Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)^kw=0,(D-l)^kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}.     

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

Polynomial Basins of Infinity

We study the projection p: M_d ? B_d{\pi : \mathcal{M}_d \rightarrow \mathcal{B}_d} which sends an affine conjugacy class of polynomial f : \mathbbC ? \mathbbC{f : \mathbb{C} \rightarrow \mathbb{C}} to the holomorphic conjugacy class of the restriction of f to its basin of infinity. When B_d{\mathcal{B}_d} is equipped with a dynamically natural Gromov–Hausdorff topology, the map π becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of π are connected. Consequently, quasiconformal and topological basin-of-infinity conjugacy classes are also connected. The key ingredient in the proof is an analysis of model surfaces and model maps, branched covers between translation surfaces which model the local behavior of a polynomial.     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d := (−π,π]^d, associated with the Hamiltonian of a system consisting of at most two particles on a d-dimensional lattice ℤ^d, interacting via both a pair contact potential (μ > 0) and creation and annihilation operators (γ > 0). We prove the existence of a unique eigenvalue of H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d , or its absence depending on both the interaction parameters γ,μ ≥ 0 and the system quasimomentum k ∈ \mathbbT^d \mathbb{T}^d . We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum k ∈ \mathbbT^d \mathbb{T}^d in the existence domain G ⊂ \mathbbT^d \mathbb{T}^d .     

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic

We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function f : V ? \mathbbC{f : V \rightarrow \mathbb{C}} on a finite-dimensional vector space V over a finite field \mathbbF{\mathbb{F}} has large Gowers uniformity norm ||f||_U^s+1(V){{\parallel{f}\parallel_{U^{s+1}(V)}}} , then there exists a (non-classical) polynomial P: V ? \mathbbT{P: V \rightarrow \mathbb{T}} of degree at most s such that f correlates with the phase e(P) = e ^2πiP . This conjecture had already been established in the “high characteristic case”, when the characteristic of \mathbbF{\mathbb{F}} is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author [22] and of Kaufman and Lovett [28].     

Dilation of Newton Polytope and <Emphasis Type="Italic">p</Emphasis>-adic Estimate

Let f(X) be a polynomial in n variables over the finite field  \mathbbF_q\mathbb{F}_{q}. Its Newton polytope Δ(f) is the convex closure in ℝ ⁿ of the origin and the exponent vectors (viewed as points in ℝ ⁿ ) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of $\mathbb{Z}_{>0}^{n}$\mathbb{Z}_{>0}^{n} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in  \mathbbF_q\mathbb{F}_{q}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.     

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbF_m,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbF_m,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbF_m,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbF_m,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?_j=1^m X_j²{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d ^2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X ₁, . . . , X _m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

The Nonexistence of Pseudoquaternions in {\mathbb{C}^{2\times 2}}

The field of quaternions, denoted by \mathbbH{\mathbb{H}} can be represented as an isomorphic four dimensional subspace of \mathbbR^4×4{\mathbb{R}^{4\times 4}}, the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in \mathbbR^4×4{\mathbb{R}^{4\times 4}} which is also a field and which has – in connection with the quaternions – many pleasant properties. This field is called field of pseudoquaternions. It exists in \mathbbR^4×4{\mathbb{R}^{4\times 4}} but not in \mathbbH{\mathbb{H}}. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in \mathbbR⁴{\mathbb{R}^4}. And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b.     

Compact Operators that Commute with a Contraction

Let T be a C₀–contraction on a separable Hilbert space. We assume that I_H − T*T is compact. For a function f holomorphic in the unit disk \mathbbD{\mathbb{D}} and continuous on [`(\mathbbD)]\overline{{\mathbb{D}}}, we show that f(T) is compact if and only if f vanishes on s(T)?\mathbbT\sigma(T)\cap{\mathbb{T}}, where σ(T) is the spectrum of T and \mathbbT{\mathbb{T}} the unit circle. If f is just a bounded holomorphic function on \mathbbD{\mathbb{D}}, we prove that f(T) is compact if and only if lim_{n? ￥}||Tⁿf(T)|| = 0\lim\limits_{n\rightarrow \infty}\|T^{n}f(T)\| = 0.     

Minimal Cubic Cones via Clifford Algebras

In this paper, we construct two infinite families of algebraic minimal cones in ^n{\mathbb{R}^{n}}. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any n ≥ 4, n ≠ 16k + 1, there is at least one minimal cone in \mathbbRⁿ{\mathbb{R}^{n}} given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in \mathbbR^m2{\mathbb{R}^{m^2}}, m ≥ 2, defined by an irreducible homogeneous polynomial of degree m. These examples provide particular answers to the questions on algebraic minimal cones in \mathbbRⁿ{\mathbb{R}^{n}} posed by Wu-Yi Hsiang in the 1960s.     

Divisibility of polynomials over finite fields and combinatorial applications

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252–260, 1998) uses trinomials over \mathbbF₂{\mathbb{F}_2} to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1–17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over \mathbbF₂{\mathbb{F}_2} . Here we first simplify the requirement in Munemasa’s approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over \mathbbF₃{\mathbb{F}_3} . Some of our results apply to any finite field \mathbbF_q{\mathbb{F}_q} with q elements.     

Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7

In this paper we prove that two complete intersections X_n(d){X_n(\underline{d})} and X_n(d￠){X_n(\underline{d}^\prime)} are homeomorphic if and only if they have the same total degree, Pontrjagin classes and Euler characteristics, provided n = 5, 6, 7. This extends earlier result of Fang and Klaus (Manuscr Math 90:139–147, 1996).     

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      19. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      20. Balanced Deep Matrix Algebras      Christopher Kennedy 《Algebras and Representation Theory》2011,14(6):1187-1202 This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksln{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl2\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl2{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksl_n{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl₂\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl₂{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.