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A singular perturbation problem called WKB equation (Eq) $h^2u(x,h)-Q(x)u(x,h)=0$ is studied. $h>0$ is a small parameter. Investigation of (Eq) has long history. Recently it has developed by a new method named "Exact WKB Analysis" based on Borel resummation method and new analytic results. Here we study (Eq) by another elementary method. We only apply advanced calculus and the theory of differential equations to (Eq). We neither assume turning points are simple nor there is no Stokes curve that connects two turning points.

Bandeira et al. (2017) show that the eigenvalues of the Kendall correlation matrix of $n$ i.i.d. random vectors in $\mathbb{R}^p$ are asymptotically distributed like $1/3 + (2/3)Y_q$, where $Y_q$ has a Marčenko-Pastur law with parameter $q=\lim(p/n)$ if $p, n\to\infty$ proportionately to one another. Here we show that another Marčenko-Pastur law emerges in the "ultra-high dimensional" scaling limit where $p\sim q'\, n^2/2$ for some $q'>0$: in this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to $(1/3)Y_{q'}$.

Finding a Hamiltonian cycle in a given graph is computationally challenging, and in general remains so even when one is further given one Hamiltonian cycle in the graph and asked to find another. In fact, no significantly faster algorithms are known for finding another Hamiltonian cycle than for finding a first one even in the setting where another Hamiltonian cycle is structurally guaranteed to exist, such as for odd-degree graphs. We identify a graph class -- the bipartite Pfaffian graphs of minimum degree three -- where it is NP-complete to decide whether a given graph in the class is Hamiltonian, but when presented with a Hamiltonian cycle as part of the input, another Hamiltonian cycle can be found efficiently. We prove that Thomason's lollipop method~[Ann.~Discrete Math.,~1978], a well-known algorithm for finding another Hamiltonian cycle, runs in a linear number of steps in cubic bipartite Pfaffian graphs. This was conjectured for cubic bipartite planar graphs by Haddadan [MSc~thesis,~Waterloo,~2015]; in contrast, examples are known of both cubic bipartite graphs and cubic planar graphs where the lollipop method takes exponential time. Beyond the lollipop method, we address

In our effort to find an arithmetically pure proof of the Bertrand postulate, we investigate and solve (using only elementary arithmetical methods) another less usual inequality in positive integers inspired by the classical proof of the postulate given by P. Erdős.

Family members' satisfaction with one another is central to creating healthy and supportive family environments. In this work, we propose and implement a novel computational technique aimed at exploring the possible relationship between the topology of a given family tree graph and its members' satisfaction with one another. Through an extensive empirical evaluation ($N=486$ families), we show that the proposed technique brings about highly accurate results in predicting family members' satisfaction with one another based solely on the family graph's topology. Furthermore, the results indicate that our technique favorably compares to baseline regression models which rely on established features associated with family members' satisfaction with one another in prior literature.

Free ribbon lemma that every free sphere-link in the 4-sphere is a ribbon sphere-link is shown in an earlier paper by the author. In this paper, another proof of this lemma is given.

We study how to extend Sanov theorem to the quantum setting. Although a quantum version of the Sanov theorem was proposed in Bjelakovic et al (Commun. Math. Phys., 260, p.659 (2005)), the classical case of their statement is not the same as Sanov theorem because Sanov theorem discusses the behavior of the empirical distribution when the empirical distribution is different from the true distribution, but they studied a problem related to quantum hypothesis testing, whose classical version can be shown by classical Sanov theorem. We propose another quantum version of Sanov theorem by considering the quantum analog of the empirical distribution.

Based on the Sinc approximation combined with the tanh transformation, Haber derived an approximation formula for numerical indefinite integration over the finite interval (-1, 1). The formula uses a special function for the basis functions. In contrast, Stenger derived another formula, which does not use any special function but does include a double sum. Subsequently, Muhammad and Mori proposed a formula, which replaces the tanh transformation with the double-exponential transformation in Haber's formula. Almost simultaneously, Tanaka et al. proposed another formula, which was based on the same replacement in Stenger's formula. As they reported, the replacement drastically improves the convergence rate of Haber's and Stenger's formula. In addition to the formulas above, Stenger derived yet another indefinite integration formula based on the Sinc approximation combined with the tanh transformation, which has an elegant matrix-vector form. In this paper, we propose the replacement of the tanh transformation with the double-exponential transformation in Stenger's second formula. We provide a theoretical analysis as well as a numerical comparison.

RAT0455+1305 was discovered during the Rapid Temporal Survey which aims in finding any variability on timescales of a few minutes to several hours. The star was found to be another sdBV star with one high amplitude mode and relatively long period. These features along with estimation of T_eff and log g makes this star very similar to Balloon 090100001. Encouraged by prominent results obtained for the latter star we have decided to perform white light photometry on RAT0455+1305. In 2009 we used the 1.5m telescope located in San Pedro Martir Observatory in Mexico. Fourier analysis confirmed the dominant mode found in the discovery data, uncovered another peak close to the dominant one, and three peaks in the low frequency region. This shows that RAT0455+1305 is another hybrid sdBV star pulsating in both p- and g-modes.

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.

One of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work[13] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem $ Tf = λA f $ is obtained. Finally a theorem of Williams[16] is generalized to obtain a translatable radius of an operator in the direction of another operator.

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated $R$-modules and $R$-linear maps over a principal ideal domain $R$, and the neighbourhood category of oriented knots in the 3-sphere.

In this paper, we study the problem of simulating a DMC channel from another DMC channel under an average-case and an exact model. We present several achievability and infeasibility results, with tight characterizations in special cases. In particular for the exact model, we fully characterize when a BSC channel can be simulated from a BEC channel when there is no shared randomness. We also provide infeasibility and achievability results for simulation of a binary channel from another binary channel in the case of no shared randomness. To do this, we use properties of Rényi capacity of a given order. We also introduce a notion of "channel diameter" which is shown to be additive and satisfy a data processing inequality.

In this paper, the decades-old clustering method k-means is revisited. The original distortion minimization model of k-means is addressed by a pure stochastic minimization procedure. In each step of the iteration, one sample is tentatively reallocated from one cluster to another. It is moved to another cluster as long as the reallocation allows the sample to be closer to the new centroid. This optimization procedure converges faster to a better local minimum over k-means and many of its variants. This fundamental modification over the k-means loop leads to the redefinition of a family of k-means variants. Moreover, a new target function that minimizes the summation of pairwise distances within clusters is presented. We show that it could be solved under the same stochastic optimization procedure. This minimization procedure built upon two minimization models outperforms k-means and its variants considerably with different settings and on different datasets.

The punctured binary Reed-Muller code is cyclic and was generalized into the punctured generalized Reed-Muller code over $\gf(q)$ in the literature. The major objective of this paper is to present another generalization of the punctured binary Reed-Muller code. Another objective is to construct a family of reversible cyclic codes that are related to the newly generalized Reed-Muller codes.

We study another realization of the Kerr/CFT correspondence. By imposing new asymptotic conditions for the near horizon geometry of Kerr black hole, an asymptotic symmetry which contains all of the exact isometries can be obtained. In particular, the Virasoro algebra can be realized as an enhancement of SL(2,R) symmetry of the AdS geometry. By using this asymptotic symmetry, we discuss finite temperature effects and show the correspondence concretely.

The Higgs mechanism is designed to generate mass for massless particles. The mass comes from the interaction of observed particles with an external field -- the Higgs field. In the past, several alternatives to the Higgs mechanism for mass generation have been proposed to avoid the postulation of the Higgs field. This article proposes yet another one. This alternative is distinctly different from the others because it considers mass generation through internal interactions of a particle rather than interactions with external fields. This requires particles to have an internal structure beyond intrinsic spin. A complete field theory of such composite particles is seen to be possible. Of course, if Higgs bosons are observed by experiment, there will be no need for any alternatives. On the other hand, if experiment fails to detect Higgs bosons, such alternate mechanisms for particle mass generation would be very useful.

We suggest that some of the remarkable results on stringy dynamics which have been found recently indicate the existence of another dynamical length scale in string theory that, at weak coupling, is much shorter than the string scale. This additional scale corresponds to a mass $\sim m_{\rm s}/g_{\rm s}$ where $m_{\rm s}$ is the square root of the string tension and $g_{\rm s}$ is the string coupling constant. In four dimensions this coincides with the Planck mass.

In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.