

Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-closed cobordisms. Similarly, Costello introduced a chain complex of black and white graphs $BW$-Graphs, as a rational homological model of mapping class groups. We use the result on admissible fat graphs to give a new integral proof of Costellos's result that $BW$-Graphs is a homological model of mapping class groups. The nature of this proof also provides a direct connection between both models which were previously only known to be abstractly equivalent. Furthermore, we endow Godin's model with a composition structure which models composition of cobordisms along their boundary and we use the connection between both models to give $BW$-Graphs a composition structure and show that $BW$-Graphs are actually a model for the open-closed cobordism category.
The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly to the exact solution. Recently, an explicit strongly convergent numerical scheme, called the tamed Euler method, has been proposed in [8] for such SDEs. Motivated by their work, we here introduce a tamed version of the Milstein scheme for SDEs with commutative noise. The proposed method is also explicit and easily implementable, but achieves higher strong convergence order than the tamed Euler method does. In recovering the strong convergence order one of the new method, new difficulties arise and kind of a bootstrap argument is developed to overcome them. Finally, an illustrative example confirms the computational efficiency of the tamed Milstein method compared to the tamed Euler method.