Existence of at least one positive solution to the nonlocal differential equation − M ( ( a ∗ ( g ∘ u ) ) ′ ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) , 0 > t > 1 , \begin{equation} -M\Big (\big (a*(g\circ u)\big )’(1)\Big )u(t)=\lambda f\big (t,u(t)\big ),\;0>t>1,\notag \end{equation} subject to the right-focal boundary data u ( 0 ) = 0 = u ′ ( 1 ) u(0)=0=u’(1) , is considered. Depending upon the choice of the kernel a a in the finite convolution, ( a ∗ ( g ∘ u ) ) ′ ( 1 ) \big (a*(g\circ u)\big )’(1) , this formulation includes as a special case the Riemann–Liouville fractional derivative. Topological fixed point theory, carried out via a specialized order cone, is employed to deduce the main result. Due to the presence of the derivative in the nonlocal coefficient, we use a nonstandard index theory applicable to unbounded sets.
If p ≥ 5 p\geq 5 is prime and k ≥ 4 k\geq 4 is an even integer with ( p − 1 ) ∤ k (p-1)\nmid k we consider the Eisenstein series G k G_k on S L 2 ( Z ) SL_2(\mathbb {Z}) modulo powers of p p . It is classically known that for such k k we have G k ≡ G k ′ ( mod p ) G_k\equiv G_{k’}\pmod p if k ≡ k ′ ( mod p − 1 ) k\equiv k’\pmod {p-1} . Here we obtain a generalization modulo prime powers p m p^m by giving an expression for G k ( mod p m ) G_k\pmod {p^m} in terms of modular forms of weight at most m p mp . As an application we extend a recent result of Ahlgren with Hanson, Raum, and Richter by showing that, modulo powers of E p − 1 E_{p-1} , every such Eisenstein series is congruent modulo p m p^m to a modular form of weight at most m p mp . We prove a similar result for the normalized Eisenstein series E k E_k in the case that ( p − 1 ) ∣ k (p-1)\mid k and m > p m>p .
We will show that the category of the unitary representations of a compact, connected, simply connected, semisimple Lie group is isomorphic to one of full holomorphic maps of a flag manifold to complex Grassmannians with gauge condition for semipositive Hermitian holomorphic homogeneous vector bundles.
We demonstrate that there exist sequences { u(n)}infinity (n=0) subset of (0, +infinity) such that u(n+1) > u(n) for each n is an element of N-0 , but yet the fractional difference of the sequence is uniformly negative. This demonstrates a pathological property of nonlocal discrete operators.
We prove that the fiber space consisting of BMOA functions that are the logarithms of derivatives of conformal homeomorphisms of the unit disk onto bounded quasidisks forms a real-analytic disk bundle over the Bers embedding of the BMO Teichmüller space. For the VMO Teichmüller space, we show that the corresponding sub-bundle consisting of VMOA functions is real-analytically trivial.
This paper studies the asymptotic behavior of solutions of the parabolic-parabolic chemotaxis model with logistic-type sources in heterogeneous bounded domains: { ut = Delta u-chi V & centerdot; (uVv) +u a0(t, x)-a(1)(t, x)u-a(2)(t, x) integral (Omega)u) , x is an element of Omega tau vt = Delta v-)v + mu u, x is an element of Omega partial derivative u/partial derivative nu = partial derivative nu /partial derivative v = 0, x is an element of partial derivative Omega(*) We find parameter regions in which the system has a unique positive entire solution, which is globally asymptotically stable. More precisely, under suitable assumptions on the model's parameters, the system has a unique positive entire solution (u*(t,x),v*(t,x)) such that for any u(0) E C-0(Omega), v(0) is an element of W-1,W-infinity(Omega ) with u(0), v(0) > 0 and u(0) not approximate to 0, the global classical solution (u(t, x; t(0), u(0), v(0)), v(t, x; t(0), u(0), v(0))) of (*) satisfies lim(t ->infinity)(sup(t0 is an element of R)& Vert;u(t+t(0),& centerdot;;t(0),u(0),v(0))-u(& lowast;)(t+t(0),& centerdot;)& Vert;C-0( Omega) +sup(t0 is an element of R)& Vert;v(t+t(0),& centerdot;;t(0),u(0),v(0))-v & lowast;(t+t(0),& centerdot;)& Vert;C-0( Omega)) = 0
Every 2-dimensional spine of an aspherical 3-manifold has the nonpositive towers property, but every collapsed 2-dimensional spine of a 3-ball containing a 2-cell has an immersed sphere.
In our previous article, “On the Cesàro operator on the Hardy space in the upper half-plane” ( arXiv: 2405.19627 ), we proved that the Cesàro operator on the Hardy space H 2 ( C + ) H^2(\mathbb {C}_+) in the upper half-plane is the sum of the identity operator and a unitary operator. In this article we investigate properties of that unitary operator. We explicitly determine its resolution of identity and show that this unitary operator is unitarily equivalent to the multiplication by a rational unimodular on the real axis function in the space of Mellin transforms of functions from H 2 ( C + ) H^2(\mathbb {C}_+) . In particular we deduce that the Cesàro operator on H 2 ( C + ) H^2(\mathbb {C}_+) is cyclic.
We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop, Carleson, Garnett, and Jones and Browder and Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi, and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist.
Edtmair and Hutchings have recently defined, using periodic Floer homology (PFH), a “ U U -cycle property” for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U U -cycle property satisfies the smooth closing lemma and also satisfies a kind of Weyl law involving the actions of certain periodic points; they show that every rational isotopy class on the two-torus satisfies the U U -cycle property. It seems that, in general, not much is known about the U-module structure on PFH. Here we consider a version of Seiberg–Witten Floer cohomology, which is known by work of Lee and Taubes to be isomorphic, as a U-module, to the periodic Floer homology in sufficiently high degree. We show that the analogous U U -cycle property holds for every rational Hamiltonian isotopy class on any closed surface and, more generally, for any nontorsion spin-c structure. On the other hand, we also show that a rational isotopy class may contain elements that are not U U -cyclic. By the Lee–Taubes isomorphism, the same results hold for PFH. Our results are some of the first computations concerning the U-module structure on these theories.
We use the slice filtration to study the M U MU -homology of the fixed points of connective models of Lubin–Tate theory studied by Hill, Hopkins, and Ravenel and Beaudry, Hill, Shi, and Zeng. We show that, unlike their periodic counterparts E O n EO_n , the M U MU homology of B P ( ( G ) ) ⟨ m ⟩ G BP^{((G))}\langle m\rangle ^G usually fails to be even and torsion free. This can only happen when the height n = m | G | / 2 n=m|G|/2 is less than 3 3 , and in the edge case n = 2 n=2 , we show that this holds for t m f 0 ( 3 ) tmf_0(3) but not for t m f 0 ( 5 ) tmf_0(5) , and we give a complete computation of the M U ∗ M U MU_*MU -comodule algebra M U ∗ t m f 0 ( 3 ) MU_*tmf_0(3) .
A monotone surjective map F F is continuous and defines a spectral family and unitary group on L 2 L^{2} . We use the H 0 \mathcal {H}_{0} splitting theorem (Int. J. Dyn. Syst. Differ. Equ. 10(2020), no. 4, 358–372) to extend the Lebesgue decomposition F = F s + F a F \!=\! F_{s} \!+\! F_{a} to the singular component of F F , F s = F m + F w s F_{s} \!=\! F_{m} + F_{w_{s}} . The components (recurrent: F m F_{m} , and stable: F w s F_{w_{s}} ) are unique up to additive constants and are characterized by their Fourier–Stieltjes transforms. Several examples are included. Section 7.2 shows that each Riesz product has a nontrivial recurrent component.
We provide in this paper a constructive proof of optimal L ∞ L_{\infty } star discrepancy values in dimension 2 for up to 21 points and up to 8 points in dimension 3. This extends work by White (Numer. Math. 27 (1976/77), no. 2, 157–164) for up to six points in dimension 2 and of Larcher and Pillichshammer (J. Comput. Appl. Math. 206 (2007), no. 2, 977–985) for two points in arbitrary dimensions. We show that these optimal sets have a far lower discrepancy than the previous references and, perhaps more importantly, present a very different structure.
In an earlier paper, Buskes and the author pointed out that, given two Archimedean Riesz spaces E E and F F , it is relatively simple to construct, from their Ogasawara-Maeda representations, a third Archimedean Riesz space G G and a bi-injective Riesz bimorphism ϕ : E × F → G \phi :E\times F\to G . They further pointed out that the Riesz subspace of G G generated by ϕ ( E × F ) \phi (E\times F) is isomorphic to the Archimedean Riesz space tensor product E ⊗ ¯ F E\overline {\otimes }F constructed by Fremlin. The proof there relies on the properties of E ⊗ ¯ F E\overline {\otimes }F proved by Fremlin. In this paper we show that this approach actually gives a simple way to construct and establish the properties of E ⊗ ¯ F E\overline {\otimes }F and that any choice of G G and ϕ \phi yield isomorphic objects.
The classical Maclaurin inequality asserts that the elementary symmetric means s k ( y ) ≔ 1 ( n k ) ∑ 1 ≤ i 1 > ⋯ > i k ≤ n y i 1 … y i k \begin{equation*} s_k(y) ≔\frac {1}{\binom {n}{k}} \sum _{1 \leq i_1 > \dots > i_k \leq n} y_{i_1} \dots y_{i_k} \end{equation*} obey the inequality s ℓ ( y ) 1 / ℓ ≤ s k ( y ) 1 / k s_\ell (y)^{1/\ell } \leq s_k(y)^{1/k} whenever 1 ≤ k ≤ ℓ ≤ n 1 \leq k \leq \ell \leq n and y = ( y 1 , … , y n ) y = (y_1,\dots ,y_n) consists of nonnegative reals. We establish a variant | s ℓ ( y ) | 1 ℓ ≪ ℓ 1 / 2 k 1 / 2 max ( | s k ( y ) | 1 k , | s k + 1 ( y ) | 1 k + 1 ) \begin{equation*} |s_\ell (y)|^{\frac {1}{\ell }} \ll \frac {\ell ^{1/2}}{k^{1/2}} \max (|s_k(y)|^{\frac {1}{k}}, |s_{k+1}(y)|^{\frac {1}{k+1}}) \end{equation*} of this inequality in which the y i y_i are permitted to be negative. In this regime the inequality is sharp up to constants. Such an inequality was previously known without the k 1 / 2 k^{1/2} factor in the denominator.
We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $\sigma_f$ on the completion $\overline{\mathcal{T}_P}$ of the associated Teichmüller space $\mathcal{T}_P$ with respect to the Weil-Petersson metric is easy to understand when $\overline{\mathcal{T}_P}$ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map $f$ has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if $P$ contains all critical points of $f$ and each point in $P$ is periodic, then such a cover of $\overline{\mathcal{T}_P}$ can be obtained from a $\sigma_f$-invariant tessellation by ideal hyperbolic triangles.
. This article examines three radii associated to bounded analytic functions on the polydisk: the well-known Bohr radius, the Bohr-Agler radius, and the Schur-Agler radius. We prove explicit upper and lower bounds for the Bohr-Agler radius, an explicit lower bound for the Schur-Agler radius, and an asymptotic upper bound for the Schur-Agler radius. The Bohr-Agler radius obeys the same (known) asymptotic as the Bohr radius while we show the Schur-Agler radius is roughly of the same growth as the Bohr radius. As a corollary, we bound the Bohr radius on the bidisk below by 0.3006. Finally, we improve some estimates of P. G. Dixon on Agler norms of homogeneous polynomials using some modern inequalities.
We study log-correlated Gibbs measures on the d-dimensional torus with weakly interacting focusing quartic potentials whose coupling constants tend to 0 as we remove regularization. In particular, we exhibit a phase transition for this model by identifying a critical threshold, separating the weakly and strongly coupling regimes; in the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized L^2-cutoff), whereas, in the strongly coupling regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence. Our result answers an open question posed by Brydges and Slade (1996).
We prove the existence of a C*-diagonal in the Cuntz algebra O_2 with spectrum homeomorphic to the Cantor space.
