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Suppose $$\{P_{n}^{(\alpha , \beta )}(x)\} _{n=0}^\infty $$ is a sequence of Jacobi polynomials with $$ \alpha , \beta >-1.$$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+k}^{(\alpha + t, \beta + s )}(x)$$ are interlacing if $$s,t >0$$ and $$ k \in {\mathbb {N}}.$$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+1}^{(\alpha , \beta + 1 )}(x),$$ $$ \alpha> -1, \beta > 0, $$ $$ n \in {\mathbb {N}},$$ are partially, but in general not fully, interlacing depending on the values of $$\alpha , \beta $$ and n. A similar result holds for the extent to which interlacing holds between the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+1}^{(\alpha + 1, \beta + 1 )}(x),$$ $$ \alpha>-1, \beta > -1.$$ It is known that the zeros of the equal degree Jacobi polynomials $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n}^{(\alpha - t, \beta + s )}(x)$$ are interlacing for $$ \alpha -t> -1, \beta > -1, $$ $$0 \le t,s \le 2.$$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n}^{(\alpha + 1, \beta + 1 )}(x),$$ when $$ \alpha> -1, \beta > -1.$$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $$\alpha = \beta = \lambda -1/2$$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $$ C_{n}^{(\lambda )}(x)$$ and $$ C_{n + 1}^{(\lambda +1)}(x),$$ $$ \lambda > -1/2,$$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $$ C_{n}^{(\lambda )}(x)$$ and $$ C_{n}^{(\lambda +3)}(x),$$ $$ \lambda > -1/2,$$ is also discussed.

We propose a method of determining temperature fields in a spatially inhomogeneous medium with temperature-dependent thermophysical properties of the material. For this purpose, point and nonlocal transformations of the nonstationary heat conduction equation are used. Examples of applying the theory for various boundary conditions in the spherical symmetric case are given.

Aggregation functions have attracted much attention in recent times because of its potential use in many areas such us data fusion and decision making. In practice, most of the aggregation functions that scientists use in their studies are constructed from very simple (usually affine or polynomial) functions. However, these are distinct in nature. In this paper, we develop a systematic study of these two classes of functions from a common point of view. To do this, we introduce the class of affine aggregation functions, which cover both the aforementioned families and most of examples of aggregation functions that are used in practice, including, by its great applicability, the symmetric case. Our study allows us to characterize when a function constructed from affine or polynomial functions is, in fact, a new aggregation function. We also study when sums or products of this kind of functions are again an aggregation function. The authors were partially supported by Spanish research project TIN2016-77356-P (AEI/FEDER,UE) and TIN2017-89517-P (AEI/FEDER,UE).

We study analytically, numerically, and experimentally the mechanical behavior of a fixed-angle bow, i.e., a slender rod with both ends rotationally fixed, where one end is completely constrained (orientation and displacements), while the other can be translated (only) along the line connecting the two ends of the bow. Understanding the behavior of fixed-angle bows is of importance and direct relevance to the design of diverse engineering applications, such as compliant mechanisms, nonlinear springs, energy absorbers and energy harvesters. The analysis, based on inextensible elastica, accounts for large rotations and large displacements. We find exact analytical solutions for the symmetric case and approximate solutions for the nonsymmetric case. These are validated by comparison with numerical results and experimental measurements. The similarities and differences between fixed-angle bows, fixed-angle circular arcs and rings are also discussed.

In wireless networks, link strengths are often affected by some topological factors such as propagation path loss, shadowing and inter-cell interference. Thus, different users in the network might experience different link strengths. In this work we consider a $K$ -user asymmetric interference channel, where the channel gains of the links connected to Receiver $k$ are scaled with $\sqrt {P^{\alpha _{k}}}$ , $k=1,2, \cdots, K$ , for $0 . For this setting, we show that the optimal sum generalized degrees-of-freedom (GDoF) is characterized as $d_{\text {sum}}= \frac { \sum _{k=1}^{K} \alpha _{k} + \alpha _{K} -\alpha _{K-1}}{2}$ which matches the existing result $d_{\text {sum}}= \frac {K}{2}$ when $\alpha _{1} = \alpha _{2} = \cdots = \alpha _{K} =1$ . The achievability is based on multi-layer interference alignment, where different interference alignment sub-schemes are designed in different layers associated with specific power levels, and successive decoding is applied at the receivers. While the converse for the symmetric case only requires bounding the sum degrees-of-freedom (DoF) for selected two users, the converse for this asymmetric case involves bounding the weighted sum GDoF for selected $J+2$ users, with corresponding weights $(2^{J}, 2^{J-1}, \cdots, 2^{2}, 2^{1})$ , a geometric sequence with common ratio 2, for the first $J$ users and with corresponding weights (1, 1) for the last two users, for $J \in \left\{{1,2, \cdots, \lceil \log \frac {K}{2} \rceil }\right\}$ .

In this paper, we focus on the issue of bid orchestration in scoring procurement auctions, which is realized by a supplier, who receives preferential evaluation of the respective bid in exchange of a bribe to the auctioneer. For first‐score and second‐score auctions, with an honest (noncorrupted) and a dishonest (corrupted) supplier, we investigate the equilibrium bidding strategy and find that an asymmetry between the two bidders is created by the presence of quality manipulation. We also explore how bidding is affected compared with the symmetric case where both opponents are honest. Our main findings indicate that both bidders (honest and dishonest) find the first‐score auction more profitable compared with the second‐score auction format. Contrary to the symmetric case, the asymmetry of quality manipulation differentiates the utility of the buyer and translates to a preference of the buyer for a second‐score auction. Finally, we also provide some discussions on policy implications of our results.

Objectives To develop a stochastic model relating measurement uncertainty, including reproducibility, to clinical accuracy, as demonstrated by the receiver operating characteristic curve. Methods A model is developed based on the symmetric case of the well-known binormal distribution. The overall distribution is partitioned further into analytical and biological components based on assumptions derived from the Cotlove criterion. Explicit mathematical solutions are derived and further verified by Monte Carlo analyses. Results The model demonstrates that tests with analytical error that conforms to the classic Cotlove criterion can achieve receiver operating characteristic curves with areas under the curve of 0.68 to 0.76 and Youden indices of 0.26 to 0.38 but have overall agreement for duplicate measurements of only 80% to 82%. Furthermore, the analytically accurate agreement is only 75% to 78%, and the clinically accurate agreement is only 50% to 60%. Conclusions The model suggests that assays may have reasonable clinical accuracy despite having reproducibility of less than 85%. Imperfect assays can substantially improve medical decision-making. The findings must be interpreted with caution given the binormal assumptions, but such assumptions are often useful as a first approximation. Practicing pathologists should feel comfortable performing semiquantitative assays shown to have a strong biological association with clinical outcome.

Sharp conditions are obtained for the unique solvability of focal boundary value problems for higher-order functional differential equations under integral restrictions on functional operators. In terms of the norm of the functional operator, unimprovable conditions for the unique solvability of the boundary value problem are established in the explicit form. If these conditions are not fulfilled, then there exists a positive bounded operator with a given norm such that the focal boundary value problem with this operator is not uniquely solvable. In the symmetric case, some estimates of the best constants in the solvability conditions are given. Comparison with existing results is also performed.

We introduce in this paper the concept of Dunkl-coherent pair of forms (linear functionals) in the symmetric case. We prove that if { u , v } is a Dunkl-symmetrically coherent pair of symmetric for...

We study gravitational wave (GW) production in strongly supercooled cosmological phase transitions, taking particular care of models featuring a complex scalar field with a U$(1)$ symmetric potential. We perform lattice simulations of two-bubble collisions to properly model the scalar field gradients, and compute the GW spectrum sourced by them using the thin-wall approximation in many-bubble simulations. We find that in the U$(1)$ symmetric case the low-frequency spectrum is $\propto\omega$ whereas for a real scalar field it is $\propto\omega^3$. In both cases the spectrum decays as $\omega^{-2}$ at high frequencies., Comment: 7 pages, 4 figures. published version

In this paper, we study the Cauchy problem for systems of 3-D quasilinear wave equations satisfying the null condition with initial data of low regularity. In the radially symmetric case, we prove the global existence for every small data in H 3 × H 2 with a low weight. To achieve this goal, we will show how to extend the global iteration method first suggested by Li and Chen (1988) [32] to the low regularity case, which is also another purpose of this paper. Finally, we apply our result to 3-D nonlinear elastic waves.

Generalized majority rules are electoral rules in which an alternative needs to obtain a fixed percentage (not necessarily 50%) of all votes in order to win. While Krishna and Morgan (Am Econ J Microeconom 7:339–375, 2015) demonstrate that simple majority maximizes expected utilitarian welfare for limiting populations regardless of the prior support for the alternatives, this paper finds that, when the prior support is known, a continuum of voting rules also achieves the same welfare. Moreover, as the population approaches the limit, every voting rule eventually becomes welfare inferior to picking the ex-ante majority without an election. Examining the properties of these optimal rules allows us to generalize the relationship between voter participation and welfare beyond the symmetric case.

There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005).

Let K be a convex body in $${\mathbb {R}}^n$$ with ellipsoid of maximal volume $$B_2^n$$ . We prove that every k-dimensional central section of K has volume at most $$\frac{\left( \sqrt{k+1}\sqrt{n}\right) ^{k+1}}{k!\sqrt{k}}.$$ In the centrally symmetric case the upper bound is $$(\frac{4n}{k})^\frac{k}{2}.$$ As an application of these inequalities we get extremal properties of cube and simplex.

Consider a generalized multiterminal source coding system, where $\ell\choose m$ encoders, each observing a distinct size-$m$ subset of $\ell$ ($\ell\geq 2$) zero-mean unit-variance symmetrically correlated Gaussian sources with correlation coefficient $\rho$, compress their observations in such a way that a joint decoder can reconstruct the sources within a prescribed mean squared error distortion based on the compressed data. The optimal rate-distortion performance of this system was previously known only for the two extreme cases $m=\ell$ (the centralized case) and $m=1$ (the distributed case), and except when $\rho=0$, the centralized system can achieve strictly lower compression rates than the distributed system under all non-trivial distortion constraints. Somewhat surprisingly, it is established in the present paper that the optimal rate-distortion performance of the afore-described generalized multiterminal source coding system with $m\geq 2$ coincides with that of the centralized system for all distortions when $\rho\leq 0$ and for distortions below an explicit positive threshold (depending on $m$) when $\rho>0$. Moreover, when $\rho>0$, the minimum achievable rate of generalized multiterminal source coding subject to an arbitrary positive distortion constraint $d$ is shown to be within a finite gap (depending on $m$ and $d$) from its centralized counterpart in the large $\ell$ limit except for possibly the critical distortion $d=1-\rho$., Comment: 12 pages, double column

In the paper, we derive Procrustes-based closed-form solutions to the point-wise weighted rigid-body transformation under different adjustment scenarios. We treat both asymmetric and symmetric case...

The Gelfand-Kirillov dimension is an invariant which can measure the size of infinite-dimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.

We consider a system of two discrete quasiperiodic 1D particles as an operator on $\ell^2(\mathbb Z^2)$ and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to those values., Comment: Some notation has been revised, and the referee's suggestions addressed. The result now covers a larger class of interaction potentials. To appear in Geometric and Functional Analysis

This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = Δu + eu in ℝN. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9, and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen (2008) in a finite ball to the whole space.

For covariate-adjusted nonparametric regression model, an adaptive estimation method is proposed for estimating the nonparametric regression function. Compared with the procedures introduced in the existing literatures, the new method needs less strict conditions and is adaptive to covariate-adjusted nonparametric regression with asymmetric variables. More specifically, when the distributions of the variables are asymmetric, the new procedures can gain more efficient estimators and recover data more accurately by elaborately choosing proper weights; and for the symmetric case, the new estimators can obtain the same asymptotic properties as those obtained by the existing method via designing equal bandwidths and weights. Simulation studies are carried out to examine the performance of the new method in finite sample situations and the Boston Housing data is analyzed as an illustration.

We study group contests where group sizes are stochastic and unobservable to participants at the time of investment. When the joint distribution of group sizes is symmetric, with expected group size k ¯ , the symmetric equilibrium aggregate investment is lower than in a symmetric group contest with commonly known fixed group size k ¯ . A similar result holds for two groups with asymmetric distributions of sizes. For the symmetric case, the reduction in individual and aggregate investment due to group size uncertainty increases with the variance in relative group impacts. When group sizes are independent conditional on a common shock, a stochastic increase in the common shock mitigates the effect of group size uncertainty unless the common and idiosyncratic components of group size are strong complements. Finally, group size uncertainty undermines the robustness of the group size paradox otherwise present in the model.

Many mathematical models of biological processes can be represented as free boundary problems for systems of PDEs. In the radially symmetric case, the free boundary is a function of \begin{document}$ r = R(t) $\end{document} , and one can generally prove the existence of global in-time solutions. However, the asymptotic behavior of the solution and, in particular, of \begin{document}$ R(t) $\end{document} , has not been explored except in very special cases. In the present paper we consider two such models which arise in cancer treatment by combination therapy with two drugs. We study the asymptotic behavior of the solution and its dependence on the dose levels of the two drugs.

In this paper, we obtain all the symmetric semi-classical linear functionals of class four taking into account the irreducible expression of the corresponding Pearson equation. We focus our attention on their integral representations. Thus, some linear functionals very well known in the literature, associated with perturbations of semi-classical linear functionals of class two at most, appear as well as new linear functionals which have not been studied.

Approximate analytical and numerical solutions of the three-dimensional Logunov–Tavkhelidze equation are found for the spherically symmetric case. Solutions are obtained in momentum and relativistic configuration representations. The wave functions in the relativistic configuration representation have additional zeroes compared to the wave functions of the nonrelativistic harmonic oscillator in the coordinate representation.

We find a class of cylindrically symmetric, static electrovacuum spacetimes generated by a non-homogeneous magnetic field and involving the cosmological constant and one additional parameter, which determine uniquely the strength of the magnetic field. We provide a simple model of a source producing the field., Comment: 7 pages, no figures, a reference added

We study the delta-T noise in the Kondo regime, which implies the charge current noise under the temperature bias for the SU(2) Kondo quantum dot. We propose an experimentally measurable quantity to quantify the low-temperature properties in the delta-T noise: $S_{\ell}=S(T_{\mathrm{L}},T_{\mathrm{R}}) - (1/2)[S(T_{\mathrm{L}},T_{\mathrm{L}}) + S(T_{\mathrm{R}},T_{\mathrm{R}})]$, which yields the shot noise expression in the noninteracting limit. We calculate this quantity for the SU(2) Kondo quantum dot in the particle-hole symmetric case. We found that the $S_{\ell}$ exhibits qualitatively the same behavior in both the electrochemical potential biased case and the temperature biased case. The quantitative difference appears as a difference of the coefficients of the noises, which reflects the difference of the Fermi distribution function: electrochemical potential biased or temperature biased., Comment: 13pages, 4 figures

This chapter is in a sense the culmination of the various topics discussed in previous chapters; it arises by the restriction of the considered locally symmetric spaces to the specific case of locally hermitian symmetric spaces. It has already been observed that the hermitian symmetric case permits much more precise results on its structure; in particular it allows making stronger contact with algebraic geometry.

In this article we discuss the hydrodynamic limit of the weakly asymmetric simple exclusion process, whose asymmetry is regulated by a factor \(N^\gamma \) with \(\gamma \ge 1\), and in contact with stochastic reservoirs, which are regulated by two factors \(N^\theta \) and \(N^\delta \) for, respectively, the symmetric and asymmetric parts of the jump rate at the boundary. Depending on the strength of the asymmetry, that is on the parameter \(\gamma \), we derive the heat equation (when \(\gamma >1\)) as in the purely symmetric case studied in [1], or the viscous Burgers equation (when \(\gamma =1\)). In both cases, the PDEs have several boundary conditions which depend on the range of the parameters \(\delta \) and \(\theta \).

We investigate the existence of solutions to the fractional nonlinear Schrodinger equation ( − Δ ) s u = f ( u ) − μ u with prescribed L 2 -norm ∫ R N | u | 2 d x = m in the Sobolev space H s ( R N ) . Under fairly general assumptions on the nonlinearity f, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.

A general result on global solvability is established for a diffusion–absorption model with memory-driven flux at the boundary. Such a boundary condition has been studied previously for application to the problem of new capillary growth as induced by a pre-metastatic tumor. In earlier results for the model without absorption, we provided a complete characterization of power law memory boundary conditions regarding either global solvability or blow-up in finite time. It turns out such results are identical to those for the corresponding model with localized power law flux conditions at the boundary. Now in the case of nonlinear absorption added to the model, which in fact better incorporates natural growth factor decay or uptake in the capillary growth application, the threshold of global solvability for the localized model is dependent upon strength of absorption in a way that is not parallel to our result for the memory-driven model. We conclude by proving blow-up results in the radially symmetric case and observing that future studies are needed to complete the full characterization of global solvability.

The gravitational field of a massive object (for instance, of a galaxy group or cluster) disturbs the Hubble stream, decreasing its speed. Dependence $v(r_0)$ of the radial velocity of the stream from the present-day radius $r_0$ can be directly observed and may provide valuable information about the cluster properties. We offer an exact analytical relationship $v(r_0)$ for a spherically-symmetric system., Comment: 6 pages, 3 figures, accepted to Physical Review D

The leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in $\mathbb{Z}^2$ and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. We compute the limit shape as a function of $d$ in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as $d\to 1$ and a diamond as $d\to\infty$. We compute the limit shape by comparing the odometer function at a site to the probability that a killed random walk dies at that site. When $d\to 1$ the Leaky-ASM converges to the abelian sandpile model (ASM) with a modified initial configuration. We also prove the limit shape is a circle when simultaneously with $n\to\infty$ we have that $d=d_n$ converges to $1$ slower than any power of $n$. To gain information about the ASM faster convergence is necessary., Comment: 30 pages, 10 figures. To be published in International Mathematics Research Notices. The proof of Lemma 3.3 has been simplified and we have corrected several typos

We consider non-topological solutions of a nonlinear elliptic system problem (see (1.4) below) derived from the S U ( 3 ) Chern-Simons models in R 2 . The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, Choe, Kim, and Lin in [7] , [8] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [7] , [8] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity.