We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over unit polydisc. say closed subspace Q H2(Dn) is if Mzj(Q)⊆Q 1≤j≤k and Mzj⁎(Q)⊆Q, k+1≤j≤n some integer k∈{1,2,…,n−1}. prove that only ifQ=ΘH2(Dk)⊗Qθ1⊗⋯⊗Qθn−k, where Θ∈H∞(Dk) inner function Qθj either Jordan block H2(D)⊖θjH2(D) θj or H2(D). Furthermore, an explicit representation commutant ...

The standard interpretation of importance-weighted autoencoders is that they maximize a tighter lower bound on the marginal likelihood than the standard evidence lower bound. We give an alternate interpretation of this procedure: that it optimizes the standard variational lower bound, but using a more complex distribution. We formally derive this result, present a tighter lower bound, and visua...

We prove the nonexistence of [gq(4, d), 4, d]q codes for (a) d = q /2−q− 2q+1 for q = 2, h ≥ 3, (b) d = 2q−3q−2q+1 for q ≥ 7, and (c) d = 2q−rq−q+1 for 3 ≤ r ≤ q − q/p, q = p with p prime, where gq(k, d) = ∑k−1 i=0 ⌈ d/q ⌉

Let PG(r, q) be the projective space of dimension r over GF (q). A k–cap K̄ in PG(r, q) is a set of k points, no three of which are collinear [10], and a k–cap is said to be complete if it is maximal with respect to set–theoretic inclusion. The maximum value of k for which there is known to exist a k–cap in PG(r, q) is denoted by m2(r, q). Some known bounds for m2(r, q) are given below. Suppose ...

We introduce the concept of weak k-majorization extending the classical notion of weak sub-majorization. For integers k and n with k n a vector x 2 R is weakly k-majorized by a vector q 2 R if the sum of the r largest components of x does not exceed the sum of the r largest components of q, for r = 1; : : : ; k. For a given q the set of vectors weakly k-majorized by q de nes a polyhedron P (q;k...

In this paper, we study the p-ary linear code Ck(n, q), q = ph, p prime, h 1, generated by the incidence matrix of points and k-dimensional spaces in PG(n, q). For k n/2, we link codewords of Ck(n, q) \ Ck(n, q)⊥ of weight smaller than 2qk to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small...

A graph G is (Kq, k) stable if it contains a copy of Kq after deleting any subset of k vertices. In a previous paper we have characterized the (Kq, k) stable graphs with minimum size for 3 6 q 6 5 and we have proved that the only (Kq, k) stable graph with minimum size is Kq+k for q > 5 and k 6 3. We show that for q > 6 and k 6 q 2 + 1 the only (Kq, k) stable graph with minimum size is isomorphi...

We give new q-(1+q)-analogue of the Gaussian coefficient, also know as the q-binomial which, like the original q-binomial [ n k ] q , is symmetric in k and n− k. We show this q-(1 + q)-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our q-(1 + q)-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birk...

The following propositions are true: (1) For every natural number a and for every real number q such that 1 < q and q = 1 holds a = 0. (2) For all natural numbers a, k, r and for every real number x such that 1 < x and 0 < k holds xa·k+r = x · xa·(k− ′1)+r. (3) For all natural numbers q, a, b such that 0 < a and 1 < q and q − 1 | q − 1 holds a | b. (4) For all natural numbers n, q such that 0 <...