Standard Model Higgs boson mass from inflation

We analyse one-loop radiative corrections to the inflationary potential in the theory, where inflation is driven by the Standard Model Higgs field. We show that inflation is possible provided the Higgs mass mH$m_H$ lies in the interval mmin<mH<mmax$m_{\min }<m_H<m_{\max }$, where mmin=[136.7+(mt−171.2)×1.95] GeV$m_{\min }=[136.7+(m_t-171.2)\times 1.95]\text{GeV}$, mmax=[184.5+(mt−171.2)×0.5] GeV$m_{\max }=[184.5+(m_t-171.2)\times 0.5]\text{GeV}$ and mt$m_t$ is the mass of the top quark. In the renormalization scheme associated with the Einstein frame the predictions of the spectral index of scalar fluctuations and of the tensor-to-scalar ratio practically do not depend on the Higgs mass within the admitted region and are equal to ns=0.97$n_s=0.97$ and r=0.0034$r=0.0034$ correspondingly.     

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.      

Intramolecular [4+2] cycloaddition of furfurylsubstituted homoallylamines to allylhalides,acryloyl chloride and maleic anhydride

Acylation of 4‐(furyl‐2)‐4‐R‐aminobut‐1‐enes and 4‐R‐4‐furfurylaminobut‐1‐enes with maleic anhydride, acryloyl chloride or allylhalides provided 3‐aza‐10‐oxatricyclo[5.2.1.0^1,5]decenes. The tricycles are formed via an initial amide formation followed by a stereoselective exo‐IMDAF (Intramolecular Diels‐Alder of Furan). In case of competing cycloaddition (for compounds possessing two furan or two dienophilic moieties) the most substituted fivemembered cycle is preferably annulated. Refluxing of 4‐R‐4‐furfurylaminobut‐1‐enes in acetic anhydride led to exo‐3‐aza‐11‐oxatricyclo[6.2.1.0^1,6]undecenes with the pseudoequatorial substituent R‐4. Treatment of 3‐aza‐10‐oxatricyclo[5.2.1.0^1,5]decenes with PPA at 90?110°C promoted cyclic ether opening, aromatization and intramolecular cyclization reactions sequence to give the corresponding tetracyclic compounds — tetrahydroisoindolo[2,1‐a]quinolines and tetrahydroisoindolo[2,1‐b][2]benzazepines in good yields. Unusual products of ipso‐substitution in aromatic ring were obtained on cyclization of N‐p‐R‐substituted 2‐allyl‐4‐oxo‐3‐aza‐10‐oxatricyclo[5.2.1.0^1,5]dec‐8‐enes.     

Refractive-index dispersion of glassy semiconductors in the pseudo-binary As2Se3–SbSI system

The bulk chalcogenide glasses in the pseudo-binary system (As₂Se₃)_100?x(SbSI)_x (with x = 20, 30, 50, 70 and 80 at.%) were prepared from high-purity elemental components by melt-quenching technique. It is a system with the variable ratio of classical amorphous compound As₂Se₃ and the molecule of antimony-sulphoiodide, SbSI, which in the monocrystal form is characterized as a ferroelectric. The refractive-index behaviour (magnitude and spectral dispersion) of investigated glasses was determined by the prism method and analyzed. To calculate and discuss the parameters of dispersion in the band gap region three different approaches were used (Cauchy, Sellmeier and Wemple–DiDomenico single-oscillator model). The comparison of the results shows good agreement between all applied models. It was found that the value of the refractive-index shows normal dispersion behaviour and increases with the increase of Sb (or SbSI) content in the investigated system, whereas the optical gap of these glasses E_g slightly decreases.     

Convolutions of cantor measures without resonance

Denote by µ _a the distribution of the random sum \((1 - a)\sum\nolimits_{j = 0}^\infty {{w_j}{a^j}} \), where P(ω _j = 0) = P(ω _j = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ _a is supported on C _a , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 ? 2a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ _a * (µ _b ° S _λ ^?1 ), where S _λ (x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ ≠ 0, then

$D({\mu _a} * ({\mu _b} \circ S_\lambda ^{ - 1})) = \min ({\dim _H}({C_a}) + {\dim _H}({C_b}),1)$

, where D denotes any of correlation, Hausdorff or packing dimension of a measure.

We also show that, perhaps surprisingly, for uncountably many values of λ the convolution µ_1/4* (µ_1/3 ° S _λ ^?1 ) is a singular measure, although dim _H (C _1/4) + dim _H (C _1/3) > 1 and log(1/3)/ log(1/4) is irrational.     

On a question of A. E. Nussbaum on measurability of families of closed linear operators in a Hilbert space

The purpose of this note is to answer a question A. E. Nussbaum formulated in 1964 about the possible equivalence between weak measurability of a family of densely defined, closed operators {T(t)} _t∈ℝ in a separable complex Hilbert space H\mathcal{H} on one hand, and the notion of measurability of the 2 × 2 operator-valued matrix of projections {(P(Γ(T(t))) _j,k )_1⩽j,k⩽2} _t∈ℝ onto the graph Γ(T(t)) of T(t) on the other, in the negative.     

Koplienko Spectral Shift Function on the Unit Circle

The note shows that, for every pair of contractions, a trace formula of Koplienko holds with a summable spectral shift function, see Theorem 1 below. This answers a question by F. Gesztesy, A. Pushnitski, B. Simon (see Zh Mat Fiz Anal Geom 4(1):63–107, 202, 2008, Open Question 11.2).     

On a possible empirical meaning of meets and joins for quantum propositions

Following up an idea of Jauch, an empirical basis is worked out for the meets (and joins) of quantum propositions through approximate measurement to an arbitrary degree of accuracy of special yes-no effects constructed from chains of yes-no measurements. This is done with a view to interpreting the meets (and joins) as Dedekind's cuts in analogy with irrationals.     

Mott's cloud-chamber theory made explicit and the relative-collapse interpretation of quantum mechanics thus obtained

It is pointed out that the concept of a shiftable split between object and subject with a well-defined subject, introduced and utilized in two preceding articles of the author, is present in Mott's historical cloud-chamber measurement theory. The crucial question of the occurrence of the subject events (the main constituents of the subject) is also given a tacit answer in Mott's text. When Mott's theory is made sufficiently explicit, the object-subject complementarity principle (an elaborated form of Bohr's macroscopic complementarity principle) emerges. The (individual-system) relative-collapse postulate, which appears as a natural completion of Born's (ensemble) postulate, takes form in it. The proposed interpretation is compared with the many-worlds and the modal approaches, which share with the former the idea of nondestruction of coherence (in contrast to interpretations that contain the idea of an absolute, i.e., observer-independent collapse, which destroys coherence irrevocably).