Reactions of tetrahalogeno-o-benzoquinones part VI: Reactions of tetrachloro-o-benzoquinone with methyl naphthols—Formation of naphthoquinone methides

The oxidation of a few methyl naphthols with tetrachloro-o-benzoquinone (1) has been described. Reaction of 1-methyl-2-naphthol (4) with the quinone (1) was found to give the spirodimer, 1,1′,2,2′-tetrahydro-3H-benzo (f) chromen-3-spiro-1′-naphthalene-2′-one (6) and a yellow solid which has been assigned the structure of 5,6,7,8-tetrachloro-1,4-benzodioxan-2-spiro-1′-naphthalen-2′-one (7) on the basis of chemical and spectral data. Among the other naphthols studied, 4-methyl-2-t-butyl-1-naphthol afforded 5,6,7,8-tetrachloro-1,4-benzodioxan-2-spiro-1′-naphthalen-3′-t-butyl-4′-one (22) while 2-methyl-1-naphthol and 4-methyl-1-naphthol gave 2-methyl-4,4-(tetrachloro-o-phenylenedioxy) naphthalen-1(4H)-one(15) and 4-methyl-2,2-(tetrachloro-o-phenylenedioxy) naphthelen-1(2H)-one (19) respectively. The present study has shown that oxidation of suitably substituted naphthols with the quinone (1) gives rise to naphthoquinone methides.     

On the collineation groups of infinite projective and affine planes

A partial plane is a triple Π=(P,L,I) whereP is the set of points,L the set of lines andI?PXL the incidence relation satisfying the axiom that $$p_i {\rm I}\ell _j (i,j = 1,2) implies p_1 = p_2 or \ell _1 = \ell _2 .$$ Using methods of E. MENDELSOHN, Z. HEDRLIN and A. PULTR we prove the followingTHEOREM. Given a subgroup G ofthe collineation group Aut Π ofsome partial plane Π, there is a projective plane Π′such that Πis invariant under the automorphisms of Π′, Aut Π′Π^′=G,and we obtain an isomorphism of Aut Πonto Aut Π′by restriction. Moreover, any 3 points (lines) of Πare collinear (concurrent) in Π iff they are so in Π′. Corollaries of this result improve some of E. Mendelsohn's theorems [6,7].     

Characterization of extendible distributions with exponential minima via processes that are infinitely divisible with respect to time

We present a stochastic representation for multivariate extendible distributions with exponential minima (exEM), whose components are conditionally iid in the sense of de Finetti’s theorem. It is shown that the “exponential minima property” is in one-to-one correspondence with the conditional cumulative hazard rate process being infinitely divisible with respect to time (IDT). The Laplace exponents of non-decreasing IDT processes are given in terms of a Bernstein function applied to the state space variable and are linear in time. Examples for IDT processes comprise killed Lévy subordinators, monomials whose slope is randomized by a stable random variable, and several combinations thereof. As a byproduct of our results, we provide an alternative proof (and a mild generalization) of the important conclusion in Genest and Rivest (Stat. Probab. Lett. 8:207211, 1989), stating that the only copula which is both Archimedean and of extreme-value kind is the Gumbel copula. Finally, we show that when the subfamily of strong IDT processes is used in the construction leading to exEM, the result is the proper subclass of extendible min-stable multivariate exponential (exMSMVE) distributions.     

A rotation theorem in the class of bounded univalent functions

Let S_M, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class S_M.     

Fibered incidence groups which are not kinematic

An incidence space \((\beta ,\mathfrak{L})\) which is obtained from an affine space \((\beta _a ,\mathfrak{L}_a )\) by omitting a hyperplane is calledstripe space. If \((\beta _a ,\mathfrak{L}_a )\) is desarguesian, then \(\beta \) can be provided with a group operation “ ○ ” such that \((\beta ,\mathfrak{L}, \circ )\) becomes a kinematic space calledstripe group. It will be shown that there are stripe groups \((\beta ,\mathfrak{L}, \circ )\) where the incidence structure \(\mathfrak{L}\) can be replaced by another incidence structure ? such that \((\beta ,\Re , \circ )\) is afibered incidence group which is not kinematic. An application on translation planes concerning the group of affinities is also given.     

Arithmetic of pall orders and representation of integers by some ternary quadratic forms

The discrete ergodic method (Yu. V. Linnik, Izv. Akad. Nauk SSSR, Ser. Mat.,4, 363–402 (1940); A. V. Malyshev, Tr. Mat. Inst. Akad. Nauk SSSR,65) is applied to the study of properties of integral points on the ellipsoids $$\sum\nolimits_{g,m} : g(x) = m,x = (x_1 ,x_2 ,x_3 ), g(x) = \bar f(Cx),$$ where \(\bar f\) is the adjoint of one of the 39 quadratic forms of Pall (Trans. Am. Math. Soc,59, 280–332 (1946);C is an integral matrix,¦detC¦?1. We construct a flow of integral points on the genus surface of the ellipsoidsg(x)=m. The ergodicity of this flow and a mixing theorem are proved. We obtain an asymptotic formula for the number of representations ofm belonging to a given domain on the ellipsoid and lying in a given residue class.     

The analogue of the law of large numbers for additive functions on sparse sets

An analog of the Turan'n-Kubilyus inequality is proved for a sufficiently wide class of sequences which contains, in particular,a _n=f (n) and a_n=f (p_n), wheref (n) is a polynomial with integral coefficients. This result helps us to obtain integral limit theorems for additive functions on the class of sequences under investigation.     

Ein Approximationssatz für holomorphe Funktionen auf nicht-kompakten Riemannschen Flächen

Let R be a non-compact Riemann surface andO(R) the algebra of all holomorphic functions on R. A subalgebraA ?O(R) is calledfull (“voll”), if (F1) for every point ??R there is a function f∈A with a simple zero at ? and no other zeros; (F2) if f, g∈A and f/g has no poles, then f/∈A. In 1971 Ian RICHARDS set the problem whether full subalgebras are dense inO(R), with respect to the topology of compact convergence. We answer this question in the positive, using a lemma of I. RICHARDS and theorems of R. ARENS and the author. Does this approximation theorem remain true for Stein manifolds of dimension n>1, if one modifies condition (F1) in a natural way? A counterexample is provided by a domain of holomorphy G??² and a full, but not dense subalgebraA ?O(G).     

Ein Axiomatischer Aufbau der Ähnlichkeitsgeometrie

Let Φ?F fields. With respect to the complex number plane we call the elements of Fpoints, the subsets Φm^+b, m≠O,b?F, lines and the bijektions z?F→zm⁺b?F (direct)similitudes. Two noncollinear point-triplets (a₁,a₂,a₃) and (b₁,b₂,b₃) are said to besimilar triangles if there exists a similitude, mapping a₁ onto b₁ for i=1,2,3. Therefore, similarity is an equivalence. relation on the set of all triangles. In this paper, we characterize these geometries axiomatically, starting from incidence structures with an abstract equivalence relation — called similarity — on the set of all triangles by imposing successively similarity-axioms for triangles.     

Optical properties of magnetic crystals

From symmetry considerations and using generalized Onsager relations, it is shown that 66 of the 90 magnetic classes, consisting of 29 single colour and 37 double colour ones, can exhibit what may be called the strain gyrotropic rotation. Similarly, 69 of the 90 magnetic classes, consisting of 21 single colour and 48 double colour ones, can exhibit what may be called the strain gyrotropic birefringence. A crystal in the class m3 or m3m is interesting as it can exhibit strain gyrotropic rotation despite its being cubic and incapable of exhibiting gyrotropic rotation in the unstressed state. Similarly, a crystal in the classm3m, is interesting as it can exhibit strain gyrotropic birefringence despite its being cubic and incapable of exhibiting gyrotropic birefringence in the unstressed state.     

An academic experiment on the use of computers in elementary school mathematics classrooms

The role of computers in elementary school math classrooms is still being determined. Although computers are promised effective visual tools to promote independent work and study; many educators neglect to use them. Since there are varying points of view, individual teachers generally decide whether to incorporate computers into their methods.Purpose: My experiment analyzes and quantifies the value of computers in elementary school math classrooms.Method: Over a course of 11 weeks, my first grade class worked with the teaching software “Mathematikus 1” (Lorenz, 2000). Using both interpersonal and video observation, I completed written evaluations of each pair of my students' will and ability to cooperate, communicate and independently solve mathematical problems.Conclusion: My results show that it is generally beneficial to use computers in elementary school math lessons. However, some elements of said software leave room for improvement.     

Interpolation by analytic functions smooth along the boundary. II

In this paper one considers the following classes of functions, analytic in the circle of the complex plane: , . One establishes that the well-known Carleson condition is necessary and sufficient for the solvability of the interpolation problems, intrinsic for the given class, on sets lying along a ray. In this connection, the class of sequences to be interpolated is described in terms of divided differences of order S. See also author's paper, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,30, 167–169 (1972).     

Ergodic properties of integral points on ellipsoids of genus

The ergodic and the intermixing theorems of Yu. V. Linnik (Mat. Sb.,43, No. 2, 257–276) are generalized to arbitrary positive quadratic forms of genus of , where Ω > 1 is an odd number, of invariants [Ω,1], defined by the character for all prime numbers ρ¦Ω. One obtains estimates for the remainder term. The method of proof is simplified.     

Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.     

Certain complexes associated to a sequence and a matrix

Let R be a commutative noetherian ring with unit. To a sequencex:=x₁,...,x_n of elements of R and an m-by-n matrix α:=(α_ij) with entries in R we assign a complex D*(x;α), in case that m=n or m=n?1. These complexes will provide us in certain cases with explicit minimal free resolutions of ideals, which are generated by the elements a_i:=∑α_ijx_j and the maximal minors of α.     

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied:

1. O _Y,y is reduced.
2. f is universally open at the generic points of the components of X_y which contain x.
3. For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to X_y, we have dim_x, (X_y')=dim_x(X_y)=d.
4. X_y is reduced at the generic points of the components of X_y which contain x and (X_y)_red is geometrically normal over K(y) in x.

Then there exist an open neighbourhood U of x in X and a subscheme U₀ of U which have the same underlying space as U such that f₀:U₀\arY is normal (i.e. f₀ is a flat morphism whose geometric fibers are normal).     

A characterization of the Grassmann embedding of H(q), with q even

In this note, we characterize the Grassmann embedding of H(q), q even, as the unique full embedding of H(q) in PG(12, q) for which each ideal line of H(q) is contained in a plane. In particular, we show that no such embedding exists for H(q), with q odd. As a corollary, we can classify all full polarized embeddings of H(q) in PG(12, q) with the property that the lines through any point are contained in a solid; they necessarily are Grassmann embeddings of H(q), with q even.     

An efficient compact quadratic convex reformulation for general integer quadratic programs

We address the exact solution of general integer quadratic programs with linear constraints. These programs constitute a particular case of mixed-integer quadratic programs for which we introduce in Billionnet et al. (Math. Program., 2010) a general solution method based on quadratic convex reformulation, that we called MIQCR. This reformulation consists in designing an equivalent quadratic program with a convex objective function. The problem reformulated by MIQCR has a relatively important size that penalizes its solution time. In this paper, we propose a convex reformulation less general than MIQCR because it is limited to the general integer case, but that has a significantly smaller size. We call this approach Compact Quadratic Convex Reformulation (CQCR). We evaluate CQCR from the computational point of view. We perform our experiments on instances of general integer quadratic programs with one equality constraint. We show that CQCR is much faster than MIQCR and than the general non-linear solver BARON (Sahinidis and Tawarmalani, User??s manual, 2010) to solve these instances. Then, we consider the particular class of binary quadratic programs. We compare MIQCR and CQCR on instances of the Constrained Task Assignment Problem. These experiments show that CQCR can solve instances that MIQCR and other existing methods fail to solve.