Un teorema sulla rappresentazione degli integrali

Let be a De Possel differentiation basis in a complete measure space , with μ≥0, μ(X)<+∞; letB be a Banach space and . When the F′ exists μ-a.e. onX, we prove that for eachp>0, the μ-integral of‖F′‖ ^p is the lower-bound of a class ofp-variations.     

Un teorema di unicità per una equazione iperbolica,non lineare

We prove the uniqueness of weak solutions of initial boundary value problems Each functiona _i is required to be sufficiently smooth and must satisfy the following conditions: \(e) \sum\limits_1^n {ij} \partial _{\eta _j \eta _h }^2 a_i (x, \ldots , \eta 1, \ldots , \eta _n )\xi _i \xi _j \leqslant 0, h = 1, \ldots , n,\) for some positive constantsK ₀, α, some non negative constantsK _i , some positive functionsH(t)∈L ¹(0,T) and for all ξ≡(ξ _i ), η≡(η _i )∈R ⁿ      

Maps between Banach function algebras satisfying certain norm conditions

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar A$ and $\bar B$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ?\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)?α‖Y=‖ρ(p)τ(p′) ? α‖ _x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M _B and a homeomorphism ?: M _B → M _A between the maximal ideal spaces of B and A such that for all f ∈ A, where $\hat \cdot$ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from $\bar A$ onto $\bar B$ inducing a homeomorphism between $M_{\bar B}$ and $M_{\bar A}$ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.     

Algebraic extensions of finite corank of hilbertian fields

We consider here a hilbertian fieldk and its Galois group (k _s/k). For a natural numbere we prove that almost all (σ) ∈ (k_s/k)^e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ₁, …, σ_e is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldk _s (σ) of 〈σ〉, …, σ_e ink _s, which containsk. Then the group (k _s/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈ (k/k), τ≠1, of finite order such that [k _s (σ):k _s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk?K?k _s (σ) such that 1<[k _s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group (k_s/k)^e.     

On some classes of contact metric manifolds

We define four new classes of contact metric manifoulds using Tanaka connection and Jacobi operators. We prove that a contact metric manifold with the structure vector field ξ belonging to thek-nullity distribution is contact metric locally ?-symmetric (in the sense of D. B. Blair) if and only if the manifold is a and space. Also, we prove that a 3-dimensional contact metric and is locally ?-symmetric (in the sense of D. E. Blair) and give counter-examples of the converse.     

Интегро-дифференциа ляные формулы сферических средних

Реэуме. Установлена формула для средних $$S(t)f(\mu ): = \frac{1}{{\left| {\sigma ^\prime } \right|}}\smallint _{\sigma (\mu )} f(\mu cos t + \nu sin t)d\nu $$ функцииf, определеннои на единичнои сфере а пространства R ⁿ ,n>2; σ(μ) — Экватор с полусом в μ, |σ′| — мера Экватора. Для проиэводнои ? _t ^(r) дано представление.     

ОцЕНкА (ε, δ)-ЁНтРОпИИ к лАссА цЕлых ФУНкцИИ В ИНтЕгРАльНОИ МЕтРИ кЕ

LetB _σ be the class of entire functions of exponential type σ, real valued and bounded in modulus by 1 in the real line. A setG of functions defined on the segment [-T-r, T+r], wherer is a fixed positive number, is called an (ε, δ)-net of the classB _σ on the segment [-т, т] if for any f?B _σ there existsg?G such that for anyx?[-T,T] $$\left| {f(x) - g(x)} \right| \leqq \frac{\varepsilon }{{2r}}\int\limits_{x - r}^{x + r} {\left| {f(t)} \right|dt + \delta .} $$ The main result consists in the following: For any positive σ, r, ε≦1, δ≦1 and sufficiently largeT we have $$H_{\varepsilon ,\delta } (B_\sigma ,T) \leqq \frac{{2\sigma T}}{\pi }\log \frac{{c(\sigma r)}}{{\max (\varepsilon ,\delta )}},$$ where c(σr) depends only on the product σr. The main tool of the proof of this inequality is the following estimate of the derivative of a polynomialP(x) with real coefficients: $$\left\| {P'(x)} \right\|_{L_p ( - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}) \leqq } c\left( {q + 1 + \sum\limits_{i = 1}^{n - q} {\frac{1}{{\left| {a_i } \right|^2 }}} } \right)\left\| {P(x} \right\|_{L_p ( - 1,1)} ,$$ whereq is the number of roots of the polynomialP(x) lying in the disk ¦z¦<1; a₁, ..., a_n?g are the other roots, с is an absolute constant, and 1≦p≦∞.     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.     

Cofiniteness with respect to a Serre subcategory

Let Φ be a system of ideals in a commutative Noetherian ring R, and let be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ . It is shown that if the functor $ D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) $ is exact, then, for any $ \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) $ ∈ for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that $ H_\mathfrak{a}^i (M,N) $ ∈ for all i < t, then $ Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) $ ∈ , provided that is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that $ H_\mathfrak{a}^i (L) $ ∈ , then $ Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) $ ∈ if and only if $ Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) $ ∈ for all j ≥ 0.     

Unitary Equivalence of Proper Extensions of a Symmetric Operator and the Weyl Function

Let A be a densely defined simple symmetric operator in ${\mathfrak{H}}$ , let ${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$ be a boundary triplet for A ^* and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A ₀}, uniquely up to the unitary similarity. Here ${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric ${A \subset A^*}$ and a special boundary triplet ${\widetilde{\Pi}}$ for{A, A} such that the corresponding Weyl function is ${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$ , where B is a non-self-adjoint bounded operator in ${\mathcal{H}}$ . We are interested in the problem whether the result on the unitary similarity remains valid for ${\widetilde{M}(\cdot)}$ in place of M(·). We indicate some sufficient conditions in terms of the operators A ₀ and ${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in ${L^2(\mathbb{R}_+)}$ , we show that ${\widetilde{M}(\cdot)}$ defined in ${\Omega_+ \subset \mathbb{C}_+}$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function ${\widetilde{M}(\cdot)}$ does not determine A _B even up to the similarity.     

Estimates related to sumfree subsets of sets of integers

A subsetA of the positive integers ?₊ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?₊ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines _k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) _k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys _k (B)>δ _k |B|, valid for any finite subsetB of ?₊, necessarilyδ _k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.     

Projective Hausdorff gaps

Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ .     

A base-matrix lemma for sets of rationals modulo nowhere dense sets

We study some properties of the quotient forcing notions ${Q_{tr(I)} = \wp(2^{< \omega})/tr(I)}$ and P _I ?= B(2 ^ω )/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2 ^ω . We show that the remainder forcing R _I =?Q _tr(I)/P _I is σ-closed in these cases. We also study the cardinal invariant of the continuum ${\mathfrak{h}_{\mathbb{Q}}}$ , the distributivity number of the quotient ${Dense(\mathbb{Q})/nwd}$ , in order to show that ${\wp(\mathbb{Q})/nwd}$ collapses ${\mathfrak{c}}$ to ${\mathfrak{h}_{\mathbb{Q}}}$ , thus answering a question addressed in Balcar et?al. (Fundamenta Mathematicae 183:59–80, 2004).     

On inductive limits of measure spaces

We introduce a category of (topological) measure spaces in which inductive limitis exist and where the Banach spaces and (1≤p≤+∞) are isometric for arbitrary inductive systems of (topological) measure spaces.     

Classi di isomorfismo delle cubiche diF q

LetF _q (q=p^r) be a field of characteristicp>3 andA the set of all elliptic cubic curves overF _q having a given absolute invariantj. Furthermore let ≈be the following equivalence relation: « if and only if and F_q are isomorphic overF _q as abelian varieties». The aim of this paper is to study the equivalence classes inA, induced by ≈, and the Frobenius' traces of the cubic curves belonging to different subclasses ofA.     

Essential dimension of involutions and subalgebras

Essential dimension is an invariant of algebraic groups G over a field F that measures the complexity of G-torsors over field extensions of F. We use theorems of N. Karpenko about the incompressibility of Severi-Brauer varieties and quadratic Weil transfers of Severi-Brauer varieties to compute the essential dimension of some closed subgroups of R _K/F (GL ₁(A)), where A is a central division K-algebra of prime power degree and K/F is a separable field extension of degree ≤ 2. In particular, we determine the essential dimension of the group Sim(A, σ) of similitudes of (A, σ), where σ is an F-involution on A, and the essential dimension of the normalizer $N_{GL_1 (A)} \left( {GL_1 \left( B \right)} \right)$ , where B is a separable subalgebra of A.     

Uniqueness of diffusion operators and capacity estimates

Let Ω be a connected open subset of R ^d . We analyse L ₁-uniqueness of real second-order partial differential operators ${H = - \sum^d_{k,l=1} \partial_k c_{kl} \partial_l}$ and ${K = H + \sum^d_{k=1}c_k \partial_k + c_0}$ on Ω where ${c_{kl} = c_{lk} \in W^{1,\infty}_{\rm loc}(\Omega), c_k \in L_{\infty,{\rm loc}}(\Omega), c_0 \in L_{2,{\rm loc}}(\Omega)}$ and C(x) = (c _kl (x)) > 0 for all ${x \in \Omega}$ . Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C ^?1 and their Lebesgue measure |B(r)|. First, we establish that if the balls B(r) are bounded, the Täcklind condition ${\int^\infty_R dr r({\rm log}|B(r)|)^{-1} = \infty}$ is satisfied for all large R and H is Markov unique then H is L ₁-unique. If, in addition, ${C(x) \geq \kappa (c^{T} \otimes c)(x)}$ for some ${\kappa > 0}$ and almost all ${x \in \Omega}$ , ${{\rm div} c \in L_{\infty,{\rm loc}}(\Omega)}$ is upper semi-bounded and c ₀ is lower semi-bounded, then K is also L ₁-unique. Secondly, if the c _kl extend continuously to functions which are locally bounded on ?Ω and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of ${\overline\Omega}$ there exist ${\eta_n \in C_c^\infty(\Omega)}$ satisfying , where ${\Gamma(\eta_n) = \sum^d_{k,l=1}c_{kl} (\partial_k \eta_n) (\partial_l \eta_n)}$ , and for each ${\varphi \in L_2(\Omega)}$ or if and only if cap(?Ω) = 0.     

Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of \({\mathcal{K}(X)}\) -convergence. Moreover, let \({\mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(A)}\) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from \({\mathfrak{M}_{\ell}(C(X, A))}\) onto \({X \times \mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(C(X, A))}\) onto \({X \times \mathfrak{M}(A)}\) . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.