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Let \mathcal {B} be the collection of rectangular parallelepipeds in \mathbb {R}^3 whose sides are parallel to the coordinate axes and such that \mathcal {B} consists of parallelepipeds with side lengths of the form s, 2^j s, t, where s, t > 0 and j lies in a nonempty subset S of the integers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha }\left (1 + \log ^+ \frac {|f|}{\alpha }\right)\; \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))} = 0. \end{equation*} On the other hand, if S is an infinite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha } \left (1 + \log ^+ \frac {|f|}{\alpha }\right)^{2} \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))^2} = 0. \end{equation*} [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]

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. In this paper, we extend Simons's lemma to closed convex functions, show its connections to Attouch–Théra duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings. [ABSTRACT FROM AUTHOR]

In this paper, we provide some algebraic structures of convex subrings of *C, a nonstandard extension of the field of complex numbers C. In particular, a detailed description of the prime spectrum of any convex subring of *C is given. To achieve our goal, first we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain. We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum; moreover, nonzero elements are a disjoint union of three subsets where one of them is strongly dense and the other two are dense in the spectrum. [ABSTRACT FROM AUTHOR]

In this paper, we establish some conformable fractional Hermite-Hadamard type integral inequalities via harmonic s-convexity, and the estimates of the products of two harmonic s-convex functions are also considered. [ABSTRACT FROM AUTHOR]

Let I ⊂ N be an infinite subset, and let {ai}i≤I be a sequence of nonzero real numbers indexed by I such that there exist positive constants m,C1 for which |ai| ≤ C1·im for all I ∈ i. Furthermore, let ci ∈ [-1, 1] be defined by ci = ai C1·im for each i ∈ I, and suppose the ci's are equidistributed in [-1, 1] with respect to a continuous, symmetric probability measure μ. In this paper, we show that if I ⊂ N is not too sparse, then the sequence {ai}i∈I fails to obey Benford's Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when μ([0, t]) is a strictly convex function of t ∈ (0, 1). Nonetheless, we also provide conditions on the density of I ⊂ N under which the sequence {ai}i∈I satisfies Benford's Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benford's Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate operator functions by making use of the operator harmonic mean ' !'. For 0 < A ≤ B, we construct a unique pair X, Y such that 0 < X ≤ Y, A = X,!Y,; B = X+Y/2. We next give a condition for operators A, B, C ≥ 0 in order that C ≤ A! B and show that g ≠ 0 is strongly operator convex on J if and only if g(t) > 0 and g (A+B/2 ≤ g(A)!g(B) for A, B with spectra in J. This inequality particularly holds for an operator decreasing function on the right half line. We also show that ƒ(t) defined on (0, b) with 0 < b ≤ ∞ is operator monotone if and only if ƒ(0+) < ∞ ƒ (A!B) ≤ 1/2(f(A) + f(B)). In particular, if ƒ > 0, then f is operator monotone if and only if ƒ (A ! B) ≤ ƒ(A) ! ƒ(B). We lastly prove that if a strongly operator convex function g(t) > 0 on a finite interval (a, b) is operator decreasing, then g has an extension g to (a, ∞) that is positive and operator decreasing. [ABSTRACT FROM AUTHOR]

We establish some \Phi-moment inequalities for noncommutative differentially subordinate martingales. Let \Phi be a p-convex and q-concave Orlicz function with 1

In this paper we prove that any C1,1/2curve in Rn is the solution of the gradient flow equation for some C¹ convex function f if and only if it is strongly self-contracted. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C¹. Moreover, applications of this result are given. [ABSTRACT FROM AUTHOR]

K. Ball proved a strengthening of the Blaschke-Santaló inequality for unconditional convex bodies in ℝn. In this paper, we characterize the uniqueness of this inequality and establish its functional version. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. One is that for a kind of selfshrinker in a manifold with warped product metric, we prove that under some conditions on ambient space, the mean convex self-shrinker must have parallel second fundamental form. The other one is a generalization of Brendle's Minkowski inequality for weighted mean curvature. [ABSTRACT FROM AUTHOR]

In this paper, we study entire translating solutions u(x) to a mean curvature flow equation in Minkowski space. We show that if Σ = {(x, u(x))∣x ϵ Rn} is a strictly spacelike hypersurface, then Σ reduces to a strictly convex rank k soliton in Rk,1 (after splitting off trivial factors) whose "blowdown" converges to a multiple λ ϵ (0, 1) of a positively homogeneous degree one convex function in Rk. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation. [ABSTRACT FROM AUTHOR]

The inequalities involving the volume of convex bodies play an important role in convex geometry. In this paper, the Rogers and Shephard inequality for the Orlicz difference body of a convex body in the two-dimensional case is established. [ABSTRACT FROM AUTHOR]

We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In a recent work of Panchenko, it was proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies the original proof and presents more general results. [ABSTRACT FROM AUTHOR]

In this paper, we establish boundary Holder gradient estimates for solutions to the linearized Monge-Ampere equations with Lp (n < p ≤ ∞) right-hand side and C¹, γ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampere measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampere equations with bounded right-hand side and C1,1 boundary data. [ABSTRACT FROM AUTHOR]

A deep result of J. Lewis (1983) shows that the polylogarithms Liɑ(z):= σk=1∞ zk/kɑ map the open unit disk D centered at the origin one-to-one onto convex domains for all ɑ ≥ 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain &Lgr;:= C\[1, ∞), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Liɑ(z) maps an arbitrary open disk or half-plane in &Lgr; one-to-one onto a convex domain for every ɑ ≥ 1. [ABSTRACT FROM AUTHOR]

We quantify Peter Scott's theorem that surface groups are locally extended residually finite (LERF) in terms of geometric data. In the process, we will quantify another result by Scott that any closed geodesic in a surface lifts to an embedded loop in a finite cover. [ABSTRACT FROM AUTHOR]

We study positive weak solutions of the quasilinear Lane-Emden equation −Qu = uα in Ω ⊂ Rn, where α ≥ n+2/n−2, for n ≥ 3, is supercritical and the operator Q, known as Finsler-Laplacian or anisotropic Laplacian, is defined by Qu ≔ ∑i=1n ∂/∂xi(F(∇ u)Fξi(∇u)). Here, Fξi = &#8706F/∂μi and F: Rn → [0,+ ∞) is a convex function of C2(Rn\{0}), that satisfies positive homogeneity of first order and other certain assumptions. We prove that the Hausdorff dimension of singular set of u is less than n−2 α+1/α−1. [ABSTRACT FROM AUTHOR]

We prove that convex viscosity solutions to the quadratic Hessian inequality σ2(D2u) ≥ 1 are strictly 2-convex. As a consequence we obtain short proofs of smoothness and interior C2 estimates for convex viscosity solutions to σ2(D2u) = 1, which were proven using different methods in recent works of Guan-Qiu, McGonagle-Song-Yuan, and Shankar-Yuan. [ABSTRACT FROM AUTHOR]

Using recent work of Bourgain-Dyatlov (2018) we show that for any convex co-compact hyperbolic surface Strichartz estimates for the Schrodinger equation hold with an arbitrarily small loss of regularity. [ABSTRACT FROM AUTHOR]

Let b ∈ ℝ, let c > 0, let x > 0, and let Gb,c(x) = e-x / xb Pc(x) with Pc(x) = ∞Σ k=0 xk / Γ(c + k) . We prove that Gb,c is completely monotonic on (0,∞) if and only if b ≥ 0 and b+c ≥ 1. Moreover, we present various functional inequalities for Pc. Among others, we show that if c ∈ (0, 1), then, for x, y > 0 we have e < Pc(1/x)x Pc(1/y)y / Pc(1/(x + y))x+y . If c > 1, then the reverse inequality holds for x, y > 0. In both cases, the constant bound e is best possible. [ABSTRACT FROM AUTHOR]

Given any continuous, lower bounded, and κ-convex function V on a metric measure space (X, d,m) which is infinitesimally Hilbertian and satisfies some synthetic lower bound for the Ricci curvature in the sense of Lott-Sturm-Villani, we prove existence and uniqueness for the (downward) gradient flow for V. Moreover, we prove Lipschitz continuity of the flow w.r.t. the starting point .... [ABSTRACT FROM AUTHOR]

Existence results for viscosity solutions to the Dirichlet problem for stationary Hamilton-Jacobi equations, the associated relaxed problem via quasiconvex envelopes, and for minimizers of the corresponding L∞ functionals are obtained for given boundary data in W1,∞. [ABSTRACT FROM AUTHOR]

We prove that in small codimensions, the sections of a convex body in Rn through its centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies. [ABSTRACT FROM AUTHOR]

For f analytic and close to convex in D = {z : ∣z∣ < 1}, we give sharp estimates for the logarithmic coefficients γn of f defined by log f(z)/z = 2Σ∝n=1 γnzn when n = 1, 2, 3. [ABSTRACT FROM AUTHOR]

Let R(x) denote the ring of polynomials in g freely noncommuting variables x = (x1, . . . , xg). There is a natural involution . on R(x) determined by x* j = xj and (pq)* = q*p*, and a free polynomial p E R(x) is symmetric if it is invariant under this involution. If X = (X1, . . . , Xg) is a g tuple of symmetric n × n matrices, then the evaluation p(X) is naturally defined and further p.(X) = p(X). In particular, if p is symmetric, then p(X)* = p(X). The main result of this article says if p is symmetric, p(0) = 0 and for each n and each symmetric positive definite n×n matrix A the set {X : A.p(X) > 0} is convex, then p has degree at most two and is itself convex, or .p is a hermitian sum of squares. [ABSTRACT FROM AUTHOR]