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Given a fixed hypergraph H, let \operatorname {wsat}(n,H) denote the smallest number of edges in an n-vertex hypergraph G, with the property that one can sequentially add the edges missing from G, so that whenever an edge is added, a new copy of H is created. The study of \operatorname {wsat}(n,H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H very little is known regarding \operatorname {wsat}(n,H), Alon proved in 1985 that for every graph H there is a limiting constant C_H so that \operatorname {wsat}(n,H)=(C_H+o(1))n. Tuza conjectured in 1992 that Alon's theorem can be (appropriately) extended to arbitrary r-uniform hypergraphs. In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime p is the same in both fields for almost all prime p. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely on the degree of the fields. More specifically, for every positive n we exhibit a positive constant c_{n} such that any two degree n number fields K and L are arithmetically equivalent if and only if the set of primes p such that the arithmetic type of p in K and L is not the same has Dirichlet density at most c_n. We in fact show that \displaystyle c_n=\frac {1}{4n^2} works and give a heuristic evidence that points to the fact that this value might be improved to \displaystyle \frac {2}{n^2}. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such number. [ABSTRACT FROM AUTHOR]

Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage's Liouville-type theorem. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists a unique weak solution to the Neumann boundary value problem for second order elliptic operators whose coefficients are signed measures. We will give a probabilistic representation of the solution. The heat kernel estimates and reflected diffusion processes play important roles. [ABSTRACT FROM AUTHOR]

Given a smooth map f_V\colon V \to K with f^*_V (\nu _K) = \nu _V, a general question arises: under which conditions there exists a smooth extension f\colon M \to N of f_V such that f is transverse to K and f^{-1}(K) = V, where M, N are smooth closed manifolds of dimension m and n, V, K are closed submanifolds of M and N, respectively, of same codimension and \nu _K, \nu _V are the normal bundles of K in N and V in M, respectively. In this paper, we give conditions to the existence of extensions, by using bordism intersection product. Moreover, we present an interesting and non-trivial example illustrating the systematic construction of such extensions, skeletonwise. [ABSTRACT FROM AUTHOR]

In this paper, a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure on the Teichmüller space of genus g\ge 2 is obtained. [ABSTRACT FROM AUTHOR]

The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to Da Prato and Röckner [ Seminar on stochastic analysis, random fields and applications V , Birkhäuser, Basel, 2008] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future. [ABSTRACT FROM AUTHOR]

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator in \mathbb {R}^4, denoted by \mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator \Delta in four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. \Delta = \mathcal {\overline {D}} \mathcal {D} to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of \mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the S-spectrum. [ABSTRACT FROM AUTHOR]

All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman [J. Topol. 2 (2009), pp. 227–248]. [ABSTRACT FROM AUTHOR]

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on \mathbb {R}^d (5 \leq d \leq 8) enjoys the same decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result by Pausader [J. Funct. Anal. 256 (2009), pp. 2473–2517]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics. [ABSTRACT FROM AUTHOR]

This paper proves the persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems with Bruno non-degeneracy conditions, whose frequency vector is a small dilation of the prescribed one. The proof is based on the stability of real roots of approximating real odd-order polynomials. [ABSTRACT FROM AUTHOR]

This paper presents a curvature-free version of the \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we study some geometric properties of a class of analytic functions which is defined from the J-fraction expansion of the ratio {zf'(z)}/{f(z)}. We find the disk domain which is mapped into a starlike domain by these functions. Moreover, we study similar results for two different normalized forms of regular Coulomb wave functions and a normalized Bessel function of the first kind by using continued fractions expansions. [ABSTRACT FROM AUTHOR]

In this paper, we study a locally constrained mean curvature flow with free boundary in a hyperbolic ball. Under the flow, the enclosed volume is preserved and the area is decreasing. We prove the long time existence and smooth convergence for such flow under certain star-shaped condition. As an application, we give a flow proof of the isoperimetric problem for the star-shaped free boundary hypersurfaces in a hyperbolic ball. [ABSTRACT FROM AUTHOR]

In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model M of PA (Peano arithmetic) is recursively saturated iff M has an expansion that satisfies the subsystem Δ11−CA0 of second order arithmetic. In this paper we identify a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, and we offer a correct proof of the problematic direction. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of a standing wave solution for the nonlinear Schrödinger equation with a real-valued linear potential in energy-subcritical. Moreover, we also prove global well-posedness to the Cauchy problem with the initial data, whose action is less than that of the ground state with the potential. Hong [Commun. Pure Appl. Anal. 15 (2016), pp. 1571-1601] and the authors showed a scattering result for the problem with the initial data, whose action is less than that of the ground state without the potential. It was noted that the action of the ground state with the potential is greater than that of the ground state without the potential. Our new contribution enables us to treat initial data, which were not treated in those papers. [ABSTRACT FROM AUTHOR]

Let \mathcal {L}=-\Delta _{\mathbb {H}^n}+V be a Schrödinger operator with the nonnegative potential V belonging to the reverse Hölder class B_{Q}, where Q is the homogeneous dimension of the Heisenberg group \mathbb {H}^n. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to \mathcal {L}. As an application, we characterize the space BMO_{\mathcal {L}}(\mathbb {H}^n), associated to the Schrödinger operator \mathcal {L}, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup \{e^{-s\mathcal {L}}\}_{s>0} and the Poisson kernel of the semigroup \{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}, respectively. At last, we pose a conjecture about the converse characterization of BMO_{\mathcal {L}}(\mathbb {H}^n). [ABSTRACT FROM AUTHOR]

In this paper we establish a new identity, and then based on this identity we derive the Maclaurin's inequality for multiplicatively convex functions. [ABSTRACT FROM AUTHOR]

In this paper, the nontriviality of homotopy elements \beta _1\varrho _s in the p-primary component of stable homotopy groups of spheres is shown by virtue of the Adams spectral sequence and the May spectral sequence, where \varrho _s is represented by b_0g_{0}\widetilde {\gamma }_{s} in the Adams spectral sequence, p is a prime number greater than 5, and 3\leq s

In this paper, we prove that if K is an o-symmetric cylinder and L is an o-symmetric convex body in \mathbb R^3, then the logarithmic Minkowski inequality \[ \frac {1}{V(K)}\int _{\mathbb S^{2}}\log \frac {h_L}{h_K}\,dV_K\geq \frac {1}{3}\log \frac {V(L)}{V(K)} \] holds, with equality if and only if K and L are relative cylinders. [ABSTRACT FROM AUTHOR]

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189–244]. [ABSTRACT FROM AUTHOR]

In this paper we completely solve the open problem of finding the fundamental solution of the semidiscrete fractional-spatial damped wave equation. We combine operator theory and Laplace transform methods with properties of Bessel functions to show an explicit representation of the solution when initial conditions are given. Our findings extend known results from the literature and also provide new insights into the qualitative behavior of the solutions for the studied model. As an example, we show the existence of almost periodic solutions as well as their profile in the homogeneous case. [ABSTRACT FROM AUTHOR]

For k\ge 1, the homogeneous symmetric functions G(k,m) of degree m defined by \sum _{m\ge 0} G(k,m) z^m=\prod _{i\ge 1} \big (1+x_iz+x^2_iz^2+\cdots +x^{k-1}_iz^{k-1}\big) are called Petrie symmetric functions. As derived by Grinberg and Fu–Mei independently, the expansion of G(k,m) in the basis of Schur functions s_{\lambda } turns out to be signed multiplicity free, i.e., the coefficients are -1, 0 and 1. In this paper we give a combinatorial interpretation of the coefficient of s_{\lambda } in terms of the k-core of \lambda and a sequence of rim hooks of size k removed from \lambda. We further study the product of G(k,m) with a power sum symmetric function p_n. For all n\ge 1, we give necessary and sufficient conditions on the parameters k and m in order for the expansion of G(k,m)\cdot p_n in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case n=2. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the L^p restriction estimates for the truncated conic surface \begin{equation*} \Sigma =\big \{(\xi ',\xi _n,-\xi _n^{-1}\langle \xi ',N\xi '\rangle): (\xi ',\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*} with N=I_{n-1-m}\oplus (-I_m) for m\leq \lfloor \tfrac {n-3}2\rfloor provided p>\tfrac {2(n+3)}{n+1}. The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps. [ABSTRACT FROM AUTHOR]

According to Tzouvaras, a set is nontypical in the Russell sense if it belongs to a countable ordinal definable set. The class \mathbf {HNT} of all hereditarily nontypical sets satisfies all axioms of \mathbf {ZF} and the double inclusion \mathbf {HOD}\subseteq \mathbf {HNT}\subseteq \mathbf {V} holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of \mathbf {ZFC} is presented in which \mathbf {HOD}\subsetneqq \mathbf {HNT}\subsetneqq \mathbf {V}, and another model of \mathbf {ZFC} in which \mathbf {HNT} does not satisfy the axiom of choice. [ABSTRACT FROM AUTHOR]

Let \psi :\mathbb {R}_+\to \mathbb {R}_+ be a non-increasing function. A real number x is said to be \psi-Dirichlet improvable if the system \begin{equation*} |qx-p|< \psi (t) \ \ {\text {and}} \ \ |q|

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia [Trans. Amer. Math. Soc. 372 (2019), pp. 6771–6803]. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzschild manifold. [ABSTRACT FROM AUTHOR]

In this paper we consider multivariate approximation by weighted polynomials of the form w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}), where p_n is a multivariate polynomial of degree at most n, w is a given nonnegative weight with nonempty zero set, and \gamma _n\uparrow \infty. We study the question if every continuous function vanishing on the zero set of w is a uniform limit of weighted polynomials w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}). It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that \gamma _n=o(n). [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any bounded set of finite perimeter \Omega \subset \mathbb {R}^n, we can choose smooth sets E_k \Subset \Omega such that E_k \rightarrow \Omega in L^1 and \begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega)+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In the above \Omega ^1 is the measure-theoretic interior of \Omega, P(\cdot) denotes the perimeter functional on sets, and C_1(n) is a dimensional constant. Conversely, we prove that for any sets E_k \Subset \Omega satisfying E_k \rightarrow \Omega in L^1, there exists a dimensional constant C_2(n) such that the following inequality holds: \begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega)+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In particular, these results imply that for a bounded set \Omega of finite perimeter, \begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*} holds if and only if there exists a sequence of smooth sets E_k such that E_k \Subset \Omega, E_k \rightarrow \Omega in L^1 and P(E_k) \rightarrow P(\Omega). [ABSTRACT FROM AUTHOR]

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if f\in A_t^p(\mathcal {U}) and \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index \alpha, then f\equiv 0, where \mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with \mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n} for j=1,\ldots, n-1 and \mathcal {L}_n = \frac {\partial }{\partial z_n}. As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a "derivative norm", which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space. [ABSTRACT FROM AUTHOR]

We prove that a parabolic subgroup P contained in another parabolic subgroup P' of an Artin group A is a parabolic subgroup of P'. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction A \to P of the inclusion map from a parabolic subgroup P into A. This retraction was implicitly constructed in a previous paper by Charney and the second author [Bull. Lond. Math. Soc. 46 (2014), pp. 1248–1255]. [ABSTRACT FROM AUTHOR]

Levin, Przytycki and Shen [Invent. Math. 205 (2016), pp. 363–382] proved for a polynomial map f_c(z)=z^d+c, d\geq 2 and c \in \mathbb C, with Julia set J(f) of positive measure that for a.e. z \in J(f) the Lyapunov exponent \chi _s(z)=0. The aim of this paper is to show that the extension to non-entire transcendental meromorphic functions is not possible. [ABSTRACT FROM AUTHOR]

We characterize the norm attainment toward vectors for vector-valued Lipschitz maps defined on a general metric space. The main theorem of the present paper states that on a large class of metric spaces including infinite subsets of finite-dimensional spaces, every Lipschitz map attains its norm toward a vector if and only if the range space is finite-dimensional. Furthermore, motivated by the first negative example given by G. Godefroy, some denseness results for norm attaining Lipschitz maps toward vectors are also presented. [ABSTRACT FROM AUTHOR]

This paper deals with the dynamics of a system governed by a multi-dimensional linear hyperbolic PDE. The dynamical behaviors of linear PDEs extremely depend on the selection of space, and a conventional way is to define a infinite-dimensional space with a tuning parameter. Thereby, the linear PDEs can exhibit chaos or stability in the different range of tuning parameter. In this work, the chaos of the C_0-semigroup corresponding to the system is established on the Banach space of multivariate analytic functions when the tuning parameter exceeds some given positive number. Based on this, both Devaney and distributional chaos of the system are further obtained. Meanwhile, the C_0-semigroup is proved to be uniformly exponentially stable when the tuning parameter is less than a certain positive number, which contributes to showing the global stability of the system. Finally, two examples are given to illustrate effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Finsler measure spaces whose weighted Ricci curvature satisfies {\mathbf {Ric}}_{\infty }=cF^{2}. This class contains all gradient Ricci solitons and Finsler Gaussian shrinking solitons. Thus Finsler measure spaces in this class are called Finsler gradient Ricci solitons. For a Randers measure space, we find sufficient and necessary conditions for this space to be a Finsler gradient Ricci soliton. In particular, we show that Randers-Finsler gradient Ricci solitons must have isotropic S-curvature. Finally, we give an equivalent condition for a Randers measure space to be a Finsler gradient Ricci soliton of constant S-curvature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second example is constructed for initial data with small energy but nontrivial topology. A counter example of maximum principle to the system is constructed by utilizing the Poiseuille flow in dimension one. [ABSTRACT FROM AUTHOR]

In this paper a multiplier theorem in the Hardy space H^1(\mathbb {T}) associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence \left \{\lambda _n\right \}_{n=-\infty }^{\infty } is a multiplier from H^1(\mathbb {T}) into the sequence space \ell ^1(\mathbb {Z}) associated with Jacobi expansions of exponential type, if \[ \sup _N\sum _{k=1}^{\infty }\left (\sum _{kN<|j|\le (k+1)N}|\lambda _j|\right)^2<\infty.\] This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space H^1(\mathbb {T}), and for \lambda _n=(|n|+1)^{-1}, a Hardy type inequality for Jacobi expansions is immediate which has ever been proved by Kanjin and Sato [Math. Inequal. Appl. 7 (2004), pp. 551–555]. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness theory for Bergman projection in the operator-valued setting. More precisely, let \mathbb {D} be the open unit disk in the complex plane \mathbb {C} and \mathcal {M} be a semifinite von Neumann algebra. We prove that \begin{equation*} \|P(f)\|_{L_{1,\infty }(\mathcal {N})}\leq C \|f\|_{L_1(\mathcal {N})}, \end{equation*} where \mathcal {N}=L_{\infty }(\mathbb {D})\bar {\otimes }\mathcal {M} and P denotes the Bergman projection. Consequently, P is bounded on L_{p}(\mathcal {N}) with 1

The theory of saturated fusion systems resembles in many parts the theory of finite groups. However, some concepts from finite group theory are difficult to translate to fusion systems. For example, products of normal subsystems with other subsystems are only defined in special cases. In this paper the theory of localities is used to prove the following result: Suppose \mathcal {F} is a saturated fusion system over a p-group S. If \mathcal {E} is a normal subsystem of \mathcal {F} over T\leq S, and \mathcal {D} is a subnormal subsystem of N_\mathcal {F}(T) over R\leq S, then there is a subnormal subsystem \mathcal {E}\mathcal {D} of \mathcal {F} over TR, which plays the role of a product of \mathcal {E} and \mathcal {D} in \mathcal {F}. If \mathcal {D} is normal in N_\mathcal {F}(T), then \mathcal {E}\mathcal {D} is normal in \mathcal {F}. It is shown along the way that the subsystem \mathcal {E}\mathcal {D} is closely related to a naturally arising product in certain localities attached to \mathcal {F}. [ABSTRACT FROM AUTHOR]

We force the existence of a non-trivial \kappa-complete ultrafilter over \kappa which fails to satisfy the Galvin property. This answers a question asked by Benhamou and Gitik [Ann. Pure Appl. Logic 173 (2022), Paper No. 103107]. [ABSTRACT FROM AUTHOR]

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at \varphi (0) = \frac {\mu }{2}. The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma). [ABSTRACT FROM AUTHOR]

This paper introduces a power-law subordinator and a power-law Lévy process whose Laplace transform and characteristic function are simply made up of power functions or the ratio of power functions, respectively, and proposes a power-law vector random field whose finite-dimensional characteristic functions consist merely of a power function or the ratio of two power functions. They may or may not have first-order moment, and contain Linnik, variance Gamma, and Laplace Lévy processes (vector random fields) as special cases. For a second-order power-law vector random field, it is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. An important feature of the power-law Lévy processes (random fields) is that they can be used as the building blocks to construct other Lévy processes (random fields), such as hyperbolic secant, cosine ratio, and sine ratio Lévy processes (random fields). [ABSTRACT FROM AUTHOR]

Let \mathcal {H} be a separable Hilbert space and let B(\mathcal {H}) be the *-algebra of all bounded linear operators on \mathcal {H}. In the present paper, we prove that a positive/regular operator from L_1(0,1) into an arbitrary separable operator ideal in B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra \mathcal {M} and an arbitrary KB -ideal C_E in B(\mathcal {H}), the predual \mathcal {M}_* of \mathcal {M} is not isomorphic to any subspace of C_E. This observation complements several earlier results. [ABSTRACT FROM AUTHOR]

Let B be an integral domain over a field K of characteristic 0. The derivation δ of B[Yd] = B[y1, . . . ,yd] is elementary if δ (B) = 0 and δ (yi) ∈ B, i = 1, . . . ,d. Then for d ≥ 2 the elements uij = δ (yi)yj-δ (yj)yi, 1 ≤ i < j ≤ d, belong to the algebra B[Yd]δ of constants of δ, and it is a natural question whether the B-algebra B[Yd]δ is generated by all uij. In this paper we consider the special case of B = K[Xd] = K[x1, . . . ,xd]. If δ (yi) = xi, i = 1, . . . ,d, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case δ (yi) = xini, ni > 0, i = 1, . . . ,d, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if δ (yi) = ƒi(xi), for any nonconstant polynomials ƒi(xi), i = 1, . . . ,d, then B[Yd]δ = K[Xd,Yd]δ is generated by Xd and Ud = uij = ƒi(xi)yj-yiƒj(xj) mid 1 ≤ i < j ≤ d. In the present paper we have found a presentation of the algebra K[Xd,Yd]δ = K[Xd,Ud mid R = S = 0],d ≥ 4, R = r(i,j,k,l)mid 1 ≥ i < j < k < l \≤ d, S = s(i,j,k)mid 1 ≤ i < j < k ≤ d, and a basis of K[Xd,Yd]δ as a vector space. As a corollary we have shown that the defining relations R ∩ S form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras K[Xd,Yd]δ, d < 4, are also described. [ABSTRACT FROM AUTHOR]

The lines of curvature of a surface embedded in R3 together with umbilic points comprise its principal foliations. Monge described the principal foliations of the triaxial ellipsoid in the eighteenth century, but few examples beyond these and surfaces of revolution have been characterized since. This paper analyzes the principal foliations of perturbations of the ellipsoid. The results are surprising. Notably, surfaces with dense lines of curvature are prevalent A geometric construction establishes an intimate relation of the principal foliation on perturbed ellipsoids with nonvanishing vector fields on the two dimensional torus T2, rather than the two sphere S2. The dynamics of nonvanishing vector fields on T2 with a global cross-section has been extensively studied. In generic one-parameter families, vector fields with dense, quasiperiodic trajectories occur at positive measure sets of parameters. This paper establishes a comparable result proving that large sets of embeddings of S3 in R3 have dense lines of curvature. [ABSTRACT FROM AUTHOR]

The goal of this paper is to address the following question: if A is an Ek-ring for some k ≥ 1 and ƒ: π0 A → B is a map of commutative rings, when can we find an Ek-ring R with an Ek-ring map g: A → R such that π0 g = ƒ? A classical result in the theory of realizing E∞-rings, due to Goerss-Hopkins, gives an affirmative answer to this question if ƒ is étale. The goal of this paper is to provide answers to this question when ƒ is ramified. We prove a non-realizability result in the K(n)-local setting for every n ≥ 1 for H∞-rings containing primitive pth roots of unity. As an application, we give a proof of the folk result that the Lubin-Tate tower from arithmetic geometry does not lift to a tower of H∞-rings over Morava E-theory. [ABSTRACT FROM AUTHOR]

Let C be a 2-Calabi-Yau triangulated category, and let T be a cluster tilting subcategory of C. An important result from Dehy and Keller tells us that a rigid object c ∈ C is uniquely defined by its index with respect to T. The notion of triangulated categories extends to the notion of (d + 2)-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a (d + 2)-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate that this result applies to higher dimensional cluster categories of Dynkin-type A. [ABSTRACT FROM AUTHOR]