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In this paper, we investigate the Monge–Ampere equation $$\text{ det }D^2u=f $$ in $${\mathbb {R}}^n_+$$ , where f is bounded, positive and $$f(x)=1+O(|x|^{-\beta })$$ for some $$\beta >2$$ at infinity. If u is a quadratic polynomial on $$\{x_n=0\}$$ and satisfies $$ \mu |x|^2\le u\le \mu ^{-1}|x|^2$$ for some $$00$$ is some constant depending only on $$\beta $$ and n. Meanwhile, the existence and uniqueness of viscosity solutions of the Dirichlet problem with prescribed asymptotic behavior at infinity will be concerned. The condition $$\beta >2$$ is sharp.

We establish the following Helly-type theorems: Let $${\mathcal {K}}$$ be a family of sets in $${\mathbb {R}}^3$$ , and let j be a fixed integer, $$0 \le j \le 3$$ . For every countable subfamily of $${\mathcal {K}}$$ , assume that the corresponding intersection is consistent relative to staircase paths, is staircase convex, and contains a convex set of dimension at least j. Then $$\cap \{K \, : \, K \text { in } {\mathcal {K}} \}$$ has these properties as well. For d a fixed integer, $$d \ge 1$$ , define a function f on $$\{0,1\}$$ by $$f(0)=d+1$$ , $$f(1)=2d$$ . Let $${\mathcal {K}}$$ be a finite family of orthogonal polytopes in $${\mathbb {R}}^d$$ . For j fixed in $$\{ 0,1\}$$ , assume that every f(j) members of $${\mathcal {K}}$$ intersect in a set that is staircase convex and contains a convex subset of dimension at least j. Then $$\cap \{K : K \text { in } {\mathcal {K}} \}$$ has these properties, too. The number f(j) is best possible for $$j=0$$ and for $$j=1$$ . There is no analogous Helly number for $$2 \le j \le d$$ .

In this note, we prove the following inequality for the norm of a convex body $K$ in $\mathbb{R}^n$, $n\geq 2$: $N(K) \leq \frac{\pi^{\frac{n-1}{2}}}{2 \Gamma \left(\frac{n+1}{2}\right)}\cdot \operatorname{length} (\gamma) + \frac{\pi^{\frac{n}{2}-1}}{\Gamma \left(\frac{n}{2}\right)} \cdot \operatorname{diam}(K)$, where $\operatorname{diam}(K)$ is the diameter of $K$, $\gamma$ is any curve in $\mathbb{R}^n$ whose convex hull covers $K$, and $\Gamma$ is the gamma function. If in addition $K$ has constant width $\Theta$, then we get the inequality $\operatorname{length} (\gamma) \geq \frac{2(\pi-1)\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\pi}\,\Gamma \left(\frac{n}{2}\right)}\cdot \Theta \geq 2(\pi-1) \cdot \sqrt{\frac{n-1}{2\pi}}\cdot \Theta$. In addition, we pose several unsolved problems., Comment: 7 pages

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret’s theorem, and the Brill-Noether Restsatz theorem.

We prove that if a convex set in $${\mathbb {C}}^{n}$$ contains two inscribed complex ellipsoids of maximal volume, then one is a translate of the other. On the other hand, the circumscribed complex ellipsoid of minimal volume is unique. As application we prove the complex analogue of Brunn’s classical characterization of ellipsoids.

In this paper, we introduce a new class of modules satisfying S-dccr (S-dccr $$^{\star })$$ condition which is a generalization of S-artinian modules. Let $$A\ $$ be a commutative ring with $$0\ne 1\ $$ and $$X\ $$ a unital A-module. Suppose that $$S\subseteq A\ $$ is a multiplicatively closed subset. $$X\ $$ is said to satisfy S-dccr (S-dccr $$^{\star })$$ condition if for each finitely generated (principal) ideal $$I\ $$ of $$A\ $$ and a submodule $$Y\ $$ of $$X,\ $$ the descending chain $$\{I^{i}Y\}_{i\in {\mathbb {N}}}$$ is S-stationary. Many examples and properties of modules satisfying S-dccr (S-dccr $$^{\star })\ $$ condition are given. Furthermore, we characterize modules satisfying dccr (dccr $$^{\star })$$ condition in terms of some known class of rings and modules. Also, we give Nakayama’s Lemma for modules satisfying S-dccr condition.

Our study in this paper is focused on mixed equilibrium problems as well as combination with and fixed point problems in real Hilbert spaces. We introduce a new extension of a skew-symmetric bi-functions, called generalized skew-symmetric bi-functions. Two iterative methods based on the subgradient extragradient method is presented and their weak convergence theorems are proved under some mild assumptions. Further, we discuss some numerical examples to demonstrate the applicability of the iterative algorithm. The methods and analysis is new and extend several related result in the literature.

In this paper, we establish the pointwise boundary differentiability on Reifenberg domains for viscosity solutions of fully nonlinear elliptic equations which extends the result under the usual $$C^{1,\mathrm {Dini}}$$ condition and generalizes the result for linear equations in Huang et al (Manuscr. Math 162(3–4):305–313, 2020). Moreover, our proofs are relatively simple.

We prove that every Jordan curve in $$\mathbb {R}^2$$ inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.

Let R be a commutative ring with unity. In this paper we introduce the concept of Nonnil–Laskerian ring that is related to the class of Laskerian rings. A ring R is said to be Nonnil–Laskerian if every nonnil ideal I of R is decomposable. We show that Nonnil–Laskerian rings enjoy analogs of many properties of Laskerian ring. We give an example of Nonnil–Laskerian ring, wich is not Laskerian. We study the Nonnil–Laskerian property over the polynomial and formel power series rings. In particular, we show that we have not an equivalence between Nonnil–Laskerian and Nonnil–Noetherian concepts in R[[X]] and R[X], contrary to the Laskerian and Noetherian concepts.

Consider an oriented curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $\Phi:D\to \boldsymbol R^3$ such that $\Gamma$ is the singular set of $\Phi$, the word "origami" coming from the Japanese term for paper folding. We call the singular set image $C:=\Phi(\Gamma)$ the crease of $\Phi$ and the singular set $\Gamma$ the crease pattern of $\Phi$. We are interested in the number of origami maps whose creases and crease patterns are $C$ and $\Gamma$, respectively. Two such possibilities have been known. In the authors' previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possibility of the number $N$ of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if $C$ is a non-closed simple arc, then $N=4$ if and only if both $\Gamma$ and $C$ do not admit any symmetries. On the other hand, when $C$ is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general., Comment: 29 pages, 7 figures

It is conjectured that the sum $$\begin{aligned} S_r(n)=\sum _{k=1}^{n} \frac{k}{k+r}\left( {\begin{array}{c}n\\ k\end{array}}\right) \end{aligned}$$ for positive integers r, n is never integral. This has been shown for $$r\leqslant 22$$ . In this note we study the problem in the “n aspect” showing that the set of n such that $$S_r(n)\in {\mathbb {Z}}$$ for some $$r\geqslant 1$$ has asymptotic density 0. Our principal tools are some deep results on the distribution of primes in short intervals.

This paper has two souls. On one side, it is a survey on (infinite) groups in which certain systems of subgroups are complemented (like for instance the abelian subgroups). On another side, it provides generalizations and new, easier proofs of some (un)known results in this area.

In this paper, by using the generalized symmetric difference $$\Delta _{h}^{m}$$ of order m and step $$h>0$$ , we obtain a generalization of Titchmarsh’s Theorem for deformed Hankel transform.

Why is it that semidefinite relaxations have been so successful in numerous applications in computer vision and robotics for solving non-convex optimization problems involving rotations? In studying the empirical performance, we note that there are few failure cases reported in the literature, in particular for estimation problems with a single rotation, motivating us to gain further theoretical understanding. A general framework based on tools from algebraic geometry is introduced for analyzing the power of semidefinite relaxations of problems with quadratic objective functions and rotational constraints. Applications include registration, hand-eye calibration, and rotation averaging. We characterize the extreme points and show that there exist failure cases for which the relaxation is not tight, even in the case of a single rotation. We also show that some problem classes are always tight given an appropriate parametrization. Our theoretical findings are accompanied with numerical simulations, providing further evidence and understanding of the results., Journal of mathematical imaging and vision, 64 (1), ISSN:0924-9907, ISSN:1573-7683

We establish the concept of r-almost Yamabe soliton immersed into a Riemannian manifold, which extends in a natural way the notion of almost Yamabe solitons introduced by Barbosa and Ribeiro in [3]. In this setting, under suitable hypothesis on the potential and soliton functions, we prove nonexistence and characterization results. Moreover, some examples of these new geometric objects are presented.

Let G be a noncompact connected simple Lie group, and $$G^\sigma $$ the subgroup of the fixed points under the action of $$\sigma $$ on G, where $$\sigma $$ is an automorphism of G of even order. This article shows a necessary condition for the pair $$(G,G^\sigma )$$ satisfying condition (D.D.); that is, there exists at lease one infinite-dimensional simple $$({\mathfrak {g}},K)$$ -module that is discretely decomposable as a $$({\mathfrak {g}}^\sigma ,K^\sigma )$$ -module.

Let $${\mathfrak {T}}={\mathfrak {T}}_3(\mathrm {R}_i, \mathrm {M}_{ij})$$ be a triangular 3-matrix ring. In the present paper, we study of multiplicative Lie triple derivation on triangular 3-matrix rings and prove that every multiplicative Lie triple derivation on triangular 3-matrix rings can be written as a sum of an additive derivation and a center valued map vanishing at each second commutator.

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces., Two minor typos are corrected. Accepted version

Asympototic formulae are obtained for the variance associated with the distribution in arithmetic progressions of the numbers that are the sum of two positive cubes of integers.

Let P(z) be a polynomial of degree at most n. We consider an operator N, which carries a polynomial P(z) into $$\begin{aligned} N[P](z):=\sum \limits _{j=0}^{m}\lambda _j\bigg (\frac{nz}{2}\bigg )^j\frac{P^{(j)}(z)}{j!}, \end{aligned}$$ where $$\lambda _0,\lambda _1,\ldots ,\lambda _m$$ are such that all the zeros of $$\begin{aligned} u(z)=\sum \limits _{j=0}^{m}\left( {\begin{array}{c}n\\ j\end{array}}\right) \lambda _jz^j \end{aligned}$$ lie in the half plane $$\begin{aligned} |z|\le \bigg |z-\frac{n}{2}\bigg |. \end{aligned}$$ In this paper, we estimate the minimum and maximum modulii of N[P(z)] on $$|z|=1$$ with restrictions on the zeros of P(z) and thereby obtain compact generalizations of some well known polynomial inequalities.

In this paper, we investigate the gap phenomena for complete Lagrangian submanifolds satisfying $$\nabla ^{*}\varvec{T}\equiv 0$$ in complex space forms $$\mathbf{N}^{n}(4c)$$ , where $$\varvec{T}=\frac{1}{n}\nabla ^{*}\varvec{B}$$ and $$\varvec{B}$$ is the Lagrangian trace-free second fundamental form. We prove that under some pointwise or integral curvature pinching condition, the submanifold is either totally geodesic or isometric to the Whitney sphere.

We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on the degree of the foliation and of the degrees of the toric homogeneous coordinates.

In the paper we begin to explore relation between the question of unitarizability of classical p-adic groups, and Arthur packets., Comment: 35 pages

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions., Comment: 21 pages. arXiv admin note: text overlap with arXiv:math/0408381 by other authors

We give a construction of p-adic Asai L-functions for cohomological cuspidal automorphic representations of $$\mathrm{GL}_2$$ over CM fields. If the base field is imaginary quadratic, Loeffler–Williams recently constructed such p-adic Asai L-functions. We generalize their construction to the case that the base fields are general CM fields.

We show that a smooth embedded paper Moebius band must have aspect ratio at least $$\begin{aligned}\lambda _1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\root 4 \of {3} \sqrt{2}+2 \sqrt{2 \sqrt{3}-3}} =1.69497\ldots \end{aligned}$$ This bound comes more than 3/4 of the way from the old known bound of $$\pi /2=1.5708\ldots $$ to the conjectured bound of $$\sqrt{3} = 1.732\ldots $$

Let E be a vector bundle generated by sections and L be an ample line bundle over a smooth projective variety X. We give here a condition for the vanishing of Dolbeault cohomology groups of the form $$H^{p,q}(X,{{\mathcal {S}}}^{\alpha }E\otimes \wedge ^{\beta } E\otimes L)$$ . In most cases the result is stronger than what has been obtained for the condition that E is nef and L is ample. In particular a result of Zhi Jiang is improved. To derive our result we introduce a new spectral sequence, which can be used in a broader context.

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, qω-compact, uω-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?

We introduce new kind of $p$-adic hypergeometric functions. We show these functions satisfy congruence relations, so they are convergent functions. And we show that there is a transformation formula between our new $p$-adic hypergeometric functions and $p$-adic hypergeometric functions of logarithmic type defined by M. Asakura in a particular case., Comment: 39 pages

We study the trigonometry of non-Euclidean tetrahedra using tools from algebraic geometry. We establish a bijection between non-Euclidean tetrahedra and certain rational elliptic surfaces. We interpret the edge lengths and the dihedral angles of a tetrahedron as values of period maps for the corresponding surface. As a corollary we show that the cross-ratio of the exponents of the solid angles of a tetrahedron is equal to the cross-ratio of the exponents of the perimeters of its faces. The Regge symmetries of a tetrahedron are related to the action of the Weyl group $$W(D_6)$$ on the Picard lattice of the corresponding surface.

We show that there exist Kahler–Einstein metrics on two exceptional Pasquier’s two-orbits varieties. As an application, we will provide a new example of K-unstable Fano manifold with Picard number one.

In this paper we consider immersions with constant anisotropic mean curvature (CAMC) of a smooth oriented connected and compact surface $$\varSigma $$ , with non-empty boundary $$\partial \varSigma $$ , in a region $$\varOmega $$ such that the boundary $$\partial \varOmega $$ is a rotational surface. We prove that, under a suitable condition on the anisotropic function, the flat disks are the only free boundary CAMC immersions in $$\varOmega $$ . Moreover, we study which disks are stable. Finally, we consider an interesting result that allows us to build a wide variety of examples of Wulff Shape.

Given a Riemannian manifold M, and an open interval $$I\subset \mathbb {R},$$ we characterize nontrivial totally umbilical hypersurfaces of the product $$M\times I$$ —as well as of warped products $$I\times _\omega M$$ —as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of M. By means of this characterization, we fully extend to $$\mathbb {S}^n\times \mathbb {R}$$ and $$\mathbb {H}^n\times \mathbb {R}$$ the results by Souam and Toubiana on the classification of totally umbilical surfaces in $$\mathbb {S}^2\times \mathbb {R}$$ and $$\mathbb {H}^2\times \mathbb {R}.$$ It is also shown that an analogous classification holds for arbitrary warped products $$I\times _\omega \mathbb {S}^n$$ and $$I\times _\omega \mathbb {H}^n.$$

We show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of Kähler balls and vice versa.

In this paper, we study the global F-splitting of varieties admitting an int-amplified endomoprhism. We prove that surfaces admitting an int-amplified endomorphism are of dense globally F-split type and, in particular, of Calabi–Yau type.

Let L be a globally generated line bundle over a smooth irreducible complex projective surface X. The syzygy bundle $$M_{L}$$ is the kernel of the evaluation map $$H^0(L)\otimes \mathcal O_X\rightarrow L$$ . We prove the L-stability of $$M_L$$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $$(-K_X)$$ -stability of syzygy bundles $$M_L$$ over del Pezzo surfaces is also obtained.

In this paper, we study the long-time dynamics of Navier–Stokes equations with a fractional operator in the memory term and external forces $$\begin{aligned} \partial _tu-\nu \Delta u +\displaystyle \int _{0}^{\infty }g(s)(-\Delta )^\alpha u(t-s)\,\mathrm{d}s +(u\cdot \nabla )u+\nabla p=\epsilon f(x) \end{aligned}$$ on a bounded domain $$\Omega $$ in $$\mathbb {R}^2$$ with smooth boundary. We establish the existence of global attractor for the associated dynamical system. We prove the continuity of global attractors with respect to forcing parameter $$\epsilon $$ in a residual dense set. Moreover, the upper-semicontinuity with respect to parameter $$\epsilon $$ of global attractors is shown.

We study the branched holomorphic projective structures on a compact Riemann surface X with a fixed branching divisor $$S\, =\, \sum _{i=1}^d x_i$$ , where $$x_i \,\in \, X$$ are distinct points. After defining branched $$\mathrm{SO}(3,{\mathbb C})$$ -opers, we show that the branched holomorphic projective structures on X are in a natural bijection with the branched $$\mathrm{SO}(3,{\mathbb C})$$ -opers singular at S. It is deduced that the branched holomorphic projective structures on X are also identified with a subset of the space of all logarithmic connections on $$J^2((TX)\otimes {\mathcal O}_X(S))$$ singular over S, satisfying certain natural geometric conditions.

Let K be a number field, and let G be a finitely generated subgroup of $$K^\times $$ . We are interested in computing the degree of the cyclotomic-Kummer extension $$K(\root n \of {G})$$ over K, where $$\root n \of {G}$$ consists of all n-th roots of the elements of G. We develop the theory of entanglements introduced by Lenstra, and we apply it to compute the above degrees.

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds., Comment: 41 pages; minor update following some comments from S. Mukai and G. Ouchi

We prove that two-dimensional Dirichlet distributions for any collection of positive parameters can be modeled by means of a sequence of distributions defined via non-negative valued multiplicative functions which satisfy some regularity conditions on prime powers.

We show that delta invariant of a log Fano pair can be approximated by lc places of plt complements if it is no greater than one. Under the assumption that delta invariant (no greater than one) of a log Fano pair can be approximated by lc places of bounded plt complements, we show the existence of divisorial valuation computing delta invariant of this log Fano pair.

A right R-module M is called: (1) retractable if $${{\,\mathrm{Hom}\,}}_R(M, N) \ne 0$$ for any non-zero submodule N of M; (2) coretractable if $${{\,\mathrm{Hom}\,}}_R(M/N, M)\ne 0$$ for any proper submodule N of M. It shows that if M is locally noetherian and every nonzero module in the category $$\sigma [M]$$ has a maximal submodule, then the retractability and coretractability of modules in $$\sigma [M]$$ coincide. Let C be a coalgebra over a field k. We prove that all right C-comodules are retractable if and only if every right C-comodule is coretractable.

We show that one can recover the local Neron-Tate height of a rational point from the data of two homomorphisms of topological groups. See Theorem 1.1.

We study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family $$H_k$$ H k in Mond’s list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.

In this paper, we use the fixed point theory to obtain the existence and uniqueness of solutions for a class of nonlinear fractional differential equations. Two examples are given to illustrate this work.

Trigonal curves provide an example of Brill–Noether special curves. Theorem 1.3 of Larson (Invent Math 224(3):767–790, 2021) characterizes the Brill–Noether theory of general trigonal curves and the refined stratification by Brill–Noether splitting loci, which parametrize line bundles whose push-forward to $$\mathbb {P}^1$$ has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill–Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected, and if furthermore the Maroni invariant is 0 or 1, they are irreducible.

Let$${\mathcal {T}}_\infty (F)$$T∞(F)be the algebra of all$${\mathbb {N}}\times {\mathbb {N}}$$N×Nupper triangular matrices defined over a fieldFof characteristic different from 2. We consider the Jordan triple homomorphisms of$${\mathcal {T}}_\infty (F)$$T∞(F), i.e. the additive maps that satisfy the condition$$\phi (xyx)=\phi (x)\phi (y)\phi (x)$$ϕ(xyx)=ϕ(x)ϕ(y)ϕ(x)for all$$x,y\in {\mathcal {T}}_\infty (F)$$x,y∈T∞(F). For the case whenFis a prime field we find the form of all such maps$$\phi $$ϕ. For the general case we present the form of the surjective maps$$\phi $$ϕ.

Let R be a prime ring of characteristic different from 2, $$Q_r$$ be the right Martindale quotient ring of R and $$C=Z(Q_r)$$ be the extended centroid of R. Suppose that $$f(x_1,\ldots ,x_n)$$ is a noncentral multilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If $$\begin{aligned} F(u)u-G(u)F(u)=0 \end{aligned}$$ for all $$u\in f(R)$$ , then one of the following holds