An eigenfunction expansion of the non-selfadjoint Sturm–Liouville operator with a singular potential

In this paper, we consider the operator $L$ L generated in $L^{2}\left( \mathbb{R }_{+}\right) $ L 2 R + by the differential expression $$\begin{aligned} l\left( y\right) =-y^{\prime \prime }+\left[ \frac{\nu ^{2}-\frac{1}{4}}{x^{2}}+q\left( x\right) \right] y,\,\,x\in \mathbb{R }_{+}:=\left( 0,\infty \right) \end{aligned}$$ l y = - y ' ' + ν 2 - 1 4 x 2 + q x y , x ∈ R + : = 0 , ∞ and the boundary condition $$\begin{aligned} \underset{x\rightarrow 0}{\lim }x^{-\nu -\frac{1}{2}}y\left( x\right) =1, \end{aligned}$$ lim x → 0 x - ν - 1 2 y x = 1 , where $q$ q is a complex valued function and $\nu $ ν is a complex number with $Re\nu >0$ R e ν > 0 . We have proved a spectral expansion of L in terms of the principal functions under the condition $$\begin{aligned} \underset{x\in \mathbb{R }_{+}}{Sup}\left\{ e^{\epsilon \sqrt{x}}\left| q(x)\right| \right\} <\infty , \epsilon >0 \end{aligned}$$ S u p x ∈ R + e ? x q ( x ) < ∞ , ? > 0 taking into account the spectral singularities. We have also investigated the convergence of the spectral expansion.     

Auswertung von EMK-Messungen zur Bestimmung thermodynamischer Exzeßgrößen im System Silberchlorid—Lithiumchlorid

The partial molar excessGibbs energies \(\Delta \overline G _{AgCl}^E \) of AgCl in the binary system AgCl?LiCl have been measured over the entire composition range at temperatures between 923.15K and 1175.15K in steps of 50K, using the reversible formation cell $${{Ag\left( s \right)} \mathord{\left/ {\vphantom {{Ag\left( s \right)} {AgCl\left( l \right)}}} \right. \kern-\nulldelimiterspace} {AgCl\left( l \right)}}---LiCl\left( l \right)/C,Cl_2 $$ The measured \(\Delta \overline G _{AgCl}^E \) values were fitted by the use of theRedlich-Kister-Ansatz for thermodynamic excess functions. The evaluatedRedlich-Kister parameters have been used to calculate the molar excessGibbs energies ΔG ^E and the partial molar excessGibbs energies \(\Delta \overline G _{LiCl}^E \) of LiCl. From the temperature dependence of theRedlich-Kister parameters for ΔG ^E the partial and integral molar heats of mixing and excess entropies were calculated. For 1073 K and the mole fractionx=0.5 the following values were obtained: $$\Delta G^E = 2130\left[ {J mol^{ - 1} } \right], \Delta H^E = 1994\left[ {J mol^{ - 1} } \right], \Delta S^E = 0.127 \left[ {J mol^{ - 1} K^{ - 1} } \right]$$      

Bounds for the Kirchhoff index via majorization techniques

Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of ${\mathbb{R}^{n}}$ , we find upper and lower bounds for the Kirchhoff index K(G) of an arbitrary simple connected graph G that improve those existing in the literature. Specifically we show that $$K(G) \geq \frac{n}{d_{1}} \left[ \frac{1}{1+\frac{\sigma}{\sqrt{n-1}}} + \frac{(n-2)^{2}}{n-1-\frac{\sigma}{\sqrt{n-1}}}\right] ,$$ where d ₁ is the largest degree among all vertices in G, $$\sigma ^{2} = \frac{2}{n} \sum_{(i, j) \in E} \frac{1}{d_{i}d_{j}} = \left( \frac{2}{n}\right) R_{-1}(G),$$ and R _?1(G) is the general Randi? index of G for ${\alpha =-1}$ . Also we show that $$K(G) \leq \frac{n}{d_{n}}\left( \frac{n-k-2}{1-\lambda _{2}}+\frac{k}{2}+\frac{1}{\theta}\right) ,$$ where d _n is the smallest degree, ${\lambda _{2}}$ is the second eigenvalue of the transition probability of the random walk on G, $$k = \left \lfloor \frac{\lambda _{2} \left( n-1\right) +1}{\lambda _{2}+1}\right\rfloor {\rm and}\quad\theta = \lambda _{2} \left( n-k-2\right) -k+2.$$      

Relation générale entre l'indice de liaison et la distance interatomique dans les molécules conjuguées,planes et linéaires

Two general relation between bond orderl and bond distance d (Å) are proposed:

1. between atomssp ₂-hybridised of second and third row: $$d_{PQ} = \left[ {0,731 + 0,3181\left( {n_P + n_Q } \right) - 0,1477\left( {\zeta _P + \zeta _Q } \right)} \right] - 0,020 + 0,0523\left( {\zeta _P + \zeta _Q } \right)l_{PQ} $$ ,ζ=Z/n,Z=Slater's effective nuclear charge of theπ-orbital).
2. between atomssp-hybridised of the second row: $$d_{PQ} = \left[ {1,904 - 0,123\left( {\zeta _P + \zeta _Q } \right)} \right] - \left[ {0,075 + 0,023\left( {\zeta _P + \zeta _Q } \right)} \right]l_{PQ} $$ (l=total bond orderπ+π′).

     

Exact transient solutions of kinetics of first‐order reactions with end effects

The finite set of rate equations C _m,n ^' =α _n,n-1 C _m,n-1 (t)+α _n,n C _m,n (t)+α _n,n+1 C _m,n+1 (t), $$0 \leqslant m \leqslant N,0 \leqslant n \leqslant N,$$ where $$\alpha _{i,j}$$ are $\alpha _{j,j - 1} = A,\alpha _{j,j} = - \left( {A + B} \right),\alpha _{j,j + 1} = B$ , with $\alpha _{0,0} = - \alpha _{1,0} = - \alpha$ and $\alpha _{N,N} = - \alpha _{N - 1,N} = - b,\alpha _{0, - 1} = \alpha _{N,N + 1} = 0$ , subject to the initial condition $C_{m,n} \left( 0 \right) = \delta _{n,m}$ (Kronecker delta) for some $m$ , arises in a number of applications of mathematics and mathematical physics. We show that there are five sets of values of $a$ and $b$ for which the above system admits exact transient solutions.     

On some multiplicative product of kth derivative of Dirac's delta in x0+|x| and x0−|x|

In this paper we give a sense to the products $${{\left| x \right|^{(n - 2)/2} }} \cdot \frac{{\delta ^{(k - 1)} (x_0 + \left| x \right|)}} {{\left| x \right|^{(n - 2)/2} }}$$ and $\delta ^{(k - 1)} (x_0 - \left| x \right|) \cdot \delta ^{(k - 1)} (x_0 + \left| x \right|)$ . The first of them is a generalization of the product $${{\left| x \right|^{(n - 2)/2} }} \cdot \frac{{\delta (x_0 + \left| x \right|)}} {{\left| x \right|^{(n - 2)/2} }}{\text{ }}$$ given in [1, p. 158].     

Bemerkung zur Rayleigh-Schrödinger-Störungstheorie

The time-independent Hamiltonians ? ₀ and ?=? ₀ + V have a discrete spectrum, eigenvalues, and eigenvectors E _s ^(o) , ¦s〉^(o) resp. E _s, ¦s〉. If the RS perturbation theory can be applied here then an operator \(\mathfrak{p}\) with the property $$\left| s \right\rangle ^{(n + 1)} = \frac{1}{{n + 1}}\mathfrak{p}\left| s \right\rangle ^{(n)} , E_s^{(n + 1)} = \frac{1}{{n + 1}}\mathfrak{p}E_s^{(n)} $$ exists where ¦s〉⁽ⁿ⁾ and E _s ⁽ⁿ⁾ denote the n-th order corrections of perturbation theory if E _s ^(o) is nondegenerate. In the case of degeneracy the operation \(\mathfrak{p}\) remains defined and can always be used todetermine perturbation corrections of quantum mechanical expressions which are invariant in zerothorder under transformations of the basis in degenerate subspaces of ? ₀. The equations $$\left| s \right\rangle = \sum\limits_n^{0,\infty } {\left| s \right\rangle ^{(n)} = e^\mathfrak{p} \left| s \right\rangle ^{(0)} } , E_s = \sum\limits_n^{0,\infty } {E_s^{(n)} } = e^\mathfrak{p} E_s^{(0)} $$ correspond to a basis transformation where nondegenerate eigenvectors ¦s∝> ^(o) and eigenvalues E _s ^(o) of ? ₀ transform into eigenvectors ¦s∝> and eigenvalues E _s of ?. Examples show the usefulness of this formulation.     

Thermodynamic Properties of Ternary Ionic Liquid Mixture Containing a Common Ion: Excess Molar Volumes,Excess Isentropic Compressibilities,Excess Molar Enthalpies and Excess Heat Capacities

In the present investigations, the excess molar volumes, \( V_{ijk}^{\text{E}} \), excess isentropic compressibilities, \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), and excess heat capacities, \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \), for ternary 1-butyl-2,3-dimethylimidazolium tetrafluoroborate (i) + 1-butyl-3-methylimidazolium tetrafluoroborate (j) + 1-ethyl-3-methylimidazolium tetrafluoroborate (k) mixture at (293.15, 298.15, 303.15 and 308.15) K and excess molar enthalpies, \( \left( {H^{\text{E}} } \right)_{ijk} \), of the same mixture at 298.15 K have been determined over entire composition range of x _i and x _j . Satisfactorily corrections for the excess properties \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) have been obtained by fitting with the Redlich–Kister equation, and ternary adjustable parameters along with standard errors have also been estimated. The \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) data have been further analyzed in terms of Graph Theory that deals with the topology of the molecules. It has also been observed that Graph Theory describes well \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) values of the ternary mixture comprised of ionic liquids.     

The Study of Solute–Solvent Interactions in 1-Butyl-3-Methylimidazolium Hexafluorophosphate + 2-Pyrrolidone from Volumetric,Acoustic, Optical and Spectral Properties

The density (ρ), speed of sound (u) and refractive index (n_D) of [Bmim][PF₆], 2-pyrrolidone and their binary mixtures were measured over the whole composition range as a function of temperature between (303.15 and 323.15)?K at atmospheric pressure. Experimental values were used to calculate the excess molar volumes \( \left( {V_{m}^{\text{E}} } \right) \), excess partial molar volumes \( \left( {\overline{V}_{m}^{\text{E}} } \right) \), partial molar volumes at infinite dilution \( \left( {\overline{V}_{m}^{{{\text{E}},\infty }} } \right) \), excess values of isentropic compressibility \( \left( {\kappa_{S}^{\text{E}} } \right) \), free length \( \left( {L_{\text{f}}^{\text{E}} } \right) \) and speeds of sound \( \left( {u^{\text{E}} } \right) \) for the binary mixtures. The calculated properties are discussed in terms of molecular interactions between the components of the mixtures. The results reveal that interactions between unlike molecules take place, particularly through intermolecular hydrogen bond formation between the C₂–H of [Bmim][PF₆] and the carbonyl group of pyrrolidin-2-one. An excellent correlation between thermodynamic and IR spectroscopic measurements was observed. The observations were further supported by the Prigogine–Flory–Patterson (PFP) theory of excess molar volume.     

Half-width and asymmetry of glow peaks and their consistent analytical representation

The kinetic equation which describes many electronic as well as atomic or chemical reactions under the condition of a steadily linear raise of the temperature, is considered in a mathematically exact and straightforward way. Therefore, the equation has been transformed into a dimensionsless form, using with profit the maximum condition for the intensity peak. The two temperatures T₁ and T₂, corresponding to the half-height of the intensity peak, are found as unique polynomials of the small argument \(\bar y \equiv {{k\bar T} \mathord{\left/ {\vphantom {{k\bar T} E}} \right. \kern-0em} E}\) only ( \(\bar T\) =temperature of peak maximum). Thereupon, further combinations give half-widthδ, peak asymmetryA=δ₂/δ₁ or \(\tilde A = {{\bar C} \mathord{\left/ {\vphantom {{\bar C} {(1 - \bar C)}}} \right. \kern-0em} {(1 - \bar C)}}\) and the maximum of the intensity peakJ; they again all depend only on¯y. In some cases this dependence is weak, so that e.g. it is deduced that the half-width energy product divided by \(\bar T^2 \) is an invariant, different for every kinetic orderπ: $$\frac{{\delta \cdot E[eV]}}{{\bar T^2 }} = \left\{ {\begin{array}{*{20}c} {{1 \mathord{\left/ {\vphantom {1 {4998 K for monomolecular process}}} \right. \kern-\nulldelimiterspace} {4998 K for monomolecular process}}} \\ {{1 \mathord{\left/ {\vphantom {1 {3542 K for bimolecular process}}} \right. \kern-\nulldelimiterspace} {3542 K for bimolecular process}}} \\ {{1 \mathord{\left/ {\vphantom {1 {2872 K for trimolecular process}}} \right. \kern-\nulldelimiterspace} {2872 K for trimolecular process}}} \\ \end{array} } \right.$$ By means of these correlations, activation energy valuesE [eV] can be determined accurately to within 0.5 %, so that for most experiments the inaccuracy of theδ values becomes dominant and limiting. A special nomogram for the express estimation ofE from experimentally observedδ and \(\bar T\) is demonstrated.     

Experimental and DFT study on complexation of Eu3+ with a macrocyclic lactam receptor

From extraction experiments and $ \gamma $ -activity measurements, the extraction constant corresponding to the equilibrium $ {\text{Eu}}^{ 3+ } \left( {\text{aq}} \right) + 3 {\text{A}}^{ - } \left( {\text{aq}} \right) + {\mathbf{1}}\left( {\text{nb}} \right) \Leftrightarrow {\mathbf{1}} \cdot {\text{Eu}}^{ 3+ } \left( {\text{nb}} \right) + 3 {\text{A}}^{ - } \left( {\text{nb}} \right) $ taking place in the two-phase water–nitrobenzene system ( $ {\text{A}}^{ - } = \text {CF}_{3} \text{SO}_{3}^{ - } $ ; 1 = macrocyclic lactam receptor—see Scheme 1; aq = aqueous phase, nb = nitrobenzene phase) was evaluated as $ { \log } K_{{{\text{ex}} }} ({\mathbf{1}} \cdot {\text{Eu}}^{ 3+ } ,{\text{ 3A}}^{ - } )\; = \; - 4. 9 \pm 0. 1 $ . Further, the stability constant of the 1·Eu³⁺ cationic complex in nitrobenzene saturated with water was calculated for a temperature of 25 °C: $ { \log } \beta_{{{\text{nb}} }} ({\mathbf{1}} \cdot {\text{Eu}}^{ 3+ } ) \; = \; 8. 2 \pm 0. 1 $ . Finally, using DFT calculations, the most probable structure of the cationic complex species 1·Eu³⁺ was derived. In the resulting 1·Eu³⁺ complex, the “central” cation Eu³⁺ is bound by five bond interactions to two ethereal oxygen atoms and two carbonyl oxygens, as well as to one carbon atom of the corresponding benzene ring of the parent macrocyclic lactam receptor 1 via cation-π interaction.

A note on the principal measure and distributional ({\varvec{p, q}})-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart in (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we prove that for any non-zero coupling constant $\varepsilon \in (0, 1)$ , this coupled map lattice system is distributionally $(p, q)$ -chaotic for any pair $0\le p\le q\le 1$ , and that its principal measure is not less than $(1-\varepsilon )\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $$\begin{aligned} (1-\varepsilon )\left( \frac{2}{3}+\sum \limits _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}\right) \end{aligned}$$ for any non-zero coupling constant $\varepsilon \in (0, 1)$ and the tent map $\Lambda $ defined by $$\begin{aligned} \Lambda (x)=1-|1-2x|,\quad x\in [0, 1]. \end{aligned}$$      

Molecular magnets based on chain polymer complexes of copper(II) bis(hexafluoroacetylacetonate) with isoxazolyl-substituted nitronyl nitroxides

First isoxazolyl-substituted nitronyl nitroxides (L and $L^{Me_2 }$ ) were synthesized and characterized. Their reactions with Cu(hfac)₂ and Mn(hfac)₂ (hfac is hexafluoroacetylacetonate) afford the heterospin complexes [Cu(hfac)₂L] _n , [Cu₂(hfac)₄L] _n , $\left[ {Cu_2 (hfac)_4 L^{Me_2 } } \right]_n$ , $\left[ {Cu(hfac)_2 L^{Me_2 } } \right]_n$ , $\left[ {Cu(hfac)_2 L^{Me_2 } _2 } \right]$ , $\left[ {Cu(hfac)_2 L^{Me_2 } (MeCN)} \right]$ , [Mn(hfac)₂]₃L₄, and $\left[ {Me(hfac)_2 L^{Me_2 } } \right]_2$ . In the ligand L, the N atom of the isoxazole ring (NIz) has weak electron-donating properties. For example, the paramagnetic ligand in the chain polymer complex [Cu(hfac)₂L] _n acts as a bidentate bridging ligand coordinated through both O atoms of the nitronyl nitroxide group (O_N-O); the N_Iz and O_Iz atoms are not involved in the coordination. The introduction of Me groups into the isoxazole substituent results in an increase in the electron density on the N_Iz atom and enables the synthesis of the chain polymer complex $\left[ {Cu(hfac)_2 L^{Me_2 } } \right]_n$ , in which the bidentate bridging ligand $L^{Me_2 }$ is coordinated through the O_N-O and N_Iz atoms. In the mononuclear complexes $\left[ {Cu(hfac)_2 L^{Me_2 } _2 } \right]$ and $\left[ {Cu(hfac)_2 L^{Me_2 } (MeCN)} \right]$ , the paramagnetic ligand is coordinated only through the N_Iz atom. The solid heterospin Mn complexes [Mn(hfac)₂]₃L₄ and $\left[ {Mn(hfac)_2 L^{Me_2 } } \right]_2$ have a molecular structure. In these complexes, strong antiferromagnetic intracluster exchange interactions arise. The residual magnetic moments of the exchange clusters in the complex [Mn(hfac)₂]₃L₄ are ferromagnetically coupled, resulting in the increase in the effective magnetic moment (??_eff) of the complex with decreasing temperature in the range of 300??30 K. The thermomagnetic study of the complexes [Cu(hfac)₂L] _n , [Cu₂(hfac)₄L] _n , and $\left[ {Cu_2 (hfac)_4 L^{Me_2 } } \right]_n$ in the range of 2?C300 K revealed the ferromagnetic ordering at temperatures below 5 K. The ESR study of the solid complex $\left[ {Cu(hfac)_2 L^{Me_2 } } \right]_n$ showed that the decrease in its ??_eff in the temperature range of 30?C300 K is associated with the direct exchange interaction between the unpaired electrons of the nitronyl nitroxides of adjacent chains, whereas at temperatures below 30 K, only Cu²⁺ ions contribute to the magnetic susceptibility of the complex.     

The principal measure and distributional (p, q)-chaos of a coupled lattice system with coupling constant \varepsilon =1 related with Belusov–Zhabotinskii reaction

We consider the following system coming from a lattice dynamical system stated by Kaneko (Phys Rev Lett, 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f\left( x_{n}^{m}\right) +\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$ where $m$ is discrete time index, $n$ is lattice side index with system size $L$ (i.e., $n=1, 2, \ldots , L$ ), $\varepsilon \ge 0$ is coupling constant, and $f(x)$ is the unimodal map on $I$ (i.e., $f(0)=f(1)=0$ , and $f$ has unique critical point $c$ with $0<c<1$ and $f(c)=1$ ). In this paper, we prove that for coupling constant $\varepsilon =1$ , this CML (Coupled Map Lattice) system is distributionally $(p, q)$ -chaotic for any $p, q\in [0, 1]$ with $p\le q$ , and that its principal measure is not less than $\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $\frac{2}{3}+\sum _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}$ for coupling constant $\varepsilon =1$ and the tent map $\Lambda $ defined by $\Lambda (x)=1-|1-2x|, x\in [0, 1]$ . So, our results complement the results of Wu and Zhu (J Math Chem, 50:2439–2445, 2012).     

Thermochemical properties of hydrazinium dipicrylamine in N-methyl pyrrolidone and dimethyl sulfoxide

The enthalpies of dissolution for Hydrazinium Dipicrylamine (HDPA) in N-methyl pyrrolidone (NMP) and dimethyl sulfoxide (DMSO) were measured using a RD496-2000 Calvet microcalorimeter at 298.15 K. Empirical formulae for the calculation of the enthalpies of dissolution (Δ_diss H) were obtained from the experimental data of the dissolution processes of HDPA in NMP and DMSO. The linear relationships between the rate (k) and the amount of substance (a) were found. The corresponding kinetic equations describing the two dissolution processes were $ {{\text{d}\alpha } \mathord{\left/ {\vphantom {{\text{d}\alpha} {\text{d}t}}} \right. \kern-0pt} {\text{d}t}} = 10^{ - 2.71}\left( {1 - \alpha } \right)^{1.23} $ d α / d t = 10 ? 2.71 ( 1 ? α ) 1.23 for the dissolution of HDPA in NMP, and $ {{\text{d}\alpha } \mathord{\left/ {\vphantom {{\text{d}\alpha} {\text{d}t}}} \right. \kern-0pt} {\text{d}t}} = 10^{ - 2.58}\left( {1 - \alpha } \right)^{0.81} $ d α / d t = 10 ? 2.58 ( 1 ? α ) 0.81 for the dissolution of HDPA in DMSO, respectively.     

Spectroscopic Studies on the Thermodynamics of the Complexation of Trivalent Curium with Propionate in the Temperature Range from 20 to 90 °C

The thermodynamics of the stepwise complexation reaction of Cm(III) with propionate was studied by time resolved laser fluorescence spectroscopy (TRLFS) and UV/Vis absorption spectroscopy as a function of the ligand concentration, the ionic strength and temperature (20–90 °C). The molar fractions of the 1:1 and 1:2 complexes were quantified by peak deconvolution of the emission spectra at each temperature, yielding the log₁₀ $ K_{n}^{\prime } $ values. Using the specific ion interaction theory (SIT), the thermodynamic stability constants log₁₀ $ K_{n}^{0} (T) $ were determined. The log₁₀ $ K_{n}^{0} (T) $ values show a distinct increase by 0.15 (n = 1) and 1.0 (n = 2) orders of magnitude in the studied temperature range, respectively. The temperature dependency of the log₁₀ $ K_{n}^{0} (T) $ values is well described by the integrated van’t Hoff equation, assuming a constant enthalpy of reaction and $ \Updelta_{\text{r}} C^\circ_{{p,{\text{m}}}} = 0, $ yielding the thermodynamic standard state $ \left( {\Updelta_{\text{r}} H^\circ_{\text{m}} ,\Updelta_{\text{r}} S^\circ_{\text{m}} ,\Updelta_{\text{r}} G^\circ_{\text{m}} } \right) $ values for the formation of the $ {\text{Cm(Prop)}}_{n}^{3 - n} $ , n = (1, 2) species.     

The multiplicative product (\delta (x_0 - \left| x \right|)/\left| x \right|^{(n - 2)/2} )(\delta (x_0 + \left| x \right|)/\left| x \right|^{(n - 2)/2} )

In this paper we give a sense to the distributional multiplicative product $(\delta (x_o - \left| x \right|)/\left| x \right|^{(n - 2)/2} )(\delta (x_o + \left| x \right|)/\left| x \right|^{(n - 2)/2} )$ . For the case n=4 we obtain a formula which is used for a perturbative calculation of Green function in quantum field theories. As an application of our product, we show the calculation of the self-energy of the theory of $\lambda \emptyset ^4 $ .     

DSC study of the decomposition of azodicarbonamide in different media

The decomposition of azodicarbonamide (Genitron AC-2) in the solid state was investigated by DSC. It was found that the decomposition under non-isothermal conditions can be described by the autocatalytic reaction scheme $$X\xrightarrow{{k_1 }}Y,X + Y\xrightarrow{{k'_2 }}2Y$$ where the following dependences hold for the rate constants: $$k_1 = 4.8 \times 10^{19} e - {{243 600} \mathord{\left/ {\vphantom {{243 600} {RT_s - 1}}} \right. \kern-\nulldelimiterspace} {RT_s - 1}}$$ and $$k'_2 = 1.0 \times 10^{13} e - {{133 500} \mathord{\left/ {\vphantom {{133 500} {RT_s - 1}}} \right. \kern-\nulldelimiterspace} {RT_s - 1}}$$ The first pre-exponential factor includes the thermal history of the sample, especially the quick heating to a certain temperature, from which normal slow heating starts. Due to this fast heating, the decomposition reaction of AZDA may be understood as the collapse of its crystal lattice into nucleation centres with critical dimensions.     

Volumetric and Viscometric Studies in Binary Liquid Mixtures of 2-Methyl-2-propanol with Some Ketones at Different Temperatures

Densities, ??, and viscosities, ??, of binary mixtures of 2-methyl-2-propanol with acetone (AC), ethyl methyl ketone (EMK) and acetophenone (AP), including those of the pure liquids, were measured over the entire composition range at 298.15, 303.15 and 308.15?K. From these experimental data, the excess molar volume $V_{\mathrm{m}}^{\mathrm{E}}$ , deviation in viscosity ????, partial and apparent molar volumes ( $\overline{V}_{\mathrm{m},1}^{\,\circ }$ , $\overline{V}_{\mathrm{m},2}^{\,\circ }$ , $\overline{V}_{\phi ,1}^{\,\circ}$ and $\overline{V}_{\phi,2}^{\,\circ} $ ), and their excess values ( $\overline{V}_{\mathrm{m},1}^{\,\circ \mathrm{E}}$ , $\overline{V}_{\mathrm{m,2}}^{\,\circ \mathrm{ E}}$ , $\overline {V}_{\phi \mathrm{,1}}^{\,\circ \mathrm{ E}}$ and $\overline{V}_{\phi \mathrm{,2}}^{\,\circ \mathrm{ E}}$ ) of the components at infinite dilution were calculated. The interaction between the component molecules follows the order of AP > AC > EMK.     

Densities,Refractive Indices,Ultrasonic Speeds and Excess Properties of Acetonitrile + Poly(ethylene glycol) Binary Mixtures at Temperatures from 298.15 to 313.15 K

The densities, ρ, refractive indices, n _D, and ultrasonic speeds, u, of binary mixtures of acetonitrile (AN) with poly(ethylene glycol) 200 (PEG200), poly(ethylene glycol) 300 (PEG300) and poly(ethylene glycol) 400 (PEG400) were measured over the entire composition range at temperatures (298.15, 303.15, 308.15 and 313.15) K and at atmospheric pressure. From the experimental data, the excess molar volumes, \( V_{\text{m}}^{\text{E}} \) , deviations in refractive indices, \( \Delta n_{\text{D}} \) , excess molar isentropic compressibility, \( K_{{s , {\text{m}}}}^{\text{E}} \) , excess intermolecular free length, \( L_{\text{f}}^{\text{E}} \) , and excess acoustic impedance, Z ^E, have been evaluated. The partial molar volumes, \( \overline{V}_{\text{m,1}} \) and \( \overline{V}_{\text{m,2}} \) , partial molar isentropic compressibilities, \( \overline{K}_{{s , {\text{m,1}}}} \) and \( \overline{K}_{{s , {\text{m,2}}}} \) , and their excess values over whole composition range and at infinite dilution have also been calculated. The variations of these properties with composition and temperature are discussed in terms of intermolecular interactions in these mixtures. The results indicate the presence of specific interactions among the AN and PEG molecules, which follow the order PEG200 < PEG300 < PEG400.