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In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974., Comment: 16 pages with large spacings, comments welcome

Let H \mathcal {H} be a separable Hilbert space and let B ( H ) B(\mathcal {H}) be the ∗ * -algebra of all bounded linear operators on H \mathcal {H} . In the present paper, we prove that a positive/regular operator from L 1 ( 0 , 1 ) L_1(0,1) into an arbitrary separable operator ideal in B ( H ) 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 M \mathcal {M} and an arbitrary KB-ideal C E C_E in B ( H ) B(\mathcal {H}) , the predual M ∗ \mathcal {M}_* of M \mathcal {M} is not isomorphic to any subspace of C E C_E . This observation complements several earlier results.

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})} . Here, f ¯ Ω \bar {f}_{\Omega } denotes the mean value of f f over Ω \Omega . In the case that Ω \Omega is a ball B r B_r of radius r r in R n \mathbb {R}^n , we calculate C ( n , B r ) = C ( n ) r C(n,B_r)=C(n)r explicitly in terms of n n and a ratio of the volumes of the unit balls in R 2 n − 1 \mathbb {R}^{2n-1} and R n \mathbb {R}^n . More generally, we prove that C ( n , B r ( Ω ) ) ≤ C ( n , Ω ) ≤ n n + 1 d i a m ( Ω ) C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega ) , where B r ( Ω ) B_{r(\Omega )} is a ball in R n \mathbb {R}^n with the same n − n- dimensional Lebesgue measure as Ω \Omega . Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

We prove that the Lipschitz-free space over a Banach space X X of density κ \kappa , denoted by F ( X ) \mathcal {F}(X) , is linearly isomorphic to its ℓ 1 \ell _1 -sum ( ⨁ κ F ( X ) ) ℓ 1 \left (\bigoplus _{\kappa }\mathcal {F}(X)\right )_{\ell _1} . This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 0 over a L p \mathcal {L}_p -space. More precisely, we establish that, for every 1 ≤ p ≤ ∞ 1\leq p\leq \infty , if X X is a L p \mathcal {L}_p -space of density κ \kappa , then L i p 0 ( X ) \mathrm {Lip}_0(X) is either isomorphic to L i p 0 ( ℓ p ( κ ) ) \mathrm {Lip}_0(\ell _p(\kappa )) if p > ∞ p>\infty , or L i p 0 ( c 0 ( κ ) ) \mathrm {Lip}_0(c_0(\kappa )) if p = ∞ p=\infty .

We study the quaternionic Monge-Ampère equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Ampère equation has always a unique solution for every basic datum. This approach includes the study of the equation on SU(3)., 12 pages, last version to appear in Proc. Amer. Math. Soc

We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes.

We force the existence of a non-trivial κ \kappa -complete ultrafilter over κ \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].

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

We introduce and study a class of abstract neutral differential equations with state-dependent delay at the temporal derivative. We study the local and global existence and uniqueness of mild solution and the existence of strict solution. Some examples concerning partial neutral integro-differential equations are presented.

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on [ 0 , 1 ] [0,1] that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of R \mathbb {R} , but in subsets of the p p -adic field Q p \mathbb {Q}_p .

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero.

In this article, we investigate the existence and uniqueness of solutions of linear and semilinear second–order equations involving time scales. To obtain such results, we make use of exponential dichotomy and fixed point results. Also, we present some examples and applications to illustrate our main results.

For a prime number p p , we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2 p 2p . In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes.

We introduce and investigate quantitative versions of the positive Schur property. We show, in particular, that each L 1 ( μ ) L_{1}(\mu ) -space( μ \mu σ \sigma -finite) enjoys the 1 1 -positive Schur property in the strongest possible way. By introducing a quantity measuring L L -weak non-compactness of sets in Banach lattices, we prove that the positive Schur property is always quantitative. We also show that this quantity is equal to several natural measures of weak non-compactness in abstract L L -spaces.

A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory with relations and applications to many other areas of mathematics as well as to theoretical computer science. A central problem in the area due to Erdős and Rado from 1960 asks for the minimum number of sets of size r r needed to guarantee the existence of a sunflower of a given size. Despite a lot of recent attention including a polymath project and some amazing breakthroughs, even the asymptotic answer remains unknown. We study a related problem first posed by Duke and Erdős in 1977 which requires that in addition the intersection size of the desired sunflower be fixed. This question is perhaps even more natural from a graph theoretic perspective since it asks for the Turán number of a hypergraph made by the sunflower consisting of k k edges, each of size r r and with common intersection of size t t . For a fixed size of the sunflower k k , the order of magnitude of the answer has been determined by Frankl and Füredi. In the 1980’s, with certain applications in mind, Chung, Erdős and Graham considered what happens if one allows k k , the size of the desired sunflower, to grow with the size of the ground set. In the three uniform case, r = 3 r=3 , the correct dependence on the size of the sunflower has been determined by Duke and Erdős and independently by Frankl and in the four uniform case by Bucić, Draganić, Sudakov and Tran. We resolve this problem for any uniformity, by determining up to a constant factor the n n -vertex Turán number of a sunflower of arbitrary uniformity r r , common intersection size t t and with the size of the sunflower k k allowed to grow with n n .

Let s ( n ) = ∑ d ∣ n , d > n d s(n)=\sum _{d\mid n,~d>n} d denote the sum of the proper divisors of n n . The second-named author proved that ω ( s ( n ) ) \omega (s(n)) has normal order log ⁡ log ⁡ n \log \log {n} , the analogue for s s -values of a classical result of Hardy and Ramanujan [The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92], AMS Chelsea Publ., Providence, RI, 2000, pp. 262–275]. We establish the corresponding Erdős–Kac theorem: ω ( s ( n ) ) \omega (s(n)) is asymptotically normally distributed with mean and variance log ⁡ log ⁡ n \log \log {n} . The same method applies with s ( n ) s(n) replaced by any of several other unconventional arithmetic functions, such as β ( n ) ≔ ∑ p ∣ n p \beta (n)≔\sum _{p\mid n} p , n − φ ( n ) n-\varphi (n) , and n + τ ( n ) n+\tau (n) ( τ \tau being the divisor function).

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).

We show that, under suitable conditions, an operator acting like a shift on some sequence space has a frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. We also apply it to every operator satisfying the Frequent Hypercyclicity Criterion, recovering a result of Murillo and Peris., Comment: 21 pages; results not valid for arbitrary F-spaces but for locally bounded or locally convex F-spaces

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t t -modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t t -modules which lack the height property. In particular, let A A be the polynomial ring in one variable θ \theta over the finite field F q \mathbb {F}_q with the quotient field K K . Let ( E , ϕ ) (E, \phi ) be an iterated extension of the tensor powers of the Carlitz module defined over K K . We show that the F q [ t ] \mathbb {F}_q[t] -module E ( K ) E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free F q [ t ] \mathbb {F}_q[t] -module of rank ℵ 0 \aleph _0 .

A classical observation of Deligne shows that, for any prime $p \geq 5$, the divisor polynomial of the Eisenstein series $E_{p-1}(z)$ mod $p$ is closely related to the supersingular polynomial at $p$, $$S_p(x) := \prod_{E/\bar{\mathbb{F}}_p \text{ supersingular}}(x-j(E)) \in \mathbb{F}_p[x].$$ Deuring, Hasse, and Kaneko and Zagier found other families of modular forms which also give the supersingular polynomial at $p$. In a new approach, we prove an analogue of Deligne's result for the Hecke trace forms $T_k(z)$ defined by the Hecke action on the space of cusp forms $S_k$. We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod $p$, and then relate their divisor polynomials to $S_p(x)$ using Deligne's observation., Minor revisions addressing a referee's comments

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term ϕ \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, ϕ \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], ϕ \phi is immediately periodic with period 2. We show that, for every positive integer π \pi , there exists a ring for which ϕ \phi is immediately periodic with period π \pi .

Let M n M^n be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere S n + 1 \mathbb {S}^{n+1} . Denote by S S and H H the squared length of the second fundamental form and the mean curvature of M M , respectively. For a fixed integer k k , where 1 ≤ k ≤ n − 2 1\leq k\leq n-2 , we prove that there exists a positive constant γ ( n ) \gamma (n) depending only on n n , such that if | H | > γ ( n ) |H|>\gamma (n) and α k ( n , H ) ≤ S ≤ α k + 1 ( n , H ) \alpha _k(n,H)\leq S \leq \alpha _{k+1}(n,H) , then either S = α k ( n , H ) S=\alpha _k(n,H) and M M must be the Clifford torus S n − k ( 1 1 + λ k 2 ) × S k ( λ k 1 + λ k 2 ) \mathbb {S}^{n-k}\big (\frac {1}{\sqrt {1+\lambda _k^2}}\big )\times \mathbb {S}^{k}\big (\frac {\lambda _k}{\sqrt {1+\lambda _k^2}}\big ) , or S = α k + 1 ( n , H ) S=\alpha _{k+1}(n,H) and M M is the Clifford torus S n − k − 1 ( 1 1 + λ k + 1 2 ) × S k + 1 ( λ k + 1 1 + λ k + 1 2 ) \mathbb {S}^{n-k-1}\big (\frac {1}{\sqrt {1+\lambda _{k+1}^2}}\big )\times \mathbb {S}^{k+1}\big (\frac {\lambda _{k+1}}{\sqrt {1+\lambda _{k+1}^2}}\big ) . Here α k ( n , H ) = n + n 3 2 k ( n − k ) H 2 − n ( n − 2 k ) 2 k ( n − k ) n 2 H 4 + 4 k ( n − k ) H 2 \alpha _k(n,H)=n+\frac {n^3}{2k(n-k)}H^2-\frac {n(n-2k)}{2k(n-k)}\sqrt {n^2H^4+4k(n-k)H^2} and λ k = n | H | + n 2 H 2 + 4 k ( n − k ) 2 ( n − k ) \lambda _k=\frac {n|H|+\sqrt {n^2H^2+4k(n-k)}}{2(n-k)} .

This paper is concerned with the monotonicity problem of traveling wave solutions for a class of nonlocal bistable system on two-dimensional lattice. We prove that any traveling wave solution, including the standing wave solution (wave speed c = 0 c=0 ), is strictly monotone increasing for this nonlocal bistable lattice system.

We consider the question of whether every collection of compact operators that is contained in a triangular operator algebra (in the sense of Kadison and Singer) must be simultaneously triangularisable. The answer is shown to be affirmative if the collection consists of finite-rank operators or is a norm-closed algebra of compact operators.

Decomposable graphical models, also known as perfect directed acyclic graph (DAG) models, play a fundamental role in standard approaches to probabilistic inference via graph representations in modern machine learning and statistics. However, such models are limited by the assumption that the data-generating distribution does not entail strictly context-specific conditional independence relations. The family of staged tree models generalizes DAG models so as to accommodate context-specific knowledge. We provide a new characterization of perfect discrete DAG models in terms of their staged tree representations. This characterization identifies the family of balanced staged trees as the natural generalization of discrete decomposable models to the context-specific setting.

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory.

In this note we give an elementary proof of the space-like real-analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type ( ∂ t − div x ⁡ ( B ( x ) ∇ x ) ) s (\partial _t - \operatorname {div}_x(B(x)\nabla _x))^s , 0 > s > 1 0>s>1 . Our primary interest is in the so-called extension variable. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where B ( x ) B(x) is a uniformly elliptic matrix-valued function with real-analytic entries.

Let μ \mu be an n n -dimensional finite positive measure on R m \mathbb {R}^m . We obtain a T 1 T1 condition sufficient for the boundedness of Calderón–Zygmund operators on R B M O ( μ ) \mathrm {RBMO}(\mu ) , the regular BMO space of Tolsa [Math. Ann. 319 (2001), pp. 89–149].

We extend Quine's bound on the number of self-intersection of curves with polynomial parameterization to the case of Laurent polynomials. As an application, we show that circle embeddings are dense among all maps from a circle to a plane with respect to an integral norm., Comment: 10 pages

Let K K be a closed convex cone in the state space R n \mathbb {R}^n . This note characterizes the K K -monotonicity of a non-autonomous dynamical system x ˙ ( t ) = f ( t , x ( t ) ) \dot x (t) = f(t,x(t)) governed by a locally Lipschitz velocity field. We deviate from the classical literature in two important ways. Firstly, the velocity field f f is not required to be differentiable with respect to the state variables. And, secondly, the closed convex cone K K is allowed to be absolutely general. In particular, we impose neither pointedness, nor solidity.

Let V be an infinite-dimensional vector space over a field of characteristic not equal to 2. We classify ideals of the Lie algebra gl(V) of all linear transformations of the space V., 9 pages

This paper provides a refinement of the study of asymptotic behaviour for a class of nonlocal in time telegraph equations with positively singular kernels. Based on fundamental properties of relaxation functions and recent representation of the fundamental solution in [Nonlinear Anal. 193 (2020), 111411], we establish the asymptotic expansions of the variance of the stochastic process for both long-time and short-time, which sharply improves the main result in [Proc. Amer. Math. Soc. 149 (2021), 2067–2080] by removing their technical conditions on the regularly varying behaviours and reformulating the asymptotic expansion in a more natural form. By analysing a new noncommutative operation on a subclass of completely positive functions, we provide a new way to construct finitely many ultraslow subdiffusion processes that are rapidly slower than a given ultraslow kernel. Consequently, we show that for a given completely monotonic ultraslow kernel, there is an induced kernel whose corresponding mean square displacement is logarithmic.

Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with non-torsion Brauer group. In contrast, we demonstrate that such examples cannot exist over the algebraic closure of a finite field., Comment: 5 pages. Comments welcome! v3: final version, to appear in Proceedings of the American Mathematical Society

We show that a weighted composition operator W ψ , φ {W_{\psi ,\varphi }} is compact on BMOA precisely when it is unconditionally converging. As a direct consequence to this result we deduce that all weakly compact or completely continuous weighted composition operators on BMOA are compact. The proof of this result is based on a new simplified function-theoretic characterization of the compactness of weighted composition operators on BMOA.

We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a $01$-matrix with many ones and without large all-one submatrices, and we put our results together and construct an $n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov rank $n-o(n)$., Comment: 9 pages

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to Sudakov and Tomon, and our proof builds up on their work. A recent construction of B. Janzer implies that our bound is tight up to an $O((\log n)^4 \log \log n)$ factor., Comment: 9 pages; corrected typos

In the present paper, we investigate the existence and multiplicity properties of the normalized solutions to the following Kirchhoff-type equation with Sobolev critical growth ( P ) { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + λ u = μ | u | p − 2 u + | u | 4 u , a m p ; in R 3 , u > 0 , ∫ R 3 | u | 2 d x = c 2 , a m p ; in R 3 , \begin{equation*} \begin {cases} -\left (a+b\int _{\mathbb {R}^3}|\nabla u|^2dx\right )\Delta u+\lambda u=\mu |u|^{p-2}u+|u|^{4}u, \quad &\text {in } \mathbb {R}^3,\\ u>0, \ \int _{\mathbb {R}^3}|u|^2dx=c^2, \quad &\text {in }\mathbb {R}^3, \end{cases} \tag {$P$}\end{equation*} where a , b , c , μ > 0 a, \ b, \ c, \ \mu >0 and 4 > p > 6 4>p>6 . We consider both the L 2 L^2 -subcritical and the L 2 L^2 -supercritical cases. Precisely, in the L 2 L^2 -subcritical case, by combining the truncation method, the concentration-compactness principle and genus theory, we obtain the multiplicity of the normalized solutions for problem ( P ) (P) . In the L 2 L^2 -supercritical case, by using a fiber map and the concentration-compactness principle, we obtain a couple of normalized solutions for problem ( P ) (P) , as well as their asymptotic behavior. These results extend and complement the existing results from Sobolev subcritical growth to the critical Sobolev setting.

Let K K and S S be locally compact Hausdorff spaces and let T T be a real linear isomorphism of an extremely regular subspace A A of C 0 ( K ) C_{0}(K) into C 0 ( S ) C_{0}(S) satisfying ‖ T ‖ ‖ T − 1 ‖ > 2 \|T \| \ \|T^{-1}\|>2 . Put L = T ‖ T − 1 ‖ L=T \ \|T^{-1}\| and pick 0 ≤ ε > 1 0 \leq \varepsilon >1 such that ‖ T ‖ ‖ T − 1 ‖ = 1 + ε \|T \| \ \|T^{-1}\|=1+\varepsilon . We prove that there exist a subset S 0 S_0 of S S , a proper map φ \varphi of S 0 S_{0} onto K K and a map λ : S 0 → { − 1 , 1 } \lambda :S_{0} \to \{-1, 1\} such that | L f ( s ) − λ ( s ) f ( φ ( s ) ) | ≤ ε ‖ f ‖ , ∀ s ∈ S 0 and f ∈ A . \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For ε > 1 / 2 \varepsilon > 1/2 the maps φ \varphi and λ \lambda are continuous, and this approximation of L L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where ε = 0 \varepsilon =0 , K K and S S are compact and A = C 0 ( K ) A=C_{0}(K) . The approximation of L L in the general case where 0 ≤ ε > 1 0 \leq \varepsilon >1 is an extension of a Benyamini’s result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz’s theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K K is a continuous image not just of a subset of S S but of a locally compact subset of S S . Moreover, when K K is compact, it is a continuous image not just a closed subset of S S but of a compact subset of S S , namely the set S 0 S_0 having the above properties.

In this note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol [Trans. Amer. Math. Soc. 368 (2016), pp. 6159–6188] to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi’s method.

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{\"a}ger \cite{shmuck} and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in \cite{Dyer}. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2102.10134

Using elliptic fibrations with specific singular fibers, we find spheres with very negative self-intersections in elliptic surfaces and in their blow-ups.

In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows. The first author was supported by the Basque Government through the BERC 2022-2025 program and by the Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project PID2020-114189RB-I00 funded by Agencia Estatal de Investigaci´on (PID2020-14189RB-I00 / AEI / 10.13039/501100011033). The second author was supported by the project “Mathematical Analysis of Fluids and Applications” Grant PID2019-109348GA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and acronym “MAFyA”. This publication is part of the project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. The second author was also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation.

We compute the optimal constant and characterise the maximisers at all spatial scales for the Agmon–Hörmander L 2 L^2 -Fourier adjoint restriction estimate on the sphere. The maximisers switch back and forth from being constants to being non-symmetric at the zeros of two Bessel functions. We also study the stability of this estimate and establish a sharpened version in the spirit of Bianchi–Egnell. The corresponding stability constant and maximisers again exhibit a curious intermittent behaviour.

In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer’s theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and Solomyak.

In this article we unconditionally show that { μ ( n ) } \{\mu (n)\} is asymptotically orthogonal to { λ π ( n ) e ( n α ) } \{\lambda _{\pi }(n)e(n\alpha )\} where λ π ( n ) \lambda _{\pi }(n) is the n n -th Dirichlet coefficient of a GL ⁡ ( m ) \operatorname {GL}(m) cusp form π \pi . Our result does not assume any unproven hypothesis.

In this paper, we establish lower bounds for the eigenvalues of even ordered quasilinear eigenvalue problems with indefinite weights. We first consider the case when the weight function is identically one and later generalize the results to any weight function within the p p -th Muckenhopt class. In the process, we obtain new Lyapunov-type inequalities for the corresponding problems and utilize them to obtain the lower bound of the k k -th eigenvalues. In particular, we show that the first eigenvalue is bounded below in terms of the integral of the weight instead of the integral of its positive part. Our results provide a much sharper bound than existing results.

We introduce toric promotion as a cyclic analogue of Sch\"utzenberger's promotion operator. Toric promotion acts on the set of labelings of a graph $G$. We discuss connections between toric promotion and previously-studied notions such as toric posets and friends-and-strangers graphs. Our main theorem provides a surprisingly simple description of the orbit structure of toric promotion when $G$ is a forest., Comment: 10 pages, 5 figures, to be published in Proceedings of the AMS