The Harmonious Chromatic Number of Deep and Wide Complete n- ary Trees

The harmonious chromatic number of a graph G is the least number of colors which can be used to color V(G) such that adjacent vertices are colored differently and no two edges have the same color pair on their vertices. Unsolved Problem 17.5 of Graph Coloring Problems by Jensen and Toft asks for the harmonious chromatic number of T_m,n the complete n-ary tree on m levels. Let q be the number of edged of T_m,n and k be the smallest positive integer such that the binomial coefficient C(k, 2) ≥ q. We show that for all sufficiently large m, n, the harmonious chromatic number of T_m,n is at most k + 1, and that many such T_m,n have harmonious chromatic number k.     

The near ultraviolet absorption spectra ofortho- andmeta-thiocresols

The absorption spectra ofortho- andmeta-thiocresols have been studied in the present investigation. The ortho-thiocresol spectrum consists of about forty-five bands of rather a diffuse nature and in general low intensity in the region from 2873 Å to 2600 Å. The maximum number of bands is obtained by using a path length of 330 cm. for absorption, the temperature of the bulb being maintained at 14°C. Several of these bands are assigned as due tov-v-transitions. The (0, 0) band is chosen at 35386 cm.^?1 which is the strongest band on the longer wavelength side. Vibrational frequencies in the excited state have values 729, 957 and 1159 and combinations and overtones of these are present. Themeta-thiocresol spectrum consists of about forty bands of rather a diffuse nature and very weak intensities in the region from 2900 Å to 2590 Å. The maximum number of bands is obtained by using a path length 200 cm. for absorption and by keeping the temperature of the bulb at 20° C. The (0, 0) band is chosen to be that at 34793 cm.^?1 which is the strongest band on the longer wavelength side. Vibrational frequencies in the excited state have values 492, 611, 720, 845, 965, 1016 and 1155 cm.^?1 and combinations and overtones of these are present.     

The near ultraviolet absorption spectrum ofpara-thio-cresol

The absorption spectrum ofpara-thio-cresol consists of about ten bands in the region 2922 to 2600 Å. The bands are quite intense but broad and fail to show any structure such as is observed in the case ofpara-cresol bands. The maximum number of bands about ten is obtained by using an absorption path length of 100 cm. The (0, 0) band is chosen to be that at 34222 cm.^?1 which is the strongest band on the longer wavelength side. Vibrational frequencies in the excited state have values 353, 752 and 1114 and combinations and overtones of these are found to be present.     

Isotope shift studies of the ultra-violet and visible bands of P16O and P18O

The spectra of P¹⁶O and P¹⁸O were excited in sealed discharge tubes containing neon (2–3 mm. pressure), oxygen gas enriched to 65 per cent. of¹⁸O and trace amounts of phosphorus vapour and photographed on a 3 m. grating spectrograph at a dispersion of 2·5 Å/mm. Isotope shift studies in theβ-bands confirmed the earlier vibrational scheme of Curryet al. and showed conclusively that the red as well as the violet degraded bands belonged to the sameβ-system. The present studies of isotope shifts also confirmed the vibrational assignments of the extensive ultraviolet bands involving the² Σ ^??X² Π transition and theγ-bands (A² Σ ⁺?X² Π). In the case of the visible bands, they provided evidence for the first time that the bands at 5585 Å, 5962 Å and 6385 Å belonged to one system and involved 0–0, 0–1 and 0–2 transitions respectively.     

On Family of Graphs with Minimum Number of Spanning Trees

We show that there is a well-defined family of connected simple graphs Λ(n, m) on n vertices and m edges such that all graphs in Λ(n, m) have the same number of spanning trees, and if ${G \in \Lambda(n, m)}$ then the number of spanning trees in G is strictly less than the number of spanning trees in any other connected simple graph ${H, H \notin \Lambda(n, m)}$ , on n vertices and m edges.     

Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

     

Note on primitive permutation groups and a diophantine equation

It is shown that there are no transitive rank 3 extensions of the projective linear groups H, PSL(m,q) ? H ? PFL(m,q), for any prime power q and integer m ? 3. In the course of the proof the diophantine equation 5^m + 11 = x^p2, where m, x are positive integers, arose. As such equations can now be solved completely we had the choice of using number theory or geometry to complete the proof.     

Characterization of Co-blockers for Simple Perfect Matchings in a Convex Geometric Graph

Consider the complete convex geometric graph on $2m$ 2 m vertices, CGG $(2m)$ ( 2 m ) , i.e., the set of all boundary edges and diagonals of a planar convex $2m$ 2 m -gon P. In (Keller and Perles, Israel J Math 187:465–484, 2012), the smallest sets of edges that meet all the simple perfect matchings (SPMs) in CGG $(2m)$ ( 2 m ) (called “blockers”) are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$ m · 2 m ? 1 . In this paper we characterize the co-blockers for SPMs in CGG $(2m)$ ( 2 m ) , that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings M in CGG $(2m)$ ( 2 m ) where all edges are of odd order, and two edges of M that emanate from two adjacent vertices of P never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with m, the number of co-blockers grows super-exponentially.     

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M₃, solute gradient, S₁, and Taylor number, T_A1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.     

The near ultraviolet absorption spectrum of tetrahydro-naphthalene

The near ultraviolet absorption spectrum of tetrahydronaphthalene commonly known as tetralin consists of sharp bands in the region from 2800 to 2500 Å. The spectrum consists of about sixty bands. The maximum number of bands is obtained by using a path length of 200 cm. at room temperature which was nearly 30° C. Several of these very sharp bands have to be assigned tov-v transitions. The 0, 0 band is chosen to be at 36790 cm^?1. Vibrational frequencies in the excited state have values 1185, 951 and 682 and combinations and overtones of these are present. Assignments of the different frequencies are discussed.     

Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.     

Quantum theory of strong localization of a quasiparticle in a linear molecular system

Conclusions Thus, quasiparticles can be localized in chains with narrow energy bands in the presence of large quantum fluctuations if the parameter satisfies the restriction ₀ < /8 < . With increasing temperature the degree of localization first increases but then decreases. At a temperature above the critical value the excitation is delocalized, and the energy of the system is above the bottom of the energy band of the free quasiparticle.In chains with broad energy bands a quasiparticle can be localized and such a state is energetically stable if the quantum fluctuations in the system are small and bounded by the inequalities (4.4). At the same time, the dependence of the degree of localization on the temperature is analogous to that for the case of narrow energy bands considered above.Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 77, No. 2, pp. 179–189, November, 1988.     

Enumerating rooted simple planar maps

The main purpose of this paper is to find the number of combinatorially distinct rooted simpleplanar maps,i.e.,maps having no loops and no multi-edges,with the edge number given.We haveobtained the following results.1.The number of rooted boundary loopless planar [m,2]-maps.i.e.,maps in which there areno loops on the boundaries of the outer faces,and the edge number is m,the number of edges on theouter face boundaries is 2,is(?)for m≥1.G_0~N=0.2.The number of rooted loopless planar [m,2]-maps is(?)3.The number of rooted simple planar maps with m edges H_m~s satisfies the following recursiveformula:(?)where H_m~(NL) is the number of rooted loopless planar maps with m edges given in [2].4.In addition,γ(i,m),i≥1,are determined by(?)for m≥i.γ(i,j)=0,when i>j.     

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt $(0<q<1)$ . This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to ${m} (m \leqslant 2)$ , the global accuracy of k level corrected approximation is $O(N^{-(2m(k+1)-\varepsilon)})$ , where N is the number of the nodes, and $\varepsilon$ is an arbitrary small positive number.     

N-dimensional Schrödinger equation at finite temperature using the Nikiforov–Uvarov method

The N-radial Schrödinger equation is analytically solved. The Cornell potential is extended to finite temperature. The energy eigenvalues and the wave functions are calculated in the N-dimensional form using the Nikiforov–Uvarov (NV) method. At zero temperature, the energy eigenvalues and the wave functions are obtained in good agreement with other works. The present results are applied on the charmonium and bottomonium masses at finite temperature. The effect of dimensionality number is investigated on the quarkonium masses. A comparison is discussed with other works, which use the QCD sum rules and lattice QCD. The present approach successfully generalizes the energy eigenvalues and corresponding wave functions at finite temperature in the N-dimensional representation. In addition, the present approach can successfully be applied to the quarkonium systems at finite temperature.     

On the distribution of integer random variables related by a certain linear inequality: I

We consider the mathematical problem of the allocation of indistinguishable particles to integer energy levels under the condition that the number of particles can be arbitrary and the total energy of the system is bounded above. Systems of integer as well as fractional dimension are considered. The occupation numbers can either be arbitrary nonnegative integers (the case of “Bose particles”) or lie in a finite set {0, 1, …, R} (the case of so-called parastatistics; for example, R = 1 corresponds to the Fermi-Dirac statistics). Assuming that all allocations satisfying the given constraints are equiprobable, we study the phenomenon whereby, for large energies, most of the allocations tend to concentrate near the limit distribution corresponding to the given parastatistics.     

On the exponent average of integer sequences

Let ω(m) denote the number of distinct prime factors of the integerm, let Ω(m) be the number of prime factors ofm counted with multiplicities. The exponent averageA(m) is defined byA(m)=Ω(m)/ω(m) form>1, andA(1)=1. If (m _n) is a sequence of positive integers, we can study the asymptotic exponent average lim _n→∞ A(m_n) (if it exists) resp. lim sup and lim inf. In this article, we consider exponent averages for general sequences, and particularly for sequences of binomial coefficients as well as the divisor function. One of the many results on binomial coefficients is that $$\mathop {\lim }\limits_{n \to \infty } A\left( {\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)} \right) = 1,$$ which shows that these binomial coefficients are almost squarefree. For the divisor functiond(n), we prove for instance $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{{A(d(n))\log \log n}}{{\log n}} = 1.$$      

Fast algorithms for determining (generalized) core groups in social networks

The structure of a large network (graph) can often be revealed by partitioning it into smaller and possibly more dense sub-networks that are easier to handle. One of such decompositions is based on ??k-cores??, proposed in 1983 by Seidman. Together with connectivity components, cores are one among few concepts that provide efficient decompositions of large graphs and networks. In this paper we propose an efficient algorithm for determining the cores decomposition of a given network with complexity ${\mathcal{O}(m)}$ , where m is the number of lines (edges or arcs). In the second part of the paper the classical concept of k-core is generalized in a way that uses a vertex property function instead of degree of a vertex. For local monotone vertex property functions the corresponding generalized cores can be determined in ${\mathcal{O}(m\cdot\max(\Delta,\log{n}))}$ time, where n is the number of vertices and ?? is the maximum degree. Finally the proposed algorithms are illustrated by the analysis of a collaboration network in the field of computational geometry.     

Floral colours and their origins

A new orientation is given to the subject of floral colours by the author’s discovery that these colours may be placed into two distinct spectral categories, which have been designated by him respectively as the spectrum of florachrome A and of florachrome B. Typical of these two categories are the colours ofDelphinium ajacis (larkspur) in the blue and pink varieties respectively, the former showing the spectrum of florachrome A and the latter that of florachrome B. As a general rule, all blue flowers exhibit the spectrum of florachrome A which consists of three distinct and clearly separated bands of absorption appearing respectively in the red at 630 mμ, in the yellow at 580 mμ and in the green at 540 mμ. The spectrum of florachrome B also consists of three distinct bands of absorption, but these now appear in the orange-yellow at 590 mμ, in the green at 545 mμ and in the blue-green at 505 mμ. Spectra exhibiting these features are reproduced with the paper. Their explanation is discussed and it is shown that they owe their origin to an electronic absorption frequency located at the first of the three bands combining with vibrational transitions, the oscillator being the CO group present in the structure of the florachrome.