Weighted forms of Euler's theorem

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's “lost” notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Pak's iterated Dyson's map and Sylvester's fish-hook bijection are the main ingredients in the weighted forms of Euler's theorem.     

New weighted Rogers-Ramanujan partition theorems and their implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

     

A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications

In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.

     

Computational analysis of supersonic turbulent boundary layers over rough surfaces using the k–ω and the stress–ω models

The influence of surface roughness in the prediction of the mean flow and turbulent properties of a high-speed supersonic (M = 2.7, Re/m = 2 × 10⁷) turbulent boundary layer flow over a flat plate is numerically investigated. In particular, the performance of the k–ω and stress–ω turbulence models is evaluated against the available experimental data. Even though the performance of these models have been proven satisfactory in the computation of incompressible boundary layer flow over rough surfaces, their validity for high-speed compressible has not been investigated yet. It is observed from this study that, for smooth surface, both k–ω and stress–ω models perform very well in predicting the mean flow and turbulence quantities in supersonic flow. For rough surfaces, both models matched the experimental data fairly well for lower roughness heights but performed unsatisfactorily for higher roughness conditions. Overall the performance of the k–ω model is better than the stress–ω model. The stress–ω model does not show any strong advantages to make up for the extra computational cost associated with it. The predictions indicate that the ω boundary conditions at the wall in both models, especially the stress–ω model, need to be refined and reconsidered to include the geometric factor for supersonic flow over surfaces with large roughness values.     

新Dirichlet级数Σn=0∞aneλns收敛横坐标

本文给出新Dirichlet级数Σ_n=0^∞a_ne^λ_ns的收敛横坐标σ_c、一致收敛横坐标σ_u和绝对收敛横坐标σ_a的定义.通过指数λ_n和系数a_n的关系去估计三个横坐标,并补充证明两类Dirichlet级数Σ_n=0^∞a_ne^λ_ns和Σ_n=0^∞a_ne^-λ_ns的收敛条件是一致的.     

On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY^′ and ZZ^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=(X_I)_κ=∏_iIX_i in the κ-box topology (that is, with basic open sets of the form ∏_iIU_i with U_i open in X_i and with U_i≠X_i for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and [β<α and λ<κ]β^λ<α) with α regular, be a set of non-empty spaces with each d(X_i)<α, π[Y]=X_J for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is J[I]^<α such that f(x)=f(y) whenever x_J=y_J, and (c) every such f extends to .     

Moments about the mean of the size of a self-conjugate (s,t)-core partition

Johnson proved that if $s,t$ are coprime integers, then the $r$th moment of the size of an $(s,t)$-core is a polynomial of degree $2r$ in $t$ for fixed $s$. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of ${\stackrel{?}{\mathfrak{S}}}_t∕{\mathfrak{S}}_t$ and $t$-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an $(s,t)$-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an $(s,t)$-core and the first nine moments about the mean of the size of an $(s,s+1)$-core using Maple. To get the moments about the mean of the size of a self-conjugate $(s,t)$-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if $s,t$ are coprime integers, then the $r$th moment about the mean of the size of a self-conjugate $(s,t)$-core is a quasipolynomial of period 2 and degree $2r$ in $t$ for fixed odd $s$. Then, based on a bijection of Ford, Mai and Sze between self-conjugate $(s,t)$-cores and lattice paths in $?\frac{s}{2}?\times ?\frac{t}{2}?$ rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate $(s,t)$-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate $(s,t)$-core.     

THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU（2,2）/S（U（2） × U（2））

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.     

An explicit factorization formula for the singular values of φ and the generalized principal ideal theorem

Let K be a quadratic imaginary number field, f∈N and let Of be the order of conductor f in K. We consider the singular values of the Kleinian normalization φ of the Weierstrass σ-function belonging to an arbitrary proper ideal of Of. The factorization of these singular values goes back to K. Ramachandra, R. Schertz and W. Bley. But the factorization formula in [W. Bley, Konstruktion von Ganzheitsbasen in abelschen Körpererweiterungen von imaginär-quadratischen Zahlkörpern, Dissertation, Universität Augsburg, 1991] is very implicit and not easy to handle in view of many practical applications. In this paper we provide an explicit factorization formula and give different tools to control this factorization. As an immediate application we prove the generalized principal ideal theorem in the ring class field situation.     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

The universal modality, the center of a Heyting algebra, and the Blok–Esakia theorem

We introduce the bimodal logic , which is the extension of Bennett’s bimodal logic by Grzegorczyk’s axiom □(□(p→□p)→p)→p and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of , thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logic WS5.C, which is the extension of WS5 by the axiom (p¬p)→(p→p), and the bimodal logic , which is the extension of Shehtman’s bimodal logic by Grzegorczyk’s axiom, and show that the lattice of normal extensions of WS5.C is isomorphic to the lattice of normal extensions of .     

El Naschie’s ε theory and effects of nanoparticle clustering on the heat transport in nanofluids

Effects of nanoparticle clustering on the heat transfer in nanofluids using the scale relativity theory in the topological dimension D_T = 3 are analyzed. In the one-dimensional differentiable case, the clustering morphogenesis process is achieved by cnoidal oscillation modes of the speed field. In such conjecture, a non-autonomous regime implies a relation between the radius and growth speed of the cluster while, a quasi-autonomous regime requires El Naschie’s ε^(∞) theory through the cluster–cluster coherence (El Naschie global coherence). Moreover, these two regimes are separated by the golden mean. In the one-dimensional non-differentiable case, the fractal kink spontaneously breaks the ‘vacuum symmetry’ of the fluid by tunneling and generates coherent structures. This mechanism is similar to the one of superconductivity. Thus, the fractal potential acts as an energy accumulator while, the fractal soliton, implies El Naschie’s ε^(∞) theory (El Naschie local coherence). Since all the properties of the speed field are transferred to the thermal one, for a certain conditions of an external load (e.g. for a certain value of thermal gradient) the soliton and fractal one breaks down (blows up) and release energy. As result, the thermal conductibility in nanofluids unexpectedly increases. Here, El Naschie’s ε^(∞) theory interferes through El Naschie global and local coherences.     

Exis tence and st ability of μ-pseudo almost automorphic solutions for stochastic evolution equations

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.     

A conjecture of Abe-Iiyori and the Ree groups 2F4(q), II

In this paper the author has solved a problem of Abe and liyori for the finite simple groups ² F ₄(q) and ² F ₄(2)′.     

Applications of operator identities to the multiple q-binomial theorem and q-Gauss summation theorem

In this paper, we first give an interesting operator identity. Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem and q-Gauss summation theorem, we obtain some transformation formulae and summation theorems of multiple basic hypergeometric series.     

A structure theorem for posets admitting a “strong” chain partition:: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)

Suppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q∈P lie in distinct chains and p<q, then every other element of P is either above p or below q.In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets such that, for 1?i?n, one of the following occurs:

(1)

R_i is disjoint from one of the chains of the partition; or

(2)

if p, q∈R_i are in distinct chains, then they are incomparable.

The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets.In 1996, a proof of the Daykin-Daykin conjecture was announced (by two other mathematicians), but their proof needs to be rectified.In this note, a generalization of the conjecture is proven that applies to finite or infinite posets partitioned into a (possibly infinite) number of chains with the same property. In particular, it is shown that a poset admits such a partition if and only if it is an ordinal sum of posets, each of which is either a width 2 poset or else a disjoint sum of chains. A forbidden subposet characterization of these partial orders is also obtained.