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Global index of real polynomials Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-18 Gabriel E. Monsalve, Mihai Tibăr
We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$: in terms of the atypical fibres of $f$, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$.
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A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: justification as a shallow water approximation Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-18 Vincent Duchêne, Tatsuo Iguchi
We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present
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Decay at infinity for solutions to some fractional parabolic equations Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-14 Agnid Banerjee, Abhishek Ghosh
For $s\in [\tfrac {1}{2},\, 1)$ , let $u$ solve $(\partial _t - \Delta )^s u = Vu$ in $\mathbb {R}^{n} \times [-T,\, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$ . We show that if for some $0<\mathfrak {K} < T$ and $\epsilon >0$ \[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \] then $u \equiv
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Eternal solutions to a porous medium equation with strong non-homogeneous absorption. Part I: radially non-increasing profiles Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-14 Razvan Gabriel Iagar, Philippe Laurençot
Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption \[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \] posed for $(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$ , with $m>1$ , $q\in (0,\,1)$ and $\sigma =\sigma _c:=2(1-q)/ (m-1)$ is proved. Looking for radially symmetric solutions of the form \[ u(t,x)={\rm e}^{-\alpha
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Necessary and sufficient conditions for ground state solutions to planar Kirchhoff-type equations Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-11 Chunyu Lei, Binlin Zhang
In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem: \[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \] where $b,\, \omega >0$ are constants, $p>2$ . Based on variational methods, regularity theory and Schwarz symmetrization, the equivalence of ground state solutions for
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Index estimates of compact hypersurfaces in smooth metric measure spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-07 Márcio Batista, Matheus B. Martins
In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced
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Optimal inverse problems of potentials for two given eigenvalues of Sturm–Liouville problems Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-07 Min Zhao, Jiangang Qi
The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.
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Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-04 Chuang Xiang, Jicai Huang, Min Lu, Shigui Ruan, Hao Wang
In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo
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Invariant measures and large deviation principles for stochastic Schrödinger delay lattice systems Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-03-04 Zhang Chen, Xiaoxiao Sun, Bixiang Wang
This paper is concerned with stochastic Schrödinger delay lattice systems with both locally Lipschitz drift and diffusion terms. Based on the uniform estimates and the equicontinuity of the segment of the solution in probability, we show the tightness of a family of probability distributions of the solution and its segment process, and hence the existence of invariant measures on $l^2\times L^2((-\rho
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Invariant set generated by a nonreal number is everywhere dense Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-29 Artūras Dubickas
A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb
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On a critical time-harmonic Maxwell equation in nonlocal media Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-29 Minbo Yang, Weiwei Ye, Shuijin Zhang
In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media \[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \] where $\Omega \subset \mathbb {R}^{3}$ is a bounded domain, either
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Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-27 Debdip Ganguly, Diksha Gupta, K. Sreenadh
The paper is concerned with positive solutions to problems of the type \[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \] where $\mathbb {B}^N$ denotes the hyperbolic space, $1< p<2^*-1:=\frac {N+2}{N-2}$ , $\;\lambda < \frac {(N-1)^2}{4}$ , and $f \in H^{-1}(\mathbb {B}^{N})$ ( $f \not \equiv 0$ ) is a non-negative functional
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A collision result for both non-Newtonian and heat conducting Newtonian compressible fluids Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-26 Šárka Nečasová, Florian Oschmann
We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature-depending viscosities.
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Common valuations of division polynomials Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-26 Bartosz Naskręcki, Matteo Verzobio
In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$ . The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue
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Nonlocal anisotropic interactions of Coulomb type Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-26 Maria Giovanna Mora
In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness
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Qualitative properties of solutions for system involving the fractional Laplacian Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-26 Ran Zhuo, Yingshu Lü
In this paper, we consider the following non-linear system involving the fractional Laplacian 0.1 \begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation} in two different types of domains, one is bounded, and the other is an infinite cylinder, where $0< s<1$ . We employ the direct sliding method for fractional
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On the Γ-convergence of the Allen–Cahn functional with boundary conditions Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-12 Dimitrios Gazoulis
We study minimizers of the Allen–Cahn system. We consider the $\varepsilon$ -energy functional with Dirichlet values and we establish the $\Gamma$ -limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves
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On the Sobolev stability threshold for shear flows near Couette in 2D MHD equations Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-12 Ting Chen, Ruizhao Zi
In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field $(\alpha,0 )^\top$ on $\mathbb {T}\times \mathbb {R}$ . More precisely, for sufficiently large $\alpha$ , we show that when the initial datum of the shear flow satisfies $\left \| U(y)-y\right \|_{H^{N+6}}\ll 1$ , with $N>1$ , and the
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Remarks on a formula of Ramanujan Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-02-06 Andrés Chirre, Steven M. Gonek
Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.
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On the existence of a nodal solution for p-Laplacian equations depending on the gradient Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-31 F. Faraci, D. Puglisi
In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$ -Laplace operator. The same conclusion is obtained
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Regular matrices of unbounded linear operators Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-31 Paolo Leonetti
Let $X,\,Y$ be Banach spaces and fix a linear operator $T \in \mathcal {L}(X,\,Y)$ and ideals $\mathcal {I},\, \mathcal {J}$ on the nonnegative integers. We obtain Silverman–Toeplitz type theorems on matrices $A=(A_{n,k}: n,\,k \in \omega )$ of linear operators in $\mathcal {L}(X,\,Y)$ , so that \[ \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\mathcal{I}\text{-}\lim \boldsymbol{x}) \] for every $X$ -valued
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Karamata's theorem for regularized Cauchy transforms Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-26 Matthias Langer, Harald Woracek
We prove Abelian and Tauberian theorems for regularized Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularized Cauchy transform.
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On generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-23 Nirjan Biswas, Firoj Sk
For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$ , we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators: \[ (-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} \] where $\Omega
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Classification of simple smooth modules over the Heisenberg–Virasoro algebra Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-17 Haijun Tan, Yufeng Yao, Kaiming Zhao
In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$ , and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules
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Nonexpansive and noncontractive mappings on the set of quantum pure states Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-17 Michiya Mori, Peter Šemrl
Wigner's theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such
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Sharp gradient estimate, rigidity and almost rigidity of Green functions on non-parabolic RCD(0, N) spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-17 Shouhei Honda, Yuanlin Peng
Inspired by a result in T. H. Colding. (16). Acta. Math.209(2) (2012), 229-263 [16] of Colding, the present paper studies the Green function $G$ on a non-parabolic $\operatorname {RCD}(0,\,N)$ space $(X,\, \mathsf {d},\, \mathfrak {m})$ for some finite $N>2$ . Defining $\mathsf {b}_x=G(x,\, \cdot )^{\frac {1}{2-N}}$ for a point $x \in X$ , which plays a role of a smoothed distance function from $x$
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Branching patterns of wave trains in mass-in-mass lattices Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-11 Ling Zhang, Shangjiang Guo
We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant
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Quasilinear duality and inversion in Banach spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-11 Jesús M. F. Castillo, Manuel González
We present a unified approach to the processes of inversion and duality for quasilinear and $1$ -quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.
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Cavitation of a spherical body under mechanical and self-gravitational forces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-08 Pablo V. Negrón–Marrero, Jeyabal Sivaloganathan
In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an
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Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-08 Hyung Jun Choi, Seonghak Kim, Youngwoo Koh
We generalize the one-dimensional population model of Anguige & Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet
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Nowhere scattered multiplier algebras Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-05 Eduard Vilalta
We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$ -algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$ , that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$ -unital $\mathrm {C}^*$ -algebra $A$ of (i) finite nuclear dimension, or (ii) real rank zero, or (iii) stable rank one with
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On Calderon's problem for the connection Laplacian Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2024-01-05 Ravil Gabdurakhmanov, Gerasim Kokarev
We consider Calderón's problem for the connection Laplacian on a real-analytic vector bundle over a manifold with boundary. We prove a uniqueness result for this problem when all geometric data are real-analytic, recovering the topology and geometry of a vector bundle up to a gauge transformation and an isometry of the base manifold.
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Various topological complexities of small covers and real Bott manifolds Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-12-27 Koushik Brahma, Bikramaditya Naskar, Soumen Sarkar, Subhankar Sau
In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices. Then we compute symmetric topological complexity of several small covers over a product of simplices. We calculate the LS one-category of real Bott manifolds and infinitely
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The lower bounds of non-real eigenvalues for singular indefinite Sturm–Liouville problems Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-12-22 Fu Sun
The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.
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Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-12-21 Konstantinos T. Gkikas, Phuoc-Tai Nguyen
Let $\Omega \subset \mathbb {R}^N$ ( $N\geq 3$ ) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$ , $0\leq k \leq N-1$ . We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$ . Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$ , where $\mu \in {\mathbb
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On stable commutator length of non-filling curves in surfaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-12-14 Max Forester, Justin Malestein
We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling $1$ –chains.
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Partitions with parts separated by parity: conjugation, congruences and the mock theta functions Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-11-30 Shishuo Fu, Dazhao Tang
Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized
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Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-11-28 Andreas Minne, David Tewodrose
The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $\mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D
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Spaces of functions and sections with paracompact domain Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-11-21 Jaka Smrekar
We study spaces of continuous functions and sections with domain a paracompact Hausdorff k-space $X$ and range a nilpotent CW complex $Y$ , with emphasis on localization at a set of primes. For $\mathop {\rm map}\nolimits _\phi (X,\,Y)$ , the space of maps with prescribed restriction $\phi$ on a suitable subspace $A\subset X$ , we construct a natural spectral sequence of groups that converges to $\pi
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Trend to equilibrium solution for the discrete Safronov–Dubovskiĭ aggregation equation with forcing Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-11-16 Arijit Das, Jitraj Saha
We consider the discrete Safronov-Dubovskiĭ aggregation equation associated with the physical condition, where particle injection and extraction take place in the dynamical system. In application, this model is used to describe the aggregation of particle-monomers in combination with sedimentation of particle-clusters. More precisely, we prove well-posedness of the considered model for a large class
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A Lebesgue–Lusin property for linear operators of first and second order Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-11-06 Adolfo Arroyo-Rabasa
We prove that for a homogeneous linear partial differential operator $\mathcal {A}$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying \[ \mathcal{A} u(x)= f(x) \qquad \text{almost everywhere}. \] This extends a result of G. Alberti for gradients on $\mathbf {R}^N$ . In particular, for
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Existence of renormalized solutions to fully anisotropic and inhomogeneous elliptic problems Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-31 Bartosz Budnarowski, Ying Li
We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper
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Smooth integers and de Bruijn's approximation Ʌ Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-31 Ofir Gorodetsky
This paper is concerned with the relationship of $y$ -smooth integers and de Bruijn's approximation $\Lambda (x,\,y)$ . Under the Riemann hypothesis, Saias proved that the count of $y$ -smooth integers up to $x$ , $\Psi (x,\,y)$ , is asymptotic to $\Lambda (x,\,y)$ when $y \ge (\log x)^{2+\varepsilon }$ . We extend the range to $y \ge (\log x)^{3/2+\varepsilon }$ by introducing a correction factor
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Arens regularity of ideals of the group algebra of a compact Abelian group Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-27 Reza Esmailvandi, Mahmoud Filali, Jorge Galindo
Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$ . We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$ . Arens regularity of $L_E^1(G)$ , the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal
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Differential Harnack estimates for a weighted nonlinear parabolic equation under a super Perelman–Ricci flow and implications Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-27 Ali Taheri, Vahideh Vahidifar
In this paper, we derive new differential Harnack estimates of Li–Yau type for positive smooth solutions to a class of nonlinear parabolic equations in the form \[ {\mathscr L}_\phi^{\mathsf a} [w]:= \left[ \frac{\partial}{\partial t} - \mathsf{a}(x,t) - \Delta_\phi \right] w (x,t) = \mathscr G(t, x, w(x,t)), \quad t>0, \] on smooth metric measure spaces where the metric and potential are time dependent
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Prescribing nearly constant curvatures on balls Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-27 Luca Battaglia, Sergio Cruz-Blázquez, Angela Pistoia
In this paper, we address two boundary cases of the classical Kazdan–Warner problem. More precisely, we consider the problem of prescribing the Gaussian and boundary geodesic curvature on a disk of $\mathbb {R}^2$ , and the scalar and mean curvature on a ball in higher dimensions, via a conformal change of the metric. We deal with the case of negative interior curvature and positive boundary curvature
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The space of commuting elements in a Lie group and maps between classifying spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-27 Daisuke Kishimoto, Masahiro Takeda, Mitsunobu Tsutaya
Let $\pi$ be a discrete group, and let $G$ be a compact-connected Lie group. Then, there is a map $\Theta \colon \mathrm {Hom}(\pi,G)_0\to \mathrm {map}_*(B\pi,BG)_0$ between the null components of the spaces of homomorphisms and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective
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Lipschitz-free spaces and subsets of finite-dimensional spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-17 Jan Bíma
We consider two questions on the geometry of Lipschitz-free $p$ -spaces $\mathcal {F}_p$ , where $0< p\leq 1$ , over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then $\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every $\alpha \in
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Spectral mapping theorems for essential spectra and regularized functional calculi Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-12 Jesús Oliva-Maza
Gramsch and Lay [8] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$ , \[ \widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), \] for several extended essential spectra $\widetilde {\sigma }_i$ . In this work, we extend such theorems for the regularized functional calculus introduced by Haase [10, 11] assuming suitable conditions on $f$ . At the same
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Fixed point sets and the fundamental group II: Euler characteristics Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-10 Sylvain Cappell, Shmuel Weinberger, Min Yan
For a finite group $G$ of not prime power order, Oliver showed that the obstruction for a finite CW-complex $F$ to be the fixed point set of a contractible finite $G$ -CW-complex is determined by the Euler characteristic $\chi (F)$ . (He also has similar results for compact Lie group actions.) We show that the analogous problem for $F$ to be the fixed point set of a finite $G$ -CW-complex of some given
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Topological regularity of isoperimetric sets in PI spaces having a deformation property Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-09 Gioacchino Antonelli, Enrico Pasqualetto, Marco Pozzetta, Ivan Yuri Violo
We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply
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Long-time dynamics and semi-wave of a delayed nonlocal epidemic model with free boundaries Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-05 Qiaoling Chen, Sanyi Tang, Zhidong Teng, Feng Wang
This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined
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From the Mayer–Vietoris spectral sequence to überhomology Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-02 Luigi Caputi, Daniele Celoria, Carlo Collari
We prove that the second page of the Mayer–Vietoris spectral sequence, with respect to anti-star covers, can be identified with another homological invariant of simplicial complexes: the $0$ -degree überhomology. Consequently, we obtain a combinatorial interpretation of the second page of the Mayer–Vietoris spectral sequence in this context. This interpretation is then used to extend the computations
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Reconstruction of topological graphs and their Hilbert bimodules Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-10-02 Rodrigo Frausino, Abraham C.S. Ng, Aidan Sims
We show that the Hilbert bimodule associated with a compact topological graph can be recovered from the $C^*$ -algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson–Katsoulis and of Davidson–Roydor on local conjugacy of topological graphs and isomorphism
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Nonlinear elasticity with vanishing nonlocal self-repulsion Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-09-28 Stefan Krömer, Philipp Reiter
We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$ -limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev–Slobodeckiĭ
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Generic ill-posedness of the energy–momentum equations and differential inclusions Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-09-26 Erik Duse
We show that the energy–momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion into the orthogonal group and in particular these solutions can be nowhere $C^1$ . We prove that these solutions are not stationary
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Sharp convergence for sequences of Schrödinger means and related generalizations Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-09-25 Wenjuan Li, Huiju Wang, Dunyan Yan
For decreasing sequences $\{t_{n}\}_{n=1}^{\infty }$ converging to zero and initial data $f\in H^s(\mathbb {R}^N)$ , $N\geq 2$ , we consider the almost everywhere convergence problem for sequences of Schrödinger means ${\rm e}^{it_{n}\Delta }f$ , which was proposed by Sjölin, and was open until recently. In this paper, we prove that if $\{t_n\}_{n=1}^{\infty }$ belongs to Lorentz space ${\ell }^{r
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On a class of special Euler–Lagrange equations Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-09-21 Baisheng Yan
We make some remarks on the Euler–Lagrange equation of energy functional $I(u)=\int _\Omega f(\det Du)\,{\rm d}x,$ where $f\in C^1(\mathbb {R}).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$
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Shape index, Brouwer degree and Poincaré–Hopf theorem Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-09-14 Héctor Barge, José M.R. Sanjurjo
In this paper, we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar é–Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of
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Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces Proc. R. Soc. Edinburgh Sect. A (IF 1.3) Pub Date : 2023-09-14 Shangquan Bu, Gang Cai
In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ( $P_4$ ): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$ , where $A$ , $B$ , $L$ and $M$ are closed linear