Radioactive decay of Eu154

The radioactive decay of Eu¹⁵⁴ (t _1/2=16y), obtained from thermal neutron irradiations of 95% enriched Eu¹⁵³, has been studied by using the Seigbahn-Slätis intermediate image spectrometer, a thin lens magnetic spectrometer and a coincidence scintillation spectrometer. The following gamma-rays (energy in kev.) have been identified as belonging to this isotope: 123, 248, 585, 717, 759, 875, 998, 1007, 1285, 1600. Beta end points of Eu¹⁵⁴ were measured at 275 (20%), 590 (45%), 890 (23%) and 1860 (12%). The K-conversion coefficient of 123 kev. gamma-ray has been experimentally determined to be ～0·5. On the basis of gamma-gamma and beta-gamma coincidences, the decay scheme of Eu¹⁵⁴ has been deduced: the resulting level-structure of Gd¹⁵⁴ conforms well with the predictions of the unified nuclear model.     

Decay of Samarium-153

Radiations emitted in the decay of Samarium-153 have been studied in the Siegbahn-Slätisbeta-ray spectrometer. Using the internal conversion electron spectrum and the photo-electron spectrum with tin as radiator, the internal conversion coefficienta _k has been determined for 102 Kev. and 70 Kev.gamma-rays. The relative intensities of the threebeta-ray branches have been determined. A weakgamma-ray of energy 83 Kev. has been found and can be interpreted as a transition to the ground state from the first rotational level in Europium-153. The multipole order and character of the 102 Kev. transition is discussed.     

Interpolation in Logiken monotoner systeme

A monotone structure ( ;μ) consists of a structure and a monotone systemμ over the domain of .L(Q ⁿ ) is , enlarged by a newn-ary quantifierQ ⁿ . says in ( ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( ;μ) for allā ∈U ⁿ . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class of all ultrafilters that interpolation with respect to holds forL(Q ⁿ ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q ⁿ ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI ⁿ instead ofQ ⁿ , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.     

The equivalence of theories that characterize ALogTime

A number of theories have been developed to characterize ALogTime (or uniform NC ¹, or just NC ¹), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory characterizes polytime functions. Among these, ALV′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S ₅ is complete for ALogTime. On the other hand, ALV (by Clote), T ⁰ NC ⁰ (by Clote and Takeuti) as well as Arai’s theory and its two-sorted version VNC ¹ (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV′ has been an open problem. Here we show that ALV′ is indeed equivalent to the other theories.      

Paramagnetic resonance of Gd3+ in Sm (NO3)3. 6H2O single crystals

The paramagnetic resonance of Gd³⁺ in Sm(NO₃)₃.6H₂O single crystals, grown from solution, is studied at room temperature. A seven line spectrum for H//Z as well as for H//X corresponding to ΔM=±1 transitions are observed. The spin Hamiltonian analysis is presented.     

EPR spectrum of gadolinium in hydrated praseodymium nitrate

The paramagnetic resonance spectrum of Gd³⁺ in Pr(NO₃)₃·6H₂O single crystals, is studied at room temperature. A seven line spectrum for H//Z as well as for H//X corresponding to ΔM=±1 transitions is observed along with a number of low field transitions (ΔM?2). The spin-Hamiltonian analyses is presented.     

Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

Model theory of the regularity and reflection schemes

Abstract  This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here is a language with a distinguished linear order <, and REF consists of formulas of the form

The region of attainability of nonlinear systems with unbounded controls

In this work, we give a characterization of the topological boundary and the interior of the reachable set for a family of vector fields of the formX ⁰+aX ⁱ , wherea andi belongs to an arbitrary set of indices. The methods considered in this article are applied to various problems in control theory.The authors wish to express their gratitude to Professor Felix Albrecht for his precious comments on the first version of this work.     

Paramagnetic resonance of Gd3+ in SmCl3·6H2O single crystals

The Paramagnetic Resonance of Gd³⁺ in SmCl₃·6H₂O single crystals, grown from solution, is studied at room temperature. A six line spectrum for H//Z and a seven line spectrum for H//X corresponding to ΔM=± 1 transitions are observed. Their angular variation in ZX plane from?=0° to?=90°, is studied and the spin-Hamiltonian analysis is presented. The probable amount of admixture of the next higher electronic state⁶P_7/2 with the ground state⁸S_7/2 is also estimated.     

Degrees of difficulty of generalized r.e. separating classes

Important examples of classes of functions are the classes of sets (elements of ^ω 2) which separate a given pair of disjoint r.e. sets: . A wider class consists of the classes of functions f ∈ ^ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: . We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let denote the Medvedev degrees of those such that no m + 1 sets among A ₀,...,A _k-1 have a nonempty intersection. It is shown that each is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions is the greatest element of . If 2 ≤ l < k, then 0 _M is the only degree in which is below a member of . Each is densely ordered and has the splitting property and the same holds for the lattice it generates. The elements of are exactly the joins of elements of for . Supported by National Science Foundation grants DMS 0554841, 0532644 and 0652732.     

Interpolation,compactness and JEP in soft model theory

Let be the following statement: “for any infinite regularκ, for any uniform ultrafilterD onκ,D isλ-descendingly incomplete for all infiniteλ”. is weaker than ?0^#. Assuming we prove the following: letL be a logic in which the class of sentences of typeτ is a set if so isτ; then: (I)L is compact iffL has JEP; (II)L satisfies Robinson Consistency Theorem iffL is compact and satisfies Craig Interpolation theorem; (III) if, in addition,L is single-sorted, thenL satisfies Robinson Consistency Theorem iffL has JEP^#. JEP (resp. JEP^#) are the natural generalizations for logicL of the familiar Joint Embedding Property of elementary (resp. complete) embeddings in first order logic. As a corollary, we characterize first order logic as the only logic having Löwenheim number equal toω together with JEP.     

Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the d-dimensional lattice ℤ^d. We find a sufficient condition for the positivity of a family of operators h(k) appearing after the “separation of the center of mass” of a system of two particles depending on the values of the total quasimomentum k ∈ T^d (where T^d is a d-dimensional torus). We use the obtained result to show that the operator h(k) has positive eigenvalues for nonzero k ∈ T^d. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 381–387, December, 2007.     

Feynman’s Operational Calculi: Disentangling Away from the Origin

In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ ⁿ . The method of disentangling follows Feynman’s heuristic rules from in (Feynman in Phys. Rev. 84:18–128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ ⁿ while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.     

A solution to Curry and Hindley’s problem on combinatory strong reduction

It has often been remarked that the metatheory of strong reduction , the combinatory analogue of βη-reduction in λ-calculus, is rather complicated. In particular, although the confluence of is an easy consequence of being confluent, no direct proof of this fact is known. Curry and Hindley’s problem, dating back to 1958, asks for a self-contained proof of the confluence of , one which makes no detour through λ-calculus. We answer positively to this question, by extending and exploiting the technique of transitivity elimination for ‘analytic’ combinatory proof systems, which has been introduced in previous papers of ours. Indeed, a very short proof of the confluence of immediately follows from the main result of the present paper, namely that a certain analytic proof system G _e [] , which is equivalent to the standard proof system CL _ext of Combinatory Logic with extensionality, admits effective transitivity elimination. In turn, the proof of transitivity elimination—which, by the way, we are able to provide not only for G _e [] but also, in full generality, for arbitrary analytic combinatory systems with extensionality—employs purely proof-theoretical techniques, and is entirely contained within the theory of combinators.      

Partitions of large Rado graphs

Let κ be a cardinal which is measurable after generically adding many Cohen subsets to κ and let be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value such that the set [κ] ^m can be partitioned into classes such that for any coloring of any of the classes C _i in fewer than κ colors, there is a copy of in such that is monochromatic. It follows that , that is, for any coloring of with fewer than κ colors there is a copy of such that has at most colors. On the other hand, we show that there are colorings of such that if is any copy of then for all , and hence . We characterize as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, and for m > 2 we have where r _m is the corresponding number of types for the countable Rado graph. Research of M. Džamonja and J. A. Larson were partially supported by Engineering and Physical Sciences Research Council and research of W. J. Mitchell was partly supported by grant number DMS 0400954 from the United States National Science Foundation.     

Balanced Metrics and Chow Stability of Projective Bundles over Kähler Manifolds II

In the previous article (Seyyedali, Duke Math. J. 153(3):573–605, 2010), we proved that slope stability of a holomorphic vector bundle E over a polarized manifold (X,L) implies Chow stability of $(\mathbb{P}E^{*},\mathcal{O}_{\mathbb{P}E^{*}}(1)\otimes\pi^{*} L^{k})$ for k?0 if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of 2πc ₁(L). In this article, using asymptotic expansions of the Bergman kernel on Sym ^d E, we generalize the main theorem of Seyyedali (Duke Math. J. 153(3):573–605, 2010) to polarizations $(\mathbb{P}E^{*},\mathcal {O}_{\mathbb{P}E^{*}}(d)\otimes\pi^{*} L^{k})$ for k?0, where d is a positive integer.     

Embedding FD(ω) into {\mathcal{P}_s} densely

Let be the lattice of degrees of non-empty subsets of 2 ^ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in . Cenzer and Hinman proved that is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any , we can lattice embed FD(ω) into strictly between and . We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made. Thanks to my adviser Peter Cholak for his guidance in my research. I also wish to thank the anonymous referee for helpful comments and suggestions. My research was partially supported by NSF grants DMS-0245167 and RTG-0353748 and a Schmitt Fellowship at the University of Notre Dame.     

Analysis of discrete techniques for image transformation

Let an image be distorted by a nonlinear transformationT, and then be restored back to its original by the inverse transformationT ^–1. Such a cycle conversion,T ^–1 T, of digital images can be facilitated by the combination (CSIM) given in [6] using the splitting-shooting method and the splitting-integrating method forT andT ^–1 respectively. Since there is no need to solve nonlinear equations, CSIM has been widely applied to specific areas of image processing and pattern recognition, even those with complicated transformations, e.g., the harmonic transformation [12]; however, no strict error analysis has been provided so far. In this paper, a priori error estimates and convergence rates are derived for pixel greyness obtained from CSIM; also the analytical results are extended to the harmonic transformation. Numerical and graphical experiments are provided to support the theoretical analysis.Some preliminary results of this paper were presented at the 40th anniversary meeting of SIAM, Los Angeles, July 20–24, 1992. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds pour la Formation de Chercheurs et l'Aide à la Recherche of Quebec by the Ministère de l'Enseignement Supérieur et de la Science (Action Structurante), and the Science Council of Republic of China under Grant No. NSC-85-2121-M-008.