Quantization causes waves: Smooth finitely computable functions are affine

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F _p = {0, 1,..., p - 1} produces a 1-Lipschitz map f _A from the space Z _p of p-adic integers to Z _p . The map fA can naturally be plotted in a unit real square I² ? R²: To an m-letter non-empty word v = γ _m-1γ _m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ _m-1γ _m-2... γ0; so to every m-letter input word w = α _m-1α _m-2 ··· α0 of A and to the respective m-letter output word a(w) = β _m-1β _m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R². Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C ²-smooth curve in R², the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T² ∈ R³, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e ^i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.     

Verbal Subgroups and Subalgebras in Skew Fields

Let D be an arbitrary skew field and K a central subfield of D. We prove that D can be embedded in a skew field Δ such that w(Δ)=Δ for every nonempty Lie word w on a set of variables y₁,y₂,.?.?. with coefficients in K; moreover, we have for the multiplicative group Δ^* that v(Δ^*)=Δ^* for every nonempty word \(v=x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\ldots x_{n}^{\varepsilon_{n}}\) (?_i=±1; i=1,2,.?.?.,n).     

Generalizing the Krein–Rutman theorem,measures of noncompactness and the fixed point index

If L : Y → Y is a bounded linear map on a Banach space Y, the “radius of the essential spectrum” or “essential spectral radius” ρ(L) of L is well-defined and there are well-known formulas for ρ(L) in terms of measures of noncompactness. Now let \({C \subset D}\) be complete cones in a normed linear space (X, || · ||) and f : C → C a continuous map which is homogeneous of degree one and preserves the partial ordering induced by D. We prove (see Section 2) that various obvious analogs of the formulas for the essential spectral radius for the case f : C → C have serious defects, even when f is linear on C. We propose (see (3.5)) a definition for ρ _C (f), the “cone essential spectral radius of f,” which avoids these difficulties. If \({{\tilde r}_{C}(f)}\) denotes the (Bonsall) cone spectral radius of f, we conjecture (see Conjecture 4.1) that if \({\rho_{C}(f) < {\tilde r}_{C}(f)}\), then there exists \({u \in C {\backslash} \, \{0\}}\) with f(u) = ru where r ? r _C (f). If f satisfies certain additional conditions (for example, if f is a compact perturbation of a map which is linear on C), we obtain the conclusion of the conjecture; but in general we observe (Remark 4.7) that the conjecture is intimately related to old and difficult conjectures in asymptotic fixed point theory. In Section 5 we briefly discuss extensions of generalized max-plus operators which were our original motivation and for which Conjecture 4.1 is already nontrivial.     

On structural decompositions of finite frames

A frame in an n-dimensional Hilbert space H _n is a possibly redundant collection of vectors {f _i } _i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f _i } _i∈I is said to be scalable if there exist nonnegative scalars {c _i } _i∈I such that {c _i f _i } _i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {f _i } _i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J?I is in the factor poset iff {f _j } _j∈J is a tight frame for H _n . We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability.     

Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Let L _∞,s ¹ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that ?f/?x _i ∈ L _s (? ^{m) for each i = 1, ...,m} . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D ^α f∥_∞ of functions f ∈ L _∞,s ¹ (? ^m ). Stechkin’s problem on approximation of unbounded operators D ^α by bounded operators on the class of functions f ∈ L _∞,s ¹ (? ^m ) such that ∥?f∥ _s ≤ 1 and the problem of optimal recovery of the operator D ^α on elements from this class given with error δ are solved.     

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: G ^m → G and word maps with constants w _Σ: G ^m → G of a simple algebraic group G, where w is a nontrivial word in the free group F _m of rank m, w _Σ = w ₁ σ ₁ w ₂ ··· w _r σ _r w _{r + 1}, w ₁, …, w _{r + 1} ∈ F _m , w ₂, …, w _r ≠ 1, Σ = {σ ₁, …, σ _r | σ _i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γ_w, G) of the group Γ_w = F _m /<w>.     

A Refined Luecking’s Theorem and Finite-Rank Products of Toeplitz Operators

For a bounded measurable function f on the open unit disk \({\mathbb{D}}\) , let T _f denote the corresponding Toeplitz operator on the Bergman space \({A^2(\mathbb{D})}\) . A recent result of Luecking shows that if T _f has finite rank, then f must be the zero function. Using a refined version of this result, we show that if all, except possibly one, of the functions f ₁,..., f _m are radial and \({T_{f_1}\cdots T_{f_m}}\) has finite rank, then one of these functions must be zero.     

The degree of biholomorphic mappings between special domains in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> preserving 0

Let G _i be a closed Lie subgroup of U(n), Ω _i be a bounded G _i -invariant domain in C ⁿ which contains 0, and \(O{\left( {{\mathbb{C}^n}} \right)^{{G_i}}} = \mathbb{C}\), for i = 1; 2. If f: Ω₁ → Ω₂ is a biholomorphism, and f(0) = 0, then f is a polynomial mapping (see Ning et al. (2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan’s theorem.     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).     

<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and _S V _R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom _R (I _V (R),?) and Hom _R (?,I _V (R)) exact exact complex \( \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots \) of V-injective modules I _i and I ⁱ , i ∈ N₀, such that N ? Im(I ₀ → I ⁰). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A _V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I _V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if \(Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0\) for all modules I with finite I _V (R)-injective dimension.     

Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V ₁,…V _t such that for all i the subtournament T[V _i] induced on T by V _i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k, t)=O(k ⁷ t ⁴) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists aninteger h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)=O(t ⁵) suffices.     

On judicious bipartitions of graphs

For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G)=V ₁?V ₂ such that G has at most s edges with both incident vertices in V _i . Alon proved that the limsup of \(f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)\) tends to infinity as m tends to infinity, establishing a conjecture of Erd?s. Bollobás and Scott proposed the following judicious version of Erd?s' conjecture: the limsup of \(m/4 + \left( {\sqrt {m/32} - g(m)} \right)\) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon's result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to k-partitions for odd integers k.     

Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Let S = K[x₁; x₂;...; x_n] be the polynomial ring in n variables over a field K; and let I be a squarefree monomial ideal minimally generated by the monomials u₁; u₂;...; u_m: Let w be the smallest number t with the property that for all integers 1 6 i₁ < i₂ <... < i _t 6 m such that \(lcm({u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_t}}}) = lcm({u_1},{u_2},...,{u_m})\) We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I: As a corollary, the projective dimension of I is bounded by the number w.     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].     

On the rank of odd hyper-quasi-polynomials

Given any nonzero entire function g: ? → ?, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+???+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ? → ?. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

On necessary and sufficient conditions for the self-normalized central limit theorem

Let X₁, X₂, … be a sequence of independent random variables and S_n = Σ _i=1 ⁿ X_i and V _n ² = Σ _i=1 ⁿ X _i ² . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n=V_n converges to a standard normal distribution if and only if max_1?i?n|X_i|/V_n→0 in probability and the mean of X₁ is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max_1?i?n|X_i|/V_n→0 in probability, then these sufficient conditions are necessary.     

W-Gröbner basis and monomial ideals under polynomial composition

The notion of weakly relatively prime and W-Gröbner basis in K[x ₁, x ₂, …, x _n ] are given. The following results are obtained: for polynomials f ₁, f ₂, …, f _m , \(\{ f_1^{\lambda _1 } ,f_2^{\lambda _2 } ,...,f_m^{\lambda _m } \} \) is a Gröbner basis if and only if f ₁, f ₂, …, f _m are pairwise weakly relatively prime with λ ₁, λ ₂, …, λ _m arbitrary non-negative integers; polynomial composition by Θ = (θ ₁, θ ₂, …, θ _n ) commutes with monomial-Gröbner bases computation if and only if θ ₁, θ ₂, …, θ _m are pairwise weakly relatively prime.     

Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T _i on U for all i ∈ I. Since T _i (f) is not contained in f, Lusztig considered two subalgebras _i f and ⁱ f of f for any i ∈ I, where _i f={x ∈ f | T _i (x) ∈ f} and \({^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\,|\,\,T^{-1}_{i}(x)\in \mathbf {f}\}\). The restriction of T _i on _i f is also denoted by \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\). The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of _i f, ⁱ f and \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) by using the language ’quiver with automorphism’ introduced by Lusztig.     

On a generalization of the Hadwiger–Nelson problem

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QG _Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ V form an edge if and only if Q(v ? w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger–Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QG _Q is infinite if and only if Q represents zero non-trivially over F. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6].