Δ2 degrees without Σ1 induction

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ₁ induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove:

1. P ^? + BΣ₁ + Exp ? There is a proper d-r.e. degree.
2. P ^? +BΣ₁+ Exp ? IΣ₁ ? There is a proper d-r.e. degree below 0′.
3. P ^? + BΣ₁ + Exp ? There is a non-r.e. degree below 0′.

     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.

     

Über Deformationen von analytischen Abbildungskeimen

The notion of deformations of germs of k-analytic mappings generalizes the one of deformations of germs of k-analytic spaces. Using algebraic terms, we prove:

1. The morphism f: A→B of analytic algebras is rigid, iff it is infinitesimally rigid. Moreover, this is equivalent to Ex_A (B,B)=0. This theorem generalizes a result of SCHUSTER [11].
2. Let A be a regular analytic algebra. Then f is rigid iff there exists a rigid analytic algebra B_o such that f is equivalent to the canonic injection A→A?B_o.
3. If f is “almost everywhere” rigid or smooth, then the injection Ext _B ^l (Ω_B|A, Bⁿ)→Ex_A(B, Bⁿ) is an isomorphism.

     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

Pancyclism in Chvátal-Erdös's graphs

LetG = (X, E) be a simple graph of ordern, of stability numberα and of connectivityk withα ≤ k. The Chvátal-Erdös's theorem [3] proves thatG is hamiltonian. We have investigated under these conditions what can be said about the existence of cycles of lengthl. We have obtained several results:

1. IfG ≠ K _k,k andG ≠ C ₅,G has aC _n?1 .
2. IfG ≠ C ₅, the girth ofG is at most four.
3. Ifα = 2 and ifG ≠ C ₄ orC ₅,G is pancyclic.
4. Ifα = 3 and ifG ≠ K _3,3,G has cycles of any length between four andn.
5. IfG has noC ₃,G has aC _n?2 .

     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

Slightly triangulated graphs are perfect

A graph istriangulated if it has no chordless cycle with at least four vertices (?k ≥ 4,C _k ?G). These graphs Jhave been generalized by R. Hayward with theweakly triangulated graphs $(\forall k \geqslant 5,C_{k,} \bar C_k \nsubseteq G)$ . In this note we propose a new generalization of triangulated graphs. A graph G isslightly triangulated if it satisfies the two following conditions;

1. G contains no chordless cycle with at least 5 vertices.
2. For every induced subgraphH of G, there is a vertex inH the neighbourhood of which inH contains no chordless path of 4 vertices.

     

Bump solutions for the mesoscopic Allen�CCahn equation in periodic media

Given a double-well potential F, a ${\mathbb{Z}^n}$ -periodic function H, small and with zero average, and ???>?0, we find a large R, a small ?? and a function H _?? which is ??-close to H for which the following two problems have solutions:

1. Find a set E _?? ,R whose boundary is uniformly close to ? B _R and has mean curvature equal to ?H _?? at any point,
2. Find u = u _?? ,R,?? solving $$ -\delta\,\Delta u + \frac{F'(u)}{\delta} +\frac{c_0}{2} H_\varepsilon = 0, $$ such that u _??,R,?? goes from a ??-neighborhood of +?1 in B _R to a ??-neighborhood of ?1 outside B _R .

     

Zur Struktur Geschlitzter Doppelräume

Let (P, \(\mathfrak{G}\) ,∥_?,∥_r) be an incidence space with two parallelisms ∥_? and ∥_r. (P, \(\mathfrak{G}\) ,∥_?,∥_r) is called double space [5], if for any two intersecting lines A and B and for any two points a ? A, b ? B the ?-parallel to B through a and the r-parallel to A through b intersect. A double space (P, \(\mathfrak{G}\) ,∥_ol,∥_r) is called h1-slit, if (P, \(\mathfrak{G}\) ) is a slit space [7] with at most one affine plane through every point of P. We show that every h1-slit double space of at least dimension three contains h1-slit double spaces of dimension three. Every h1-slit double space (P, \(\mathfrak{G}\) , ∥_?,∥_r) with ∥_? ≠ ∥_r has dimension three. The h1-slit double spaces of dimension three are characterized.     

О рядах по системе Фра нклина

In this paper some basis properties are proved for the series with respect to the Franklin system, which are analogous to those of the series with respect to the Haar system. In particular, the following statements hold:

1. The Franklin series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) converges a.e. onE if and only if \(\mathop \Sigma \limits_{n = 0}^\infty a_n^2 f_n^2 (x)< + \infty \) a.e. onE;
2. If the series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) , with coefficients ¦a _n ¦↓0, converges on a set of positive measure, then it is the Fourier-Franklin series of some function from \(\bigcap\limits_{p< \infty } {L_p } \) ;
3. The absolute convergence at a point for Fourier—Franklin series is a local property;
4. If an integrable function (fx) has a discontinuity of the first kind atx=x ₀, then its Fourier-Franklin series diverges atx=x ₀.

     

On certain properties of linear summation methods of Fourier series on the classesC(ɛ)

Для линейных методов суммирования рядов Ф урье (1) $$L_n (f;x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2} + \sum\limits_{k = 1}^n {\lambda _{k,n} } \cos kt} \right)dt$$ на классах $$C(\varepsilon ) = \{ f:E_n (f) \leqq \varepsilon _n ;\forall n \geqq 0\} ,\varepsilon = \{ \varepsilon _n \} _{n = 0.}^\infty \varepsilon _n \downarrow 0,$$ доказываются:

1. оценки для порядка р оста норм ∥{L_n∥, если из вестен порядок приближения операторами (1) некоторого классаС (?) (при этом, если опера торы (1) приближают класс С(е) с наилучшим порядком, то находится точная а симптотика возрастания норм {∥ L_n∥);
2. сравнительные оцен ки порядков приближе ния классовС(?) операторами (1), если известен порядок при ближения ими некотор ого более узкого класса С(?*).

В том случае, когда опе раторы (1) приближают кл асс С(?*) с наилучшим порядком, получаются точные по рядковые оценки для л юбого более широкого класса С(?).     

On the representation of measurable functions by martingales

Пусть (gW, ?,P) - вероятност ное пространство, ?₁??₂?...?? _n ?...,? _n ?? -последовательност ь σ-алгебр и ?_∞ - порожден ная ими минимальная σ-алгебра. В статье указано необ ходимое и достаточно е условие на последовательность σ-алгебр {? _n }, при выполнении кото рого для любой ?_∞-измер имой функцииf(x) существует ряд \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) центрированных отн осительно {? _n } функций {? _n } _n=1 ^∞ такой, что

1. \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) абсолютно почти вс юду сходится кf(x) на множестве {x: ¦f(x)¦<+∞};
2. \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) сходится по мере кf(x) на множестве {х: ¦f(х)¦=+∞ }.

Полученные результа ты представляют обоб щения и усиления доказанных ранее теорем Р. Ганди и Г. Ламба о пре дставлении ?_∞-измерим ых функций мартингалам и {? _n ,? _n } (см. [1] и [2]).     

On an Ulam's problem

The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? ⁿ ,n≥3, such that:

1. X has not any stationary points on ? ⁿ ;
2. all orbits ofX are bounded;
3. there exists a filtration forX are proved in the present note.

     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Die Endlichen Polardreiseitgeometrien

LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∈ S. Assume thatS contains three elements which generate a four-group. IfS contains four elements of which no three have a product of order two, then one of the following occurs.

1. G?PGL(2,n), n≡1 (mod 2).
2. G?PSL(2,n), n≡1 (mod 2) and n≥5.
3. G?PSU(3,16).
4. G/Z(G)?PSL(2,9) with ¦Z(G)¦=3.

     

New families of graceful banana trees

Consider a family of stars. Take a new vertex. Join one end-vertex of each star to this new vertex. The tree so obtained is known as abanana tree. It is proved that the banana trees corresponding to the family of stars

1. (K_1,1, K_1,2,…, K_{1,t ?1}, (α + l) K_1,t, K_{1,t + 1}, …, K_1,n), α ? 0
2. (2K_1,1, 2K_1,2,…, 2K_{1,t? 1}, (α + 2)K_1,t, 2K_{1,t + 1, …}, 2K_1,n), 0 ? α <t and
3. (3K_1,t, 3K_1,2, …, 3K_1,n) are graceful.