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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]

It is natural to ask whether solvsolitons are global maxima for the Ricci pinching functional F := scal2/|Ric|2 on the set of all left-invariant metrics on a given solvable Lie group S, as it is to ask whether they are the only global maxima. A positive answer to both questions was given in a recent paper by the same authors when the Lie algebra s of S is either unimodular or has a codimension-one abelian ideal. In the present paper, we prove that this also holds in the following two more general cases: 1) s has a nilradical of codimension-one; 2) the nilradical n of s is abelian and the functional F is restricted to the set of metrics such that a ⊥ n, where s = a ⊕ n is the orthogonal decomposition with respect to the solvsoliton. [ABSTRACT FROM AUTHOR]

In this paper we consider the class of l-bijective C-systems, i.e., C-systems for which the length function is a bijection. The main result of the paper is a construction of an isomorphism between two categories: the category of l-bijective C-systems and the category of Lawvere theories. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer. [ABSTRACT FROM AUTHOR]

In his recent Proceedings of the AMS paper, ''Gossez's skew linear map and its pathological maximally monotone multifunctions'', Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is not convex whenever the scalar is between 0 and 4. The problem of the convexity of that range when the scalar is equal to 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4. We derive this result from an abstract framework that allows us to also obtain a corresponding result for the Fitzpatrick-Phelps integral operator. [ABSTRACT FROM AUTHOR]

In this paper, we prove Lp decay estimates for multilinear oscillatory integrals in R2, establishing sharpness through a scaling argument. The result in this paper is a generalization of previous work by Gressman and Xiao [J. Funct. Anal. 271 (2016), pp. 3695-3726]. [ABSTRACT FROM AUTHOR]

In this paper, we prove Lp decay estimates for multilinear oscillatory integrals in R2, establishing sharpness through a scaling argument. The result in this paper is a generalization of previous work by Gressman and Xiao [J. Funct. Anal. 271 (2016), pp. 3695-3726]. [ABSTRACT FROM AUTHOR]

In this paper we study the b-ary expansions of the square roots of the function defined by the recurrence ƒb(n) = b ƒb(n−1) + n with initial value ƒ(0) = 0 taken at odd positive integers n, of which the special case b = 10 is often referred to as the ''schizophrenic'' or ''mock-rational'' numbers. Defined by Darling in 2004 and studied in more detail by Brown in 2009, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 by formally defining the schizophrenic pattern present in the b-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within. [ABSTRACT FROM AUTHOR]

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark. [ABSTRACT FROM AUTHOR]

The Long-time behavior of orbits is one of the most fundamental properties in dynamical systems. Poincaré studied the Poisson stability, which satisfies a time-reversal symmetric condition, to capture the property of whether points return arbitrarily near the initial positions after a sufficiently long time. Birkhoff introduced and studied the concept of non-wandering points, which is one of the time-reversal symmetric conditions. Moreover, minimality and pointwise periodicity satisfy the time-reversal symmetric condition for limit sets. This paper characterizes flows with the time-reversal symmetric condition for limit sets, which refine the characterization of irrational or Denjoy flows by Athanassopoulos. Using the description, we construct flows on a sphere with Lakes of Wada attractors and with an arbitrarily large number of complementary domains, which are flow versions of such examples of spherical homeomorphisms constructed by Boroński, Činč, and Liu. [ABSTRACT FROM AUTHOR]

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real (resp. complex) analytic spaces to \mathbb {R}^m (resp. \mathbb {C}^m). It is shown that the flatness of analytic maps from germs of real (resp. complex) analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection (resp. Cohen-Macaulay) analytic germs can be approximated arbitrarily well by polynomial (resp. Nash) maps in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone. [ABSTRACT FROM AUTHOR]

Let x_1, x_2, ..., x_n be n points on the sphere S^2. Determining the value \inf \sum _{1\leq k

In this paper, we study complete self-shrinkers in Euclidean space and prove that an n-dimensional complete self-shrinker in Euclidean space \mathbb {R}^{n+1} is isometric to either \mathbb {R}^{n}, S^{n}(\sqrt {n}), or S^k (\sqrt {k})\times \mathbb {R}^{n-k}, 1\leq k\leq n-1, if the squared norm S of the second fundamental form, f_3 are constant and S satisfies S<1.83379. We should remark that the condition of polynomial volume growth is not assumed. [ABSTRACT FROM AUTHOR]

This paper is devoted to the limit cycle bifurcation problem for some cubic polynomial systems, whose unperturbed systems have a period annulus and two invariant lines. Using the first order Melnikov function and Chebyshev criterion, we obtain the maximum number of limit cycles bifurcating from the period annulus. It improves a known result given by Sui and Zhao [Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28 (2018), 1850063]. [ABSTRACT FROM AUTHOR]

In this paper we prove supercongruence relations for truncated N-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogues of some Ramanujan-type supercongruences. [ABSTRACT FROM AUTHOR]

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound \sum _{n=1}^\infty \frac {\mu (n)}{n^2} \exp \left (- \frac {x}{n^2} \right) = O_{\epsilon } \left (x^{-\frac {3}{4} + \epsilon } \right), as x \rightarrow \infty, for any \epsilon >0. Around the same time, Hardy and Littlewood gave another equivalent criterion for the Riemann Hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann Hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood. [ABSTRACT FROM AUTHOR]

In this paper, we prove a power-law version dynamical localization for a random operator \mathrm {H}_{\omega } on \mathbb {Z}^d with long-range hopping. In brief, we show that the Sobolev norm of the discrete linear Schrödinger equation \mathrm {i}\partial _{t}u=\mathrm {H}_{\omega }u with well-localized initial state is bounded for any t\geq 0. [ABSTRACT FROM AUTHOR]

Ritt's theorem of zeroes and Siedenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs. [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]

The isochronicity problem is a classical problem in the qualitative theory of planar vector fields which consists in characterizing whether a center is isochronous or not, that is, if all the trajectories in a neighborhood of the center have the same period. This problem is usually investigated by means of the so-called period function. In this paper, we are interested in exploring the isochronicity problem for tangential centers of planar Filippov vector fields. By computing the period function for planar Filippov vector fields around tangential centers, we show that such centers are never isochronous. [ABSTRACT FROM AUTHOR]

This paper focuses on a two-dimensional incompressible Oldroyd-B model with mixed partial dissipation. The goal here is to establish the small data global existence and stability in the Sobolev space H^2(\mathbb R^2). The velocity equation itself, without coupling with the equation of the non-Newtonian stress tensor, is an anisotropic 2D Navier-Stokes whose solutions are not known to be stable in Sobolev spaces due to potential rapid growth in time. By unearthing the hidden wave structure of the system and exploring the smoothing and stabilizing effect of the non-Newtonian stress tensor on the fluid, we are able to solve the desired global existence and stability problem. [ABSTRACT FROM AUTHOR]

The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we introduce the notion of 'finite splitting height' which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include relatively quasiconvex subgroups of relatively hyperbolic groups with virtually polycyclic peripheral subgroups. In particular, our results extend Stallings' theorem and generalise a theorem of Sageev on decomposing a hyperbolic group by quasiconvex subgroups. [ABSTRACT FROM AUTHOR]

This paper is concerned with propagation dynamics in a nonlocal dispersal HIV infection model. The existence and asymptotic behavior of traveling waves with wave speeds not less than a critical speed were derived in the recent work of Wang and Ma [J. Math. Anal. Appl. 457 (2018), pp. 868–889]. However, the asymptotic behavior of the critical traveling wave and minimum wave speed were not clarified completely. In this article, we first affirm the asymptotic behavior of the critical traveling wave at negative infinity. Then we prove the non-existence of traveling waves when either the basic reproduction number \mathcal {R}_0<1 or the wave speed is less than the critical spreed and \mathcal {R}_0>1. Our result provides a complete complement for the wave propagation in the infection model. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded. [ABSTRACT FROM AUTHOR]

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous-trace C^*-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arises naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish two gap theorems for ends of smooth metric measure space (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor \operatorname {Ric}_{f}\!\ge -(n-1) in a geodesic ball B_{o}(R) with radius R and center o\in M^n. When \operatorname {Ric}_{f}\ge 0 and f has some degeneration outside B_{o}(R), we show that there exists an \epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such that such a space has at most two ends if R\le \epsilon. When \operatorname {Ric}_{f}\ge \frac 12 and f(x)\le \frac 14d^2(x,B_{o}(R))+c for some constant c>0 outside B_{o}(R), we can also get the same gap conclusion. [ABSTRACT FROM AUTHOR]

The Bowen topological entropy was introduced by Bowen in way which resembles the Hausdorff dimension. It is well-known that the Bowen topological entropy and the topological entropy of subsets are not equal in general. We show in the paper that the supremum of the Bowen topological entropy of the dynamical balls is the same as that of the topological entropy of the dynamical balls, which naturally implies that every positively countably expansive map is also positively entropy expansive. This answers a question posed by Artigue, Carvalho, Cordeiro, and Vieitez [Proc. Amer. Math. Soc. 150 (2022), pp. 3369–3378]. Besides, our main result is applied to find some other interesting phenomena. [ABSTRACT FROM AUTHOR]

Let (X,d, \mu) be an Ahlfors n-regular metric measure space. Let \mathcal {L} be a non-negative self-adjoint operator on L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F(\mathcal {L}) satisfy an appropriate weighted L^2 estimate. By the spectral theory, we can define the imaginary power operator \mathcal {L}^{is}, s\in \mathbb R, which is bounded on L^2(X). The main aim of this paper is to prove that for any p \in (0,\infty), \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1/p-1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H^p_\mathcal {L}(X) is the Hardy space associated to \mathcal {L}, and C is a constant independent of s. Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on \mathbb {R}^n with n\ge 2. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the nth-logarithmically improved regularity criterion of smooth solutions for the incompressible Leray-\alpha-MHD model in terms of the magnetic field B. Meanwhile, the new logarithmically improved regularity criterion for the 3D Navier-Stokes equation in terms of the pressure \pi and gradient of velocity \nabla u can also be established. Especially, we explore a new logarithmically improved Serrin's criterion for the 3D Navier-Stokes equations, which improves the results of Lei and Zhou [Commn. Pure Appl. Anal. 12 (2013), pp. 2715–2719]. [ABSTRACT FROM AUTHOR]

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two. [ABSTRACT FROM AUTHOR]

In this paper, q-Gaussian distribution, q-analogy of Gaussian distribution, is introduced to characterize the prior information of unknown parameters for inverse problems. Based on q-Hermite polynomials, we propose a spectral likelihood approximation (SLA) algorithm of Bayesian inversion. Convergence results of the approximated posterior distribution in the sense of Kullback–Leibler divergence are obtained when the likelihood function is replaced with the SLA and the prior density function is truncated to its partial sum. In the end, two numerical examples are displayed, which verify our results. [ABSTRACT FROM AUTHOR]

In this paper, regarding the Riemannian product \mathbb {S}^2\times \mathbb {S}^2 of two unit 2-spheres as a Kähler surface, we study its real hypersurfaces with typical geometric properties. First, we classify the real hypersurfaces of \mathbb {S}^2\times \mathbb {S}^2 with isometric Reeb flow and then, by using a Simons' type inequality, a characterization of these compact real hypersurfaces is provided. Next, we classify Hopf hypersurfaces of \mathbb {S}^2\times \mathbb {S}^2 with constant product angle function. Finally, we classify Hopf hypersurfaces of \mathbb {S}^2\times \mathbb {S}^2 with parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Laurent biorthogonal polynomials with a weight function of three parameters, i.e. z^\alpha e^{-t_1z-\frac {t_2}{z}}, z\in (0,+\infty), (t_1>0,\ t_2>0,\ \alpha \in \mathbb {R}). First, the structure relation of the Laurent biorthogonal polynomials is found with the aid of biorthogonality. Then we derive an alternate discrete Painlevé II by considering the compatibility condition of the three-term recurrence relation and the structure relation. In addition, we make use of the relativistic Toda chains and nonlinear difference system to obtain two continuous Painlevé-type differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a family of stochastic partial differential equations with nonlinear multiplicative white noise. The existence of Lipschitz mean-square random invariant unstable manifolds for these equations has been obtained by Wang [Discrete Contin. Dyn. Syst. 41 (2021), pp. 1449–1468]. Based on this result, we investigate the convergence of Lipschitz mean-square random unstable manifolds for these equations which are defined in varying phase spaces. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

In this paper, Multiplicative Ergodic Theorem (MET) on manifolds with semi-discrete time variable is proved. Considering that there is no cocycle property with any semi-discrete time variable t\in \mathbb {T}, we define the quasi-cocycle property on forward and backward time scales. We obtain the skew-product quasi-flow with semi-discrete time variable t\in \mathbb {T}. For dynamic equations with \Delta-derivative and \nabla-derivative on \mathbb {T}, we present a more generalized version about MET of semi-discrete system. The result is more suitable for treating models with nonuniform time difference and studying the stability of systems induced by both differential and difference operators. [ABSTRACT FROM AUTHOR]

Let S= \{ a_{1}, a_{2}, \dots, a_{n} \} be a finite set of non-zero integers. R. Balasubramanian, F. Luca and R. Thangadurai [Proc. Amer. Math. Soc. 138 (2010), pp. 2283–2288] gave an exact formula for the degree of the multi-quadratic field \mathbb {K}= \mathbb {Q}(\sqrt {a_{1}}, \sqrt {a_{2}}, \dots, \sqrt {a_{n}}) over \mathbb {Q}. In this paper, we calculate the explicit structure of the Galois group \operatorname {Gal}(\mathbb {K}/\mathbb {Q}) in terms of its action on \sqrt {a_{i}} for 1 \leq i \leq n. [ABSTRACT FROM AUTHOR]

Let \lambda \in (1,\sqrt {2}] be an algebraic integer with Mahler measure 2. A classical result of Garsia shows that the Bernoulli convolution \mu _\lambda is absolutely continuous with respect to the Lebesgue measure with a density function in L^\infty. In this paper, we show that the density function is continuous. [ABSTRACT FROM AUTHOR]

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

Spun normal surfaces are a useful way of representing proper essential surfaces, using ideal triangulations for 3-manifolds with tori and Klein bottle boundaries. In this paper, we consider spinning essential surfaces in an irreducible P^2-irreducible, anannular, atoroidal 3-manifold with tori and Klein bottle boundary components. We can assume that such a 3-manifold is equipped with an ideal 1-efficient triangulation. In particular, we prove that for a given choice of a set of boundary slopes for a proper essential surface, there is a set of essential vertex solutions for the projective solution space at these boundary slopes, answering a question of Dunfield and Garoufalidis [Trans. Amer. Math. Soc. 364 (2012), pp. 6109–6137], so long as the slopes are not of a fiber of a bundle structure. [ABSTRACT FROM AUTHOR]

In this paper, we prove constant rank theorems for Li-Yau-Hamilton type matrices of heat equations on Riemannian manifolds as well as on Kähler manifolds under some curvature conditions. [ABSTRACT FROM AUTHOR]

Let n, r and f be positive integers. Let p be a prime number and \psi be an arbitrary fixed nontrivial additive character of the finite field \mathbb F_q with q=p^f elements. Let F be a polynomial in \mathbb F_q[x_1,\dots,x_n] and V be the affine algebraic variety defined over \mathbb {F}_q by the simultaneous vanishing of the polynomials \{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots,x_n]. Let \mathbb {Z}_{\ge 0} stand for the set of all nonnegative integers and A be an arbitrary nonempty subset of \{1,\dots,n\}. For a polynomial H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with {\mathbf {d}}=(d_1,\dots,d_n)\in \mathbb {Z}_{\ge 0}^n, X^{\mathbf {d}}=x_1^{d_1}\dots x_n^{d_n} and \alpha _{\mathbf {d}}\in \mathbb {F}_q^*, we define \deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\} to be the A-degree of H. In this paper, for the exponential sum S(F,V,\psi)=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X)) with V(\mathbb {F}_q) being the set of the \mathbb {F}_q-rational points of V, we show that \begin{equation*} \mathrm {ord}_q S(F,V,\psi)\ge \frac {|A|-\sum _{i=1}^r\deg _A(F_i)} {\max _{1\le i\le r}\{\deg _A(F),\deg _A(F_i)\}} \end{equation*} if \deg _A(F)>0 or \deg _A(F_i)>0 for some i\in \{1,\dots,r\}. This estimate improves Sperber's theorem obtained in 1986. This also leads to an improvement of the p-adic valuation of the number N(V) of \mathbb {F}_q-rational points on the variety V which strengthens the Ax-Katz theorem. Moreover, we use the A-degree and p-weight A-degree to establish p-adic estimates on multiplicative character sums and twisted exponential sums which improve Wan's results gotten in 1995. [ABSTRACT FROM AUTHOR]