Some characterizations of orthant monotonic norms

In real n-space the orthant monotonic norms of Gries [5] can be given a new characterization similar to one for monotonic norms: a norm is orthant monotonic if and only if for every D=diag(δ₁,δ₂,…,δ_n)?0, the operator norm of D equals max δ_i. This gives an alternative proof to Gries's: a norm is orthant monotonic if and only if its dual norm is orthant monotonic. Also, it follows that the principal axis vectors are self-dual for orthant monotonic norms.     

The circuit complexity of checking polynomiality for functions over residue ring modulo a composite number is linear

It is proved that for each fixed composite number k, the circuit complexity of the problem which is to check if an arbitrary function f(x ₁, ..., x _n ) over a residue ring modulo k given by its value vector with length N = k ⁿ and, if so, to construct its polynomial representation is linear.     

A characterization of P1 spaces

A Banach space X is $\text{P}$₁ if and only if the radius of each bounded set in X is half its diameter.     

On a problem concerning generalizations of realcompact spaces

PF-compact spaces are defined. Every almost realcompact space is PF-compact and every separable PF-compact space is almost realcompact. Nonetheless in the constructible universe V = L, a space of cardinality ?₁ is PF-compact if, and only if, the closure of each of its countable subsets is almost realcompact, and ω₁ is hereditarily PF-compact.     

Score vectors of Kotzig tournaments

An n-tournament T_n is said to be a Kotzig tournament if the n subtournaments of T_n of order n − 1 are isomorphic. And a nonnegative integral vector R is said to be potentially Kotzig if there exists some Kotzig tournament T_n such that its score vector is R. In this paper, a criterion is found for determining whether a non-negative integral vector R is potentially Kotzig.     

Sequences with constant number of return words

An infinite word has the property R _m if every factor has exactly m return words. Vuillon showed that R ₂ characterizes Sturmian words. We prove that a word satisfies R _m if its complexity function is (m ? 1)n + 1 and if it contains no weak bispecial factor. These conditions are necessary for m = 3, whereas for m = 4 the complexity function need not be 3n + 1. A new class of words satisfying R _m is given.     

New characteristics of infinitesimal isometry and Ricci solitons

We prove that a vector field X on a compact Riemannian manifold (M, g) with Levi-Cività connection ? is an infinitesimal isometry if and only if it satisfies the system of differential equations: trace _g (L _X ?) = 0, trace _g (L _X Ric) = 0, where L _X is the Lie derivative in the direction of X and Ric is the Ricci tensor. It follows from the second assertion that the Ricci soliton on a compact manifold M is trivial if its vector field X satisfies one of the following two conditions: trace _g (L _X Ric) ≤ 0 or trace _g (L _X Ric) ≥ 0.     

Dual operator norms and the spectra of matrices

Let ∥·∥ be an operator norm and ∥·∥^D its dual. Then it is shown that ∥A∥^D? ∑|λ_i(A)|, where λ_i(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm.     

Injective coloring of planar graphs

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic numberχ_i(G) of a graph G is the least k such that there is an injective k-coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree Δ, then (1) χ_i(G)=Δ if either g≥20 and Δ≥3, or g≥7 and Δ≥71; (2) χ_i(G)≤Δ+1 if g≥11; (3) χ_i(G)≤Δ+2 if g≥8.     

Equal signs additive sequences in Banach spaces

A sequence (e_n) spanning a Banach space E is called ESA or equal signs additive if the norm of a linear combination of the e_i's does not change when adjacent coefficients of equal sign are combined. Call the sequence (e_n) regular if neither E nor its dual contain an isomorphic copy of c₀. It is shown that a regular ESA sequence is a boundedly complete and 1-shrinking basis for its span which is thus quasi-reflexive. It is further possible to replace a regular ESA norm by an equivalent ESA norm rendering E isometrically isomorphic to its second dual. A sequence (e_n) is called IS or invariant under spreading if the norm of a linear combination of the e_i's does not change when the mutual distances of the terms in the sequence (but not their relative positions) change. We give a simple construction of an unconditional norm for an IS sequence, hence, in particular, for an ESA sequence. Also, an inverse construction is obtained: We prove that each unconditional IS basis gives rise to an ESA basis by means of an inversion formula; to nonequivalent IS unconditional bases correspond nonequivalent ESA bases. It follows that nonisomorphic ESA bases are plentiful.     

Construction of pure states in mean field models for spin glasses

If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda–Guerra identities, and if the law of the overlaps has a point mass at the largest point q* of its support, we prove that one can decompose the configuration space into a sequence of sets (A _k ) such that, generically, the overlap of two configurations is equal to q* if and only if they belong to the same set A _k . For the study of the overlaps each set A _k can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda–Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.     

New Effects of an Old Z. Zalcwasser's Theorem

It is well known that the strong (C, 1)-summability of an orthogonal series does not imply its very strong (C, 1)-summability, generally. For a given index-sequence {v _n }, first, Z. Zalcwasser gave an interesting condition implying the strong (C, 1)-summability of these partial sums s _{v n} (x). We show that Zalcwasser's condition on {v _n } holds if and only if the subsequence {v ₂ n} is quasi geometrically increasing. Utilizing this fact and known theorems several strong summability results are presented for given index-sequences {v _n }.     

Uncountable products of determined groups need not be determined

If H is a dense subgroup of G, we say that H determines G if their groups of characters are topologically isomorphic when equipped with the compact open topology. If every dense subgroup of G determines G, then we say that G is determined. The importance of this property is justified by the recent generalizations of Pontryagin-van Kampen duality to wider classes of topological Abelian groups. Among other results, we show (a) _⊕i∈IR determines the product _∏i∈IR if and only if I is countable, (b) a compact group is determined if and only if its weight is countable. These answer questions of Comfort, Raczkowski and the third listed author. Generalizations of the above results are also given.     

Products of EP matrices

Necessary and sufficient conditions are given for a matrix to be a product of an EP_r matrix by an EP_s matrix. It is shown that a given square matrix is a product of more than two EP matrices of specified ranks (and hence nullities) if and only if its rank is less than or equal to the minimum of the given ranks and its nullity is less than or equal to the sum of the given nullities. It is also shown that given two EP matrices, the rank of their product is independent of the order of the factors.     

On indecomposable subcones

Let K₁,K₂ be cones. We say that K₁ is a subcone of K₂ if ExtK₁?ExtK₂. Furthermore, if K₁≠K₂, K₁ is called a proper subcone; if dimK₁=dimK₂, K₁ is called a non-degenerate subcone. We first prove that every n-dimensional indecomposable cone, n?3, contains a non-degenerate indecomposable subcone which has no more than 2n-2 extremals. Then we construct for each n?3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable.     

On absolute valued algebras with involution

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247-258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eA_s = A_s, where e denotes the unique nonzero self-adjoint idempotent of A, and A_s stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay’s algebra [A.A. Albert, A note of correction, Bull. Amer. Math. Soc. 55 (1949) 1191], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

ENTIRE CHROMATIC NUMBER AND △-MATCHING OF OUTERPLANE GRAPHS

Let G be an outerplane graph with maximum degree A and the entire chromatic number Xvef（G）. This paper proves that if △ ≥6, then △＋ 1≤Xvef（G）≤△＋ 2, and Xvef （G） = △＋ 1 if and only if G has a matching M consisting of some inner edges which covers all its vertices of maximum degree.