Generalizations of some conjectures of Sun on the relations between N(a, b, c, d; n) and T(a, b, c, d; n)

The Ramanujan Journal - We derive some new finite sums involving the sequence $$s_{2}\left( n\right) ,$$ the sum-of-digits of the expansion of n in base 2. These functions allow us to generalize...     

Fluid thermodynamics,energy redistribution law,two-dimensional condensate,and <Emphasis Type="Italic">T</Emphasis> map

In detail, we consider the concurrence of experimental data with the results of calculations in accordance with formulas we previously derived for a real gas. We formulate the energy redistribution law following from the occurrence of dimers. We state a theorem on the two-dimensional condensate for any gas and describe the process of transition to equilibrium thermodynamics as the pressure under the piston increases.     

Area,coarea, and approximation in <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript>

Let Ω⊂ℝ ⁿ be an arbitrary open set. We characterize the space W ^1,1 _loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space W ^1,1(ℝ ⁿ ).     

Constructions for generalized Steiner systems <Emphasis Type="Italic">GS</Emphasis>(3, 4, <Emphasis Type="Italic">v</Emphasis>, 2)

Generalized Steiner systems GS (3, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance three, in which each codeword has length v and weight four. Not much is known for GS (3, 4, v, 2)s except for a recursive construction and two small designs for v = 8,10 given by Etzion. In this paper, more small designs are found by computer search and also given are direct constructions based on finite fields and rotational Steiner quadruple systems and recursive constructions using three-wise balanced designs. Some infinite families are also obtained.      

Invariants of knots,surfaces in R<Superscript>3</Superscript>, and foliations

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1239–1252, September, 2007.     

p-Helson sets, 1<p<2

Ap-Helson set is defined to be a closed subsetE of the circle groupT with the property that every continuous function onE can be extended to the full circle in such a way that this extension has its sequence of Fourier coefficients inl ^p. For 1<p<2, the union of two such sets is again ap-Helson set. It is shown that thep-Helson sets (p>1) differ from the Helson sets and also that the notion really depends on the indexp. An analogue of H. Helson’s result is given: ap-Helson set supports no nonzero measure with Fourier-Stieltjes transform inl ^q, 1/p+1/q=1.     

Generalized convex spaces, <Emphasis Type="Italic">L</Emphasis>-spaces,and <Emphasis Type="Italic">FC</Emphasis>-spaces

We show that FC-spaces due to Ding are particular types of L-spaces due to Ben-El-Mechaiekh et al., and hence particular types of G-convex spaces. Some counter-examples are given and related matters are also discussed.     

χ‐bounds,operations, and chords

A long unichord in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is long unichord free if it does not contain any long unichord. We prove a structure theorem for long unichord free graph. We give an time algorithm to recognize them. We show that any long unichord free graph G can be colored with at most colors, where ω is the maximum number of pairwise adjacent vertices in G.     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/∞ system revisited: finiteness,summability, long range dependence,and reverse engineering

We explore M/G/∞ systems ‘fed’ by Poissonian inflows with infinite arrival rates. Three processes – corresponding to the system's state, workload, and queue-size – are studied and analyzed. Closed form formulae characterizing the system's stationary structure and correlation structure are derived. And, the issues of queue finiteness, workload summability, and Long Range Dependence are investigated. We then turn to devise a ‘reverse engineering’ scheme for the design of the system's correlation structure. Namely: how to construct an M/G/∞ system with a pre-desired ‘target’ workload/queue auto-covariance function. The ‘reverse engineering’ scheme is applied to various examples, including ones with infinite queues and non-summable workloads. AMS Subject Classifications Primary: 60K25; Secondary: 60G55, 60G10     

Spencer <Emphasis Type="Italic">δ</Emphasis>-cohomology,Restrictions, Characteristics and Involutive Symbolic PDEs

We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating Spencer δ-cohomology of a symbolic system and its restriction to a non-characteristic subspace.      

Über ein Turánsches problem für radiale, positiv definite Funktionen, II

Turán's problem is to determine the greatest possible value of the integral for positive definite functions f(x), , supported in a given convex centrally symmetric body , . We consider the problem for positive definite functions of the form f(x)=(x₁), , with supported in [0,π], extending results of our first paper from two to arbitrary dimensions.Our two papers were motivated by investigations of Professor Y. Xu and the 2nd named author on, what they called, ℓ-1 summability of the inverse Fourier integral on . Their investigations gave rise to a pair of transformations (h_d,m_d) on which they studied using special functions, in particular spherical Bessel functions.To study the d-dimensional Turán problem, we had to extend relevant results of B. & X., and we did so using again Bessel functions. These extentions seem to us to be equally interesting as the application to Turán's problem.     

Fredholm determinants,Jimbo‐Miwa‐Ueno τ‐functions,and representation theory

The authors show that a wide class of Fredholm determinants arising in the representation theory of “big” groups, such as the infinite‐dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann‐Hilbert problems. © 2002 Wiley Periodicals, Inc.