Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V?H^O(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E₁→E₂→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.     

Bounded solutions of the Golab—Schinzel equation

Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)ⁿ y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .     

A Liouville theorem for weighted p−Laplace operator on smooth metric measure spaces

We prove that on a smooth metric measure space with m ?Bakry–Émery curvature bounded from below by ?(m ? 1)K for some constant K ≥0 (i.e., Ric_{f ,m} ≥?(m ? 1)K ), the following degenerate elliptic equation (0.1) has no nonconstant positive solution when p > 1 and constant λ _{f ,p} satisfies Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng‐Yau's method, which was used by Wang‐Zhu in 2012 to derive a same Liouville theorem when 1 < p ≤2, Ric_{f ,m} ≥?(m ? 1)K and the sectional curvature is bounded from below. Copyright © 2016 John Wiley & Sons, Ltd.     

Clique Vertex Magic Cover of a Graph

Let G admit an H-edge covering and f : V èE ? {1,2,?,n+e}{f : V \cup E \to \{1,2,\ldots,n+e\}} be a bijective mapping for G then f is called H-edge magic total labeling of G if there is a positive integer constant m(f) such that each subgraph H _i , i = 1, . . . , r of G is isomorphic to H and f(H_i)=f(H)=S_{v ? V(Hi)}f(v)+S_{e ? E(Hi)} f(e)=m(f){f(H_i)=f(H)=\Sigma_{v \in V(H_i)}f(v)+\Sigma_{e \in E(H_i)} f(e)=m(f)}. In this paper we define a subgraph-vertex magic cover of a graph and give some construction of some families of graphs that admit this property. We show the construction of some C _n - vertex magic covered and clique magic covered graphs.     

Semistability and finite maps

Let ${f : Y \longrightarrow M}Let f : Y ? M{f : Y \longrightarrow M} be a surjective holomorphic map between compact connected K?hler manifolds such that each fiber of f is a finite subset of Y. Let ω be a K?hler form on M. Using a criterion of Demailly and Paun (Ann. Math. 159 (2004), 1247–1274) it follows that the form f*ω represents a K?hler class. Using this we prove that for any semistable sheaf E ? M{E\, \longrightarrow\,M} , the pullback f*E is also semistable. Furthermore, f*E is shown to be polystable provided E is reflexive and polystable. These results remain valid for principal bundles on M and also for Higgs G-sheaves.     

A representation theorem and Z-cyclic whist tournaments

Let E denote the group of units (i.e., the reduce set of residues) in the ring Z. Here we consider q,p to be primes, q ≡ 3 (mod 4), q ? 7, p ≡ 1 (mod 4). Let W denote a common primitive root of 3, q, and p². If H denotes the (normal) subgroup of E that is generated by {?1, W}, we show that the factor group E/H is cyclic by demonstrating the existence of an element x in E such that the coset xH has order equal to |E/H|. This order is given by gcd(p^n?1(p ? 1),q ? 1). This representation of E/H is exploited via an appropriate construction to produce Z-cyclic whist tournaments for 3qpⁿ players. Consequently these results extend those of an early study of Wh(3qpⁿ) that was restricted to gcd(p^n?1(p ? 1),q ? 1) = 2. © 1995 John Wiley & Sons, Inc.     

Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels

We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner-Riesz multiplier operator. The convolution kernel K(x) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve L^p -boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator Kf(x) = sup_{r>0 r}ⁿ∣K(r) * f(x)∣ associated with such a kernel. The maximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, and strong type bounded between. This together with our assumptions on K(x) leads to the pointwise convergence result lim_γ→ γⁿ K(γ·) * f(x) = cf(x) a. e. for radial f ? L^P(?ⁿ), a ≥ p > b.     

Cokernel bundles and Fibonacci bundles

We are interested in those bundles C on ?^N which admit a resolution of the form 0 → ?^s ? E ?^t ? F → C → 0. In this paper we prove that, under suitable conditions on (E, F), a generic bundle with this form is either simple or canonically decomposable. As applications we provide an easy criterion for the stability of such bundles on ?² and we prove the stability when E = ??, F = ??(1) and C is an exceptional bundle on ?^N for N ≥ 2. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Infinite games in the Cantor space and subsystems of second order arithmetic

In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA₀ ? ‐Det* ? ‐Det* ? WKL₀. 2. RCA₀ ? ( )2‐Det* ? ACA₀. 3. RCA₀ ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR₀. 4. For 1 < k < ω, RCA₀ ? ( )_k ‐Det* ? ( )_{k –1}‐Det. 5. RCA₀ ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )_k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

On the Injective Tensor Product of Distinguished Frechet Spaces

An example of two distinguished Fréchet spaces E, F is given (even more, E is quasinormable and F is normable) such that their completed injective tensor product E?F is not distinguished. On the other hand, it is proved that for arbitrary reflexive Fréchet space E and arbitrary compact set K the space of E - valued continuous functions C(K, E) is distinguished and its strong dual is naturally isomorphic to ? where L¹(μ) = C(K)¹.     

Hilbert polynomials and powers of an ideal

Sunto Nel n1. viene definito il complesso generalizzato di Koszul K(A; E; t) di R-moduli associato ad una matrice A sopra un anello R e un R-modulo E. Si studia poi K(A; E; t) nel caso particolare in cui A sia una matrice della forma B(m, s) data nel n.3. Si dimostra infine che, sotto certe condizioni di finitezza, la lunghezza di ogni modulo d'omologia à una funzione polinomiale in m per m grande, e che per ogni m ≥1 la caratteristica di Euler-Poincaré è il prodotto di un coefficiente binomiale per la caratteristica del complesso di Koszul K(B(1, s); E;0).

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Entrata in Redazione il 18 novemb e 1972.     

Characterization of field homomorphisms and derivations by functional equations

Summary. Let K and [`(K)] \overline K be fields containing \Bbb Q {\Bbb Q} . We characterize pairs of additive functions f,g: K ?[`(K)] f,g: K \to \overline K satisfying a functional equation¶¶ g(x^ln) = f(x^l)ⁿ     \textrespectively        g(x^ln) = Ax^ln + x^ln-lf(x^l) g(x^{ln}) = f(x^l)^n \quad \text{respectively} \qquad g(x^{ln}) = Ax^{ln} + x^{ln-l}f(x^l) ,¶where n ? \Bbb Z \{0,1} n \in {\Bbb Z} \setminus \{0,1\} , l ? \Bbb N l\in {\Bbb N} and A ? K A \in K .     

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means

We indicate criteria for the coincidence of the Knopp kernels K(f) K(A _f), and K (R _f) of bounded functions f(t); here,

The ACL property of homeomorphisms under weak conditions

Suppose thatD is a domain in andy=f(x)∶D→R ⁿ is a homeomorphism. We prove that if the modulus dilatationK(x, f) satisfies the condition A thenf(x) is ACL. Project supported by the National Natural Science Foundation of China and JTU     

Coincidence criteria for the kernel of a function and the kernel of its integral almost positive means

We establish necessary and sufficient conditions for a point A of the Knopp kernelK(f) of a functionf to belong to the kernelK(M) of a functionM(t):=∫ _S fdμ _t , where the so-called almost positive measures μ _t determine a regular method of summation. In particular, this gives coincidence criteria for the kernelsK(f) andK(M). National Pedagogical University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1267–1275, September, 1999.     

On functional which are orthogonally additive modulo Z

Let E be a real inner product space with dimension at least 2, D ? E, f: E → R with f(x+y)?f(x)?f(y) ∈ Z for all orthogonal x,y ∈ E, and f(D) ? (?γ,γ)+Z witn some real γ > 0. We prove that, under some additional assumptions, there are a unique linear functional A: E → R and a unique constant d ∈ R with f(x)?d∥x∥²?A(x) ∈ Z for x ∈ E. We also show some applications of this result to the determination of solutions F: E → C of the conditional equation: F(x+y) = F(x)F(y) for all orthogonal x,y ∈ E.     

On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation

We completely answer the question which positive solutions of the Emden-Fowler equation , n > ½, m > ? 1, m+n > 0 induce models of polytropic gas spheres of finite radius or finite mass. We continue and complete the investigation of the radius continuity of these models initiated by van den Broek and Verhulst⁸ in connection with a numerical analysis by Hénon⁷.     

Über das Verhalten der Lösungen der Maxwellschen Randwertaufgabe in Gebieten mit Kegelspitzen

We study the behavior of the solution E of the Maxwell's boundary value problem ? × ? × E + λE = F, n × E|r = 0 in domains Ω which have conical boundary points. In a neighbourhood K(R) = B(a,R) ∩ Ω of a singular boundary point a the field E is expanded using a theorem of N. Weck. It e.g. turns out that the solution lies in H₁⁽³⁾(K(R)) if K(R) is convex.