Miracle-dz/UniFAR-FTUs huggingface dataset:
Multiplication in Sobolev spaces, revisited

In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev-Slobodeckij spaces, and along the way we cite a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev-Slobodeckij spaces when a bounded domain is replaced by Rn. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation carefully, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and we carefully distinguish between the case of spaces defined on the all of Rn and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood-Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in the way presented here. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev-Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for the construction of counter-examples such as the one included here.
Fisher Information and Exponential Families Parametrized by a Segment of Means

We consider natural and general exponential families $(Q_m)_{m\in M}$ on $\mathbb{R}^d$ parametrized by the means. We study the submodels $(Q_{\theta m_1+(1-\theta)m_2})_{\theta\in[0,1]}$ parametrized by a segment in the means domain, mainly from the point of view of the Fisher information. Such a parametrization allows for a parsimonious model and is particularly useful in practical situations when hesitating between two parameters $m_1$ and $m_2$. The most interesting examples are obtained when $\mathbb{R}^d$ is a linear space of matrices, in particular for Gaussian and Wishart models.
Tensor decompositions and tensor equations over quaternion algebra

In this paper, we investigate and discuss in detail the structures of quaternion tensor SVD, quaternion tensor rank decomposition, and $\eta$-Hermitian quaternion tensor decomposition with the isomorphic group structures and Einstein product. Then we give the expression of the Moore-Penrose inverse of a quaternion tensor by using the quaternion tensor SVD. Moreover, we consider a generalized Sylvester quaternion tensor equation. We give some necessary and sufficient conditions for the existence of a solution to the generalized Sylvester quaternion tensor equation in terms of the Moore-Penrose inverses of the quaternion tensors. We also present the expression of the general solution to this tensor equation when it is solvable. As applications of this generalized Sylvester quaternion tensor equation, we derive some necessary and sufficient conditions for the existences of $\eta$-Hermitian solutions to some quaternion tensor equations. We also provide some numerical examples to illustrate our results.
Emergent Spacetime and Cosmic Inflation

We present a novel background-independent framework for cosmic inflation, starting with a matrix model. In this framework, inflation is portrayed as a dynamic process responsible for the generation of both space and time. This stands in contrast to conventional inflation, which is characterized as a mere (exponential) expansion of an already existing spacetime, driven by the vacuum energy associated with an inflaton field. We observe that the cosmic inflation is triggered by the condensate of Planck energy into a vacuum and responsible for the dynamical emergence of spacetime. The emergent spacetime picture admits a background-independent formulation so that the inflation is described by a conformal Hamiltonian system which requires neither an inflaton field nor an ad hoc inflation potential. This implies that the emergent spacetime may incapacitate all the rationales to introduce the multiverse hypothesis.
Single observation unbiased priors

This paper studies a class of default priors, which we call single observation unbiased priors (SOUP). A prior for a parameter is a SOUP if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter. We prove that, under mild regularity conditions, a default prior for a convolution parameter is noninformative in the sense of yielding a posterior inference invariant under amalgamation only if it is a SOUP. Therefore, when amalgamation invariance is desirable, as in our motivating example of performing imputation for census undercount, SOUP is the only possible coherent noninformative prior for Bayesian predictions that will be utilized under aggregation. The use of SOUP also mutually calibrates Bayesian and frequentist inferences for aggregates of convolution parameters across many small areas. We describe approaches that identify SOUPs in many common models, in particular a constructive duality method that identifies SOUPs in pairs of distribution families. We introduce O-completeness, a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean. Uniqueness of the SOUP is determined by the O-completeness of the dual family. O-completeness of a natural exponential family is implied by its completeness. Hence, the Diaconis-Ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families. For most of the examples we have examined, the inverse of the variance function is the SOUP for the mean parameter of the corresponding family, suggesting that Hartigan's results on asymptotic unbiasedness can be generalized to some families with discrete parameters. We also discuss a possible extension of Berger's result on the inadmissibility of unbiased estimators, as the nonexistence of SOUP can be a first step in establishing such inadmissibility.
Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(\epsilon)$, the number of prime cubic residues $q < p^{1/2+\epsilon}$ exceeds $p^{\epsilon/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{1/9}$ prime quadratic residues $q
DUALITIES OF ARTINIAN COALGEBRAS WITH APPLICATIONS TO NOETHERIAN COMPLETE ALGEBRAS

A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let A be a noetherian com- plete basic semiperfect algebra over an algebraically closed field, and C be its dual coalgebra. If A is Artin-Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule 1C� � witha coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twisted bimodule � � A1. Ifis an inner automorphism, then A is Calabi-Yau.
Topology of time-reversal invariant energy bands with adiabatic structure

We classify the topology of bands defined by the energy states of quantum systems with scale separation between slow and fast degrees of freedom, invariant under fermionic time reversal. Classical phase space transforms differently from momentum space under time reversal, and as a consequence the topology of adiabatic bands is different from that of Bloch bands. We show that bands defined over a two-dimensional phase space are classified by the Chern number, whose parity must be equal to the parity of the band rank. Even-rank bands are equivalently classified by the Kane–Mele index, an integer equal to one half the Chern number.
Wall-to-wall optimal transport in two dimensions

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of the velocity fields by a Péclet number $Pe$ proportional to their root-mean-square rate of strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e. the Nusselt number $Nu$ up to $Pe\approx 10^{5}$. The resulting transport exhibits a change of scaling from $Nu-1\sim Pe^{2}$ for $Pe<10$ in the linear regime to $Nu\sim Pe^{0.54}$ for $Pe>10^{3}$. Optimal fields are observed to be approximately separable, i.e. products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound ${\lesssim}Pe^{6/11}=Pe^{0. \overline{54}}$ as $Pe\rightarrow \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh–Bénard convection are discussed.
Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions

We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to $1$. We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the properties $F_n$ and $FP_n$ from its group of units. We also obtain results which relate the geometric and cohomological dimensions of the monoid to those of its group of units. As an application of these results we show that all monoids defined by one-relator presentations of the form $\langle A \mid r=1 \rangle$ are of type left- and right-$F_{\infty}$ (and $FP_{\infty}$), resolving an important case of a question of Kobayashi from 2000 on homological finiteness properties of one-relator monoids. Moreover, in the case that the relator $r$ is not a proper power, we prove that the right and left geometric and cohomological dimensions of the monoid are bounded above by $2$. We also show how our topological approach can be used to prove results about the closure properties of right and left-$F_n$, bi-$F_n$, right and left geometric and cohomological dimension, and geometric and Hochschild cohomological dimension of monoids, for amalgamated free products of monoids and HNN extension constructions for monoids. To prove these results we introduce new methods for constructing equivariant, and bi-equivariant, classifying spaces for monoids, as well as developing a Bass-Serre theory for free constructions of monoids. Additionally, we give a topological proof that a free inverse monoid on one or more generators is neither of type left-$FP_2$ nor right-$FP_2$. This generalises a classical result of Schein that such monoids are not finitely presented.
Statistical inference for inequality measures based on semi-parametric estimators

Measures of inequality, also used as measures of concentration or diversity, are very popular in economics and especially in measuring the inequality in income or wealth within a population and between populations. However, they have applications in many other fields, e.g. in ecology, linguistics, sociology, demography, epidemiology and information science. A large number of measures have been proposed to measure inequality. Examples include the Gini index, the generalized entropy, the Atkinson and the quintile share ratio measures. Inequality measures are inherently dependent on the tails of the population (underlying distribution) and therefore their estimators are typically sensitive to data from these tails (nonrobust). For example, income distributions often exhibit a long tail to the right, leading to the frequent occurrence of large values in samples. Since the usual estimators are based on the empirical distribution function, they are usually nonrobust to such large values. Furthermore, heavy-tailed distributions often occur in real life data sets, remedial action therefore needs to be taken in such cases. The remedial action can be either a trimming of the extreme data or a modification of the (traditional) estimator to make it more robust to extreme observations. In this thesis we follow the second option, modifying the traditional empirical distribution function as estimator to make it more robust. Using results from extreme value theory, we develop more reliable distribution estimators in a semi-parametric setting. These new estimators of the distribution then form the basis for more robust estimators of the measures of inequality. These estimators are developed for the four most popular classes of measures, viz. Gini, generalized entropy, Atkinson and quintile share ratio. Properties of such estimators are studied especially via simulation. Using limiting distribution theory and the bootstrap methodology, approximate confidence intervals were derived. Through the various simulation studies, the proposed estimators are compared to the standard ones in terms of mean squared error, relative impact of contamination, confidence interval length and coverage probability. In these studies the semi-parametric methods show a clear improvement over the standard ones. The theoretical properties of the quintile share ratio have not been studied much. Consequently, we also derive its influence function as well as the limiting normal distribution of its nonparametric estimator. These results have not previously been published. In order to illustrate the methods developed, we apply them to a number of real life data sets. Using such data sets, we show how the methods can be used in practice for inference. In order to choose between the candidate parametric distributions, use is made of a measure of sample representativeness from the literature. These illustrations show that the proposed methods can be used to reach satisfactory conclusions in real life problems. iii University of Stellenbosch http://scholar.sun.ac.za
New Method in the Theory of Imperfect Gases and Liquids

We derive a new expansion method for the thermodynamic properties and correlation functions of imperfect gases. When the interparticle potential is repulsive, we obtain an alternating series of upper and lower bounds for the activity, free energy, pressure, entropy, internal energy, and for the correlation functions. These bounds are valid even if there is a transition. If the potential has an attractive part, we can obtain upper bounds for the activity, free energy, internal energy, and the correlation functions.
A property of finite simple non-abelian groups

It has been proved by Frohlich [1] that the collection of inner automorphisms of a finite simple non-abelian group G generates the group (under pointwise multiplication) of all functions from G into G leaving the identity fixed, and, conversely, that the only finite groups with this property are the simple non-abelian groups, Z2, and [e]. It follows directly that the inner automorphisms and the constant functions of a simple non-abelian group into itself generate the entire group of functions from G into G. The purpose of this note is to prove the following generalization of this theorem.
Dual-band infrared metallodielectric photonic crystal filters

We have designed a metallodielectric photonic crystal with planar metallic scatterers to produce high reflectance in both the 3–5 and 8–12 μm atmospheric windows over a wide range of angles of incidence. We have fabricated similar crystals using photolithography with Al scatterers in a polymer dielectric medium and measured their optical transmission, confirming the results of our design simulations. A fabricated filter produced a reflection band with >30 dB rejection and a fractional bandwidth of 0.38. Our results show that it is possible to fabricate infrared photonic crystal filters comparable in performance to those with thick metal elements, but with the ease of manufacture of thin metal layers.
Hausdorff dimension of weighted singular vectors

Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $\mathbb R^2$.
THE SCHWARZSCHILD BLACK HOLE AS A GRAVITATIONAL MIRROR

The gravitational field outside of a nonrotating black hole is described using the Schwarzschild metric. The geodesic equations of the Schwarzschild metric are derived and those describing null and circular timelike orbits are discussed. Some numerical solutions of the null geodesic equations are shown. These depict photon trajectories which circle the black hole one or two times and then terminate at their emission points. Thus a sequence of ring‐shaped mirror images is produced. An equation which gives the angle between the photon’s trajectory and the radial direction at the emitter is derived and applied to the numerical solutions. These results serve to illustrate how an observer ‘‘passes through’’ his or her mirror image at r=3 MG/c2, as he or she moves toward a Schwarzschild black hole.
BULANIK ANALİTİK HİYERARŞİK PROSES VE ÜNİVERSİTE TERCİH SIRALAMASINDA UYGULANMASI

Bu calismada, bulanik analitik hiyerarsik proses arastirilarak universite tercih siralamasinda uygulanmasi yapilmistir. Ayrica, uygulamada uc farkli metot (Chang, Liou-Wang, Kareli Ortalama) kullanilarak bunlarin karsilastirmalari yapilmistir. Chang’in metodunda tutarlilik oraninin hesaplanabilmesi mumkun olmamaktadir. Bu nedenle calismada Chang’in metodunun yaninda Liou ve Wang’in yontemi de kullanilmis ve tutarlilik orani hesaplanarak tutarlilik test edilebilmistir. Ayrica bu calismada Kareli Ortalama Yontemi ismiyle yeni bir yontem onerisi yapilmis ve bu yontemle yapilan uygulama sonucunda bulunan tutarlilik oraninin, Liou ve Wang’in yontemine gore bulunan degere cok yakin oldugu gorulmustur
Motivic integration and Milnor fiber

We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called "nonarchimedean Milnor fiber" that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. We also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert, Loeser, and Merle. Finally, applying $T$-convex integration after descent, matching the Euler Characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries.
Exploding Stars Point to a Universal Repulsive Force

ASTRONOMYStudies of distant, exploding stars called supernovae are suggesting that the universe could be permeated by a large-scale repulsive force. The reservoir of energy fueling that force could be anything from a quantum-mechanical shimmer in empty space, called the cosmological constant, to even more exotic possibilities that go by names like X-matter and quintessence.
The nature of dielectric breakdown

Dielectric breakdown is the process of local materials transiting from insulating to conductive when the dielectric is submerged in a high external electric field environment. We show that the atomistic changes of the chemical bonding in a nanoscale breakdown path are extensive and irreversible. Oxygen atoms in dielectric SiO2 are washed out with substoichiometric silicon oxide (SiOx with x<2) formation, and local energy gap lowering with intermediate bonding state of silicon atoms (Si1+, Si2+, and Si3+) in the percolation leakage path. Oxygen deficiency within the breakdown path is estimated to be as high as 50%–60%.
Does Bladder Cancer with Inchworm Sign Indicate Better Prognosis after TURBT?

Simple Summary Inchworm sign is considered to be a characteristic finding in non-muscle invasive bladder cancer. We retrospectively investigated the factors related to muscle invasive status in bladder cancer associated with inchworm sign and the role of inchworm sign in tumor outcomes following transurethral resection of bladder tumor. Of the 109 patients with inchworm sign, 94 (86.2%) were non-muscle invasive bladder cancer. Non-papillary tumors and tumors located in the bladder neck were significant predictors of muscle invasive bladder cancer with inchworm sign. Additionally, inchworm sign was not a prognostic factor in patients with non-muscle invasive bladder cancer in this study. Abstract Background: Inchworm sign is considered to be a characteristic finding in non-muscle invasive bladder cancer (NMIBC). Nevertheless, pathologically diagnosed muscle invasive bladder cancers (MIBCs) are occasionally diagnosed from tissue obtained by transurethral resection of bladder tumor (TURBT) in patients with inchworm sign. Methods: We retrospectively investigated the factors related to muscle invasive status in bladder cancer associated with inchworm sign and the role of inchworm sign in tumor outcomes following TURBT. Results: Of the 109 patients with inchworm sign, 94 (86.2%) and 15 (13.8%) were NMIBC and MIBC, respectively. Non-papillary tumors (hazard ratio (HR): 9.55, 95% confidence interval (CI): 2.07–44.10; p < 0.01) and tumors located in the bladder neck (HR: 7.73, 95% CI: 1.83–32.76; p < 0.01) were significant predictors of MIBC in bladder cancer with inchworm sign. Furthermore, recurrence-free survival (RFS) and progression-free survival were compared between patients with NMIBC with and without inchworm sign; however, no significant differences were found. In patients with NMIBC with inchworm sign, positive urine cytology was a prognostic factor for RFS (HR: 1.90, 95% CI: 1.04–3.48; p = 0.04). Conclusions: In bladder cancer with inchworm sign, 86.2% were NMIBC. Even in the case of inchworm sign, the presence of a non-papillary tumor or a bladder neck tumor before TURBT should be noted because of the possibility of MIBC. In this study, the inchworm sign was not a prognostic factor in patients with NMIBC.